1 Linear Transceiver design for Downlink Multiuser MIMO Systems: Downlink-Interference Duality Approach Tadilo Endeshaw Bogale, Student Member, IEEEand Luc Vandendorpe Fellow, IEEEAbstract— Thi s paper con sid ers linear tra nsc ei ver des ign for downlink multiuser multiple-input multiple-output (MIMO) systems. We examine different transceiver design problems. We focus on two groups of design problems. The first group is the weighted sum mean-square-error (WSMSE) (i.e., symbol-wise or user-wise WSMSE) minimization problems and the second group is the minimi zat ion of the max imum we ighted me an- squ ar e- error (WMS E) (symb ol-wise or user -wis e WMSE ) prob lems. The problems are examined for the practically relevant scenario where the power constraint is a combination of per base station (BS ) ant enna and per symbol (user ), and the noi se vec tor ofeach mobile station is a zero-mean circularly symmetric complex Gaussian random variable with arbitrary covariance matrix. For each of these problems, we propose a novel downlink-interference duality based iterative solution. Each of these problems is solved as fol lows . Firs t, we estab lish a new mean-squ are -err or (MSE) downlink-interf erence duality . Second, we formulate the power all oca tion par t of the prob lem in the downlink cha nne l as a Geometric Program (GP). Third, using the duality result and the solut ion of GP , we utili ze alternatin g optimizati on tech nique to solve the original downlink problem. For the first group of prob- lems , we have establis hed symbo l-wi se and user -wis e WSMSE downlink-interf erence duality . These duality are established by for mulat ing the noise covaria nce matr ices of the inte rfer ence cha nne ls as fixe d poi nt fun cti ons. On the other hand, for the second group of problems, we have established symbol-wise and user-wise MSE downlink-interference duality. These duality are established by formulating the noise covariance matrices of the interference channels as marginally stable (convergent) discrete- time-switched syst ems. The prop osed duali ty based iter ative solutions can be extended straightforwardly to solve many other linear transceiver design problems. We also show that our MSE down link- inter fer ence duali ty unify all exist ing MSE dualit y. In our simulatio n res ults, we hav e obse rve d that the pro posed duality based iterative algorithms utilize less total BS power than that of the existing algorithms. Index Terms— Mul tiuser MI MO, MSE , Downl ink -up link dualit y, Down link-interference duali ty , fixed point func tion, discrete-time-switched system and convex optimization. I. I NTRODUCTION Mul tip le- inp ut mul tip le- out put (MI MO) is a promis ing technique to exploit the spectral efficiency of wireless chan- nels. Thi s spe ctr al ef ficienc y can be exp loited by app lyi ng The authors would like to thank BELSPO for the financ ial suppor t ofthe IAP pr oj ect BESTCOM in the fr amewor k of whic h this work has been achi ev ed. Pa rt of this work has been publ is hed in the ICASSP , Kyo to, Jap an, Mar . 2012. Tadi lo Endesh aw Bogale and Luc V anden- dorp e are wit h the ICTEAM Ins tit ute , Uni ver sit ´ e ca thol ique de Lou- va in, Pl ace du Le va nt 2, 134 8 - Louvai n La Ne uve, Belgium. Emai l: {tadilo.bogale, luc.vandendorpe }@uclo uvain .be, Phone: +3210478 071, Fax: +3210472089. signa l proce ssing at the trans mitter (precode r) and recei ver (decoder). Signal processing is performed to meet a certain design criterion. It is well known that most practically relevant des ign proble ms suc h as wei ght ed sum rat e max imi zat ion , rate or signal-to-interference-plus -noise-ratio (SINR) balanc- ing and rat e or SINR con str ain ed power minimi zat ion can be equivalently expressed as mean-square-error (MSE) based problems (see for example [1]). Because of this, the current paper examines MSE-based problems. In general, the uplinkchannel MSE-based problems are better understood than those of the downl ink cha nne l. Due to this fact, mos t lit era tur es foc us on sol vin g the downlink MSE-ba sed pro blems. The downlink MSE-bas ed problems can be solved by direc t ap- proach as in [2], [3] or by uplink-downlink duality approach as in [4]–[6]. For a given downlink channel system model and its MSE- based problem, the idea behind uplink-downlink duality is first to create the virtual uplink channel by exchanging the roles of the trans mit ter and rec ei ve r, and then to ena ble the pre - coder/decoder transformation from uplink to downlink channel and vice versa by ensuring the same MSE in both channels. Once the se two tas ks are per for med , the do wnl ink MSE- bas ed pro ble ms are exami ned as fol lows: Whe n the glo bal optimality of the dual uplink channel MSE-based problem is guaranteed, the duality approach simply transfers the optimal uplink channel precoder/decoder pairs from uplink to downlinkchannel (see for example the sum MSE minimization problem in [5]). When the global optimal solution of the dual uplinkcha nne l MSE-ba sed pro ble m can not ens ure d, the dua lit y app roa ch exami nes the do wnl ink MSE-ba sed proble ms by iteratively switching between the uplink and downlink channel problems (see for example the problems in [7]). Several MSE-based problems have been examined by dual- ity approach [4]–[8]. However, the duality of these papers are able to solve total BS power constrained MSE-based problems only . In a prac tical multi-antenna base stati on (BS) system, the maximum power of eac h BS antenna is limite d [9]. In some scenario allocatin g diff erent powers to diff erent users (symbols) according to their priority or protection level has some interest. This motivates [10] to solve (robust) sum MSE- bas ed proble ms wit h per ant enn a, use r and symbol power constraints by duality approach. However, since the problems in [10] allocate the same MSE weight to all symbols (users), [10] ignores priority and fairness issues in terms of MSE. In a mul timedi a commun ica tion, dif ferent typ es of in- for mation (for examp le, audio and video inf ormation) can
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8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
ApproachTadilo Endeshaw Bogale Student Member IEEE and Luc Vandendorpe Fellow IEEE
Abstractmdash This paper considers linear transceiver designfor downlink multiuser multiple-input multiple-output (MIMO)systems We examine different transceiver design problems Wefocus on two groups of design problems The first group is theweighted sum mean-square-error (WSMSE) (ie symbol-wise oruser-wise WSMSE) minimization problems and the second groupis the minimization of the maximum weighted mean-square-error (WMSE) (symbol-wise or user-wise WMSE) problemsThe problems are examined for the practically relevant scenario
where the power constraint is a combination of per base station(BS) antenna and per symbol (user) and the noise vector of each mobile station is a zero-mean circularly symmetric complexGaussian random variable with arbitrary covariance matrix Foreach of these problems we propose a novel downlink-interferenceduality based iterative solution Each of these problems is solvedas follows First we establish a new mean-square-error (MSE)downlink-interference duality Second we formulate the powerallocation part of the problem in the downlink channel as aGeometric Program (GP) Third using the duality result and thesolution of GP we utilize alternating optimization technique tosolve the original downlink problem For the first group of prob-lems we have established symbol-wise and user-wise WSMSEdownlink-interference duality These duality are established byformulating the noise covariance matrices of the interference
channels as fixed point functions On the other hand for thesecond group of problems we have established symbol-wise anduser-wise MSE downlink-interference duality These duality areestablished by formulating the noise covariance matrices of theinterference channels as marginally stable (convergent) discrete-time-switched systems The proposed duality based iterativesolutions can be extended straightforwardly to solve many otherlinear transceiver design problems We also show that our MSEdownlink-interference duality unify all existing MSE dualityIn our simulation results we have observed that the proposedduality based iterative algorithms utilize less total BS power thanthat of the existing algorithms
Index Termsmdash Multiuser MIMO MSE Downlink-uplinkduality Downlink-interference duality fixed point function
discrete-time-switched system and convex optimization
I INTRODUCTION
Multiple-input multiple-output (MIMO) is a promising
technique to exploit the spectral efficiency of wireless chan-
nels This spectral efficiency can be exploited by applying
The authors would like to thank BELSPO for the financial support of the IAP project BESTCOM in the framework of which this work hasbeen achieved Part of this work has been published in the ICASSPKyoto Japan Mar 2012 Tadilo Endeshaw Bogale and Luc Vanden-dorpe are with the ICTEAM Institute Universite catholique de Lou-vain Place du Levant 2 1348 - Louvain La Neuve Belgium Emailtadilobogale lucvandendorpeuclouvainbe Phone +3210478071
Fax +3210472089
signal processing at the transmitter (precoder) and receiver
(decoder) Signal processing is performed to meet a certain
design criterion It is well known that most practically relevant
design problems such as weighted sum rate maximization
rate or signal-to-interference-plus-noise-ratio (SINR) balanc-
ing and rate or SINR constrained power minimization can
be equivalently expressed as mean-square-error (MSE) based
problems (see for example [1]) Because of this the current
paper examines MSE-based problems In general the uplink
channel MSE-based problems are better understood than those
of the downlink channel Due to this fact most literatures
focus on solving the downlink MSE-based problems The
downlink MSE-based problems can be solved by direct ap-
proach as in [2] [3] or by uplink-downlink duality approach
as in [4]ndash[6]
For a given downlink channel system model and its MSE-
based problem the idea behind uplink-downlink duality is first
to create the virtual uplink channel by exchanging the roles
of the transmitter and receiver and then to enable the pre-
coderdecoder transformation from uplink to downlink channel
and vice versa by ensuring the same MSE in both channelsOnce these two tasks are performed the downlink MSE-
based problems are examined as follows When the global
optimality of the dual uplink channel MSE-based problem is
guaranteed the duality approach simply transfers the optimal
uplink channel precoderdecoder pairs from uplink to downlink
channel (see for example the sum MSE minimization problem
in [5]) When the global optimal solution of the dual uplink
channel MSE-based problem can not ensured the duality
approach examines the downlink MSE-based problems by
iteratively switching between the uplink and downlink channel
problems (see for example the problems in [7])
Several MSE-based problems have been examined by dual-
ity approach [4]ndash[8] However the duality of these papers are
able to solve total BS power constrained MSE-based problems
only In a practical multi-antenna base station (BS) system
the maximum power of each BS antenna is limited [9] In
some scenario allocating different powers to different users
(symbols) according to their priority or protection level has
some interest This motivates [10] to solve (robust) sum MSE-
based problems with per antenna user and symbol power
constraints by duality approach However since the problems
in [10] allocate the same MSE weight to all symbols (users)
[10] ignores priority and fairness issues in terms of MSE
In a multimedia communication different types of in-
formation (for example audio and video information) can
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Fig 1 Multiuser MIMO system model (a) downlink channel (b) virtualinterference channel
denoting the symbol intended for the k th user as dk isin CS ktimes1
and S =sumK
k=1 S k the entire symbol can be written in a data
vector d isin CS times1 as d = [dT 1 middot middot middot dT
K ]T The BS precodes d
into an N length vector by using its overall precoder matrix
B = [b11 middot middot middot bKS K ] where bks isin CN times1
is the precodervector of the BS for the kth MS sth symbol The kth MS
employs a receiver wks to estimate the symbol dks We follow
the same channel matrix notations as in [8] The estimates of
the k th MS sth symbol (dks) and k th user (dk) are given by
dks =wH ks(HH
k
K 991761i=1
Bidi + nk) = wH ks(HH
k Bd + nk) (1)
dk =WH k (HH
k Bd + nk) (2)
where HH k isin CM ktimesN is the channel matrix between the BS
and kth MS Wk = [wk1 middot middot middotwkS k ] Bk = [bk1 middot middot middotbkS k ] and
nk is the kth MS additive noise Without loss of generality
we can assume that the entries of dk are independent and
identically distributed (iid) ZMCSCG random variables all
with unit variance ie EdkdH k = IS k Edkd
H i = 0
foralli = k and EdknH i = 0 foralli k The kth MS noise vector is
a ZMCSCG random variable with covariance matrix Rnk isinCM ktimesM k
To establish our MSE downlink-interference duality we
model the virtual interference channel (Fig 1(b)) is modeled
by introducing precoders Vk = [vk1 middot middot middot vkS k ]K k=1 and de-
coders Tk = [tk1 middot middot middot tkS k ]K k=1 where vks isin CM ktimes1 and
tks isin CN times1 forallk s In this channel it is assumed that the k th
userrsquos sth symbol (dks) is an iid ZMCSCG random variable
with variance ζ ks and estimated independently by tks isin C N times1
ie EdksdH ks = ζ ks EdksdH
ij = 0 forall(i j) = (k s) and
EdknH i = 0 foralli k Moreover nI
ks forallsK k=1 (Fig 1(b))
are also ZMCSCG random variables with covariance matrices
∆ks isin realN timesN = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1 and the
channels between the k th transmitter and all receivers are the
same (ie Hkjs = Hk forall j sK k=1) Note that from the system
model aspect the current paper and [10] share the same idea
As can be seen from Fig 1 the outputs of Fig 1(a) and
Fig 1(b) are not the same However since Fig 1(b) is a
rdquovirtualrdquo interference channel which is introduced just to solve
the downlink MSE-based problems by duality approach the
output of Fig 1(b) is not required in practice For this reason
the difference in the outputs of the downlink and interference
channels of Fig 1 will not affect the downlink MSE-based
problem formulations and the duality based solutions
For the downlink system model of Fig 1(a) the symbol-
wise and user-wise MSEs can be expressed as
ξ DLks =Ed(dks minus dks)(dks minus dks)H
=wH ks(HH
k BBH Hk + Rnk)wks minus wH ksH
H k bksminus
bH ksHkwks + 1 (3)
ξ DLk =Ed(dk minus dk)(dk minus dk)H
=trIS k + WH k (HH
k BBH Hk + Rnk)Wkminus
WH k H
H k Bk minus BH
k HkWk (4)
Using these two equations the symbol-wise and user-wise
WSMSEs can be expressed as
ξ DLws =
K 991761k=1
S k991761s=1
ηksξ DLks = trη + ηWH HH BBH HW+
ηWH RnW minus ηWH HH B minus ηBH HW (5)
ξ DLwu =
K 991761k=1
ηkξ DLk = trη + ηWH HH BBH HW
+ ηWH RnW minus ηWH HH B minus ηBH HW (6)
where Rn = blkdiag(Rn1 middot middot middot RnK ) η =diag(η11 middot middot middot η1S 1 middot middot middot ηK 1 middot middot middot ηKS K ) and η =blkdiag(η1IS 1 middot middot middot ηK IS K ) with ηks and ηk are the
MSE weights of the kth user sth symbol and kth user
respectively Like in the downlink channel the interference
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
λ = blkdiag(λ1IS 1 middot middot middot λK IS K ) and Γc =sumK i=1
sumS ij=1 ζ ijHivijv
H ijH
H i with λks and λk are the
MSE weights of the kth user sth symbol and kth user
respectively
III PROBLEM F ORMULATION
The aforementioned MSE-based optimization problems
can be formulated as
P 1 minBkWkKk=1
K
991761k=1
S k
991761s=1 ηksξ
DL
ks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (11)
P 2 minBkWk
Kk=1
K 991761k=1
ηkξ DLk
st [BBH ](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (12)
P 3 minBkWkKk=1
max ρksξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (13)
P 4 minBkWkKk=1
max ρkξ DLk
st [BBH
](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (14)
where ρk(ˆ pk) and ρks(˘ pks) are the MSE balancing weights
(maximum available power) of the kth user and kth user sth
symbol respectively and ˘ pn denotes the maximum transmitted
power by the nth antenna
For both the WSMSE minimization and min max WMSE
problems different weights are given to different symbols
(users) However at optimality the solutions of these two
problems are not necessarily the same This is due to the
fact that the aim of the WSMSE minimization problem is
just to minimize the WSMSE of all symbols (users) (ie in
such a problem the minimized WMSE of each symbol (user)
depends on its corresponding channel gain) whereas the aim
of min max WMSE problem is to minimize and balance the
WMSE of each symbol (user) simultaneously (ie in such
a problem all symbols (users) achieve the same minimized
WMSE [13]) Moreover as will be clear later the solution ap-
proach of WSMSE minimization problem can not be extended
straightforwardly to solve the min max WMSE problem Due
to these facts we examine the WSMSE minimization and min
max WMSE problems separatelySince the problems P 1 - P 4 are not convex convex
optimization framework can not be applied to solve them To
the best of our knowledge duality based solutions for these
problems are not known In the following we present an MSE
downlink-interference duality based approach for solving each
of these problems which is shown in Algorithm I2
Algorithm I
Initialization For each problem initialize Bk = 0K k=1
such that the power constraint functions are satisfied3
Then update WkK k=1 by using minimum mean-
square-error (MMSE) receiver approach ie
Wk = (HH k BBH Hk + Rnk)minus1HH k Bk forallk (15)
Repeat Interference channel
1) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from downlink to interference channel
2) Update the receivers of the interference channel
tks forallsK k=1 using MMSE receiver technique
Downlink channel
3) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from interference to downlink channel
4) Update the receivers of the downlink channel WkK k=1
by MMSE receiver approach (15)
Until convergence
The above iterative algorithm is already known in [5] [8] and
[10] However the approaches of these papers can not ensure
the power constraints of P 1 - P 4 at step 3 of Algorithm I
Hence one can not apply the approaches of these papers to
solve P 1 - P 4 In the following sections we establish our
MSE downlink-interference duality
IV SYMBOL-WISE WSMSE DOWNLINK-INTERFERENCE
DUALITY
This duality is established to solve symbol-wise WSMSE-
based problems (for example P 1)
A Symbol-wise WSMSE transfer (From downlink to interfer-
ence channel)
In order to use this WSMSE transfer for solving P 1 we set
the interference channel precoder decoder noise covariance
input covariance and MSE weight matrices as
V = β W T = Bβ ζ = η λ = I ∆ks = Ψ + microksI
(16)
2As will be clear later in Section VIII to solve P 3 and P 4 (and moregeneral MSE-based problems) an additional power allocation step is requiredIn Algorithm I this step is omitted for clarity of presentation
3For the simulation we use Bk = [Hk](1Sk)Kk=1 followed by the
appropriate normalization of BkKk=1 to ensure the power constraints
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(x) [ f 1 middot middot middot f N f 11 middot middot middot f 1S 1 middot middot middot f K 1 middot middot middot f KS K ]with xn = ψn isin [ϵ (β 2τ minus ϵ
sumN i=1 i=n pim)pnm]N
n=1
and xrS +N r=N +1 = microks =isin [ϵ (β 2τ minus
ϵsumK
i=1
sumS ij=1(ij)=(ks) pijm)pksm] forallsK
k=14 As we can
see from (27) when ∥(x1) minus(x2)∥2 = 0 with x1 = x2 or
x1 = x2 one can set κ(κ) = 0 and χ(χ) = 0 to satisfy thisinequality And when ∥(x1) minus(x2)∥2 gt 0 (ie x1 = x2)
one can select appropriate κ(κ) isin [0 1) and χ(χ) ge 0 such
that (27) is satisfied This is due to the fact that in the latter
case ∥x2 minus (x1)∥2 gt 0 andor ∥x1 minus (x2)∥2 gt 0 and
∥x1 minus x2∥2 gt 0 are positive and bounded This explanation
shows the existence of κ(κ) isin [0 1) and χ(χ) ge 0 ensuring
(27) for any ∥(x1) minus (x2)∥2 x1x2 isin X Consequently
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Fig 1 Multiuser MIMO system model (a) downlink channel (b) virtualinterference channel
denoting the symbol intended for the k th user as dk isin CS ktimes1
and S =sumK
k=1 S k the entire symbol can be written in a data
vector d isin CS times1 as d = [dT 1 middot middot middot dT
K ]T The BS precodes d
into an N length vector by using its overall precoder matrix
B = [b11 middot middot middot bKS K ] where bks isin CN times1
is the precodervector of the BS for the kth MS sth symbol The kth MS
employs a receiver wks to estimate the symbol dks We follow
the same channel matrix notations as in [8] The estimates of
the k th MS sth symbol (dks) and k th user (dk) are given by
dks =wH ks(HH
k
K 991761i=1
Bidi + nk) = wH ks(HH
k Bd + nk) (1)
dk =WH k (HH
k Bd + nk) (2)
where HH k isin CM ktimesN is the channel matrix between the BS
and kth MS Wk = [wk1 middot middot middotwkS k ] Bk = [bk1 middot middot middotbkS k ] and
nk is the kth MS additive noise Without loss of generality
we can assume that the entries of dk are independent and
identically distributed (iid) ZMCSCG random variables all
with unit variance ie EdkdH k = IS k Edkd
H i = 0
foralli = k and EdknH i = 0 foralli k The kth MS noise vector is
a ZMCSCG random variable with covariance matrix Rnk isinCM ktimesM k
To establish our MSE downlink-interference duality we
model the virtual interference channel (Fig 1(b)) is modeled
by introducing precoders Vk = [vk1 middot middot middot vkS k ]K k=1 and de-
coders Tk = [tk1 middot middot middot tkS k ]K k=1 where vks isin CM ktimes1 and
tks isin CN times1 forallk s In this channel it is assumed that the k th
userrsquos sth symbol (dks) is an iid ZMCSCG random variable
with variance ζ ks and estimated independently by tks isin C N times1
ie EdksdH ks = ζ ks EdksdH
ij = 0 forall(i j) = (k s) and
EdknH i = 0 foralli k Moreover nI
ks forallsK k=1 (Fig 1(b))
are also ZMCSCG random variables with covariance matrices
∆ks isin realN timesN = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1 and the
channels between the k th transmitter and all receivers are the
same (ie Hkjs = Hk forall j sK k=1) Note that from the system
model aspect the current paper and [10] share the same idea
As can be seen from Fig 1 the outputs of Fig 1(a) and
Fig 1(b) are not the same However since Fig 1(b) is a
rdquovirtualrdquo interference channel which is introduced just to solve
the downlink MSE-based problems by duality approach the
output of Fig 1(b) is not required in practice For this reason
the difference in the outputs of the downlink and interference
channels of Fig 1 will not affect the downlink MSE-based
problem formulations and the duality based solutions
For the downlink system model of Fig 1(a) the symbol-
wise and user-wise MSEs can be expressed as
ξ DLks =Ed(dks minus dks)(dks minus dks)H
=wH ks(HH
k BBH Hk + Rnk)wks minus wH ksH
H k bksminus
bH ksHkwks + 1 (3)
ξ DLk =Ed(dk minus dk)(dk minus dk)H
=trIS k + WH k (HH
k BBH Hk + Rnk)Wkminus
WH k H
H k Bk minus BH
k HkWk (4)
Using these two equations the symbol-wise and user-wise
WSMSEs can be expressed as
ξ DLws =
K 991761k=1
S k991761s=1
ηksξ DLks = trη + ηWH HH BBH HW+
ηWH RnW minus ηWH HH B minus ηBH HW (5)
ξ DLwu =
K 991761k=1
ηkξ DLk = trη + ηWH HH BBH HW
+ ηWH RnW minus ηWH HH B minus ηBH HW (6)
where Rn = blkdiag(Rn1 middot middot middot RnK ) η =diag(η11 middot middot middot η1S 1 middot middot middot ηK 1 middot middot middot ηKS K ) and η =blkdiag(η1IS 1 middot middot middot ηK IS K ) with ηks and ηk are the
MSE weights of the kth user sth symbol and kth user
respectively Like in the downlink channel the interference
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
λ = blkdiag(λ1IS 1 middot middot middot λK IS K ) and Γc =sumK i=1
sumS ij=1 ζ ijHivijv
H ijH
H i with λks and λk are the
MSE weights of the kth user sth symbol and kth user
respectively
III PROBLEM F ORMULATION
The aforementioned MSE-based optimization problems
can be formulated as
P 1 minBkWkKk=1
K
991761k=1
S k
991761s=1 ηksξ
DL
ks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (11)
P 2 minBkWk
Kk=1
K 991761k=1
ηkξ DLk
st [BBH ](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (12)
P 3 minBkWkKk=1
max ρksξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (13)
P 4 minBkWkKk=1
max ρkξ DLk
st [BBH
](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (14)
where ρk(ˆ pk) and ρks(˘ pks) are the MSE balancing weights
(maximum available power) of the kth user and kth user sth
symbol respectively and ˘ pn denotes the maximum transmitted
power by the nth antenna
For both the WSMSE minimization and min max WMSE
problems different weights are given to different symbols
(users) However at optimality the solutions of these two
problems are not necessarily the same This is due to the
fact that the aim of the WSMSE minimization problem is
just to minimize the WSMSE of all symbols (users) (ie in
such a problem the minimized WMSE of each symbol (user)
depends on its corresponding channel gain) whereas the aim
of min max WMSE problem is to minimize and balance the
WMSE of each symbol (user) simultaneously (ie in such
a problem all symbols (users) achieve the same minimized
WMSE [13]) Moreover as will be clear later the solution ap-
proach of WSMSE minimization problem can not be extended
straightforwardly to solve the min max WMSE problem Due
to these facts we examine the WSMSE minimization and min
max WMSE problems separatelySince the problems P 1 - P 4 are not convex convex
optimization framework can not be applied to solve them To
the best of our knowledge duality based solutions for these
problems are not known In the following we present an MSE
downlink-interference duality based approach for solving each
of these problems which is shown in Algorithm I2
Algorithm I
Initialization For each problem initialize Bk = 0K k=1
such that the power constraint functions are satisfied3
Then update WkK k=1 by using minimum mean-
square-error (MMSE) receiver approach ie
Wk = (HH k BBH Hk + Rnk)minus1HH k Bk forallk (15)
Repeat Interference channel
1) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from downlink to interference channel
2) Update the receivers of the interference channel
tks forallsK k=1 using MMSE receiver technique
Downlink channel
3) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from interference to downlink channel
4) Update the receivers of the downlink channel WkK k=1
by MMSE receiver approach (15)
Until convergence
The above iterative algorithm is already known in [5] [8] and
[10] However the approaches of these papers can not ensure
the power constraints of P 1 - P 4 at step 3 of Algorithm I
Hence one can not apply the approaches of these papers to
solve P 1 - P 4 In the following sections we establish our
MSE downlink-interference duality
IV SYMBOL-WISE WSMSE DOWNLINK-INTERFERENCE
DUALITY
This duality is established to solve symbol-wise WSMSE-
based problems (for example P 1)
A Symbol-wise WSMSE transfer (From downlink to interfer-
ence channel)
In order to use this WSMSE transfer for solving P 1 we set
the interference channel precoder decoder noise covariance
input covariance and MSE weight matrices as
V = β W T = Bβ ζ = η λ = I ∆ks = Ψ + microksI
(16)
2As will be clear later in Section VIII to solve P 3 and P 4 (and moregeneral MSE-based problems) an additional power allocation step is requiredIn Algorithm I this step is omitted for clarity of presentation
3For the simulation we use Bk = [Hk](1Sk)Kk=1 followed by the
appropriate normalization of BkKk=1 to ensure the power constraints
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(x) [ f 1 middot middot middot f N f 11 middot middot middot f 1S 1 middot middot middot f K 1 middot middot middot f KS K ]with xn = ψn isin [ϵ (β 2τ minus ϵ
sumN i=1 i=n pim)pnm]N
n=1
and xrS +N r=N +1 = microks =isin [ϵ (β 2τ minus
ϵsumK
i=1
sumS ij=1(ij)=(ks) pijm)pksm] forallsK
k=14 As we can
see from (27) when ∥(x1) minus(x2)∥2 = 0 with x1 = x2 or
x1 = x2 one can set κ(κ) = 0 and χ(χ) = 0 to satisfy thisinequality And when ∥(x1) minus(x2)∥2 gt 0 (ie x1 = x2)
one can select appropriate κ(κ) isin [0 1) and χ(χ) ge 0 such
that (27) is satisfied This is due to the fact that in the latter
case ∥x2 minus (x1)∥2 gt 0 andor ∥x1 minus (x2)∥2 gt 0 and
∥x1 minus x2∥2 gt 0 are positive and bounded This explanation
shows the existence of κ(κ) isin [0 1) and χ(χ) ge 0 ensuring
(27) for any ∥(x1) minus (x2)∥2 x1x2 isin X Consequently
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Fig 1 Multiuser MIMO system model (a) downlink channel (b) virtualinterference channel
denoting the symbol intended for the k th user as dk isin CS ktimes1
and S =sumK
k=1 S k the entire symbol can be written in a data
vector d isin CS times1 as d = [dT 1 middot middot middot dT
K ]T The BS precodes d
into an N length vector by using its overall precoder matrix
B = [b11 middot middot middot bKS K ] where bks isin CN times1
is the precodervector of the BS for the kth MS sth symbol The kth MS
employs a receiver wks to estimate the symbol dks We follow
the same channel matrix notations as in [8] The estimates of
the k th MS sth symbol (dks) and k th user (dk) are given by
dks =wH ks(HH
k
K 991761i=1
Bidi + nk) = wH ks(HH
k Bd + nk) (1)
dk =WH k (HH
k Bd + nk) (2)
where HH k isin CM ktimesN is the channel matrix between the BS
and kth MS Wk = [wk1 middot middot middotwkS k ] Bk = [bk1 middot middot middotbkS k ] and
nk is the kth MS additive noise Without loss of generality
we can assume that the entries of dk are independent and
identically distributed (iid) ZMCSCG random variables all
with unit variance ie EdkdH k = IS k Edkd
H i = 0
foralli = k and EdknH i = 0 foralli k The kth MS noise vector is
a ZMCSCG random variable with covariance matrix Rnk isinCM ktimesM k
To establish our MSE downlink-interference duality we
model the virtual interference channel (Fig 1(b)) is modeled
by introducing precoders Vk = [vk1 middot middot middot vkS k ]K k=1 and de-
coders Tk = [tk1 middot middot middot tkS k ]K k=1 where vks isin CM ktimes1 and
tks isin CN times1 forallk s In this channel it is assumed that the k th
userrsquos sth symbol (dks) is an iid ZMCSCG random variable
with variance ζ ks and estimated independently by tks isin C N times1
ie EdksdH ks = ζ ks EdksdH
ij = 0 forall(i j) = (k s) and
EdknH i = 0 foralli k Moreover nI
ks forallsK k=1 (Fig 1(b))
are also ZMCSCG random variables with covariance matrices
∆ks isin realN timesN = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1 and the
channels between the k th transmitter and all receivers are the
same (ie Hkjs = Hk forall j sK k=1) Note that from the system
model aspect the current paper and [10] share the same idea
As can be seen from Fig 1 the outputs of Fig 1(a) and
Fig 1(b) are not the same However since Fig 1(b) is a
rdquovirtualrdquo interference channel which is introduced just to solve
the downlink MSE-based problems by duality approach the
output of Fig 1(b) is not required in practice For this reason
the difference in the outputs of the downlink and interference
channels of Fig 1 will not affect the downlink MSE-based
problem formulations and the duality based solutions
For the downlink system model of Fig 1(a) the symbol-
wise and user-wise MSEs can be expressed as
ξ DLks =Ed(dks minus dks)(dks minus dks)H
=wH ks(HH
k BBH Hk + Rnk)wks minus wH ksH
H k bksminus
bH ksHkwks + 1 (3)
ξ DLk =Ed(dk minus dk)(dk minus dk)H
=trIS k + WH k (HH
k BBH Hk + Rnk)Wkminus
WH k H
H k Bk minus BH
k HkWk (4)
Using these two equations the symbol-wise and user-wise
WSMSEs can be expressed as
ξ DLws =
K 991761k=1
S k991761s=1
ηksξ DLks = trη + ηWH HH BBH HW+
ηWH RnW minus ηWH HH B minus ηBH HW (5)
ξ DLwu =
K 991761k=1
ηkξ DLk = trη + ηWH HH BBH HW
+ ηWH RnW minus ηWH HH B minus ηBH HW (6)
where Rn = blkdiag(Rn1 middot middot middot RnK ) η =diag(η11 middot middot middot η1S 1 middot middot middot ηK 1 middot middot middot ηKS K ) and η =blkdiag(η1IS 1 middot middot middot ηK IS K ) with ηks and ηk are the
MSE weights of the kth user sth symbol and kth user
respectively Like in the downlink channel the interference
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
λ = blkdiag(λ1IS 1 middot middot middot λK IS K ) and Γc =sumK i=1
sumS ij=1 ζ ijHivijv
H ijH
H i with λks and λk are the
MSE weights of the kth user sth symbol and kth user
respectively
III PROBLEM F ORMULATION
The aforementioned MSE-based optimization problems
can be formulated as
P 1 minBkWkKk=1
K
991761k=1
S k
991761s=1 ηksξ
DL
ks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (11)
P 2 minBkWk
Kk=1
K 991761k=1
ηkξ DLk
st [BBH ](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (12)
P 3 minBkWkKk=1
max ρksξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (13)
P 4 minBkWkKk=1
max ρkξ DLk
st [BBH
](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (14)
where ρk(ˆ pk) and ρks(˘ pks) are the MSE balancing weights
(maximum available power) of the kth user and kth user sth
symbol respectively and ˘ pn denotes the maximum transmitted
power by the nth antenna
For both the WSMSE minimization and min max WMSE
problems different weights are given to different symbols
(users) However at optimality the solutions of these two
problems are not necessarily the same This is due to the
fact that the aim of the WSMSE minimization problem is
just to minimize the WSMSE of all symbols (users) (ie in
such a problem the minimized WMSE of each symbol (user)
depends on its corresponding channel gain) whereas the aim
of min max WMSE problem is to minimize and balance the
WMSE of each symbol (user) simultaneously (ie in such
a problem all symbols (users) achieve the same minimized
WMSE [13]) Moreover as will be clear later the solution ap-
proach of WSMSE minimization problem can not be extended
straightforwardly to solve the min max WMSE problem Due
to these facts we examine the WSMSE minimization and min
max WMSE problems separatelySince the problems P 1 - P 4 are not convex convex
optimization framework can not be applied to solve them To
the best of our knowledge duality based solutions for these
problems are not known In the following we present an MSE
downlink-interference duality based approach for solving each
of these problems which is shown in Algorithm I2
Algorithm I
Initialization For each problem initialize Bk = 0K k=1
such that the power constraint functions are satisfied3
Then update WkK k=1 by using minimum mean-
square-error (MMSE) receiver approach ie
Wk = (HH k BBH Hk + Rnk)minus1HH k Bk forallk (15)
Repeat Interference channel
1) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from downlink to interference channel
2) Update the receivers of the interference channel
tks forallsK k=1 using MMSE receiver technique
Downlink channel
3) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from interference to downlink channel
4) Update the receivers of the downlink channel WkK k=1
by MMSE receiver approach (15)
Until convergence
The above iterative algorithm is already known in [5] [8] and
[10] However the approaches of these papers can not ensure
the power constraints of P 1 - P 4 at step 3 of Algorithm I
Hence one can not apply the approaches of these papers to
solve P 1 - P 4 In the following sections we establish our
MSE downlink-interference duality
IV SYMBOL-WISE WSMSE DOWNLINK-INTERFERENCE
DUALITY
This duality is established to solve symbol-wise WSMSE-
based problems (for example P 1)
A Symbol-wise WSMSE transfer (From downlink to interfer-
ence channel)
In order to use this WSMSE transfer for solving P 1 we set
the interference channel precoder decoder noise covariance
input covariance and MSE weight matrices as
V = β W T = Bβ ζ = η λ = I ∆ks = Ψ + microksI
(16)
2As will be clear later in Section VIII to solve P 3 and P 4 (and moregeneral MSE-based problems) an additional power allocation step is requiredIn Algorithm I this step is omitted for clarity of presentation
3For the simulation we use Bk = [Hk](1Sk)Kk=1 followed by the
appropriate normalization of BkKk=1 to ensure the power constraints
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(x) [ f 1 middot middot middot f N f 11 middot middot middot f 1S 1 middot middot middot f K 1 middot middot middot f KS K ]with xn = ψn isin [ϵ (β 2τ minus ϵ
sumN i=1 i=n pim)pnm]N
n=1
and xrS +N r=N +1 = microks =isin [ϵ (β 2τ minus
ϵsumK
i=1
sumS ij=1(ij)=(ks) pijm)pksm] forallsK
k=14 As we can
see from (27) when ∥(x1) minus(x2)∥2 = 0 with x1 = x2 or
x1 = x2 one can set κ(κ) = 0 and χ(χ) = 0 to satisfy thisinequality And when ∥(x1) minus(x2)∥2 gt 0 (ie x1 = x2)
one can select appropriate κ(κ) isin [0 1) and χ(χ) ge 0 such
that (27) is satisfied This is due to the fact that in the latter
case ∥x2 minus (x1)∥2 gt 0 andor ∥x1 minus (x2)∥2 gt 0 and
∥x1 minus x2∥2 gt 0 are positive and bounded This explanation
shows the existence of κ(κ) isin [0 1) and χ(χ) ge 0 ensuring
(27) for any ∥(x1) minus (x2)∥2 x1x2 isin X Consequently
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
λ = blkdiag(λ1IS 1 middot middot middot λK IS K ) and Γc =sumK i=1
sumS ij=1 ζ ijHivijv
H ijH
H i with λks and λk are the
MSE weights of the kth user sth symbol and kth user
respectively
III PROBLEM F ORMULATION
The aforementioned MSE-based optimization problems
can be formulated as
P 1 minBkWkKk=1
K
991761k=1
S k
991761s=1 ηksξ
DL
ks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (11)
P 2 minBkWk
Kk=1
K 991761k=1
ηkξ DLk
st [BBH ](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (12)
P 3 minBkWkKk=1
max ρksξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks (13)
P 4 minBkWkKk=1
max ρkξ DLk
st [BBH
](nn) le ˘ pn trBH k Bk le ˆ pk foralln k (14)
where ρk(ˆ pk) and ρks(˘ pks) are the MSE balancing weights
(maximum available power) of the kth user and kth user sth
symbol respectively and ˘ pn denotes the maximum transmitted
power by the nth antenna
For both the WSMSE minimization and min max WMSE
problems different weights are given to different symbols
(users) However at optimality the solutions of these two
problems are not necessarily the same This is due to the
fact that the aim of the WSMSE minimization problem is
just to minimize the WSMSE of all symbols (users) (ie in
such a problem the minimized WMSE of each symbol (user)
depends on its corresponding channel gain) whereas the aim
of min max WMSE problem is to minimize and balance the
WMSE of each symbol (user) simultaneously (ie in such
a problem all symbols (users) achieve the same minimized
WMSE [13]) Moreover as will be clear later the solution ap-
proach of WSMSE minimization problem can not be extended
straightforwardly to solve the min max WMSE problem Due
to these facts we examine the WSMSE minimization and min
max WMSE problems separatelySince the problems P 1 - P 4 are not convex convex
optimization framework can not be applied to solve them To
the best of our knowledge duality based solutions for these
problems are not known In the following we present an MSE
downlink-interference duality based approach for solving each
of these problems which is shown in Algorithm I2
Algorithm I
Initialization For each problem initialize Bk = 0K k=1
such that the power constraint functions are satisfied3
Then update WkK k=1 by using minimum mean-
square-error (MMSE) receiver approach ie
Wk = (HH k BBH Hk + Rnk)minus1HH k Bk forallk (15)
Repeat Interference channel
1) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from downlink to interference channel
2) Update the receivers of the interference channel
tks forallsK k=1 using MMSE receiver technique
Downlink channel
3) Transfer the symbol-wise (user-wise) WSMSE or
WMSE from interference to downlink channel
4) Update the receivers of the downlink channel WkK k=1
by MMSE receiver approach (15)
Until convergence
The above iterative algorithm is already known in [5] [8] and
[10] However the approaches of these papers can not ensure
the power constraints of P 1 - P 4 at step 3 of Algorithm I
Hence one can not apply the approaches of these papers to
solve P 1 - P 4 In the following sections we establish our
MSE downlink-interference duality
IV SYMBOL-WISE WSMSE DOWNLINK-INTERFERENCE
DUALITY
This duality is established to solve symbol-wise WSMSE-
based problems (for example P 1)
A Symbol-wise WSMSE transfer (From downlink to interfer-
ence channel)
In order to use this WSMSE transfer for solving P 1 we set
the interference channel precoder decoder noise covariance
input covariance and MSE weight matrices as
V = β W T = Bβ ζ = η λ = I ∆ks = Ψ + microksI
(16)
2As will be clear later in Section VIII to solve P 3 and P 4 (and moregeneral MSE-based problems) an additional power allocation step is requiredIn Algorithm I this step is omitted for clarity of presentation
3For the simulation we use Bk = [Hk](1Sk)Kk=1 followed by the
appropriate normalization of BkKk=1 to ensure the power constraints
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(x) [ f 1 middot middot middot f N f 11 middot middot middot f 1S 1 middot middot middot f K 1 middot middot middot f KS K ]with xn = ψn isin [ϵ (β 2τ minus ϵ
sumN i=1 i=n pim)pnm]N
n=1
and xrS +N r=N +1 = microks =isin [ϵ (β 2τ minus
ϵsumK
i=1
sumS ij=1(ij)=(ks) pijm)pksm] forallsK
k=14 As we can
see from (27) when ∥(x1) minus(x2)∥2 = 0 with x1 = x2 or
x1 = x2 one can set κ(κ) = 0 and χ(χ) = 0 to satisfy thisinequality And when ∥(x1) minus(x2)∥2 gt 0 (ie x1 = x2)
one can select appropriate κ(κ) isin [0 1) and χ(χ) ge 0 such
that (27) is satisfied This is due to the fact that in the latter
case ∥x2 minus (x1)∥2 gt 0 andor ∥x1 minus (x2)∥2 gt 0 and
∥x1 minus x2∥2 gt 0 are positive and bounded This explanation
shows the existence of κ(κ) isin [0 1) and χ(χ) ge 0 ensuring
(27) for any ∥(x1) minus (x2)∥2 x1x2 isin X Consequently
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(x) [ f 1 middot middot middot f N f 11 middot middot middot f 1S 1 middot middot middot f K 1 middot middot middot f KS K ]with xn = ψn isin [ϵ (β 2τ minus ϵ
sumN i=1 i=n pim)pnm]N
n=1
and xrS +N r=N +1 = microks =isin [ϵ (β 2τ minus
ϵsumK
i=1
sumS ij=1(ij)=(ks) pijm)pksm] forallsK
k=14 As we can
see from (27) when ∥(x1) minus(x2)∥2 = 0 with x1 = x2 or
x1 = x2 one can set κ(κ) = 0 and χ(χ) = 0 to satisfy thisinequality And when ∥(x1) minus(x2)∥2 gt 0 (ie x1 = x2)
one can select appropriate κ(κ) isin [0 1) and χ(χ) ge 0 such
that (27) is satisfied This is due to the fact that in the latter
case ∥x2 minus (x1)∥2 gt 0 andor ∥x1 minus (x2)∥2 gt 0 and
∥x1 minus x2∥2 gt 0 are positive and bounded This explanation
shows the existence of κ(κ) isin [0 1) and χ(χ) ge 0 ensuring
(27) for any ∥(x1) minus (x2)∥2 x1x2 isin X Consequently
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(x) [ f 1 middot middot middot f N f 11 middot middot middot f 1S 1 middot middot middot f K 1 middot middot middot f KS K ]with xn = ψn isin [ϵ (β 2τ minus ϵ
sumN i=1 i=n pim)pnm]N
n=1
and xrS +N r=N +1 = microks =isin [ϵ (β 2τ minus
ϵsumK
i=1
sumS ij=1(ij)=(ks) pijm)pksm] forallsK
k=14 As we can
see from (27) when ∥(x1) minus(x2)∥2 = 0 with x1 = x2 or
x1 = x2 one can set κ(κ) = 0 and χ(χ) = 0 to satisfy thisinequality And when ∥(x1) minus(x2)∥2 gt 0 (ie x1 = x2)
one can select appropriate κ(κ) isin [0 1) and χ(χ) ge 0 such
that (27) is satisfied This is due to the fact that in the latter
case ∥x2 minus (x1)∥2 gt 0 andor ∥x1 minus (x2)∥2 gt 0 and
∥x1 minus x2∥2 gt 0 are positive and bounded This explanation
shows the existence of κ(κ) isin [0 1) and χ(χ) ge 0 ensuring
(27) for any ∥(x1) minus (x2)∥2 x1x2 isin X Consequently
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
the second inequality is due to the fact that |||XY|||1 le|||X|||1|||Y|||1 [17] (page 290) the third equality is obtained
by applying Theorem 2 and the last inequality employs thefollowing facts Using the definition (73) (see Appendix A)
one can get ||| ˇΩ|||1 le 1 by applying (46) and (51) and
||| ˇP|||1 le 1 by applying (13) (41) and (51)
Thus maxn ∥ (xprimen)∥⋆ = 1 holds true and (54) is guar-
anteed to converge As we can see (54) is derived by using
(41) and (46) Thus the solution of (54) also satisfies (41)
and (46) Moreover for any initial xprime0 gt 0 since (xprimen) foralln is
positive the solution of (54) is strictly positive and [ψ micro]T =
( ˜P)minus1xprime gt 0 which is the desired result
7Since (xprimen) is the products of stochastic matrices (see the proof of Theorem 2) (xprimen) is a bounded matrix for any xprime gt 0 Thus the assumption
(xprimen) = Fσn foralln holds true
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
(ie minimizes the maximum WMSE of all symbols (users))
each iteration of this algorithm is not able to guarantee
balanced WMSEs of all symbols (users) On the other hand
for an MSE constrained total BS power minimization problem
(for example P 7 in Section IX) an iterative algorithm that
can provide a non increasing sequence of total BS power is
required This shows that Algorithm I also can not solve the
latter problem In the following we address the drawbacks of Algorithm I just by including a power allocation step into
Algorithm I as explained below
In [8] for fixed transmit and receive filters the power
allocation parts of total BS power constrained MSE-based
problems have been formulated as GPs by employing the
approach and system model of [1] under the assumption that
all symbols are strictly active8 For this assumption in [8] we
show that the system model of [1] is appropriate to solve any
kind of total BS power constrained MSE-based problems using
duality approach (alternating optimization) This motivates us
to utilize the system model of [1] in the downlink channel
only and then include the power allocation step (ie GP) into
Algorithm I Towards this end we decompose the precoders
and decoders of the downlink channel as
Bk =GkP12k Wk = UkαkP
minus12k forallk (64)
where Pk = diag( pk1 middot middot middot pkS k) isin realS ktimesS k Gk =[gk1 middot middot middot gkS k ] isin CN timesS k Uk = [uk1 middot middot middot ukS k ] isin CM ktimesS k
and αk = diag(αk1 middot middot middot αkS k) isin realS ktimesS k are the transmit
power unity norm transmit filter unity norm receive filter and
receiver scaling factor matrices of the kth user respectively
ie gH ksgks = uH
ksuks = 1 forallsK k=1
By employing (64) and stacking ξ = [ξ DL11 middot middot middot ξ DL
KS K]T
= [ξ DL1 middot middot middot ξ DL
S ]T = [ξ DLl S
l=1]T the l th downlink symbol
MSE can be expressed as (see [1] and [10] for more detailsabout (64) and the above descriptions)
ξ DLl = pminus1l [(D + α2ΦT )p]l + pminus1l α2
l uH l Rnul (65)
where
Φ(lj) =
983163 |gH
l Huj |2 for l = j0 for l = j
(66)
D(ll) =α2l |gH
l Hul|2 minus 2αlreal(uH l H
H gl) + 1 (67)
1 le l( j) le S P = blkdiag(P1 middot middot middot PK ) =diag( p1 middot middot middot pS ) p = [ p1 middot middot middot pS ]
T G = [G1 middot middot middot GK ] =[g1 middot middot middot gS ] U = blkdiag(U1 middot middot middot UK ) = [u1 middot middot middot uS ]
and α = blkdiag(α1 middot middot middot αK ) = diag(α1 middot middot middot αS ) with∥gl∥2 = ∥ul∥2 = 1 Using (65) for fixed GU and α the
power allocation part of P 1 can be formulated as
min plSl=1
S 991761l=1
ηlξ DLl st ς T
np le ˘ pn pl le ˘ pl foralln l (68)
where ς T n isin real1timesS = |[G(ni)|2S
i=1 [η1 middot middot middot ηS ]T =
[η11 middot middot middot ηKS K ]T and [ ˘ pl middot middot middot ˘ pS ]T = [ ˘ p11 middot middot middot ˘ pKS K ]T As
ξ DLl is a posynomial (where plS
l=1 are the variables) (68)
8Note that this assumption is not always true for all MSE-based problemsHowever as mentioned in [1] in practice replacing zero powers by a smallvalue will not affect the overall optimization Due to this reason we replace
zero powers by 10minus6 in the simulation section
is a GP for which global optimality is guaranteed Thus it
can be efficiently solved using interior point methods with a
worst-case polynomial-time complexity [18]
For fixed GU and α the power allocation parts of
P 2 minus P 4 can be formulated as GPs like in P 1 Our duality
based algorithm for each of these problems including the
power allocation step is summarized in Algorithm II
Algorithm II
Initialization Like in Algorithm I
Repeat Interference channel
1) For P 1 and P 2 set V = WT = B (ie β = β = 1)
then compute ψn microks forallksn and ψn microk forallk nusing (25) and (38) respectively For P 3 and P 4 first
compute ψn microks forallksn and ψn microk forallk n using
(54) and (63) respectively then transfer each symbol
and user MSE from downlink to interference channels
by (39) and (56) respectively
2) Update the MMSE receivers of the interference channel
for P 1 P 2 P 3 and P 4 using (19) (32) (44) and (59)
respectivelyDownlink channel
3) Transfer the MSE (weighted sum user or symbol MSE)
from interference to downlink channel using (20) (33)
(45) and (60) for P 1 P 2 P 3 and P 4 respectively
4) For each of the problems P 1 minus P 4 decompose the
precoder and decoder matrices of each user as in (64)
Then formulate and solve the GP power allocation part
For example the power allocation part of P 1 can be
expressed in GP form as (68)
5) For each of the problems P 1minusP 4 by keeping PkK k=1
constant update the receive filters UkK k=1 and scal-
ing factors αkK
k=1 by applying downlink MMSE
receiver approach ie Ukαk = (HH k GPGH Hk +
Rnk)minus1HH k GkPkK
k=1 Note that in these expressions
αkK k=1 are chosen such that each column of UkK
k=1
has unity norm Then compute BkWkK k=1 by (64)
Until convergence
Convergence It can be shown that at each iteration
of this algorithm the objective function of each of the
problems P 1 - P 4 is non-increasing [4] [7] [19] Thus
the above iterative algorithm is convergent However
since P 1 - P 4 are non-convex this iterative algorithm
is not guaranteed to converge to the global optimum
In this algorithm we stop iteration (ie our convergence
condition) when the difference between the objectivefunctions in two consecutive iterations is smaller than
some small value ϵ (we use ϵ = 10minus6 for the simulation)
Computational complexity As can be seen from this
algorithm when we increase the number of users andor
(BS andor MS antennas) the number and size of
optimization variables increase Because of this the
computational complexity of Algorithm II increases as
K andor N andor M increases However studying the
complexity of this algorithm as a function of K N and
M needs effort and time And such a task is beyond the
scope of this work and is an open research topic
The power allocation step of Algorithm II has thus the
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
following benefits (1) For BS power constrained WSMSE
minimization problems this step improves the convergence
speed of Algorithm II compared to that of Algorithm I9 (for
example in P 1 minus P 2) The degree of improvement depends on
different parameters (for example Hk ∆ks forallk s etc) Thus
the theoretical comparison of these two algorithms in terms of
convergence speed requires time and effort And this task is
beyond the scope of this work and it is an open research topic(2) For symbol-wise (user-wise) WMSE balancing problems
this step helps to balance the WMSE of all symbols (users)
(for example in P 3 minus P 4) (3) For MSE constrained total
BS power minimization problems this step ensures a non
increasing total BS power at each iteration of Algorithm II
IX APPLICATION OF THE PROPOSED DUALITY BASED
ALGORITHM FOR OTHER PROBLEMS
A MSE based problem with entry-wise power constraint
The symbol-wise WSMSE minimization constrained with
entry wise power ie bH
ksnb
ksn le macr p
ksn forallksn problem is
formulated as
P 5 minBkWkKk=1
K 991761k=1
S k991761s=1
ηksξ DLks st bH
ksnbksn le macr pksn (69)
It can be shown that this problem can be solved by Algorithm
II with ∆ks = diag(δ ks1 middot middot middot δ ksN ) forallsK k=1
B Weighted sum rate optimization constrained with per an-
tenna and symbol power problem
By employing the approach of [11] (see (16) of [11]) one
can equivalently express the weighted sum rate maximizationconstrained with per antenna and symbol power problem as
P 6 (70)
minτ ksν ksbkswksforallsK
k=1
K 991761k=1
S k991761s=1
θks1
τ ksν γ ks
ks +K 991761
k=1
S k991761s=1
ηksξ DLks
st [BBH ](nn) le ˘ pnbH ksbks le ˘ pks
K prodk=1
S kprods=1
ν ks = 1 τ ks gt 0
where 0 lt ωks lt 1 forallsK k=1 are the rate weighting factors
for all symbols ηks = τ microksks γ ks = 11minusωks
microks = 1ωks
minus 1 and
macrθks = ωksmicro
(1minusωks)
ks For fixed τ ks ν ks foralls
K
k=1 the aboveoptimization problem has the same mathematical structure
as that of P 1 Thus by keeping τ ks ν ks forallsK k=1 constant
bkswks forallsK k=1 can be optimized by applying the MSE
duality discussed in Section IV Moreover τ ks ν ks forallsK k=1
and the power allocation part of the above problem can be
optimized by a GP method like in (25) of [20] Consequently
we can apply Algorithm II to solve (70) The detailed expla-
nations are omitted for conciseness The following problems
9This is at the expense of additional computation However as mentionedin [1] (see Appendix A of [1]) a small desktop computer can solve a GP of 100 variables and 10000 constraints by standard interior point method under aminute Thus we believe that the complexity of Algorithm I and Algorithm
II are almost the same
can also be solved by simple modification of Algorithm II
P 7 minBkWk
Kk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
SINRks ge ϱks forallnks
equiv minBkWkKk=1
K 991761k=1
trBkBH k
st [BBH ](nn) le ˘ pn
trBH k Bk le ˆ pk
ξ DLks le (1 + ϱks)minus1 forallnks
P 8 maxBkWk
Kk=1
min Rks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
equiv minBkWkKk=1
max ξ DLks
st [BBH ](nn) le ˘ pn bH ksbks le ˘ pks forallnks
where SINRks(Rks) is the SINR (rate) of the kth user sth
symbol and we use the fact that Rks = log(1 + SINRks)and ξ DL
ks = (1 + SINRks)minus1 [1] It is clearly seen that the
application of Algorithm II is not limited to the problems of
this paper
Note that under imperfect channel state information (CSI)
condition the stochastic robust design versions of P 1 - P 5 can
be solved like in [10] However to the best of our knowledge
the relationship between rate (SINR) and MSE is not known
when the CSI is imperfect [7] Hence solving the rate (SINR)-
based robust design problems (for example robust versions of
P 6 - P 8) by our duality approach is an open problem
X SIMULATION R ESULTS
In this section we present simulation results for P 1 minus P 4
All of our simulation results are averaged over 100 randomly
chosen channel realizations We set K = 2 N = 4 and
M k = S k = 2 ηks = ρks = ηk = ρk = 1 forallsK k=1
It is assumed that Rn1 = σ21IM 1 Rn2 = σ2
2IM 2 and
σ22 = 2σ2
1 The maximum power of each BS antenna is set
to ˘ pn = 25mW N n=1 And the maximum power allocated
to each symbol and user are set to ˘ pks
= 25mW forallsK
k=1and ˆ pk = 5mW K k=1 respectively For better exposition we
define the Signal-to-noise ratio (SNR) as P maxKσ2av and it
is controlled by varying σ2av where P max = 10mW is the
total maximum BS power and σ2av = (σ2
1 + σ22)2 We also
compare Algorithm II and the algorithm in [2]10
Note that the algorithm in [2] is designed for coordinated
BS systems scenario And the iterative algorithm of [2] is
based on the per BS power constraint However according
to [2] and [21] B coordinated BS systems each with Z
10As will be clear in the sequel the proposed algorithm and the algorithmof [2] may not utilize the entire 10mW for all noise levels Thus the afore-mentioned SNR can be considered as the maximum SNR of the transmitted
signal
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
From this equation we can see that if ρ(A) lt 1 the
nonnegativity of Aminus1 can be ensured Next we show that ρ(A)is indeed less than 1 For any n times n matrix X we have [17]
(pages 294 and 297 of [17])
ρ(X) le|||X||| |||X||1 max1lejlen
n991761i=1
|xij | (73)
where |||||| is any matrix norm and ||||||1 is a matrix onenorm By using (73) we get the following bound [17]
ρ(A) le |||A|||1 lt 1 (74)
Since Aminus1 has nonnegative elements A is also an M-matrix
[22] By defining S Aminus1 and e 1ntimes1 we get
eT A = eT rArr eT = eT S =[n991761
j=1
Sj1 middot middot middot n991761
j=1
Sjn]
rArr |||S|||1 =1 (75)
where the third equality follows from the fact that S is a
nonnegative matrix
REFERENCES
[1] S Shi M Schubert and H Boche ldquoRate optimization for multiuserMIMO systems with linear processingrdquo IEEE Tran Sig Proc vol 56no 8 pp 4020 ndash 4030 Aug 2008
[2] S Shi M Schubert N Vucic and H Boche ldquoMMSE optimizationwith per-base-station power constraints for network MIMO systemsrdquoin Proc IEEE International Conference on Communications (ICC)Beijing China 19 ndash 23 May 2008 pp 4106 ndash 4110
[3] P Ubaidulla and A Chockalingam ldquoRobust transceiver design formultiuser MIMO downlinkrdquo in Proc IEEE Global Telecommunications
Conference (GLOBECOM) Nov 30 ndash Dec 4 2008 pp 1 ndash 5[4] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiver
optimization for multiuser MIMO systems Duality and sum-MSEminimizationrdquo IEEE Tran Sig Proc vol 55 no 11 pp 5436 ndash 5446
Nov 2007[5] R Hunger M Joham and W Utschick ldquoOn the MSE-duality of the
broadcast channel and the multiple access channelrdquo IEEE Tran Sig
Proc vol 57 no 2 pp 698 ndash 713 Feb 2009[6] M Schubert S Shi E A Jorswieck and H Boche ldquoDownlink sum-
MSE transceiver optimization for linear multi-user MIMO systemsrdquo inConference Record of the Thirty-Ninth Asilomar Conference on Signals
Systems and Computers Oct 28 ndash Nov 1 2005 pp 1424 ndash 1428[7] T E Bogale B K Chalise and L Vandendorpe ldquoRobust transceiver
optimization for downlink multiuser MIMO systemsrdquo IEEE Tran Sig
Proc vol 59 no 1 pp 446 ndash 453 Jan 2011[8] T E Bogale and L Vandendorpe ldquoMSE uplink-downlink duality of
MIMO systems with arbitrary noise covariance matricesrdquo in 45th Annual
conference on Information Sciences and Systems (CISS) Baltimore MDUSA 23 ndash 25 Mar 2011
[9] W Yu and T Lan ldquoTransmitter optimization for the multi-antenna
downlink with per-antenna power constraintsrdquo IEEE Trans Sig Procvol 55 no 6 pp 2646 ndash 2660 Jun 2007[10] T E Bogale and L Vandendorpe ldquoRobust sum MSE optimization
for downlink multiuser MIMO systems with arbitrary power constraintGeneralized duality approachrdquo IEEE Tran Sig Proc Dec 2011
[11] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO coordinated base station systems Centralizedand distributed algorithmsrdquo IEEE Tran Sig Proc Dec 2011
[12] H Sampath P Stoica and A Paulraj ldquoGeneralized linear precoderand decoder design for MIMO channels using the weighted MMSEcriterionrdquo IEEE Tran Sig Proc vol 49 no 12 pp 2198 ndash 2206 Dec2001
[13] S Shi M Schubert and H Boche ldquoDownlink MMSE transceiveroptimization for multiuser MIMO systems MMSE balancingrdquo IEEE
Tran Sig Proc vol 56 no 8 pp 3702 ndash 3712 Aug 2008[14] V Berinde ldquoGeneral constructive fixed point theorems for Ciric-type
almost contractions in metric spacesrdquo Carpathian J Math vol 24 no
2 pp 10 ndash 19 2008
[15] V Berinde ldquoApproximation fixed points of weak contractions using thePicard iterationrdquo Nonlinear Analysis Forum vol 9 no 1 pp 43 ndash 532004
[16] Z Sun ldquoA note on marginal stability of switched systemsrdquo IEEE Tran
Auto Cont vol 53 no 2 pp 625 ndash 631 2008[17] R H Horn and C R Johnson Matrix Analysis Cambridge University
Press Cambridge 1985[18] S Boyd and L Vandenberghe Convex optimization Cambridge
University Press Cambridge 2004[19] T E Bogale and L Vandendorpe ldquoSum MSE optimization for downlink
multiuser MIMO systems with per antenna power constraint Downlink-uplink duality approachrdquo in 22nd IEEE International Symposium on
Personal Indoor and Mobile Radio Communications (PIMRC) TorontoCanada 11 ndash 14 Sep 2011
[20] T E Bogale and L Vandendorpe ldquoWeighted sum rate optimization fordownlink multiuser MIMO systems with per antenna power constraintDownlink-uplink duality approachrdquo in IEEE International Conference
On Acuostics Speech and Signal Processing (ICASSP) Kyoto Japan25 ndash 30 Mar 2012
[21] T Tamaki K Seong and J M Cioffi ldquoDownlink MIMO systems usingcooperation among base stations in a slow fading channelrdquo in Proc
IEEE International Conference on Communications (ICC) GlasgowUK 24 ndash 28 Jun 2007 pp 4728 ndash 4733
[22] G Poole and T Boullion ldquoA survey on M-matricesrdquo Society for
Industrial and Applied Mathematics (SIAM) vol 16 no 4 pp 419ndash 427 1974
Tadilo Endeshaw Bogale was born in GondarEthiopia He received his BSc and MSc degreein Electrical Engineering from Jimma UniversityJimma Ethiopia and Karlstad University KarlstadSweden in 2004 and 2008 respectively From 2004-2007 he was working in Ethiopian Telecommuni-cations Corporation (ETC) in mobile project depart-ment
Since 2009 he has been working towards his PhDdegree and as an assistant researcher at the ICTEAMinstitute University Catholique de Louvain (UCL)
Louvain-la-Neuve Belgium His reasearch interests include robust (non-robust) transceiver design for multiuser MIMO systems centralized anddistributed algorithms and convex optimization techniques for multiuser
systems
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE
8142019 Linear Transceiver design for Downlink Multiuser MIMO Systems Downlink-Interference Duality Approach
Luc Vandendorpe (Mrsquo93-SMrsquo99-Frsquo06) was bornin Mouscron Belgium in 1962 He received theElectrical Engineering degree (summa cum laude)and the PhD degree from the Universit Catholiquede Louvain (UCL) Louvain-la-Neuve Belgium in1985 and 1991 respectively Since 1985 he has beenwith the Communications and Remote Sensing Lab-oratory of UCL where he first worked in the fieldof bit rate reduction techniques for video coding In1992 he was a Visiting Scientist and Research Fel-low at the Telecommunications and Traffic Control
Systems Group of the Delft Technical University The Netherlands where heworked on spread spectrum techniques for personal communications systemsFrom October 1992 to August 1997 he was Senior Research Associate of the Belgian NSF at UCL and invited Assistant Professor He is currentlya Professor and head of the Institute for Information and CommunicationTechnologies Electronics and Applied Mathematics
His current interest is in digital communication systems and more pre-cisely resource allocation for OFDM(A)-based multicell systems MIMO anddistributed MIMO sensor networks turbo-based communications systemsphysical layer security and UWB based positioning
Dr Vandendorpe was corecipient of the 1990 Biennal Alcatel-Bell Awardfrom the Belgian NSF for a contribution in the field of image coding In2000 he was corecipient (with J Louveaux and F Deryck) of the BiennalSiemens Award from the Belgian NSF for a contribution about filter-bank-based multicarrier transmission In 2004 he was co-winner (with J Czyz)
of the Face Authentication Competition FAC 2004 He is or has beenTPC member for numerous IEEE conferences (VTC Globecom SPAWCICC PIMRC WCNC) and for the Turbo Symposium He was Co-TechnicalChair (with P Duhamel) for the IEEE ICASSP 2006 He was an Editorfor Synchronization and Equalization of the IEEE TRANSACTIONS ONCOMMUNICATIONS between 2000 and 2002 Associate Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS between 2003 and2005 and Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING between 2004 and 2006 He was Chair of the IEEE Benelux
joint chapter on Communications and Vehicular Technology between 1999 and2003 He was an elected member of the Signal Processing for Communicationscommittee between 2000 and 2005 and an elected member of the SensorArray and Multichannel Signal Processing committee of the Signal ProcessingSociety between 2006 and 2008 Currently he is an elected member of theSignal Processing for Communications committee He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networking LVandendorpe is a Fellow of the IEEE