8 September, 2015 Coordinated Multiantenna Interference Management in 5G Networks 1 Coordinated Multiantenna Interference Management in 5G Networks Antti T¨ olli [email protected]Department of Communications Engineering (DCE) University of Oulu, Finland 8 September, 2015 c Antti T¨olli, Department of Comm. Engineering
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OutlineEvolution of multiantenna systemsMIMO with large antenna arraysLinear transceiver design and resource allocation
I Introduction to convex optimisationI Resource allocation and linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming
Minimum power multicell beamforming with QoS constraintsI Centralised solutionI Decentralised solution via optimisation decompositionI Large system approximation
Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilotI Bidirectional signalling strategies for dynamic TDDI Mode selection and transceiver design in underlay D2D MIMO
General ObjectiveGoal: Design dynamic multi-dimensional radio resource managementacross time, frequency, and space (location)Assumption: Heterogeneous network composed of
I Large macro cells with massive MIMO antenna arrays,I Small cells and relays with small or distributed MIMO arrays, andI D2D communication with macro cell coordination
Coherent (joint) multi-cell transmissionI Each data stream may be transmitted from multiple nodesI Tight synchronisation across the transmitting nodes (common carrier
phase reference)I A high-speed backbone network, e.g. Radio over Fibre
I Dynamic multi-cell scheduling and inter-cell interference avoidanceI Coordinated precoder design and beam allocationI Each data stream is transmitted from a single BS nodeI No carrier phase coherence requirementI Looser requirement on the coordination and the backhaul →
Interference AlignmentExact capacity characterization of the K-user interference channel isunknownDegrees of freedom (or multiplexing gain)DoF = lim
SNR→∞sum ratelog2 SNR
Achievable via interference alignment (IA) 1
Feasibility conditions for IA (for constant MIMO channel, K ≥ 3)234
(K + 1)d ≤ 2M =⇒ Total DoF ≤ 2MK
K + 1≤ 2M (1)
1V.R. Cadambe and S.A. Jafar. ”Interference Alignment and Degrees of Freedom of the K-User Interference Channel.”IEEE Trans. Inform. Theory, August 2008.
2C. Yetis, T. Gou, S. Jafar, and A. Kayran, ”On feasibility of interference alignment in MIMO interference networks,” IEEETrans. Signal Process., 2010.
3M. Razaviyayn, G. Lyubeznik, and Z.Q. Luo, ”On the Degrees of Freedom Achievable Through Interference Alignment ina MIMO Interference Channel,” IEEE Trans. Signal Process., 2012
4G. Bresler, D. Cartwright, and D. Tse, Feasibility of Interference Alignment for the MIMO Interference Channel,” in IEEETrans. Info. Theory, 2014
OutlineEvolution of multiantenna systemsMIMO with large antenna arraysLinear transceiver design and resource allocation
I Introduction to convex optimisationI Resource allocation and linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming
Minimum power multicell beamforming with QoS constraintsI Centralised solutionI Decentralised solution via optimisation decompositionI Large system approximation
Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilotI Bidirectional signalling strategies for dynamic TDDI Mode selection and transceiver design in underlay D2D MIMO
D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge,England: Cambridge University Press, 2005.
A. M. Tulino and Sergio Verd, Random Matrix Theory and Wireless Communications,Foundations and Trends in Communications and Information Theory, vol. 1, no. 1, pp1-182.
Rusek, F.; Persson, D.; Buon Kiong Lau; Larsson, E.G.; Marzetta, T.L.; Edfors, O.;Tufvesson, F., ”Scaling Up MIMO: Opportunities and Challenges with Very LargeArrays,” Signal Processing Magazine, IEEE , vol.30, no.1, pp.40–60, Jan. 2013
Foundations and Trends in Communications and Information Theory, Vol. 1, Issue 1,”Random Matrix Theory and Wireless Communications” by A. Tulino and S Verdu
R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications, 1sted. Cambridge University Press, 2011.
Point-to-point MIMO Architecture334 MIMO II: capacity and multiplexing architectures
Figure 8.1 The V-BLASTarchitecture for communicatingover the MIMO channel.
+
Pnt
P1
Qx[m]
H[m]
w[m]
y[m] Jointdecoder
AWGN coderrate R1
AWGN coderrate Rnt
····
········
coordinate system given by a unitary matrix Q, not necessarily dependent onthe channel matrix H. This is the V-BLAST architecture. The data streamsare decoded jointly. The kth data stream is allocated a power Pk (such thatthe sum of the powers, P1+· · ·+Pnt
, is equal to P, the total transmit powerconstraint) and is encoded using a capacity-achieving Gaussian code with rateRk. The total rate is R=!nt
k=1Rk.As special cases:
• If Q=V and the powers are given by the waterfilling allocations, then wehave the capacity-achieving architecture in Figure 7.2.
• If Q= Inr , then independent data streams are sent on the different transmitantennas.
Using a sphere-packing argument analogous to the ones used in Chapter 5,we will argue an upper bound on the highest reliable rate of communication:
R < logdet"Inr +
1N0
HKxH!#bits/s/Hz! (8.2)
Here Kx is the covariance matrix of the transmitted signal x and is a functionof the multiplexing coordinate system and the power allocations:
Kx "=Q diag#P1$ % % % $Pnt&Q!! (8.3)
Considering communication over a block of time symbols of length N , thereceived vector, of length nrN , lies with high probability in an ellipsoid ofvolume proportional to
det'N0Inr +HKxH!(N ! (8.4)
This formula is a direct generalization of the corresponding volume for-mula (5.50) for the parallel channel, and is justified in Exercise 8.2. Sincewe have to allow for non-overlapping noise spheres (of radius
"N0 and,
hence, volume proportional to NnrN0 ) around each codeword to ensure reliable
[D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.]
Generalized architecture to multiplex nt independent data streams
The choice of Q depends on the CSITI Q = V requires full CSIT → capacity achieving scheme with WF
power allocationI Q = I requires no CSIT → independent streams sent on each TX
Large Antenna Array RegimeFocus on the square channel n = nt = nr
Cnn(SNR) = E
[n∑
i=1
log
(1 + SNR
λ2in
)](5)
where λi/√n are the singular values of H/
√n
When n→∞, thedistribution of λi/
√n
becomes deterministic
f∗(x) =
{1π
√4− x2 0 ≤ x ≤ 2,
0 else
342 MIMO II: capacity and multiplexing architectures
due to Marcenko and Pastur [78], the empirical distribution of the singularvalues of H/
!n converges to a deterministic limiting distribution for almost
all realizations of H. Figure 8.4 demonstrates the convergence. The limitingdistribution is the so-called quarter circle law.3 The corresponding limitingdensity of the squared singular values is given by
f "!x"=
!"
#
1#
$1x# 1
4 0 $ x $ 4$
0 else%(8.23)
Hence, we can conclude that, for increasing n,
1n
n%
i=1
log&1+ SNR
&2i
n
'%
( 4
0log!1+ SNRx"f "!x"dx% (8.24)
If we denote
c"!SNR" '=( 4
0log!1+ SNRx"f "!x"dx$ (8.25)
Figure 8.4 Convergence of theempirical singular valuedistribution of H/
!n. For
each n, a single randomrealization of H/
!n is
generated and the empiricaldistribution (histogram) of thesingular values is plotted. Wesee that as n grows, thehistogram converges to thequarter circle law.
0 0.5 1 1.5 20
1
2
3
4n = 32
0 0.5 1 1.5 20
2
4
6
8
10n = 64
0 0.5 1 1.5 20
5
10
15
20n = 128
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7Quarter circle law
3 Note that although the singular values are unbounded, in the limit they lie in the interval(0$2) with probability 1.
[D. Tse and P. Viswanath, Fundamentals of Wireless Communication,Cambridge University Press, 2005]
Capacity grows linearly in n at any SNR5More in: Foundations and Trends in Communications and Information Theory, Vol. 1, Issue 1, ”Random Matrix Theory
and Wireless Communications” by A. Tulino and S Verdu
we can solve the integral for the density in (8.23) to arrive at (see Exer-cise 8.17)
c!!SNR"= 2 log!1+ SNR" 1
4F!SNR"
"" log e
4SNRF!SNR"# (8.26)
where
F!SNR" $=##
4SNR+1"1$2
% (8.27)
The significance of c!!SNR" is that
limn$%
Cnn!SNR"n
= c!!SNR"% (8.28)
So capacity grows linearly in n at any SNR and the constant c!!SNR" is therate of the growth.We compare the large-n approximation
Cnn!SNR"& nc!!SNR"# (8.29)
with the actual value of the capacity for n = 2#4 in Figure 8.5. We see theapproximation is very good, even for such small values of n. In Exercise 8.7,we see statistical models other than i.i.d. Rayleigh, which also have a linearincrease in capacity with an increase in n.
Linear scaling: a more in-depth lookTo better understand why the capacity scales linearly with the number ofantennas, it is useful to contrast the MIMO scenario here with three otherscenarios:
Figure 8.5 Comparisonbetween the large-napproximation and the actualcapacity for n= 2! 4.
–5 0 10 15 20SNR (dB)
25 30
Approximate capacity c!
–10
9
8
7
6
5
4
3
2
1
0
Rat
e(b
its /s
/ Hz)
5
Exact capacity 14 C44
Exact capacity 12 C22
Figure: Comparison between the large-n approximation and the actual capacityfor n = 2, 4. [D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005]
Uplink System ModelAssume time-invariant uplink channel with K single-antenna usersand a single BS with nr receive antennas.
426 MIMO IV: multiuser communication
the downlink. We conclude in Section 10.5 with a discussion of the systemimplications of using MIMO in cellular networks; this will link up the newinsights obtained here with those in Chapters 4 and 6.
10.1 Uplink with multiple receive antennas
We begin with the narrowband time-invariant uplink with each user havinga single transmit antenna and the base-station equipped with an array ofantennas (Figure 10.1). The channels from the users to the base-station aretime-invariant. The baseband model is
y!m"=K!
k=1
hkxk!m"+w!m"# (10.1)
with y!m" being the received vector (of dimension nr , the number of receiveantennas) at time m, and hk the spatial signature of user k impinged on thereceive antenna array at the base-station. User k’s scalar transmit symbol attime m is denoted by xk!m" and w!m" is i.i.d. !" $0#N0Inr% noise.
10.1.1 Space-division multiple access
In the literature, the use of multiple receive antennas in the uplink is oftencalled space-division multiple access (SDMA): we can discriminate amongstthe users by exploiting the fact that different users impinge different spatialsignatures on the receive antenna array.An easy observation we can make is that this uplink is very similar to
the MIMO point-to-point channel in Chapter 5 except that the signals sent
Figure 10.1 The uplink withsingle transmit antenna at eachuser and multiple receiveantennas at the base-station.
out on the transmit antennas cannot be coordinated. We studied preciselysuch a signaling scheme using separate data streams on each of the transmitantennas in Section 8.3. We can form an analogy between users and transmitantennas (so nt , the number of transmit antennas in the MIMO point-to-pointchannel in Section 8.3, is equal to the number of users K). Further, theequivalent MIMO point-to-point channel H is !h1# & & & #hK", constructed fromthe SIMO channels of the users.Thus, the transceiver architecture in Figure 8.1 in conjunction with the
receiver structures in Section 8.3 can be used as an SDMA strategy. Forexample, each of the user’s signal can be demodulated using a linear decorre-lator or an MMSE receiver. The MMSE receiver is the optimal compromisebetween maximizing the signal strength from the user of interest and sup-pressing the interference from the other users. To get better performance, onecan also augment the linear receiver structure with successive cancellationto yield the MMSE–SIC receiver (Figure 10.2). With successive cancella-tion, there is also a further choice of cancellation ordering. By choosing a
[D. Tse and P.Viswanath,
Fundamentals ofWireless Communication,
Cambridge UniversityPress, 2005]
The received signal vector at symbol time m isdescribed by
y[m] =
K∑
k=1
hkxk[m] + n[m]
= Hx[m] + n[m]
(14)
whereI xk is the TX symbol of user k, subject to
E[|x|2] ≤ Pk,I y ∈Cnr is the RX signal,I n ∼ CN (0, N0Inr
) complex white Gaussian noise,I hk =
√akhk ∈Cnr is the channel vector of user k,
where ak is the large scale fading factor and hk is thenormalized channel
Sum capacity expression for SDMA is equal to SU-MIMO withoutCSIT
log
∣∣∣∣∣Inr +
K∑
k=1
PkN0
hkhHk
∣∣∣∣∣ = log
∣∣∣∣Inr +1
N0HKxH
H
∣∣∣∣
=
K∑
i=1
log(1 + γmmse−sick ) (15)
where H = [h1, . . . ,hK ], Kx = diag(P1, . . . , PK)427 10.1 Uplink with multiple receive antennas
MMSE Receiver 2
MMSE Receiver 1
y[m]
User 2Decode User 2
Subtract User 1
User 1Decode User 1
different order, users are prioritized differently in the sharing of the commonFigure 10.2 The MMSE–SICreceiver: user 1’s data is firstdecoded and then thecorresponding transmit signalis subtracted off before the nextstage. This receiver structure,by changing the ordering ofcancellation, achieves the twocorner points in the capacityregion.
resource of the uplink channel, in the sense that users canceled later are treatedbetter.
Provided that the overall channel matrix H is well-conditioned, all ofthese SDMA schemes can fully exploit the total number of degrees of free-dom min!K"nr# of the uplink channel (although, as we have seen, differentschemes have different power gains). This translates to being able to simul-taneously support multiple users, each with a data rate that is not limitedby interference. Since the users are geographically separated, their trans-mit signals arrive in different directions at the receive array even whenthere is limited scattering in the environment, and the assumption of a well-conditionedH is usually valid. (Recall Example 7.4 in Section 7.2.4.) Contrastthis to the point-to-point case when the transmit antennas are co-located, anda rich scattering environment is needed to provide a well-conditioned channelmatrix H.
Given the power levels of the users, the achieved SINR of each user canbe computed for the different SDMA schemes using the formulas derived inSection 8.3 (Exercise 10.1). Within the class of linear receiver architecture,we can also formulate a power control problem: given target SINR require-ments for the users, how does one optimally choose the powers and linearfilters to meet the requirements? This is similar to the uplink CDMA powercontrol problem described in Section 4.3.1, except that there is a furtherflexibility in the choice of the receive filters as well as the transmit powers.The first observation is that for any choice of transmit powers, one alwayswants to use the MMSE filter for each user, since that choice maximizes theSINR for every user. Second, the power control problem shares the basicmonotonicity property of the CDMA problem: when a user lowers its transmitpower, it creates less interference and benefits all other users in the system.As a consequence, there is a component-wise optimal solution for the pow-ers, where every user is using the minimum possible power to support theSINR requirements. (See Exercise 10.2.) A simple distributed power controlalgorithm will converge to the optimal solution: at each step, each user firstupdates its MMSE filter as a function of the current power levels of the otherusers, and then updates its own transmit power so that its SINR requirementis just met. (See Exercise 10.3.)
[D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005]
Downlink System ModelAssume time-invariant downlink channel with K single-antennausers and a single BS with nt transmit antennas.
448 MIMO IV: multiuser communication
10.3 Downlink with multiple transmit antennas
We now turn to the downlink channel, from the base-station to the multiple
Figure 10.15 The downlinkwith multiple transmit antennasat the base-station and singlereceive antenna at each user.
users. This time the base-station has an array of transmit antennas but eachuser has a single receive antenna (Figure 10.15). It is often a practicallyinteresting situation since it is easier to put multiple antennas at the base-station than at the mobile users. As in the uplink case we first consider thetime-invariant scenario where the channel is fixed. The baseband model of thenarrowband downlink with the base-station having nt antennas and K userswith each user having a single receive antenna is
yk!m"= h!kx!m"+wk!m"# k= 1# $ $ $ #K# (10.31)
where yk!m" is the received vector for user k at time m, h!k is an nt dimen-
sional row vector representing the channel from the base-station to user k.Geometrically, user k observes the projection of the transmit signal in thespatial direction hk in additive Gaussian noise. The noise wk!m"" !" %0#N0&and is i.i.d. in time m. An important assumption we are implicitly makinghere is that the channel’s hk are known to the base-station as well as to theusers.
10.3.1 Degrees of freedom in the downlink
If the users could cooperate, then the resulting MIMO point-to- point channelwould have min%nt#K& spatial degrees of freedom, assuming that the rank ofthe matrix H= !h1# $ $ $ #hK" is full. Can we attain this full spatial degrees offreedom even when users cannot cooperate?Let us look at a special case. Suppose h1# $ $ $ #hK are orthogonal (which is
only possible if K # nt). In this case, we can transmit independent streams ofdata to each user, such that the stream for the kth user 'xk!m"( is along thetransmit spatial signature hk, i.e.,
x!m"=K!
k=1
xk!m"hk) (10.32)
The overall channel decomposes into a set of parallel channels; user k receives
yk!m"= $hk$2xk!m"+wk!m") (10.33)
Hence, one can transmit K parallel non-interfering streams of data to theusers, and attain the full number of spatial degrees of freedom in the channel.What happens in general, when the channels of the users are not orthogonal?
Observe that to obtain non-interfering channels for the users in the exampleabove, the key property of the transmit signature hk is that hk is orthogonal
[D. Tse and P.Viswanath,
Fundamentals ofWireless Communication,
Cambridge UniversityPress, 2005]
The received signal vector at symbol time m is
yk[m] = hHk x[m] + wk[m]
= hHkuk√pkdk[m] +
K∑
i=1,i 6=khHkui√pidi[m] + wk[m]
(18)
I x ∈Cnt is the TX signal vector, subject to powerconstraint E[Tr(xxH)] =
∑Kk=1 pk ≤ P ,
I uk ∈Cnt is the normalised beamformer, ‖uk‖ = 1I dk ∈C is the normalised data symbol, E
[|dk|2
]= 1
I yk ∈C is the RX signal,I wk ∼ CN (0, N0) complex white Gaussian noise,I hk =
Sum Capacity for the Multiuser DownlinkMaximisation of the DL sum rate via dual uplink reformulationOptimal solution from constrained optimisation problem
maxqk
log2
∣∣∣∣∣Int +1
N0
K∑
k=1
qkhkhHk
∣∣∣∣∣
subject to
K∑
k=1
qk ≤ P, qk ≥ 0, k = 1, . . . ,K (19)
where qk is the dual UL power such that∑K
k=1 qk =∑K
k=1 pk = PWhen nt >> K, the objective of (19) is simplified to
maxqk
log
∣∣∣∣IK +1
N0KxH
HH
∣∣∣∣ ≈ maxqk
K∑
k=1
log(1 +qkntakN0
) (20)
where Kx = diag(q1, . . . , qK) and HHHnt≈ diag(a1, . . . , aK)
Linear MMSE Receiver – DerivationConsider the uplink system model y[m] = Hx[m] + n[m] (41),where the decision variables are generated as x = WHyAssume n ∼ CN (0,Kn) and x ∼ CN (0,Kx) are independent, theoptimal linear MMSE receiver is found by minimizing
W = arg minW
E[‖x−WHy‖2
]
︸ ︷︷ ︸MSE
(21)
After differentiation with respect to W and setting the gradient tozero, the optimal MMSE filter is
Lemma 1: Assume a vector x ∈ CN with i.i.d elements which havezero mean and variance equal to 1
N . Also consider a Hermitianmatrix A ∈ CN×N with elements independent of x, then
xHAx− 1
Ntr(A)
N→∞−−−−→ 0 (26)
Lemma 2:Let AN be a complex N ×N matrix with uniformlybounded spectral norm. Also, consider random Hermitian matrixCN such that for smallest eigenvalue of CN there exist an ε withprobability one such that λmin > ε for all large N , then
Theorem 3: Stieltjes transform of Gram matrix YYH. Consider aN × n random matrix denoted by Y, such that elements of Y areindependent and zero mean. The variance of entries is given byE[|yi,j |2] = α2
i,j
mYYH(z) =
1
Ntr(YYH − zIN)−1 − 1
Ntr(Θ(z))
n→∞, Nn→ c−−−−−−−−→ 0
(28)where Θ(z) = diag(θ1(z), ..., θN (z)). The entries θi can be foundby fixed point iteration
OutlineEvolution of multiantenna systemsMIMO with large antenna arraysLinear transceiver design and resource allocation
I Introduction to convex optimisationI Resource allocation and linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming
Minimum power multicell beamforming with QoS constraintsI Centralised solutionI Decentralised solution via optimisation decompositionI Large system approximation
Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilotI Bidirectional signalling strategies for dynamic TDDI Mode selection and transceiver design in underlay D2D MIMO
Engineering designs are often posed as constrained optimisationproblems:
minimise f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p(34)
whereI x is a vector of decision variablesI f0 is the objective functionI fi(x), i = 1, . . . ,m are the inequality constraint functionsI hi(x), i = 1, . . . , p are the equality constraint functions
Hard to solve in generalI especially when the number of variables in x is largeI the problem might have multiple local minimaI difficult to find a feasible solutionI possibly poor convergence rate
Convex Optimisation ProblemIf f0, f1, . . . , fm in
minimise f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p(35)
are convex and hi, i = 1, . . . , p are affine (hi(x) = aTi x− bi), then
I any locally optimal point is globally optimalI feasibility can be determined unambiguouslyI can be solved efficiently using, e.g. interior point methods
incorporated in generic convex optimisation tools
A function f is convex if its domain dom(f) is convex andf(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y) ∀ x, y ∈ dom(f), θ ∈ [0, 1]
then so are the sets
f!1(S) = {x | Ax + b ! S}f(T ) = {Ax + b | x ! T }
An example is coordinate projection {x | (x, y) !S for some y}. As another example, a constraint of theform
"Ax + b"2 # cT x + d,
where A ! Rk"n, a second-order cone constraint, sinceit is the same as requiring the affine expression (Ax +b, cT x + d) to lie in the second-order cone in Rk+1.Similarly, if A0, A1, . . . , Am ! Sn, solution set of thelinear matrix inequality (LMI)
F (x) = A0 + x1A1 + · · · + xmAm $ 0
is convex (preimage of the semidefinite cone under anaffine function).A linear-fractional (or projective) function f : Rm %
Rn has the form
f(x) =Ax + b
cT x + d
and domain dom f = H = {x | cT x + d > 0}. If Cis a convex set, then its linear-fractional transformationf(C) is also convex. This is because linear fractionaltransformations preserve line segments: for x, y ! H,
f([x, y]) = [f(x), f(y)]
PSfrag replacementsx1 x2
x3x4
PSfrag replacementsf(x1) f(x2)
f(x3)f(x4)
Two further properties are helpful in visualizing thegeometry of convex sets. The first is the separatinghyperplane theorem, which states that if S, T & Rn
are convex and disjoint (S ' T = (), then there exists ahyperplane {x | aT x ) b = 0} which separates them.
PSfrag replacements ST
a
The second property is the supporting hyperplane the-orem which states that there exists a supporting hy-perplane at every point on the boundary of a convex
set, where a supporting hyperplane {x | aT x = aT x0}supports S at x0 ! !S if
x ! S * aT x # aT x0
PSfrag replacementsS
x0
a
III. CONVEX FUNCTIONS
In this section, we introduce the reader to someimportant convex functions and techniques for verifyingconvexity. The objective is to sharpen the reader’s abilityto recognize convexity.
A. Convex functionsA function f : Rn % R is convex if its domain dom f
is convex and for all x, y ! dom f , " ! [0, 1]
f("x + (1 ) ")y) # "f(x) + (1 ) ")f(y);
f is concave if )f is convex.PSfrag replacements
xxx
convex concave neither
Here are some simple examples on R: x2 is convex(dom f = R); log x is concave (dom f = R++); andf(x) = 1/x is convex (dom f = R++).It is convenient to define the extension of a convex
function f
f(x) =
!f(x) x ! dom f++ x ,! dom f
Note that f still satisfies the basic definition for allx, y ! Rn, 0 # " # 1 (as an inequality in R - {++}).We will use the same symbol for f and its extension,i.e., we will implicitly assume convex functions areextended.The epigraph of a function f is
A Simple Exampleinfeasible). A point x ! C is an optimal point if f(x) =f! and the optimal set is Xopt = {x ! C | f(x) = f!}.As an example consider the problem
minimize x1 + x2
subject to "x1 # 0"x2 # 01 " $
x1x2 # 0
0 1 2 3 4 50
1
2
3
4
5
PSfrag replacements
x1
x2
CCC
The objective function is f0(x) = [1 1]T x; the feasibleset C is half-hyperboloid; the optimal value is f ! = 2;and the only optimal point is x! = (1, 1).In the standard problem above, the explicit constraints
are given by fi(x) # 0, hi(x) = 0. However, there arealso the implicit constraints: x ! dom fi, x ! domhi,i.e., x must lie in the set
D = dom f0 % · · ·%dom fm %domh1 % · · ·%domhp
which is called the domain of the problem. For example,
minimize " log x1 " log x2
subject to x1 + x2 " 1 # 0
has the implicit constraint x ! D = {x ! R2 | x1 >0, x2 > 0}.A feasibility problem is a special case of the standard
problem, where we are interested merely in finding anyfeasible point. Thus, problem is really to
• either find x ! C• or determine that C = & .
Equivalently, the feasibility problem requires that weeither solve the inequality / equality system
fi(x) # 0, i = 1, . . . , mhi(x) = 0, i = 1, . . . , p
or determine that it is inconsistent.An optimization problem in standard form is a convex
optimization problem if f0, f1, . . . , fm are all convex,and hi are all affine:
minimize f0(x)subject to fi(x) # 0, i = 1, . . . , m
aTi x " bi = 0, i = 1, . . . , p.
This is often written asminimize f0(x)subject to fi(x) # 0, i = 1, . . . , m
Ax = b
where A ! Rp!n and b ! Rp. As mentioned in theintroduction, convex optimization problems have threecrucial properties that makes them fundamentally moretractable than generic nonconvex optimization problems:
1) no local minima: any local optimum is necessarilya global optimum;
2) exact infeasibility detection: using duality theory(which is not cover here), hence algorithms areeasy to initialize;
3) efficient numerical solution methods that can han-dle very large problems.
Note that often seemingly ‘slight’ modifications ofconvex problem can be very hard. Examples include:
• convex maximization, concave minimization, e.g.
maximize 'x'subject to Ax ( b
• nonlinear equality constraints, e.g.
minimize cT xsubject to xT Pix + qT
i x + ri = 0, i = 1, . . . , K
• minimizing over non-convex sets, e.g., Booleanvariables
find xsuch that Ax ( b,
xi ! {0, 1}
To understand global optimality in convex problems,recall that x ! C is locally optimal if it satisfies
y ! C, 'y " x' # R =) f0(y) * f0(x)
for some R > 0. A point x ! C is globally optimalmeans that
y ! C =) f0(y) * f0(x).
For convex optimization problems, any local solution isalso global. [Proof sketch: Suppose x is locally optimal,but that there is a y ! C, with f0(y) < f0(x). Thenwe may take small step from x towards y, i.e., z =!y +(1"!)x with ! > 0 small. Then z is near x, withf0(z) < f0(x) which contradicts local optimality.]There is also a first order condition that characterizes
optimality in convex optimization problems. Suppose f0
is differentiable, then x ! C is optimal iff
y ! C =) +f0(x)T (y " x) * 0
So "+f0(x) defines supporting hyperplane for C at x.This means that if we move from x towards any otherfeasible y, f0 does not decrease.
9
x1[A. Hindi, ”A Tutorial on Convex Optimization”, Proc. of the 2004 American Control Conference Boston, Massachusetts,June, 2004]
Multi-cell MIMO System ModelB BSs, NT TX antennas per BS and NRk RX antennas per user kA user k is served by Mk = |Bk| BSs from the joint processing setBk, Bk ⊆ B = {1, . . . , B}6
yk =∑
b∈Bab,kHb,kx
′b + nk (36)
=∑
b∈Bk
ab,kHb,kxb,k +∑
b∈Bk
ab,kHb,k
∑
i6=kxb,i
+∑
b∈B\Bk
ab,kHb,kx′b + nk
whereI ab,kHb,k ∈CNRk
×NT channel from BS b to user k
I x′b ∈CNT total TX signal from BS b, and
I xb,k is the transmitted data vector from BS b to user k6Extension to multicarrier systems is straightforward – add sub-carrier index c to every variable
Linear Transceiver DesignEntire capacity region of multiuser MIMO DL has been recentlydiscovered
I Also with individual peak power constraint per BS antenna78
I Require complex nonlinear precoding based on dirty paper coding
I Sub-optimal but less complex transmission methods are neededLinear beamforming is usually remarkably simpler in practice
I Dimensionality contraint per BS:
0 ≤∑
k∈Ubmk ≤ NT, 0 ≤ mk ≤ NRk
. (38)
I Dimensionality constraint in the multi-cell network: Upper bound∑k∈U mk ≤ BNT
I Very difficult in general (feasibility conditions for interferencealignment in high SNR)
7W. Yu and T. Lan, ”Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEETransactions on Signal Processing, vol. 55, no. 6, part 1, pp. 2646–2660, Jun. 2007.
8H. Weingarten, Y. Steinberg, and S. Shamai, ”The capacity region of the Gaussian multiple-input multiple-outputbroadcast channel,” IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 3936–3964, Sep. 2006.
Coherent Multi-cell versus Coordinated Single-cellBeamforming
A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and CoordinatedMulti-cell Transmission”, The International Workshop on LTE Evolution in conjunctionwith the International Conference on Communications (ICC’09), Dresden, Germany,June 2009
A. Tolli, M. Codreanu, and M. Juntti, ”Linear multiuser MIMO transceiver design withquality of service and per antenna power constraints,” IEEE Transactions on SignalProcessing, vol. 56, no. 7, pp. 3049 – 3055, Jul. 2008.
A. Tolli, M. Codreanu, and M. Juntti, ”Cooperative MIMO-OFDM cellular system withsoft handover between distributed base station antennas,” IEEE Transactions onWireless Communications, vol. 7, no. 4, pp. 1428–1440, Apr. 2008.
Coordinated single-cell beamformingEach stream is transmitted from a single BS, |Bs| = 1 ∀ sA user ks is typically allocated to arg max
b∈Bab,ks
Near the cell edge, the optimal beam allocation strategy depends onthe the channel Hb,k.
Large gains from fast beam allocation (cell selection) availableI A difficult combinatorial problem → exhaustive searchI Sub-optimal allocation algorithms
Figure: Ergodic sum of user rates of {K,B,NT, NRk} = {4, 2, 2, 1} system, 0
dB single link SNR. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and CoordinatedMulti-cell Transmission”, IEEE ICC’09, Dresden, Germany, June 2009]
Figure: Ergodic sum of user rates of {K,B,NT, NRk} = {4, 2, 2, 1} system, 20
dB single link SNR. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and CoordinatedMulti-cell Transmission”, IEEE ICC’09, Dresden, Germany, June 2009]
Figure: Ergodic sum rate of {K,B,NT, NRk} = {2, 2, 2, 1} system at 20 dB
single link SNR. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and Coordinated Multi-cellTransmission”, IEEE ICC’09, Dresden, Germany, June 2009]
Figure: Ergodic sum rate of {K,B,NT, NRk} = {4, 2, 2, 1} system at 20 dB
single link SNR. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and Coordinated Multi-cellTransmission”, IEEE ICC’09, Dresden, Germany, June 2009]
OutlineEvolution of multiantenna systemsMIMO with large antenna arraysLinear transceiver design and resource allocation
I Introduction to convex optimisationI Resource allocation and linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming
Minimum power multicell beamforming with QoS constraintsI Centralised solutionI Decentralised solution via optimisation decompositionI Large system approximation
Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilotI Bidirectional signalling strategies for dynamic TDDI Mode selection and transceiver design in underlay D2D MIMO
Minimum Power Multi-cell Beamforming with UserSpecific QoS Constraints
D. N. C. Tse and P. Viswanath, Fundamentals of Wireless Communication. CambridgeUniversity Press, 2005, Chapter 10
H. Dahrouj and W. Yu, ”Coordinated beamforming for the multicell multi-antennawireless system”, IEEE Transactions on Wireless Communications, vol. 9, no. 5, pp.1748–1759, 2010.
A. Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cellBeamforming with Limited Backhaul Signalling”, IEEE Trans. on Wireless Comm., vol.10, no. 2, pp. 570 - 580, February 2011
H. Pennanen, A. Tolli and M. Latva-aho, ”Decentralized Coordinated DownlinkBeamforming via Primal Decomposition”, IEEE Signal Processing Letters, vol. 8, no.11,pp. 647 - 650, November 2011
H. Pennanen, A. Tolli and M. Latva-aho, ”Multi-Cell Beamforming with DecentralizedCoordination in Cognitive and Cellular Networks”, IEEE Transactions on SignalProcessing, vol. 62, no. 2, pp. 295 - 308, January 2014
Second order cone is associated withthe Euclidian norm ‖x‖2Important constraint in manyprecoding design applications
2.2 Some important examples 31
x1x2
t
!1
0
1
!1
0
10
0.5
1
Figure 2.10 Boundary of second-order cone in R3, {(x1, x2, t) | (x21+x2
2)1/2 !
t}.
It is (as the name suggests) a convex cone.
Example 2.3 The second-order cone is the norm cone for the Euclidean norm, i.e.,
C = {(x, t) " Rn+1 | #x#2 ! t}
=
!"xt
# $$$$$
"xt
#T "I 00 $1
#"xt
#! 0, t % 0
%.
The second-order cone is also known by several other names. It is called the quadraticcone, since it is defined by a quadratic inequality. It is also called the Lorentz coneor ice-cream cone. Figure 2.10 shows the second-order cone in R3.
2.2.4 Polyhedra
A polyhedron is defined as the solution set of a finite number of linear equalitiesand inequalities:
P = {x | aTj x " bj , j = 1, . . . ,m, cT
j x = dj , j = 1, . . . , p}. (2.5)
A polyhedron is thus the intersection of a finite number of halfspaces and hyper-planes. A!ne sets (e.g., subspaces, hyperplanes, lines), rays, line segments, andhalfspaces are all polyhedra. It is easily shown that polyhedra are convex sets.A bounded polyhedron is sometimes called a polytope, but some authors use theopposite convention (i.e., polytope for any set of the form (2.5), and polyhedron
Boundary of second-order cone in IR3,
{(x1, x2, t) |√
(x21 + x22) ≤ t}Canonical form of SOCP
minimise cTxsubject to ‖Aix + bi‖2 ≤ cT
i x + di, i = 1, . . . ,mFx = g,
(57)
where x ∈ IRn is the opt. variable, Ai ∈ IRni×n and F ∈ IRp×n
Decentralised Solution via Optimisation DecompositionProposed distributed solution
Beamformers are designed locally relying on limited informationexchanged between adjacent BSs
The coupled terms are decoupled by a dual decomposition10,Alternating direction method of multipliers (ADMM)11, or primaldecomposition12 approach
I Decentralized algorithm
The approach is able to guarantee always feasible solutions evenwith low feedback rate
Allows for a number of special cases with reduced backhaulinformation exchange
10A. Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with LimitedBackhaul Signalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011
11C. Shen, T. H. Chang, K. Y. Wang, Z. Qiu, and C. Y. Chi, ”Distributed robust multi-cell coordinated beamforming withimperfect CSI: An ADMM approach,” IEEE Trans. Signal Processing, vol. 60, no. 6, pp. 2988 - 3003, Jun. 2012.
12H. Pennanen, A. Tolli and M. Latva-aho, ”Decentralized Coordinated Downlink Beamforming via Primal Decomposition”,IEEE Signal Processing Letters, vol. 8, no.11, pp. 647 - 650, November 2011
Decentralised Solution via Optimisation Decomposition
Now, (42) can be reformulated for the special case |Bk| = 1 ∀ k as:
min.B∑b=1
∑k∈Ub
∥∥mk
∥∥22
s. t. Γk ≥ γk,∀ k∑i∈Ub
∣∣hb,kmi
∣∣2 ≤ ζ2b,k, ∀ k 6∈ Ub,∀ b(61)
where the variables are mk and ζb,k.
Inter-cell interference generated from a given base station b cannotexceed the user specific thresholds ζb,k ∀ k 6∈ UbBSs are coupled by the interference terms ζb,k. For fixed ζb,k, theproblem would be decoupled between BSs
Introduce local copies ζ(b)b,k of the interference terms ζb,k
Introduce additional equality constraintsI Each ζb,k couples exactly two (adjacent) base stations, i.e., the
serving BS bk and the interfering BS b.I Enforce the two local copies to be equal ζ
(b)b,k = ζ
(bk)b,k ∀ k, b ∈ Bk,
where Bk = B \ bk.
min.B∑b=1
∑k∈Ub
∥∥mk
∥∥22
s. t. Γ(b)k ≥ γk,∀ k ∈ Ub ∀ b∑
i∈Ub
∣∣hb,kmi
∣∣2 ≤ ζ(b)2b,k , ∀ k 6∈ Ub∀ b
ζ(b)b,k = ζ
(bk)b,k , ∀ k, b ∈ Bk
(62)
where the variables mk, and ζ(b)b,k ∀ k, b ∈ Bk are local for each BS b
13S. Boyd, L. Xiao, A. Mutapcic, and J. Mattingley, ”Notes on decomposition methods: course reader for convexoptimization II, Stanford,” 2008. Available online: http://www.stanford.edu/class/ee364b/
Decentralised Solution via Dual DecompositionDual decomposition approach: the consistency constraints in (62) arerelaxed by forming the partial Lagrangian as
L(M1, . . . ,MB , ζ
(1), . . . , ζ(B),ν1, . . . ,νB
)(63)
=B∑b=1
∑k∈Ub
∥∥mk
∥∥2
2+
K∑k=1
∑b∈Bk
νb,k(ζ(b)b,k − ζ
(bk)b,k ) =
B∑b=1
∑k∈Ub
∥∥mk
∥∥2
2+
B∑b=1
νTb ζ(b)
where νb,k ∀ k, b ∈ Bk are real valued Lagrange multipliers associated withthe consistency constraints
The dual function can now be written as
g(ν1, . . . ,νB) =∑B
b=1gb(νb) (64)
where gb(νb) is the minimum value of the partial Lagrangian solved for agiven νb
Figure: Suboptimality of the distributed algorithm versus the iteration number tfor 0 dB and 10 dB SINR targets. [Tolli, H. Pennanen, and P. Komulainen, ”Decentralized MinimumPower Multi-cell Beamforming with Limited Backhaul Signalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 -580, February 2011]
Figure: Sum power of {K,B,NT} = {4, 2, 4} system with 0 dB SINR target.[Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with Limited BackhaulSignalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011]
Figure: Sum power of {K,B,NT} = {4, 2, 4} system with 20 dB SINR target.[Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with Limited BackhaulSignalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011]
coordinated, per user constr.coordinated (ideal), per user constr.ZF for inter−cell interference
Figure: Time evolution of the distributed algorithm with 0 dB SINR target,TSfd = 0.1 (e.g., 30 km/h with 2 ms reporting period). [Tolli, H. Pennanen, and P.Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with Limited Backhaul Signalling”, IEEE Trans. onWireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011]
TS is the signalling period and fd is the maximum Doppler shift.
ExtensionsCognitive underlay cellular network14 - a sum interference constraintis imposed to every primary user k ∈ UP from the secondary BSs BS
∑i∈US
b
∣∣hb,kmi
∣∣2 ≤ φb,k,∀b ∈ BS,∀k ∈ UP
∑b∈BS
φb,k ≤ Φk, ∀k ∈ UP
(68)
(69)
Worst case beamformer design with ellipsoid CSIT uncertainty15,16
hb,k = hb,k + ub,k ∀ b ∈ B, k ∈ UEb,k = {ub,k : ub,kEb,ku
Hb,k ≤ 1} ∀ b ∈ B, k ∈ U
(70)
(71)
where hb,k and ub,k are the estimated channel at the BS and the CSIerror, respectively, and PSD matrix Eb,k defines the CSI accuracy.
14H. Pennanen, A. Tolli and M. Latva-aho, ”Multi-Cell Beamforming with Decentralized Coordination in Cognitive andCellular Networks”, IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 295 - 308, January 2014
15C. Shen, T. H. Chang, K. Y. Wang, Z. Qiu, and C. Y. Chi, ”Distributed robust multi-cell coordinated beamforming withimperfect CSI: An ADMM approach,” IEEE Trans. Signal Processing, vol. 60, no. 6, pp. 2988 - 3003, Jun. 2012.
16H. Pennanen, A. Tolli and M. Latva-aho, ”Decentralized Robust Beamforming for Coordinated Multi-Cell MISONetworks”, IEEE Signal Processing letters, vol. 21, no. 3, pp. 334 - 338, March 2014
Decentralising the Optimal Multi-cell Beamforming viaLarge System Analysis
H. Asgharimoghaddam, A. Tolli & N. Rajatheva, Decentralizing the Optimal Multi-cellBeamforming via Large System Analysis, in Proc. IEEE ICC 2014, Sydney, Australia,June, 2014
H. Asgharimoghaddam, A. Tolli & N. Rajatheva, ”Decentralized Multi-cell BeamformingVia Large System Analysis in Correlated Channels”, in Proc. EUSIPCO 2014, Lisbon,Portugal, September, 2014
H. Asgharimoghaddam, A. Tolli & N. Rajatheva, ”Decentralizing the Optimal Multi-cellBeamforming in Correlated Channels via Large System Analysis”, submitted to IEEETrans on Signal Prog., July 2015
Large Dimension Approximation of (75)As N,K →∞ with K/N ∈ (0, B)
qk − qk −→ 0 (80)
almost surely, where the approximated power qk is given as theresult of the convergence of the following iterative formula
q(t+1)k = q
(t)k
γk
e(t)bk,k
. (81)
e(t)bk,k
is the large system approximation of kth user’s SINR atiteration t. The functions ebk,1, ..., ebk,n are given as the uniquenonnegative solution of the following system of equation,
e(t+1)bk,i
=1
NTr{q(t)i θbk,i(
1
N
∑
l∈U
q(t)l θbk,l
1 + e(t)bk,l
+ µbkIN )−1} ∀i. (82)
where θbk,k is the correlation matrix from user k to BS bk,
Similarly, the entries of matrix G can be approximated by ,19
Gl,k =
γk(λk)2
l = k
− 1N
e′bk,l
λk(1+e∗bk,l
)2l 6= k
(83)
where, e′bk,l(−1) is the differential of ebk,l(z) with respect to z atpoint z = −1
19H. Asgharimoghaddam, A. Tolli & N. Rajatheva, ”Decentralizing the Optimal Multi-cell Beamforming in CorrelatedChannels via Large System Analysis”, submitted to IEEE Trans on Signal Processing, July 2015
OutlineEvolution of multiantenna systemsMIMO with large antenna arraysLinear transceiver design and resource allocation
I Introduction to convex optimisationI Resource allocation and linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming
Minimum power multicell beamforming with QoS constraintsI Centralised solutionI Decentralised solution via optimisation decompositionI Large system approximation
Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilotI Bidirectional signalling strategies for dynamic TDDI Mode selection and transceiver design in underlay D2D MIMO
Throughput Optimal Linear Transmitter-Receiver Design
S. S. Christensen, R. Agarwal, E. Carvalho, and J. Cioffi, ”Weighted sum-rate maximization using weighted MMSEfor MIMO-BC beamforming design,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 47924799, Dec. 2008.
Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weighted MMSE approach to distributed sum-utilitymaximization for a MIMO interfering broadcast channel, IEEE Trans. Signal Processing, vol. 59, no. 9, pp. 4331 –4340, Sep. 2011
Kaleva, J.; Tolli, A.; Juntti, M.; , ”Weighted Sum Rate Maximization for Interfering Broadcast Channel viaSuccessive Convex Approximation”, Global Communications Conference, 2012. GLOBECOM 2012. IEEE , Dec. 2012
P. Komulainen, A. Tolli & M. Juntti, Effective CSI Signaling and Decentralized Beam Coordination in TDDMulti-Cell MIMO Systems, IEEE Transactions on Signal Processing, vol. 61, no. 9, pp. 2204 – 2218, May 2013
J. Kaleva, A. Tolli & M. Juntti, ”Primal Decomposition based Decentralized Weighted Sum Rate Maximization withQoS Constraints for Interfering Broadcast Channel”, in Proc. IEEE SPAWC 2013, Darmstadt, Germany, June, 2013
J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Beamforming for Weighted Sum Rate Maximization with RateConstraints”, in Proc. IEEE PIMRC 2013 - Workshop on Cooperative and Heterogeneous Cellular Networks, London,UK, Sep. 2013
J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Sum Rate Maximization with QoS Constraints for InterferingBroadcast Channel via Successive Convex Approximation”, IEEE Transactions on Signal Processing, submitted Feb2014, major revision May 2015
MSE ReformulationFor fixed Mk, the rate maximizing Uk are solved from the roots ofthe Lagrangian of (93) as Uk = R−1k Hbk,kMk, k = 1, . . . ,KFor fixed receive beamformers Uk, the concave objective function isiteratively linearised w.r.t Ek.21
The linearised convex subproblem in ith iteration is given as
min .Mk,E
ik
K∑
k=1
µkTr(Wi
kEik
)
s. t. Ek 4 Eik, k = 1, . . . ,K,
Mk ∈ Pbk , k = 1, . . . ,K,
(94)
where Wik = Gi
k and Gik = ∇Ei−1
k
(log det
(Ei−1k
))= [Ei−1
k ]−1 for
all k = 1, . . . ,K.Monotonic improvement of the objective of (93) on every iteration.
21This method in the context of weighted sum rate maximisation was established in [Shi et al, TSP’11], where it wasreferred to as (iteratively) weighted MMSE minimization.
2. Compute the optimal LMMSE receivers Uk ∀k, for given Mi ∀ i3. Compute the MSE weights Wk, for given Uk,Mk ∀b, k4. Compute Mk ∀k, for given Ui,Wi ∀i5. Repeat steps 2-4 until convergence
Every step can be calculated locally → decentralised design
Implementation challenges:I Uk∀ k needs to be conveyed to the BSs → precoded UL pilot
I Wk∀ k needs to be shared among BSs → backhaul exchange
23Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weighted MMSE approach to distributed sum-utilitymaximization for a MIMO interfering broadcast channel, IEEE Trans. Signal Processing, vol. 59, no. 9, pp. 4331 – 4340,Sep. 2011
Figure: Impact of the limited number of iterations to the achievable sum ratewith NT = 4, NR = 2, K = 8. [Kaleva, J.; Tolli, A.; Juntti, M.; , ”Weighted Sum Rate Maximization forInterfering Broadcast Channel via Successive Convex Approximation”, Global Communications Conference, 2012.GLOBECOM 2012. IEEE , Dec. 2012]
[3] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO
interfering broadcast channel, IEEE Trans. Signal Processing, Sep. 2011.
[4] T. Bogale and L. Vandendorpe, Weighted sum rate optimization for downlink multiuser MIMO coordinated base station systems: Centralized and
distributed algorithms, IEEE Trans. Signal Processing, Dec. 2011.
24J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Beamforming for Weighted Sum Rate Maximization with RateConstraints”, in Proc. IEEE PIMRC 2013 - Workshop on Cooperative and Heterogeneous Cellular Networks, London, UK,Sep. 2013
10: Solve transmit beamformers mk,l ∀ (k, l).11: until Desired level of convergence has been reached or i > Imax.12: until Desired level of convergence has been reached or j > Jmax.
Figure: Behaviour of the unconstrained users at SNR = 5dB with 3dB cellseparation, NT = 4, NR = 2, Kb = 2 and β = 10. [J. Kaleva, A. Tolli & M. Juntti,”Decentralized Beamforming for Weighted Sum Rate Maximization with Rate Constraints”, in Proc. IEEE PIMRC 2013 -Workshop on Cooperative and Heterogeneous Cellular Networks, London, UK, Sep. 2013]
Figure: Convergence at SNR = 15dB with 3dB cell separation, NT = 4,NR = 2, Kb = 3 and β = 4. [J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Beamforming for WeightedSum Rate Maximization with Rate Constraints”, in Proc. IEEE PIMRC 2013 - Workshop on Cooperative and HeterogeneousCellular Networks, London, UK, Sep. 2013]
Each user is associated tosingle BS (non-cooperative)
TDD and perfect CSI
Precoded UL pilot sequences
1
2
1 BK-1
K
Goal: Scheduling & TX – RX design to control interferencea
CSI acquisition: Pilot & Backhaul signalling
Decentralized, practical methods, based on locally available CSI
Support for independent user scheduling by BSs
aP. Komulainen, A. Tolli & M. Juntti, Effective CSI Signaling and Decentralized Beam Coordination in TDD Multi-CellMIMO Systems, IEEE Transactions on Signal Processing, vol. 61, no. 9, pp. 2204 – 2218, May 2013
Strategy A: Global AlgorithmEach BS b calculates own weights Wk ∀ k ∈ Ub : distribute viabackhaulEach BS calculates own precoders Mk ∀ k ∈ Ub : use for datatransmission
Each UE calculates own receiver Uk : use for reception and ULsounding
Figure: Average convergence of the sum rate at 0dB cell separation, at 25dBSNR. [P. Komulainen, A. Tolli & M. Juntti, Effective CSI Signaling and Decentralized Beam Coordination in TDDMulti-Cell MIMO Systems, IEEE Transactions on Signal Processing, vol. 61, no. 9, pp. 2204 – 2218, May 2013]
Bidirectional Signalling Strategies for Dynamic TDD
P. Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directionalSignaling for Dynamic TDD with Decentralized Beamforming” ICCWorkshop on Small Cell and 5G Networks (SmallNets), London, UK, June,2015
P. Jayasinghe, A. Tolli, and M. Latva-aho, ”Bi-directional SignalingStrategies for Dynamic TDD Networks” IEEE SPAWC, Stockholm,Sweden, July, 2015
METIS Deliverable D3.3 on ”Final performance results and consolidatedview on the most promising multi-node/multi-antenna transmissiontechnologies” (A. Tolli & P. Jayasinghe), https://www.metis2020.com/
Figure: Backward-Forward training structure [Changxin Shi; Berry, R.A.; Honig, M.L.,”Bi-Directional Training for Adaptive Beamforming and Power Control in Interference Networks,” Signal Processing, IEEETransactions on , vol.62, no.3, pp.607–618, Feb.1, 2014]
Bidirectional training(BIT) phase is used in the beginning of eachTDD frame to speed up the convergence of the iterative algorithms
A number of blocks of pilots are alternately transmitted in thedownlink and the uplink
Figure: TDD frame structure with two bi-directional beamformer signalingiterations. [P. Jayasinghe, A. Tolli, and M. Latva-aho, ”Bi-directional Signaling Strategies for Dynamic TDD Networks”IEEE SPAWC, Stockholm, Sweden, July, 2015]
4-antenna BSs, 4 2-antenna UEs per BSThe signaling overhead per one signaling iteration is γ and the totalsignaling overhead is ρ = BIT× γ. The actual throughput is(1− ρ)R, where R is the achieved WSR from the iterative algorithm.
Figure: Actual Sum rate vs overhead with different SNR (10, 20 dB) values. [P.Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directional Signaling for Dynamic TDD with DecentralizedBeamforming” ICC Workshop on Small Cell and 5G Networks (SmallNets), London, UK, June, 2015]
Figure: Actual sum rate vs overhead at SNR = 20 dB, with diferent α, β, δ. [P.Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directional Signaling for Dynamic TDD with DecentralizedBeamforming” ICC Workshop on Small Cell and 5G Networks (SmallNets), London, UK, June, 2015]
Figure: Average sum rate over time-correlated channel at SNR = 20 dB withγ = 0.01. [P. Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directional Signaling for Dynamic TDD withDecentralized Beamforming” ICC Workshop on Small Cell and 5G Networks (SmallNets), London, UK, June, 2015]
Mode Selection and Transceiver Design in Underlay D2DMIMO Systems
A. Tolli, J. Kaleva and P. Komulainen, ”Mode Selection and TransceiverDesign for Rate Maximization in Underlay D2D MIMO Systems”, IEEEICC, London, UK, June, 2015
A. Ghazanfari, A. Tolli and J. Kaleva, ”Joint Power Loading and ModeSelection for Network-assisted Device-to-Device Communication”, IEEEICC, London, UK, June, 2015
Reformulation via MSE minimizationIntroducing new optimization variables for the RX filters as well asupper bounds for the MSE matrices, (104) can be written as
min .∑L
l=1µl
(log det
(E
(1)l
)+ log det
(E
(2)l
)− tl
)(105a)
s. t. log det(E
(s)B,l
)≤ −tl, s = 1, 2 (105b)
E(s)l 4 E
(s)l , E
(s)B,l 4 E
(s)B,l ∀ l, s = 1, 2 (105c)
(103b), (103c), (103d). (105d)
where the variables are, ∀ l = 1, . . . , L, s = 1, 2
Numerical ExamplesL = 4 D2D pairs, each with 2 antennasBS is equipped with 8 antennasFrequency flat Rician fading with factor κ
[A. Tolli, J. Kaleva and P. Komulainen, ”Mode Selection and Transceiver Design for Rate Maximization in Underlay D2DMIMO Systems”, IEEE ICC, London, UK, June, 2015]
Figure: Average achieved sum rate peruser with NB = 8, Nk = 2 ∀ k, L = 4,κ = 10, SNR = 15dB and D2Ddistance of 20m.
20 25 30 35 40 45 50 55 607
8
9
10
11
12
13
14
15
Ave
rag
e s
um
ra
te [
bits/s
ec/H
z].
Cell radius.
Cellular Only
D2D + Cellular
D2D Only
Figure: Average achieved sum rate peruser with NB = 8, Nk = 2 ∀ k, L = 4,κ = 0, SNR = 15dB and D2D distanceof 20m.
26A. Tolli, J. Kaleva and P. Komulainen, ”Mode Selection and Transceiver Design for Rate Maximization in Underlay D2DMIMO Systems”, IEEE ICC, London, UK, June, 2015