Edinburgh Research Explorer Energy Efficient Resource Allocation for Multiuser Relay Networks Citation for published version: Singh, K, Gupta, A & Ratnarajah, T 2016, 'Energy Efficient Resource Allocation for Multiuser Relay Networks' IEEE Transactions on Wireless Communications. DOI: 10.1109/TWC.2016.2641961 Digital Object Identifier (DOI): 10.1109/TWC.2016.2641961 Link: Link to publication record in Edinburgh Research Explorer Document Version: Peer reviewed version Published In: IEEE Transactions on Wireless Communications General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 29. Aug. 2018
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Edinburgh Research Explorer · 1 Energy Efficient Resource Allocation for Multiuser Relay Networks Keshav Singh, Member, IEEE, Ankit Gupta, and Tharmalingam Ratnarajah, Senior Member,
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Edinburgh Research Explorer
Energy Efficient Resource Allocation for Multiuser RelayNetworks
Citation for published version:Singh, K, Gupta, A & Ratnarajah, T 2016, 'Energy Efficient Resource Allocation for Multiuser RelayNetworks' IEEE Transactions on Wireless Communications. DOI: 10.1109/TWC.2016.2641961
Digital Object Identifier (DOI):10.1109/TWC.2016.2641961
Link:Link to publication record in Edinburgh Research Explorer
Document Version:Peer reviewed version
Published In:IEEE Transactions on Wireless Communications
General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.
Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact [email protected] providing details, and we will remove access to the work immediately andinvestigate your claim.
for Multiuser Relay NetworksKeshav Singh, Member, IEEE, Ankit Gupta, and Tharmalingam Ratnarajah, Senior Member, IEEE
Abstract—In this paper, a novel resource allocation algorithmis investigated to maximize the energy efficiency (EE) in multiuserdecode-and-forward (DF) relay interference networks. The EEoptimization problem is formulated as the ratio of the spectrumefficiency (SE) over the entire power consumption of the networksubject to total transmit power, subcarrier pairing and allocationconstraints. The formulated problem is a nonconvex fractionalmixed binary integer programming problem, i.e., NP-hard tosolve. Further, we resolve the convexity of the problem by aseries of convex transformations and propose an iterative EEmaximization (EEM) algorithm to jointly determine the optimalsubcarrier pairing at the relay, subcarrier allocation to each userpair and power allocation to all source and the relay nodes.Additionally, we derive an asymptotically optimal solution byusing the dual decomposition method. To gain more insights intothe obtained solutions, we further analyze the resource allocationalgorithm in a two-user case with interference-dominated andnoise-dominated regimes. In addition, a suboptimal algorithmis investigated with reduced complexity at the cost of acceptableperformance degradation. Simulation results are used to evaluatethe performance of the proposed algorithms and demonstrate theimpacts of various network parameters on the attainable EE andSE.
Index Terms—Energy efficiency, resource allocation, multiuser,decode-and-forward, relay networks.
I. INTRODUCTION
The cooperative communication and small cell have
emerged as promising future technologies for improving the
network throughput, enlarging the transmission range of wire-
less networks and enhancing the link reliability [1]. The relay
networks can swiftly increase the spectral efficiency (SE) of
the network. However, the power dissipation, which is not
only due to transceiver but also due to complete radio access
network, increases significantly and are predicted to surge
rapidly and reach to the current level of the total electricity
consumption in the next 20-25 years [2]. To enhance the
energy efficiency (EE) of wireless networks is of paramount
importance in realizing 5G radio access solutions. Conse-
quently, it is urgent to investigate energy-aware architecture
and resource allocation techniques that prolong the network
Manuscript received June 4, 2016; revised September 25, 2016; acceptedDecember 06, 2016. The associate editor coordinating the review of this paperand approving it for publication was Prof. Shi Jin.
This work was supported by the U.K. Engineering and Physical SciencesResearch Council (EPSRC) under Grant EP/L025299/1.
Keshav Singh and Tharmalingam Ratnarajah are with the Institute for Digi-tal Communications, the University of Edinburgh, Kings Building, Edinburgh,UK, EH9 3JL. E-mails: K.Singh; [email protected].
Ankit Gupta is with Aricent Technologies Limited (Holdings), Gurgaon,India. E-mail: [email protected].
The corresponding author of this paper is Keshav Singh.
lifespan or provide significant energy savings under the um-
brella of the green communications [3].
Various relaying protocols have been proposed for coop-
erative networks, among which amplify-and-forward (AF),
decode-and-forward (DF) [1] and compress-and-forward (CF)
[4] are three common ones. In AF protocol, the relay re-
transmits the amplified signal to the destination, whereas
in the DF protocol, the relay first attempts to decode the
received signal and then forwards the re-encoded information
bits to the destination. In CF protocol, the relay compresses
the received signal and sends the compressed signal to the
destination. The AF protocol has an advantage over others
in terms of low implementation complexity. However, the DF
protocol performs better than other two protocols when the
channel quality of forward links, i.e., source-to-relay (SR)
links is good enough. Another advantage of the DF protocol
is that it is possible to use different channel coding schemes
at the source and the relay nodes and the transmission can
be optimized for both links. i.e., SR and relay-to-destination
Fig. 5. Effect of number of users on the average EE and SE (Nsc = 15and dSR = dRD = 200 m).
a reverse trend. The EEM algorithm gives highest average
EE(SE) when relay node is in middle and it’s performance
decreases(increases) with increasing N .
IX. CONCLUSIONS
In this paper, we investigated the problem of joint power
and subcarrier allocation in relay-assisted multiuser networks
from a green energy perspective. To improve the EE of the
network a penalty based approach has been adopted and a
balance is created between the maximum achievable network
EE and SE. The primal problem was nonconvex due to a form
of fractional and mixed binary integer nonlinear programming.
We transformed this problem into a convex problem by a series
of transformations. Further, the dual decomposition method is
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.065
rd
Ave
rage
EE
[bits
/mJo
ule/
Her
tz]
EEM, N
sc=10
EEM, Nsc
=12
EEM, Nsc
=15
(a) Average EE versus Pmax
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91.0
1.5
2.0
2.5
rd
Ave
rage
SE
[bits
/sec
/Her
tz]
EEM, Nsc
=10
EEM, Nsc
=12
EEM, Nsc
=15
(b) Average SE versus Pmax
Fig. 6. Effect of the relay position with different Nsc on the average EEand SE (N = 5 and Pmax = 15 dBm).
adopted to find the optimal power and subcarrier pairing and
allocation matrices, respectively. Moreover, the convergence
behaviour of the EEM algorithm is proved theoretically. To
reduce the complexity of the proposed iterative EEM algo-
rithm, we further demonstrated a suboptimal algorithm by
exchanging minimal network performance. We compared the
performance of the proposed algorithms with that of the ES
and SEM algorithms and demonstrated the effects of various
network parameters such as number of subcarriers, users, and
the relay’s position, on the average EE and SE performance.
Simulation results confirmed that a higher number of subcar-
riers, although marginally decreases the SE, can improves the
EE. Improvements in a SE come by increasing the number of
users at a certain expense of the EE of the network. This
15
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
rd
Ave
rage
EE
[bits
/mJo
ule/
Her
tz]
EEM, N=3EEM, N=5EEM, N=8
(a) Average EE versus Pmax
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
rd
Ave
rage
SE
[bits
/sec
/Her
tz]
EEM, N=3EEM, N=5EEM, N=8
(b) Average SE versus Pmax
Fig. 7. Effect of the relay position with different N on the average EE andSE (Nsc = 10 and Pmax = 15 dBm).
design framework provides a useful model for developing
future energy efficient multiuser relay netwroks.
APPENDIX A
PROOF OF THEOREM 1
The objective function in (P5) can be rewrite as:
FLB
(
P(m)
S , P(n)
R ,ρ,Φ, Γ(m)
,α(m),β(m))
=
Nsc∑
m=1
Nsc∑
n=1
Λm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
, (A.1)
where the term Λm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
is ex-
plicitly defined as
Λm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
=
N∑
i=1
1
2ρm,nΦi,(m,n)
(
α(m)i
ln(2)Γ(m)i + β
(m)i
)
−Ψ
N∑
i=1
ρm,nΦi,(m,n)
(
eP(m)S,i + eP
(n)R,i
)
−Ψ
N2sc
Pc , (A.2)
wherein P(m)
S , P(n)
R , Γ(m) ∈ W
N and Λm,n (·) : WN → R,
are not necessarily convex in nature.
In a similar manner, the constraints (C.1) − (C.5), (C.7)and (C.8) can be expressed as
Nsc∑
m=1
Nsc∑
n=1
Ωm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
6 0 , (A.3)
where Ωm,n (·) : WN → RT and T = 7 is the total number
of constraints limiting the optimization problem (P5). Hence,
the optimiztion problem (P5) can be written as
OP ⋆ ,
maxP(m)S ,P
(n)R ,Γ
(m)
Nsc∑
m=1
Nsc∑
n=1
Λm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
s.t.
Nsc∑
m=1
Nsc∑
n=1
Ωm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
60, (A.4)
where 0 ∈ RL. Next, we define a perturbation function ω(U)
to prove DG ≈ 0, as follows:
ω(U) ,
maxP(m)S ,P
(n)R ,Γ
(m)
Nsc∑
m=1
Nsc∑
n=1
Λm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
s.t.
Nsc∑
m=1
Nsc∑
n=1
Ωm,n
(
P(m)
S , P(n)
R , Γ(m)
,α(m),β(m))
6U, (A.5)
where U ∈ RL is a perturbation vector. From [26] and [27],
it is evident that duality gap tends to zero DG ≈ 0, when the
time-sharing condition is satisfied. Moreover, it is also shown
that the time-sharing condition will be satisfied if the optimal
value of the optimization problem (P5) is a concave function
of the constraints. Therefore, if ω(U) is a concave function of
U, then DG ≈ 0. Next, we define time sharing property as
follows:
Definition 3: If(
P(m)⋆
Si, P
(n)⋆
Ri, Γ
(m)⋆
i
)
, i = 1, 2, denotes
the optimal solution of (A.5) and represented by ω(U1)and ω(U2), respectively, then there always exists a solution
16
(
P(m)⋆
S3, P
(n)⋆
R3, Γ
(m)⋆
3
)
that satisfies the following condition:
Nsc∑
m=1
Nsc∑
n=1
Ωm,n
(
P(m)
S3, P
(n)
R3, Γ
(m)
3 ,α(m),β(m))
6 ∆U1 + (1−∆)U1 ; (A.6)
Nsc∑
m=1
Nsc∑
n=1
Λm,n
(
P(m)
S3, P
(n)
R3, Γ
(m)
3 ,α(m),β(m))
> ∆Λm,n
(
P(m)⋆
S1, P
(n)⋆
R1, Γ
(m)⋆
1
)
+ (1−∆)Λm,n
(
P(m)⋆
S2, P
(n)⋆
R2, Γ
(m)⋆
2
)
, (A.7)
where 0 ≤ ∆ ≤ 1.
Next, we need to prove that ω(U) is a concave function
of U. For some ∆, it is easy to find out U3 that satisfies
U3 = ∆U1+(1−∆)U2. We assume(
P(m)⋆
S1, P
(n)⋆
R1, Γ1
(m)⋆)
,(
P(m)⋆
S2, P
(n)⋆
R2, Γ2
(m)⋆)
and(
P(m)⋆
S3, P
(n)⋆
R3, Γ3
(m)⋆)
are the
optimal solutions subject to the constraints of ω(U1), ω(U2)and ω(U3), respectively. Time-sharing condition points out
that there exists a value(
P(m)
S3, P
(n)
R3, Γ3
(m),α(m),β(m)
)
that
satisfying (A.6) and (A.8).(
P(m)⋆
S3, P
(n)⋆
R3, Γ3
(m)⋆)
is the
optimal solution to ω(U3), thus
Nsc∑
m=1
Nsc∑
n=1
Λm,n
(
P(m)⋆
S3, P
(n)⋆
R3, Γ3
(m)⋆)
>
Nsc∑
m=1
Nsc∑
n=1
Λm,n
(
P(m)
S3, P
(n)
R3, Γ3
(m),α(m),β(m)
)
> ∆Λm,n
(
P(m)⋆
S1, P
(n)⋆
R1, Γ1
(m)⋆)
+ (1−∆)Λm,n
(
P(m)⋆
S2, P
(n)⋆
R2, Γ2
(m)⋆)
, (A.8)
Then, the concavity of ω(U) is proved.
Since, ω(U) is concave, it is possible to show that (A.4)
satisfies the time-sharing property. When the number of
subcarriers goes to infinity, the time-sharing condition al-
ways holds for multicarrier systems [29]. Let us assume(
P(m)⋆
S1, P
(n)⋆
R1, Γ1
(m)⋆)
and(
P(m)⋆
S2, P
(n)⋆
R2, Γ2
(m)⋆)
be two
feasible solutions. Then, ∆ percentage of total subcarriers
Nsc are allocated with the solution(
P(m)⋆
S1, P
(n)⋆
R1, Γ1
(m)⋆)
,
whereas the rest of (1−∆) percentage of the total subcarriers
Nsc are allocated with the solution(
P(m)⋆
S2, P
(n)⋆
R2, Γ2
(m)⋆)
.
Further,Nsc∑
m=1
Nsc∑
n=1Λm,n
(
P(m)
S3, P
(n)
R3, Γ3
(m))
is a linear com-
bination, expressed as ∆Λm,n
(
P(m)⋆
S1, P
(n)⋆
R1, Γ1
(m)⋆)
+
(1−∆)Λm,n
(
P(m)⋆
S2, P
(n)⋆
R2, Γ2
(m)⋆)
. Hence the constraints
are linear combinations, therefore, it is proved that (A.4)
satisfies the time-sharing property. Henceforth, ω(U) is a
concave function of U, and the duality gap DG ≈ 0. This
proof of theorem in completed.
APPENDIX B
PROOF OF THEOREM 2
This theorem is proved in two steps: 1) we need to prove
that the network EE performance is increased monotonically
along with the update of two coefficients α(m)i (t) and β
(m)i (t);
and 2) the optimal solution for (P5) is also a local maximizer
for (P1) when the update has converged. To prove first part,
let(
P(m)⋆
S (t), P(n)⋆
R (t),ρ⋆(t),Φ⋆(t), Γ(m)⋆
(t))
is the optimal
solution in the t-th iteration for the coefficients α(m)i (t) and
β(m)i (t). If we update the coefficients in accordance with (35)
and (36), then from (12)-(14), we have (B.1) and also from the
problem (P5) it directly implies the condition defined in (B.2).
From (B.1) and (B.2), we can conclude that the lower bound
performance improves with the update of coefficients α(m)i (t)
and β(m)i (t). Next, we need to prove that the optimal solution
of (P5) is a local maximizer for (P1) when the update has con-
verged. This can be done by proving that the optimal solutions
which satisfies the KKT conditions of (P5) also satisfies the
KKT conditions of (P1). Therefore, we take the partial deriva-
tive of the Lagrangian function in (18) with respect to P(m)S,i ,
P(n)R,i and Γ
(m)i at
(
P(m)⋆
S , P(n)⋆
R ,ρ⋆,Φ⋆,Γ(m)⋆ , λ⋆,µ⋆,ν⋆)
and equate the results to zero. Furthermore, by using Lemma
1, we get
P(m)⋆
S,i =
µ(m)⋆
i
N∑
j=1,j 6=i
Γ(m)⋆
i P(m)⋆
S,j
∣∣∣h
(m)SjR
∣∣∣
2
P(m)⋆
S,i
∣∣∣h
(m)SiR
∣∣∣
2 +Γ(m)⋆
i
(
σ(m)R
)2
P(m)⋆
S,i
∣∣∣h
(m)SiR
∣∣∣
2
Nsc∑
n=1ρ⋆m,nΦ
⋆i,(m,n) (λ
⋆ +Ψ)
;
(B.3)
P(n)⋆
R,i =
ν(n)⋆
i
N∑
j=1,j 6=i
Γ(n)⋆
i P(n)⋆
R,j
∣∣∣h
(n)RDj
∣∣∣
2
P(n)⋆
R,i
∣∣∣h
(n)RDi
∣∣∣
2 +Γ(n)⋆
i
(
σ(n)Di
)2
P(n)⋆
R,i
∣∣∣h
(n)RDi
∣∣∣
2
Nsc∑
m=1ρ⋆m,nΦ
⋆i,(m,n) (λ
⋆ +Ψ)
;
(B.4)
Γ(m)⋆
i can also be computed in a similar manner. For the
optimal solutions obtained in (B.3) and (B.4), the dual function
is given as
g(
λ⋆,µ(m)⋆ ,ν(m,n)⋆)
= maxρ,Φ
N∑
i=1
Nsc∑
m=1
Nsc∑
n=1
ρm,nΦi,(m,n)B′i,(m,n) + C
′
s.t. (C.1) − (C.5) , (B.5)
where B′i,(m,n) and C′ are defined as
B′i,(m,n) =
(
α(m)⋆
i Γ(m)⋆
i + ϕβ(m)⋆
i
)
− (Ψ + λ⋆)(
P(m)⋆
S,i + P(n)⋆
R,i
)
, (B.6)
17
FLB
(
P(m)⋆
S (t), P(n)⋆
R (t),ρ⋆(t),Φ⋆(t),Γ(m)⋆(t),α(m)(t),β(m)(t))
≤ F(
P(m)⋆
S (t), P(n)⋆
R (t),ρ⋆(t),Φ⋆(t),Γ(m)⋆(t))
= FLB
(
P(m)⋆
S (t), P(n)⋆
R (t),ρ⋆(t),Φ⋆(t),Γ(m)⋆(t), α(m)i (t+ 1), β
(m)i (t+ 1)
)
; (B.1)
FLB
(
P(m)⋆
S (t), P(n)⋆
R (t),ρ⋆(t),Φ⋆(t),Γ(m)⋆(t), α(m)i (t+ 1), β
(m)i (t+ 1)
)
(B.2)
≤ FLB
(
P(m)⋆
S (t+ 1), P(n)⋆
R (t+ 1),ρ⋆(t+ 1),Φ⋆(t+ 1),Γ(m)⋆(t+ 1), α(m)i (t+ 1), β
(m)i (t+ 1)
)
,
C′ = −ΨPc + λ⋆Pmax
−N∑
i=1
Nsc∑
k=1
µ(k)⋆
i
N∑
j=1,j 6=i
(
Γ(k)⋆
i + P(k)⋆
S,j
) ∣∣∣h
(k)SjR
∣∣∣
2
P(k)⋆
S,i
∣∣∣h
(k)SiR
∣∣∣
2
+Γ(k)⋆
i
(
σ(k)R
)2
P(k)⋆
S,i
∣∣∣h
(k)SiR
∣∣∣
2 + 1
−N∑
i=1
Nsc∑
k=1
ν(k)⋆
i
N∑
j=1,j 6=i
(
Γ(k)⋆
i + P(k)⋆
R,j
) ∣∣∣h
(k)RDj
∣∣∣
2
P(k)⋆
R,i
∣∣∣h
(k)RDi
∣∣∣
2
+Γ(k)⋆
i
(
σ(k)Di
)2
P(k)⋆
R,i
∣∣∣h
(k)RDi
∣∣∣
2 + 1
, (B.7)
Similar to (29) the optimal subcarrier allocation is determined
for a given subcarrier pairing as
Φ⋆i,(m,n) =
1, for i = argmaxi B′i,(m,n) ,
0, otherwise(B.8)
and the optimal subcarrier pairing matrix for obtained power
and subcarrier allocation can be found similar to (31) where
B′ is defined as
B′ =
B′i⋆,(1,1) · · · B′
i⋆,(1,K)
.... . .
...
B′i⋆,(K,1) · · · B′
i⋆,(K,K)
(B.9)
Moreover, the optimal solution of (P1) can be derived from the
KKT conditions that the partial derivatives of the Lagrangian
function of the problem (P1) with respect to P(m)S,i and P
(n)R,i
are identical to (B.3) and (B.4), respectively. This is also true
for the subcarrier pairing and allocation matrix. Hence, the
theorem is proved.
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Keshav Singh (S’12, M’16) received the degreeof Master of Technology (with first-class hon-ors) in Computer Science from Devi Ahilya Vish-wavidyalaya, Indore, India, in 2006, the M.Sc. inInformation & Telecommunications Technologiesfrom Athens Information Technology, Greece, in2009, and the Ph.D. degree in Communication En-gineering from National Central University, Taiwan,in 2015. Since 2016, he has been with Institutefor Digital Communications, School of Engineering,University of Edinburgh, where he is currently a
Postdoctoral Research Associate. He is a memeber of IEEE. He also hasserved as a Technical Program Committee Member for numerous IEEEconferences. His current research interests are in the areas of green com-munications, resource allocation, full-duplex radio, cooperative and energyharvesting networks, multiple-input multiple-output (MIMO) systems, andoptimization of radio access.
Ankit Gupta received the Bachelor of Technology(B.Tech) degree in Electronics and CommunicationEngineering from Guru Gobind Singh IndraprasthaUniversity, Delhi, India, in 2014. He is currentlywith Aricent Technologies Limited (Holdings), Gu-rugram, India. His current research interests in-clude 5G, cooperative communications, multiple-input multiple-output (MIMO) networks and opti-mization methods in signal processing and commu-nications.
Tharmalingam Ratnarajah (A’96-M’05-SM’05) iscurrently with the Institute for Digital Communica-tions, University of Edinburgh, Edinburgh, UK, asa Professor in Digital Communications and SignalProcessing and the Head of Institute for DigitalCommunications. His research interests include sig-nal processing and information theoretic aspects of5G and beyond wireless networks, full-duplex radio,mmWave communications, random matrices theory,interference alignment, statistical and array signalprocessing and quantum information theory. He has
published over 300 publications in these areas and holds four U.S. patents.He is currently the coordinator of the FP7 project ADEL (3.7Me) in thearea of licensed shared access for 5G wireless networks. Previously, hewas the coordinator of the FP7 project HARP (3.2Me) in the area ofhighly distributed MIMO and FP7 Future and Emerging Technologies projectsHIATUS (2.7Me) in the area of interference alignment and CROWN (2.3Me)in the area of cognitive radio networks. Dr Ratnarajah is a Fellow of HigherEducation Academy (FHEA), U.K., and an associate editor of the IEEETransactions on Signal Processing.