Joint Power Allocation and Beamforming for Energy-Efficient Two-Way Multi-Relay Communications

0.25 MB·

by  Zhichao Sheng

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Joint Power Allocation and Beamforming for
Energy-Efficient Two-Way Multi-Relay
Communications
Zhichao Sheng, Hoang D. Tuan, Trung Q. Duong and H. Vincent Poor
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n Abstract
a
J
5 This paper considers the joint design of user power allocation and relay beamforming in relaying
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communications,in which multiple pairs of single-antenna users exchange informationwith each other

] via multiple-antenna relays in two time slots. All users transmit their signals to the relays in the first
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I timeslotwhiletherelaysbroadcastthebeamformedsignalstoallusersinthesecondtimeslot.Theaim
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c is to maximize the system’s energy efficiency (EE) subject to quality-of-service (QoS) constraints in
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terms of exchange throughput requirements. The QoS constraints are nonconvex with many nonlinear
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v cross-terms, so finding a feasible point is already computationally challenging. The sum throughput
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8 appears in the numerator while the total consumption power appears in the denominator of the EE
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objective function.The formeris a nonconcavefunctionand the latter is a nonconvexfunction,making
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fractionalprogramminguselessforEEoptimization.Nevertheless,efficientiterationsoflowcomplexity
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0 toobtainitsoptimizedsolutionsaredeveloped.Theperformancesofthemultiple-userandmultiple-relay
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networks under various scenarios are evaluated to show the merit of the paper development.
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:
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X Index Terms
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Two-way relaying, information exchange (IE), energy efficiency (EE), quality-of-service (QoS),
relay beamforming, power allocation, joint optimization, path-following algorithms.
ThisworkwassupportedinpartbytheAustralianResearchCouncilsDiscoveryProjectsunderProjectDP130104617, inpart
bytheU.K.Royal AcademyofEngineeringResearchFellowshipunder GrantRF1415/14/22, andinpartbytheU.S.National
Science Foundation under Grants CCF-1420575 and CNS-1456793.
ZhichaoShengandHoangD.TuanarewiththeFacultyofEngineeringandInformationTechnology,UniversityofTechnology
Sydney, Broadway, NSW 2007, Australia (email: [email protected], [email protected])
Trung Q. Duong is with Queen’s University Belfast, Belfast BT7 1NN, UK (email: [email protected])
H. Vincent Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (e-mail:
[email protected])
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I. INTRODUCTION
Two-way relaying (TWR) [1]–[3] has been the focus of considerable research interest in
recent years due to its potential in offering higher information exchange throughput for cognitive
communications such as device-to-device (D2D) and machine-to-machine (M2M) communi-
cations [4], [5]. Unlike the conventional one-way relaying, which needs four time slots for
information exchange between a pair of users (UEs), TWR needs just two time slots for this
exchange [2], [6], [7]. In the first time slot, known as the multiple access (MAC) phase, all
UEs simultaneously transmit their signals to the relays. In the second time slot, also known as
the broadcast (BC) phase, the relays broadcast the beamformed signals to the all UEs. Offering
double fast communication, TWR obviously suffers from double multi-channel interference as
compared to one-way relaying [8], [9]. Both optimal control of UEs’ transmit power and TWR
beamforming are thus very important in exploring the spectral efficiency of TWR.
There are various scenarios of TWR considered in the literature. The most popular scenario is
single-antenna relays serving a pair of single-antenna UEs [7], [10], [11]. The typical problems
are to design the TWR weights to either maximize the throughput or minimize the relay power
subject to signal-to-interference-plus-noise ratio (SINR) constraints at UEs. A branch-and-bound
(BB) algorithm of global optimization [12] was used in [10] for sum throughput maximization.
Its computational complexity is already very high in very low-dimensional problems. Semi-
definite relaxation of high computational complexity followed by bisection used in [7] works
strictly under a single total relay power constraint. Furthermore, the scenario of single-antenna
relays serving multiple pairs of UEs was addressed in [13] by a polyblock algorithm of global
optimization [12] and in [13] by local linearization based iteration. The mentioned semi-definite
relaxation was also used in [14]–[16] in designing TWR beamformer for the scenario of a multi-
antenna relay serving multiple pairs of UEs. It should be realized that most of relay beamformer
optimization problems considered in all these works are not more difficult computationally than
their one-way relaying counterparts, which have been efficiently solved in [9].
The fixed power allocation to UEs does not only miss the opportunity of power distribution
within a network but can also potentially increase interference to UEs of other networks [17],
[18]. The joint design of UE power allocation and TWR weights for single antenna relays
serving a pair of single antenna UEs to maximize the minimum SINR was considered in [19].
Under the strict assumption that the complex channel gains from UEs to the relays are the same
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as those from the relays to the UEs, its design is divided into two steps. The first step is to
optimize the beamformer weights with UEs’ fixed power by sequential second-order convex
cone programming (SOCP). The second step performs an exhaustive grid search for UE power
allocation. Joint optimization of UE precoding and relay beamforming for a multi-antenna relay
serving a pair of multi-antenna UEs could be successfully addressed only recently in [20]. An
efficient computation for joint UE power allocation and TWR beamforming to maximize the
worst UE throughput for multiple antenna relays serving multiple pairs of single antenna UEs
was proposed in [21]. The reader is also referred to [22] for joint UE and relay power allocation
in MIMO OFDM system of one relay and one pair of UEs.
Meanwhile,theaforementionedclassicalspectral efficiency(SE)in termsofhighthroughputis
now only one among multiple driving forces for the development of the next generation commu-
nication networks (5G) [23]. The energy consumption of communication systems has become
sizable, raising environmental and economic concerns [24]. Particularly, the network energy
efficiency (EE) in terms of the ratio of the sum throughput and the total power consumption,
which counts not only the transmission power but also the drain efficiency of power amplifiers,
circuit power and other power factors in supporting the network’s activities, is comprehensively
pushed forward in 5G to address these concerns [25]. EE in single-antenna TWR has been
considered in [26] for single-antenna OFDMA in assisting multiple pairs of single-antenna UEs
and in [27] and [28] for multi-antenna relays in assisting a pair of multi-antenna UEs. Again,
the main tool for computational solution in these works is semi-definite relaxation, which not
only significantly increases the problem dimension but also performs unpredictably [29]. Also,
the resultant Dinkelbach’s iteration of fractional programming invokes a logarithmic function
optimization, which is convex but still computationally difficult with no available algorithm of
polynomial complexity.
The above analysis of the state-of-the-art TWR motivates us to consider the joint design of
single-antenna UE power allocation and TWR beamformers in a TWR network to maximize
its EE subject to UE QoS constraints. We emphasize that both sum throughput maximization
(for spectral efficiency) and EE maximization are meaningful only in the context of UE QoS
satisfaction, without which they will cause the UE service discrimination. Unfortunately, to our
best knowledge such UE QoS constraints were not addressed whenever they are nonconvex
[25]–[28]. The nonconvexity of these QoS constraints implies that even finding their feasible
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pointsisalready computationallydifficult.QoS constraintsin terms ofUEs’exchangethroughput
requirementsaremuchmoredifficultthanthatin termsofindividualUEthroughputrequirements
because the former cannot be expressed in terms of individual SINR constraints as the latter.
To address the EE maximization problem, we first develop a new computational method for
UE exchange throughput requirement feasibility, which invokes only simple convex quadratic
optimization. A new path-following computational procedure for computational solutions of the
EE maximization problem is then proposed.
The rest of the paper is organized as follows. Section II formulates two optimization problems
of EE maximization and UE QoS optimization. Two path-following algorithms are developed in
Section III and IV for their computation. In contrast to fractional programming, these algorithms
invoke only a simple convex quadratic optimization of low computational complexity at each
iteration. Section V provides simulation results to verify the performance of these algorithms.
Finally, concluding remarks are given in Section VI
Notation. Vectors and matrices are represented by boldfaced lowercase and uppercase, re-
spectively. xxx(n) is the nth entry of vector xxx, while XXX(n,.) and XXX(n,m) are the nth row
and (n,m)th entry of a matrix XXX. xxx,yyy = xxxHyyy is the inner product between vectors xxx and
h i
yyy. . is either the Euclidean vector squared norm or the Frobenius matrix squared norm,
|| ||
and RN = xxx RN : xxx(n) 0, n = 1,2,...,N . XXX is the trace of matrix XXX and
+ { ∈ ≥ } h i
diag[XXX ] is a block diagonal matrix with diagonal blocks XXX . Lastly, xxx (xxx¯,RRR )
m m=1,...,M m xxx
∼ CN
means xxx is a vector of Gaussian random variables with mean xxx¯ and covariance RRR .
xxx
II. TWO-WAY RELAY NETWORKS WITH MULTIPLE MIMO RELAYS AND MULTIPLE
SINGLE-ANTENNA USERS
Fig.1illustratesaTWRnetworkinwhichK pairsofsingle-antennaUEsexchangeinformation
witheach other.NamelythekthUE(UEk)andthe(K+k)thUE(UEK+k),withk = 1,...,K,
exchange information with each other via M relays designated as relay m, m = 1,...,M,
equipped with N antennas.
R
Denote by sss = (s ,...,s )T C2K the vector of information symbols transmitted by the
1 2K
∈
UEs, whose entries are independent and have unit energy, i.e., E[ssssssH] =III , where I is the
2K 2K
identity matrix of size (2K) (2K). Let hhh CNR be the vector of channels from UE ℓ to
ℓ,m
× ∈
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N antennas
R
h h
1 K+1
g
1
UE R UE
1 1 g K+1
K+1
h
K h
2K
gK g2K
UE R UE
K M 2K
Fig. 1. Two-way relay networks with multiple single-antenna users and multiple multi-antenna relays.
relay m. The received signal at relay m is
2K
rrr = √p hhh s +nnn , (1)
m ℓ ℓ,m ℓ R,m
ℓ=1
X
wherennn (0,σ2III ) is the background noise, andppp = (p ,...,p )T R2K represents
R,m ∼ CN R NR 1 2K ∈ +
the powers allocated to the UEs.
Relay m performs linear processing on the received signal by applying the beamforming
matrix WWW NR×NR. The beamformed signals are
m
∈ C
2K
rrr =WWW rrr = √p WWW hhh s +WWW nnn ,m = 1,...,M. (2)
m,b m m ℓ m ℓ,m ℓ m R,m
ℓ=1
X
The transmit power at relay m is calculated as
2K
E[ rrr 2] = p WWW hhh 2 +σ2 WWW 2.
|| m,b|| ℓ|| m ℓ,m|| R|| m||
ℓ=1
X
Relay m transmits the beamformed signal rrr to the UEs. Let ggg CNR be the vector of
m,b m,k
∈
channels from the relay m to UE k. The received signal at UE k is given by
M
y = gggT rrr +n
k m,k m,b k
m=1
X
M 2K
= gggT ( √p WWW hhh s +WWW nnn )+n , (3)
m,k ℓ m ℓ,m ℓ m R,m k
m=1 ℓ=1
X X
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where n (0,σ2) is the background noise, which can be written as
k ∼ CN k
M M
y = √p gggT WWW hhh s +√p gggT WWW hhh s
k χ(k) m,k m χ(k),m χ(k) k m,k m k,m k
m=1 m=1
X X
desiredsignal self-interference
(4)
M 2K
|+ gggT ( {z √p WWW }hhh |s +WWW nnn{z )+n . }
m,k ℓ m ℓ,m ℓ m R,m k
m=1 ℓ=1,ℓ6=k,χ(k)
X X
inter-pairinterference
In the above equation, (k|,χ(k)) is a pair{ozf UEs that exch}ange information with each other, so
K +k for k = 1,...,K
χ(k) = (5)

 k K for k = K +1,...,2K

−
Assuming that the channel stateinformation (CSI) of the forward and backward channels and

the beamforming matrices is available, UE k effectively subtracts the self-interference term in
(4) to have the SINR:
2
M
p fffH WWW hhh
χ(k) m,k m χ(k),m
(cid:12) (cid:12)
γ (ppp,WWW) = (cid:12)mX=1 (cid:12) , (6)
k (cid:12) 2 (cid:12)
2K M M
(cid:12) (cid:12)
p fffH (cid:12)WWW hhh +σ2 (cid:12) fffH WWW 2 +σ2
ℓ(cid:12) m,k m ℓ,m(cid:12) R || m,k m|| k
ℓ=1,Xℓ6=k,χ(k) (cid:12)mX=1 (cid:12) mX=1
(cid:12) (cid:12)
where fff =ggg∗ , so fffH =gggT . (cid:12) (cid:12)
m,k m,k m,k m,k (cid:12) (cid:12)
Under the definitions
M
(WWW) , fffH WWW hhh ,
Lk,ℓ m,k m ℓ,m (7)
m=1
X
(WWW) , fffH WWW fffH WWW ... fffH WWW ,
Lk 1,k 1 2,k 2 M,k M
h i
it follows that
2K
γ (ppp,WWW) = p (WWW) 2/[ p (WWW) 2 +σ2 (WWW) 2 +σ2]. (8)
k χ(k)|Lk,χ(k) | ℓ|Lk,ℓ | R||Lk || k
ℓ=1,ℓ6=k,χ(k)
X
In TWR networks, the UEs exchange information in bi-direction fashion in one time slot. Thus,
the throughput at the kth UE pair is defined by the following function of beamforming matrix
WWW and power allocation vector ppp:
R (ppp,WWW) = ln(1+γ (ppp,WWW))+ln(1+γ (ppp,WWW)), k = 1,...,K. (9)
k k K+k
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Accordingly, the problem of maximizing the network EE subject to UE QoS constraints is
formulated as:
K [ln(1+γ (ppp,WWW))+ln(1+γ (ppp,WWW))]
max F (w,p) , k=1 k K+k (10a)
WWW∈CN×N,ppp∈R2+K EE Pζ(PsUum(p)+PsRum(p,W))+MPR +2KPU
subject to 0 p PU,max, k = 1,...,2K, (10b)
≤ k ≤ k
2K
p PU,max, (10c)
k ≤ sum
k=1
X
2K
p WWW hhh 2 +σ2 WWW 2 PA,max,m = 1,...,M, (10d)
ℓ|| m ℓ,m|| R|| m|| ≤ m
ℓ=1
X
M 2K
( p WWW hhh 2 +σ2 WWW 2) PR,max, (10e)
ℓ|| m ℓ,m|| R|| m|| ≤ sum
m=1 ℓ=1
X X
R (ppp,WWW) r ,k = 1, ,K, (10f)
k k
≥ ···
where ζ, PR and PU are the reciprocal of drain efficiency of power amplifier, the circuit powers
of the relay and UE, respectively. PR = N P and P is the circuit power for each antenna
R r r
in relay. (10f) is the exchange throughput QoS requirement for each pair of UEs. Constraints
(10b) and (10d) cap the transmit power of each UE k and each relay m at predefined values
PU,max and PA,max, respectively. On the other hand, constraints (10c) and (10e) ensure that the
k m
total transmit power of UEs and the total transmit power of the relays not exceed the allowed
power budgets PU,max and PR,max, respectively.
sum sum
Notethat(10)isaverydifficultnonconvexoptimizationproblembecausethepowerconstraints
(10d) and (10e), the exchange throughput QoS constraints (10f), and the objective function (10a)
are nonconvex. Moreover, the exchange throughput QoS constraints (10f) are much harder than
the typical individual throughput QoS constraints
ln(1+γ (ppp,WWW)) r˜ ,k = 1,...,2K, (11)
k min
≥
which are equivalent to the computationally easier SINR constraints
γ (ppp,WWW) er˜min 1,k = 1,...,2K. (12)
k
≥ −
Note that even finding feasible point of the EE problem (10) is already difficult as it must be
based on the following UEs QoS optimization problem
max ϕ(ppp,WWW) , min ln(1+γ (ppp,WWW))1/rk +ln(1+γ (ppp,WWW))1/rk (13a)
k K+k
WWW∈CN×N,ppp∈R2K k=1,...,K
+
(cid:2) (cid:3)
subject to (10b),(10c),(10d),(10e), (13b)
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which is still highly nonconvex because the objective function in (13a) is nonsmooth and
nonconcave while the joint power constraints (10d) and (10e) in (13b) are nonconvex. Only
a particular case of r r was addressed in [21], under which (13) is then equivalent to the
k min
≡
SINR multiplicative maximization
max min [(1+γ (ppp,WWW))(1+γ (ppp,WWW)] (14a)
k K+k
WWW∈CN×N,ppp∈R2K k=1,...,K
+
subject to (10b),(10c),(10d),(10e), (14b)
which can be solve by d.c. (difference of two convex functions) iterations [30]. To the authors’
best knowledge there is no available computation for (13) in general. The next section is devoted
to its solution.
III. MAXIMIN EXCHANGE THROUGHPUT OPTIMIZATION
To address (13), our first major step is to transform the nonconvex constraints (10d) and (10e)
to convex ones through variable change as follows.
Following [21], make the variable change
1
β = 0, k = 1,...,2K. (15)
k p2 ≥
k
For α > 0 and β > 0, define the functions
(WWW) 2
Ψ (WWW,α,β) , |Lk,ℓ | , k,ℓ = 1,...,2K,
k,ℓ
√αβ
(WWW) 2
Υ (WWW,α) , ||Lk || , k = 1,...,2K, (16)
k
√α
hhhH WWW 2
Φ (WWW ,α,β) , || ℓ,m m|| , ℓ = 1,...,2K;m = 1,...,M.
ℓ,m m
√αβ
which are convex [31].
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Theorem 1: The optimization problem (13) can be equivalently rewritten as
1 (WWW) 2
max f(WWW,ααα,βββ) , min ln(1+ |Lk,K+k | )
WWW∈CN×N,ααα∈R2+K,βββ∈R2+K k=1,...,K rk " αkβK+k
p (WWW) 2
+ln(1+ |LK+k,k | ) (17a)
αK+kβk #
2K 1 p
subject to Ψ (WWW,α ,β )+σ2Υ (WWW,α )+σ2 1, k = 1,...,2K(17b)
k,ℓ k ℓ R k k k√α ≤
ℓ=1,ℓ6=k,χ(k) k
X
β 1 , k = 1,...,2K, (17c)
k ≥ (PU,max)2
k
2K
1
PU (βββ) := PU,max, (17d)
sum √β ≤ sum
k
k=1
X
2K
Φ (WWW ,1,β )+σ2 WWW 2 PA,max,m = 1,...,M, (17e)
ℓ,m m ℓ R|| m|| ≤ m
ℓ=1
X
M 2K
[ Φ (WWW ,1,β )+σ2 WWW 2] PR,max. (17f)
ℓ,m m ℓ R|| m|| ≤ sum
m=1 ℓ=1
X X
Proof: One can see that γ (ppp,WWW) in (8) is expressed in terms of functions in (16) as
k
(WWW) 2
k,χ(k)
γ (ppp,WWW) = |L | (18)
k
2K
1 [ Ψ (WWW,1,1/p2)+σ2Υ (WWW,1)+σ2]
pχ(k) k,ℓ ℓ R k k
ℓ=1,ℓ6=k,χ(k)
X
which is
(WWW) 2
k,χ(k)
|L | (19)
α β
k χ(k)
for p
β = 1/p2,k = 1,...,2K (20)
k k
and
2K
α = Ψ (WWW,1,β )+σ2Υ (WWW,1)+σ2. (21)
k k,ℓ ℓ R k k
ℓ=1,ℓ6=k,χ(k)
X
Therefore,
ϕ(p,W) = f(W,ααα,βββ)
for ααα and βββ defined by (20)-(21), which is also feasible for (17) whenever (p,W) is feasible
for (13). We thus have proved that
max (13) max (17). (22)
≤
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Note that (17b) is the same as
2K
Ψ (WWW,1,β )+σ2Υ (WWW,1)+σ2 √α . (23)
k,ℓ ℓ R k k ≤ k
ℓ=1,ℓ6=k,χ(k)
X
The point (p,W) with p = 1/√β , k = 1,...,2K is feasible for (13) whenever (W,ααα,βββ) is
k k
feasible for (17). Using (18) and (23), one can see
f(WWW,ααα,βββ) ϕ(ppp,WWW),
≤
implying
max (17) max (13).
≤
The last inequality together with (22) yield
max (17) max (13),
≤
completing the proof of Theorem 1. (cid:3)
The benefit of expressing (13) by (17) is that all constraints in the latter are convex so the
computational difficulty is concentrated in its objective function, which is lower bounded by a
concave function based on the following result.
Theorem 2: At (WWW(κ),ααα(κ),βββ(κ)) it is true that
(WWW) 2
ln(1+ |Lk,χ(k) | ) f(κ) (WWW,α ,β )
α β ≥ k,χ(k) k χ(k)
k χ(k)
p , a(κ) b(κ) α(κ)β(κ) [2 (WWW)( (WWW(κ)))∗
k,χ(k) − k,χ(k) k χ(k) ℜ{Lk,χ(k) Lk,χ(k) }
1 q
(WWW(κ)) 2(α /α(κ) +β /β(κ) )]−1 (24)
−2|Lk,χ(k) | k k χ(k) χ(k)
over the trust region
1
2 (WWW)( (WWW(κ)))∗ (WWW(κ)) 2(α /α(κ) +β /β(κ) ) > 0, (25)
ℜ{Lk,χ(k) Lk,χ(k) }− 2|Lk,χ(k) | k k χ(k) χ(k)
for x(κ) = (WWW(κ)) 2/ α(κ)β(κ) ,
k,χ(k) |Lk,χ(k) | k χ(k)
q
(κ)
x
(κ) (κ) k,χ(k)
a = ln(1+x )+ > 0,
k,χ(k) k,χ(k) (κ)
x +1
k,χ(k) (26)
(x(κ) )2
(κ) k,χ(k)
b = > 0.
k,χ(k) (κ)
x +1
k,χ(k)