Energy-Efficient Resource Allocation for Massive MIMO ... - IEEE Xplore

Received April 18, 2016, accepted May 3, 2016, date of publication May 19, 2016, date of current version June 24, 2016. Digital Object Identifier 10.1109/ACCESS.2016.2570805

Energy-Efficient Resource Allocation for Massive MIMO Amplify-and-Forward Relay Systems HUI GAO1 , (Member, IEEE), TIEJUN LV1 , (Senior Member, IEEE), XIN SU2 , (Member, IEEE), HONG YANG3 , AND JOHN M. CIOFFI4 , (Fellow, IEEE)

1 School

of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China of Networks and Communications Research Department, Nokia Bell Laboratories, Murray Hill, NJ 07974, USA 4 Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA 2 Tsinghua

3 Mathematics

Corresponding author: T. Lv ([email protected]) This work was supported in part by the National Natural Science Foundation of China under Grant 61271188 and Grant 61401041, in part by the National High Technology Research and Development Program of China (863 Program) under Grant 2015AA01A706 and Grant 2015AA01A7096, in part by the Beijing Municipal Science and Technology Commission Research Fund Project under Grant D151100000115002, in part by the National Science and Technology Major Project under Grant 2016ZX03001017, in part by the Scientific and Technological Cooperation Project under Grant 2015DFT10160B, and in part by the State Key Laboratory of Wireless Mobile Communications, China Academy of Telecommunications Technology.

ABSTRACT Energy-efficient resource allocation is investigated for multi-pair massive MIMO amplifyand-forward relay systems, where a dedicated relay assists pairwise information exchange among many pieces of single-antenna user equipment (UE). The system energy efficiency (EE) is theoretically analyzed by employing large system analysis and random matrix theory. This analytical result provides excellent approximation for the system with a moderate number of antennas, and it also enables several efficient algorithms, working with a different knowledge of channel state information (CSI), to maximize the system EE by scheduling the optimal numbers of relay antennas and UE pairs as well as the corresponding relay transmission power. In contrast to the conventional resource allocation schemes, the proposed algorithms avoid complicated matrix calculations and the instantaneous CSI of small-scale fading; therefore, they are computationally efficient with low CSI overhead. The proposed optimization framework sheds light on the optimized system configurations, and it also offers an efficient way to achieve EE-oriented resource allocation for the multi-pair massive MIMO relay systems. INDEX TERMS Energy-efficient, low-complexity, massive MIMO relay, resource allocation.

I. INTRODUCTION

Accompanied by the explosive growth of data traffic and ubiquitous connectivity, energy efficiency (EE) has become one of the key design objectives for the fifth generation (5G) wireless communication systems [1]. In order to improve EE, various methods have been developed in the last few years [2]–[6]. Recently, as one of the major candidate technologies for 5G systems, massive MIMO [7]–[10] has been proved to bear the potential in achieving unprecedented EE gains. It is shown in [11] that massive MIMO systems can increase the EE by three orders of magnitude as compared to single antenna systems. The energy-efficient design of massive MIMO systems has become a hot topic. For example, [12]–[15] study the EE performance of centralized massive MIMO systems, and [16]–[18] discuss energy-efficient transmission schemes for distributed massive MIMO systems. VOLUME 4, 2016

Relay techniques have also drawn great attentions for 5G wireless communication systems. Considering the scenarios where devices are densely deployed with very heavy single-hop traffics [19], the idea of shared relaying in LTE-A [20] can be employed to solve the severe interference problem. More specifically, a relay with multiple antennas, known as MIMO relay, can be placed among the devices, changing the single-hop transmission into two-hop transmission. Employing an appropriate MIMO relay transceiver, e.g., zero-forcing (ZF) transceiver, the interference among different data streams/UEs can be significantly mitigated. Furthermore, it is recently proved in [21] that if a relay is equipped with a large-scale antenna array, even with a simple ZF relay transceiver, the system spectral efficiency can scale up with the number of relay antennas. Therefore, the massive MIMO relay working with a simple ZF transceiver is attractive for very good

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performance-complexity tradeoff [21]. Currently, the researches on massive MIMO relay are just at the initial stage [21]–[23]. For example, the capacity of multi-hop relay systems is addressed in [22], and the spectral efficiency of full-duplex massive MIMO relay systems is studied in [23]. Unfortunately, unlike the energy-efficient researches on single-hop massive MIMO systems, the topic of energyefficient transmission schemes for massive MIMO relay systems is still widely open. In general, the existing energyefficient studies on single-hop massive MIMO systems [13], [14], [24], [25] cannot be directly extended to massive MIMO relay systems, because the design of the relay processing matrix and the system performance analysis are all coupled with the two-hop channels. As compared to the energy-efficient solutions for conventional MIMO relay systems [26]–[28], the designs tailored for massive MIMO relay systems face new challenges. (i) Complicated matrix calculations are usually required to design transceivers and/or obtain resource allocation metric. For large channel matrices, huge amounts of time, power and hardware resources would be consumed, which might be unaffordable for practical systems. (ii) Most of the energy-efficient scheduling schemes rely on instantaneous small-scale fading (SSF) and large-scale fading (LSF) channel state information (CSI) of all pieces of the candidate user equipment (UEs). However, in massive MIMO systems there is a limited number of users whose instantaneous CSI can be estimated within each channel coherence block; the pilot overhead for CSI acquisition should be taken into account. As a result, aiming to develop energy-efficient resource allocation schemes for massive MIMO relay systems, the above two challenges should be properly addressed, reducing the computational complexity and limiting the pilot overhead. In this paper, we propose an energy-efficient resource allocation scheme for a multi-pair amplify-and-forward (AF) massive MIMO relay system. In particular, the UE pair selection, the number of relay antennas and the relay transmit power allocation are jointly optimized. The minimum mean square error (MMSE) channel estimation and ZF transmission are performed at the relay. Leveraging the tools of random matrix theory (RMT), especially the free probability theory (FPT) [29]–[31], we analyze the system performance in large-system regime and derive an asymptotic expression of the system EE in order to obtain an tractable optimization objective. Applying this analytical result, we propose an energy-efficient resource allocation scheme based on LSF CSI. Furthermore, we introduce the energy-efficient system configurations in terms of the relay transmit power, the number (density) of the active UE pairs and the number of relay antennas. The main contributions of our work are summarized as follows. • Aiming to enable an efficient EE resource allocation, we derive a tight approximation of the system EE for the considered massive MIMO AF relay system and verify the tightness of this approximation by Monte-Carlo simulations. Thanks to the channel hardening phenomenon 2772

in massive MIMO [32], the impacts of SSF on EE performance are averaged out, and the approximation is only related to system parameters and LSF CSI of active UE pairs. Therefore, we can evaluate the system EE performance and design resource allocation schemes without the instantaneous CSI of SSF. The complicated calculations of large-dimensional channel matrices and the acquisitions of the instantaneous CSI of SSF for candidate UE pairs are avoided, significantly saving resources for signal processing and data transmission. • We formulate the EE resource allocation problem as a joint optimization of UE pair selection, relay antenna activating and relay transmit power allocation. Moreover, we decompose the complicated joint optimization problem into two tractable sub-problems. First, the UE pair selection and the number of active relay antennas are iteratively optimized with an equal relay power allocation strategy. Then, the optimal relay power allocation is performed for the actually selected UE pairs. In particular, both sub-problems are formulated with the CSI of LSF, therefore their solutions are simple and of practical interests. To be more specific, we use the Dinkelbach method [33] to solve the fractional programming problem of power allocation; and we employ the max-min criterion to select the active UE pairs. Although the max-min criterion has been used to improve the outage or rate performance of relay systems [34], [35], it is the first time that max-min criterion is employed to select UE pairs for improving EE performance in massive MIMO relay systems. • By averaging out the LSF, we further analyze the system EE performance with random user locations and study energy-efficient resource allocation from the view of system configurations. The optimization of system configurations is different from the resource allocation for specific active UE pairs. In particular, the explicit CSI of LSF regarding each UE pair is not known by the relay. Only assuming the knowledge of UE distribution, the density of active UE pairs, the number of relay antennas and the relay transmit power are optimized to improve the system EE. Unlike the conventional optimization of system configuration through extensive Monte-Carlo simulations, we optimize the system parameters by an analytical approach, reducing the implementation complexity. The remainder of this paper is organized as follows. We describe the system model in Section II. In Section III, we analyze the asymptotic EE of the considered massive MIMO relay systems and propose the LSF-based low-complexity energy-efficient resource allocation scheme. In Section IV, the framework of EE-optimized system configuration is presented. Numerical results are provided in Section V. Finally, the conclusions are drawn in Section VI. Notations: We use uppercase and lowercase boldface letters for matrices and vectors, respectively. (·)H , (·)† , tr(·) and E[·] denote the conjugate transpose, pseudo-inverse, trace and VOLUME 4, 2016


the expectation, respectively. [A]i,j represents the element at the i-th row and j-th column of a matrix A. CN (m, 2) denotes the circularly-symmetric complex Gaussian distribution with mean vector m and covariance matrix 2. a.s. −−→ denotes the almost sure convergence. 2 F1 (·) indicates the hypergeometric function. |A| denotes the cardinality of a set A. A × B indicates the Cartesian product of set A and set B. II. SYSTEM MODEL

As shown in Fig.1, we consider a multi-pair two-hop one-way AF relay system which consists of an M -antenna relay and N pairs of candidate single-antenna UE. We assume the system operates over a bandwidth of B Hz and the user channels stay static within a time-frequency coherence block of T = Bc Tc symbols. Bc and Tc indicate the coherence bandwidth and coherence time of the channel, respectively. At each coherence block, the set of active UE pairs is denoted by S, |S| = K , and K is specified only after S is determined; the K (K < M ) pairs of source-destination UE (i.e., active UE pairs) transmit data simultaneously. We focus on the active UE pairs hereinafter and suppose the k-th UE (source node) intends to transmit information to the (k + K )-th UE (destination node). In addition, the direct link between any source node and destination node is ignored. The relay works on time-division duplexing mode. Each coherence interval is divided into three phases, i.e., the channel estimation phase (CE), the source to relay data phase (S → R) and the relay to destination data phase (R → D).

√ with [DS ]k,k = dk indicates the LSF of the k-th active source node in Phase S → R. The channel from the relay to the K destinations within Phase R → D is indicated by a K × M dimensional matrix GD , i.e., G D = DD H D , where DD √∈ RK ×K is also a diagonal matrix with [DD ]k,k = dk+K representing the LSF of the k-th active destination node, and HD = [hT1+K , · · · , hT2K ]T ∈ CK ×M is the SSF channel matrix. A. CHANNEL ESTIMATION

In Phase CE, we assume all the K pairs of active UEs transmit orthogonal pilot sequences to the relay and the relay acquires the CSI of active UEs via training-based MMSE channel estimation [12]. Therefore, we have hi = hˆ i + h˜ i , i = 1, . . . , 2K , (1) τr ρr di I and the estimawhere the estimate hˆ i ∼ CN 0, 1+τ r ρr di 1 tion error h˜ i ∼ CN 0, 1+τr ρr di I are mutually independent. Here, τr is the length of pilot sequence; ρr is the normalized reverse link power that is proportional to the radiated power of the i-th UE divided by the noise power at the relay receiver. Assuming the LSF CSI (i.e., long-term CSI) matrices DS and DD are known at the relay through measuring over frequency and tracking over time, we have the expressions of GS , GD with channel estimation and corresponding estimation error as ˆS+G ˜S =H ˆ S DS + H ˜ S DS , GS = G ˆD+G ˜ D = DD H ˆ D + DD H ˜ D, GD = G ˆ S = [hˆ T ,· · ·, hˆ T ], H ˜ S = [h˜ T ,· · ·, h˜ T ], where H K K 1 1 ˜ D = [h˜ T ,· · ·, h˜ T ]T . [hˆ T1+K ,· · ·, hˆ T2K ]T and H 1+K 2K

(2) (3) ˆD = H

B. DATA TRANSMISSION

In Phase S → R, K sources transmit a symbol vector xU ∈ CK ×1 to the relay with power Pt for each source,1 i.e., p xU = Pt s, (4)

FIGURE 1. The multi-pair two-hop relay scenario, where K pairs of UEs communicate with the help of a MIMO relay.

Assuming hi ∈ = 1, . . . , 2K ) with independent and identically distributed (i.i.d.) CN (0, 1) elements models the SSF between the i-th UE and the relay, we denote the M × K dimensional channel matrix from the K sources to the relay as C1×M (i

GS = HS DS , [hT1 , · · ·

where HS = , hTK ] ∈ CM ×K is the SSF channel matrix of Phase S → R. The diagonal matrix DS ∈ RK ×K VOLUME 4, 2016

in which s = [s1 , . . . , sK ]T is the information-bearing symbol vector with E(ssH ) = IK , and sk is the symbol delivered from the k-th UE to the (k + K )-th UE. The received signal yR ∈ CM ×1 at the relay is yR = GS xU + nR ,

(5)

where nR ∈ CM ×1 is the zero-mean additive white Gaussian noise (AWGN) at the relay with a variance 1 We mainly focus on the optimization for the massive MIMO relay, and the assumption of fixed source transmission power eases the subsequent theoretical derivations to offer clear insights. It is also noted that for non-massive MIMO relay some existing work suggests a joint relay and UE power allocation for EE [28]. The joint power allocation among relay and UE pairs may further improve the EE performance of the massive MIMO relay systems as well, but due to limited space, we defer this interesting work as a future study.

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2 E nR nH R = σr IM . The relay performs ZF processing on yR and obtains the filtered signal vector xR ∈ CM ×1 as xR = FyR ,

(6)

where F ∈ CM ×M is the relay processing matrix designed as ˆ †. ˆ † PG F=G S D

(7)

The diagonal matrix P ∈ RK ×K is the relay power allocation √ matrix, where [P]k,k = pk denotes the relay transmit power allocated to the k-th active UE pair. The long-term power constraint at the relay can be written as ESSF [tr(xR xH R )] ≤ Pr ,

2 ESSF [tr(xR xH R )] = Pt C1 + σr C2 + Pt C3 + Pt C4 + Pt C5 ,

(9) where C1 , C2 , C3 , C4 and C5 are defined as h i ˆ DG ˆ H )−1 , C1 = ESSF tr P2 (G D h i H ˆ −1 ˆ ˆ DG ˆ H )−1 , C2 = ESSF tr P(GS GS ) P(G D !# " H H −1 ˜ ˆ ˆ ˆ ˆ DG ˆ H )−1 GS GS (GS GS ) P(G D C3 = ESSF tr , ˆ HG ˆ −1 ˆ H ˜ ×P(G S S ) GS GS h i ˜ S P(G ˆ DG ˆ H )−1 P(G ˆ HG ˆ −1 ˆ H , C4 = ESSF tr G D S S ) GS h i ˆ S (G ˆ HG ˆ −1 ˆ ˆ H −1 ˜ H , C5 = ESSF tr G S S ) P(GD GD ) PGS

(10)

and ESSF [·] represents the averaging over SSF. During Phase R → D, the relay amplifies and forwards xR to all the K active destinations. The received signal yU ∈ CK ×1 is given by yU = GD xR + nU , where nU is the zero-mean AWGN at the destinations with a 2 I . In particular, the received signal variance E nU nH = σ K u U at the k-th destination is p p √ † † ˜ yk = pk Pt sk + pk gˆ S,k nR + pk Pt gˆ S,k G Ss + g˜ D,k xR + nk , (11) where yk , sk and nk are the k-th element of yU , s and nU , † ˆ† respectively. gˆ S,k and g˜ D,k are the k-th row of G S ˜ D , respectively. The signal-to-interference-plus-noise and G ratio (SINR) of the k-th data stream is characterized by

where ζk =

pk σr2

h

pk Pt i ˆ HG ˆ S )−1 (G S

† ˜ ˜ H †H ˆ S,k pk Pt gˆ S,k G S GS g

k,k

+ ζk + σu2

k=1

2774

,

(12)

˜H + g˜ D,k xR xH Rg D,k .

The sum rate is then given by K 2K X B R= 1− log2 (1 + γk ). 2 T

C. POWER CONSUMPTION

We employ a practical power consumption model similar to the models proposed in [14], [17], and [36]. The total power consumption of the considered system can be modeled as

(8)

where Pr is the maximum transmit power available at the relay. With the help of (2) – (7), we have

γk =

It is noted that we assume a coherence frequency-time block consists of T symbols and each active UE is assigned an orthogonal pilot sequence of duration τr = 2K symbols. Therefore, there exists a pre-log factor (1 − 2K /T ) in (13) representing the desired data transmission part of each coherence interval. In addition, B/2 indicates the half-duplex mode of the considered system.

P = Ptx + Pc , where −1 ηPA,U

2K 1− KPt 2 T −1 −1 ηPA,U 4K 2 ηPA,R 2K 2 + ρr σr + 1− Ptx,R , T 2 T Pc = Pfix + Ptc + Psig . (15)

Ptx =

Here, Ptx is the power consumed by power amplifiers (PAs), in which Pt is the data transmit power at each active UE source, ρr σr2 is the pilot transmit power at each active UE, and Ptx,R = tr xR xH R is the transmit power of the relay. ηPA,U ∈ (0, 1) and ηPA,R ∈ (0, 1) are the PA efficiency factors at the UEs and the relay, respectively. Pc denotes the total circuit power consumption, where Pfix is a constant accounting for the fixed power consumption required for controlling, site-cooling, and the load-independent power of baseband processors, Ptc indicates the power consumption of transceiver chains, and Psig accounts for the power consumption of load-dependent signal processing. To be more specific, following the results on computational complexities related to MMSE channel estimation, vector multiplication and channel inversion [14], we have Ptc = MPR + 2KPU + PSYN , B 8MK 2 2K 4MK Psig = +B 1− T LR T LR B 2 3 2 + (K + 9MK + 3MK ), (16) T 3LR in which PR and PU indicate the power required to run the circuit components attached to each antenna at the relay and the UEs, respectively. PSYN is the power consumed by the oscillator. Psig denotes the power consumption of channel estimation (the first term) and linear signal processing (the remaining two terms) at the relay. LR denotes the computational efficiency in complex-valued operations per Joule at the relay. The EE E in bit/Joule is defined as E(P, S, M ) =

(13)

(14)

R(P, S, M ) , P(P, S, M )

(17)

in which R(P, S, M ) denotes the system sum rate as a function of the transmit power allocation P at the relay, the VOLUME 4, 2016


selection S of active UE pairs, and the number M of the relay antennas. Similarly, P(P, S, M ) denotes the total power consumption of the considered system. III. LSF-BASED ENERGY-EFFICIENT RESOURCE ALLOCATION

B. ASYMPTOTIC PERFORMANCE ANALYSIS

In this section, we propose an energy-efficient resource allocation scheme, in which the selection of active UE pairs, the amount of relay antennas and the allocation of transmit power at the relay are carefully designed to optimize the system EE performance. After formulating the EE optimization problem, we first perform the asymptotic analysis to obtain a tight approximation of the original EE objective. The new objective function is only related to the LSF CSI. Then, we develop a low-complexity alternative optimization framework to solve EE maximization problem. A. PROBLEM FORMULATION

The original optimization problem can be formulated as max E(P, S, M ),

(18)

P,S ,M

s.t. (8), S ∈ U,

(19)

M ∈ {1, 2, . . . , Mmax },

(20)

0 ≤ pk ≤ Pr ,

(21)

γk ≥ γ0 ,

k = 1, . . . , |S|,

k = 1, . . . , |S|.

(22)

where U denotes all the possible sets of active UE pairs. The objective function E is defined by (17). (8) specifies the available transmit power constraint for the relay. (19) is the user selection constraint. The constraint of available number of relay antennas is presented as (20), in which Mmax is the largest available number. (21) is the boundary constraint for the relay power allocation variables. The inequalities in (22) are the quality-of-service (QoS) constraints, which imply that the SINR of each active UE pair is lower bounded by γ0 . It is noted that the above problem (18) is non-convex. In order to find the global optimal solution, we could solve it with brute force searching, but the complexity is too high. First, the search space is huge even withfixed power allocation. Roughly speaking, there are 2N −2 non-empty subsets of U, i.e., the maximum number of choices on S is 2N −2 , and Mmax available choices of the number of relay antennas for each possible user selection S. Hence, the maximum size of the search space is 2N −2 Mmax . Second, the evaluation of EE depends on instantaneous CSI (i.e., CSI of both SSF and LSF). Complicated matrix calculations are required. For the massive MIMO relay systems, the channel matrix is huge, leading to very high computational complexity. Therefore, instead of solving (18) directly, we propose a suboptimal, low-complexity user selection and power allocation scheme. The complexity is reduced from two aspects. First, we design (P, S, M ) based on the large-system approximation instead of the exact value, avoiding complicated matrix computations. Second, we decompose the VOLUME 4, 2016

optimization problem of (P, S, M ) into two parts: 1) the allocation of relay transmit power (i.e., P), and 2) the selection of active UE pairs and the optimization of the active relay antenna number (i.e., S, M ).

In this subsection, we analyze the asymptotic EE performance in the large-system regime (M , K → ∞, M /K > 1) with the tools of RMT, especially FPT. ˆ S, G ˜ S, G ˆ D and According to (1) and (2), we can rewrite G ˜ D as G ˆ S = Z1 M1 , G ˜ S = Z2 M2 , G ˆ D = M3 Z3 and G ˜ D = M4 Z4 , where {Mm , m = 1, . . . , 4} are K × K diagG onal matrices with diagonal elements mm,k , (k = 1, . . . , K ) given by s s 2K ρr dk2 dk , m2,k = , m1,k = 1 + 2K ρr dk 1 + 2K ρr dk s s 2 2K ρr dk+K dk+K m3,k = , m4,k = . 1 + 2K ρr dk+K 1 + 2K ρr dk+K (23) The matrices Z1 ∈ CM ×K , Z2 ∈ CM ×K , Z3 ∈ CK ×M and Z4 ∈ CK ×M with i.i.d. CN (0, 1) entries are independent of each other. The key result of the large-system approximation of system EE is presented in the following theorem. Theorem 1: For M → +∞, K = |S| → +∞ and M /K > 1, the system EE can be approximated as a function of P, S and M , i.e., ¯ ¯ S, M ) = R(P, S, M ) , E(P, ¯ P(P, S, M )

(24)

¯ ¯ where R(P, S, M ) and P(P, S, M ) are the large-system approximations of the system sum rate and power consump¯ tion, respectively. In particular, R(P, S, M ) is given by K B 2K X ¯ 1− log2 (1 + γ¯k ), R(P, S, M ) = 2 T

(25)

k=1

and γ¯k is calculated as γ¯k =

pk σr2 A1,k

+ pk Pt A1,k m−2

pk Pt PK

i=1 A2,i

, + Ptx,R A4,k + σu2 (26)

1,k where A1,k = M −K , A2,k = m22,k , A3,k = 2 A4,k = m4,k . ¯ P(P, S, M ) is given by

¯ P(P, S, M ) = +

−1 ηPA,R

2

m−2 3,k M −K

and

−1 ηPA,U

−1 ηPA,U 4K 2 2K 1− KPt + ρr σr2 2 T T 2K 1− Ptx,R + Pc (27) T 2775


where Ptx,R denotes the transmit power at the relay expressed as Ptx,R = Pt

K X

A3,k pk + σr2

k=1 K X

+ Pt

K X

A1,k A3,k pk

k=1

A2,i

i=1

K X

A1,k A3,k pk .

(28)

k=1

Algorithm 1 LSF-Based Optimal Relay Power Allocation 1) Initialize the maximum number of iterations Nloop and the maximum tolerance . 2) Set the energy efficiency q and the iteration index n as q = 0, n = 0, respectively. 3) Solve the problem (31). ¯ ¯ a) If n > Nloop or R(P) − qP(P) < , go to 4). b) Else, update

Proof: Please see Appendix I. Similar to the derivation of (28) in the above theorem, the transmit power constraint (8) of the relay can be rewritten as Pt

K X

A3,k pk + σr2

k=1

+Pt

K X

K X

q=

¯ R(P) . ¯ P(P)

Restart 3). 4) Get the final solution P∗ .

A1,k A3,k pk

k=1

A2,i

i=1

K X

A1,k A3,k pk ≤ Pr

(29)

k=1

Remark 1: Applying Theorem 1, we can evaluate the performance of system EE without any instantaneous CSI of ¯ and P¯ only depend on the LSF CSI of active SSF, because R UE pairs and the system parameters. In the following subsection, we will use these approximations to design energy-efficient resource allocation. C. LSF-BASED RESOURCE ALLOCATION

In this subsection, we develop an LSF-based resource allocation scheme with the analytical results given by Theorem 1. First, given a fixed UE pair selection and a number of relay antennas, the optimal relay power allocation and the equal relay power allocation are proposed, which will be used in the ending step and the iterative steps of our practical alterative optimization framework. Then, given a fixed UE pair selection and power allocation, the optimal number of relay antennas is studied. Finally, we propose a LSF-based maxmin UE pair selection along with the optimization of the number of relay antennas and transmit power to achieve the optimized system EE. 1) OPTIMAL RELAY POWER ALLOCATION

With fixed S and M , the power allocation problem based on large-system approximation is given by ¯ max E(P), P

s.t. (29), 0 ≤ pk ≤ Pr , k = 1, . . . , K , γ¯k ≥ γ0 , k = 1, . . . , K ,

(30)

¯ where Ptx,R , γ¯k are defined in (26), and E(P) is given by (24). Obviously, the optimization problem (30) is a non-linear fractional program. We solve it by using the Dinkelbach method [33]. The details are shown in Algorithm 1. At each iteration, we need to solve the following problem with a given parameter q, i.e., ¯ ¯ max R(P) − qP(P), P s.t. (29), 0 ≤ pk ≤ Pr , k = 1, . . . , K , γ¯k ≥ γ0 , k = 1, . . . , K . 2776

n = n + 1,

¯ is concave over pk , For (31), it is easy to check that R P¯ is linear over pk , and all the constraints are linear. As a consequence, (31) is a convex optimization problem, and it can be solved efficiently using standard interior point method software tools, e.g., SeDuMi [37]. 2) EQUAL RELAY POWER ALLOCATION

When the time and computing resource are limited, it might be difficult to solve the optimization problem (30) at low cost. It is thus necessary to use some other approaches with reduced complexity for the relay power allocation. An intuitive choice is the equal relay power allocation among active UE pairs. Considering the equal power allocation condition with (28), for any k = 1, . . . , K , the power allocation coefficient can be calculated as Ptx,R (M − K )2 , (32) Pt (M − K )B3 + σr2 B13 + Pt B2 B13 PK PK −2 −2 −2 where B3 = k=1 m3,k , B13 = k=1 m1,k m3,k and PK B2 = k=1 m22,k . Substituting (32) into (26), we obtain the following corollary about the system EE with equal relay power allocation. Corollary 1: For M → +∞, K → +∞, M /K > 1, the approximate EE E¯¯ with equal power allocation can be calculated as pk =

¯¯ R(P tx,R , S, M ) ¯¯ , E(P tx,R , S, M ) = ¯ P(Ptx,R , S, M )

(33)

¯ tx,R , S, M ) is defined by (27). The approximate where P(P ¯¯ sum rate R(P tx,R , S, M ) is K B 2K X ¯ ¯ R(Ptx,R , S, M ) = 1− log2 (1 + γ¯¯k ). 2 T

(34)

k=1

where the SINR γ¯¯k is given by

(31)

γ¯¯k =

Ptx,R Pt (M − K )2 2 Ptx,R m−2 1,k (M − K )(σr + Pt B2 ) + ψk

,

(35)

VOLUME 4, 2016


where

ψk = Pt (M + Pt B13 B2 × Ptx,R m24,k + σu2 . By using the above corollary, for fixed S and M , the resource allocation problem with equal relay power allocation can be formulated as ¯¯ max E(P ), (36) Ptx,R

− K )B3 + σr2 B13

tx,R

s.t. 0 ≤ Ptx,R ≤ Pr , γ¯¯k ≥ γ0 , k = 1, . . . , K .

(37) (38)

Compared to the optimal power allocation (30) for each UE pair, we only need to optimize the total transmit power Ptx,R with respect to the equal power allocation scheme. To solve the above optimization problem, we first find the feasible region of Ptx,R and then find the global extrema. The detailed steps are shown as follows. First, we solve the equations γ¯¯k (Ptx,R ) = γ0 and get the solution Ptx,R,k for k = 1, 2, . . . , K , which corresponds to the non-optimized power for each UE. It can be easily observed from (35) that γ¯¯k is a monotonically increasing function for Ptx,R . Hence, the QoS constraints (38) can be rewritten as Ptx,R ≥ Ptx,R,k , k = 1, 2, . . . , K , i.e., Ptx,R ≥ Ptx,R,max ,

(39)

in which Ptx,R,max = max{Ptx,R,1 , Ptx,R,2 , . . . , Ptx,R,K }. Considering both (37) and (39), the feasible region of Ptx,R for (36) becomes [Ptx,R,max , Pr ]. It should be noted that, if Ptx,R,max > Pr , the optimization problem becomes infeasible. In that case, we are not able to find a solution of Ptx,R satisfying the QoS constraint for the given S and M , so we should move to a larger M or another S. If we have Ptx,R,max ≤ Pr , (36) is feasible on the interval [Ptx,R,max , Pr ]. Once feasible, we can find the global maximum ¯¯ of E(P tx,R ). To be more specific, it can be verified that ¯¯ the function E(P tx,R ) and its first-order derivative are all continuous with respect to Ptx,R on the finite closed interval [Ptx,R,max , Pr ]. As a result, according to the extreme ¯¯ value theorem [38], E(P tx,R ) has an absolute maximum on [Ptx,R,max , Pr ]. To find the global maximum, we need to check all the critical points in the given interval (i.e., solve the ¯¯ equation d E(P tx,R )/dPtx,R = 0) and evaluate the function ¯ ¯ E(Ptx,R ) at these critical points and at the endpoints of the interval. The largest of the values in the previous steps is ¯¯ the global maximum value of E(P tx,R ) on [Ptx,R,max , Pr ], and the corresponding value of Ptx,R is the solution of (36) (i.e., the optimal value P∗tx,R ) for the given S and M . Remark 2: It is noted that both the optimal and equal relay power allocation algorithms will be used in our alternative optimization framework. In particular, during the iteration steps for choosing the optimal numbers of relay antennas and UE pairs, the relay transmission power will be optimized according to the equal power allocation algorithm within each iteration; when the iteration is finished with the final results of VOLUME 4, 2016

the optimal numbers of relay antennas and UE pairs, the optimal power allocation is used. Our approach can avoid complicated optimal power allocation during the iteration steps, and it will be shown later via numerical results that such optimization framework achieves attractive performance-complexity tradeoff. 3) OPTIMAL NUMBER OF RELAY ANTENNAS

After the discussion of the relay power allocation, we study the optimal number of relay antennas. Given S and P, the optimization problem is ¯¯ ), max E(M M

s.t. M ∈ {1, . . . , Mmax }, γ¯¯k ≥ γ0 , k = 1, . . . , K . (40) ¯ ¯ )/dM here cannot Unlike the solution of M ∗ in [13], d E(M be reformulated to the standard format introduced in [39], so it is not possible to present M ∗ with a closed-form lambertW expression. Fortunately, the feasible region of (40) is {1, 2, . . . , Mmax }, which is discrete and finite. Therefore, although we cannot find a closed-form solution of (40), we can solve it efficiently with the help of standard searching techniques, e.g. the bisection method [40]. 4) MAX-MIN USER SELECTION

Theoretically speaking, we could solve (18) by performing the optimization of P and M over all possible sets of candidate UE pairs and choosing the best set as the active UE pairs, i.e., selecting UE pairs through exhaustive searching. However, as mentioned before, the total number of possible sets (i.e., the size of search space) would be very large, which leads to prohibitive implementation complexity in practice. Here we propose a much simpler method to select active UE pairs. Of particular note, we reduce the computational complexity from the following two aspects, i.e., narrowing the selection space and simplifying the selection metric. (i) It can be observed that the considered system performance is closely related to the LSF CSI of both the S-R phase and the R-D phase. Neither of these two phases could be ignored during the user selection. Noting this, we propose an LSF-based max-min user selection scheme working with LSF CSI. More specifically, we sort the candidate UE pairs in descending order with respect to the smaller one of its S-R LSF gain dk and R-D LSF gain dk+K , and denote the first n UE pairs of the sorted set of candidate UE pairs as Sn . Then we select UE pairs successively, i.e., check S1 , S2 , . . . to find the final active UE pairs maximizing the system EE. The searching size is dramatically reduced to N . (ii) During each calculation of the EE for selecting an active UE pair, assume that the equal power allocation is employed at the relay. In other words, we exploit E¯¯ to evaluate the EE during the user selection procedure, avoiding the iterative calculations of the optimal power allocation during the solving of ¯ Only after the whole selection of active selection metric E. UE pairs is completed, the optimal relay power allocation is carried out. 2777


Algorithm 2 LSF-Based max-min User Selection Algorithm 1) Sort the candidate UE pairs in descending order with respect to min{dk , dk+K }. a) Denote the first n pairs of UEs as Sn . ¯¯ ) = 0. b) Set k = 0, E(S 0 2) For k ≤ min{M , N }: a) Set k = k + 1. b) Solve P∗tx,R and M ∗ for Sk . i) Optimize Ptx,R for a fixed M using (36). ii) Optimize M for a fixed Ptx,R using (40). iii) Repeat until convergence is achieved. ¯¯ ) with (35). c) Calculate the EE of E(S k ¯¯ ) ≤ E(S ¯¯ If E(S k k−1 ), go to 3). Else, restart 2). 3) Get the final selection of active UE pairs Sk and the number of relay antennas M ∗ .

The LSF-based resource allocation scheme can be summarized as follows. Step 1. Optimize S and M using Algorithm 2. Obtain the optimized selection of active UE pairs and the number of relay antennas (S ∗ , M ∗ ). Step 2. Optimize P using Algorithm 1 with the result (S ∗ , M ∗ ) obtained in Step 1, and get the optimal relay power allocation P∗ . Hence, we get the final resource allocation solution (P∗ , S ∗ , M ∗ ). Remark 3: We use the LSF CSI to perform UE pair selection. To this end, we are about to select the UE pairs located near to the relay, and the power allocation is designed to maximize overall network EE. Therefore, the fairness (in terms of individual rate) is not guaranteed. We also note there are some recent works assuming equal-rate power allocation to ensure the fairness for the one-hop scenarios [14]. Extending such fairness-aware power allocation to the relay network is an interesting future work. Remark 4: We follow an alternative optimization framework, therefore we choose to do the optimization in a decoupled and suboptimal way. To this end, the relay transmission power is optimized assuming fixed S and M , and then the optimal number of relay antennas M is optimized assuming fixed (S and P∗tx,R ). Finally, S is optimized in a greedy fashion with its corresponding P∗tx,R and M ∗ . In addition, regarding the convergence of the proposed iterative algorithm, we have performed extensive numerical simulations, where the convergence is empirically achieved and proved as shown in Fig. 4 in Section V. Remark 5: One of the key differences on resource allocation between non-massive MIMO and massive MIMO relay systems is the role of the SSF. For non-massive MIMO relay systems [26]–[28], the CSI of SSF is more important for resource allocation to combat/exploit fading. As the relay is equipped with large-antenna array, ‘‘channel hardening’’ phenomenon arises [8], and the instantaneous SINR converge to a deterministic equivalent according the large system analysis. 2778

To this end, the CSI of SSF is not as important as the case in non-massive MIMO systems. Therefore, low-complexity large-scale fading (LSF) based resource management can be employed to achieve very good performance in massive MIMO systems. IV. OPTIMAL DESIGN OF SYSTEM PARAMETERS

The optimization framework in the above section is userspecific, namely, the resource allocation is designed for each specific realization of long-term channel (i.e., LSF CSI of UEs). Obviously, any change of LSF CSI of UEs lead to recalculation of (P, S, M ). In this section, we consider the energy-efficient resource allocation from the view of system configuration. The motivation behind our work is to improve system EE performance without any instantaneous CSI (i.e., without any SSF CSI and LSF CSI) of UE pairs and to provide some insights on the EE-optimum system parameter selection for practical deployment. To this end, we first derive a tractable and computable expression of the system EE as a function of system parameters (e.g., the relay transmit power Ptx,R , the number K of active UE pairs and the number of relay antennas M ), and then use this expression to optimization system parameters for better EE performance. It should be mentioned that the randomness of SSF CSI has already been averaged out in the RMT-based result in Theorem 1. Hence, we focus on the randomness of LSF CSI in this subsection. Assume that the relay is located at the center point and covers a disk of radius Rmax . Furthermore, all the active source and destination UEs are supposed to be independent uniformly distributed (i.u.d.) within the relay coverage area. The LSF CSI of the k-th active UE is modeled as dk = clk−α , where lk is the distance between the k-th UE and the relay, α is the path loss exponent, and c is the path-loss at the reference distance Rmin . The PDF of lk is fL (lk ) =

2lk , − R2min

R2max

Rmin ≤ lk ≤ Rmax ,

(41)

in which Rmin is the reference distance in the user location model [41]. Next, we discuss the optimization problem of the energy-efficient system parameters. First, we obtain an approximation of the system EE as shown in Theorem 2. Theorem 2: Considering both the SSF and LSF CSI, the system EE can be approximated as ˜ ˜ tx,R , K , M ) = R(Ptx,R , K , M ) . E(P ¯ tx,R , K , M ) P(P

(42)

˜ tx,R , K , M ) is the approximate system sum rate given by R(P ˜ tx,R , K , M ) = BK 1 − 2K I(Ptx,R , K , M ), R(P (43) 2 T where ZZ

Rmax

I=

log(1 + γ˜k )fL (lk )fL (lk+K )dlk dlk+K ,

(44)

Rmin VOLUME 4, 2016


Algorithm 3 Optimization of (Ptx,R , K , M ) 1) For M = 1, 2, . . . , Mmax { For K = 1, 2, . . . , M − 1{ Optimize Ptx,R for (K , M ) as

γ˜k , k = 1, . . . , K , is given by γ˜k =

Ptx,R Pt (M − K )2 2 ˜ χk + Ptx,R m−2 1,k (M − K )(σr + Pt B2 )

,

! α+2 2(Rα+2 max −R0 ) + , α+2 !2 2(α+1) 2(α+1) Rmax −R0 2(Rα+2 −Rα+2 ) 1 K max 0 + , B˜ 13 = c(α + 1) α+2 c2 42R 2ρr K 1 α+2 Rαmax 2 ˜B2 = c Rmax 2 F1 1, ; ;− 2ρr 4R α α 2cρr K Rα0 1 α+2 · − R20 2 F1 1, ; (45) ;− α α 2cρr K and 4R = R2max − R2min and χk = (Ptx,R m24,k + σu2 ) Pt (M − K )B˜ 1 + σr2 B˜ 13 + Pt B˜ 13 B˜ 2 . Proof: Please see Appendix II. Using Theorem 2, we are able to evaluate the system EE performance without any instantaneous CSI. Next, we turn to the QoS constraint γk ≤ γ0 , k = 1, 2, . . . , K . It it clear that γ˜k in (45) is still related to specific LSF CSI of a given UE pair, which does not satisfy our requirement of CSIindependency. Hence, we need to find a further processing which makes the QoS constraint independent of instantaneous CSI. Assume the total number of active UE pairs within the coverage area is K , and let lmax = max{l1 , . . . , lK }, the PDF fL (lmax ) of lmax can be written as 2(α+1)

2(α+1)

K 1 Rmax −R0 B˜ 1 = c4R 2ρr K c(α + 1)

fL (lmax ) =

2 − R2 )K −1 2lmax (lmax min

K (R2max − R2min )K

.

(46)

Then, we can relax the QoS constraints as follows γ˜˜min ≥ γ0 ,

(47)

where γ˜˜min is an approximation of the minimum value of SINRs among all the active UE pairs, i.e., γ˜˜min ≈ min{γ˜0 , γ˜1 , . . . , γ˜K }. Additionally, γ˜˜min is given by Z Z Rmax γ˜k fL (lmax )fL (lk+K )dlmax dlk+K . (48) γ˜˜min = Rmin

The detailed derivations of (47) and (48) are also given in Appendix II. Given the relay coverage π (R2max − R2min ), the density of active UE pairs can be calculated as ρUE =

K . − R2min )

π (R2max

The problem on the density of active UE pairs ρUE could be transformed into the problem on the number of active UE pairs K . Therefore, we formulate the energy-efficient system parameters optimization problem as max

Ptx,R ,K ,M

˜ tx,R , K , M ), E(P

s.t. M ∈ {1, 2, . . . , Mmax }, VOLUME 4, 2016

(49) (50)

˜ tx,R ), max E(P Ptx,R

(54)

s.t. 0 ≤ Ptx,R ≤ Pr , γ˜˜min ≥ γ0 . ˜ ∗ Obtain P∗tx,R|KM and E(P tx,R|KM , K , M ). } } 2) Find the optimal value of (K , M ) as (K ∗ , M ∗ ) =

arg max M ∈{1,...,Mmax }, K ∈{1,...,M −1}

˜ ∗ E(P tx,R|KM , K , M ).

(55)

Hence, the optimal value of (Ptx,R , ρUE , M ) is ∗ (P∗tx,R , ρUE , M ∗)

=

P∗tx,R|K ∗ M ∗ ,

! K∗ ∗ ,M . π (R2max − R2min )

K ∈ {1, 2, . . . , M − 1}, 0 ≤ Ptx,R ≤ Pr , γ˜˜min ≥ γ0 .

(51) (52) (53)

In the above optimization problem, (50) is the constraint on the number of relay antennas, which implies that the optimal number of relay antennas, M , must be a positive integer chosen from the feasible set {1, 2, . . . , Mmax }. (51) is the constraint on the number of active UE pairs K . For each given number of relay antennas, the feasible set of K is a finite set of positive integers smaller than M , where K < M ensures the feasibility of ZF transceiving at the relay. (52) is the boundary constraint for the relay transmit power. (53) specifies the QoS requirement. To solve the optimization problem (49), as presented in Algorithm 3, we first find the value of relay transmit power maximizing the EE for every possible choice of (K , M ). The solution of Ptx,R and the corresponding EE are denoted as ˜ ∗ P∗tx,R|KM and E(P tx,R|KM , K , M ), respectively. Particularly, we solve the problem (54) in Algorithm 3 with the method employed in (36), which guarantees the global optimality of the solution P∗tx,R|KM for the given (K , M ). Then the optimal value of (K , M ) can be obtained by searching throughout the feasible set {1, 2, . . . , Mmax } × {1, 2, . . . , M − 1} for max˜ ∗ imizing E(P tx,R|KM , K , M ). Thus, the global optimal value of the relay transmit power is given by P∗tx,R = P∗tx,R|K ∗ M ∗ , The optimal density of active UE pairs can be calculated as ∗ = K ∗ /(π(R2 2 ρUE max − Rmin )). Remark 6: The impact of LSF CSI is also averaged out in the derived expressions (42) – (48). In other words, (49) only depends on the system parameters. Therefore, we can optimize the system parameters in terms of EE maximization without any instantaneous CSI. 2779


V. NUMERICAL SIMULATIONS

In this section, we evaluate the performance of the proposed energy-efficient resource allocation schemes via numerical simulations. The parameters on energy cost introduced in [14] are employed in our simulations. For the channel model, the reference distance Rmin is set to be 35 m and the LSF model is given by 10−0.53 /l 3.76 . The transmission bandwidth of the system is B = 20 MHz, the channel coherence time is Tc = 10 ms and the coherence bandwidth is Bc = 180 kHz.2 The total noise power is −96 dBm. For the power consumption model, the computational efficiency at the relay is LR = 12.8 Gflops/W. PA efficiency at the relay and UEs are ηPA,R = 0.39 and ηPA,U = 0.3, respectively. The fixed power consumption is Pfix = 18 W. The circuit power consumption at the relay and UEs are PR = 1 W and PU = 0.1 W, respectively.

FIGURE 2. EE and SINR vs. K , M, Ptx,R . We assume M = 128 and Ptx,R = 30 dBm in sub-figure (a) and (d), K = 32 and Ptx,R = 30 dBm in sub-figure (b) and (e), and M = 128, K = 40 in sub-figure (c) and (f). Two scenarios, i.e., ρr = 100 (in red color) and ρr = 0.1 (in blue color), are presented in each sub-figure.

usually on the boundary. However, the SINR statistically increases with M and Ptx,R , but decreases when K increases. Therefore, in order to improve the SINR of a certain UE pair, we should use more relay antennas, increase the transmit power at the relay, or serve fewer pairs of UEs simultaneously. In Fig. 3, we further verify the accuracy of our approximations for EE and SINR. Sub-figure (a) shows the numerical results of EE and the analytical approximations (24). Specifically, for each given number of relay antennas M , we fix the LSF channel and use 100 realizations of randomly generated SSF channel. Each red dot presents an EE value calculated by (17) for a realization of SSF channel. The blue line is the analytical results calculated according to our approximation (24). Sub-figure (b) presents the numerical simulation results of EE and the analytical approximation (42) when the the randomness of LSF and SSF is considered. For each M , 100 randomly generated channel realizations are used, i.e., both the LSF CSI and the SSF CSI are random. In order to measure the accuracy of approximations, we adopt the nor1 P100 malized MSE metric in the expression with 100 n=1 kEn − ¯ 2 /kEk ¯ 2 to describe the deviation between the numeriEk cal results and the approximation (24) in sub-figures (c). In sub-figure (d), we give the normalized MSE between the numerical results and the approximation (42), defined 1 P100 ˜ 2 ˜ 2 as 100 n=1 kEn − Ek /kEk . Based on the observation on normalized MSE, we show that (24) and (42) provide tight approximations of the system EE performances. Moreover, the deviation between the numerical result and the approximation is getting smaller as the number of relay antennas M increases, which indicates the instantaneous EE with random channel realization can be finely approximated by the deterministic result obtained with large-system analysis.

A. LSF-BASED RESOURCE ALLOCATION

In Fig. 2, the EE E and SINR γk are evaluated with Rmax = 250 m and Pt = 0.1 W. The simulated EE and SINR (marked as ‘Sim.’ in the figure) are obtained by averaging over 10000 independent channel realizations. The approximations in (33), (35) (marked as ‘Appx 1.’) and in (42), (45) (marked as ‘Appx 2.’) are also shown in Fig. 1. It is clear that our large system approximations are accurate even for finite-size systems. Moreover, the EE performances versus the number of active UE pairs K , the number of relay antennas M and the transmit power Ptx,R at the relay are presented in sub-figures (a), (b) and (c), respectively. The corresponding SINRs are shown in sub-figures (d), (e) and (f). From the sub-figures (a), (b) and (c) in Fig. 2, we observe that given fixed system parameters, whether the channel estimation is good (e.g., ρr = 100 shown in red in the figure) or not (e.g., ρr = 0.1 shown in blue in the figure), the system EE is not a monotonic increasing/decreasing function of K , M or Ptx,R . The optimal value of K , M or Ptx,R maximizing EE is not 2 In our simulations, we assume a symbol covers 1 ms in time domain

and 1 kHz in frequency domain. Hence, one coherence frequency-time block contains 10 × 180 = 1800 symbols, i.e., T = 1800. 2780

FIGURE 3. EE vs the number of relay antennas M. K = 30, ρr = 100, Rmax = 250 m, Pt = 0.1W and Ptx,R = 1 W. Sub-figure (a) and (c) are for channel realizations of random SSF with fixed LSF. Sub-figure (b) and (d) are for channel realizations of both random SSF and random LSF.

Fig. 4 illustrates the convergence of the proposed algorithms with Rmax = 250 m, M = 64, Pr = 1 W, Pt = 0.01 W, and a fixed set of active UE pairs S, K = |S| = 20. VOLUME 4, 2016


By contrast, when ‘AVG+OPT’ is used, we only need to do the optimal relay power allocation once. The simulation parameters are set as Mmax = 16, 32, 64 and 128, Pt = 0.1 W, Pr = 20 W and γ0 = 1. The number of candidate UE pairs is set to be N = M . The implementation complexity measured with the average CPU time of these two schemes are also presented in Fig. 5(b). It is shown that compared with the ‘OPT’ scheme, the complexity is greatly reduced by the proposed scheme (i.e., the ‘AVG+OPT’ scheme), whereas the EE performance is only slightly decreased.

FIGURE 4. EE vs the number of iterations. Rmax = 250 m, M = 64, Pr = 1 W, Pt = 0.01 W with a fixed set of active UE pairs S, K = |S| = 20.

The QoS constraint is γ0 = 1. The value of the system EE versus the number of iterations of the relay power allocation algorithm in Section III-C.1 is shown and marked as ‘OPT’. It can be investigated that the proposed ‘OPT’ algorithm reaches a steady value in about three iterations in both good (i.e., ρr = 100) and bad (i.e., ρr = 0.1) channel estimation scenarios. The equal relay power allocation scheme in Section III-C.2 (marked as ‘AVG’) is also given in Fig. 3 to show the advantage of ‘OPT’ scheme in terms of EE enhancement.

FIGURE 5. Performance comparisons among various resource allocation schemes. Rmax = 250m, Pt = 0.1 W, Pr = 20 W and γ0 = 1.

In sub-figure (a) of Fig. 5, we compare the EE performance of two resource allocation schemes, i.e., the max-min user selection with optimal relay power allocation at each selection of active user pair (marked as ‘OPT’), and the proposed LSF-based resource allocation scheme in which the optimal relay power allocation is only carried out with the final results of S and M (marked as ‘AVG+OPT’). It is note that, when ‘OPT’ is used, K times of optimal relay power allocation are needed to complete one selection of active UE pairs. VOLUME 4, 2016

FIGURE 6. The optimization of (Ptx,R , K , M). Rmax = 250m, ρr = 100, Mmax = 256, γ0 = 1, Pr = 50 W and Pt = 0.1 W.

∗ , ρ∗ , M ∗) B. OPTIMAL SYSTEM PARAMETERS (Ptx,R UE

In Fig. 6, we show the procedure for finding the global optimal value of (Ptx,R , K , M ). The optimization process of Ptx,R with a given pair of (K , M ) is shown in sub-figure (a). Three (K , M ) scenarios are taken as examples, i.e., (79, 140), (16, 32) and (224, 256). We note that the global maximum for (79, 140) is obtained at the critical point P∗tx,R|KM = 30.1 W and the corresponding value of EE is ˜ E(30.1, 79, 140) = 20.9 Mbits/Joule. The global maximum is at the point P∗tx,R|KM = 8.8 W for (16, 32), which is also the critical point. The corresponding maximum EE is ˜ E(8.8, 16, 32) = 12.1 Mbits/Joule. Regarding the curves of (79, 140) and (16, 32), before reaching the maximum points the system is still energy-limited, after reaching the maximum the system is energy inefficient. When it comes to (K , M ) = (224, 256), the global maximum 11.8 Mbits/Joule is obtained at the right endpoint of the close interval of Ptx,R , i.e., P∗tx,R|KM = 50 W. This observation indicates that the degree-of-freedoms are almost occupied with (K , M ) = (224, 256), so the system is working in the energy-limited region for the considered Ptx,R|KM range; EE monotonically increases and reaches the maximum at the boundary. Subfigure (b) is the contour figure of EE on (K , M ); it is a vertical projection of the 3-dimensional surface corresponding to ∗ ˜ the function E Ptx,R|KM (K , M ) , K , M as defined in (54), and the color of the contour indicates the value of EE, the darker (purple) the higher (EE). It can be observed that the among all the feasible sets of (K , M ), there exists one point ∗ ∗ ˜ ∗ maximizing the value of E(P tx,R|KM , K , M ), i.e., (K , M ) = (79, 140). Considering sub-figures (a) and (b), we find the 2781


As expected, it is shown in sub-figure (d) that good channel estimation leads to high system EE performance. Observing sub-figures (a), (b) and (c), we note that under poor channel estimation scenarios, more relay antennas, more active users and lower relay transmit power are energy-efficient. Moreover, it is also noted that with the increasing of ρr , the system EE growth slows down and converges to the value with perfect CSI estimation. This implies though the system with high-quality channel estimation (i.e., ρr = 100) is able to achieve better EE performance than that of the system with the imperfect channel estimation (i.e., ρr = 0.1), the pursuit of extremely high accuracy of channel estimation is unnecessary (e.g., ρr ≥ 10 is enough in our simulation). ∗ and M ∗ for different R ∗ , ρUE FIGURE 7. The optimal EE, Ptx,R max . ρr = 100, γ0 = 1, Mmax = 256, Pr = 50 W and Pt = 0.1 W.

global optimal solution (P∗tx,R , K ∗ , M ∗ ) = (30.1, 79, 140), ∗ , M ∗ ) = (30.1, 410.4, 140). i.e., (P∗tx,R , ρUE In Fig. 7, we investigate how the optimal parameters ∗ , M ∗ ) change with the increases of the coverage (P∗tx,R , ρUE area Rmax in sub-figures (a), (b) and (c), respectively. The corresponding optimal EE is shown in sub-figure (d). We can see that with the increasing of Rmax , P∗tx,R and M ∗ increase, ∗ , decreases. meanwhile the density of active UE pairs, ρUE In other words, concerning the system EE, we should increase the relay transmit power, use more relay antennas, and decrease the user density for a larger coverage area of the relay. Additionally, the optimum EE declines with the growth of the coverage area. In fact, it is easy to obtain E ∗ → 0 as Rmax → +∞.

∗ ∗ and M ∗ vs. QoS constraint γ . FIGURE 9. The optimal EE, Ptx,R , ρUE 0 Rmax = 250 m , ρr = 100, Mmax = 256, Pr = 50 W and Pt = 0.1 W.

In Fig. 9, we evaluate the optimal parameters ∗ , M ∗ ) and the corresponding EE for various QoS (P∗tx,R , ρUE constraints γ0 . It is shown that for γ0 ≤ 6, the value ∗ , M ∗ ) stays unchanged, but for γ of (P∗tx,R , ρUE 0 > 6 ∗ ∗ (Ptx,R , ρUE , M ∗ ) changes with the increase of γ0 . To be more specific, for our simulation, the global optimum without QoS ∗ , M ∗ ) = (30.1, 410.4, 140), and the constraints is (P∗tx,R , ρUE corresponding QoS is γmin = 6.42. Therefore, in the case of γ0 > 6.42, the optimal solution would be found along the edges of the feasible region affected by the QoS constraint. VI. CONCLUSIONS

∗ ∗ and M ∗ vs. channel estimation FIGURE 8. The optimal EE, Ptx,R , ρUE quality ρr . Rmax = 250 m , γ0 = 1, Mmax = 256, Pr = 50W and Pt = 0.1 W.

Fig. 8 shows the energy-efficient system parameters for various quality of channel estimation ρr . The relay coverage radius, the transmit power at each source user and the QoS constraint are assumed to be Rmax = 250 m, Pr = 50 W, Pt = 0.1 W, Mmax = 256 and γ0 = 1, respectively. 2782

We analyzed the system energy efficiency (EE) of the massive MIMO AF relay systems by employing random matrix theory in large-system regime. These analytical results provide excellent approximations to enable LSF-based lowcomplexity resource allocation. In particular, the transmission power allocation at the relay, the max-min user selection and the number of relay antennas were optimized to achieve high EE without complicated matrix calculations and the CSI of instantaneous SSF. We further pursued the EE optimization by averaging out the influence of LSF. Such optimization framework only requires the distribution of LSF within a target coverage area; therefore, it enjoys low implementation VOLUME 4, 2016


complexity and system overhead. The framework also sheds light on the EE-optimum system configuration in terms of the numbers of relay antenna and active UE-pair as well as the relay transmission power. It is noted that the massive MIMO relay can be considered as a base station in the cellular work, and the pair-wise information exchange can be a possible transmission mode of the base station. The extension of the this work to the more general multi-hop/multi-way relay networks is an interesting future direction.

easily get h i −1 Ez M−2 W 1 1

where, Ez [·] indicates the expectation regarding the randomness of z, M13 = M1 P−1 M3 . Applying Jensen’s inequality E[log2 (1 + 1/x)] ≥ log2 (1 + ¯ of the ergodic sum rate 1/E[x]), a lower bound R ¯ E[R(P, S, M )] (i.e., Ez [R(P, S, M )] ≥ R(P, S, M )) is given by K 2K X B ¯ 1− log2 (1 + γ¯k ). R(P, S, M ) = 2 T

(60)

K X

−1 m−2 3,i Ez [W3 ]i,i

(57)

PK =

(58)

in which

SW1 /K (z) = SW3 /K (z) =

T1,k T2,k

h i 2 H †H † = Ez m−2 M Z z z Z , 2 1,k 2 2 1,k 1,k h i H = Ez m24,k z4,k xR xH R z4,k .

k,k

(61)

1 . z+β

(62)

SW1 M13 W3 M13 /K 2 (z) = SW1 /K (z)SM13 W3 M13 /K (z) =

1 S 2 (z), (z + β)2 M13

(63)

where SM2 (z) is the S-transform of the l.s.d. of M213 . 13 Furthermore, for the matrix M213 , using the definition of ϒ-transform [22], we have ϒM2 (s) = 13

K 2 2 1 X sm1,k m3,k . K 1 − sm21,k m23,k k=1

(64)

Moreover, the relationship of S-transform and Stieltjes transform is given by [29]

13

,

.

Referring to the free Hermitian random matrices results in [29], we know the matrices W1 /K and M13 W3 M13 /K are asymptotically free almost everywhere, and the matrices M213 and W2 /K are also asymptotically free almost everywhere. Using the multiplicative free convolution, the S-transform of the l.s.d. of W1 M13 W3 M13 /K 2 is given by

SM2 (z) =

h i −1 = Ez M−2 W 1 1

M −K

(l.s.d.) functions of W1 /K , W3 /K follow the Mar˘cenkoPasture Law, we have the S-transforms of the l.s.d. of W1 /K and W3 /K as

γ¯k is an approximation of γk by taking the expectation of the denominator part in (12), i.e., pk Pt , + pk Pt T2,k + T3,k + σu2

−2 i=1 m3,i

Next, we apply the FTP to find a closed-form expression −1 −1 −1 of tr M−1 . As the limiting spectral density 13 W3 M13 W1

k=1

z + 1 −1 ϒ 2 (z), M13 z

ϒM2 (s) = 13

GM2 (−s−1 ) 13

−s

− 1, (65)

in which GM2 (s) denotes the Stieltjes transform of M213 . 13

(59)

In order to obtain γ¯k , we need to calculate the value of Ti,k and Cj , i = 1, 2, 3, j = 1, 2, 3, 4. First, for the −1 inverse-Wishart random matrix W−1 1 ∼ WK (M , I), we can VOLUME 4, 2016

.

i=1

H H Let us define W1 = ZH 1 Z1 , W2 = Z2 Z2 , W3 = Z3 Z3 H and W4 = Z4 Z4 . Obviously, W1 , W2 , W3 and W4 are all K × K dimensional Wishart matrices, i.e., W1 , W2 , W3 , W4 ∼ WK (M , I). Therefore, C1 , C2 , C3 , C4 and C5 in (10) can be rewritten as h i −1 C1 = Ez tr P2 M−2 W , 3 3 h i −1 −1 −1 C2 = Ez tr M−1 , 13 W3 M13 W1 h i −1 −1 −1 −1 −1 H C3 = Ez tr M22 ZH Z W M W M W Z Z , 1 2 2 1 1 13 3 13 1 h i −1 −1 −1 H C4 = Ez tr Z2 M2 PM−1 , 3 W3 M13 W1 Z1 h i −1 −1 −1 H C5 = Ez tr Z1 W−1 , (56) 1 M13 W3 M3 PM2 Z2

T3,k

M −K

i=1

APPENDIX I PROOF OF THEOREM 1

pk σr2 T1,k

k,k

m−2 1,k

−1 Similarly, because of W−1 3 ∼ WK (M , I), we have " K # h i X −2 −1 −2 −1 Ez tr M3 W3 = Ez m3,i [W3 ]i,i

=

γ¯k =

=

Substituting (62) into (63) and substituting (63), (65) into (64), we can get an equation about the Stieltjes transform G(s) of the matrix W1 M13 W3 M13 /K 2 as follows K X (sG(s)+G(s)(sG(s) − 1+β)2 m21,k m23,k )−1 = K .

(66)

k=1 2783


of lk , k = 1, . . . , 2K , as

Therefore, we have h i −1 −1 −1 Ez tr M−1 W M W 13 3 13 1 " # W1 M13 W3 M13 −1 1 tr = Ez K2 K2 = =

1 G(0) K PK −2 −2 i=1 m1,i m3,i (M − K )2

fL (lk ) =

,

(67)

k=1 a.s.

K X

h i −1 −1 −1 m22,k Ez tr M−1 W M W 13 3 13 1

k=1 PK 2 a.s. k=1 m2,k

−−→

(M

−2 −2 i=1 m1,i m3,i . − K )2

(68)

Then, because xR xH R has uniformly bounded spectral norm and is independent of z4,k , applying [42, Corollary 1], we have h i a.s. a.s. H Ez m24,k z4,k xR xH z −→ m24,k Ez [tr(xR xH −→ m24,k Pr . R 4,k − R )] − (69)

k,k

i=1 a.s.

−−→

m−2 1,k

M −K

.

# z2,i (70)

Additionally, since Z1 , Z3 are independent of Z2 , we have h n oi a.s. −1 −1 −1 −1 Ez tr Re M12 ZH M W M W M Z −−→ 0. 1 2 1 1 13 3 3 (71) So far, Ti,k and Cj are all represented in deterministic ¯ can expressions with the help of (60) – (71). Therefore, R be formulated as (26) in Theorem 1. Similarly, the total power can be approximated as (27) in Theorem1. Appendix II PROOF OF THEOREM 2

Applying strong law of large numbers (SLLN) and Jensen’s inequality, we obtain the probability density function (PDF) 2784

B˜ 1 = K Elk [m−2 3,k ] Z Rmax K 1 2α = lk + lkα fL (lk )dlk c Rmin cτr ρr ( 2(α+1) 2(α+1) 1 Rmax − Rmin K = c4R τr ρr c(α + 1) ) α+2 2(Rα+2 max − Rmin ) + α+2

(73)

−2 B˜ 13 = K Elk ,lk+K [m−2 1,k m3,k ] −2 = K Elk [m−2 1,k ]Elk+K [m3,k ] ( 2(α+1) 2(α+1) 1 Rmax − Rmin K = c(α + 1) c2 42R τr ρr )2 α+2 2(Rα+2 max − Rmin ) + α+2

(74)

a.s. Further, B2 in (32) can be approximated as B2 −−→ B˜ 2 , which is given by

PK

2 i=1 m2,i

(72)

where 4R = (R2max − R2min ). a.s. Similarly, B13 in (32) can be approximated as B13 −−→ B˜ 13 , where B˜ 13 is

PK

As Z1 is independent of Z2 , we attain h i 2 H †H † Ez m−2 tr M Z z z Z 2 2 2 1,k 1,k 1,k " K h i X −1 2 H = m−2 E m z W z 2,i 2,i 1,k 1

Rmin ≤ lk ≤ Rmax .

As we assume the LSF CSI of all the UEs are i.i.d., the value of B1 in (32) converges almost surely to its expected a.s. value according to the SLLN, i.e., B1 −−→ B˜ 1 . And B˜ 1 is given by

where G(0) is easily obtained by setting s = 0 in (66). −1 −1 −1 −1 H Similarly, as Z1 W−1 1 M13 W3 M13 W1 Z1 has uniformly bounded spectral norm and is independent of z2,k , we have h i −1 −1 −1 −1 −1 H Ez tr M22 ZH Z W M W M W Z Z 1 2 2 1 1 13 3 13 1 " K # X −1 −1 −1 −1 −1 H 2 H = Ez m2,k z2,k Z1 W1 M13 W3 M13 W1 Z1 z2,k −−→

2lk , − R2min

R2max

B˜ 2 = K Elk [m22,k ] Z Rmax 1 = Kc α + cτ ρ fL (lk )dlk l r r Rmin k 1 α+2 Rα Kc = R2max 2 F1 1, ; ; − max τr ρr 4R α α cτr ρr Rαmin 1 α+2 2 −Rmin 2 F1 1, ; ;− (75) α α cτr ρr ˜ tx,R , K , M ). Next, we derive the approximation R(P First, for the equal relay power allocation, we find ¯¯ ESSF [R(Ptx,R , K , M )] ≥ R(P tx,R , K , M ) as a special case of (57), in which ESSF [·] denotes only the randomness of SSF LSF is considered. Then, we take random LSF CSI on top of the SSF CSI into account, i.e., ESSF,LSF [R(Ptx,R , K , M )] = ELSF ESSF [R(Ptx,R , K , M )] h i ¯¯ ≥E R(P , K, M) (76) LSF

tx,R

VOLUME 4, 2016


Substituting the SLLN results (73) - (75) into B1 , B13 and B2 in (35), we have h i ¯¯ = K 1 − 2K E ELSF R l1 ,...,l2K log2 (1 + γ¯¯k ) 2 T 2K a.s. K −−→ 1− Elk ,lk+K log2 (1 + γ˜k ) , 2 T (77) where γ˜k is defined as

 1

γ˜˜min = Elk+K ,lmax 

ac

α lk+K +cτr ρr

+b

2α lmax c2 τr ρr

+

α lmax c

. +e (82)

Therefore, we can relax the QoS constraints as follows γ˜˜min ≥ γ0 .

− K )2

, (78) 2 ˜ χk + Ptx,R m−2 1,k (M − K )(σr + Pt B2 ) χk = Pt (M − K )B˜ 1 + σr2 B˜ 13 + Pt B˜ 13 B˜ 2 (Ptx,R m24,k +σu2 ). With (76) and (77), we can find a lower bound ˜ tx,R , K , M ) of the ergodic sum rate E[R(Ptx,R , K , M )], R(P i.e., ˜ tx,R , K , M ), E[R(Ptx,R , K , M )] ≥ R(P where X K ˜ tx,R , K , M ) = B 1 − 2K Elk ,lk+K log2 (1 + γ˜k ) R(P 2 T k=1 BK 2K = 1− I Ptx,R , K , M , 2 T RR Rmax where I = Rmin log(1+ γ˜k )fL (lk )fL (lk+K )dlk dlk+K . Considering the QoS constraints γk ≥ γ0 , k = 1, . . . , K , they are equivalent to γmin ≥ γ0 , where γmin = min γk . k=1,...,K

With respect to the SLLN-based result γ˜k , we have ! 1 γ˜min = min k=1,...,K m2 a + m−2 b + e 4,k 1,k 1 = max m24,k a + m−2 1,k b + e k=1,...,K

1

≤ m24,k a +

, max m−2 b + e 1,k

(79)

k=1,...,K

where B˜ 1 σr2 B˜ 13 B˜ 13 B˜ 2 + + , 2 M −K Pt (M − K ) (M − K )2 B˜ 2 σr2 + , b= (M − K )Pt M −K σu B˜ 1 σu2 B˜ 13 B˜ 2 σu2 σr2 B˜ 13 e= + + . Ptx,R (M − K ) Ptx,R Pt (M − K )2 Ptx,R (M − K )2 (80) a=

Taking the expectation of the above expression with respect to lk , we have E γ˜min ≤ γ˜˜min , (81) VOLUME 4, 2016



The proof of Theorem 2 is finished. Ptx,R Pt (M

γ˜k =

where

ACKNOWLEDGMENT

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HUI GAO (S’10–M’13) received the B.Eng. degree in information engineering and the Ph.D. degree in signal and information processing from the Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2007 and 2012, respectively. From 2009 to 2012, he served as a Research Assistant with the Wireless and Mobile Communications Technology R&D Center, Tsinghua University, Beijing. In 2012, he visited the Singapore University of Technology and Design (SUTD), Singapore, as a Research Assistant. From 2012 to 2014, he was a Post-Doctoral Researcher with SUTD. He is currently with the School of Information and Communication Engineering, BUPT, as an Assistant Professor. His research interests include massive MIMO systems, cooperative communications, and ultrawideband wireless communications.

TIEJUN LV (M’08–SM’12) received the M.S. and Ph.D. degrees in electronics engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1997 and 2000, respectively. He was a Post-Doctoral Fellow with Tsinghua University, Beijing, China, from 2001 to 2003. In 2005, he became a Full Professor with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications. From 2008 to 2009, he was a Visiting Professor with the Department of Electrical Engineering, Stanford University, Stanford, CA, USA. He has authored over 200 published technical papers on the physical layer of wireless mobile communications. His current research interests include signal processing, communications theory, and networking. Dr. Lv is also a Senior Member of the Chinese Electronics Association. He was a recipient of the Program for New Century Excellent Talents in University Award from the Ministry of Education, China, in 2006.

XIN SU (M’02) received the M.S. and Ph.D. degrees in electronics engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1996 and 1999, respectively. He is currently a Full Professor with the Research Institute of Information Technology, Tsinghua University, Beijing, China. He is also the Chairman of the IMT-2020 (5G) Wireless Technology Work Group in the Ministry of Industry and Information Technology of China and the Vice Chairman of the Innovative Wireless Technology Work Group of the China Communications Standards Association. He has authored over 100 papers in the core journals and important conferences and is the holder of more than 30 patents. His research interests include broadband wireless access, wireless and mobile network architecture, self-organizing network, software-defined radio, and cooperative communications.

VOLUME 4, 2016


HONG YANG received the Ph.D. degree in applied mathematics from Princeton University, Princeton, NJ. He is a member of the Technical Staff with the Mathematics of Networks and Communications Research Department, Nokia Bell Laboratories, Murray Hill, NJ, where he conducts research in wireless communications networks. He has over 15 years of industrial research and development experience, and was with the Radio Frequency Technology Systems Engineering Department and the Wireless Design Center at Alcatel-Lucent (USA), and for a start-up networking company. He has published research papers in wireless communications, applied mathematics, and financial economics; and holds several U.S. patents.

VOLUME 4, 2016

JOHN M. CIOFFI (F’96) received the B.S. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Urbana, IL, USA, in 1978, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 1984. He was with Bell Laboratories in 1978–1984 and IBM Research in 1984–1986. Since 1986, he has been with Stanford University, where he was a Professor in electrical engineering and is currently an Emeritus Professor. He founded Amati Communications Corporation in 1991 (purchased by TI in 1997) and was an Officer/Director from 1991 to 1997. He is also an Adjunct Professor of Computing/Information Technology with King Abdulaziz University, Jeddah, Saudi Arabia. He is also with the Board of Directors of ASSIA (Chairman and CEO), Alto Beam, and the Marconi Foundation. He has authored more than 600 papers and holds more than 100 patents, of which many are heavily licensed, including key necessary patents for the international standards in ADSL, VDSL, DSM, and WiMAX. His specific interests are in the area of high-performance digital transmission. He was a recipient of the IEEE Alexander Graham Bell and Millennium Medals (2010 and 2000, respectively), Economist Magazines 2010 Innovations Award, the IEEE Kobayashi and Kirchmayer Awards (2001 and 2014, respectively), the IEE J.J. Tomson Medal (2000), the IEEE Communications Magazine Best Paper Awards in 1991 and 2007, and numerous conference best paper awards. He was a member of the Internet Hall of Fame in 2014, became a member of the U.S. National and U.K. Royal Academies of Engineering in 2001 and 2009, respectively, and was named an International Marconi Fellow in 2014.

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