Game-Theoretic Precoding for SWIPT in the DF-based MIMO Relay ...

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Game-Theoretic Precoding for SWIPT in the DF-based MIMO Relay Networks Bing Fang, Wei Zhong, Shi Jin, Zuping Qian, and Wei Shao

Abstract—In this paper, we study the distributed precoding problem for simultaneous wireless information and power transfer (SWIPT) in a decode-and-forward (DF)-based multiple-input multiple-output (MIMO) relay network. The system model considered here consists of a source, a relay, an information decoding (ID) receiver and an energy harvesting (EH) receiver, all mounted with multiple antennas. Since full CSI is usually unavailable, a practical scenario, where only local CSI is required, is considered in this work. Such a practical scenario naturally constitutes a noncooperative game, where the source and the relay can be regarded as two rational game players. With properly designed utilities, it can be further shown that the existence and uniqueness of the pure-strategy Nash equilibrium (NE) of the proposed game can be both guaranteed under some mild conditions. Therefore, a distributed iterative precoding algorithm can be developed based on the best-response dynamic to obtain the unique NE solution for the proposed game. Moreover, a proximal-pointbased regularization approach is also pursued to ensure the convergence of the proposed algorithm without requiring any special restrictions on the channel ranks. Numerical simulations are also provided to demonstrate the proposed algorithm. Results show that our algorithm can converge fast to a satisfactory solution with guaranteed convergence. Keywords—MIMO relay, precoder design, energy harvesting, convex optimization, game theory.

I. I NTRODUCTION LECTROMAGNETIC waves can be used not only for wireless information transmission (WIT), but also for wireless power transfer (WPT). WPT is not a new concept but becomes a hot-topic recently with the potential to power ubiquitous wireless terminals in the current communication systems [1]. Thus, the idea of simultaneous wireless information and power transfer (SWIPT) become more and more attractive due to the following two contradictory reasons: (a) wireless communication is to be everywhere and cannot be deprived

E

Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. The work of W. Zhong was supported by the Natural Science Foundation of China under Grant 61201218, by the Jiangsu Province Natural Science Foundation under Grant BK2012056, and by the Project funded by China Postdoctoral Science Foundation under Grants 2013M532207 and 2014T70966. The work of S. Jin was supported by the National Natural Science Foundation of China under Grants 61531011 and 61450110445. The work of W. Shao was supported by the Natural Science Foundation of China under Grant 61201241. B. Fang, W. Zhong, Z. Qian, and W. Shao are with the College of Communications Engineering, PLA University of Science and Technology, Nanjing, 210007, China. (e-mail: bingfang [email protected], [email protected], [email protected], [email protected]). S. Jin is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing, China, 210096. (e-mail: [email protected]).

from life; (b) all user terminals are poorly battery-powered and not truly “wireless” [2]. With great progress of rectifying antennas (rectennas) for efficient RF-to-DC conversion [3], SWIPT has become a new research field that attracts increasing attention [4]- [11]. More recently, the researches concerned on precoding for SWIPT in the MIMO systems have invoked considerable interests [7]- [11]. For example, the authors in [7] focused on SWIPT in a MIMO broadcasting system, and studied the optimal precoder design problem for rateenergy tradeoff via convex optimization. In addition, they also investigated two possible methods for practical receiver design intended for simultaneously information decoding and energy harvesting. Authors in [8] concerned with SWIPT in a MIMOOFDMA system and proposed a SWIPT-enabled architecture to study the power control problem in the broadband wireless communication systems. It is known that advanced MIMO relay networks can further improve spectrum utilization and enhance link reliability at the same time [12] [13]. Moreover, they are also efficient ways to provide high quality service with low power cost to mobile users even at the cell edges [12]- [16]. More important, the concept of SWIPT is more applicable for short-distance applications since WPT will suffer from high attenuation due to long-distance path loss. Therefore, it is reasonable to predict that MIMO relay networks will become beneficial and indispensable supplements for existing cellular network architecture to provide fair service to all mobile users in the upcoming future. In addition, such network architectures, i.e., base station plus relay, will be more applicable for SWIPT, especially when MIMO technologies are employed. As a pioneer work, authors in [9] have studied the optimal performance boundaries for SWIPT in an amplify-and-forward (AF)-based MIMO relay network. However, the mentioned work considers only one data stream. Such a simplified approach has significantly limited the capacity of MIMO relay networks intended for data transmission. Thus, the study of SWIPT in MIMO relay networks is still lacking. In this paper, we focus on the distributed precoding problem for SWIPT in a decode-and-forward (DF)-based MIMO relay network, where a direct link from the source to the information decoding (ID) receiver is considered. Specifically, the system model considered here consists of a source, a relay, an energy harvesting (EH) receiver and an ID receiver, all mounted with multiple antennas. This scenario may possibly arise in future MIMO cellular networks, especially when the mobile users are served at the edges of cellular networks. With the help of the infrastructure-based relays, these base stations can serve the cell-edge mobile users fairly well. In such a scenario, it is also assumed that the mobile devices in future can either work

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in a communication mode with enough battery, or in an energy harvesting mode when the battery level is low. In practice, the source and the relay usually only have local CSI, and cannot cooperate efficiently with each other to achieve a common goal. In such case, game theory, which has been widely used to study the distributed decision strategies in wireless networks [18]- [20], has become a suitable tool to study the distributed precoding problem for SWIPT in the DF-based MIMO relay networks. Such a practical scenario naturally constitutes a noncooperative game, where the source and the relay are regarded as two rational game players. With properly designed utilities, it can be further shown that the existence and uniqueness of the pure-strategy Nash equilibrium (NE) of the proposed game can be both guaranteed under some mild conditions. Thus, a distributed precoding algorithm, based on the best-response dynamic, can be developed to obtain the unique NE solution. Moreover, a proximal-point-based regularization approach is also pursued in this work to ensure the convergence of the proposed algorithm without requiring any special restrictions on the channel ranks. Numerical simulations are further provided to demonstrate the proposed algorithm. Results show that our algorithm can converge fast to a satisfactory solution with guaranteed convergence. All in all, our approach can be of some significance to the practical system design intended for SWIPT. Note that, the present paper is a extended version of the conference paper [11], but some substantial contributions are also made in this work. In summary, the main contributions of the present paper can be listed as follows. • With only local CSI required, a noncooperative game for the distributed precoding problem is established. We further prove that the existence and uniqueness of the pure-strategy NE of the proposed game can be both guaranteed with the properly designed utilities under some mild conditions. • Based on the best-response dynamic, a distributed iterative precoding algorithm is developed to obtain the unique NE solution for the proposed SWIPT system, the convergence behavior of which is also theoretically analyzed. • A proximal-point-based regularization approach is also pursued to ensure the convergence of the proposed algorithm without requiring any special restrictions on the channel ranks. Numerical simulations are also provided to demonstrate our algorithm. The rest of this paper is organized as follows: in Section II, we introduce the system model; in Section III, a noncooperative precoding game is established; a distributed precoding algorithm is developed in Section IV; in Section V, the proposed algorithm are verified via numerical simulations; the paper is concluded in Section VI. Notations: Bold uppercase letters denote matrices and bold lowercase letters denote vectors; Cm×n defines the space of all m×n complex matrices; Sn , Sn+ and Sn++ stand for the sets of n × n complex Hermitian, positive semi-definite and positive definite matrices, respectively; In is an identity matrix of size n×n; A ≽ 0 and B ≽ C mean that the matrices A and B−C are positive semi-definite. Transpose and Hermitian transpose

EH H re

H se

Source

Relay

H sr H sd

H rd

ID

Fig. 1. System model of simultaneous wireless information and power transfer (SWIPT) in a DF-based MIMO relay network.

of matrix A are represented by AT and AH , respectively; |A|, ||A||F and Tr(A) means the determinant, the Frobenius norm, and the trace of matrix A, respectively; and log(·) denotes the base-2 logarithm. II. S YSTEM M ODEL In this paper, we focus on the distributed precoding problem for SWIPT in a DF-based MIMO relay network. As shown in Fig. 1, the system model considered here consists of a source, a relay, an EH receiver and an ID receiver, all mounted with multiple antennas. The number of antennas employed at the source, the relay, the EH receiver and ID receiver are assumed to be Ns , Nr , Ne and Nd , where s, r, e and d stand for the source, the relay, the EH receiver, and ID receiver, respectively. Moreover, Nr ≥ Ns , Nd ≥ Ns are assumed throughout the paper to ensure full ranks of all data transmission links. In addition, the source and the relay are assumed to be infrastructure-based with constant power supply. As the previous works done in [10] [11], the relay employed in this work also works in a half-duplex mode. So that, the relay cannot transmit and receive signals at the same time [16], and each round of transmission from the source to the receivers must take two time slots. In the first time slot, the relay keeps silent, the source broadcasts signals to the relay, the ID receiver, and the EH receiver; while in the second time slot, the source keeps silent, and the relay decodes the received signals and retransmits them to the ID receiver and the EH receiver. Thus, the ID receiver can obtain the desired information through decoding the combined signals received over the aforementioned two time slots; and the EH receiver can harvest the wireless energy either from the source or from the relay or both in the two time slots. In the first time slot, the source multiplies the complex information vector x by a precoding matrix Ws ∈ CNs ×Ns before transmission. Thus, the vector-valued signals received at the relay, the ID receiver, and EH receiver can be given by yr,1 = Hsr Ws x + nr,1 , yd,1 = Hsd Ws x + nd,1 , ye,1 = Hse Ws x + ne,1 ,

(1)

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respectively, where yr,1 , yd,1 and ye,1 are the received signals at r, d, and e, respectively; nr,1 , nd,1 and ne,1 are the received additive noise at r, d, and e, respectively; Hsj ∈ CNj ×Ns denotes the complex channel matrix between s and j ∈ {r, d, e}. In the second time slot, the relay decodes the signal and retransmits it after multiplying it by a precoding matrix Wr ∈ CNr ×Ns . Here, we assume that the information decoded at the relay is free of error as in [15]. Thus, the vector-valued signals received by the ID receiver and the EH receiver in the second time slot can be written as yd,2 = Hrd Wr x + nd,2 , ye,2 = Hre Wr x + ne,2 ,

(2)

where yd,2 , ye,2 are the received signals at d and e, respectively; nd,2 , ne,2 are the received additive noise at d and e, respectively; Hrj ∈ CNj ×Nr denotes the complex channel matrix between r and j ∈ {d, e}. It is assumed that the covariance matrix of the information vector x is E{xxH } = INs . Thus, the corresponding transmit covariance matrices for both the source and relay are given by Qs = Ws WsH and Qr = Wr WrH . Indeed, once the Nr s covariance matrix Qs ∈ SN + and Qr ∈ S+ are computed, the precoding matrices Ws , Wr can be obtained immediately by Cholesky factorization [17]. Therefore, we use Qs and Qr as variables and regard them as the precoding matrices for simplicity. It is further assumed that nr,1 ∼ CN (0, σ 2 INr ), nd,1 ∼ CN (0, σ 2 INd ) and nd,2 ∼ CN (0, σ 2 INd ) throughout this work. Therefore, the mutual information I(yr,1 , x) between yr,1 and x can be written as I(yr,1 , x) = log |INr + ρHsr Qs HH sr |,

(3)

and the mutual information I(yd,2 , x) between yd,2 and x can be written as I(yd,2 , x) = log |INd +

ρHrd Qr HH rd |,

(4)

where ρ = σ −2 is defined as the averaged signal-to-noise ratio (SNR). Moreover, the signals received by the ID receiver in the two time slots can be combined as [ ] [ ] [ ] yd,1 Hsd Ws nd,1 = x+ , (5) yd,2 Hrd Wr nd,2 which can be further reduced to yd = Hx + nd ,

(6)

T T T with yd = [yd,1 , yd,2 ] , H = [(Hsd Ws )T , (Hrd Wr )T ]T and T T T nd = [nd,1 , nd,2 ] . Hence, the mutual information I(yd , x) between yd and x can be written as I(yd , x) = log I2Nd + ρHHH , (7) H = log IN + ρHsd Qs HH sd + ρHrd Qr Hrd . d

Then, the achievable data rate for the DF-based MIMO relay network can be given as [15] RDF =

1 min{I(yr,1 , x), I(yd , x)}, 2

(8)

where the factor 21 is due to the half-duplex loss in this twohop MIMO relay network. Moreover, the transmit power constraint of the source can be given by Tr(Qs ) ≤ Ps , (9) where Ps is the maximum transmit power available for the source, and the transmit power constraint of the relay is given as Tr(Qr ) ≤ Pr , (10) where Pr is the maximum transmit power allowed for the relay. According to [7], the power received by the EH receiver in the first time slot can be given by P1 = µTr(Hse Qs HH se ),

(11)

and in the second time slot is given as P2 = µTr(Hre Qr HH re ),

(12)

where 0 < µ ≤ 1 is a constant that accounts for the loss for energy transduction. Without loss of generality, it is assumed that µ = 1 for convenience in the rest of this paper. III. G AME - THEORETIC F ORMULATION In practice, the source and the relay usually have only local CSI and a distributed precoding algorithm is often more desired, which suggests a game-theoretic approach. If we regard the source and the relay as two game players, then, each player can play a noncooperative game, where only local CSI is required, to maximize its own utility. In this section, we will provide such a game-theoretic formulation, which eventually gives rise to a distributed precoding algorithm. A. Game-theoretic formulation For the DF-based MIMO relay network as shown in Fig. 1, CSI of the channels used for data transmission can be easily obtained at both the source and the relay, by using the techniques as channel training, feedback or channel reciprocity exploiting, etc. However, CSI of the channels used for energy transfer can only be known by their own transmitter, i.e., Hse can only be known by the source through channel feedback, and Hre can only be known by the relay. Thus, due to the uncoordinated CSI, the source and the relay, both selfishly select their own precoding matrix to maximize their own utility, are sure to have some conflict interests, and such a scenario naturally constitutes a noncooperative game. In this work, the game players, i.e., the source and the relay, are assumed to be strictly rational, and there is no cooperation between them. Thus, the noncooperative game for SWIPT in the proposed system can be formally formulated as G = [N , {Qi }i∈N , {ui }i∈N ],

(13)

where N = {s, r} is the set of game players, Qi is the set of pure strategies of player i, ∀i ∈ N . For the game player s, the utility function us is defined as us = I(yd , x) + ηTr(HH se Hse Qs ),

(14)

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where the first term I(yd , x) is the mutual information between the source and the ID receiver, the second term ηTr(HH se Hse Qs ) is the energy transferred from the source to the EH receiver, and η is a nonnegative scalar with a unit stated as bps/Hz/watt. For the game player r, the utility function ur is defined as ur = min[I(yr,1 , x), I(yd,2 , x)] + ζTr(HH re Hre Qr ),

(15)

where the first term min[I(yr,1 , x), I(yd,2 , x)] is interpreted as the received mutual information of relay can not be suppressed by the maximum delivery mutual information from the relay to the ID receiver, the second term ζTr(HH re Hre Qr ) is the energy transferred from the relay to the EH receiver, and ζ is also a nonnegative scalar with a unit stated as bps/Hz/watt. Note that, η and ζ defined in the above utility functions are energy-sensitive parameters. They can be used to exploit the “near-far” based power allocation scheduling between the two time slots, that is, more energy can be transferred from the “near” transmitter to the EH receiver. The parameters also have a function of unit transfer to keep the unit of the second term to be the same with the first term included in the utility functions. In game G, the source and the relay compete against each other by choosing their most favorable precoding matrix to maximize their own utility. Formally, this concept can be formulated as max ui (Qi , Q−i ) Qi G: ∀i ∈ N . (16) s.t. Qi ∈ Qi Here, Qs is the feasible strategy set of the source, which is defined as } ∆ { Qs = Qs ∈ CNs ×Ns : Qs ≽ 0, Tr (Qs ) ≤ Ps , (17) Qr is the feasible strategy set of the relay, which is defined as } ∆ { Qr = Qr ∈ CNr ×Nr : Qr ≽ 0, Tr (Qr ) ≤ Pr . (18) Obviously, both Qs and Qr are nonempty compact convex subsets of the Euclidean space. The most famous and widely used solution to the gametheoretic problems is the NE solution [18], which can reflect the characteristic of the proposed game. When a NE solution is reached, no players will get an increase in his own utility by unilateral changing his own strategy, given that the strategies of the others remains unchanged. For the proposed game G, the definition of the pure strategy NE is formally defined as bellows. Definition 1: (Pure-strategy NE) The strategy profile Q∗ = (Q∗s , Q∗r ) is a pure strategy NE of game G, if the following conditions are satisfied ui (Q∗i , Q∗−i ) ≥ui (Qi , Q∗−i ), (19) ∀ Qi ∈ Qi ,i ∈ N . Further by defining Q , (Qs , Qr ) and Q , Qs × Qr , the problem of finding a pure-strategy NE of game G consists in finding a point Q∗ ∈ Q, such that Tr{(Q − Q∗ )F(Q∗ )} ≤ 0, ∀Q ∈ Q,

∂ui r ) (intended to be a column where F is defined as F = ( ∂Q i i=s vector). Therefore, the NE problem of the proposed game can also be viewed a partitioned variational inequality (VI) problem [21].

B. Analysis of the pure-strategy NE In this subsection, such important properties as the existence and uniqueness the pure-strategy NE of the proposed game will be theoretically analysed. 1) Existence: Given that the utility functions being defined as us and ur , the existence of the pure-strategy NE of game G is investigated in the following theorem. Lemma 1: Game G always possesses at least one purestrategy NE. Proof: Obviously, the feasible strategy sets Qs and Qr are both convex cones, and us is concave over Qs and ur is concave over Qr . Then, according to Theorem 11.2 in [22], it can be concluded that the existence of the pure-strategy NE of game G is always ensured. According to Lemma 1, it can be concluded that the NE solution to game G must exist. That is to say, the source and the relay can always reach a stable operating point if they are noncooperative and selfishly compete against each other. However, it is still unclear that the number of the pure-strategy NE(s) that game G can admit. This issue, the uniqueness of the pure-strategy NE of game G, is also very important and will be studied in the following theorem. 2) Uniqueness: The uniqueness of the pure-strategy NE of game G is given in the following theorem, which is based on the matrix generalization of Rosen’s condition of diagonally strict condition (DSC) [23] (i.e., Definition 2). Definition 2: (Generalized DSC Condition [24]) The matrix generalization of Rosen’s DSC condition for game G can be formally stated as { } Tr (Q0 − Q1 )[F(Q1 ) − F(Q0 )] > 0, (21) ∀ Q1 , Q0 ∈ Q, and Q1 ̸= Q0 , where the superscript 0, 1 index two different points in Q. Theorem 1: The generalized DSC condition always holds for game G and the pure-strategy NE of game G is unique. Proof: For simplicity, we redefine As = σ −2 Hsd Qs HH sd , Ar = σ −2 Hrd Qr HH rd , and it is obvious that As ≻ 0 and Ar ≻ 0 for any nonsingular precoding matrices Qs and Qr . From the source’s utility function us , we have ∂us −1 = (INs + As + Ar ) + A0 , ∂As

(22)

where A0 ≻ 0 is a constant Hermitian matrix comes from the differential of the second term of (14). From the relay’s utility function ur , we have ∂ur −1 = (INr + Ar ) + A1 , ∂Ar

(23)

where A1 ≻ 0 is also a constant Hermitian matrix comes from the differential of the second term of (15).

(20)

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With two different points (A0s , A0r ) and (A1s , A1r ), we can further define P = A0s − A1s , (24) ∂us ∂us (A1 , A1 ) − (A0 , A0 ) ∂As s r ∂As s r = (I + A1s + A1r )−1 − (I + A0s + A0r )−1 ,

R=

(25)

(b)

The complementarity condition: ∀i ∈ N , there must exei , λ bi ≥ 0, such that, for one NE solution (Q e s, Q e r ), ist λ we have ei [Tr(Q e i ) − Pi ] = 0, (31) λ b s, Q b s ), we have and for the other NE solution (Q bi [Tr(Q b i ) − Pi ] = 0. λ

from the formula (22). Likewise, we can also define M=

A0r

−

A1r ,

∂ur ∂ur N= (A1r ) − (A0 ) ∂Ar ∂Ar r −1 −1 = (I + A1r ) − (I + A0r ) .

(26) (27)

Then, according to [25], it can be concluded that the matrix trace inequality defined in (28) is always true under the condition that A0r ̸= A1r and A0s ̸= A1s . Note that, there is still another particular case to mention that the right part of equation (23) is possible to equal zero due to the minimum operation defined in (15). But this case is trivial and can be omitted here, since the inequality (27) already implies such a circumstance. Therefore, it can be concluded that the DSC condition defined in (21) always holds for game G. Then, based on the generalized DSC condition, the uniqueness of the pure-strategy NE of game G can be proved. Suppose that there exist at least two different NE solutions, e Q. b Then, the generalized DSC condition for game G i.e., Q, e and Q1 = must be met for the particular choice that Q0 = Q b Q. On the other hand, by the definition of the pure-strategy e s, Q e r ) and NE defined in (19), the precoding strategies (Q b b (Qs , Qs ) are both solutions of the NE problem defined in (16). This is to say, the three optimality conditions, also known as the Kuhn-Tucker condition [26], must be satisfied by both e s, Q e r ) and (Q b s, Q b s ), which are listed as follows: (Q (a) The zero gradient condition: ∀i ∈ N , for one NE e s, Q e r ), we have solution (Q ∂ui e ei IN , (Q) = λ (29) i ∂Qi b s, Q b s ), we have and for the other NE solution (Q ∂ui b bi IN . (Q) = λ i ∂Qi

e i, Q bi (c) The centrality condition: ∀i ∈ N , the strategies Q must be in the feasible strategy set, that is, for one NE e s, Q e r ), we have soluiton (Q e i ) ≤ Pi , Q e i ≽ 0, Tr(Q

from the formula (23). Then, according to the formulae defined in (24), (25), (26) and (27), the generalized DSC condition for game G can be reduced to tr(MN + PR) > 0. (28)

(30)

(32)

(33)

b s, Q b s ), we have and for the other NE solution (Q b i ) ≤ Pi , Q b i ≽ 0. Tr(Q

(34)

Then, the generalized DSC condition under the assumption that there exist at least two different NE solutions, can be deduced as e − Q)[F( b b − F(Q)]} e Tr{(Q Q) e Q) b − QF( b Q) b + QF( b Q) e − QF( e Q)} e = Tr{QF( ∑ (a) bi Q bi Q ei Q ei Q ei − λ bi + λ bi − λ e i} = Tr{λ ∑ (b) ∑ ei {Tr(Q bi {Tr(Q b i ) − Pi } e i ) − Pi } + λ λ =

(35)

(c)

≤ 0.

However, the inequality deduced in (35) contradicts the generalized DSC condition defined in (21), which means that the assumption that there exist at least two different NE solutions for game G cannot be true. Therefore, it can be concluded that the pure-strategy NE of game G is unique. Based on Lemma 1 and Theorem 1, it is known that the purestrategy NE of game G always exists and is unique. Thus, we can predict that the performance of the NE solution of the proposed game. Moreover, we also can get a better trade off between the energy transfer and the information transmission by properly changing the energy-sensitive parameters, i.e. η and ζ, defined in the utility functions. IV. D ISTRIBUTED P RECODING A LGORITHM In this section, a distributed iterative precoding algorithm is developed based on the best-response dynamic. Moreover, a proximal-point-based regularization approach is also pursued to ensure the convergence of the proposed algorithm.

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Since the NE solution Q∗ is unique, then from (48), we have [ ] Tr (Qv+1 − Qv )F(Qv ) > 0, (42)

Algorithm 1: Distributed Precoding Algorithm (DPA). 1: 2: 3: 4:

initially set η, ζ, Hsr , Hse , Hsd , Hre , and Hrd . initialize δ, v = 1, Q0s and Q0r . compute u0s and u0r with (Q0s , Q0r ). compute Q1s = arg max us for fixed Q0r with CVX solver [27].

5:

compute Q1r = arg max ur for fixed Q0s with CVX solver.

6: 7: 8: 9:

compute u1s and u1r with (Q1s , Q1r ). v−1 v−1 | > δ) while (|uv | > δ or |uv r − ur s − us update v = v + 1. v−1 compute Qv . s = arg max us for fixed Qr

10:

v−1 compute Qv . r = arg max ur for fixed Qs

11: 12: 13:

Qs

Qr

Qs

Qr

v v v compute uv s and ur with (Qs , Qr ). end v v v return uv s , ur , Qs , and Qr .

A. Algorithm design and convergence analysis According to the theoretical analysis in section III, a distributed precoding algorithm can be developed to obtain the unique NE solution of game G, which is formally presented as Algorithm 1. In Algorithm 1, the source and the relay iteratively maximizes their own utility until the convergence condition is met. Moreover, the convergence behavior of Algorithm 1 can be studied in the following theorem. Theorem 2: Suppose that the utility functions us and ur are strict concave over their own feasible sets, then the distributed precoding algorithm, presented as Algorithm 1, will always converge to the unique NE solution of game G with a feasible initial point. Proof: Suppose that Q∗ = (Q∗s , Q∗r ) is the unique NE solution of game G. Then, according to the generalized NE definition, which is defined in (20), we have Tr[(Q − Q∗ )F(Q∗ )] ≤ 0, ∀Q ∈ Q.

(36)

From the generalized DSC condition, which is defined in (21), we have Tr {(Q − Q∗ )[F(Q∗ ) − F(Q)]} > 0, (37) ∀Q ∈ Q, Q ̸= Q∗ . Then, we can combine the above two inequalities as Tr[(Q∗ − Q)F(Q)] > 0, ∀Q ∈ Q, Q ̸= Q∗ .

(38)

The inequality defined in (38) indicates that there always exists a impetus to move from any point Q to the unique NE solution Q∗ , if Q ̸= Q∗ . On the other hand, from the best-response dynamic employed by Algorithm 1, we have [ ] v+1 v ∂us v Tr (Qs − Qs ) (Q ) ≥ 0, (39) ∂Qs for the source. And for the relay, we similarly have [ ] v+1 v ∂ur v Tr (Qr − Qr ) (Q ) ≥ 0. (40) ∂Qr Likewise, the above two inequalities can also be combined as the following inequality [ ] Tr (Qv+1 − Qv )F(Qv ) ≥ 0. (41)

if Qv ̸= Q∗ . The inequality in (42) indicates that one can always move closer to the unique NE solution Q∗ by employing the best-response dynamic and this process is strictly non-decreasing. Then, it can be concluded that from any feasible initial point Q ∈ Q, Algorithm 1 will always move towards the unique NE solution Q∗ closer by maximizing each player’s utility. Because this process is strictly non-decreasing, and the feasible strategy sets are finite, it can be concluded that Algorithm 1 must reach the unique NE solution Q∗ in finite number of iterations. According to Theorem 2, it can be concluded that the convergence of Algorithm 1 is always guaranteed under the condition that the utility functions us and ur are strict concave over their own feasible sets. However, such a condition is not always satisfied, especially when the MIMO channels intended for information transmission are strongly spatially correlated. Therefore, a proximal-point-based regularization approach [28] [29] is pursued next to ensure the convergence of the proposed algorithm without requiring any special restrictions on the channel ranks. B. Proximal-point-based regularization The basic idea of a proximal-point-based regularization [30] consists in penalizing the utility functions us and ur of game G using a quadratic regularization term. So that, we can redefine the utility function us as u es = us − τ (||Qs − Qv−1 ||2F ), s

(43)

and redefine the utility function ur as u er = ur − τ (||Qr − Qv−1 ||2F ), r

(44)

where τ > 0 is a small value to force (Qs , Qr ) stay “close” to (Qv−1 , Qv−1 ) obtained in the previous iteration. With the s r utility functions redefined as u es and u er , the proposed game will be reformulated as max u ei (Qi , Q−i ) Qi ∀i ∈ N . (45) GP : s.t. Qi ∈ Qi The iterative precoding algorithm with proximal-point-based regularization can also be given as Algorithm 1, just with u es and u er replacing us and us . However, in order to avoid the ambiguity, the proximal-point-based regularization algorithm will be hereafter referred as Algorithm 1(P), and the convergence behavior of it is analyzed in the following theorem. Theorem 3: Without any special restrictions required on the channel ranks of the proposed MIMO relay system, Algorithm 1(P) will always converge to the unique NE solution of game G given that the initial point is feasible. Proof: According to the strict convexity of the Frobenius norm, it can be concluded that the strict concavity of the utility

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8.9 8.8 8.7

Utility of the relay (bps/Hz)

Utility of the source (bps/Hz)

8

7.8

7.6

7.4

τ=0.1 τ=0.5 τ=1.0

7.2

7

8.6 8.5 8.4 8.3 8.2

τ=0.1 τ=0.5 τ=1.0

8.1 8

6.8

1

2

3

4

5

6

7

8

9

7.9

10

1

2

3

4

Iterations

(a) utility of the source Fig. 2.

5

6

7

8

9

10

Iterations

(b) utility of the relay

Convergence behavior of Algorithm 1(P) under different τ (with η = 1 and ζ = 1).

8.2

8.9 8.8 8.7

Utility of the relay (bps/Hz)

Utility of the source (bps/Hz)

8

7.8

7.6

7.4

point 1 point 2 point 3

7.2

7

1

2

3

4

5

6

7

8

9

8.6 8.5 8.4 8.3 8.2

point 1 point 2 point 3

8.1 8 7.9

10

1

2

3

4

Iterations

(a) utility of the source Fig. 3.

5

6

7

8

9

10

Iterations

(b) utility of the relay

Convergence behavior of Algorithm 1(P) under different initial points (with τ = 0.5, η = 1, and ζ = 1).

functions u es and u er is always guaranteed. Then, according to the analysis process of Algorithm 1 in Theorem 2, it can be concluded that the convergence of Algorithm 1(P) is always guaranteed with τ > 0. Further suppose (Qvs , Qvr ) is the sequence generated by employing Algorithm 1(P), then according to the analysis process of Algorithm 1 in Theorem 2, it can be concluded that the limit point of sequence (Qvs , Qvr ) must constitutes the unique NE solution of game G P . Moreover, we have the following formula lim ||Qv+1 − Qvs ||2F = 0, s

v→∞

lim ||Qv+1 − Qvr ||2F = 0, r

(46)

v→∞

which further means that the limit point of the sequence (Qvs , Qvr ) simultaneously constitutes the unique NE solution of game G, because regularization terms are equal to zero at the limit point. Therefore, it can be concluded that Algorithm 1(P) always converges to the unique NE solution of game G.

According to Theorem 3, it can be concluded that Algorithm 1(P) always converges to the unique NE solution of game G without requiring any special restrictions on the channel ranks. However, the price to pay for the guaranteed convergence is that a possibly slower convergence rate is common to the proximal-point-based regularization algorithm. V. N UMERICAL S IMULATIONS In this section, the proposed distributed precoding algorithm is evaluated via numerical simulations. During the simulations, the elements of all channel matrices between i ∈ {s, r} and j ∈ {r, d, e} are modeled as independent identically distributed (i.i.d.) zero-mean circularly symmetric complex Gaussian (ZMCSCG) random variables 2 2 2 2 2 }, i.e., the , σre , σsd with variances denoted as {σse , σsr , σrd channel matrices have been modeled as Hij = |σij |H, where the elements of H are i.i.d. ZMCSCG random variables

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4.7 4.6

ζ=1 ζ=5 ζ=10

6

4.5

Energy harvested (Watt)

Achievable data rate (bps/Hz)

6.5

5.5

5

4.4 4.3 4.2 4.1

ζ=1 ζ=5 ζ=10

4

4.5

3.9 4 −3

−2

−1

0

η (dB)

1

2

3

(a) achievable data rate of ID Fig. 4.

3.8 −3

−2

−1

0

η (dB)

1

2

3

(b) energy harvested by EH

Effect of the resource allocation parameter η (with τ = 0.5).

2 with unit variance, and σij is the channel fading factor. The maximum transmit power of the source and the relay are set to be equal as Ps = Pr = 1. It also assumed that the number of antennas employed by the all nodes are equal and denoted as Ns = Nr = Nd = Ne = 3, the channel fading factors are 2 2 2 2 2 set as {σse , σsr , σrd , σre , σsd } = {−6, −5, −5, −4, −15}dB, and the averaged SNR is set to ρ = 20dB. The convergence behavior of Algorithm 1(P) (with η = ζ = 1) is shown in Figs. 2 and 3. Specifically, Fig. 2 is obtained under different τ , while Fig. 3 is obtained with different initial points. From Fig. 2, it can be seen that Algorithm 1(P) always converges to a single point (us = 7.8489, ur = 8.7334) under different τ , i.e., τ = 0.1, τ = 0.5, and τ = 1.0, as illustrated by Theorem 3. Under three different initial points, i.e., point 1 (Q0s = I/6, Q0r = I/6), point 2 (Q0s = I/15, Q0r = I/15), and point 3 (Q0s = I/30, Q0r = I/30), Algorithm 1(P) also converges the single point (us = 7.8489, ur = 8.7334), which is shown in Fig. 3 and follows the result shown by Theorem 2. Note that, Fig. 2 is obtained with the initial point (Q0s = I/30, Q0r = I/30) and Fig. 3 is obtained with τ = 0.5. The effect of the energy-sensitive parameter η, defined in the utility us , is shown in Fig. 4, which is obtained with τ = 0.5 and presented under three different cases, i.e., ζ = 1, ζ = 5 and ζ = 10. With a fixed ζ, it can be seen from Fig. 4(a) that the achievable data rate of the ID receiver is monotonically nonincreasing with the increase of η; while the energy that can be harvested by the EH receiver is monotonically non-decreasing with the increase of η as shown by Fig. 4(b), which means that the energy-sensitive parameter η is tunable. However, there also exists a upper limit for the energy that can be harvested by the EH receiver and a lower limit for the data rate that can be achieved by the ID receiver.

VI. C ONCLUSION In this paper, we have studied the distributed precoding problem for SWIPT in a DF-based MIMO relay network. With only local CSI required, a noncooperative game is established for the proposed system, of which the existence and uniqueness

of the pure-strategy NE are theoretically proved. Based on the best-response dynamic, a distributed precoding algorithm is developed to obtain the unique NE solution. Furthermore, a proximal-point-based regularization approach is also pursued to ensure the convergence of the proposed algorithm. Results show that our algorithm can converge fast to a satisfactory solution with guaranteed convergence, and the energy-sensitive parameters defined in the utilities can guide the available transmit power to a desired NE. All in all, our approach can be of some significance to the practical MIMO relay network design intended for SWIPT. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which have greatly helped us to improve the quality of this paper. R EFERENCES [1] R. M. Dickinson. “Power in the sky,” IEEE Microwave Mag., vol. 14, no. 2, pp. 35 - 47, Mar./Apr. 2013. [2] Z. Popvic. “Cut the cord,” IEEE Microwave Mag., vol. 14, no. 2, pp. 55 - 62, Mar./Apr. 2013. [3] S. ladan, N. Ghassemi, A. Ghiotto, and K. Wu. “Highly efficient compact rectenna for wireless energy harvesting application,” IEEE Microwave Mag., vol. 14, no. 1, pp. 117 - 122, Jan./Feb. 2013. [4] L. R.Varshney, “Transporting information and energy simultaneously,” Proc. IEEE Int. Symp. Inf. Theory (ISIT), Toronto, Canada, pp. 1612 1616, Jul. 2008. [5] P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and power transfer,” Proc. IEEE Int. Symp. Inf. Theory (ISIT), Austin, TX, USA, pp. 2363 - 2367, Jun. 2010. [6] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus and A. Yener. “Transmission with energy harvesting nodes in fading wireless channels: Optimal policies,” IEEE J. Sel. Areas Commun., vol. 29, no. 8, pp. 1732-1743, Sep. 2011. [7] R. Zhang, C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989 - 2001, May 2013.

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[8] K. Huang, E. Larsson, “Simultaneous information and power transfer for broadband wireless systems,” IEEE Trans. Signal Process., vol. 61, no. 23, pp. 5972 - 5986, Dec. 2013. [9] B. K. Chalise, W.-K. Ma, Y. D. Zhang, H. A. Suraweera, and M. G. Amin, “Optimum performance boundaries of OSTBC based AF-MIMO relay system with energy harvesting receiver,” IEEE Trans. Signal Process., vol. 61, no. 17, pp. 4199 - 4213, Sep. 2013. [10] B. Fang, W. Zhong, Z. Qian, S. Jin, J. Wang, and W. Shao, “Optimal precoding for simultaneous information and power transfer in MIMO relay networks,” in Proc. IEEE Chinacom 2014, pp. 462-467, Aug. 2014. [11] B. Fang, W. Zhong, Z. Qian, S. Jin, J. Wang, and W. Shao, “Distributed precoding for wireless information and power transfer in MIMO DF relay networks,” in Proc. IEEE WCSP 2014, pp. 1-6, Oct. 2014. [12] B. Wang, J. Zhang, and A. H. Madsen, “On the capacity of MIMO relay channels,” IEEE Trans. Inf. Theory., vol. 51, no. 1, pp. 29 - 43, Jan. 2005. [13] Y. Kim, and H. Liu, “Infrastructure relay transmission with cooperative MIMO,” IEEE Trans. Vehicular Tech., vol. 57, no. 4, pp. 2180-2188, July 2008. [14] W. Park, S. Jeong, H.-Y. Song, and Chungyong Lee, “The global optimality of the MIMO cooperative system with source and relay precoders for capacity maximization,” IEEE Trans. Commun., vol. 60, no. 10, pp. 2886 - 2892, Oct. 2012. [15] J. Y. Ryu, and W. Choi, “Balanced linear precoding in decode-andforward based MIMO relay communications,” IEEE Trans. Wireless Commun., vol. 10, no. 7, pp. 2390 - 2400, Jul. 2011. [16] K. Xiong, P. Fan, Z. Xu, H.-C Yang, and K. B. Letaief, “Optimal cooperative beamforming design for MIMO decode-and-forward relay channels,” IEEE Trans. Signal Process., vol. 62, no. 6, pp. 1476-1489, Mar. 2014. [17] X. Zhang, Matrix Analysis And Applications, Beijing: Tsinghua University Press, pp. 225-226, Sep. 2004. [18] S. Lasaulce, M. Debbah, and E. Altman, “Methodologies for analyzing equilibria in wireless games,” IEEE Signal Proces. Mag., pp. 41 - 52, Sep. 2009. [19] W. Zhong, Y. Xu, and H. Tianfield, “Game-theoretic opportunistic spectrum sharing strategy selection for cognitive MIMO multiple access channels,” IEEE Trans. Signal Process., vol. 59, no. 6, pp. 2745-2759, Jun. 2011. [20] W. Zhong, and J. Wang, “Energy efficient discrete spectrum sharing strategy selection in cognitive MIMO interference channels,” IEEE Trans. Signal Process., vol. 61, no. 14, pp. 3705-3717, Jul. 2013. [21] G. Scutari, D. P. Palomar, J.-S. Pang, and F. Facchinei, “Convex optimization, game theory, and variational inequality theory,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 35-49, May 2010. [22] D. P. Palomar, and Y. C. Eldar, Convex optimization in signal processing and communications, Cambridge, U.K.: Cambridge Univ. Press, pp. 396397, 2009. [23] J. Rosen, “Existence and uniqueness of equilibrium points for concave n-person games,” Econometrica, vol. 33, pp. 520-534, 1965. [24] E.-V. Belmega, S. Lasaulce, and M. Debbah, “Power allocation games for MIMO multiple access channels with coordination,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 3182 - 3192, Jun. 2009. [25] E. V. Belmega, S. Lasaulce and M. Debbah, “A trace inequality for positive definite matrices,” Journal of Inequalities in Pure and Applied Mathematics (JIPAM), vol. 10, No. 1, pp. 1-4, 2009. [26] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [27] CVX Research, Inc. CVX: Matlab software for disciplined convex programming, version 2.0. http://cvxr.com/cvx.

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Bing Fang (S’13) received the B.S. and M.S. degrees from the College of Communications Engineering, PLA University of Science and Technology (PLAUST), Nanjing, China, in 2003 and 2007, respectively. He is currently towards his Ph.D. degree at the College of Communications Engineering, PLAUST. He is a recipient of the Best Paper Award at the 2014 International Conference on Wireless Communications and Signal Processing (WCSP 2014). His current research interests include optimization theory, machine learning and game theory for signal processing and wireless communication.

Wei Zhong (S’07-M’11) received the B.Sc. degree from Guangzhou Institute of Communications, Guangzhou, China, in 2003, the M.Sc. degree from the PLA University of Science and Technology (PLAUST), Nanjing, China, in 2006, and the Ph.D. degree from Shanghai Jiao Tong University (SJTU), Shanghai, China, in 2011. He currently holds a position as an Associate Professor at the College of Communications Engineering, PLAUST. He is a co-recipient of the Best Paper Award at the 2014 International Conference on Wireless Communications and Signal Processing (WCSP 2014). His current research interests include game theory, signal processing, wireless communication, machine learning, and computing.

Shi Jin (S’06-M’07) received the B.S. degree in communications engineering from Guilin University of Electronic Technology, Guilin, China, in 1996, the M.S. degree from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2003, and the Ph.D. degree in communications and information systems from the Southeast University, Nanjing, in 2007. From June 2007 to October 2009, he was a Research Fellow with the Adastral Park Research Campus, University College London, London, U.K. He is currently with the faculty of the National Mobile Communications Research Laboratory, Southeast University. His research interests include space time wireless communications, random matrix theory, and information theory. He serves as an Associate Editor for the IEEE Transactions on Wireless Communications, and IEEE Communications Letters, and IET Communications. Dr. Jin and his co-authors have been awarded the 2011 IEEE Communications Society Stephen O. Rice Prize Paper Award in the field of communication theory and a 2010 Young Author Best Paper Award by the IEEE Signal Processing Society.

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Zuping Qian (M’01) received the B.S. and M.S. degrees in applied mathematics from Hunan University, Changsha, China, in 1982 and 1985, respectively. He received the Ph.D. degrees in microwave techniques from Southeast University, Nanjing, China, in 2000. During 1985-1999, he worked at Institute of Communications Engineering (ICE) as a Lecturer and later as an Associate Professor. Since 2000, he is a Professor at the College of Communications Engineering, PLAUST. He published several books like Electromagnetic Compatibility, Antenna and Propagation. He published over 80 international and regional refereed journal papers. His current research interests include antenna, metamaterials, computational electromagnetics, array signal processing and EMI/EMC.

Wei Shao (M’13) received the B.S., M.S. and Ph.D. degrees from the College of Communication Engineering, PLAUST, Nanjing, China, in 2001, 2004, and 2007 respectively. He is a co-recipient of the Best Paper Award at the 2014 International Conference on Wireless Communications and Signal Processing (WCSP 2014). He currently holds a position as an Associate Professor at the College of Communication Engineering, PLAUST. His research interests mainly include spectrum management, signal processing and wireless communication.

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