Wireless Information and Energy Transfer in Multi-Antenna Interference Channel

Transcription

1 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Wireless Information and Energy Transfer in Multi-Antenna Interference Cannel Cao Sen, Wei-Ciang Li and Tsung-ui Cang arxiv:8.88v [cs.it] Aug Abstract Tis paper considers te transmitter design for wireless information and energy transfer WIET in a multiple-input single-output MISO interference cannel IFC. Te design problem is to imize te system trougput i.e., te weigted sum rate subject to individual energy arvesting constraints and power constraints. Different from te conventional IFCs witout energy arvesting, te cross-link signals in te considered scenario play two opposite roles in information detection and energy arvesting. It is observed tat te ideal sceme, were te receivers can simultaneously perform and from te received signal, may not always acieve te best tradeoff between information transfer and energy arvesting, but simple practical scemes based on time splitting may perform better. We terefore propose two practical time splitting scemes, namely time division mode switcing TDMS and time division multiple access TDMA, in addition to a power splitting PS sceme wic separates te received signal into two parts for and, respectively. In te two-user scenario, we sow tat beamforming is optimal to all te scemes. Moreover, te design problems associated wit te TDMS and TDMA scemes admit semi-analytical solutions. In te general K-user scenario, a successive convex approximation metod is proposed to andle te WIET problems associated wit te ideal sceme and te PS sceme, wic are known to be NP-ard in general. Te K-user TDMS and TDMA scemes are sown efficiently solvable as convex problems. Simulation results sow tat stronger cross-link cannel powers actually improve te information sum rate under energy arvesting constraints. Moreover, none of te scemes under consideration can dominate anoter in terms of te sum rate performance. Index terms wireless energy transfer, energy arvesting, interference cannel, beamforming, convex optimization EDICS: SPC-APPL, SPC-INTF, SPC-CCMC, SAM-BEAM Te work of Cao Sen is supported by te Opening Project of Te State Key Laboratory of Integrated Services Networks, Xidian University Grant No. ISN4-9, te Cina Postdoctoral Science Foundation Grant No. M559, te National Natural Science Foundation of Cina Grant No. 65, te Key Project of State Key Lab of Rail Traffic and Control Grant No. RCSZZ4, Beijing Jiaotong University, te Key grant Project of Cinese Ministry of Education No. 6, and te Fundamental Researc Funds for te Central Universities Grant No. JBZ8 and YJS7. Te work of Tsung-ui Cang is supported by National Science Council, Taiwan R.O.C., by Grant NSC --E--5-MY. Part of tis work was presented in IEEE GLOBECOM []. Cao Sen is wit te State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, Cina and te State Key Laboratory of Integrated Services Networks, Xidian University, Xi an, Cina. caosen@ieee.org. Wei-Ciang Li is wit Institute of Communications Engineering, National Tsing ua University, sincu, Taiwan, R.O.C.. weiciangli@gmail.com. Tsung-ui Cang is te corresponding autor. Address: Department of Electronic and Computer Engineering, National Taiwan University of Science and Tecnology, Taipei 67, Taiwan, R.O.C.. tsungui.cang@ieee.org. I. INTRODUCTION Recently, scavenging energy from te environment as been considered as a potential approac to prolonging te lifetime of battery-powered sensor networks and to implementing selfsustained communication systems. For example, te base stations may be powered by wind mills or solar potovoltaic PV arrays, and can arvest energy for providing services to te mobile users. Tis idea as motivated considerable researc endeavors in te past few years, investigating wireless systems wit energy-arvesting transmitters; see, e.g., [ 6]. In tese works, optimal transmission strategies under energy-arvesting constraints are studied from single-input single-output SISO cannels to complex interference cannels IFCs. In contrast to te base stations, it may be difficult for te mobile devices and sensor nodes to arvest energy from te sun and wind effectively. One possible solution to tis issue is wireless energy transfer WET, tat is, te power-connected transmitters transfer energy wirelessly to carge te mobile devices. A successful application of WET is te radio frequency identification RF system were te receiver wirelessly carges energy from te transmitter troug induction coupling and use te energy to communicate wit te transmitter. Te works in [7, 8] sowed tat, using coupled magnetic resonances, energy can be wirelessly transferred for two meters wit over 5% energy conversion efficiency. WET can also be acieved via te RF electromagnetic signals; see [9, ] for recent developments of RF-based energy arvesting circuits. Compared to te tecniques based on induction and magnetic resonance coupling, RF signals can acieve long-distance WET; owever, te energy conversion efficiency is in general low. Tis calls for advanced signal processing tecniques, suc as beamforming, to improve te energy conversion efficiency. Since te RF signals can carry bot information and energy, in recent years, it as been of great interest to study wireless communication systems were te receivers can not only decode information bits but also arvest energy from te received RF signals, i.e., wireless information and energy transfer WIET systems [ 7]. Specifically, in [], te optimal tradeoff between information capacity and energy transfer of te WIET system was studied for a SISO flat fading cannel. In [], te optimal power allocation strategy for a SISO frequency-selective fading cannel was derived under a receiver energy arvesting constraint. Te work in [] furter extends tese studies to te multiple access cannel MAC and two-op relay network wit an energy arvesting relay. It was sown tat in general tere exist

2 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING nontrivial tradeoffs between information transfer and energy arvesting. Te works in [ ] assume te ideal receivers wic can decode information bits and arvest energy from te received RF signals simultaneously. Unfortunately, current circuit tecnologies cannot acieve tis yet. In view of tis, practical WIET scemes are proposed. In particular, Zou et al. proposed in [4, 5] a dynamic power splitting PS sceme for a SISO flat fading cannel, werein, te received RF signal is eiter used for information detection, energy arvesting, or is split into two parts, one for and te oter for. Considering a multiple-input multiple-output MIMO flat-fading cannel, in addition to te PS sceme, te autors in [6] furter proposed a time switcing sceme were te receiver performs in one time slot wile in te oter time slot. In [7], te dynamic PS sceme was extended to a multi-user multiple-input single-output MISO broadcast cannel, and te optimal transmit beamforming and power splitting coefficients are jointly optimized to minimize te transmission power subject to information rate and energy arvesting constraints. In tis paper, we consider a K-user MISO interference cannel and study te optimal transmission strategies for WIET. We first consider te ideal receivers, and formulate te design problem as a weigted sum rate imization problem subject to individual energy arvesting constraints and power constraints. It is interesting to note tat, different from te conventional IFCs witout energy arvesting, te cross-link signals in te considered scenario can degrade te information sum rate on one and, but, at te same time, boost energy arvesting of te receivers on te oter and. And it turns out tat te ideal sceme wit ideal receivers may not always perform best in te complex interference environment, but simple practical scemes based on time splitting may instead yield better sum rate performance. Tis is in sarp contrast to te scenarios studied in [4 7] were time splitting scemes usually exibit poorer performance. Tis intriguing observation motivates us to propose two practical WIET scemes for te MISO IFC, namely, te time division mode switcing TDMS sceme and te time division multiple access TDMA sceme, in addition to te PS sceme [5]. In te TDMS sceme, te transmission time is divided into two time slots. All receivers perform in te first time slot and subsequently perform in te second time slot. Te TDMA sceme divides te transmission time into K time slots, and in eac time slot, one receiver performs wile te oters perform. We analytically sow ow te design problems associated wit te tree scemes can be efficiently andled. Specifically, for te two-user scenario, we sow tat transmit beamforming is an optimal transmission strategy for all scemes. Moreover, te design problems associated wit te TDMS and TDMA scemes admit semi-analytical solutions in te two-user scenario and can be solved as convex problems in te general K-user scenario. Since te WIET design problems associated wit te ideal sceme and te PS sceme in te K-user scenario are NP-ard in general, we As will be sown in Section IV-A, te proposed TDMA sceme is similar to but not completely te same as te TDMA sceme in conventional IFCs witout energy arvesting. furter present an efficient approximation metod based on te log-exponential reformulation and successive convex approximation tecniques [8]. Te presented simulation results will sow tat stronger cross-link cannel powers actually improve te information sum rate under energy arvesting constraints. Moreover, te tree scemes do not dominate eac oter in terms of sum rate performance. Rougly speaking, if te crosslink cannel powers are not strong or te energy arvesting constraints are not stringent, te PS sceme can outperform TDMS and TDMA scemes; oterwise, te TDMS sceme can perform best. In some interference dominated scenarios, te TDMS sceme and TDMA sceme even outperform te ideal sceme. Te rest of tis paper is organized as follows. In Section II, te signal model of te MISO interference cannel is presented. Starting wit te two-user scenario, in Section III, te optimal WIET transmission strategy for ideal receivers is analyzed. Te result motivates te developments of te practical TDMS and TDMA scemes, wic are presented in Section IV. Section V extends te study to te general K-user scenario; te design problem of te PS sceme is also presented in tat section. Simulation results are presented in Section VI. Te conclusions and discussion of future researces are given in Section VII. Notations: Column vectors and matrices are written in boldfaced lowercase and uppercase letters, e.g., a and A. Te superscripts T, and represent te transpose, ermitian conjugate transpose and matrix inverse, respectively. ranka and TrA represent te rank and trace of matrix A, respectively. A means tat matrix A is positive semidefinite positive definite. a denotes te Euclidean norm of vector a. Te ortogonal projection onto te column space of a tall matrix A is denoted by Π A AA A A. Moreover, te projection onto te ortogonal complement of te column space of A is denoted by Π A I Π A were I is te identity matrix. II. SIGNAL MODEL AND PROBLEM STATEMENT We consider a multi-user interference cannel wit K pairs of transmitters and receivers communicating over a common frequency band. Eac of te transmitters is equipped wit N t antennae, wile eac of te receivers as single antenna. Let x i C Nt be te signal vector transmitted by transmitter i, and ik C Nt be te cannel vector from transmitter i to receiver k, for all i, k {,,..., K}. Te received signal at receiver i is given by y i = ii x i + kix k + n i, i =,..., K, k=,k i were n i CN, σi is te additive Gaussian noise at receiver i. Unlike te conventional MISO IFC [9] were te receivers focus only on extracting information, we consider in tis paper tat te receivers can also scavenge energy from te received signals [,, 6], i.e, energy arvesting. Terefore, in addition to information, te transmitters can also wirelessly transfer energy to te receivers. We call te two operation modes te information detection mode and te energy arvesting mode, respectively.

3 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Assume tat x i contains te information intended for receiver i wic is Gaussian encoded wit zero mean and covariance matrix S i, i.e., x i CN, S i for i =,..., K. Moreover, assume tat eac receiver i decodes x i by single user detection in te mode. Ten te acievable information rate of receiver i is given by R i S,..., S K = log ii + S i ii k i ki S k ki + σi, for i =,..., K. Alternatively, te receiver may coose to arvest energy from te received signal. It can be assumed tat te total arvested RF-band energy during a transmission interval is proportional to te power of te received baseband signal [6]. Specifically, for receiver i, te arvested energy, denoted by E i, can be expressed as E i = γ kis k ki, i =,..., K, k= were γ is a constant accounting for te energy conversion loss in te transducer [6]. Suppose tat te receivers desire to arvest certain amounts of energy. We are interested in investigating te optimal transmission strategies of S i, i =,..., K, so tat te information trougput of te K-user IFCs can be imized wile te energy arvesting requirements of te receivers are satisfied at te same time. One sould note tat current energy arvesting receivers are not yet able to decode te information bits simultaneously [6]. In subsequent sections, we will first study an ideal scenario were te receivers can simultaneously operate in te mode and mode. Ten, we furter investigate some practical scemes were te receivers operate eiter in te mode or mode at any time instant. In order to gain more insigts, we will begin our investigation wit te two-user scenario K =, and later extend te studies to te general K-user case in Section V. III. OPTIMAL WIET DESIGN FOR EAL SCEME Let us assume tat K = and consider ideal receivers wic can simultaneously decode te information bits and arvest te energy from te received signals. Suppose tat te two receivers desire to arvest total amounts of energy E and E, respectively. We are interested in te following transmitter design problem for WIET: P w R S, S + w R S, S S, S S + S E, S + S E, TrS P, TrS P, 4a 4b 4c 4d 4e were w, w > are positive weigts, and P > and P > in 4d and 4e represent te individual power constraints. Te constraints in 4b and 4c are te energy arvesting constraints were we ave set γ = = for notational simplicity. Note tat, in te absence of 4b and 4c, problem P reduces to te classical sum rate imization problem in MISO IFC [9]: w R S, S + w R S, S S, S TrS P, TrS P. 5a 5b 5c It can be observed from 4 and 5 tat te energy arvesting constraints 4b and 4c would trade te imum acievable sum rate for energy arvesting; i.e., te imum sum rate in 4a is in general no larger tan tat in 5a. To see wen tis would appen, let S, S be an optimal solution to problem 5. One can verify from te rate function in and problem 5 tat S, S must satisfy [ ] { S [E ] T S Ω E E = S,TrS P, S E [ ] { S [E ] T S Ω E E = S,TrS P, S E S, E P }, 6 S, E P }. 7 Tat is, te energies arvested at te two receivers due to S, S must lie in Ω +Ω. It can be sown tat in Ω +Ω, S + S S + S min E,E Ω,E,E Ω E + E = P ĥ, 8a min E,E Ω,E,E Ω E + E = P ĥ, 8b were ĥ ij Π ii ij Π ij ii. Equations in 8 implies tat te two receivers can at lease arvest energies P ĥ and P ĥ, respectively. Te minimum amounts of energies are acieved wen E = P ĥ, E =, E = P ĥ and E = ; tat is, wen eac of te transmitters only focus on transmitting signals to its own receiver, witout allowing any leakage of energy to te oter receiver. According to 8, we ave tat Property Te energy arvesting constraints 4b and 4c are inactive at te optimum if E P ĥ and E P ĥ ; ence, P reduces to te conventional MISO IFC problem 5 under tis condition. owever, wen E > P ĥ or E > P ĥ, te imum information trougput may ave to be compromised wit energy arvesting. Interestingly, te following proposition sows tat te optimal transmit structure of P is still similar to problem 5 wic does not ave te energy arvesting constraints.

4 4 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Proposition Assume tat problem P is feasible, and tat and witout loss of generality. Let S, S denote te optimal solution to problem P. Ten, TrS = P and TrS = P. Moreover, tere exist a i R, b i C, i =,, suc tat S = a + b a + b, S = a + b a + b. 9a 9b Te proof is given in Appendix A. Proposition implies tat beamforming is an optimal transmission strategy of P. Moreover, te beamforming direction of transmitter i sould lie in te range space of [ i, i ], for i =,, wic is te same as te optimal beamforming direction of problem 5 in te conventional IFCs [9]. Given 9, te searc of S and S in P reduces to te searc of a i and b i over te ellipsoids a i i + b i i = P i for all i =,. owever, unlike problem 5, optimizing te coefficients a i, b i, i =,, for problem P ave to take into account bot te needs of energy arvesting and information transfer. Remark It is important to remark tat, wile P is ideal in te sense tat te receivers can simultaneously operate in te and modes, P does not necessarily perform best in terms of sum rate imization. Te reason is tat te cross-link signal power ik S i ik plays two completely opposite roles in te considered scenario It can boost te energy arvesting of receiver k on one and, but also degrades te acievable information rate on te oter and. Terefore, wen te cross-link cannel power is strong e.g., te interference dominated scenario and wen te energy arvesting constraints are not negligible e.g., te conditions in Property do not old, te transmitters ave to compromise te acievable information rate for energy arvesting. Under suc circumstances, it migt be a wiser strategy to split te and modes in time. To furter look into tis aspect, we present in Fig. two simulation examples for te -user scenario. Te detailed setting of te simulations are presented in Section VI. Fig. a sows te sum rate-versus-energy requirement regions for two randomly generated cannel realizations. Te curves are obtained by exaustively solving P for various values of symmetric energy requirement E E = E. Te average powers of te direct link cannels are normalized to one, wile te average powers of te cross-link cannels are measured by te parameter η. As one can observe from tis figure, for η =, te rate-energy region is not convex for tis randomly generated cannel realization. Moreover, for some values of E, te receivers may acieve a iger sum rate troug time saring between te mode and mode see te dased line between point A and point B. Fig. b displays te rate region R versus R of te two users. Analogously, we observe tat time saring for multiple access may acieve a iger sum rate see te dased line between points A and B. Te two simulation results in Fig. imply tat te ideal sceme P may not always acieve te best tradeoff between information transfer and energy arvesting, but, instead, time saring for / mode switcing or time saring for multiple Sum Rate bps/z A η =5. η =. Ideal sceme Time saring sceme B E Joule/s a Sum rate vs. requirement E, for N t = 4 and SNR = db. Parameter η measures te cross-link cannel power. R bps/z 5 A Ideal sceme Time saring sceme R bps/z b Acievable rate region R, R, for N t = 4, E =, E =, η = and SNR = db. Fig. : Motivating simulation examples for te -user scenario. access may yield iger information sum rate. Tis motivates us to develop two practical scemes, namely, te time-division mode switcing TDMS sceme and te time-division multiple access TDMA sceme, in te next section. It is wortwile to note tat, in tese time saring scemes, te receivers operate eiter in te mode or mode at eac time instant, and tus are more practical tan te ideal receivers. IV. PRACTICAL WIET SCEMES AND OPTIMAL TRANSMISSION STRATEGIES A. Time Division Mode Switcing TDMS Sceme In te first practical sceme, we divide te transmission interval into two time slots. In one time slot, bot receivers operate in te mode, wereas, in te oter time slot, bot receivers switc to te mode. Te two receivers tus coerently switc between te and modes, i.e., mode switcing. Suppose tat α fraction of te time is for mode and α fraction of te time is for mode. Te TDMS sceme is described as follows: Time slot mode: Te two receivers focus on arvesting te required energy E and E in α fraction of te time, i.e., α S + S E, α S + S E. B a b

5 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 5 Time slot mode: Bot te two receivers operate in te mode and imize te information trougput in te remaining fraction of te time, i.e., α w R S, S +w R S, S a S, S TrS P, TrS P. b Problem in te mode is te classical sum rate imization problem in te MISO IFC [see 5], wic can be efficiently andled by existing metods in [9 ]. Note tat it as been sown in [, ] tat beamforming is an optimal transmission sceme for problem. We now focus on te mode in time slot. Since time slot does not contribute to te information trougput, it is desirable to spend as least as possible time for te mode, i.e., to use a minimal time fraction α to fulfill te energy arvesting task. Matematically, we can write it as te following optimization problem β a β R, S, S S + S βe, S + S βe, TrS P, TrS P, b c d were β /α. Note tat if te optimal β of is less tan one i.e., optimal α >, ten it implies tat te energy arvesting requirements cannot be satisfied even if te receivers dedicate temselves to arvesting energy trougout te wole transmission interval. In tat case, we declare tat te TDMS sceme is not feasible. Wile problem is a convex semidefinite program SDP, wic can be solved by te off-te-self solvers, we sow tat actually admits a semi-analytical solution: Proposition Assume tat i and i are linearly independent but not ortogonal to eac oter, for i =,. Te optimal solution to problem is given by S µ = P v µ v µ, S µ = P v µ v µ, { βµ = min S µ + S µ, E S µ + S µ E a }, b were µ is te optimal dual variable associated wit constraint b, and v i µ is te principal eigenvector of µ i i + µ E E i i for i =,. Moreover, µ can be efficiently obtained using a simple bisection searc. Te proof of Proposition is given in Appendix B. Te assumptions on i and i, for i =,, old wit probability one for random continuous fading cannels. Note tat Proposition also implies tat beamforming is optimal for te mode of te TDMS sceme. B. TDMA Sceme Unlike TDMS sceme, in eac time slot of TDMA sceme, one receiver operates in te mode and te oter receiver operates in te mode. Assume tat te time fraction of te first time slot is α. Time slot : Receiver operates in te mode and receiver operates in te mode. Te objective is to imize te information rate of receiver and guarantee te energy arvesting requirement of receiver at te same time. Te design problem is given by + S S + σ S + S E /α, S, S α log TrS P, TrS P, 4a 4b 4c Time slot : Te operation modes of te two receivers are excanged: α log + S S, S S + σ 5a S + S E / α, 5b TrS P, TrS P. 5c By intuition, tis TDMA sceme would be of interest wen te two receivers ave asymmetric energy arvesting requirements and asymmetric cross-link cannel powers. Moreover, like te conventional interference cannel witout energy arvesting, te TDMA sceme may outperform te spectrum saring scemes in interference dominated scenarios. It is not difficult to sow tat: Lemma Te TDMA sceme is feasible if and only if E P + P + E P. 6 + P Proof: Te TDMA sceme is feasible if and only if bot 4 and 5 are feasible. Problem 4 is feasible if and only if tere exists some α [, ] suc tat E α S,S S + S TrS P,TrS P = α P + P, 7 were te equality is obtained by applying te result in [6, Proposition.]. Similarly, one can sow tat 5 is feasible if and only if E α P + P. 8 Combining 7 and 8 gives rise to 6. Conversely, given E 6, let α= E P +P, and tus P +P α, wic are 7 and 8, respectively. ence, wen 6 is true, te TDMA sceme is feasible. According to 7 and 8, a feasible time fraction α must lie in te interval E P +P α E P +P. 9 Interestingly, given a feasible α, bot problems 4 and 5 can be efficiently solved semi-analytically. Since problems

6 6 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 4 and 5 are similar to eac oter, we take 4 as te example. Proposition Let te time fraction α satisfy 9. Ten, an optimal solution to problem 4, denoted by S, S, is given by were S = v y v y /y, S = v y v y /y, v y = v y = yp, if gy/y E /α P, ye/α gy + yp ye/α gy, oterwise, yσ + yp yσ, y = arg y v y P + σ in wic ij = ij, ij ĥ ij = Π y σ, ii ij Π ii ij for i =,, and gy = v y. Problem is a convex problem, and tus y can be obtained by a bisection searc. Te proof is presented in Appendix C. We see from tat beamforming is also optimal to te TDMA sceme. By Proposition, given a feasible time fraction α, one can efficiently solve problems 4 and 5 and tus evaluate te acievable sum rate of te two users. Ten, te optimal time fraction α tat imizes te sum rate of te two users can be obtained by line searc over te interval in 9. C. TDMA via Deterministic Signal for Energy arvesting It sould be noticed tat, wile Gaussian signaling is optimal for information transfer, it may not be necessary for energy transfer. In particular, if one user operates in te mode, te transmitter may simply transmit some deterministic signals e.g., training/pilot signals known to bot receivers. Consider te TDMA sceme in te previous subsection, and assume tat, in te first time slot, transmitter operating in te mode transmits deterministic signals x wic are known to receiver operating in te mode. Under suc circumstances, receiver can actually remove x from te received signal before information detection, i.e., removing te cross-link interference. Te design problem in te st time slot tereby reduces to S, S α log + σ S S + S E /α, TrS P, TrS P. a b c Problem is easier to andle tan its counterpart in 4. Clearly, given α satisfying 9, optimal S is given by S = P, Terefore, boils down to S S a S E /α P, TrS P, b wic admits a closed-form solution for S according to [6, Proposition.]. Analogously, te design problem for te second time slot can be simplified. In tis paper, we refer to tis sceme as te TDMA D sceme. Since te receivers are free from cross-link interference, it is anticipated tat te TDMA D sceme performs no worse tan te TDMA sceme. owever, it sould be noted tat, in order to do so, te two receivers require perfect knowledge of te cross-link cannels and, respectively; oterwise, te receivers may suffer performance degradation due to imperfect interference cancelation. We remark tat, in addition to te above time saring based scemes, it is also possible for te receivers to split te received signals into two parts, one for and te oter for, i.e., power splitting PS [6]. Tis sceme will be studied in Section V-C. V. WIET DESIGN FOR K-USER MISO IFC In tis section, we consider te WIET problem for te K- user MISO IFC scenario. We begin wit te ideal sceme, and in te second subsection, we extend te TDMS and TDMA scemes in Section IV to te K-user scenario. In te last subsection, we furter investigate te PS sceme. A. Transmitter Optimization for Ideal Receivers By te signal model in,, and P in 4, te K- user WIET problem is formulated as w i log ii + S i ii k i ki S k ki + σi 4a S i i=,...,k i= kis k ki E i, i =,..., K, k= TrS i P i, i =,..., K, 4b 4c were E i is te energy requirement of user i, for i =,..., K. Since problem 4 is NP-ard in general [4], our interest for te K-user WIET problem lies in efficient approaces to finding an approximate solution. We propose an efficient algoritm based on successive convex approximation SCA [5] by adopting te log-exponential reformulation idea in [8]. Compared to te metods in [9 ], te proposed metod can work for scenarios wit a medium to large number of users. Specifically, by introducing slack variables {x i, y i }, we can reformulate problem 4 as w i x i y i log e 5a S i, x i, y i i=,...,k i= kis k ki + σi e xi i, 5b k= kis k ki + σi e yi i, 5c k i 4b, 4c. 5d As seen, te rate functions in 4a are equivalently decomposed into te objective function in 5a and te two constraints in 5b and 5c. In particular, one can verify

7 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 7 tat constraints 5b and 5c will old wit equality at te optimum, implying tat 5 is equivalent to 4. Problem 5 as a linear objective function and convex constrains, except for constraint 5c. We propose to linearly approximate constraint 5c in an iterative manner. Suppose tat, at iteration n, we are given S[n ],..., SK [n ]. K Let ȳ i [n] = ln k i ki S k [n ] ki + σi, i =,..., K. We solve te following problem at te nt iteration {Si [n]} K i= = arg w i x i y i log e 6a S i, x i, y i i=,...,k i= kis k ki +σi e xi i, 6b k= kis k ki +σi eȳi[n] y i ȳ i [n]+ i, k i 4b, 4c. 6c 6d Note tat constraint 6c is convex; it is a conservative approximation to 5c since it olds tat e yi eȳi[n] y i ȳ i [n]+ y i due to te convexity of e yi. As a result, problem 6 is a convex SDP wic can be solved efficiently by offte-self solvers, e.g., CVX [6]. Detailed steps of te proposed algoritm is summarized in Algoritm. Algoritm SCA algoritm for problem 4 : Find initial variables by solving te feasibility problem {Si []} K i= = find {S,..., S K} kis k iki E i i, k= TrS i P i, S i i. If te problem is infeasible, ten declare infeasibility of 4; oterwise, set n = and perform te following steps. : repeat : n := n +. K 4: ȳ i[n] = ln k i kisk[n ] ki + σi i. 5: Solve problem 6 to obtain {S [n],..., SK[n]}. 6: until te stopping criterion is met. 7: Output S [n],..., SK[n] as an approximate solution. It can be sown tat Algoritm belongs to te category of te successive upper-bound minimization SUM metod proposed in [7] and can converge to a stationary point of problem 4, as stated in Proposition 4. Te details are relegated to Appendix D. Proposition 4 Any limit point of te sequence {S [n],..., S K [n]} n= generated by Algoritm is a stationary point of problem 4. B. Practical K-User WIET Scemes We extend te TDMS and TDMA scemes in Section IV to te general K-user scenario in tis subsection. K-user TDMS sceme: Tis sceme is similar to te TDMS sceme presented in Section IV-A. In te st time slot, all users operate in te mode, and in te nd time slot, all RX RX RX a TDMS RX RX RX b TDMA Fig. : Illustration of te proposed TDMS and TDMA scemes for WIET in a -user scenario. users operate in te mode; see Fig. a. In te st time slot, te optimal time fraction α and te associated optimal signal covariance matrices {Sk }K k= for energy arvesting can be obtained by solving a convex problem analogous to problem. In te nd time slot, one as to solve te classical sum rate imization problem α w i log + ii S i ii K k i ki S 7a k ki +σi S i, i=,...,k i= TrS i P i, i. 7b Problem 7 is NP-ard, but can be efficiently andled by Algoritm by letting E i = i or existing block coordinate descent based metods [7]. K-user TDMA D sceme: Te transmission interval is divided into K time slots, eac of wic as a time fraction α l, satisfying K l= α l = ; see Fig. b for te case of K =. In te lt time slot, user l operates in te mode; wile te oter K users operate in te mode. ere we assume tat transmitters operating in te mode send deterministic signals so tat receivers operating in te mode can remove te cross-link signals see Section IV-C. Let S kl be te signal covariance matrix employed by transmitter i in te lt time slot, for k, l =,..., K. Te design problem of tis TDMA D sceme can be formulated as α,...,α K Ω,S kl k,l=,...,k l= l i w l α l log + ll S ll ll σ l α l K k= kis kl ki E i, i, 8a 8b TrS kl P k, k, l, 8c were Ω = {{α l } K l= α l [, ], K l= α l }, and 8b denotes te energy arvesting constraints of all users. Note tat in 8 we not only optimize te signal covariance matrices in all time slots but also optimize te time fractions {α l }. Problem 8 can be reformulated as a convex problem. To sow tis, define W kl = α l S kl, k, l =,..., K. 9 Ten, 8 can be rewritten as α,...,α K Ω,W kl k,l=,...,k l= l i k= w l α l log + ll W ll ll α l σl kiw kl ki E i, i, TrW kl α l P k, k, l. a b c

8 8 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING RX i CN; ~¾ i Power Splitter p ½i CN; ^¾ i p ½i Fig. : Diagram of te power splitting receiver for WIET. In, all te constraints are linear. Besides, te function α l log + ll W ll ll /α l σ l is concave since it is te perspective of te concave function log + ll W ll ll /σ l. Terefore, problem 8 is a convex optimization problem. C. Practical Sceme by Power Splitting Oter tan te TDMS and TDMA scemes, anoter practical sceme, called power splitting PS [6], splits te received signal into two parts for simultaneous and ; see Fig.. In tis subsection, we extend tis sceme to te K-user interference cannel. Specifically, suppose tat receiver i splits ρ i [, ] fraction of power for and ρ i fraction of power for. Te associated WIET design problem is given by ρ i ii S i ii S i, ρ i, i=,...,k S i, ρ i, θ i, i=,...,k i= k= w i log + ρ i k i ki S k ki +ρ i σ i +ˆσ i a kis k ki E i i =,..., K, b ρ i TrS i P i i =,..., K, i= c were σ i denotes te noise power at te RF end wile ˆσ i denotes te processing noise power. Note tat, in problem, we not only optimize te signal covariance matrices S,..., S K, but also te power splitting fractions ρ,..., ρ K in te receivers. Firstly, it is not difficult to infer from Proposition tat transmit beamforming is optimal to problem as K =. Secondly, for te general K-user case, we sow tat problem can be efficiently andled in a manner similar to Algoritm. By introducing slack variables θ i = /ρ i, i =,..., K, one can write as w i log + ii S i ii k i ki S k ki + θ iˆσ i + σ i a kis k ki E i, i =,..., K, b ρ i k= θ i /ρ i, i =,..., K, TrS i P i, i =,..., K, c d were c would old wit equality at te optimum. Note tat bot constraints b and c are convex. As a result, like problem 4, te non-convexity of is mainly due to te sum rate function. Terefore, we can apply te logexponential reformulation and SCA metod in Section V-A to. In particular, like 6, at te nt iteration, one solves te following approximation problem {S[n],..., SK[n], θ[n],..., θk[n]} = arg w i x i y i log e a S i,x i,y i,θ i, i=,...,k i= kis k ki + θ iˆσ i + σ i e xi i, b k= kis k ki +θ iˆσ i + σ i eȳi[n] y i ȳ i [n]+ i, c k i b, c and d, d K were ȳ i [n]=ln k i ki S k [n ] ki + θi [n ]ˆσ i + σ i, i =,..., K. VI. SIMULATION RESULTS AND DISCUSSIONS In tis section, simulation results are presented to examine te performance of te proposed WIET scemes. Trougout te simulations, we assumed tat eac transmitter as identical, unit power budget, i.e. P P = = P K =, and tat te receiver noise powers are te same and equal to., i.e., σ σ = = σk =.. Te signal-to-noise ratio SNR, defined as SNR P/σ, is tus equal to db. Te cannel vectors { ki } were randomly generated following te complex Gaussian distribution ki CN, Q ki, were te cannel covariance matrices Q ki were randomly generated. We normalized te imum eigenvalue of Q ii, i.e., λ Q ii, to one for all i, and normalized λ Q ki to a value η > for all k i, i =,..., K. Te parameter η tereby represents te relative cross-link cannel power. All te results presented in tis section were obtained by averaging over 5 independent cannel realizations. For Algoritm, te stopping criterion was set to Rate[n] Rate[n ] Rate[n ], were Rate[n] denotes te acieved sum rate at iteration n. Te Matlab package CVX [6] was used to solve te convex approximation problems 6, and. Example Impact of cross-link cannel power: We investigate ow te cross-link cannel power i.e., η can affect te performance of te proposed WIET scemes in te interference cannels. We first consider te feasibility rate, defined as te ratio of te total number of cannel realizations for wic te energy requirement E E = E can be satisfied to te 5 randomly generated cannel realizations, of te te ideal sceme, TDMS, TDMA, and PS scemes. Fig. 4a sows te results for K =, N t = 4 and E {, }. Notice from 4, and tat te ideal sceme, TDMS and PS scemes intrinsically ave te same feasibility rate. Terefore, in Fig. 4a, only te results of TDMS and TDMA are displayed. One can observe tat te feasibility rates of all scemes improves as η increases. Tis is owing to te fact tat te cross-link interference signals can benefit energy arvesting. We also observe tat te TDMS sceme is more likely to be feasible tan te TDMA sceme.

9 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 9 Feasibility Rate % E= E= TDMS E= TDMS E= TDMA E= TDMA E= Average Sum Rate bps/z Ideal sceme Power splitting TDMS TDMA D TDMA η a Feasibility rate vs. η, for E {, } E Joule/s a Average sum rate vs. E, for N t = 4. Average Sum Rate bps/z Ideal sceme Power splitting TDMS TDMA D TDMA Average Sum Rate bps/z Ideal sceme Power splitting TDMS TDMA D TDMA η b Average sum rate vs η, for E=. Fig. 4: Simulation results for te scenario wit K =, N t =4 and SNR = db. Fig. 4b sows te average sum rate versus η acieved by te five scemes under consideration. Note tat wenever a sceme is infeasible, te acievable sum rate was set to zero. Te results were obtained by averaging over 5 cannel realizations. Firstly, one can see tat all scemes ave improved sum rates as η increases. Tis is because, from Fig. 4a, te larger η is, te easier for te receivers to arvest te energy; all scemes can terefore allocate more time and power resources for information transfer as η increases. Secondly, one observes tat te ideal sceme, TDMS and PS scemes all outperform te TDMA and TDMA D scemes. Tis is because, given N t = 4 and K =, te cross-link interference can in general be well controlled, and tus tese spectrum saring scemes admit iger data trougput. Tirdly, one can observe from Fig. 4b tat, wen η., te PS sceme outperforms te TDMS sceme; wereas, wen η >., te TDMS sceme can yield iger sum rate. Tis is due to te fact tat, wen η is large, te TDMS sceme will spend only a negligible fraction of time in energy arvesting, and use most of te time in information transfer. Since te mode of te TDMS sceme is free from any energy arvesting constraint, it can yield iger sum rate tan te PS sceme. In fact, wen bot η and E are large, te TDMS sceme may even outperform te ideal sceme, as illustrated in te next example. Example Impact of te requirement: Fig. 5a E Joule/s b Average sum rate vs. E, for E =, N t =. Fig. 5: Simulation results for te scenario wit K =, η = 4 and SNR = db. sows te average sum rate versus te energy requirement E E = E, for N t = 4 and η = 4. As expected, te acievable sum rate decreases as te requirement increases. Moreover, wen E is small E, te ideal sceme can perform best; tis is consistent wit Property. owever, wen E >, te TDMS sceme outperforms te ideal sceme. It is also noted tat wen E.7, te PS sceme exibits te poorest sum rate performance. In Fig. 5b, we sow te simulation results under an asymmetric energy requirement setting. In particular, we plot te average sum rate versus te energy requirement of receiver E, given tat te energy requirement of receiver was fixed to E =. Interestingly, we see from Fig. 5b tat wen E is large, te TDMA and TDMA D scemes can outperform te ideal sceme and perform best. Example Performance for te K-user scenario: In tis example, we consider an interference dominated scenario by setting N t = and K = 4. Fig. 6a displays te average sum rate versus E, for η =. It can be observed from tis figure tat, except te ideal sceme, te TDMA D sceme outperforms te TDMS and PS scemes wen E.. Fig. 6b sows te simulation results for η = 4. We observe tat te TDMA D sceme instead yields igest sum rates wen E. Moreover, te TDMS sceme becomes to perform better tan te ideal sceme and PS sceme wen E.8.

10 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Average Sum Rate bps/z Average Sum Rate bps/z Ideal sceme Power splitting TDMS TDMA D E Joule/s a Average sum rate vs. E, for η =.. Ideal sceme Power splitting TDMS TDMA D E Joule/s b Average sum rate vs. E, for η = 4. Fig. 6: Simulation results for te scenario wit K = 4, N t = and SNR = db. VII. CONCLUSIONS AND FUTURE WORKS In tis paper, we ave considered te WIET problem in a multi-user MISO interference cannel. In addition to te ideal sceme, we ave proposed tree practical scemes, namely, te TDMS, TDMA and PS scemes. Starting wit te two-user scenario, we ave analyzed te optimal transmission strategy of te ideal sceme as well as semi-analytical solutions to te TDMS and TDMA scemes. It is sown tat beamforming is optimal to tese scemes. Te proposed scemes ave also been extended to te general K-user scenario. Specifically, we ave sown tat te design problems of te ideal sceme and te PS sceme can be efficiently andled by te proposed SCA metod Algoritm. Te optimal transmit signal covariance matrices and optimal time fractions of te TDMA D sceme energy arvesting using deterministic signals can be obtained by solving a convex problem [i.e., ]. Te simulation results ave revealed interesting tradeoffs between and in te complex IFC. In particular, it as been observed tat strong cross-link cannel power is not detrimental under energy arvesting constraints; instead, te acievable sum rate can be improved wit stronger crosslink cannel powers. We ave also observed tat none of te considered scemes can always dominate anoter in terms of te sum rate performance. For te tree practical scemes, we ave observed tat wen N t K, and η and E are not large, te PS sceme performs better tan te TDMS and TDMA sceme on average; wen N t K, but η and E are large, te TDMS sceme in general performs best and can even outperform te ideal sceme P; wen N t < K and E is large, te TDMA sceme in general can yield te igest sum rate. Te current work may motivate several interesting directions for future researc. Firstly, it is easy to see tat, oter tan te considered K-user TDMS and TDMA scemes, tere exist oter possible ways to separating te and modes of te K receivers across te time. It would be interesting to see ow te corresponding design problems can be efficiently solved and teir performance compared to te scemes presented in tis paper. Secondly, since none of te considered scemes can always perform best, it is wort formulating a design formulation tat unifies all tese practical scemes. Tirdly, based on some insigts gained from te current work, it is wortwile to furter study te WIET problems for some more complex interference cannels, suc as te broadcast interference cannels [8] and te MIMO interference cannels [9]. A. Proof of Proposition APPENDIX We prove by contradiction tat TrS i = P i for i =,. Suppose tat TrS < P, ten tere exists some ɛ > and S = S + ɛĥ ĥ suc tat TrS = P, were ĥ Π Π. Note tat S, S is feasible to P. Moreover, since, we ave R S, S > R S, S and R S, S = R S, S, wic contradicts te optimality of S, S. ence, it must be tat TrS = P ; similarly, one can sow tat TrS = P. Next, we sow tat S and S lie in te range space of [ ] and [ ], respectively, i.e., Π i S i Π i = for i =,. One can see tat, for any S, ikπ i S Π i ik = iks ik, TrΠ i S Π i TrS, A. A. for i, k {, }, were te equality in A. olds because Π X X = X for all X C m n. Terefore, S, S is an optimal solution to problem P only if Π SΠ, Π SΠ is optimal to P. Now suppose tat S does not lie in te range space of, i.e., TrΠ SΠ >. Ten, TrΠ S Π =TrS TrΠ S Π <TrS P, wic implies tat Π SΠ is not optimal, and tereby S is not optimal to P. Analogously, one can sow tat S must lie in te range space of.

11 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Wat remains to prove 9 is to sow tat tere exists a pair of S, S tat are of rank one. It is not difficult to see tat P is equivalent to te following problems log + ii S i ii Γ ki + σ i iks i ik + Γ kk E k, S i Γ ki + ii S i ii E i, iks i ik Γ ik, TrS i P i, A.a A.b A.c A.d A.e were Γ ki = ki S k ki, i, k {, } and i k. Let us focus on te case of i =, k =, and rewrite A. as S S S E Γ, S Γ, S E Γ, TrS P. A.4a A.4b A.4c A.4d A.4e Suppose tat Γ = E Γ. Ten A.4b and A.4c merges to one equality constraint. In tat case, A.4 as only tree inequality constraints. According to [, Teorem.], problem A.4 ten as an optimal solution S suc tat ranks. On te oter and, if Γ > E Γ, ten one of te two constrains A.4b and A.4c must be inactive for S. Terefore, te effective number of inequalities in A.4 is again tree. It ten follows from [] tat ranks. Te above results imply tat optimal S is of te form S = a + b a + b, A.5 were a, b C. Since any pase rotation of a +b is invariant to S, we witout loss of generality can let a R. Analogously, for te case of i =, k =, one can sow tat A. as an optimal S = a +b a +b, were a R and b C. Te proof is tus complete. B. Proof of Proposition Firstly, note tat problem is equivalent to te main-fairness problem { min i= i S } i i i=, i S i i A.6a S,S E E TrS P, TrS P. A.6b ence, given optimal S and S, te optimal β of is given as in b: { β =min S + S, S + } S. E E A.7 Secondly, problem satisfies te Slater s condition, so one can solve by andling its Lagrange dual problem. Let µ and η be te Lagrange dual variables associated wit constraints b and c, respectively. Te dual problem of can be sown as TrS µ + η min S,S + TrS η + µ µ,η TrS P, TrS P, E µ E η =, { = min TrS } Ψ µ+trs Ψ µ S,S µ TrS P, TrS P, A.8 were Ψ µ = µ + µe E and Ψ µ = µe E + µ. It is not difficult to sow [6, Proposition.] tat S µ = P v µv µ, S µ = P v µv µ A.9 are optimal to te inner imization problem of A.8, were v i µ C Nt is a principal eigenvector of Ψ i µ, for i =,. As will be sown later, for i =,, under te assumption tat i and i are linearly independent but not ortogonal to eac oter, Ψ i µ as a unique imum eigenvalue for any µ. ence, te solutions in A.9 are unique. According to te duality teory [], if µ is dual optimal i.e., optimal to A.8, ten te unique S µ, S µ in A.9 and β in A.7 are optimal to problem. Te optimal µ can be obtained troug a bisection searc using te dual gradient, wic is given by g = S µ E E S µ E E S µ + S µ. Lastly, we sow tat if i and i are linearly independent and Ψ i µ as two equal eigenvalues, ten i and i must be ortogonal. First note tat RangeΨ i µ = Range[ i, i ] for linearly independent i and i. Secondly, note tat any principal eigenvector v of Ψ i µ belongs to RangeΨ i µ. If Ψ i µ as two equal eigenvalues te dimension of te principal eigenspace is two, ten te principal eigenspace is exactly Range[ i, i ]. ence, i = i / i, i = i / i, i = Π i i / Π i i and i = Π i i / Π i i are all principal eigenvectors of Ψ i µ. Let λ denote te principal eigenvalue of Ψ i µ, and now consider i =. We ave Ψ µ = µ + η = λ, A.a Ψ µ = µ + η = λ, A.b Ψ µ = η = λ, Ψ µ = µ = λ, were η = Eµ E. By A.c and A.d, we ave η = µ. Furter combining A.a wit A.c yields µ + η = η, A.c A.d µ + η = η, µ + η = η = µ + η. A.

12 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Since bot µ, η are nonnegative, and,, te equality in A. implies tat = and =, i.e., and are ortogonal to eac oter. owever, tis contradicts te assumption tat is not ortogonal to. ence, te principal eigenvector of Ψ µ is unique. Similarly, te principal eigenvector of Ψ µ can be sown unique. C. Proof of Proposition Problem 4 is a quasi-convex problem. We first apply te idea of te Carnes-Cooper transformation [] to recast problem 4 as a convex problem. To illustrate tis, consider te following convex semidefinite program SDP α log + X X, X, y A.a X + yσ =, A.b X + X ye /α, A.c TrX yp, TrX yp. A.d Note tat te optimal y of A. must be positive; oterwise we ave X = X = wic violates A.b. Moreover, consider te following correspondence: y = / S + σ >, X = ys, X = ys. A.a A.b Ten, one can sow tat S, S is feasible to 4 if and only if X, X, y is feasible to A.. Furtermore, te objective value acieved by S, S in 4 is te same as te objective value acieved by X, X, y in A.. Terefore, te two problems 4 and A. are equivalent, and one can obtain S, S of 4 by solving te convex problem A.. To sow ow problem A. can be efficiently solved, we rewrite A. as X,X,y X X + yσ, X + X y E α, TrX yp, TrX yp, A.4a A.4b A.4c A.4d were te inequality constraint A.4b olds wit equality at te optimum. Te variable y as a feasible region of y /σ. We assume tat a feasible y is given and investigate te associated optimal X and X of problem A.4, wic are denoted by X y and X y, respectively. One key observation is tat X y can be obtained by solving te following problem X y = arg X X A.5a X yσ, A.5b TrX yp. A.5c Following [6, Proposition.], problem A.5 as a closed-form solution as X y=v yv y, A.6 yp, if yp < yσ, yσ v y=, + yp yσ oterwise. A.7 Notice tat, if yp < yσ, ten X y < yσ, and tus X y, y won t be optimal to problem A.4 since A.4b sould old wit equality at te optimum. Terefore, we can focus on te case of P +σ y /σ. Let gy X y = v y. Given y /σ and X y, A.4 reduces to P +σ X y = arg X X X y E α gy, TrX yp. A.8a A.8b A.8c Again, using [6, Proposition.], problem A.8 as te optimal solution given by X y=v yv y, A.9 infeasible, if y E α gy > yp, yp, if y E α gy yp, v y= ye/α gy, oterwise. + yp ye/α gy Terefore, given a P y /σ +σ, one can efficiently obtain X y and X y by A.9 and A.6, respectively. Te optimal y of problem A.4 ten can be obtained by solving te following one-dimensional problem y = arg y X y P + σ A.a y σ. A.b Te function X y is in fact concave in y, and ence A. can be solved via bisection. To sow tis, note from A.4 tat X y = X,X X X + yσ, A.a A.b X + X y E α, A.c TrX yp, TrX yp. A.d Since problem A.4 is convex jointly in X, X, y, and X y is a point-wise imum of te jointly concave linear X over all X, X feasible to A.. By [], X y is concave in y. Te proof is tus complete.

13 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING D. Proof of Proposition 4 We sow tat Algoritm essentially belongs to te SUM metod in [7]. Note tat, at te optimum, te inequalities in 6b and 6c of problem 6 will old wit equality. Terefore, problem 6 can be equivalently expressed as {Si [n]} K k= = arg US,..., S K {ȳ i [n]} K i= A.a {S i } K i= 4b, 4c, A.b were U S,..., S K {ȳ i [n]} K i= K = w i log i= ik S i ik + σi ]. i= exp[ k i ki S k ki +σ i e ȳi[n] +ȳ i [n] By te fact of e yi eȳi[n] y i ȳ i [n] + y i e ey i e ȳ i [n] +ȳ i[n] e yi y i, we see tat exp k i ki S k ki + σi e ȳ i[n] + ȳ i [n] k i ki S k ki + σi, and tus US,..., S K {ȳ i [n]} K i= log ii + S i ii k i ki S k ki + σi US,..., S K, i.e., US,..., S K {ȳ i [n]} K i= is a universal lower bound of te original objective function US,..., S K. 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