Wireless Information and Energy Transfer in Multi-Antenna Interference Channel

Transcription

1 Wireless Information and Energy Transfer in Multi-Antenna Interference Channel Journal: Transactions on Signal Processing Manuscript ID: Draft Manuscript Type: Regular Paper Date Submitted by the Author: n/a Complete List of Authors: Shen, Chao; Beijing Jiaotong University, the State Key Laboratory of Rail Traffic Control and Safety Li, Wei-Chiang; National Tsing Hua University, Department of Electrical Engineering Chang, Tsung-Hui; National Taiwan University of Science and Technology, Department of Electronic and Computer Engineering; EDICS:. SPC-APPL Applications involving signal process. for communications < SPC SIGNAL PROCESSING FOR COMMUNICATIONS,. SPC-INTF Interference management techniques < SPC SIGNAL PROCESSING FOR COMMUNICATIONS,. SPC-CCMC Cooperative and coordinated multicell techniques < SPC SIGNAL PROCESSING FOR COMMUNICATIONS,. SAM-BEAM Beamforming < SAM SENSOR ARRAY AND MULTICHANNEL PROCESSING

2 Page of 0 0 Response to Reviewers Comments for the Manuscript Wireless Information and Energy Transfer in Multi-Antenna Interference Channel (T-SP-0-0) by Chao Shen, Wei-Chiang Li and Tsung-Hui Chang June, 0 We would like to express our sincere gratitude to the two anonymous reviewers for their time and constructive comments, and to the associate editor for handling our paper. All the major revisions are marked in blue color in the revised manuscript. Note that all the page numbers mentioned in the response refer to the one-column version of the revised manuscript. Response to AE s Comments AE.) Based on the enclosed set of reviews (**See note below about attachments), I have decided that the manuscript be REVISED AND RESUBMITTED (RQ). Although the reviewers had many positive comments on the submission, a number of matters must be addressed. See the attached reviewers comments, but please focus on, in particular, reviewing and including the prior literature and making the various sections consistent. Answer: We have carefully revised the manuscript by taking into account all the reviewers comments. Let us briefly summarize our major revisions below.. Following Reviewer s comment, the consistency of the manuscript has been improved by adding the -user power splitting (PS) scheme in Section IV-D (page of the revised manuscript) and the K-user TDMA scheme in Section V-B (pages and of the revised manuscript). Now the WIET schemes for the -user case and the K-user case are consistent. Moreover, we have removed the log-exp approximation method in the previous Section V, and replace it by presenting a new approximation method based on the the successive upper-bound minimization (SUM) method []. This method can handle not only the K-user ideal scheme, but also the K-user TDMA scheme and the PS scheme. Please see the blue context from page to page 0 of the revised manuscript.. The references suggested by Reviewer have been incorporated into the revised manuscript. Besides, some other recent and relevant references are also included. Due to these changes, the introduction section is also greatly revised; please see the blue context on pages, and in the revised manuscript.. Following both reviewers suggestions, we have completely revised the simulation section by presenting a whole new set of simulation results with practical channel model, system parameters and RF-DC conversion efficiency. Please see Section VI (from page to ) of the revised manuscript. Due to the new results, the abstract, introduction and conclusion sections are also revised accordingly. Our detailed response to the reviewers comments are given below.

3 Page of 0 0 Response to Reviewer s Comments.) This paper studies the transmit design for wireless information and energy transfer in the MISO interference channel. The ideal receivers are first considered, and some practical schemes are proposed, e.g., TDMS, TDMA and PS. For each scheme, the two-user case is studied where semi closed-form solution can be obtained. Then for the K-user interference channel, efficient algorithms are proposed. This is one of the initial works of wireless information and energy transfer in the interference channel. However, the reviewer has the following concerns. Answer: We appreciate very much for your detailed review and constructive comments. Our point-by-point response to your comments is given below..) Too many schemes are proposed: idea case, TDMS, TDMA, and PS, and for each scheme, the -user case and K-user case are studied respectively. This makes it difficult to follow what is the main focus and contribution of this paper. Maybe it is better to just focus on the practical schemes, and two schemes can be ignored from the practical point of view. One is Section IV-B because deterministic energy signals in Section IV-C must be better than Gaussian signals because the interference can be cancelled. If the former must be worse than the latter, why should we study it? (The authors do not study the Gaussian energy signal case in the K-user case.) One is the ideal case because in practice the receiver cannot do ID and EH at the same time. Maybe the authors want to use the ideal case to show that time sharing can perform better, which is the motivation of the TDMS and TDMA schemes, as stated at the end of Section III. However, the logic is not correct. On one hand, it is not surprising that time sharing achieves larger sum-rate in the interference channel with strong interfering links even if the energy harvesting constraints are not considered. On the other hand, the motivation of TDMS, TDMA and PS is that they can be implemented in practical systems. Even if their performance is worse than the ideal case, we need to study it rather than the ideal case, as []-[]. Answer: We mostly agree with your comments. However, we think that the TDMA scheme (in Section IV-B) and the ideal scheme (in Section III) are interesting from their own perspective and should be presented in the manuscript. Firstly, while, as you mentioned, the TDMA (D) scheme (which uses deterministic signals for energy transfer) cannot perform worse than the original TDMA scheme, one should note that the TDMA (D) scheme requires additional assumption that the receivers can accurately acquire the channel state information of other users (see bottom paragraph of page in the revised manuscript). Therefore, the two schemes actually belong to different scenarios with different system setups, depending on whether more resources for training were invested. Secondly, more than being a motivating example, we believe that our study of the ideal scheme (e.g., Property and Proposition ) provides some interesting insights. For example, the result of Proposition also infers that

4 Page of 0 0 beamforming is optimal to the PS scheme when K =. Besides, it is interesting from an optimization perspective that (P) preserves the rank-one solution structure even it has two additional EH constraints than problem (). However, we do agree with you that the paper may be difficult to follow due to many schemes presented. To alleviate this situation, we have followed your comment in R. to further include the -user PS scheme in Section IV-D (page of the revised manuscript) and the K- user TDMA scheme in Section V-B (pages, and of the revised manuscript) to make the presentation for the -user case and K-user case more consistent. Due to these changes, we have also revised the introduction; please see the blue context in the first paragraph of page..) Why do the authors first study the -user case, and then the K-user case? From the algorithm design point of view, there is no need to study the -user case. On one hand, any problem that cannot be solved for the K-user case also cannot be solved for the -user case, e.g., ideal case, TDMS, and PS. On the other hand, any problem that can be solved for the -user case can also be solved in the K-user case, e.g., TDMA with deterministic energy signals. So why not just propose optimal or suboptimal solutions to the K-user case, and the -user case is just a special case of the K-user scenario? The only difference is that for the problem that can be solved numerically in the K-user case, maybe a semi closed-form solution can be obtained in the -user case. However, these semi closed-form solutions do not provide much insight, and the optimal solutions still need to be obtained numerically, because they are not in closed-form. If the authors want to study the -user case first, and then the K-user case, then make sure they are consistent. For example, TDMA with Gaussian energy signals is studied in the -user case but not in the Kuser case, and the PS is studied in the K-user case but not in the -user case Answer: Since our work involves presenting different transmission schemes and their design algorithms, our thought is to have a simple system model (i.e., the two-user case) in the beginning so that readers can easily catch the ideas and features of various schemes. Then, for the K-user case, where the problem formulations are more complex, we can put more emphasis on the approximation methods. Following your suggestion, we have tried to improve the consistence by adding the -user PS scheme in Section IV-D and the K-user TDMA scheme (using Gaussian energy signals) in Section V-B. Moreover, we have revised almost the whole Section V so that the ideal scheme, TDMA scheme and the PS scheme are all handled following a common approximation technique (i.e., the successive upper bound minimization (SUM) method []). Since this is different from the methods presented in the previous manuscript, we have also updated all the simulation results..) In Appendix A, the proof of rank-one solution to (A.) is not so rigorous. Note that although

5 Page of 0 0 only one of (A.b) and (A.c) is active, it does not mean that if we totally throw away the inactive constraint, the obtained problem has the same solution as the original problem. For example, if (A.b) is not active, we cannot say that we can obtain the solution to this problem only by maximizing (A.a) subject to (A.c)-(A.e), which has a rank-one optimal solution. However, the results in [] can still be applied to show the existence of a rank-one solution to (A.) in the following way. Given any feasible value of t, we use the constraint h H S h = t to replace constraints (A.b) and (A.c). Then the new problem only has three constraints and thus has a rank-one solution. Then we search t over [E γ, γ ], there must exist a t such that the new problem has the same solution as (A.). I think this is the rigorous way. Answer: Thanks for your valuable comment on the proof of Proposition. Inspired by your comment, we have found that the proof of rank-one solution of problem (A.) may be made simpler. Specifically, since constraint (A.e) specifies the lower bound for the objective value, if one drops (A.e), the solution of the relaxed problem must satisfy (A.e), implying that the relaxed problem has the same optimal solution set as (A.). Since the relaxed problem admits a rank-one solution according to [, Theorem.], (A.) has a rank-one solution as well. Please see the blue context on page and page. -) How do the authors obtain the curve of the ideal scheme in Fig.? Proposition only reduces the search dimension to two real numbers of a, a, and two complex numbers of b and b. The complexity to do exhaustive search over a i s and b i s is too high. So I do not understand how problem (P ) is solved exhaustively. Answer: We actually used an exhaustive search method presented in [, ]. Let us briefly explain this method here. For every Pareto optimal transmit covariance matrices (Ŝ, Ŝ), i.e., (R (Ŝ, Ŝ), R (Ŝ, Ŝ)) lies on the Pareto boundary of the achievable rate region of problem (P), the Pareto optimal rate tuple (R (Ŝ, Ŝ), R (Ŝ, Ŝ)) can also be achieved by the optimal solutions of the following two convex problems and S = arg max S 0 hh S h s.t. h H S h = ˆΓ h Ŝ h, Tr(S ) P, S = arg max S 0 hh S h s.t. h H S h = ˆΓ h Ŝ h, Tr(S ) P. (a) (b) (c) (a) (b) (c)

6 Page of 0 0 This can be shown as follows. Since Ŝ and Ŝ are feasible to () and (), respectively, we have h H S h h H Ŝh and h H S h h H Ŝh. () By (b), (b) and (), we obtain that (S, S ) is feasible to problem (P) and that R (S, S ) R (Ŝ, Ŝ) and R (S, S ) R (Ŝ, Ŝ). () The inequalities in () must hold with equality due to the Pareto optimality of (Ŝ, Ŝ), so (S, S ) achieves the Pareto optimal rate tuple. Therefore, we can obtain all Pareto optimal rate tuples by sampling the interference powers ˆΓ ki [0, P k h ki ], i, k {, }, k i, solving problems () and () for each sample, and then computing the achieved rate tuples of all combinations of the obtained transmit covariance matrices. Some tuples generated by the above procedure are not Pareto optimal or not feasible to (P) due to the energy-harvesting constraints of (P), and they need to be abandoned properly. Finally, the curve in Fig. (a) can be obtained by picking up the rate tuples that achieve the highest weighted sum rate for each E = E = E. On the bottom paragraph of page, we have mentioned that Fig. are obtained using the search method in [, ]. -) As to Proposition, the reviewer has the following concerns. First, it is stated that y can be obtained by bisection method. What is the gradient to update y? Otherwise, the algorithm is not completed. Second, why constraint (A.b) must be tight at the optimum? Last, the solution to (A. ) is obtained by Theorem. in [], rather than Proposition. in []. Answer: Since the subproblem of y (i.e., problem () on page ) is a convex problem, the optimal y can also be obtained by using the golden search method [] which does not need to evaluate the subgradient of h H v (y) with respect to y. Please see the revised statement of Proposition on the page. We have revised the proof of Proposition. Specifically, in the revised version, we no longer relax problem (A.). The previous wrong citation of [] (reference [] in the previous manuscript) is also corrected. Please see the blue context on pages and of the revised manuscript. -) For the K-user case, for the ideal case and the PS scheme, since the problem is not convex, Algorithm is used. However, this algorithm is just a repeat of []. For example, for the ideal receiver case, it can be seen that () is just the traditional weighted sum-rate maximization problem with energy harvesting constraints. However, the energy harvesting constraints are convex, so any algorithm designed to the traditional weighted sum-rate maximization problem can be used here. In other words, the newly energy harvesting constraints do not change the problem fundamentally. So the results do not seem interesting for the K-user case. The authors need to clarify the difference to [] from technical point of view.

7 Page of 0 0 Answer: We agree with your comment. In retrospect, we found that the log-exponential technique in [] (removed from the revised manuscript) may not be the most suitable approach to our problem () as it introduces K slack variables, i.e., x i, y i, i =,..., K. To fix this issues and also to have a unified approach that can be used not only for the K-user ideal scheme, but also for the (newly added) K-user TDMA scheme (using Gaussian energy signal) and the PS scheme, we have proposed to handle () based on the successive upper-bound minimization (SUM) method []. The same approach is also extended to approximate the design problems of the K-user TDMA scheme and the PS scheme, respectively. Please see the blue context from page to page 0 of the revised manuscript. We also agree with you that some of the existing methods that can handle the traditional weighted sum-rate maximization problem can also be used for problem (). To explicitly mention this, we have revised the paragraph below () on page and the paragraph below () on page. -) The setup for the simulation is not that practical. For example, the channels are not based on the pass loss model, and the receiver noise is so large. It is interesting to know under a practical setup of interference channel, how much energy can be harvested and how much rate can be achieved. So it is better to do the simulation under practical setups. Answer: Following your suggestion, we have set up a new simulation scenario considering practical channel effects such as large scale fading and small scale fading [0] as well as practical RF-DC conversion efficiency coefficient []. As a result, the simulation section has been completely revised with new simulation results and discussions. In particular, we have consider a network topology as depicted in Fig., where the kth transmitter and receiver are placed in polar coordinates ( D, π(k ) ) K with D = DT and D R meters, respectively. Hence, RX RX TX RX TX TX d TX D R RX TX D RX T TX RX Figure. Simulation setting with K = user pairs. the parameter D T can be used to control the strength of the direct-link and the cross-link channel powers. Based on this setting, we have examined the sum rate performance of all

8 Page of 0 0 the WIET schemes versus D T and energy harvesting requirement, for different numbers of antennas and numbers of users. Please see Section VI (from page to ) of the revised manuscript. -) It can be seen from Fig. (b) that in the case of K = and N =, strong interference can improve the sum rate. However, there is a trade-off. Strong interference is beneficial for energy transfer, but not so good for information transfer. In this case, since many antennas are equipped at each transmitter, and the number of users is, the interference can be handled, and the positive role of interference plays the dominant role. How about the case when there are a lot of users, but the number of transmit antenna is small? This is the most practical system. Does the sum-rate still increase with eta? A more general analysis of the role of interference needs to be presented. Answer: As mentioned in the previous comment, we have conducted new simulations. Following your suggestion, the new simulation section presents the performance of all schemes under consideration for various numbers of antennas and users. Specifically, the comparison result for N t = and K = is shown in Fig. (page ) and comparison results for N t = and K {,, } are shown in Fig. (page ). Discussions for these new results are also presented. Please see Section VI (from page to ) of the revised manuscript. -) For Example, what is the insight? For the most general interference channel, at which case, which scheme is best? What is the reason? This is a key question of this paper needs to be analyzed. Otherwise, we do not know in a practical system consisting of many users, which scheme should we select. Answer: As mentioned in the previous two comments, the simulation section has been fully updated. According to the new simulation results, we observed that the PS scheme in general performs best when the direct-link channel power is relatively stronger than the cross-link channel power and when N t K (i.e., interference can be well controlled by the spatial degree of freedom). When the cross-link channel power is relatively larger than the directlink channel power and N t K, the TDMS scheme in general performs best and better than the PS scheme. If the interference is overwhelming and the EH requirement is also large, then the TDMA (D) scheme may outperform the TDMS scheme. We have also revised the conclusion section to summarize these observations. -) A typo: In the single column, on Page, the fifth line of Example, there are two the. Answer: Fixed. We thank you again for your valuable comments.

9 Page of 0 0 Response to Reviewer s Comments.) This paper considers the joint information and energy transfer problem in a MISO interference channel. Specifically, the authors consider a two-user interference channel where the two transmitters wish to send information to their respective receivers, and the same time, the two receivers wish to harvest a pre-specified amount of energy from the incoming signals. The transmitters have multiple antennas while the receivers have a single antenna each. The paper first treats the two-user case, then generalizes the problem to the K-user case. The paper makes the interesting observation that, the signals going from the first transmitter to the second receiver may be interpreted as interference from the point of view of information transfer, while they may be interpreted as incoming energy from the point of view of wireless energy transfer. This creates an interesting trade-off for transmitted signals. The paper expresses the achievable schemes based on single-user achievable rates, where transmitters send Gaussian signals and the interference is treated as Gaussian noise. In this case, the achievable rates take the form of log of SINRs (signal-to-interference-plus-noiseratios). Both the rates and received energies are expressed in terms of the covariance matrices of the transmitters. The rest of the paper is based on optimization of these expressions and finding the optimal transmit covariance matrices. The paper also gives simpler time-sharing based solutions. The paper is generally well written, coherent and easy to follow. The paper adds to the recent literature on wireless energy and information transfer by focusing on the encountered optimization problems in a MISO interference channel setting. The main contribution of the paper is in terms of the optimization of the covariance matrices. Answer: We appreciate very much for your positive and constructive comments. Our pointby-point response to your comments is given below..) Please give a more thorough literature survey in the introduction section. There are several relevant papers missing. For instance, the following two papers considered information and energy transfer in relay, two-way and multiple-access channels by power splitting (i.e., orthogonal transmission) between information and energy transfer. These two papers must be included in the discussion in the second paragraph of the introduction section (top of page ). Please also provide a short discussion. B. Gurakan, O. Ozel, J. Yang and S. Ulukus, Energy Cooperation in Energy Harvesting Wireless Communications, IEEE International Symposium on Information Theory, Cambridge, MA, July 0. B. Gurakan, O. Ozel, J. Yang and S. Ulukus, Two-Way and Multiple-Access Energy Harvesting Systems with Energy Cooperation, th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, November 0. In addition, there are some other MIMO energy and information transfer papers such as the following one that the authors should refer to.

10 Page of 0 0 J. Park and B. Clerckx, Joint wireless information and energy transfer in a two-user mimo interference channel, IEEE Trans. Wireless Commun., vol., no., pp. V, Aug. 0. Answer: Thank you for pointing out theses references. We have cited them properly and revised the introduction section accordingly. Please see the blue context from page and page in the revised manuscript..) Please explain how the rate regions achieved, such as the ones shown in Figure, can be non-convex. Rate regions can be convexified by time-sharing. Answer: Mathematically, the achievable sum rate-versus-eh requirement E = E = E region in Fig. (a) and the achievable rate region R -versus-r in Fig. can be expressed as R R +R vse { [R + R, E] T R i R i (S, S ), E i = E, S i 0, i =,, and (b)-(e) }, R R vsr (E, E ) { [R, R ] T R i R i (S, S ), S i 0, i =,, and (b)-(e) }, respectively. The two regions are not guaranteed to be convex since R ( ) and R ( ) (see () on page of the revised manuscript) are not jointly concave in (S, S ). Yes, the non-convex regions may be convexified by considering time sharing. However, finding two feasible transmission schemes and their time sharing to optimize the system sum rate would be a too difficult design problem to tackle. Therefore, instead of tackling the general time-sharing transmitter design problem, we propose two simple but practical time sharing schemes, i.e., TDMS and TDMA, which exploit the dual behavior of the interference powers in terms of information detection and energy harvesting to improve the weighted sum rate performance. The simulation results in Section V show that these two schemes indeed provide nontrivial performance gains in some interference dominated scenarios..) Please comment on the practicality of wireless energy transfer in the introduction. What are some practical γ and values in practice. In addition, I think it is a better idea not to take these variables as unity as you do right after (). It is a better idea to use some realistic values for them. Answer: In the revised manuscript, we have added one more reference [] to mention that RF based energy harvester has been prototyped and tested in the great London area. In practice, the energy conversion efficiency γ depends on the input power of the RF front-end and the operating frequency band. It can be up to according to the state-of-the-art circuit design, see the references [R] (i.e., the reference []) and [R] below [R] C. Valenta and G. Durgin, Harvesting Wireless Power: Survey of Energy-Harvester Conversion Efficiency in Far-Field, Wireless Power Transfer Systems, IEEE Microwave Magazine, vol., no., pp.-, June 0

11 Page of 0 0 [R] S. Ladan, N. Ghassemi, A. Ghiotto, and Ke Wu, Highly Efficient Compact Rectenna for Wireless Energy Harvesting Application, IEEE Microwave Magazine, vol., no., pp.-, Jan.-Feb. 0 Some technical specifications are available at the data sheets of the chips developed by the Powercast Corporation ( In the revised simulation section, we have set γ = 0.. In fact, following your and the other reviewer s comments, we have completely revised the simulation section and presented a whole new set of simulation results with practical channel models. Please see Section VI (from page to ) of the revised manuscript. The time duration for energy harvesting highly depends on the applications of the receiver. For some current or potential sensor application scenarios, can be several seconds. In the manuscript, however, we prefer to simply set γ = and = for ease of presentation. We have revised the paragraph below () on page to explicitly mention that the notations are simplified but a practical value of γ will be considered in the simulation section..) Please choose practically relevant values for the parameters in the simulation results section. For instance, is it practical to chose σ to be 0.? Answer: As mentioned, following your and the other reviewer s comments, we have completely revised the simulations and practical parameters are used. Specifically, the noise power is set to be -0 dbm, which can be viewed as the noise power with noise power density - dbm/hz and MHz bandwidth. Please see the first paragraph of Section VI on page. We thank you again for your valuable comments.

12 Page of 0 0 Wireless Information and Energy Transfer in Multi-Antenna Interference Channel Chao Shen, Wei-Chiang Li and Tsung-Hui Chang Abstract This paper considers the transmitter design for wireless information and energy transfer (WIET) in a multiple-input single-output (MISO) interference channel (IFC). The design problem is to maximize the system throughput (i.e., the weighted sum rate) subject to individual energy harvesting constraints and power constraints. It is observed that the ideal scheme, where the receivers simultaneously perform information detection (ID) and energy harvesting (EH) from the received signal, may not always achieve the best tradeoff between information transfer and energy harvesting, but simple practical schemes based on time splitting may perform better. We therefore propose two practical time splitting schemes, namely the time division mode switching (TDMS) and time division multiple access (TDMA), in addition to the existing power splitting (PS) scheme which separates the received signal into two parts for ID and EH, respectively. In the two-user scenario, we show that beamforming is optimal to all the schemes. Moreover, the design problems associated with the TDMS and TDMA schemes admit semi-analytical solutions. In the general K-user scenario, a successive convex approximation method is proposed to handle the WIET problems associated with the ideal scheme, the PS scheme and the TDMA scheme, which are known NP-hard in general. Simulation results show that none of the schemes under consideration can always dominate another in terms of the sum rate performance. Specifically, it is observed that stronger crosslink channel power improves the achievable sum rate of time splitting schemes under EH constraints, but degrades the sum rate performance of the ideal scheme and PS scheme. As a result, time splitting schemes can outperform the ideal scheme and the PS scheme in interference dominated scenarios. Index terms wireless energy transfer, energy harvesting, interference channel, beamforming, convex optimization EDICS: SPC-APPL, SPC-INTF, SPC-CCMC, SAM-BEAM The work of Chao Shen is supported by the Opening Project of The State Key Laboratory of Integrated Services Networks, Xidian University (Grant No. ISN-0), the China Postdoctoral Science Foundation (Grant No. 0M), the Fundamental Research Funds for the Central Universities (Grant No. 0JBM), the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (No. RCS0ZQ00), the Natural Science Foundation of China (Grant No. U), and the Key Grant Project of Chinese Ministry of Education (No. 0). The work of Tsung-Hui Chang is supported by National Science Council, Taiwan (R.O.C.), by Grant NSC --E-0-00-MY. Part of this work was presented in IEEE GLOBECOM 0 []. Chao Shen is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China and the State Key Laboratory of Integrated Services Networks, Xidian University, Xi an, China. chaoshen@ieee.org. Wei-Chiang Li is with Institute of Communications Engineering, National Tsing Hua University, Hsinchu 0, Taiwan, (R.O.C.). weichiangli@gmail.com. Tsung-Hui Chang is the corresponding author. Address: Department of Electronic and Computer Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, (R.O.C.). tsunghui.chang@ieee.org. June, 0 DRAFT

13 Page of 0 0 I. INTRODUCTION Recently, scavenging energy from the environment has been considered as a potential approach to prolonging the lifetime of battery-powered sensor networks and to implementing self-sustained communication systems. For example, the base stations may be powered by wind mills or solar photovoltaic (PV) arrays, and can harvest energy for providing services to the mobile users. This idea has motivated considerable research endeavors in the past few years, investigating wireless systems with energy-harvesting transmitters; see, e.g., [ ]. In these works, optimal transmission strategies under energy-harvesting constraints are studied from single-input single-output (SISO) channels to complex interference channels (IFCs). In contrast to the base stations, it may be difficult for the mobile devices and sensor nodes to harvest energy from the sun and wind effectively. One possible solution to this issue is wireless energy transfer (WET), that is, the power-connected transmitters transfer energy wirelessly to charge the mobile devices. A successful application of WET is the radio frequency identification (RFID) system where the receiver wirelessly charges energy from the transmitter (through induction coupling) and use the energy to communicate with the transmitter. The works in [, ] showed that, using coupled magnetic resonances, energy can be wirelessly transferred for two meters with over 0% energy conversion efficiency. WET can also be achieved via the RF electromagnetic signals; for example, see [ ] for recent developments of RF-based energy harvesting (EH) circuits and also [] for prototype EH receivers which were tested for harvesting energy from GSM and G signals in the great London area. Compared to the techniques based on induction and magnetic resonance coupling, RF signals can achieve long-distance WET; however, the energy conversion efficiency is in general low. This calls for advanced signal processing techniques, such as beamforming, to improve the energy conversion efficiency. Since the RF signals can carry both information and energy, in recent years, it has been of great interest to study wireless communication systems where the receivers can not only decode information data but also harvest energy from the received RF signals, i.e., wireless information and energy transfer (WIET) systems [ 0]. Specifically, in [], the optimal tradeoff between information capacity and energy transfer of the WIET system was studied for a SISO flat fading channel. In [], the optimal power allocation strategy for a SISO frequency-selective fading channel was derived under a receiver EH constraint. The work in [] studied the tradeoff between achievable information rate and harvested energy for the multiple access channel (MAC) and a two-hop relay network with an energy harvesting relay; while the works [, ] respectively derived optimal power allocation for an one-way relay channel, two-way channel and the MAC, assuming that transmitters can share energy by wireless energy transfer. DRAFT June, 0

14 Page of 0 0 It was shown that there exist nontrivial tradeoffs between information transfer and energy harvesting in general, and thus pertinent transmitter optimization and resource allocation (power/time/space/frequency) are needed. The works in [ ] have assumed the ideal receivers which can decode information bits and harvest energy from the received RF signals simultaneously. Unfortunately, current circuit technologies cannot achieve this yet. In view of this, practical WIET schemes are proposed. In particular, Zhou et al. proposed in [] a dynamic power splitting (PS) scheme for a SISO flat fading channel, wherein the received RF signal is either used for information detection (ID) and EH, or is split into two parts, one for ID and the other for EH. Considering a multiple-input multiple-output (MIMO) flat-fading channel, in addition to the PS scheme, the authors in [] further proposed a time switching scheme where the receiver performs ID in one time slot while EH in the other time slot. In [0], the dynamic PS scheme was extended to a multi-user multiple-input single-output (MISO) broadcast channel where one multiantenna transmitter employs transmit beamforming for serving multiple receivers for ID and EH. The works [, ] respectively considered the WIET problem in a two-user MIMO IFC and general K-user MIMO IFC under the assumption that the receivers perform either EH or ID. In this paper, we consider a K-user MISO interference channel and study the optimal transmission and resource allocation strategies for WIET. It is interesting to note that, different from the conventional IFCs without energy harvesting, the cross-link signals in the considered scenario can degrade the information sum rate on one hand, but, at the same time, boost energy harvesting of the receivers on the other hand. Different from [, ], our interests not only lie in the optimal transmission structure of the transmitters but also in the optimal time switching, time sharing and power splitting strategies of practical receivers. To this end, we first consider an IFC with ideal receivers, and formulate the WIET design problem as a weighted sum rate maximization problem subject to individual EH constraints and transmit power constraints. Interestingly, we will see that the ideal scheme with ideal receivers may not always perform best in the complex interference environment, but simple practical schemes based on time splitting and time sharing may instead yield better sum rate performance. This is in sharp contrast to the singletransmitter scenarios studied in [ 0] where time splitting schemes usually exhibit poorer performance. This intriguing observation motivates us to propose two practical WIET schemes for the MISO IFC, namely, the time division mode switching (TDMS) scheme and the time division multiple access (TDMA) scheme. In the TDMS scheme, the transmission time is divided into two time slots. All receivers perform As will be shown in Section IV-A, the proposed TDMA scheme is similar to but not completely the same as the TDMA scheme in conventional IFCs without energy harvesting. June, 0 DRAFT

15 Page of 0 0 EH in the first time slot and subsequently perform ID in the second time slot. The TDMA scheme divides the transmission time into K time slots, and in each time slot, one receiver performs ID while the others perform EH. In addition to the two time splitting based schemes, we also study the PS scheme [, 0] for WIET in the K-user MISO IFC. We will show how the design problems associated with the three practical schemes can be efficiently handled. Specifically, for the two-user scenario, we show that the design problems associated with the TDMS and TDMA schemes admit semi-analytical solutions. Since for the general K-user scenario the TDMA scheme and the PS scheme are nonconvex problems (NP-hard in general []), we further propose efficient approximation methods based on the successive upper-bound minimization (SUM) method []. The presented simulation results will show that stronger cross-link channel power improves the information sum rate of TDMS and TDMA schemes under energy harvesting constraints, but degrades the sum rate performance of the ideal and PS schemes. As a result, if the crosslink channel powers are not strong or the energy harvesting constraints are not stringent, the PS scheme can outperform TDMS and TDMA schemes; otherwise, the TDMS and TDMA schemes can perform better. The rest of this paper is organized as follows. In Section II, the signal model of the MISO interference channel is presented. Starting with the two-user scenario, in Section III, the optimal WIET transmission strategy for ideal receivers is analyzed. The result motivates the developments of the practical TDMS, TDMA and PS schemes, which are presented in Section IV. Section V extends the study to the general K-user scenario. Simulation results are presented in Section VI. The conclusions are given in Section VII. Notations: Column vectors and matrices are written in boldfaced lowercase and uppercase letters, e.g.,a and A. The superscripts ( ) T, ( ) H and ( ) represent the transpose, (Hermitian) conjugate transpose and matrix inverse, respectively. rank(a) and Tr(A) represent the rank and trace of matrix A, respectively. A 0 ( 0) means that matrix A is positive semidefinite (positive definite). a denotes the Euclidean norm of vector a. The orthogonal projection onto the column space of a tall matrix A is denoted by Π A A(A H A) A H. Moreover, the projection onto the orthogonal complement of the column space of A is denoted by Π A I Π A where I is the identity matrix with proper dimension. II. SIGNAL MODEL AND PROBLEM STATEMENT We consider a multi-user interference channel withk pairs of transmitters and receivers communicating over a common frequency band. Each of the transmitters is equipped with N t antennae, while each of the receivers has single antenna. Let x i C Nt be the signal vector transmitted by transmitter i, and DRAFT June, 0

16 Page of 0 0 h ik C Nt be the channel vector from transmitter i to receiver k, for all i,k {,,...,K}. The received signal at receiver i is given by y i = h H iix i + k=,k i h H ki x k +n i, i =,...,K, () where n i CN(0,σi ) is the additive Gaussian noise at receiver i. Unlike the conventional MISO IFC [] where the receivers focus only on extracting information, we consider in this paper that the receivers can also scavenge energy from the received signals [,, ], i.e., energy harvesting. Therefore, in addition to information, the transmitters can also wirelessly transfer energy to the receivers. We call the two operation modes the information detection (ID) mode and the energy harvesting (EH) mode, respectively. Assume that x i contains the information intended for receiver i which is Gaussian encoded with zero mean and covariance matrix S i 0, i.e., x i CN(0,S i ) for i=,...,k. Moreover, assume that each receiver i decodes x i by single user detection in the ID mode. Then the achievable information rate of receiver i is given by h R i (S,...,S K ) = log (+ H ii S ih ii k i hh ki S kh ki +σi ), () for i =,...,K. Alternatively, the receiver may choose to harvest energy from the received signal. It can be assumed that the total harvested RF-band energy during a transmission interval is proportional to the power of the received baseband signal []. Specifically, for receiver i, the harvested energy, denoted by E i, can be expressed as E i = γ h H ki S kh ki, i =,...,K, () k= where γ is a constant accounting for the energy conversion loss in the transducer []. Suppose that the receivers desire to harvest certain amounts of energy. We are interested in investigating the optimal transmission strategies of S i, i =,...,K, so that the information throughput of the K- user IFCs can be maximized while the energy harvesting requirements of the receivers are satisfied at the same time. In subsequent sections, we will first study an ideal scenario where the receivers can simultaneously operate in the ID mode and EH mode. One should note that current energy harvesting receivers are not yet able to realize such ideal receiver []. However, the study can provide insights for understanding the tradeoff between ID and EH as well as for the design of practical schemes. In particular, in Section IV and Section V, we further investigate some practical schemes where the receivers operate either in the ID mode or EH mode at any time instant. In order to gain more insights, we will June, 0 DRAFT

17 Page of 0 0 begin our investigation with the two-user scenario (K = ), and later extend the studies to the general K-user case (in Section V). III. OPTIMAL WIET DESIGN FOR IDEAL SCHEME Let us assume thatk= and consider ideal receivers which can simultaneously decode the information bits and harvest the energy from the received signals. Suppose that the two receivers desire to harvest total amounts of energy E and E, respectively. We are interested in the following transmitter design problem for WIET: (P) max w R (S,S )+w R (S,S ) S 0,S 0 s.t. h H S h +h H S h E, h H S h +h H S h E, Tr(S ) P, Tr(S ) P, where w,w > 0 are positive weights, and P > 0 and P > 0 in (d) and (e) represent the individual power constraints. The constraints in (b) and (c) are the energy harvesting constraints where we have set γ= and = for notational simplicity. In Section VI, practical values of γ will be considered for performance evaluation. Note that, in the absence of (b) and (c), problem (P) reduces to the classical sum rate maximization problem in MISO IFC []: max w R (S,S )+w R (S,S ) S 0,S 0 s.t. Tr(S ) P, Tr(S ) P. Since the EH constraints (b) and (c) are convex constraints, the existing methods for problem () (e.g., [ ]) can be directly used for handling (). Here, we are interested in understanding the impacts of (b) and (c) on the sum rate performance and the optimal transmitter structure. It can be observed from () and () that the EH constraints (b) and (c) would trade the maximum achievable sum rate for energy harvesting; i.e., the maximum sum rate in (a) is in general no larger than that in (a). To see when this would happen, let ( S, S ) be an optimal solution to problem (). One can verify from the DRAFT June, 0 (a) (b) (c) (d) (e) (a) (b) (c)

18 Page of 0 0 rate function in () and problem () that ( S, S ) must satisfy hh S h {[ ] } E h H S Ω E = max h H S h, 0 E P h, () h E S 0,Tr(S ) P, h H Sh E hh S h {[ ] } E h H S Ω E = max h H S h, 0 E P h. () h E S 0,Tr(S ) P, h H S h E That is, the energies harvested at the two receivers due to ( S, S ) must lie in Ω +Ω. Thus, we have that h H S h +h H S h h H S h +h H S h min E +E = P h H h, (a) (E,E ) Ω, (E,E ) Ω min E +E = P h H h, (b) (E,E ) Ω, (E,E ) Ω where h ij Π h ij h ii Π h ij h ii. Equations in () imply that the two receivers can at lease harvest energies P h H h and P h H h, respectively. The minimum amounts of energies are achieved when E = P h H h, E = P h H h and E = E = 0; that is, when each of the transmitters only focus on transmitting signals to its own receiver, without allowing any leakage of energy to the other receiver. According to (), we have that Property The energy harvesting constraints (b) and (c) are inactive at the optimum if E P h H h and E P h H h ; hence, (P) reduces to the conventional MISO IFC problem () under this condition. However, when E > P h H h or E > P h H h, the maximum information throughput may have to be compromised with energy harvesting. Interestingly, the following proposition shows that the optimal transmit structure of (P) remains the same as that of problem () which does not consider the EH constraints. Proposition Assume that problem (P) is feasible, and that h h and h h without loss of generality. Then there exists a rank-one optimal solution, denoted by (S,S ), to problem (P) such that S = (a h +b h )(a h +b h ) H, S = (a h +b h )(a h +b h ) H, for some a i R, b i C, and Tr(S i ) = P i, for i =,. The proof is given in Appendix A. Proposition implies that beamforming is an optimal transmission strategy of (P). Moreover, the beamforming direction of transmitter i, that lies in the range space of June, 0 (a) (b) DRAFT

19 Page of 0 0 [h i,h i ] for i =,, is the same as the optimal beamforming direction of problem () in the conventional IFCs []. However, unlike problem (), the optimal coefficients a i,b i, i =,, for (P) have to account for both the needs of energy harvesting and information transfer. Remark It is important to remark that, while (P) is ideal in the sense that the receivers can simultaneously operate in the ID and EH modes, (P) does not necessarily perform best in terms of sum rate maximization. The reason is that the cross-link signal power h H ik S ih ik plays two completely opposite roles in the considered scenario It can boost the energy harvesting of receiver k on one hand, but also degrades the achievable information rate on the other hand. Therefore, when the cross-link channel power is strong (e.g., the interference dominated scenario) and when the energy harvesting constraints are not negligible (e.g., the conditions in Property do not hold), the transmitters have to compromise the achievable information rate for energy harvesting. Under such circumstances, it might be a wiser strategy to split the ID and EH modes in time. To further look into this aspect, we present in Fig. two simulation examples for a two-user scenario with EH constraints. The detailed setting of the simulations are presented in Section VI. Fig. a shows the achievable sum rate of (P) versus energy requirement E E = E for two randomly generated channel realizations. The achievable region is drawn by using an exhaustive search method from [, ]. ZSis SwA}R\<D 0 D T = O D T =.0 m Ideals cheme Time sharing scheme oswgdsr I)Sd (a) Sum rate vs. EH requirement E. R (bps) Ideal scheme Time sharing scheme 0 0 R (bps), seed=, E=[0., 0.] mw, D T =.0m, D R =.0m,N t =.0 (b) Achievable rate region (R,R ) for E = 0. mw, E = 0. mw, D T = m. Fig. : Motivating simulation examples for the -user scenario (N t =, K = ). DRAFT June, 0

20 Page of 0 0 In Fig. a, the parameter D T is used to control the direct-link and cross-link channel powers, as shown in Fig. in Section VI. Simply speaking, when D T increases, the direct-link channel power increases whereas the cross-link channel power decreases. One can observe from Fig. a that the rate-energy regions are not convex for these two randomly generated channel realizations. Moreover, for some values of E, the receivers may achieve a higher sum rate through time switching between the EH mode and ID mode, especially in the case D T = 0. Fig. b displays the rate region (R versus R ) of the two users. Analogously, we observe that time sharing for multiple access may achieve a higher sum rate. The two simulation results in Fig. imply that the ideal scheme (P) may not always achieve the best tradeoff between information transfer and energy harvesting, but, instead, time sharing for EH/ID mode switching or time sharing for multiple access may yield higher information sum rate. It is worthwhile to note that these time sharing schemes are practical as the receivers operate either in the EH mode or ID mode at each time instant. These motivate us to develop two practical time sharing schemes, namely, the time-division mode switching (TDMS) scheme and the time-division multiple access (TDMA) scheme, in the next section. IV. PRACTICAL WIET SCHEMES AND OPTIMAL TRANSMISSION STRATEGIES A. Time Division Mode Switching (TDMS) Scheme In the first practical scheme, we divide the transmission interval into two time slots. In one time slot, both receivers operate in the EH mode, whereas, in the other time slot, both receivers switch to the ID mode. The two receivers thus coherently switch between the EH and ID modes, i.e., mode switching. Suppose that α fraction of the time is for EH mode and ( α) fraction of the time is for ID mode. The TDMS scheme is described as follows. Time slot (EH mode): The two receivers focus on harvesting the required energy E and E in α fraction of the time, i.e., α (h H S h +h H S h ) E, α (h H S h +h H S h ) E. (a) (b) Time slot (ID mode): Both the two receivers operate in the ID mode and maximize the information June, 0 throughput in the remaining fraction of the time, i.e., max ( α)(w R (S,S )+w R (S,S )) S 0,S 0 s.t. Tr(S ) P, Tr(S ) P. (a) (b) DRAFT

21 Page 0 of 0 0 Problem () in the ID mode is the classical sum rate maximization problem in the MISO IFC [see ()], which can be efficiently handled by existing methods in [,, ]. Note that it has been shown in [, ] that beamforming is an optimal transmission scheme for problem (). We now focus on the EH mode in time slot. Since time slot does not contribute to the information throughput, it is desirable to spend as least as possible time for the EH mode, i.e., to use a minimal time fraction α to fulfill the energy harvesting task. Mathematically, we can write it as the following optimization problem max β β R,S 0,S 0 s.t. h H S h +h H S h βe, h H S h +h H S h βe, Tr(S ) P, Tr(S ) P, (a) (b) (c) (d) where β /α. Note that if the optimal β of () is less than one (i.e., optimal α > ), then it implies that the energy harvesting requirements () cannot be satisfied even if the receivers dedicate themselves to harvesting energy throughout the whole transmission interval. In that case, we declare that the TDMS scheme is not feasible. While problem () is a convex semidefinite program (SDP), which can be solved by the off-the-shelf solvers, we show that () actually admits a semi-analytical solution. Proposition Assume that h i and h i are linearly independent but not orthogonal to each other, for i =,. The optimal solution to problem () is given by S (µ ) = P v (µ )v H (µ ), S (µ ) = P v (µ )v H (µ ), { h β(µ H ) = min S (µ )h +h H S (µ )h, E h H S (µ )h +h H S (µ )h E (a) }, (b) where µ 0 is the optimal dual variable associated with constraint (b), and v i (µ ) is the principal eigenvector of µ h i h H i + ( µ E ) E h i h H i for i =,. Moreover, µ can be efficiently obtained using a simple bisection search. The proof of Proposition is given in Appendix B. The assumptions on h i and h i, for i =,, hold with probability one for random (continuous) fading channels. Note that Proposition also implies that beamforming is optimal for the EH mode of the TDMS scheme. DRAFT June, 0

22 Page of 0 0 B. TDMA Scheme Unlike TDMS scheme, in each time slot of TDMA scheme, one receiver operates in the ID mode and the other receiver operates in the EH mode. Assume that the time fraction of the first time slot is α. Time slot : Receiver operates in the ID mode and receiver operates in the EH mode. The objective is to maximize the information rate of receiver and guarantee the energy harvesting requirement of receiver at the same time. The design problem is given by ( max αlog + hh S ) h S 0,S 0 h H S h +σ s.t. h H S h +h H S h E /α, Tr(S ) P, Tr(S ) P. Time slot : The operation modes of the two receivers are exchanged: ( max ( α)log + hh S ) h S 0,S 0 h H S h +σ s.t. h H S h +h H S h E /( α), Tr(S ) P, Tr(S ) P. (a) (b) (c) (a) (b) By intuition, this TDMA scheme would be of interest when the two receivers have asymmetric energy harvesting requirements and asymmetric cross-link channel powers. Moreover, like the conventional interference channel without energy harvesting, the TDMA scheme may outperform the spectrum sharing schemes in interference dominated scenarios. It is not difficult to show that: Lemma The TDMA scheme is feasible if and only if E (c) P h +P h + E P h. () +P h Proof: The TDMA scheme is feasible if and only if both () and () are feasible. Problem () is feasible if and only if there exists some α [0,] such that max h H E α S h +h H S h S 0,S 0 Tr(S ) P,Tr(S ) P = α (P h +P h ), () where the equality is obtained by applying the result in [, Proposition.]. Similarly, one can show that () is feasible if and only if June, 0 E ( α) (P h +P h ). () DRAFT

23 Page of 0 0 Combining () and () gives rise to (). Conversely, given (), let α = E P h +P h, and thus E P h +P h α, which are () and (), respectively. Hence the TDMA scheme is feasible if and only if () is true. According to () and (), a feasible time fraction α must lie in the interval E P h +P h α E P h +P h. () Interestingly, given a feasible α, both problems () and () can be efficiently solved (semi-analytically). Since () and () are similar to each other, we take () as the example. Proposition Let the time fraction α satisfy (). Then, an optimal solution (S, S ) to problem () is given by where S = v (y )v H (y )/y, S = v (y )v H (y )/y, (0) yp h, if g(y)/y E /α P h H h, v (y) = ye /α g(y) h H h h H h h + yp ye /α g(y) h h, otherwise, v (y) = yσ h H h ( h H h ) h + yp yσ h h, in which h ij = hij h ij, h ij Π h ij h ii Π h ij h ii for i =,, and g(y) = hh v (y). The optimal y can be obtained by solving the convex problem y = argmax h H v (y) s.t. y T max (α)+σ y T min (α)+σ, () through the golden section search [], where P h E if α T max (α) = P h +P h H h, (φ (α)+φ (α)) otherwise, E 0 if α T min (α) = P h +P Π h h h, (φ (α) φ (α)) otherwise, E and φ (α) = α P h hh h h, φ (α) = P h +P h E Π h h. α h The proof is presented in Appendix C. We see from (0) that beamforming is also optimal to the TDMA scheme. By Proposition, given a feasible time fraction α, one can efficiently solve problems DRAFT June, 0

24 Page of 0 0 () and () and thus evaluate the achievable sum rate of the two users. Then, the optimal time fraction α that maximizes the sum rate of the two users can be obtained by line search over the interval in (). C. TDMA via Deterministic Signal for Energy Harvesting It should be noticed that, while Gaussian signaling is optimal for information transfer, it may not be necessary for energy transfer. In particular, if one user operates in the EH mode, the transmitter may simply transmit some deterministic signals (e.g., training/pilot signals) known to both receivers. Consider the TDMA scheme in the previous subsection, and assume that, in the first time slot, transmitter operating in the EH mode transmits deterministic signals x which are known to receiver operating in the ID mode. Under such circumstances, receiver can actually remove h H x from the received signal before information detection, i.e., removing the cross-link interference. The design problem in the st time slot thereby reduces to max S 0,S 0 αlog ( ) +σ hh S h s.t. h H S h +h H S h E /α, Tr(S ) P, Tr(S ) P. (a) (b) (c) Problem () is easier to handle than its counterpart in (). Clearly, given α satisfying (), optimal S is given by S = P h hh, Therefore, () boils down to max S 0 hh S h which admits a closed-form solution for S s.t. h H S h E /α P h, Tr(S ) P, (a) (b) (c) according to [, Theorem.]. Analogously, the design problem for the second time slot can be simplified. In this paper, we refer to this scheme as the TDMA (D) scheme. Since the receivers are free from cross-link interference, it is anticipated that the TDMA (D) scheme performs no worse than the TDMA scheme. However, it should be noted that, in TDMA (D), the two receivers require perfect knowledge of the cross-link channels h and h, respectively; otherwise, the receivers may suffer performance degradation due to imperfect interference cancelation. In Section V-B, it will be further shown that it is possible to jointly find the optimal time fraction α and transmit signal covariance matrices S,S of the TDMA (D) scheme. June, 0 DRAFT

25 Page of 0 0 RX i D. Power Splitting Scheme CN(0; ~¾ i ) Power Splitter p ½i CN(0; ^¾ i ) p ½i ID EH Fig. : Diagram of the power splitting receiver for WIET. Other than the TDMS and TDMA schemes, another practical scheme, called power splitting (PS) [], splits the received signal into two parts for simultaneous EH and ID; see Fig.. Specifically, suppose that receiver i splits ρ i [0,] fraction of power for ID and ρ i fraction of power for EH. The associated WIET design problem is given by ρ h max w log (+ H S ) h S 0,S 0, ρ h H S h +ρ σ + 0 ρ ˆσ,ρ ρ h +w log (+ H S ) h ρ h H S h +ρ σ + ˆσ s.t. h H S h +h H S h E, ρ h H S h +h H S h E, ρ Tr(S ) P, Tr(S ) P, where σ i denotes the noise power at the RF end while ˆσ i (a) (b) (c) (d) denotes the processing noise power. Note that, in problem (), we not only optimize the signal covariance matrices {S,S }, but also the power splitting fractions {ρ,ρ } in the receivers. Since problem () has the same problem structure as (P) when {ρ,ρ } are given, one can infer from Proposition that transmit beamforming is an optimal solution structure to problem () as K =. In Section V-C, efficient approximation method for handling problem () and its extension to the general K-user scenario will be further studied. V. WIET DESIGN FOR K-USER MISO IFC In this section, we consider the WIET problem for the K-user MISO IFC scenario. In the first subsection, we begin with the ideal scheme. In the second subsection, we extend the TDMS and TDMA schemes to the K-user scenario. In the last subsection, we investigate the K-user PS scheme. DRAFT June, 0

26 Page of 0 0 A. K-User Ideal Scheme By the signal model in ()-() and (P) in (), the K-user WIET problem is formulated as ) h w i log (+ H ii S ih ii k i hh ki S kh ki +σi max S i 0 i=,...,k s.t. i= h H ki S kh ki E i, i =,...,K, k= Tr(S i ) P i, i =,...,K, (a) (b) where E i 0 is the energy requirement of user i, for i =,...,K. Since problem () is NP-hard in general [], our interest for the K-user WIET problem lies in efficient algorithms for finding an approximate solution. One should note that many of the existing approximation methods that can handle the conventional K-user IFC (without EH) can also be used for problem () as the EH constraints in (b) are convex and do not complicate the problem fundamentally. (c) Here, we propose an efficient algorithm based on successive upper-bound minimization (SUM) [], which is similar to the method used in [] for IFC without EH constraints. As will be seen shortly, this method not only can be used for problem () but also can be conveniently extended to handle the design problems of the K-user TDMA scheme and the PS scheme. To proceed, let us write problem () as [ ( K ) ( )] w i log h H ki S kh ki +σi log k ih H ki S kh ki +σi () max (S,...,S K) S i= k= where S = {S i 0,i =,...,K K k= hh ki S kh ki E i,tr(s i ) P i, i =,...,K}. The idea is to iteratively approximate () by considering a locally tight lower bound of the weighted sum rate. Specifically, suppose that we have (S [n ],...,S [n ] K ) at the (n )th iteration. Then, due to the concavity of the logarithm function, the second logarithmic term in () is upper bounded by its firstorder approximation, i.e., ( ) ( log k ih H ki S kh ki +σi log k i h H ki S[n ] k h ki +σi The bound above is tight when(s,...,s K ) = (S [n ] where τ [n ] i {S [n] i } K i= = argmax {S i} K i= S ) +,...,S [n ] K [ ( K w i log h H ki S kh ki +σi i= k= k i hh ki (S k S [n ] k )h ki ( () k i hh ki S[n ] k h ki +σi )ln. ). Thus, solving the following problem ) τ [n ] i h H ki S kh ki ], () ( k i hh ki S[n ] k h ki +σi )ln, is equivalent to maximizing a locally tight lower bound of problem () (up to a constant). Note that problem () is a convex SDP which can be solved efficiently by June, 0 k i DRAFT

27 Page of 0 0 Algorithm SUM algorithm for problem () : Find initial variables by solving the feasibility problem { [0]} K S i i= = find {S,...,S K }, s.t. h H ki S kh iki E i, Tr(S i ) P i, S i 0 i. k= If infeasible, then declare infeasibility of (); otherwise, set n = 0 and perform the following steps. : repeat : n := n+. : Solve problem () to obtain S [n],...,s[n] K. : until the stopping criterion is met. : Output (S [n],...,s[n] ) as an approximate solution. K off-the-shelf solvers, e.g., CVX []. Steps of the SUM method for solving problem () are summarized in Algorithm. Convergence of Algorithm is a direct consequence of [, Theorem ], which is stated in the following proposition. Proposition Every limit point of the sequence {S [n],...,s[n] K } n= stationary point of problem (). B. K-User TDMS and TDMA Schemes generated by Algorithm is a We extend the practical TDMS and TDMA schemes in Section IV to the general K-user scenario. ) K-user TDMS scheme: This scheme is similar to the TDMS scheme presented in Section IV-A. In the st time slot, all users operate in the EH mode, and in the nd time slot, all users operate in the ID mode; see Fig. a. In the st time slot, the optimal time fraction α and the associated optimal signal covariance matrices {Sk }K k= for energy harvesting can be obtained by solving a convex problem analogous to problem (). In the nd time slot, one has to solve the classical sum rate maximization problem max ( α) S i 0, i=,...,k ) h w i log (+ H ii S ih ii K k i hh ki S kh ki +σi i= s.t. Tr(S i ) P i, i =,...,K. (a) (b) Problem () is NP-hard, but can be efficiently handled by Algorithm (by letting E i = 0 i) or existing block coordinate descent based methods [, ]. DRAFT June, 0

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29 Page of 0 0 where the objective function is a locally tight lower bound of that in () ( ) ( U l {S kl } K k=,α l {S [n ] K ) kl } K k= α l log h H kl S klh kl +σl in which η [n ] l α l [ log ( k l k= h H kl S[n ] kl h kl +σl = α l log ( K k= hh kl S klh kl +σ l η [n ] l ) + k l hh kl S[n ] kl h kl +σl, for all l =,...,K. k l hh kl (S kl S [n ] kl ( k l hh kl S[n ] kl h kl +σl ) α l( k l hh kl S klh kl +σ l ) η [n ] l ln )h kl ) ln ] + α l ln, () Problem () is not a convex optimization problem in its current form, but this can be overcome by proper change of variables. To show this, define Then, problem () can be equivalently reformulated as ({W [n] kl }K k,l=,{α[n] l } K l= ) = W kl = α l S kl, k,l =,...,K. () argmax {α l} K Ω,Wkl 0 l= k,l=,...,k s.t. l= l i k= w l U l ({W kl } K k=,α l η [n ] l ) (a) h H ki W klh ki E i, i, Tr(W kl ) α l P k, k,l, where ( U l ({W kl } K k=,α K [n ] l η k= l )=α l log hh kl W klh kl +α l σ ) l k l η [n ] hh kl W klh kl +α l σl l α l η [n ] + α l l ln ln. () Notice that, in (), all the constraints are linear. Besides, the function U l ({W kl } K k=,α [n ] l η ) is concave because α l log ( K k= hh kl Wklhkl+αlσ l η [n ] l (b) (c) ) is concave with respect to ({Wkl } K l α l k=,α l) (which is the ) ). Therefore, problem () is a convex ( K k= perspective of the concave function log hh kl Wklhkl + σ l η [n ] l η [n ] l optimization problem and is efficiently solvable. We summarize the above algorithm in Algorithm. Analogous to Proposition, Algorithm is guaranteed to converge to a stationary point of problem (). It is interesting to note that problem () can be greatly simplified if one considers the TDMA (D) scheme discussed in Section IV-C. In particular, assume that transmitters operating in the EH mode send deterministic signals so that receivers operating in the ID mode can remove the cross-link signals. Under this setting, the objective function of () reduces to K l= w ( lα l log + h ) H ll Sllhll σ. By applying the l DRAFT June, 0

30 Page of 0 0 Algorithm SUM algorithm for problem () : Find initial variables by solving the feasibility problem ({W [0] kl }K k,l=,{α[0] l }K l= ) = find {W kl} K k,l=,{α l} K l= s.t. {α l } K l= Ω, l i k= h H ki W klh ki E i, i, Tr(W kl ) α l P k, W kl 0, k,l. If infeasible, then declare infeasibility of (); otherwise, set n = 0 and perform the following steps. : repeat : n := n+. ( : Compute η [n ] l = α [n ] k l hh kl W[n ] kl h kl )+σl, l =,...,K. l : Solve () to obtain ({W [n] kl }K k,l=,{α[n] l } K l= ). : until the stopping criterion is met. : Output ({S [n] kl = W[n] kl /α[n] l } K k,l=,{α[n] l } K l= ) as an approximate solution. same change of variables as in (), the design problem of the K-user TDMA (D) scheme is given by the following convex problem max (α,...,α K) Ω, W kl 0,k,l=,...,K s.t. ( w l α l log + hh ll W ) llh ll α l σl l= l i k= h H ki W klh ki E i, i, Tr(W kl ) α l P k, k, l. (a) (b) (c) For the case of K=, Ω is characterized by α +α =. For any given α, a closed-from solution of {W (α ),W (α )} can be readily obtained as mentioned in Section IV-C; moreover, the optimal α can also be efficiently obtained by golden section search over [0,]. June, 0 DRAFT

31 Page of C. K-User PS Scheme In this subsection, we extend the PS scheme in Section IV-D to the K-user interference channel. By extending (), the design problem of the K-user PS scheme can be written as ρ i h H ii w i log + S ih ii ρ i h H ki S kh ki +ρ i σ i +ˆσ i max S i 0,0 ρ i i=,...,k s.t. i= k= k i h H ki S kh ki E i ρ i i =,...,K, Tr(S i ) P i i =,...,K, (a) (b) where ρ i [0,] is the power splitting fraction of user i, for all i =,...,K. By introducing slack variables θ i = /ρ i, i =,...,K, one can write () as ) h max w i log (+ H ii S ih ii S i 0,0 ρ i, k i hh ki S kh ki +θ iˆσ i + σ i θ i 0, i=,...,k s.t. i= h H ki S kh ki E i, i =,...,K, ρ i k= θ i /ρ i, i =,...,K, Tr(S i ) P i, i =,...,K, (c) (a) (b) (c) (d) where (c) would hold with equality at the optimum. Note that both constraints (b) and (c) are convex. As a result, like problem (), the non-convexity of () is mainly due to the sum rate function. Therefore, we can handle problem () in the same fashion as Algorithm. In particular, like (), at the nth iteration, one solves the following approximation problem ( [n] {S i } K i=,{θ [n] i } K i=,{ρ [n] i } K i=) = [ ( K arg max w i log S i 0,0 ρ i, i= k= θ i 0, i=,...,k h H ki S kh ki +θ iˆσ i + σ i ) δ [n ] i ( ] k ih H ki S kh ki +θ iˆσ i) s.t. (b), (c) and (d), () ( ) where δ [n ] i h H ki S k [n ]h ki+θi [n ]ˆσ i + σ i ln, i =,...,K. k i VI. SIMULATION RESULTS AND DISCUSSIONS In this section, simulation results are presented to examine the performance of the proposed WIET schemes. Throughout the simulations, we assumed that each transmitter has identical, unit power budget, DRAFT June, 0

32 Page of 0 0 RX RX TX RX TX TX TX D R RX TX D RX T TX RX Q +* Fig. : Simulation setting with K = user pairs. i.e. P max P k = watt, and that the receiver noise powers are the same and equal to 0 dbm, i.e., σ σ k = watt for all k [0]. For the PS schemes, the powers of the RF noise and the processing noise are set the same as σ k = ˆσ k = σ for all k. The conversion efficiency for energy harvesting is set to γ = 0% [] (see ()). All transmission pairs have the equal weight of w k = for all k. The network topology is depicted in Fig., where the kth transmitter and receiver are placed in polar coordinates ( D, π(k ) ) K with D = DT and D R meters, respectively. The channel realization is generated by h ij = G A G ij g ij, where G A = dbi is the antenna gain, G ij is the large-scale fading, defined by G ij = d ij [0] with d ij in meters the distance between the ith transmitter and jth receiver, and g ij CN(0,I Nt ) represents the small scale fading. Whenever a scheme is infeasible, the achievable sum rate was set to zero. All the results presented in this section were obtained by averaging over 00 independent channel realizations. For the line search of α in the -user TDMA case, the feasible region in () is divided into 0 equally spaced time fractions, while for the SUM-based iterative algorithms, the stopping criterion was set to Rate[n] Rate[n ], where Rate[n] denotes the achieved sum rate at iteration n. The Matlab package CVX [] was used to solve the convex problems involved in the SUM-based iterative algorithms and problem () if K >. Example (Impact of cross-link channel power): We fix D R = meters and control the average relative cross-link channel power by varying D T, in order to investigate the impact of interference on the system performance. We first study the feasibility rate, defined as the ratio of the total number of channel realizations for which the energy requirement E E = E can be satisfied to the 00 randomly generated channel realizations, of the ideal scheme, PS scheme, TDMS, TDMA (D) and TDMA schemes. Note that D T < 0 means that the transmitters are located at the point ( D T, π(k ) K +π ). June, 0 DRAFT

33 Page of 0 0 sr$a$j$ksis Sw0D E=.0 mw E=0. mw Ideal scheme Power splitting TDMS TDMA (D) TDMA D T meters (a) Feasibility rate vs. D T, for E {0.,} mw. V% s SZSis SwA}R\<D 0 Ideal scheme Power splitting TDMS TDMA (D) TDMA D T meters (b) Average sum rate vs. D T, for E = 0. mw. Fig. : Simulation results for the scenario with E = E, N t = and K =. Fig. a shows the results for K =, N t = and E {0.,.0} mw. As expected from (), () and (), the ideal scheme, PS scheme and TDMS schemes have the same feasibility rate, while TDMA (D) and TDMA have the same feasibility rate. Moreover, one can observe from Fig. a that the feasibility rates of all schemes achieve their minimum when D T = 0, and improves as D T decreases or increases. This is owing to the fact that path loss has a direct impact on the received signal power strength and that both the direct-link signal and cross-link interference signal are beneficial for EH. Fig. b shows the average sum rate versus D T achieved by the five schemes under consideration. The EH requirement is set to E = 0. mw. Firstly, as expected, the ideal scheme always outperforms the power splitting scheme; similarly, the TDMA (D) scheme is always better than the TDMA scheme. Secondly, when D T 0 and increases, all schemes have improved sum rate performance. This is because the direct-link channel power is helpful for both ID and EH. Thirdly, when D T 0 and decreases, the sum rates of the ideal scheme and PS scheme decrease as the increased cross-link channel power has a direct, negative impact on ID. However, the sum rate performance of the time sharing schemes, i.e., TDMS, TDMA (D) and TDMA, instead improves with decreased D T < 0, since these schemes have the additional degree of freedom in time to optimize the tradeoff between ID and EH when the cross-link channel power is stronger. Fourthly, among practical schemes, if the direct-link channel power is stronger than the cross-link channel power (i.e., D T 0), the PS scheme performs best, then the TDMS scheme, and the TDMA scheme is the last; whereas, when the cross-link channel power is relatively stronger DRAFT June, 0

34 Page of 0 0 V% s SZSis SwA}R\<D 0 Ideal scheme Power splitting TDMS TDMA (D) TDMA D T meters Fig. : Average sum rate vs. D T, for E =. mw, N t = and K =. (i.e., D T 0), we observe that the TDMS scheme performs best. As strong cross-link channel powers are helpful for EH, the TDMS scheme can spend a less fraction of time operating in the EH mode when D T 0, and use most of the time for information transfer. When comparing to the PS scheme, the ID mode of TDMS scheme (see ()) is free from the EH constraint and thus may outperform the PS scheme if the optimal time fraction α 0. Since under the setting of N t = K =, the cross-link interference can still be well controlled, the TDMS scheme can perform better than TDMA (D) and TDMA schemes. We will see next that if there are more users in the network then TDMA based schemes may perform better. Let us consider another example with N t =, the number of users K increased to and the EH requirement increased to. mw. The results of average sum rate versus D T are shown in Fig.. One can observe that, except for D T. meters, the TDMA (D) scheme outperforms all the other schemes. Besides, the PS scheme yields higher sum rate than the TDMS and TDMA scheme when D T > 0; whereas the TDMS and TDMA schemes perform better when D T 0. We see that the gain of the TDMS scheme over the TDMA scheme when D T 0 is not significant as both schemes suffer from strong interference signals. Example (Impact of the energy demand and the number of users): Fig. shows the feasibility rates and average sum rates versus the energy requirement E = E = = E K, for N t = and K {,,}. All transmitters are placed at the origin, i.e., D T = 0. From Fig. a, one can observe that the feasibility rate decreases quickly as E increases. However, the overall performance is improved if there are more users (K from to to ) since the interference contribute EH. As the interference June, 0 DRAFT

35 Page of 0 0 can be well controlled when K = N t, we observe from Fig. b that the ideal scheme and the PS scheme exhibit better sum rate performance than the time sharing schemes. However, there is no significant performance difference between all methods when E is large (e.g., E mw). From Fig. c and Fig. d where the number of users are respectively increased to and, one can see that the TDMS scheme in general performs better; and the performance advantage is more significant when K =. This is because the EH constraint can be fulfilled more easily when there are more users; thus the TDMS scheme can allocate more time for information transfer. One can also observe from Fig. c and Fig. d that the TDMA (D) and TDMA schemes do not exhibit much performance advantage except when E is relatively larger. The simulation results in Fig., Fig. c and Fig. d imply that the rate loss of TDMA based schemes due to time sharing is significant and they can gain performance advantage only when the cross-link interference is overwhelming or when the EH requirement is large. On the contrary, the TDMS scheme is a better choice when the interference is strong but the EH requirement is not large. VII. CONCLUSIONS AND FUTURE WORKS In this paper, we have considered the WIET problem in a multi-user MISO interference channel. In addition to the ideal scheme, we have proposed four practical schemes, namely, the TDMS, TDMA, TDMA (D) and the PS schemes. Starting with the two-user scenario, we have analyzed the optimal transmission strategy of the ideal scheme, PS scheme as well as presenting semi-analytical solutions to the TDMS and TDMA schemes. It is shown that beamforming is optimal to all the schemes when K =. The proposed schemes have also been extended to the general K-user scenario. Specifically, we have shown that the ideal scheme, the PS scheme and the TDMA scheme can be efficiently handled by the proposed SUM method (e.g., Algorithm and Algorithm ); while the design problem of the TDMA (D) scheme can be obtained by solving a convex problem [i.e., ()]. The simulation results have revealed interesting tradeoffs between EH and ID in the complex IFC. In particular, it has been observed that strong cross-link channel power is not necessarily detrimental under energy harvesting constraints; instead, it is beneficial for the time sharing based schemes. We have also observed that none of the considered schemes can always dominate another in terms of the sum rate performance. The PS scheme in general performs best when the direct-link channel power is relatively stronger than the cross-link channel power and when N t K (i.e., interference can be well controlled by the spatial degree of freedom). When the cross-link channel power is relatively larger than the direct-link channel power and N t K, the TDMS scheme in general performs best and better than the PS scheme. If the interference is overwhelming and the EH requirement is also large, then the TDMA (D) scheme DRAFT June, 0

36 Page of 0 0 sr$a$j$ksis Sw0D V% s SZSis SwA}R\<D K= K= Ideal scheme Power splitting TDMS TDMA (D) TDMA K= 0 0 E (mw) 0 0 (a) Feasibility rate vs. E, for K {,,} E (mw) (c) Average sum vs. E, for K =. Ideal scheme Power splitting TDMS TDMA (D) TDMA V% s SZSis SwA}R\<D V% s SZSis SwA}R\<D 0 Ideal scheme Power splitting TDMS TDMA (D) TDMA E (mw) 0 0 (b) Average sum vs. E, for K =. Ideal scheme Power splitting TDMS TDMA (D) TDMA 0 0 E (mw) (d) Average sum vs. E, for K =. Fig. : Simulation results for the scenario with D T = 0 m, N t = and K {,,}. may outperform best. The current work may motivate several interesting directions for future research. Firstly, it is easy to see that, besides the considered TDMS and TDMA schemes, there exist other possible ways to separating the EH and ID modes of the K receivers across the time. It would be interesting to see how the corresponding design problems can be efficiently solved and their performance compared to the schemes presented in the current paper. Secondly, since none of the considered schemes can always perform best, it is worth formulating a design formulation that unifies all these practical schemes. Thirdly, based on some insights gained from the current work, it is worthwhile to further study the WIET problems for some more complex interference channels, such as the broadcast interference channels [] and the MIMO June, 0 DRAFT

37 Page of 0 0 interference channels []. A. Proof of Proposition APPENDIX We prove by contradiction that Tr(S i ) = P i for i =,. Suppose that Tr(S ) < P, then there exists some ǫ > 0 and S = S +ǫ h ( h ) H such that Tr(S )=P, where h Π h h Π h h. Note that (S,S ) is feasible to (P). Moreover, since h h, we have R (S,S ) > R (S,S ) and R (S,S ) = R (S,S ), which contradicts the optimality of (S,S ). Hence, it must be that Tr(S )=P ; similarly, one can show that Tr(S )=P. Next, we show that S and S lie in the range space of H [h h ] and H [h h ], respectively, i.e., Π H i S i Π H i = 0 for i =,. One can see that, for any S 0, h H ik (Π H i SΠ H H i )h ik = h H ik Sh ik, Tr(Π Hi SΠ H H i ) Tr(S), (A.) (A.) for i,k {,}, where the equality in (A.) holds because Π X X =X for all X C m n. Therefore, (S,S ) is an optimal solution to problem (P) only if (Π H S Π H,Π H S Π H ) is optimal to (P). Now suppose that S does not lie in the range space of H, i.e., Tr(Π H S Π H ) > 0. Then, Tr(Π H S Π H H )=Tr(S ) Tr(Π H S Π H )<Tr(S ) P, which implies that Π H S Π H is not optimal, and thereby S can show that S must lie in the range space of H. is not optimal to (P). Analogously, one What remains to prove is to show that there exists a pair of (S,S ) that are of rank one. To this end, we note that for any given {Si }, (P) is equivalent to the following problems for i,k {,} and i k, ( max log + hh ii S ) ih ii S i 0 Γ (A.a) ki +σ i s.t. h H ik S ih ik +Γ kk E k, Γ ki +hh iis i h ii E i, h H ik S ih ik = Γ ik, Tr(S i ) P i, (A.b) (A.c) (A.d) (A.e) DRAFT June, 0

38 Page of 0 0 where Γ ki =hh ki S k h ki. Let us focus on the case of i =, k =, and rewrite (A.) as f max S 0 hh S h s.t. h H S h E Γ, h H S h = Γ, Tr(S ) P, h H S h E Γ. (A.a) (A.b) (A.c) (A.d) (A.e) It is obvious to see that problem (A.) is feasible given that (P) is feasible. Thus, we must have f E Γ. Therefore, if one drops the constraint (A.e), the relaxed problem has the same optimal solution set as (A.). Since according to [, Theorem.], the relaxed problem (which has constraints (A.b), (A.c) and (A.d)) always admits a rank-one optimal solution, problem (A.) also has a rank-one solution. The above results imply that there exists an optimal S such that S = (a h +b h )(a h +b h ) H, where a,b C. Since any phase rotation of a h + b h is invariant to S, we without loss of (A.) generality can let a R. Analogously, for the case of i =, k =, one can show that problem (A.) has an optimal S = (a h +b h )(a h +b h ) H, where a R and b C. The proof is thus complete. B. Proof of Proposition Firstly, note that problem () is equivalent to the max-main-fairness problem { h H max min S h +h H S h h H, S h +h H S h S 0,S 0 E E max S 0,S 0 { h H min S h +h H S h, E s.t. Tr(S ) P, Tr(S ) P. h H S h +h H S h E } } (A.a) (A.b) (A.c) Hence, given optimal S and S, the optimal β of () is given as in (b): { h H β = min S h +h H S h, hh S h +h H S } h. (A.) E E Secondly, problem () satisfies the Slater s condition, so one can solve () by handling its Lagrange dual problem. Let µ 0 and η 0 be the Lagrange dual variables associated with constraints (b) and June, 0 DRAFT

39 Page of 0 0 (c), respectively. The dual problem of () can be shown as min µ,η 0 = min 0 µ /E max Tr(S (µh h H +ηh h H ))+Tr(S (ηh h H +µh h H )) S 0,S 0 s.t. Tr(S ) P, Tr(S ) P, s.t. E µ E η = 0, where Ψ (µ) = µh h H + µe E show that [, Proposition.] max Tr(S Ψ (µ))+tr(s Ψ (µ)) S 0,S 0 s.t. Tr(S ) P, Tr(S ) P, (A.) h h H and Ψ (µ) = µe E h h H +µh h H. It is not difficult to S (µ) = P v (µ)v H (µ), S (µ) = P v (µ)v H (µ) (A.) are optimal to the inner maximization problem of (A.), where v i (µ) C Nt is a principal eigenvector of Ψ i (µ), for i =,. As will be shown later, for i =,, under the assumption that h i and h i are linearly independent but not orthogonal to each other, Ψ i (µ) has a unique maximum eigenvalue for any µ. Hence, the solutions in (A.) are unique. According to the duality theory [], if µ is dual optimal (i.e., optimal to (A.)), then the unique S (µ), S (µ) in (A.) and β in (A.) are optimal to problem (). The optimal µ can be obtained through a bisection search using the dual gradient, which is given by g = h H S (µ)h E E h H S (µ)h E E h H S (µ)h +h H S (µ)h. Lastly, we show that if h i and h i are linearly independent and Ψ i (µ) has two equal eigenvalues, then h i and h i must be orthogonal. First note that Range(Ψ i (µ)) = Range([h i,h i ]) and rank(ψ i (µ)) for linearly independent h i and h i. Secondly, note that any principal eigenvector v of Ψ i (µ) belongs to Range(Ψ i (µ)). If Ψ i (µ) has two equal eigenvalues (the dimension of the principal eigenspace is two), then the principal eigenspace is exactly Range([h i,h i ]). Hence, h i = h i / h i, h i = h i / h i, h i = Π h i h i / Π h i h i and h i = Π h i h i / Π h i h i are all principal eigenvectors of Ψ i (µ). Let λ max denote the principal eigenvalue of Ψ i (µ), and now consider i =. We have h H Ψ (µ) h = µ h H h +η h H h = λ max, h H Ψ (µ) h = µ h H h +η h H h = λ max, ( h ) H Ψ (µ) h = η ( h ) H h = λ max, ( h ) H Ψ (µ) h = µ ( h ) H h = λ max, (A.a) (A.b) (A.c) (A.d) DRAFT June, 0

40 Page of 0 0 where η = Eµ E. By (A.c) and (A.d), we have Further combining (A.a) with (A.c) yields η ( h ) H h = µ ( h ) H h. µ h H h +η h H h = η ( h ) H h, µ h +η( h ( h ) H h ) = η ( h ) H h µ h +η h = η ( h ) H h = µ ( h ) H h +η ( h ) H h. (A.) Since both µ, η are nonnegative, and ( h )H h h, ( h )H h h, the equality in (A.) implies that ( h )H h = h and ( h )H h = h, i.e., h and h are orthogonal to each other. However, this contradicts the assumption that h is not orthogonal to h. Hence, the principal eigenvector of Ψ (µ) is unique. Similarly, the principal eigenvector of Ψ (µ) can be shown unique. C. Proof of Proposition Problem () is equivalent to a quasi-convex problem. We first apply the idea of the Charnes-Cooper transformation [] to recast problem () as a convex problem. To illustrate this, consider the following convex semidefinite program (SDP) max X 0,X 0,y 0 αlog ( +h H X h ) (A.a) s.t. h H X h +yσ =, (A.b) h H X h +h H X h ye /α, Tr(X ) yp, Tr(X ) yp. Note that the optimal y of (A.) must be positive; otherwise we have X = X (A.b). Moreover, consider the following correspondence: y = /(h H S h +σ ) > 0, X = ys, X = ys. (A.c) (A.d) = 0 which violates (A.a) (A.b) Then, one can show that (S,S ) is feasible to () if and only if (X,X,y) is feasible to (A.). Furthermore, the objective value achieved by (S,S ) in () is the same as the objective value achieved by (X,X,y) in (A.). Therefore, the two problems () and (A.) are equivalent, and one can obtain (S,S ) of () by solving the convex problem (A.). June, 0 DRAFT

41 Page of 0 0 To show how (A.) can be efficiently solved, let us consider the equivalent problem to (A.) max X 0,X 0,y 0 hh X h s.t. h H X h +yσ =, h H X h +h H X h y E α, Tr(X ) yp, Tr(X ) yp. (A.a) (A.b) (A.c) (A.d) Denote by Y(α) the feasible region of y in (A.) for a given α. Suppose that a feasible y is given and let the associated optimal X and X of problem (A.) be X (y) and X (y), respectively. One key observation is that X (y) can be obtained by solving the following problem X (y) = argmax X 0 h H X h s.t. h H X h = yσ, Tr(X ) yp. According to [, Theorem.], problem (A.) has a closed-form solution as X (y) = v (y)v H (y), yσ v (y) = h H h ( h H h ) h + yp yσ h h, (A.a) (A.b) (A.c) (A.) (A.) where h ij = hij h ij, h ij Π h ij h ii Π h ij h ii for i =,. Therefore, given y Y(α) and X (y) above, problem (A.) reduces to X (y) = argmax X 0 h H X h s.t. h H X h y E α g(y), Tr(X ) yp, (A.a) (A.b) (A.c) where g(y) h H X (y)h = h H v (y). Again, by applying [, Theorem.], problem (A.) has the optimal solution given by X (y) = v (y)v H (y), yp h, if y E v (y) = α g(y) yp h H h, ye/α g(y) h H h h + yp ye/α g(y) h h, otherwise. h H h (A.) (A.0) DRAFT June, 0

42 Page of 0 0 Hence, for any given y Y(α), one can efficiently obtain X (y) and X (y) by (A.) and (A.), respectively. The optimal y of problem (A.) then can be obtained by solving the following onedimensional problem y = argmax y Y(α) h H X (y)h. (A.) Problem (A.) is in fact a convex problem and thus is efficiently solvable by the golden section search method []. To show this, note from (A.) that h H X (y)h = max X 0,X 0 hh X h s.t. h H X h +yσ =, h H X h +h H X h y E α, Tr(X ) yp, Tr(X ) yp. (A.a) (A.b) (A.c) (A.d) Since problem (A.) is convex jointly in (X,X,y), and h H X (y)h is a point-wise maximum of the jointly concave (linear) h H X h over all (X,X ) feasible to (A.), by [], h H X (y)h is concave with respect to y. Finally, let us specify the feasible region of y, i.e., Y(α). Since by (A.) y = /(h H S h +σ ), it [ ] must lie in the region of Y(α),, where T max(α)+σ T min(α)+σ T max (α) max S hh S h s.t. (b) and (c), 0,S 0 T min (α) min S hh S h s.t. (b) and (c), 0,S 0 provided that problem () is feasible. Problem (A.) is further equivalent to T max (α) = max S h H S h, s.t. h H S h E /α P h, Tr(S ) P, S 0, T min (α) = min S h H S h, s.t. h H S h E /α P h, Tr(S ) P, S 0, (A.a) (A.b) both of which admit closed-form solutions [, Theorem.], stated as P h E if α T max (α) = P h +P h H h, (φ (α)+φ (α)) otherwise, E 0 if α T min (α) = P h +P Π h h h, (φ (α) φ (α)) otherwise, E where φ (α) = α P h hh h h, φ (α) = P h +P h E Π h h. The α h proof is thus complete. June, 0 DRAFT

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45 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Page of 0 0 Wireless Information and Energy Transfer in Multi-Antenna Interference Channel Chao Shen, Wei-Chiang Li and Tsung-Hui Chang Abstract This paper considers the transmitter design for wireless information and energy transfer (WIET) in a multipleinput single-output (MISO) interference channel (IFC). The design problem is to maximize the system throughput (i.e., the weighted sum rate) subject to individual energy harvesting constraints and power constraints. It is observed that the ideal scheme, where the receivers simultaneously perform information detection (ID) and energy harvesting (EH) from the received signal, may not always achieve the best tradeoff between information transfer and energy harvesting, but simple practical schemes based on time splitting may perform better. We therefore propose two practical time splitting schemes, namely the time division mode switching (TDMS) and time division multiple access (TDMA), in addition to the existing power splitting (PS) scheme which separates the received signal into two parts for ID and EH, respectively. In the two-user scenario, we show that beamforming is optimal to all the schemes. Moreover, the design problems associated with the TDMS and TDMA schemes admit semi-analytical solutions. In the general K-user scenario, a successive convex approximation method is proposed to handle the WIET problems associated with the ideal scheme, the PS scheme and the TDMA scheme, which are known NP-hard in general. Simulation results show that none of the schemes under consideration can always dominate another in terms of the sum rate performance. Specifically, it is observed that stronger cross-link channel power improves the achievable sum rate of time splitting schemes under EH constraints, but degrades the sum rate performance of the ideal scheme and PS scheme. As a result, time splitting schemes can outperform the ideal scheme and the PS scheme in interference dominated scenarios. Index terms wireless energy transfer, energy harvesting, interference channel, beamforming, convex optimization EDICS: SPC-APPL, SPC-INTF, SPC-CCMC, SAM-BEAM The work of Chao Shen is supported by the Opening Project of The State Key Laboratory of Integrated Services Networks, Xidian University (Grant No. ISN-0), the China Postdoctoral Science Foundation (Grant No. 0M), the Fundamental Research Funds for the Central Universities (Grant No. 0JBM), the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (No. RCS0ZQ00), the Natural Science Foundation of China (Grant No. U), and the Key Grant Project of Chinese Ministry of Education (No. 0). The work of Tsung- Hui Chang is supported by National Science Council, Taiwan (R.O.C.), by Grant NSC --E-0-00-MY. Part of this work was presented in IEEE GLOBECOM 0 []. Chao Shen is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China and the State Key Laboratory of Integrated Services Networks, Xidian University, Xi an, China. chaoshen@ieee.org. Wei-Chiang Li is with Institute of Communications Engineering, National Tsing Hua University, Hsinchu 0, Taiwan, (R.O.C.). weichiangli@gmail.com. Tsung-Hui Chang is the corresponding author. Address: Department of Electronic and Computer Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, (R.O.C.). tsunghui.chang@ieee.org. I. INTRODUCTION Recently, scavenging energy from the environment has been considered as a potential approach to prolonging the lifetime of battery-powered sensor networks and to implementing selfsustained communication systems. For example, the base stations may be powered by wind mills or solar photovoltaic (PV) arrays, and can harvest energy for providing services to the mobile users. This idea has motivated considerable research endeavors in the past few years, investigating wireless systems with energy-harvesting transmitters; see, e.g., [ ]. In these works, optimal transmission strategies under energy-harvesting constraints are studied from single-input single-output (SISO) channels to complex interference channels (IFCs). In contrast to the base stations, it may be difficult for the mobile devices and sensor nodes to harvest energy from the sun and wind effectively. One possible solution to this issue is wireless energy transfer (WET), that is, the power-connected transmitters transfer energy wirelessly to charge the mobile devices. A successful application of WET is the radio frequency identification (RFID) system where the receiver wirelessly charges energy from the transmitter (through induction coupling) and use the energy to communicate with the transmitter. The works in [, ] showed that, using coupled magnetic resonances, energy can be wirelessly transferred for two meters with over 0% energy conversion efficiency. WET can also be achieved via the RF electromagnetic signals; for example, see [ ] for recent developments of RF-based energy harvesting (EH) circuits and also [] for prototype EH receivers which were tested for harvesting energy from GSM and G signals in the great London area. Compared to the techniques based on induction and magnetic resonance coupling, RF signals can achieve long-distance WET; however, the energy conversion efficiency is in general low. This calls for advanced signal processing techniques, such as beamforming, to improve the energy conversion efficiency. Since the RF signals can carry both information and energy, in recent years, it has been of great interest to study wireless communication systems where the receivers can not only decode information data but also harvest energy from the received RF signals, i.e., wireless information and energy transfer (WIET) systems [ 0]. Specifically, in [], the optimal tradeoff between information capacity and energy transfer of the WIET system was studied for a SISO flat fading channel. In [], the optimal power allocation strategy for a SISO frequency-selective fading channel was derived under a receiver EH constraint. The work in [] studied the

46 Page of SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 0 0 tradeoff between achievable information rate and harvested energy for the multiple access channel (MAC) and a twohop relay network with an energy harvesting relay; while the works [, ] respectively derived optimal power allocation for an one-way relay channel, two-way channel and the MAC, assuming that transmitters can share energy by wireless energy transfer. It was shown that there exist nontrivial tradeoffs between information transfer and energy harvesting in general, and thus pertinent transmitter optimization and resource allocation (power/time/space/frequency) are needed. The works in [ ] have assumed the ideal receivers which can decode information bits and harvest energy from the received RF signals simultaneously. Unfortunately, current circuit technologies cannot achieve this yet. In view of this, practical WIET schemes are proposed. In particular, Zhou et al. proposed in [] a dynamic power splitting (PS) scheme for a SISO flat fading channel, wherein the received RF signal is either used for information detection (ID) and EH, or is split into two parts, one for ID and the other for EH. Considering a multiple-input multiple-output (MIMO) flat-fading channel, in addition to the PS scheme, the authors in [] further proposed a time switching scheme where the receiver performs ID in one time slot while EH in the other time slot. In [0], the dynamic PS scheme was extended to a multi-user multiple-input single-output (MISO) broadcast channel where one multi-antenna transmitter employs transmit beamforming for serving multiple receivers for ID and EH. The works [, ] respectively considered the WIET problem in a twouser MIMO IFC and general K-user MIMO IFC under the assumption that the receivers perform either EH or ID. In this paper, we consider a K-user MISO interference channel and study the optimal transmission and resource allocation strategies for WIET. It is interesting to note that, different from the conventional IFCs without energy harvesting, the cross-link signals in the considered scenario can degrade the information sum rate on one hand, but, at the same time, boost energy harvesting of the receivers on the other hand. Different from [, ], our interests not only lie in the optimal transmission structure of the transmitters but also in the optimal time switching, time sharing and power splitting strategies of practical receivers. To this end, we first consider an IFC with ideal receivers, and formulate the WIET design problem as a weighted sum rate maximization problem subject to individual EH constraints and transmit power constraints. Interestingly, we will see that the ideal scheme with ideal receivers may not always perform best in the complex interference environment, but simple practical schemes based on time splitting and time sharing may instead yield better sum rate performance. This is in sharp contrast to the single-transmitter scenarios studied in [ 0] where time splitting schemes usually exhibit poorer performance. This intriguing observation motivates us to propose two practical WIET schemes for the MISO IFC, namely, the time division mode switching (TDMS) scheme and the time division multiple access (TDMA) scheme. In the TDMS scheme, the transmission time is divided into two As will be shown in Section IV-A, the proposed TDMA scheme is similar to but not completely the same as the TDMA scheme in conventional IFCs without energy harvesting. time slots. All receivers perform EH in the first time slot and subsequently perform ID in the second time slot. The TDMA scheme divides the transmission time into K time slots, and in each time slot, one receiver performs ID while the others perform EH. In addition to the two time splitting based schemes, we also study the PS scheme [, 0] for WIET in the K-user MISO IFC. We will show how the design problems associated with the three practical schemes can be efficiently handled. Specifically, for the two-user scenario, we show that the design problems associated with the TDMS and TDMA schemes admit semi-analytical solutions. Since for the general K-user scenario the TDMA scheme and the PS scheme are nonconvex problems (NP-hard in general []), we further propose efficient approximation methods based on the successive upper-bound minimization (SUM) method []. The presented simulation results will show that stronger cross-link channel power improves the information sum rate of TDMS and TDMA schemes under energy harvesting constraints, but degrades the sum rate performance of the ideal and PS schemes. As a result, if the cross-link channel powers are not strong or the energy harvesting constraints are not stringent, the PS scheme can outperform TDMS and TDMA schemes; otherwise, the TDMS and TDMA schemes can perform better. The rest of this paper is organized as follows. In Section II, the signal model of the MISO interference channel is presented. Starting with the two-user scenario, in Section III, the optimal WIET transmission strategy for ideal receivers is analyzed. The result motivates the developments of the practical TDMS, TDMA and PS schemes, which are presented in Section IV. Section V extends the study to the general K- user scenario. Simulation results are presented in Section VI. The conclusions are given in Section VII. Notations: Column vectors and matrices are written in boldfaced lowercase and uppercase letters, e.g., a and A. The superscripts ( ) T, ( ) H and ( ) represent the transpose, (Hermitian) conjugate transpose and matrix inverse, respectively. rank(a) and Tr(A) represent the rank and trace of matrix A, respectively. A 0 ( 0) means that matrix A is positive semidefinite (positive definite). a denotes the Euclidean norm of vector a. The orthogonal projection onto the column space of a tall matrixais denoted byπ A A(A H A) A H. Moreover, the projection onto the orthogonal complement of the column space of A is denoted by Π A I Π A where I is the identity matrix with proper dimension. II. SIGNAL MODEL AND PROBLEM STATEMENT We consider a multi-user interference channel with K pairs of transmitters and receivers communicating over a common frequency band. Each of the transmitters is equipped with N t antennae, while each of the receivers has single antenna. Let x i C Nt be the signal vector transmitted by transmitter i, and h ik C Nt be the channel vector from transmitter i to receiver k, for all i,k {,,...,K}. The received signal at receiver i is given by y i = h H iix i + k=,k i h H kix k +n i, i =,...,K, ()

47 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Page of 0 0 where n i CN(0,σi ) is the additive Gaussian noise at receiver i. Unlike the conventional MISO IFC [] where the receivers focus only on extracting information, we consider in this paper that the receivers can also scavenge energy from the received signals [,, ], i.e., energy harvesting. Therefore, in addition to information, the transmitters can also wirelessly transfer energy to the receivers. We call the two operation modes the information detection (ID) mode and the energy harvesting (EH) mode, respectively. Assume that x i contains the information intended for receiver i which is Gaussian encoded with zero mean and covariance matrix S i 0, i.e., x i CN(0,S i ) for i=,...,k. Moreover, assume that each receiver i decodes x i by single user detection in the ID mode. Then the achievable information rate of receiver i is given by ) h R i (S,...,S K ) = log (+ H ii S ih ii k i hh ki S kh ki +σi, () for i =,...,K. Alternatively, the receiver may choose to harvest energy from the received signal. It can be assumed that the total harvested RF-band energy during a transmission interval is proportional to the power of the received baseband signal []. Specifically, for receiver i, the harvested energy, denoted by E i, can be expressed as E i = γ h H kis k h ki, i =,...,K, () k= where γ is a constant accounting for the energy conversion loss in the transducer []. Suppose that the receivers desire to harvest certain amounts of energy. We are interested in investigating the optimal transmission strategies of S i, i =,...,K, so that the information throughput of the K-user IFCs can be maximized while the energy harvesting requirements of the receivers are satisfied at the same time. In subsequent sections, we will first study an ideal scenario where the receivers can simultaneously operate in the ID mode and EH mode. One should note that current energy harvesting receivers are not yet able to realize such ideal receiver []. However, the study can provide insights for understanding the tradeoff between ID and EH as well as for the design of practical schemes. In particular, in Section IV and Section V, we further investigate some practical schemes where the receivers operate either in the ID mode or EH mode at any time instant. In order to gain more insights, we will begin our investigation with the two-user scenario (K = ), and later extend the studies to the general K-user case (in Section V). III. OPTIMAL WIET DESIGN FOR IDEAL SCHEME Let us assume thatk= and consider ideal receivers which can simultaneously decode the information bits and harvest the energy from the received signals. Suppose that the two receivers desire to harvest total amounts of energy E and E, respectively. We are interested in the following transmitter design problem for WIET: (P) max w R (S,S )+w R (S,S ) S 0,S 0 s.t. h H S h +h H S h E, h H S h +h H S h E, Tr(S ) P, Tr(S ) P, (a) (b) (c) (d) (e) where w,w > 0 are positive weights, and P > 0 and P > 0 in (d) and (e) represent the individual power constraints. The constraints in (b) and (c) are the energy harvesting constraints where we have set γ= and = for notational simplicity. In Section VI, practical values of γ will be considered for performance evaluation. Note that, in the absence of (b) and (c), problem (P) reduces to the classical sum rate maximization problem in MISO IFC []: max w R (S,S )+w R (S,S ) S 0,S 0 s.t. Tr(S ) P, Tr(S ) P. (a) (b) (c) Since the EH constraints (b) and (c) are convex constraints, the existing methods for problem () (e.g., [ ]) can be directly used for handling (). Here, we are interested in understanding the impacts of (b) and (c) on the sum rate performance and the optimal transmitter structure. It can be observed from () and () that the EH constraints (b) and (c) would trade the maximum achievable sum rate for energy harvesting; i.e., the maximum sum rate in (a) is in general no larger than that in (a). To see when this would happen, let ( S, S ) be an optimal solution to problem (). One can verify from the rate function in () and problem () that ( S, S ) must satisfy [ ] { h H S h [E ] T h H S h Ω E E = H max h S 0,Tr(S ) P, h H Sh E [ ] { h H S h [E ] T h H S h Ω E E = H max h S 0,Tr(S ) P, h H Sh E S h, 0 E P h }, () S h, 0 E P h }. () That is, the energies harvested at the two receivers due to ( S, S ) must lie in Ω +Ω. Thus, we have that h H S h +h H S h h H S h +h H S h min (E,E ) Ω,(E,E ) Ω E +E = P h H h, (a) min (E,E ) Ω,(E,E ) Ω E +E = P h H h, (b) where h ij Π h h ii ij Π h h ij ii. Equations in () imply that the two receivers can at lease harvest energies P h H h and P h H h, respectively. The minimum amounts of energies

48 Page of SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 0 0 are achieved when E = P h H h, E = P h H h and E = E = 0; that is, when each of the transmitters only focus on transmitting signals to its own receiver, without allowing any leakage of energy to the other receiver. According to (), we have that Property The energy harvesting constraints (b) and (c) are inactive at the optimum if E P h H h and E P h H h ; hence, (P) reduces to the conventional MISO IFC problem () under this condition. However, when E > P h H h or E > P h H h, the maximum information throughput may have to be compromised with energy harvesting. Interestingly, the following proposition shows that the optimal transmit structure of (P) remains the same as that of problem () which does not consider the EH constraints. Proposition Assume that problem (P) is feasible, and that h h and h h without loss of generality. Then there exists a rank-one optimal solution, denoted by (S,S ), to problem (P) such that S = (a h +b h )(a h +b h ) H, S = (a h +b h )(a h +b h ) H, for some a i R, b i C, and Tr(S i ) = P i, for i =,. (a) (b) The proof is given in Appendix A. Proposition implies that beamforming is an optimal transmission strategy of (P). Moreover, the beamforming direction of transmitter i, that lies in the range space of [h i,h i ] for i =,, is the same as the optimal beamforming direction of problem () in the conventional IFCs []. However, unlike problem (), the optimal coefficients a i,b i, i =,, for (P) have to account for both the needs of energy harvesting and information transfer. Remark It is important to remark that, while (P) is ideal in the sense that the receivers can simultaneously operate in the ID and EH modes, (P) does not necessarily perform best in terms of sum rate maximization. The reason is that the cross-link signal power h H ik S ih ik plays two completely opposite roles in the considered scenario It can boost the energy harvesting of receiver k on one hand, but also degrades the achievable information rate on the other hand. Therefore, when the cross-link channel power is strong (e.g., the interference dominated scenario) and when the energy harvesting constraints are not negligible (e.g., the conditions in Property do not hold), the transmitters have to compromise the achievable information rate for energy harvesting. Under such circumstances, it might be a wiser strategy to split the ID and EH modes in time. To further look into this aspect, we present in Fig. two simulation examples for a two-user scenario with EH constraints. The detailed setting of the simulations are presented in Section VI. Fig. a shows the achievable sum rate of (P) versus energy requirement E E = E for two randomly generated channel realizations. The achievable region is drawn by using an exhaustive search method from [, ]. In Fig. a, the parameter D T is used to control the direct-link and crosslink channel powers, as shown in Fig. in Section VI. Simply speaking, when D T increases, the direct-link channel power increases whereas the cross-link channel power decreases. One can observe from Fig. a that the rate-energy regions are not convex for these two randomly generated channel realizations. Moreover, for some values of E, the receivers may achieve a higher sum rate through time switching between the EH mode and ID mode, especially in the case D T = 0. Fig. b displays the rate region (R versus R ) of the two users. Analogously, we observe that time sharing for multiple access may achieve a higher sum rate. The two simulation results in Fig. imply that the ideal scheme (P) may not always achieve the best tradeoff between information transfer and energy harvesting, but, instead, time sharing for EH/ID mode switching or time sharing for multiple access may yield higher information sum rate. It is worthwhile to note that these time sharing schemes are practical as the receivers operate either in the EH mode or ID mode at each time instant. These motivate us to develop two practical time sharing schemes, namely, the time-division mode switching ZSis SwA}R\<D R (bps) 0 D T = O D T =.0 m Ideals cheme Time sharing scheme oswgdsr I)Sd (a) Sum rate vs. EH requirement E. Ideal scheme Time sharing scheme 0 0 R (bps), seed=, E=[0., 0.] mw, D T =.0m, D R =.0m,N t =.0 (b) Achievable rate region (R,R ) for E = 0. mw, E = 0. mw, D T = m. Fig. : Motivating simulation examples for the -user scenario (K =, N t = ).

49 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Page of 0 0 (TDMS) scheme and the time-division multiple access (TDMA) scheme, in the next section. IV. PRACTICAL WIET SCHEMES AND OPTIMAL TRANSMISSION STRATEGIES A. Time Division Mode Switching (TDMS) Scheme In the first practical scheme, we divide the transmission interval into two time slots. In one time slot, both receivers operate in the EH mode, whereas, in the other time slot, both receivers switch to the ID mode. The two receivers thus coherently switch between the EH and ID modes, i.e., mode switching. Suppose that α fraction of the time is for EH mode and ( α) fraction of the time is for ID mode. The TDMS scheme is described as follows. Time slot (EH mode): The two receivers focus on harvesting the required energy E and E in α fraction of the time, i.e., α (h H S h +h H S h ) E, α (h H S h +h H S h ) E. (a) (b) Time slot (ID mode): Both the two receivers operate in the ID mode and maximize the information throughput in the remaining fraction of the time, i.e., max ( α)(w R (S,S )+w R (S,S )) (a) S 0,S 0 s.t. Tr(S ) P, Tr(S ) P. (b) Problem () in the ID mode is the classical sum rate maximization problem in the MISO IFC [see ()], which can be efficiently handled by existing methods in [,, ]. Note that it has been shown in [, ] that beamforming is an optimal transmission scheme for problem (). We now focus on the EH mode in time slot. Since time slot does not contribute to the information throughput, it is desirable to spend as least as possible time for the EH mode, i.e., to use a minimal time fraction α to fulfill the energy harvesting task. Mathematically, we can write it as the following optimization problem max β β R,S 0,S 0 s.t. h H S h +h H S h βe, h H S h +h H S h βe, Tr(S ) P, Tr(S ) P, (a) (b) (c) (d) where β /α. Note that if the optimal β of () is less than one (i.e., optimal α > ), then it implies that the energy harvesting requirements () cannot be satisfied even if the receivers dedicate themselves to harvesting energy throughout the whole transmission interval. In that case, we declare that the TDMS scheme is not feasible. While problem () is a convex semidefinite program (SDP), which can be solved by the off-the-shelf solvers, we show that () actually admits a semi-analytical solution. Proposition Assume that h i and h i are linearly independent but not orthogonal to each other, for i =,. The optimal solution to problem () is given by S (µ ) = P v (µ )v H (µ ),S (µ ) = P v (µ )v H (µ ), { h β(µ H ) = min S (µ )h +h H S (µ )h, E h H S (µ )h +h H S (µ )h E (a) }, (b) where µ 0 is the optimal dual variable associated with constraint (b), and v i (µ ) is the principal eigenvector of µ h i h H i + ( µ E ) E h i h H i for i =,. Moreover, µ can be efficiently obtained using a simple bisection search. The proof of Proposition is given in Appendix B. The assumptions on h i and h i, for i =,, hold with probability one for random (continuous) fading channels. Note that Proposition also implies that beamforming is optimal for the EH mode of the TDMS scheme. B. TDMA Scheme Unlike TDMS scheme, in each time slot of TDMA scheme, one receiver operates in the ID mode and the other receiver operates in the EH mode. Assume that the time fraction of the first time slot is α. Time slot : Receiver operates in the ID mode and receiver operates in the EH mode. The objective is to maximize the information rate of receiver and guarantee the energy harvesting requirement of receiver at the same time. The design problem is given by ( ) max αlog + hh S h S 0,S 0 h H S h +σ (a) s.t. h H S h +h H S h E /α, (b) Tr(S ) P, Tr(S ) P. (c) Time slot : The operation modes of the two receivers are exchanged: ( ) max ( α)log + hh S h S 0,S 0 h H S h +σ (a) s.t. h H S h +h H S h E /( α), (b) Tr(S ) P, Tr(S ) P. (c) By intuition, this TDMA scheme would be of interest when the two receivers have asymmetric energy harvesting requirements and asymmetric cross-link channel powers. Moreover, like the conventional interference channel without energy harvesting, the TDMA scheme may outperform the spectrum sharing schemes in interference dominated scenarios. It is not difficult to show that: Lemma The TDMA scheme is feasible if and only if E P h +P h + E P h. () +P h

50 Page of SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 0 0 Proof: The TDMA scheme is feasible if and only if both () and () are feasible. Problem () is feasible if and only if there exists some α [0,] such that ( ) max h H E α S 0,S 0 S h +h H S h Tr(S ) P,Tr(S ) P = α (P h +P h ), () where the equality is obtained by applying the result in [, Proposition.]. Similarly, one can show that () is feasible if and only if E ( α) (P h +P h ). () Combining () and () gives rise to (). Conversely, given (), let α = E P h +P h, and thus E P h +P h α, which are () and (), respectively. Hence the TDMA scheme is feasible if and only if () is true. According to () and (), a feasible time fraction α must lie in the interval E P h +P h α E P h +P h. () Interestingly, given a feasible α, both problems () and () can be efficiently solved (semi-analytically). Since () and () are similar to each other, we take () as the example. Proposition Let the time fraction α satisfy (). Then, an optimal solution (S, S ) to problem () is given by where S = v (y )v H (y )/y, S = v (y )v H (y )/y, (0) yp h, if g(y)/y E /α P h H h, ( ye /α g(y) v (y) = h H h h H h h ) + yp ye /α g(y) h h, otherwise, yσ v (y) = h H h ( h H h ) h + yp yσ h h, in which h ij = hij h, h ij ij Π h h ii ij Π h h ij ii for i =,, and g(y) = h H v (y). The optimal y can be obtained by solving the convex problem y = argmax h H v (y) (a) y s.t. T max (α)+σ y T min (α)+σ, (b) through the golden section search [], where P h E if α T max (α) = P h +P h H h, (φ (α)+φ (α)) otherwise, E 0 if α T min (α) = P h +P Π h h h, (φ (α) φ (α)) otherwise, and φ (α) = P h +P h E α E α P h hh h h, φ (α) = Π h h h The proof is presented in Appendix C. We see from (0) that beamforming is also optimal to the TDMA scheme. By Proposition, given a feasible time fraction α, one can efficiently solve problems () and () and thus evaluate the achievable sum rate of the two users. Then, the optimal time fraction α that maximizes the sum rate of the two users can be obtained by line search over the interval in (). C. TDMA via Deterministic Signal for Energy Harvesting It should be noticed that, while Gaussian signaling is optimal for information transfer, it may not be necessary for energy transfer. In particular, if one user operates in the EH mode, the transmitter may simply transmit some deterministic signals (e.g., training/pilot signals) known to both receivers. Consider the TDMA scheme in the previous subsection, and assume that, in the first time slot, transmitter operating in the EH mode transmits deterministic signals x which are known to receiver operating in the ID mode. Under such circumstances, receiver can actually remove h H x from the received signal before information detection, i.e., removing the cross-link interference. The design problem in the st time slot thereby reduces to ( ) max +σ hh S h (a) S 0,S 0 αlog s.t. h H S h +h H S h E /α, Tr(S ) P, Tr(S ) P.. (b) (c) Problem () is easier to handle than its counterpart in (). Clearly, given α satisfying (), optimal S is given by S = P h hh, Therefore, () boils down to max S 0 hh S h s.t. h H S h E /α P h, Tr(S ) P, (a) (b) (c) which admits a closed-form solution for S according to [, Theorem.]. Analogously, the design problem for the second time slot can be simplified. In this paper, we refer to this scheme as the TDMA (D) scheme. Since the receivers are free from cross-link interference, it is anticipated that the TDMA (D) scheme performs no worse than the TDMA scheme. However, it should be noted that, in TDMA (D), the two receivers require perfect knowledge of the cross-link channels h and h, respectively; otherwise, the receivers may suffer performance degradation due to imperfect interference cancelation. In Section V-B, it will be further shown that it is possible to jointly find the optimal time fractionαand transmit signal covariance matrices S,S of the TDMA (D) scheme. D. Power Splitting Scheme Other than the TDMS and TDMA schemes, another practical scheme, called power splitting (PS) [], splits the received

51 SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING Page 0 of 0 0 RX i CN(0; ~¾ i ) Power Splitter p ½i CN(0; ^¾ i ) p ½i Fig. : Diagram of the power splitting receiver for WIET. signal into two parts for simultaneous EH and ID; see Fig.. Specifically, suppose that receiver i splits ρ i [0,] fraction of power for ID and ρ i fraction of power for EH. The associated WIET design problem is given by ) max S 0,S 0, 0 ρ,ρ ID EH ρ h w log (+ H S h ρ h H S h +ρ σ + ˆσ ρ h +w log (+ H ) S h ρ h H S h +ρ σ + (a) ˆσ s.t. h H S h +h H S h E ρ, h H S h +h H S h E, ρ Tr(S ) P, Tr(S ) P, (b) (c) (d) where σ i denotes the noise power at the RF end while ˆσ i denotes the processing noise power. Note that, in problem (), we not only optimize the signal covariance matrices {S,S }, but also the power splitting fractions {ρ,ρ } in the receivers. Since problem () has the same problem structure as (P) when {ρ,ρ } are given, one can infer from Proposition that transmit beamforming is an optimal solution structure to problem () as K =. In Section V-C, efficient approximation method for handling problem () and its extension to the general K-user scenario will be further studied. V. WIET DESIGN FOR K-USER MISO IFC In this section, we consider the WIET problem for the K- user MISO IFC scenario. In the first subsection, we begin with the ideal scheme. In the second subsection, we extend the TDMS and TDMA schemes to the K-user scenario. In the last subsection, we investigate the K-user PS scheme. A. K-User Ideal Scheme By the signal model in ()-() and (P) in (), the K-user WIET problem is formulated as ) h w i log (+ H ii S ih ii k i hh ki S kh ki +σi (a) max S i 0 i=,...,k s.t. i= h H kis k h ki E i, i =,...,K, k= Tr(S i ) P i, i =,...,K, (b) (c) where E i 0 is the energy requirement of user i, for i =,...,K. Since problem () is NP-hard in general [], our interest for the K-user WIET problem lies in efficient algorithms for finding an approximate solution. One should note that many of the existing approximation methods that can handle the conventional K-user IFC (without EH) can also be used for problem () as the EH constraints in (b) are convex and do not complicate the problem fundamentally. Here, we propose an efficient algorithm based on successive upper-bound minimization (SUM) [], which is similar to the method used in [] for IFC without EH constraints. As will be seen shortly, this method not only can be used for problem () but also can be conveniently extended to handle the design problems of the K-user TDMA scheme and the PS scheme. To proceed, let us write problem () as [ ( K ) w i log h H kis k h ki +σi max (S,...,S K ) S i= k= log ( k ih H kis k h ki +σ i )] () where S = {S i 0,i =,...,K K k= hh ki S kh ki E i,tr(s i ) P i, i =,...,K}. The idea is to iteratively approximate () by considering a locally tight lower bound of the weighted sum rate. Specifically, suppose that we have ) at the (n )th iteration. Then, due to the concavity of the logarithm function, the second logarithmic term in () is upper bounded by its first-order approximation, i.e., (S [n ],...,S [n ] K log ( k ih H kis k h ki +σ i + ) log ( k i ) h H kis [n ] k h ki +σi k i hh ki (S k S [n ] k )h ki ( () k i hh ki S[n ] k h ki +σi )ln. The bound above is tight when (S,...,S K ) = (S [n ],...,S [n ] K ). Thus, solving the following problem [ ( K ) {S [n] i } K i= = argmax w i log h H kis k h ki +σi {S i} K i= S i= τ [n ] i k= h H kis k h ki ], () k i where τ [n ] i ( k i hh ki S[n ] k h ki +σi )ln, is equivalent to maximizing a locally tight lower bound of problem () (up to a constant). Note that problem () is a convex SDP which can be solved efficiently by off-the-shelf solvers, e.g., CVX []. Steps of the SUM method for solving problem () are summarized in Algorithm. Convergence of Algorithm is a direct consequence of [, Theorem ], which is stated in the following proposition. Proposition Every limit point of the sequence {S [n],...,s[n] K } n= generated by Algorithm is a stationary point of problem (). B. K-User TDMS and TDMA Schemes We extend the practical TDMS and TDMA schemes in Section IV to the general K-user scenario.

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