Wireless Information and Energy Transfer in Multi-Antenna Interference Channel

Wireless Information and Energy Transfer in Multi-Antenna Interference Channel

Chao Shen, Wei-Chiang Li and Tsung-Hui Chang
The work of Chao Shen is supported by the Opening Project of The State Key Laboratory of Integrated Services Networks, Xidian University (Grant No. ISN14-09), the China Postdoctoral Science Foundation (Grant No. 2013M530519), the National Natural Science Foundation of China (Grant No. 61222105), the Key Project of State Key Lab of Rail Traffic and Control (Grant No. RCS2012ZZ004), Beijing Jiaotong University, the Key grant Project of Chinese Ministry of Education (No. 313006), and the Fundamental Research Funds for the Central Universities (Grant No. 2010JBZ008 and 2012YJS017). The work of Tsung-Hui Chang is supported by National Science Council, Taiwan (R.O.C.), by Grant NSC 102-2221-E-011-005-MY3. Part of this work was presented in IEEE GLOBECOM 2012 [1].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. E-mail: chaoshen@ieee.org.Wei-Chiang Li is with Institute of Communications Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan, (R.O.C.). E-mail: 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 10607, Taiwan, (R.O.C.). E-mail: tsunghui.chang@ieee.org.
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. Different from the conventional IFCs without energy harvesting, the cross-link signals in the considered scenario play two opposite roles in information detection (ID) and energy harvesting (EH). It is observed that the ideal scheme, where the receivers can simultaneously perform ID and 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 time division mode switching (TDMS) and time division multiple access (TDMA), in addition to a 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 -user scenario, a successive convex approximation method is proposed to handle the WIET problems associated with the ideal scheme and the PS scheme, which are known to be NP-hard in general. The -user TDMS and TDMA schemes are shown efficiently solvable as convex problems. Simulation results show that stronger cross-link channel powers actually improve the information sum rate under energy harvesting constraints. Moreover, none of the schemes under consideration can dominate another in terms of the sum rate performance.

Index terms wireless energy transfer, energy harvesting, interference channel, beamforming, convex optimization
EDICS: SPC-APPL, SPC-INTF, SPC-CCMC, SAM-BEAM

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., [2, 3, 4, 5, 6]. 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 [7, 8] showed that, using coupled magnetic resonances, energy can be wirelessly transferred for two meters with over energy conversion efficiency. WET can also be achieved via the RF electromagnetic signals; see [9, 10] for recent developments of RF-based energy harvesting circuits. 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 bits but also harvest energy from the received RF signals, i.e., wireless information and energy transfer (WIET) systems [11, 12, 13, 14, 15, 16, 17]. Specifically, in [11], the optimal tradeoff between information capacity and energy transfer of the WIET system was studied for a SISO flat fading channel. In [12], the optimal power allocation strategy for a SISO frequency-selective fading channel was derived under a receiver energy harvesting constraint. The work in [13] further extends these studies to the multiple access channel (MAC) and two-hop relay network with an energy harvesting relay. It was shown that in general there exist nontrivial tradeoffs between information transfer and energy harvesting. The works in [11, 12, 13] assume 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 [14, 15] a dynamic power splitting (PS) scheme for a SISO flat fading channel, wherein, the received RF signal is either used for information detection (ID), energy harvesting (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 [16] further proposed a time switching scheme where the receiver performs ID in one time slot while EH in the other time slot. In [17], the dynamic PS scheme was extended to a multi-user multiple-input single-output (MISO) broadcast channel, and the optimal transmit beamforming and power splitting coefficients are jointly optimized to minimize the transmission power subject to information rate and energy harvesting constraints.

In this paper, we consider a -user MISO interference channel and study the optimal transmission strategies for WIET. We first consider the ideal receivers, and formulate the design problem as a weighted sum rate maximization problem subject to individual energy harvesting constraints and power constraints. 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. And it turns out 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 may instead yield better sum rate performance. This is in sharp contrast to the scenarios studied in [14, 15, 16, 17] 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) scheme111As 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., in addition to the PS scheme [15]. In the TDMS scheme, the transmission time is divided into two 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 time slots, and in each time slot, one receiver performs ID while the others perform EH. We analytically show how the design problems associated with the three schemes can be efficiently handled. Specifically, for the two-user scenario, we show that transmit beamforming is an optimal transmission strategy for all schemes. Moreover, the design problems associated with the TDMS and TDMA schemes admit semi-analytical solutions in the two-user scenario and can be solved as convex problems in the general -user scenario. Since the WIET design problems associated with the ideal scheme and the PS scheme in the -user scenario are NP-hard in general, we further present an efficient approximation method based on the log-exponential reformulation and successive convex approximation techniques [18]. The presented simulation results will show that stronger cross-link channel powers actually improve the information sum rate under energy harvesting constraints. Moreover, the three schemes do not dominate each other in terms of sum rate performance. Roughly speaking, 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 scheme can perform best. In some interference dominated scenarios, the TDMS scheme and TDMA scheme even outperform the ideal scheme.

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 and TDMA schemes, which are presented in Section IV. Section V extends the study to the general -user scenario; the design problem of the PS scheme is also presented in that section. Simulation results are presented in Section VI. The conclusions and discussion of future researches are given in Section VII.

Notations: Column vectors and matrices are written in boldfaced lowercase and uppercase letters, e.g., and . The superscripts , and represent the transpose, (Hermitian) conjugate transpose and matrix inverse, respectively. and represent the rank and trace of matrix , respectively. () means that matrix is positive semidefinite (positive definite). denotes the Euclidean norm of vector . The orthogonal projection onto the column space of a tall matrix is denoted by . Moreover, the projection onto the orthogonal complement of the column space of is denoted by where is the identity matrix.

Ii Signal Model and Problem Statement

We consider a multi-user interference channel with pairs of transmitters and receivers communicating over a common frequency band. Each of the transmitters is equipped with antennae, while each of the receivers has single antenna. Let be the signal vector transmitted by transmitter , and be the channel vector from transmitter to receiver , for all . The received signal at receiver is given by

(1)

where is the additive Gaussian noise at receiver . Unlike the conventional MISO IFC [19] where the receivers focus only on extracting information, we consider in this paper that the receivers can also scavenge energy from the received signals [11, 12, 16], 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 contains the information intended for receiver which is Gaussian encoded with zero mean and covariance matrix , i.e., for . Moreover, assume that each receiver decodes by single user detection in the ID mode. Then the achievable information rate of receiver is given by

(2)

for . 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 [16]. Specifically, for receiver , the harvested energy, denoted by , can be expressed as

(3)

where is a constant accounting for the energy conversion loss in the transducer [16].

Suppose that the receivers desire to harvest certain amounts of energy. We are interested in investigating the optimal transmission strategies of , so that the information throughput of the -user IFCs can be maximized while the energy harvesting requirements of the receivers are satisfied at the same time. One should note that current energy harvesting receivers are not yet able to decode the information bits simultaneously [16]. In subsequent sections, we will first study an “ideal” scenario where the receivers can simultaneously operate in the ID mode and EH mode. Then, 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 (), and later extend the studies to the general -user case (in Section V).

Iii Optimal WIET Design for Ideal Scheme

Let us assume that 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 and , respectively. We are interested in the following transmitter design problem for WIET:

(4a)
(4b)
(4c)
(4d)
(4e)

where are positive weights, and and in (4d) and (4e) represent the individual power constraints. The constraints in (4b) and (4c) are the energy harvesting constraints where we have set for notational simplicity. Note that, in the absence of (4b) and (4c), problem (P) reduces to the classical sum rate maximization problem in MISO IFC [19]:

(5a)
(5b)
(5c)

It can be observed from (4) and (5) that the energy harvesting constraints (4b) and (4c) would trade the maximum achievable sum rate for energy harvesting; i.e., the maximum sum rate in (4a) is in general no larger than that in (5a). To see when this would happen, let be an optimal solution to problem (5). One can verify from the rate function in (2) and problem (5) that must satisfy

(6)
(7)

That is, the energies harvested at the two receivers due to must lie in . It can be shown that in ,

(8a)
(8b)

where . Equations in (8) implies that the two receivers can at lease harvest energies and , respectively. The minimum amounts of energies are achieved when , , and ; 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 (8), we have that

Property 1

The energy harvesting constraints (4b) and (4c) are inactive at the optimum if and ; hence, (P) reduces to the conventional MISO IFC problem (5) under this condition.

However, when or , the maximum information throughput may have to be compromised with energy harvesting. Interestingly, the following proposition shows that the optimal transmit structure of (P) is still similar to problem (5) which does not have the energy harvesting constraints.

Proposition 1

Assume that problem (P) is feasible, and that and without loss of generality. Let denote the optimal solution to problem (P). Then, and . Moreover, there exist , , such that

(9a)
(9b)

The proof is given in Appendix -A. Proposition 1 implies that beamforming is an optimal transmission strategy of (P). Moreover, the beamforming direction of transmitter should lie in the range space of , for , which is the same as the optimal beamforming direction of problem (5) in the conventional IFCs [19]. Given (9), the search of and in (P) reduces to the search of and over the ellipsoids for all However, unlike problem (5), optimizing the coefficients , , for problem (P) have to take into account both the needs of energy harvesting and information transfer.

Remark 1

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 plays two completely opposite roles in the considered scenario – It can boost the energy harvesting of receiver 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 1 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.

(a) Sum rate vs. EH requirement , for and dB. Parameter measures the cross-link channel power.
(b) Achievable rate region , for , , , and dB.
Fig. 3: Motivating simulation examples for the 2-user scenario.

To further look into this aspect, we present in Fig. 3 two simulation examples for the 2-user scenario. The detailed setting of the simulations are presented in Section VI. Fig. (a)a shows the sum rate-versus-energy requirement regions for two randomly generated channel realizations. The curves are obtained by exhaustively solving (P) for various values of symmetric energy requirement . The average powers of the direct link channels are normalized to one, while the average powers of the cross-link channels are measured by the parameter . As one can observe from this figure, for , the rate-energy region is not convex for this randomly generated channel realization. Moreover, for some values of , the receivers may achieve a higher sum rate through time sharing between the EH mode and ID mode (see the dashed line between point A and point B). Fig. (b)b displays the rate region ( versus ) of the two users. Analogously, we observe that time sharing for multiple access may achieve a higher sum rate (see the dashed line between points A and B).

The two simulation results in Fig. 3 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. This motivates us to develop two practical schemes, namely, the time-division mode switching (TDMS) scheme and the time-division multiple access (TDMA) scheme, in the next section. It is worthwhile to note that, in these time sharing schemes, the receivers operate either in the EH mode or ID mode at each time instant, and thus are more practical than the ideal receivers.

Iv Practical WIET Schemes and Optimal Transmission Strategies

Iv-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 fraction of the time is for EH mode and fraction of the time is for ID mode. The TDMS scheme is described as follows:

  • Time slot 1 (EH mode): The two receivers focus on harvesting the required energy and in fraction of the time, i.e.,

    (10a)
    (10b)
  • Time slot 2 (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.,

    (11a)
    (11b)

Problem (11) in the ID mode is the classical sum rate maximization problem in the MISO IFC [see (5)], which can be efficiently handled by existing methods in [19, 20, 21]. Note that it has been shown in [22, 23] that beamforming is an optimal transmission scheme for problem (11).

We now focus on the EH mode in time slot 1. Since time slot 1 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 to fulfill the energy harvesting task. Mathematically, we can write it as the following optimization problem

(12a)
(12b)
(12c)
(12d)

where . Note that if the optimal of (12) is less than one (i.e., optimal ), then it implies that the energy harvesting requirements (10) 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 (12) is a convex semidefinite program (SDP), which can be solved by the off-the-shelf solvers, we show that (12) actually admits a semi-analytical solution:

Proposition 2

Assume that and are linearly independent but not orthogonal to each other, for . The optimal solution to problem (12) is given by

(13a)
(13b)

where is the optimal dual variable associated with constraint (12b), and is the principal eigenvector of for . Moreover, can be efficiently obtained using a simple bisection search.

The proof of Proposition 2 is given in Appendix -B. The assumptions on and , for , hold with probability one for random (continuous) fading channels. Note that Proposition 2 also implies that beamforming is optimal for the EH mode of the TDMS scheme.

Iv-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 .

  • Time slot 1: Receiver 1 operates in the ID mode and receiver 2 operates in the EH mode. The objective is to maximize the information rate of receiver 1 and guarantee the energy harvesting requirement of receiver 2 at the same time. The design problem is given by

    (14a)
    (14b)
    (14c)
  • Time slot 2: The operation modes of the two receivers are exchanged:

    (15a)
    (15b)
    (15c)

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 1

The TDMA scheme is feasible if and only if

(16)

Proof: The TDMA scheme is feasible if and only if both (14) and (15) are feasible. Problem (14) is feasible if and only if there exists some such that

(17)

where the equality is obtained by applying the result in [16, Proposition 2.1]. Similarly, one can show that (15) is feasible if and only if

(18)

Combining (17) and (18) gives rise to (16). Conversely, given (16), let , and thus , which are (17) and (18), respectively. Hence, when (16) is true, the TDMA scheme is feasible.

According to (17) and (18), a feasible time fraction must lie in the interval

(19)

Interestingly, given a feasible , both problems (14) and (15) can be efficiently solved (semi-analytically). Since problems (14) and (15) are similar to each other, we take (14) as the example.

Proposition 3

Let the time fraction satisfy (19). Then, an optimal solution to problem (14), denoted by (, ), is given by

(20)

where

(21)

in which , for and . Problem (21) is a convex problem, and thus can be obtained by a bisection search.

The proof is presented in Appendix -C. We see from (20) that beamforming is also optimal to the TDMA scheme. By Proposition 3, given a feasible time fraction , one can efficiently solve problems (14) and (15) and thus evaluate the achievable sum rate of the two users. Then, the optimal time fraction that maximizes the sum rate of the two users can be obtained by line search over the interval in (19).

Iv-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 2 operating in the EH mode transmits deterministic signals which are known to receiver 1 operating in the ID mode. Under such circumstances, receiver 1 can actually remove from the received signal before information detection, i.e., removing the cross-link interference. The design problem in the 1st time slot thereby reduces to

(22a)
(22b)
(22c)

Problem (22) is easier to handle than its counterpart in (14). Clearly, given satisfying (19), optimal is given by Therefore, (22) boils down to

(23a)
(23b)

which admits a closed-form solution for according to [16, Proposition 2.1]. 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 order to do so, the two receivers require perfect knowledge of the cross-link channels and , respectively; otherwise, the receivers may suffer performance degradation due to imperfect interference cancelation.

We remark that, in addition to the above time sharing based schemes, it is also possible for the receivers to split the received signals into two parts, one for EH and the other for ID, i.e., power splitting (PS) [16]. This scheme will be studied in Section V-C.

V WIET Design for -user MISO IFC

In this section, we consider the WIET problem for the -user MISO IFC scenario. We begin with the ideal scheme, and in the second subsection, we extend the TDMS and TDMA schemes in Section IV to the -user scenario. In the last subsection, we further investigate the PS scheme.

V-a Transmitter Optimization for Ideal Receivers

By the signal model in (1), (2), (3) and (P) in (4), the -user WIET problem is formulated as

(24a)
(24b)
(24c)

where is the energy requirement of user , for . Since problem (24) is NP-hard in general [24], our interest for the -user WIET problem lies in efficient approaches to finding an approximate solution.

We propose an efficient algorithm based on successive convex approximation (SCA) [25] by adopting the log-exponential reformulation idea in [18]. Compared to the methods in [19, 20, 21], the proposed method can work for scenarios with a medium to large number of users. Specifically, by introducing slack variables , we can reformulate problem (24) as

(25a)
(25b)
(25c)
(25d)

As seen, the rate functions in (24a) are equivalently decomposed into the objective function in (25a) and the two constraints in (25b) and (25c). In particular, one can verify that constraints (25b) and (25c) will hold with equality at the optimum, implying that (25) is equivalent to (24).

Problem (25) has a linear objective function and convex constrains, except for constraint (25c). We propose to linearly approximate constraint (25c) in an iterative manner. Suppose that, at iteration , we are given . Let , . We solve the following problem at the th iteration

(26a)
(26b)
(26c)
(26d)

Note that constraint (26c) is convex; it is a conservative approximation to (25c) since it holds that due to the convexity of . As a result, problem (26) is a convex SDP which can be solved efficiently by off-the-shelf solvers, e.g., CVX [26]. Detailed steps of the proposed algorithm is summarized in Algorithm 1.

1:  Find initial variables by solving the feasibility problem
If the problem is infeasible, then declare infeasibility of (24); otherwise, set and perform the following steps.
2:  repeat
3:     .
4:     
5:     Solve problem (26) to obtain .
6:  until the stopping criterion is met.
7:  Output as an approximate solution.
Algorithm 1 SCA algorithm for problem (24)

It can be shown that Algorithm 1 belongs to the category of the successive upper-bound minimization (SUM) method proposed in [27] and can converge to a stationary point of problem (24), as stated in Proposition 4. The details are relegated to Appendix -D.

Proposition 4

Any limit point of the sequence generated by Algorithm 1 is a stationary point of problem (24).

V-B Practical -User WIET Schemes

We extend the TDMS and TDMA schemes in Section IV to the general -user scenario in this subsection.

1) -user TDMS scheme: This scheme is similar to the TDMS scheme presented in Section IV-A. In the 1st time slot, all users operate in the EH mode, and in the 2nd time slot, all users operate in the ID mode; see Fig. (a)a. In the 1st time slot, the optimal time fraction and the associated optimal signal covariance matrices for energy harvesting can be obtained by solving a convex problem analogous to problem (12). In the 2nd time slot, one has to solve the classical sum rate maximization problem