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Robust Transceiver Optimization for Power-Splitting Based Downlink MISO SWIPT Systems Feng Wang, Tao Peng, Yongwei Huang, Member, IEEE, and Xin Wang, Senior Member, IEEE
Abstract—This letter considers a downlink multi-input single-out (MISO) system where each user performs simultaneous wireless information and power transfer (SWIPT) based on a power splitting receiver architecture. Assuming imperfect channel state information (CSI) at the base station, we develop two robust joint beamforming and power splitting (BFPS) designs that minimize the transmission power under both the signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constraints per user. In the first design, we consider the worst-case (WC) SINR and EH constraints, and show that the WC-BFPS problem can be relaxed as a semidefinite program (SDP) through a linear matrix inequality representation for (infinitely many) robust quadratic matrix inequality constraints. In the second design, we consider the chance constraints (CCs) for SINR and EH, and resort to both semidefinite relaxation and Bernstein-type inequality restriction to transform the CC-BFPS problem into another convex SDP. Based on these convex reformulations, the (near-)optimal robust BFPS designs can be efficiently solved. Numerical results are provided to demonstrate the merit of the proposed robust designs. Index Terms—Bernstein-type inequality, energy harvesting, power splitting, robust beamforming, semidefinite program.
I. INTRODUCTION
E
NERGY HARVESTING (EH) wireless communications have received growing attentions from both industry and academia. Besides harvesting from renewable sources such as solar and wind, a new emerging technique is to harvest energy from the ambient radio-frequency (RF) signals [1]. Since RF signals can carry both energy and information at the same time, simultaneous wireless information and power transfer (SWIPT) has been recently proposed [2]–[12]. The studies on SWIPT are mainly inspired by the information-theoretic work [2], where the fundamental capacity-energy tradeoffs were addressed. As simultaneous information processing (IP) and EH by the same circuit are not possible with existing technologies, two practical receiver architectures, i.e., time-switching (TS) and power-
Manuscript received December 27, 2014; revised February 24, 2015; accepted March 03, 2015. Date of publication March 09, 2015; date of current version March 11, 2015. This work was supported in part by the China Recruitment Program of Global Young Experts, the Program for New Century Excellent Talents in University, and by the Innovation Program of Shanghai Municipal Education Commission. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Michael Rabbat. F. Wang, T. Peng, and X. Wang are with the the Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, 200433 Shanghai, China (e-mail:
[email protected]). Y. Huang is with the Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2015.2410833
splitting (PS), were proposed in [3], [4]. The TS-based schemes were studied for the information and energy beamforming designs in [5], [6]. For the PS-based architecture, optimal joint downlink beamforming and PS (BFPS) designs were pursued for multi-input single-out (MISO) downlink [7] and interference channels [8], [9]. For these nonconvex BFPS problems, [7], [8] adopted semidefinite relaxation techniques, while [9] resorted to an alternative second-order cone program relaxation method. It is worth noting that TS can be regarded as a special form of PS with only binary PS ratios [3]. All works in [3]–[9] assumed that perfect channel state information (CSI) is available at the base station (BS). However, perfect CSI is rarely available in practical systems. Taking into account the imperfect CSI, robust beamforming design for a three-node SWIPT system was investigated in [10]. The robust SWIPT designs were also recently pursued for MISO multicast [11] and secrecy communications [12]. Based on the practical PS receiver architecture, this letter considers a downlink MISO SWIPT system. Assuming imperfect CSI at the BS, we develop two robust BFPS designs that minimize the transmission power subject to both the signal-to-interference-plus-noise ratio (SINR) and EH constraints per user. In the first design, we consider the worst-case (WC) constraints for SINR and EH. Quite different from the approaches in [7]–[9], we reformulate the WC-BFPS problem into a quadratic matrix inequality (QMI) problem by the Schur complement, and then further rely on the linear matrix inequality (LMI) reformulation for the semi-infinite QMI constraints to construct a tractable rank-constrained SDP. In the second design, we consider the chance constraints (CCs) for SINR and EH. Relying on both semidefinite relaxation and Bernstein-type inequality restriction, we show that the CC-BFPS problem can be relaxed into another SDP. Building on these convex reformulations, the (near-)optimal robust BFPS schemes can be efficiently obtained in polynomial time. The reminder of this letter is organized as follows. Section II introduces the system model. In Section III, we develop the SDP-based solutions for the WC-BFPS and CC-BFPS designs. The proposed schemes are numerically tested in Section IV, followed by the conclusion in Section V. II. SYSTEM MODEL Consider a downlink MISO SWIPT scenario where the BS equipped with antennas transmits to single-antenna users. With the linear beamforming strategy, the transmitted signal is (1) where and are the complex beamforming vector and the intended signal for user , respectively. Without , where loss of generality, we assume
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denotes a circularly symmetric complex Gaussian (CSCG) variand able with mean and covariance . Let denote the conjugated complex downlink channel vector and the noise for user , respectively. Then, the received signal at user is (2) where denotes the Hermitian transpose. With a power splitter, each SWIPT user splits its received signal into two streams: one stream is directed towards the IP circuit for information decoding, and the other is driven to the EH circuit for denote the PS ratio per user power scavenging. Let . The signal for the IP circuit is (3)
where and are the prescribed SINR target and EH target for user , respectively. The goal of (9) is to optiand PS ratios such that the mize the beamformers total transmission power is minimized while both the SINR and EH per user are always kept above certain levels. Problem (9) is nonconvex. It is particularly challenging since we have infinitely many quadratic inequalities for SINR and EH constraints. To make (9) computationally tractable, we recast the SINR and EH constraints into QMIs, and then obtain the LMI representations for these QMIs. , To this end, let , , and . The SINR constraints in (9) can be then rewritten as the following QMI constraints , such that by the Schur complement:
where is the noise introduced by the IP circuit. The SINR measurement for IP at user is given by (4)
(10) where indicates that the matrix is positive semidefinite. Likewise, the EH constraints in (9) are given by
where denotes the absolute value. On the other hand, the signal for the EH circuit of user is (5) The power from the EH circuit at user
is then given by
(11)
(6) is the ensemble average over baseband symbol duwhere is the EH conversion efficiency. ration, and
To get the LMI representations for (10) and (11), we resort to [13, Theorem 4.2] and cite it as a lemma. , then the Lemma 1: (Theorem 4.2, [13]) If following QMI system
III. ROBUST JOINT BEAMFORMING AND POWER SPLITTING DESIGN In practical systems, perfect CSI is rarely available at the BS due to the estimation errors, limited CSI feedback quantization and delays, etc. For this reason, we consider the following additive CSI uncertainty model:
(12) is equivalent to the LMI system:
, such that
(7) where is the actual channel vector, is the is the CSI error. estimated channel vector, and A. The WC-BFPS Design In the CSI uncertainty model (7), for simplicity we further assume that the CSI error is norm-bounded by , i.e.,
(13) where is the identity matrix of an appropriate dimension. To proceed, we can set
(8)
in (12). Based on Lemma 1, it then follows , such that that the LMIs for (10) are:
Considering the worst-case SINR and EH constraints, the WC-BFPS design is formulated as the following robust optimization problem:
(14)
Similarly, we obtain the LMIs for (11):
(9)
, such that
(15)
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With the LMIs in (14) and (15), we reformulate (9) as1:
(20) (21) (22) (16)
where denotes the trace of a matrix. Observe that (16) . is nonconvex only due to the rank-one constraints for Dropping these rank-one constraints, (16) becomes a convex SDP which can be efficiently solved via an interior-point method [18]. The optimal value of the relaxed (16) actually provides a lower bound for the original (16). Denote the optimal solution . If , to the relaxed (16) by then this relaxation is tight and one can obtain the optimal to (16) by performing eigenvalue decomposition for , i.e., . Otherwise, the lower bound is unachievable and only approximate solutions can be obtained for (16). In this case, we resort to a Gaussian randomization procedure (cf. [17]) . to generate an approximate solution of (16) from B. The CC-BFPS Design We next consider another robust BFPS design, i.e., the CC-BFPS design, where both SINR and EH constraints are expressed in the CC forms. Suppose that the channel error probability distribution in (7) is known. In particular, we assume that (17) is the given covariance matrix for where CC-BFPS design is then formulated as:
. The
where
and
. With these terms, the SDR of (18) is given by:
(23) denotes the real part of a complex where number. Clearly, one can get the equivalent problem , of (18) by adding rank-one constraints for in (23). Step 2- Bernstein-type Inequality Restriction: Noting and that are quadratic with respect to the Gaussian random vector , the Bernstein-type inequality for Gaussian quadratic forms can be used to construct an efficiently computable convex restriction for the CCs in (23). To proceed, we cite the result in [15] as a lemma. Lemma 2: (Bernstein-type Inequality Restriction,[15]) For and , any ( the sufficient condition for (24)
(18)
is the following system of convex SDP inequalities:
(18a) (18b) (18c) (18d) where and are the prescribed maximal outage probabilities of SINR and EH for user , respectively. The parameter governs the SINR service fidelity, making sure that user is served with a satisfiable SINR no less than of the time. Likewise, governs the EH ( service fidelity. Problem (18) is nonconvex and hard to solve since the probability functions in (18a) and (18b) do not yield closed-form expressions. To circumvent this dilemma, we next develop a conservative approximation for (18). The key to our approach is a relaxation-and-restriction (RAR) methodology [14], [15]. This RAR approach consists of two steps–semidefinite relaxation (SDR) and Bernstein-type inequality restriction of (18a) and (18b). , and Step 1- SDR: Let
(25) denotes the set of all Hermitian matrices, where denotes the vector by stacking columns of a matrix, and denotes the Euclidean norm of a vector. Upon applying Lemma 2 to (23), we obtain:
(19) 1Note
that another way to reformulate (9) could be: replacing and respectively with auxiliary variables and , adding convex and , and transforming the robust constraints SINR and EH constraints into LMIs via S-procedure (see e.g. [13]). This formulation involves more optimization variables and constraints but with the size of each LMI constraint one less, in contrast to (16).
(26)
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Problem (26) is a convex SDP and thus can be solved efficiently be the optimal solutions of (26). If rank( [18]. Let , , then we can perform eigenvalue decomposition to obtain for (18). Otherwise, a Gaussian randomization procedure can be then employed to yield an approximate solution for (18). Before leaving this section, we highlight that the performance of the CC-BFPS design (18) and the WC-BFPS design (9) can ) be compared by exploring the relationship between ( and , as shown in a different context [15]. IV. NUMERICAL RESULTS In this section, numerical results are provided for an MISO antennas SWIPT system where the BS is equipped with and each of the users has single antenna. Assume that . For the WC-BFPS design, the channel . For the CC-BFPS error norm-bounds are design, the covariance matrices . In the simu. For simplicity, all SWIPT users have lations, we set the same SINR and EH targets, i.e., . We set the noise variances dBm, dBm and . The SINR the energy convention efficiency and EH outage probabilities in the CC-BFPS design are set as . All results are averaged over 2000 independent channel realizations and 1000 randomizations are generated in the Gaussian randomization procedure for the relaxed (16) and (26). Fig. 1 shows the performance of the average minimal transmission power versus the EH target with and dB for different . For comparison, we include the performance of the nonrobust BFPS design [7], which is based on the estimated channels , . As can be seen, the proposed WC and CC schemes achieve almost the same performance as their relaxed SDPs, respectively. Among the numerical results, we observe that about 80% of simulation runs provide rank-one solutions for the relaxed (16) and (26). This implies that the tightness of SDP relaxation in SWIPT contexts is not always guaranteed in general. In fact, it was shown in [16] that when the CSI uncertainties are sufficiently small, the SDP relaxation of (16) without EH constraints is tight. However, it remains to be analytically understood the conditions that the SDP relaxation is tight in robust SWIPT contexts. Compared to the nonrobust design, the proposed robust schemes require more transmission power, since the robust ones account for the channel errors when meeting the required constraints. Besides, when increases from 4 to 8, Fig. 1 shows that the BS transmission power significantly decreases for all schemes. This demonstrates the benefit by employing more transmit antennas in MISO SWIPT systems. In Fig. 2, the cumulative distribution functions (CDFs) of the achieved SINR and EH for users are provided for dB and dBm. It is observed that the proposed robust schemes achieve higher SINR and EH values than the prescribed targets with a very high probability, while the nonrobust one only meets these targets with about 50%. This implies that the less transmission power of the nonrobust scheme, shown in Fig. 1, is at the price of severely violating the constraints. As expected, Fig. 2 shows that the more conservative WC scheme provides higher SINR and EH than the CC one. Also, we observe that the conservatism of the robust schemes is
Fig. 1. The average transmission power v.s.
with
Fig. 2. CDFs of the achieved SINR and EH for dBm.
and
,
dB.
dB, and
safely relieved when increases from 4 to 8. This again demonstrates the benefit by applying more antennas for efficiently implementing MISO SWIPT systems in practice. V. CONCLUSION Based on the PS receiver architecture, we developed the optimal robust WC-BFPS and CC-BFPS designs for downlink MISO SWIPT systems. Resorting to the LMI representations for the infinitely many robust QMIs of SINR and EH, we showed that the WC-BFPS problem can be relaxed as an SDP. Relying on both SDR and Bernstein-type inequality restriction for chance constraints, the CC-BFPS problem was recast as another SDP. By solving these convex SDPs, the (near-)optimal robust BFPS designs can be then efficiently obtained. Simulations showed the tightness of the proposed SDP relaxation is not always guaranteed in general. It would be interesting to analytically derive the conditions that these relaxations are tight. Generalizations of the proposed approaches to multi-cell MIMO channels might be an also interesting direction to pursue in our future work.
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