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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 9, SEPTEMBER 2014

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Robust Transceiver Design for Wireless Information and Power Transmission in Underlay MIMO Cognitive Radio Networks Canhao Xu, Qi Zhang, Quanzhong Li, Yizhi Tan, and Jiayin Qin

Abstract—In this letter, we investigate the robust transceiver design problem for simultaneous wireless information and power transfer in multiple-input-multiple-output underlay cognitive radio networks where the channel uncertainties are modeled by the worst-case model. Our objective is to maximize the sum harvested power at energy harvesting receivers while guaranteeing the required minimum mean-square-error at the secondary information-decoding (ID) receiver and the interference constraints at the primary receivers. We propose to alternatively optimize the transmit covariance matrix at secondary transmitter and the preprocessing matrix at secondary ID receiver. Simulation results have shown that the robust transceiver design has significant performance gain over the non-robust one. Index Terms—Cognitive radio (CR), energy harvesting (EH), multiple-input-multiple-output (MIMO), robustness, simultaneous wireless information and power transfer (SWIPT).

I. I NTRODUCTION

C

OGNITIVE RADIO (CR) is a promising technology to alleviate the spectrum shortage problem. In [1], [2], considering that the secondary user (SU) transmitter knows the imperfect channel state information (CSI) from itself to the primary user (PU) receiver where the channel uncertainties are modeled by the worst-case model, the robust transmission schemes for CR networks were proposed. The aforementioned works have not considered the energy harvesting (EH) problems. The simultaneous wireless information and power transfer (SWIPT), which is belong to EH techniques, is promising to solve the energy scarcity problem in energy-constrained wireless networks. In practice, numerous renewable energy sources can be exploited for EH, including solar, tide, geothermal, and wind. However, these natural energy sources are usually location, weather, or climate dependent and may not always be available in enclosed/indoor environments or suitable for mobile devices [3]. The SWIPT schemes for multiple-input-multiple-output (MIMO) channels have been investigated in [4], [5]. The SWIPT scheme for MIMO relay networks was studied in [6]. Considering that the transmitter knows the imperfect CSI, the robust transmission schemes with SWIPT for multiple-input-single-output (MISO) channels were proposed in [7], [8] where the channel uncertainties are

Manuscript received April 16, 2014; accepted July 12, 2014. Date of publication July 18, 2014; date of current version September 8, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61173148 and Grant 61202498 and in part by the Scientific and Technological Project of Guangzhou City under Grant 12C42051578. The associate editor coordinating the review of this paper and approving it for publication was H. A. Suraweera. The authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, China (e-mail: xucanhao@ mail2.sysu.edu.cn; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/LCOMM.2014.2340851

modeled by the worst-case model. The robust secure beamforming for SWIPT in MISO channels was studied in [9], [10]. Considering underlay CR networks, the robust secure underlay beamforming for SWIPT in MISO channels was investigated in [11]. It is noted that in [11], the MISO channels are considered. To our best knowledge, the research on the robust transceiver design for SWIPT in the underlay MIMO CR networks is missing. In this letter, we investigate the robust transceiver design problem for SWIPT in the underlay MIMO CR networks where the channel uncertainties are modeled by the worst-case model. Our objective is to maximize the sum harvested power at the EH receivers while guaranteeing the required minimum meansquare-error (MSE) at the SU information-decoding (SU-ID) receiver and the interference constraints at the PU receivers. We propose to alternatively optimize the transmit covariance matrix at the SU transmitter and the preprocessing matrix at the SU-ID receiver. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The AT , A† , AF , and tr(A) denote the transpose, conjugate transpose, Frobenius norm and trace of the matrix A, respectively. By A  0, we mean that A is positive semidefinite. a2 denotes the Euclidean norm of vector a. Re{·} denotes the real part of the variable in the bracket. II. S YSTEM M ODEL Consider an underlay MIMO CR network which consists of one SU transmitter, one SU-ID receiver, K SU-EH receivers, and J PU receivers. The SU transmitter is equipped with M antennas. The SU-ID receiver, each SU-EH receiver, and each PU receiver are equipped with N1 , N2 , and N3 antennas, respectively. The channel responses from the SU transmitter to the SU-ID receiver, the kth, k ∈ K = {1, 2, . . . , K}, SUEH receiver, and the jth, j ∈ J = {1, 2, . . . , J}, PU receiver are denoted as H ∈ CN1 ×M , Gk ∈ CN2 ×M and Qj ∈ CN3 ×M , respectively. We assume that the SU transmitter knows the imperfect CSI on H, Gk , and Qj where the channel uncertainties are modeled by the worst-case model as in [1]. The channel uncertainties of H, Gk , and Qj are bounded by the regions, denoted as H, Gk , and Qj , respectively,   ˆ + ΔH, ΔHF ≤ δH , H = H|H = H (1)   ˆ k + ΔGk , ΔGk  ≤ δG , (2) Gk = Gk |Gk = G k F   ˆ j + ΔQj , ΔQj  ≤ δQ Qj = Qj |Qj = Q (3) j F ˆ G ˆ k , and Q ˆ j denote the for k ∈ K and j ∈ J , where H, channel estimates of H, Gk and Qj , respectively; ΔH, ΔGk , and ΔQj denote the channel uncertainties of H, Gk and

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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 9, SEPTEMBER 2014

Qj , respectively; δH , δGk , and δQj denote the radii of the uncertainty regions for H, Gk , and Qj , respectively. It is noted that if the communications between the PUs are two-way communications, the SU transmitter may know the channel estimates from itself to the PU receivers because of the property of channel reciprocal. In [12], the effect of channel estimation errors was investigated due to practical difficulties in obtaining perfect CSI on the aforementioned channels. From the feedbacks of the SU-EH receivers, the SU transmitter may know the channel estimates from itself to the SU-EH receivers. Let x ∈ CM ×1 denote the transmitted signal. The received signal at the SU-ID receiver, denoted as y ∈ CN1 ×1 , is expressed as y = Hx + z

(4)

N1 ×1

where z ∈ C denotes the additive Gaussian noise vector whose entries are with zero mean and variance σz2 . In (4), the covariance matrix of the transmitted signal, x, is S = E[xx† ]. After receiving y, the SU-ID receiver estimates the transmitted ˆ = Dy, where D is the preprocessing signal x by employing x matrix at the SU-ID receiver. In this letter, our objective is to investigate the robust transceiver design problem which maximizes the sum harvested power at the SU-EH receivers while guaranteeing the required minimum MSE at the SU-ID receiver and the interference constraints at the PU receivers. The MSE of SU-ID receiver is given by     MSE = E ˆ x − x22 = E (DH − I)x + Dz22 . (5) The harvested power at the kth SU-EH receiver is given by    (6) E Gk x22 = tr Gk SG†k . Considering the worst-case channel uncertainties, we formulate the robust transceiver design optimization problem as max S,D

K

k=1

 min tr Gk SG†k

ΔGk

s.t. tr(S) ≤ P, max MSE ≤ μ,  ΔH max tr Qj SQ†j ≤ Ij , ∀ j ∈ J ΔQj

(7)

where P is the transmit power constraint, μ is the required minimum MSE at the SU-ID receiver and Ij is the interference constraint at the jth PU receiver. The problem (7) is equivalent to the following optimization problem max S,D

K

τk

(8a)

k=1

s.t. tr(S) ≤ P, MSE  ≤ μ, ∀ ΔH ∈ H, tr Qj SQ†j ≤ Ij , ∀ ΔQj ∈ Qj , j ∈ J ,  tr Gk SG†k ≥ τk , ∀ ΔGk ∈ Gk , k ∈ K.

(8b) (8c) (8d) (8e)

The problem (8) has semi-infinite constraints (8c)–(8e), which are intractable. To make the problem tractable, we employ the S-Procedure [13] to convert the constraints (8c)–(8e). To apply the S-Procedure on the interference constraint at the jth PU receiver (8d), we rewrite it as follows  (9) tr Qj SQ†j = vec(Qj )† Fp vec(Qj ) where Fp is a Hermitian matrix, F p = S T ⊗ IN3 .

(10)

Substituting (3) into (9), we have  tr Qj SQ†j ˆ j )† Fp vec(Q ˆ j )+vec(Q ˆ j )† Fp vec(ΔQj ) = vec(Q † ˆ j )+vec(ΔQj )† Fp vec(ΔQj ). (11) + vec(ΔQj ) Fp vec(Q 2 . We rewrite From (3), we have vec(ΔQj )† vec(ΔQj ) ≤ δQ j the interference constraint at the jth PU receiver (8d) as  2 ∀ ΔQj : vec(ΔQj )† vec(ΔQj ) − δQ ≤ 0, j  (12) † tr Qj SQj − Ij ≤ 0.

Applying the S-Procedure [13], we convert (8d) into

ˆ j) −Fp vec(Q λj I − Fp 0 ˆ j )† F p ξj −vec(Q

(13)

ˆ j )† F p where λj ≥ 0 is a slack variable and ξj = −vec(Q 2 ˆ vec(Qj ) + Ij − λj δQj . Similarly, applying the S-Procedure on the EH constraint at the kth SU-EH receiver (8e), we have

ˆ k) Fg vec(G η k I + Fg 0 (14) ˆ k )† F g ζk vec(G where ηk ≥ 0 is a slack variable, Fg = ST ⊗ IN2 , and ζk = ˆ k ) − τk − η k δ 2 . ˆ k )† Fg vec(G vec(G Gk To apply the S-Procedure on (8c), we rewrite the expression of MSE as follows   MSE = tr (DH − I)S(DH − I)† + σz2 tr(DD† ) = vec(DH − I)† Fs vec(DH − I) + σz2 vec(D)† vec(D) (15) where Fs = ST ⊗ IM . Using the similar method, we convert (8c) into

ρI − R† Fs R R† Fs v 0 (16) κ v † Fs R where ρ ≥ 0 is a slack variable, R = IM ⊗ D, v = ˆ and κ = −v† Fs v − σ 2 vec(D)† vec(D) + μ − vec(I − DH), z 2 ρδH . Thus, we transform the problem (8) into the following optimization problem, max

S,D,τk ≥0,ρ≥0,ηk ≥0,λj ≥0

K

τk s.t. tr(S) ≤ P, (13), (14), (16).

k=1

(17) III. T HE ROBUST T RANSCEIVER D ESIGN In this section, we present an alternative optimization algorithm for the robust transceiver design.

The problem (17) is still non-convex because the constraint (16) contains the multiplication of S and D. We propose an alternating optimization algorithm to solve the problem (17).

XU et al.: ROBUST TRANSCEIVER DESIGN FOR WIRELESS INFORMATION AND POWER TRANSMISSION

When the preprocessing matrix, D, is obtained, the problem (17) is a convex semidefinite programming (SDP), which can be solved effectively using the interior point method [14]. When the covariance matrix of the transmitted signal, S, is obtained, the problem (17) cannot be solved directly. We propose the algorithm to solve the optimization problem with the known S as follows. Given S, the optimal D should minimize the MSE. The expression of MSE can be rewritten as   1 2  MSE = (DH − I)S 2  + σz2 D2F F

    2  ˆ 12 + vec DΔHS 12 − vec S 12  = vec DHS  2

+

σz2 vec(D)22 .

Considering that vec(ΔH)2 ≤ δH , we can rewrite the MSE constraint (24) as

Θ − αψ † ψ −δH Φ† 0 (28) −δH Φ αI where α is a nonnegative slack variable. It is noted that in the left-hand side of (28), only Θ and Φ contain D. Furthermore, Θ and Φ are linear functions of D. Thus, given S, (28) is convex. After obtaining the covariance matrix of the transmitted signal, S, we can further minimize the MSE to obtain the matrix D. Therefore, the problem with the obtained S is rewritten as

(18)

min

D,ϕ≥0,α≥0

Thus, the MSE can be obtained as the norm-squared form of a vector w, w = u + Lvec(ΔH) where

 u=

  ˆ 12 vec DHS

(19)



 1  vec S 2 − , 0(M N1 )×1 

σz vec(D)   T 1 S2 ⊗D . L= 0(M N1 )×(M N1 )

(20)

(21)

(22)

where ϕ is a slack variable which should be minimized. The (22) can be rewritten as a linear matrix inequality (LMI) as follows

ϕ w†  0. (23) w I Substituting (19) into (23), we have Θ  Φ† vec(ΔH)ψ + ψ † vec(ΔH)† Φ

(24)

where Φ = [0M N1 ×1 L† ], ψ = [−1 01×(M M +M N1 ) ], and

ϕ u† Θ= . (25) u I The following derivation requires the lemma below, whose proof can be found in [15]. Lemma 1: Let A, B, and C be given matrices, with A = A† . The relation A  B† ΩC + C† Ω† B, ∀ Ω : ΩF ≤ ε is valid, if and only if  A − λC† C ∃λ ≥ 0, −εB

−εB λI

 †

 0.

(26)

(27)

ϕ

s.t. (28).

(29)

The optimization problem (29) is a convex SDP with linear objective function and LMI constraint. Therefore, we propose to alternatively optimize S and D to obtain the robust transceiver design. Remark: Because both the optimization problems (17) and (29) are convex, alternative optimization of S and D will only increase or maintain the objective value of (8). By the alternative optimization of S and D, we obtain a monotonically increasing sequence of the objective values of (8) which has the upper bound due to transmit power constraint. Therefore, the alternative optimization converges to a local optimum of the problem (8).

Since given D, the optimization of S by solving (17) ensures that the obtained MSE is less than or equal to μ. Given S, the optimization of D is that w22 ≤ ϕ

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IV. S IMULATION R ESULTS In this section, we evaluate the performance of the proposed robust transceiver design through computer simulations. We assume that the underlay MIMO CR network consists of one SU transmitter, one SU-ID receiver, K = 3 SU-EH receivers, and J = 2 PU receivers. The SU transmitter, the SU-ID receiver, each SU-EH receiver, and each PU receiver are equipped with M = N1 = N2 = N3 = 3 antennas. The entries in the channel ˆ G ˆ k , k ∈ K, and Q ˆ j , j ∈ J , are independent and estimates H, identically distributed (i.i.d.) complex random variables whose amplitudes are Rician distributed with Rician factor 3 dB. The signal attenuations from the SU transmitter to the SU-ID receiver, to each SU-EH receiver and to each PU receiver are 80 dB, 20 dB, and 85 dB, respectively. We assume that the transmit power constraint, P , is 30 dBm and the noise covariance, σz2 , is −60 dBm. The interference constraint to noise power ratio, Ij /σz2 , j ∈ J , is 0 dB. Our simulation results are averaged over 1000 randomly generated channel realizations. We define the normalized radii of the uncertainty regions for H, Gk , and Qj as δ˜H , δ˜Gk , δ˜Qj , respectively, which are 2 = δ˜H

2 2 δ2 δG δH k ˜2 = ˜2 =  Qj  . , δ , δ G Q j k E [H2F ] E [Gk 2F ] E Qj 2F (30)

In Fig. 1, we compare the worst-case sum harvested power obtained by our proposed robust transceiver design (denoted as “Robust” in the legend) and that obtained by the non-robust one (denoted as “Non-Robust” in the legend) for different

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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 9, SEPTEMBER 2014

In Fig. 2, we present the percentage of outage for different required minimum MSE at the SU-ID receiver. It is observed from Fig. 2 that the percentage of outage of our proposed robust transceiver design is always 0. When the required minimum MSE is μ = 0.25, the percentage of outage of non-robust transceiver design is higher than 0.78. V. C ONCLUSION

Fig. 1. Worst-case sum harvested power versus the required minimum MSE at the SU-ID receiver, μ; performance comparison of our proposed robust transceiver design and the non-robust one.

Fig. 2. Percentage of outage versus the required minimum MSE at the SU-ID receiver, μ; performance comparison of our proposed robust transceiver design and the non-robust one.

δ = δ˜H = δ˜Gk = δ˜Qj , k ∈ K, j ∈ J . The non-robust transceiver design is obtained by solving (8) where ΔH = 0, ΔGk = 0, and ΔQj = 0. After obtaining S and D, we compute the worstcase sum harvested power. If the required minimum MSE at the SU-ID receiver or the interference constraint at the PU receiver is not satisfied, an outage occurs. In Fig. 1, the worst-case sum harvested power obtained by using the perfect CSI (denoted as “Perfect CSI” in the legend) is also presented, which serves as the performance upper bound for the our proposed robust transceiver design. From Fig. 1, it is found that without considering the required minimum MSE and the interference constraints, the non-robust transceiver design obtains higher worst-case sum harvested power than the robust one.

In this letter, we propose the robust transceiver design for SWIPT in the MIMO CR networks where the channel uncertainties are modeled by the worst-case model. We employ the alternative optimization scheme to alternatively optimize the transmit covariance matrix at the SU transmitter and the preprocessing matrix at the SU-ID receiver. Simulation results have shown that the robust transceiver design has significant performance gain over the non-robust one. Further work may be carried out on robust transceiver design for SWIPT in underlay MIMO cognitive relay networks. R EFERENCES [1] L. Zhang, Y. C. Liang, Y. Xin, and H. V. Poor, “Robust cognitive beamforming with partial channel state information,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4143–4153, Aug. 2009. [2] G. Zheng, K. K. Wong, and B. Ottersten, “Robust cognitive beamforming with bounded channel uncertainties,” IEEE Trans. Signal Process., vol. 57, no. 12, pp. 4871–4881, Dec. 2009. [3] D. W. K. Ng, E. S. Lo, and R. Schober, “Wireless information and power transfer: Energy efficiency optimization in OFDMA systems,” IEEE Trans. Wireless Commun., vol. 12, no. 12, pp. 6352–6370, Dec. 2013. [4] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989–2001, May 2013. [5] C. Xing, N. Wang, J. Ni, Z. Fei, and J. Kuang, “MIMO beamforming designs with partial CSI under energy harvesting constraints,” IEEE Signal Process. Lett., vol. 20, no. 4, pp. 363–366, Apr. 2013. [6] B. K. Chalise, W. K. Ma, Y. D. Zhang, H. A. Suraweera, and M. G. Amin, “Optimum performance boundaries of OSTBC based AF-MIMO relay system with energy harvesting receiver,” IEEE Trans. Signal Process., vol. 61, no. 17, pp. 4199–4213, Sep. 2013. [7] Z. Xiang and M. Tao, “Robust beamforming for wireless information and power transmission,” IEEE Wireless Commun. Lett., vol. 1, no. 4, pp. 372– 375, Aug. 2012. [8] S. X. Wu, Q. Li, W.-K. Ma, and A. M.-C. So, “Robust transmit designs for an energy harvesting multicast system,” in Proc. IEEE ICASSP, 2014, pp. 4748–4752. [9] D. W. K. Ng, E. S. Lo, and R. Schober, “Robust beamforming for secure communication in systems with wireless information and power transfer,” IEEE Trans. Wireless Commun., to be published. [10] Q. Li, W.-K. Ma, and A. M.-C. So, “Robust artificial noise-aided transmit optimization for achieving secrecy and energy harvesting,” in Proc. IEEE ICASSP, 2014, pp. 1596–1600. [11] D. W. K. Ng, L. Xiang, and R. Schober, “Multi-objective beamforming for secure communication in systems with wireless information and power transfer,” in Proc. IEEE PIMRC, 2013, pp. 7–12. [12] H. A. Suraweera, P. J. Smith, and M. Shafi, “Capacity limits and performance analysis of cognitive radio with imperfect channel knowledge,” IEEE Trans. Veh. Technol., vol. 59, no. 4, pp. 1811–1822, May 2010. [13] A. Beck and Y. C. Eldar, “Strong duality in nonconvex quadratic optimization with two quadratic constraints,” SIAM J. Opt., vol. 17, no. 3, pp. 844–860, Jul. 2006. [14] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [15] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985.