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Robust Beamforming for Cognitive Multi-Antenna Relay Networks with Bounded Channel Uncertainties Quanzhong Li, Qi Zhang, Member, IEEE, and Jiayin Qin
Abstract—In cognitive relay networks, the interferences from secondary users (SUs) and relays to primary users are constrained to be lower than a threshold. The interference constraints are difficult to satisfy when the channel state information (CSI) is imperfect. In this paper, we propose a robust beamforming scheme for the multi-antenna non-regenerative cognitive relay network where the multi-antenna relay with imperfect CSIs helps the communication of single-antenna SU. Our objective is to design a robust beamforming scheme which maximizes the system capacity subject to transmit power constraint and interference constraints. The bounded channel uncertainties are modeled using the worst-case model. The robust beamforming problem, neglecting the correlation of channel uncertainties, is reformulated as a convex semidefinite programming (SDP) by rank-one relaxation. This convex SDP is related with the worstcase relay transmit power minimization problem, which is further reformulated as a convex SDP, whose rank-one solution is proved to exist. Thus, we propose the suboptimal solution to the robust beamforming problem which is found effectively by solving two convex SDPs. Simulation results are provided to demonstrate the effectiveness of the proposed scheme. Index Terms—Cognitive radio, relay networks, robust beamforming, worst-case model.
I. I NTRODUCTION OGNITIVE radio (CR) [1] is a promising technology to alleviate the spectrum shortage problem and to improve the spectrum utilization. In CR networks, the secondary user (SU) is allowed to access the same spectrum owned by the primary user (PU) subject to the interference constraint that the interference power from the SU to the PU is below a threshold. Thus, the CR networks can achieve higher spectrum utilization [2]. To satisfy the interference constraint, the SU transmitter (SU-Tx) should know the perfect channel state information (CSI) from the SU-Tx to the PU. In practice, however, the perfect CSI from the SU-Tx to the PU is seldom perfectly known because of the loose or usually no cooperation between PUs and SUs [3]–[5], [7], [8].
C
Manuscript received June 13, 2013; revised November 11, 2013. The editor coordinating the review of this paper and approving it for publication was O. Simeone. 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 authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail:
[email protected], {zhqi26, issqjy}@mail.sysu.edu.cn). Digital Object Identifier 10.1109/TCOMM.2014.011014.130437
In general, the channel uncertainty is characterized by two different models: the stochastic and deterministic (or worst-case) models [9]. In the stochastic model, the channel uncertainties are modeled as Gaussian random variables and the system design is then based on optimizing the average or outage performance [10]. Alternatively, the worst-case model assumes that the channel uncertainties, though not exactly known, are bounded by possible values [11]. In this case, the system is optimized to achieve a given quality of service (QoS) for every possible channel uncertainty if the problem is feasible, thereby, achieving absolute robustness. It was also shown in [12] that a bounded worst-case model is able to cope with quantization errors in CSIs. For CR network, the stochastic model was considered in [3] and the worst-case model was considered in [4]–[8]. For example, in [4], by assuming that SU-Tx knows the perfect CSI from the SU-Tx to the SU receiver (SU-Rx) and the imperfect CSI from the SU-Tx to the PU, Zhang et al proposed a robust cognitive beamforming scheme for CR networks where the channel uncertainty is modeled by the worst-case model. In CR network, because of the interference constraint, the QoS of the SU is difficult to guarantee. To improve the QoS of SU, cognitive relay networks (CRNs) have been proposed in [13]–[15], where both the regenerative and non-regenerative relaying schemes were considered. The CRNs were also studied in [16], [17], where it was shown that exploiting multiple antennas at the SU effectively enhances the system capacity as well as ensures the interference constraint. However, few studies have been reported on the robust beamforming design for the CRNs. In [18], Ubaidulla et al proposed the robust cognitive beamforming scheme for CRN where the channel uncertainties are modeled by the worst-case model. The CRN in [18] consists an SU-Tx, an SU-Rx, a PU transmitter, a PU receiver and multiple cognitive relays. Each relay is equipped with single antenna which makes the formulated robust beamforming simple. Furthermore, the channel uncertainty regions from the SU-Tx to multiple relays, from multiple relays to the SU-Rx and from multiple relays to the PU receiver were modeled as spheres which significantly simplifies the robust beamforming problem. The sphere channel uncertainty region assumption is restricted because it means that all the relays have the same channel uncertainties. It is worth noting that the robust beamforming design for the conventional relay networks was considered in [19]–[21],
c 2014 IEEE 0090-6778/14$31.00
LI et al.: ROBUST BEAMFORMING FOR COGNITIVE MULTI-ANTENNA RELAY NETWORKS WITH BOUNDED CHANNEL UNCERTAINTIES
where both the stochastic model and the worst-case model were considered. However, the robust beamforming design for the conventional relay networks cannot be applied directly in the CRNs. This is because that in CRNs, the channel uncertainties from the relay to the PU should be considered which makes the robust beamforming design problem much more difficult than that in conventional relay networks where no PU exists. In this paper, we propose a robust beamforming scheme for the multi-antenna non-regenerative CRN where the multiantenna cognitive relay with imperfect CSIs from the relay to the PUs (relay-to-PU CSIs) helps the communication of single-antenna SU. We also assume that the relay has perfect CSI from the SU-Tx to the relay (SU-Tx-to-relay CSI) and imperfect CSI from the relay to the SU-Rx (relay-to-SU-Rx CSI) as in [20], [21]. This is reasonable because the SU-Tx-torelay CSI is directly estimated by the relay whereas the relayto-SU-Rx CSI, in practice, is estimated by the SU-Rx and then fed back to the relay. In this paper, the bounded channel uncertainties are characterized by the worst-case model as in [4], [5]. Our objective is to design a robust beamforming scheme which maximizes the system capacity subject to the transmit power constraint and the interference constraints. The robust beamforming problem, neglecting the correlation of channel uncertainties, is reformulated as a convex semidefinite programming (SDP) by rank-one relaxation. This convex SDP is related with the worst-case relay transmit power minimization problem, which is further reformulated as a convex SDP, whose rank-one solution is proved to exist. Thus, the suboptimal solution of the robust beamforming problem can be found effectively by solving two convex SDPs. It is noted that the robust beamforming scheme for pointto-point CR communication was considered in [4]–[6]. In [4], the CR network with a multi-antenna SU-Tx, a single-antenna SU-Rx and a single-antenna PU was considered where the optimization problem is convex. In [6], the CR network with a multi-antenna SU-Tx, multiple single-antenna SU-Rxs and multiple single-antenna PUs was considered. In [5], the noncooperative game was considered to optimize the CR network where the resource allocation problem for each game player is convex. Compared with the optimization problems in [4], [5], the robust beamforming problem for CRN is more challenging because it is non-convex while the problems in [4], [5] are convex. Both our proposed scheme and the scheme proposed in [6] employ the S-procedure to transform the constraints into linear matrix inequalities (LMIs). The difference between our proposed scheme and the scheme proposed in [6] is that in our proposed scheme, the optimization problem is transformed to a single SDP while in [6], the optimization problem is solved using bisection search where in each iteration an SDP is solved. Furthermore, after the rank-one relaxation, the scheme proposed in [6] does not recover the rank-one constraint to find the rank-one solution while our proposed scheme recovers the rank-one constraint to find the rank-one solution. The rest of this paper is organized as follows. Section II describes the system model. In Section III, we propose the robust beamforming scheme. Simulation results are provided in Section IV. We conclude our paper in Section V. Notations: Boldface lowercase and uppercase letters denote
¢¢¢
PU1
479
PUM
y g21
g11
y g2M
g1M
h1 SU-Tx
¢¢¢
time slot 1
Fig. 1.
hy2
Relay
SU-Rx
time slot 2
The system model for the CRN.
vectors and matrices, respectively. The AT , A∗ , A† , ||A||, and Tr(A) denote the transpose, conjugate, conjugate transpose, Frobenius norm and trace of the matrix A, respectively. The vec(A) denotes to stack the columns of a matrix A into a single vector a while MAT(a) denotes the reverse operation. The ⊗ denotes the Kronecker product. By A 0 or A 0, we mean that the matrix A is positive semidefinite or positive definite, respectively. II. S YSTEM M ODEL We consider a non-regenerative CRN as shown in Fig. 1, where an SU transmitter (SU-Tx), an SU receiver (SU-Rx) and a cognitive relay are allowed to share the same spectrum with M PUs. Each of the SU-Tx, SU-Rx, and PUs is equipped with a single antenna and the cognitive relay is equipped with N antennas. We assume that there is no direct link between the SU-Tx and the SU-Rx, where the reliable communication link is established by the relay. The scenario is typical for deviceto-device communications [22] where two mobile phones in a underlaying cellular system communicate directly with the help of a femtocell or a laptop. It is noted that in practical CR system, there is usually a bunch of secondary users trying to access the network. To generalize the point-to-point CRN communication to the multi-user CRN communication is an interesting future work. The network operates in the time division duplex (TDD) mode. During the first time slot, the SU-Tx transmits signals to the relay and during the second time slot, the relay forwards the received signals, which is multiplied by a beamforming matrix, to the SU-Rx. The received signal at the SU-Rx is expressed as y = h†2 Fh1 x + h†2 Fnr + z
(1)
where F ∈ CN ×N denotes the linear beamforming matrix at the relay, h1 ∈ CN ×1 and h†2 ∈ C1×N denote the channel response from the SU-Tx to relay and that from the relay to SU-Rx, respectively, x ∈ C1×1 is the transmit symbol at the SU-Tx with E[|x|2 ] = σx2 , nr ∼ CN (0, σr2 I) is the additive Gaussian noise vector at the relay, and z ∼ CN (0, σd2 ) is the additive Gaussian noise at the SU-Rx. The received signal-to-
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noise ratio (SNR) at the SU-Rx can be expressed as SNR =
σx2 |h†2 Fh1 |2 . σr2 ||h†2 F||2 + σd2
(2)
The actual transmit power of the relay is σx2 ||Fh1 ||2 + σr2 ||F||2 .
(3)
To protect the communication of the primary network, we consider that the interference power from the SU-Tx and the relay to the mth PU should not exceed a threshold, denoted as Im . Then during the first time slot, the interference power from the SU-Tx to the mth PU is limited by σx2 |g1m |2 ≤ Im
(4)
where g1m ∈ C1×1 denotes the channel response from the SU-Tx to the mth PU. From (4), the maximum instantaneous transmit power of the SU-Tx is determined by m σx2 = min(Ps , |gI1m |2 ) where Ps is the maximum allowable transmit power of the SU-Tx. During the second time slot, the interference power from the relay to the mth PU is limited by † † σx2 |g2m Fh1 |2 + σr2 ||g2m F||2 ≤ Im
(5)
† where g2m ∈ C1×N represents the channel response from the relay to the mth PU. The objective of this paper is to optimize the relay beamforming matrix such that the system capacity, subject to the interference constraints and the transmit power constraint, is maximized, i.e., 1 σx2 |h†2 Fh1 |2 log2 1 + max F 2 σr2 ||h†2 F||2 + σd2
s.t. σx2 ||Fh1 ||2 + σr2 ||F||2 ≤ PR , † Fh1 |2 σx2 |g2m
† σr2 ||g2m F||2
+ m = 1, 2, · · · , M
(6) ≤ Im ,
where PR is the maximum allowable transmit power at the relay. It is noted that in (6) only the relay beamforming optimization is considered. A joint optimization of the source and relay beamforming for CRN would provide additional degrees-of-freedom and better system performance. However, the joint optimization problem is challenging whose solution would be an interesting future work. III. ROBUST B EAMFORMING WITH B OUNDED C HANNEL U NCERTAINTIES A. Worst-Case Model and Robust Beamforming Problem We employ the worst-case model as in [4], [5] to characterize the imperfect relay-to-PU and relay-to-SU-Rx CSIs at the relay. As in [4], [5], we assume that the actual channel is within the neighborhood of a nominal channel which is obtained from estimation (for relay-to-PU CSI) or feedback information (for relay-to-SU-Rx CSI). Specifically, the relay is assumed to know the noisy versions of actual channel ˘2 , respectively, ˘2m and h responses g2m and h2 , denoted as g g2m = g2m − Δg2m , m = 1, 2, · · · , M ˘
(7)
˘ 2 = h2 − Δh2 h
(8)
where Δg2m and Δh2 denote the CSI uncertainties for g2m and h2 , respectively. The Δg2m and Δh2 are bounded by the elliptical regions, † Gm = Δg2m | Δg2m Tm Δg2m ≤ 1 , m = 1, 2, · · · , M (9) and (10) H = Δh2 | Δh†2 QΔh2 ≤ 1 , M
respectively, where the matrices {Tm 0}m=1 and Q 0, assumed to be known, determine the qualities of CSIs. The CSIs become perfect when the smallest eigenvalues of {Ti 0}M m=1 and Q 0 approach infinity [4], [5], [7], [8]. The above model also embraces the Frobenius norm bounded error. It is noted that the SU-Tx also has the imperfect SU-Tx-toPU CSI, which can be expressed as g1m = g˘1m + δm where g˘1m is the estimated channel and |δm |2 ≤ m . Thus, the transmit power at the SU-Tx is σx2 = min(Ps , |˘g1mI|m ), 2 + m which leaves the robust beamforming problem essentially invariant. Using (7)-(10) and the monotonicity of the function 1 log 2 (·), the robust beamforming problem is expressed as 2 max
min
σx2 |h†2 Fh1 |2
σr2 ||h†2 F||2 + σd2 s.t. σx2 ||Fh1 ||2 + σr2 ||F||2 ≤ PR , F
Δh2 ∈H
† Fh1 |2 σx2 |g2m
+
† σr2 ||g2m F||2
(11) ≤ Im , ∀Δg2m ∈ Gm ,
m = 1, 2, · · · , M The robust problem (11) is non-convex and the optimal solution is very difficult to obtain. We will propose a suboptimal solution to the robust problem (11) by solving two convex SDPs. B. Equivalent Worst-Case Constraints Define W = ff † where f = vec(F). Applying the identity vec(ABC) = (CT ⊗ A)vec(B), the problem (11) is equivalently rewritten as max min
W0 Δh2 ∈H
σx2 h† Wh σr2 Tr(H† WH) + σd2
s.t. σx2 Tr(H†1 WH1 ) + σr2 Tr(W) − PR ≤ 0,
(12)
Tr(G†m WGm ) − Im ≤ 0, ∀Δg2m ∈ Gm , ∀m rank(W) = 1
where h = h∗1 ⊗ h2 , H = I ⊗ h2 , H1 = h∗1 ⊗ I, Gm = H∗2 ⊗ g2m , and H2 H†2 = σx2 h1 h†1 + σr2 I. We require the following lemma to obtain the equivalent transformation of the worst-case constraints. Lemma1: For the following two sets: (a) P1 = Δp | Δp† Q1 Δp ≤ 1, Q1 0 , (b) P2 = ΔP | Tr(ΔPQ2 ΔP† ) ≤ Tr(A† A), Q2 = I ⊗ Q1 , if ΔP = A ⊗ Δp and Tr(A† A) > 0, P1 = P2 . Proof: Applying the matrix identities (A ⊗ C)(B ⊗ D) = AB ⊗ CD and Tr(A ⊗ B) = Tr(A)Tr(B), we can obtain Tr(ΔPQ2 ΔP† ) = Tr((Δp† ⊗ A† )(Q1 ⊗ I)(Δp ⊗ A)) (13) = Tr((Δp† Q1 Δp) ⊗ (A† A)) = Δp† Q1 Δp · Tr(A† A)
LI et al.: ROBUST BEAMFORMING FOR COGNITIVE MULTI-ANTENNA RELAY NETWORKS WITH BOUNDED CHANNEL UNCERTAINTIES
From (13), it is found that if Δp† Q1 Δp ≤ 1, then Tr(ΔPQ2 ΔP† ) ≤ a, and vice versa. Using Lemma 1, the transformed channel vector/matrices in (12) can be modeled as ⎧ ˘ + Δh, H = H ˘ + ΔH, Gm = G ˘ m + ΔGm ⎪ h=h ⎪ ⎪ ⎪ ⎨ Δh ∈ H ¯ ¯ 1 = Δh | Δh† QΔh ≤ θ1 †¯ ¯ ⎪ ΔH ∈ H2 = {ΔH | Tr(ΔH QΔH) ≤ θ2 } ⎪ ⎪ ⎪ ⎩ ¯ m ΔGm ) ≤ θ3 }, ∀m ΔGm ∈ G¯m = {ΔGm | Tr(ΔG†m T (14) ∗ ˘2, H ˘ 2, G ˘ = h∗ ⊗ h ¯ ˘ ˘ where h = I ⊗ h = H ⊗ g ˘ , Q = m 2m 1 2 ¯ m = I ⊗ Tm , θ1 = ||h1 ||2 , θ2 = Tr(I), and θ3 = I ⊗ Q, T Tr(H2 H†2 ). Neglecting the correlation between Δh and ΔH, the rankone relaxation of the problem (12) can be formulated as the following fractional semidefinite programming (SDP), max
min
¯ 1 ,ΔH∈H ¯2 W0 Δh∈H
s.t.
σx2 h† Wh σr2 Tr(H† WH) + σd2
σx2 Tr(H†1 WH1 ) + σr2 Tr(W) − PR ≤ Tr(G†m WGm ) − Im ≤ 0, ∀ΔGm ∈
0, G¯m , ∀m
which can be further converted into a convex SDP with the following inequality constraints. It is noted that by neglecting the correlation between Δh and ΔH, the obtained scheme is suboptimal. Proposition 1: The problem (15) is equivalent to a convex SDP only with inequality constraints, given by max
s.t.
τ
S0,ν≥0,τ ≥0 ¯1 σx2 h† Sh ≥ τ, ∀Δh ∈ H σr2 Tr(H† SH) + σd2 ν ≤ 1, Tr(G†m SGm ) − νIm ≤ 0, σx2 Tr(H†1 SH1 ) + σr2 Tr(S)
it as follows ¯ − θ1 ≤ 0, ∀Δh : Δh† QΔh ˘ † Δh} − σx2 h ˘+τ ≤0 ˘ † Sh − Δh† (σx2 S)Δh − 2Re{(σx2 Sh) (17) Applying the above S-Procedure, we convert (17) into an LMI as follows Υ1 (S, λ1 , τ )
¯ + σ2 S ˘ λ1 Q σx2 Sh x ˘† S ˘ † Sh ˘ 0 σx2 h −τ − λ1 θ1 + σx2 h
(16a) (16b) ¯2 ∀ΔH ∈ H (16c) ¯ ∀ΔGm ∈ Gm , ∀m (16d) − νPR ≤ 0
(16e)
and the optimal solution to problem (15) is Wo = So /νo , where (So , νo ) is the optimal solution to problem (16). Proof: See Appendix A. The problem (16) is a convex semidefinite programming (SDP). However, the problem has semi-infinite constraints (16b)-(16d), which are intractable. To make the problem tractable, we convert the constraints (16b)-(16d) into linear matrix inequalities (LMIs) [24] equivalently, using the following S-Procedure. Lemma 2 (S-Procedure [25]): Define the functions fj (x) = x† Aj x + 2Re{b†j x} + cj , j = 1, 2 where Aj = A†j ∈ Cn×n , bj ∈ Cn , cj ∈ R. The implication f1 (x) ≤ 0 ⇒ f2 (x) ≤ 0 holds if and only if there exists λ ≥ 0 such that
A2 b2 A1 b1 − 0 λ b†1 c1 b†2 c2 provided that there exists a point x0 such that f1 (x0 ) < 0. To apply the S-Procedure to the constraint (16b), we ˘ + Δh into (16b) and rewrite substitute the expression h = h
(18)
for some λ1 ≥ 0. The condition f1 (x0 ) < 0 in the SProcedure is readily satisfied since θ1 > 0. Using the identity Tr(ABCD) = vecT (DT )(CT ⊗ A)vec(B), the constraint (16c) is equivalently expressed as ⎧ ¯ ¯† ¯ ¯ ⎪ ⎨ ∀Δh : Δh (I ⊗ Q)Δh − θ2 ≤ 0, ¯ † (σ 2 I ⊗ S)Δh ¯ + 2Re{((σ 2 I ⊗ S)h) ˆ † Δh} ¯ (19) Δh r r ⎪ ⎩ † 2 2 ˆ ˆ + h (σ I ⊗ S)h + σ ν − 1 ≤ 0 r
(15)
481
d
¯ = vec(ΔH) and h ˆ = vec(H). ˘ Applying the Swhere Δh Procedure, the above constraint is rewritten as follows Υ2 (S, λ2 , ν) ¯ − σr2 I ⊗ S λ2 I ⊗ Q ˆ † (σ 2 I ⊗ S) −h r
ˆ −(σr2 I ⊗ S)h † 2 ˆ ˆ α − h (σr I ⊗ S)h
0
(20)
for some λ2 ≥ 0 and α = 1 − σd2 ν − λ2 θ2 . The condition f1 (x0 ) < 0 in the S-Procedure is readily satisfied since θ2 > 0. Similarly, the constraint (16d) is also equivalently expressed as follows Υ3m (S, μm , ν) ¯m − I ⊗ S μm I ⊗ T † −ˆ gm (I ⊗ S)
−(I ⊗ S)ˆ gm † ˆm βm − g (I ⊗ S)ˆ gm
0, ∀m (21)
ˆm = for some μm ≥ 0, where βm = νIm − μm θ3 and g ˘ m ). vec(G C. SDP Reformulation and Rank-One Solution Using (18), (20)-(21), a suboptimal solution to the rankone relaxation of the original robust beamforming problem (11) can be found by solving the following problem, max
τ
s.t.
Υ1 (S, λ1 , τ ) 0, Υ2 (S, λ2 , ν) 0,
S,τ,ν,{λi },{μm }
Υ3m (S, μm , ν) 0, ∀m σx2 Tr(H†1 SH1 )
(22)
σr2 Tr(S)
+ − νPR ≤ 0, S 0, τ ≥ 0, ν ≥ 0, λ1 ≥ 0, λ2 ≥ 0,
μm ≥ 0, ∀m. The problem (22) is a convex SDP with a linear objective function and LMI constraints, which can be effectively solved by the numerical solvers such as CVX [26]. Denote the optimal solution to the SDP (22) as (So , νo ) and the optimal objective function value of (22) as τo . Thus, the optimal objective function value of the problem (15), denoted as γ,
.
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is γ = τo /νo . If the optimal solution So has rank of one, a suboptimal solution to the original problem (11) is found. Unfortunately, whether So has rank of one is unknown. If the rank of So is larger than one, we propose a method to find the rank-one optimal solution to the SDP (22) when the optimal objective function value of the problem (15), γ, is known. It is noted that the optimal solution to the SDP (22) may not be unique. From (15), we formulate the worst-case relay transmit power minimization problem as follows min S0
s.t.
σx2 Tr(H†1 SH1 )
+
σr2 Tr(S)
(23)
σx2 h† Sh ¯ 1 , ΔH ∈ H ¯2 ≥ γ − Δγ, ∀Δh ∈ H σr2 Tr(H† SH) + σd2 Tr(G†m SGm ) − Im ≤ 0, ∀ΔGm ∈ G¯m , ∀m
where Δγ is a non-negative constant with 0 ≤ Δγ < γ. We have the following result. Proposition 2: If So is the optimal solution of the problem (23), So achieves at least (1 − Δγ γ )-optimal to the problem (15). Especially, when Δγ = 0, So is also optimal to the problem (15). Proof: See Appendix B. From Proposition 2, we observe that if the worst-case relay transmit power minimization problem (23) has an optimal rank-one solution, the SDP (22) combining with the SDP (23) is arbitrarily close to the robust beamforming problem (15) with rank-one constraint. To make the problem (23) tractable, we also convert it into a convex SDP with LMIs. Lemma 3: The worst-case relay transmit power minimization problem (23) is equivalent to min
S,{λi },{μm }
σx2 Tr(H†1 SH1 ) + σr2 Tr(S)
s.t. Υ0 (S, λ1 , λ2 ) 0, ˆ 3m (S, μm ) 0, ∀m Υ
(24)
where ⎤ 0 ⎥ ˆ † (I ⊗ S) −ˆ γ σr2 h ⎦ λ2 ¯ − γˆ σ 2 I ⊗ S I ⊗ Q r θ2 (25)
ˆ ˆ † (I ⊗ S)h, ˘ † Sh ˘ − γˆ σr2 h with ρ = −λ1 θ1 − λ2 − γˆ σd2 + σx2 h γˆ = γ − Δγ, and ˆ 3m (S, μm ) Υ ¯m − I ⊗ S μm I ⊗ T † −ˆ gm (I ⊗ S)
−(I ⊗ S)ˆ gm † ˆm
m − g (I ⊗ S)ˆ gm
0
Algorithm 1 : Finding the suboptimal solution to the robust beamforming problem (11) 1: Find the optimal objective function value τo and optimal solution (So , νo ) to the SDP (22) using interior point method [24] or CVX [26]; 2: If rank(So ) = 1, decompose So /νo = fo fo† and go to step 4; ˇ o to the 3: If rank(So ) ≥ 2, find the optimal solution S SDP (24) using γ = τo /νo and a small positive Δγ, and ˇ o = fo f † ; decompose S o 4: Obtain the suboptimal solution to the original robust beamforming problem (11) as F = MAT(fo ). The small positive value of Δγ in Step 3 of Algorithm 1 can be set to be δγ, e.g., δ = 10−6 . To ensure the SDP (24) has a rank-one solution, Δγ should be sufficient small but not be zero. Please refer to Appendix D for the reason. The computational complexity of Algorithm 1 is mainly from the computation of the SDPs (22) and (24). From [27], the computational complexity for solving an SDP within a 2 2.5 3 0.5 tolerance is O((msdp n3.5 sdp + msdp nsdp + msdp nsdp ) · log(1/)), where nsdp is the dimension of the semidefinite cone and msdp is the number of linear constraints. Thus, the computational complexity of the SDPs (22) and (24) are both O(((M + 1)N 14 + (M + 1)2 N 10 + (M + 1)3 N 2 ) log(1/)). IV. S IMULATION R ESULTS
S 0, λ1 ≥ 0, λ2 ≥ 0, μm ≥ 0, ∀m Υ0 (S, λ1 , λ2 ) ⎡ ¯ + σ2 S ˘ λ1 Q σx2 Sh x ⎢ 2 ˘† σx h S ρ ⎣ ˆ 0 −ˆ γ σr2 (I ⊗ S)h
Proof: See Appendix D. From Proposition 3, we can obtain the optimal rank-one solution to the SDP (24) with any small positive Δγ. Thus, according to Proposition 2, a suboptimal rank-one solution to the original robust beamforming problem (11) is achieved when Δγ → 0. Now, we summary the proposed robust beamforming method in the following algorithm.
(26)
with m = Im − μm θ3 . Proof: See Appendix C. The SDP (24), as an equivalent form of the problem (23), can be also effectively solved by CVX [26]. In addition, the expressions in the SDP (24) enable us to conveniently analyze the rank of the optimal solution to the problem (23) by using the Karush-Kuhn-Tucker (KKT) conditions. Proposition 3: For any small positive value of Δγ, the optimal solution to the problem (24) must be rank-one.
In this section, we investigate the performance of proposed robust beamforming scheme through computer simulations. We assume that in the CRN, all the entries in the channel ˘ 2 , and ˘ g2m are independent and identically responses h1 , h distributed (i.i.d) complex Gaussian random variables with zero-mean and unit variance. We also assume the noise variances σr2 = σd2 = σ 2 and the maximum allowable transmit power at the SU-Tx σx2 /σ 2 = 15 dB. For simplicity, the uncertainty regions are assumed to be the norm-bounded, i.e., Q = (1/ω1 )I and Tm = (1/ω2 )I for m = 1, 2, · · · , M , where (ω1 , ω2 ) determines the quality of the CSIs. In all the simulations, the worst-case capacity, expressed as follows ˘ 2 + Δh2 )† Fh1 |2 1 σx2 |(h log2 1 + , (27) min ˘ 2 + Δh2 )† F||2 + σ 2 Δh2 ∈H 2 σr2 ||(h d is plotted by taking an average over 1000 randomly generated channel realizations. In Fig. 2, we present the average worst-case capacity versus the relay transmit power limit (denoted by PR /σ 2 ) of the scheme with perfect CSI, the suboptimal scheme by optimizing based on the lower or upper bounds of the constraints proposed in [8], [23] (denoted as “Sub-bound” in the legend), and the proposed robust beamforming scheme (denoted as
LI et al.: ROBUST BEAMFORMING FOR COGNITIVE MULTI-ANTENNA RELAY NETWORKS WITH BOUNDED CHANNEL UNCERTAINTIES
3.5
3 −4
−3
−3
−2
−2
−1
Pefect CSI
Sub−prop (10 ,10 )
3
Average Worst−Case Capacity (bps/Hz)
Average Worst−Case Capacity (bps/Hz)
Pefect CSI Sub−prop (10 ,10 ) Sub−prop (10 ,10 ) 2.5
Sub−bound (10−4,10−3) −3
−2
Sub−bound (10 ,10 ) 2
Sub−bound (10−2,10−1)
1.5
1
0.5
0
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0
5
10
15
−3 −2
Sub−prop (10 ,10−1) 2
1.5
1
0.5
1
2
2
PR/σ (dB)
Fig. 2. Average capacity versus the relay transmit power limit PR /σ2 ; N = 4, M = 2, I1 = I2 = I, I/σ2 = −10 dB.
−2
Sub−prop (10 ,10 )
0
20
Sub−prop (10−4,10−3)
2.5
3 N
4
5
Fig. 4. Average capacity versus the number of the relay antennas under different uncertainty levels; M = 2, I1 = I2 = I, PR /σ2 = 15dB, I/σ2 = −10 dB.
3.5
Average Worst−Case Capacity (bps/Hz)
Average Worst−Case Capacity (bps/Hz)
2.5
2 Pefect CSI −4
−3
−3
−2
−2
−1
Sub−prop (10 ,10 ) 1.5
Sub−prop (10 ,10 ) Sub−prop (10 ,10 ) −4
1
−3
Sub−bound (10 ,10 ) Sub−bound (10−3,10−2) −2
−1
Sub−bound (10 ,10 ) 0.5
0 −20
−15
−10
−5
0
2
I/σ (dB)
Fig. 3. Average capacity versus the interference constraint; N = 4, M = 2, I1 = I2 = I, PR /σ2 = 15 dB.
“Sub-prop” in the legend) with different channel uncertainties indicated by (ω1 , ω2 ), where N = 4, M = 2, I1 = I2 = I, and I/σ 2 = −10 dB. From Fig. 2, it is found that our proposed robust scheme has substantial capacity improvement over the suboptimal scheme proposed in [8], [23]. From Fig. 2, it is also observed that with the increases of ω1 and ω2 , the performance gaps between the scheme with perfect CSI and the proposed robust beamforming scheme become larger, especially at the high PR /σ 2 . This is because when the relayto-PU CSIs is imperfect, the relay is difficult to eliminate the interference from the relay to the PUs. Thus, the relay should reduce the transmit power to satisfy the interference constraints. Furthermore, the imperfect relay-to-SU-Rx CSI causes that the relay is difficult to steer its antenna beam towards the direction which increases the capacity. When the quality of CSIs are poor enough (e.g., ω1 = 0.01 and ω2 = 0.1), the average capacity does not increase with the
I=0 dB I=−10 dB I=−20 dB
3
2.5
2
1.5
1
0.5
0
1
2
3 N
4
5
Fig. 5. Average capacity versus the number of the relay antennas under different interference constraints; M = 2, I1 = I2 = I, PR /σ2 = 20 dB.
increase of PR /σ 2 . In Fig. 3, we present the average worst-case capacity versus the interference constraint where N = 4, M = 2, I1 = I2 = I, and PR /σ 2 = 15 dB. From Fig. 3, it is found that our proposed robust scheme outperforms the suboptimal scheme proposed in [8], [23]. From Fig. 3, it is also observed that with perfect relay-to-PUs CSIs, the interference constraint has almost no impact on the average worst-case capacity. This is because with perfect relay-to-PUs CSIs, the relay can effectively eliminate the interference to the PUs by exploiting multiple antennas. When the CSIs are imperfect, the average worst-case capacity is lower than that when the CSIs are perfect. If the channel uncertainties are the same, when the interference constraint becomes stricter, i.e., I/σ 2 becomes lower, the average worst-case capacity loss due to the interference constraint increases. This indicates that the effect of channel uncertainties on the average worst-case capacity in
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strict interference constraint regime, i.e., the low I/σ 2 regime, is much severer than that in loose interference constraint regime, i.e., the high I/σ 2 regime. In Fig. 4 and Fig. 5, we illustrate the effect of the number of the relay antennas on the average worst-case capacity obtained by our proposed robust scheme under different channel uncertainty levels and under different interference constraints. It is observed from Fig. 4 and Fig. 5 that as the number of the relay antennas increases, the average worst-case capacity improves. When the number of the relay antennas is larger than the number of PUs, the average worst-case capacity improves rapidly. This is because that more transmit antennas allow the relay to steer its antenna beam towards the SU-Rx and eliminate the interferences to the PUs. V. C ONCLUSIONS In this paper, we propose a robust beamforming scheme for the multi-antenna non-regenerative CRN where the multiantenna cognitive relay with imperfect CSIs helps the communication of single-antenna SU. The bounded channel uncertainties are modeled by the worst-case model. The robust beamforming problem, through the equivalent worst-caseconstraint transformation, is reformulated as a convex SDP by rank-one relaxation. This convex SDP is related with the worst-case relay transmit power minimization problem, which is further reformulated as a convex SDP, whose rank-one solution is proved to exist. Thus, the suboptimal solution to the robust beamforming problem can be found effectively by solving two convex SDPs. It is shown from simulation results that our proposed robust beamforming scheme is robust against the CSI uncertainties. A PPENDIX A P ROOF OF P ROPOSITION 1 Let W = S/ν, 1/ν = max
¯2 ΔH∈H
(28) σr2 Tr(H† WH) + σd2 .
(29)
The problem (15) is equivalently transformed into max
min σx2 h† Sh
s.t.
max σr2 Tr(H† SH) + σd2 ν = 1,
max σr2 Tr(H† So H) + σd2 νo = 1.
¯2 ΔH∈H
− νPR ≤ 0, ∀ΔGm ∈ G¯m , ∀m. (30)
We first prove the following lemma, which is needed for the proof of Proposition 1. Lemma 4: The optimum (So , νo ) of the following SDP only with inequality constraints min σx2 h† Sh max ¯1 S0,ν≥0 Δh∈H 2 s.t. σr Tr(H† SH) + σd2 ν ≤ 1, σx2 Tr(H†1 SH1 ) + σr2 Tr(S) Tr(G†m SGm ) − νIm ≤ 0,
¯2 ∀ΔH ∈ H − νPR ≤ 0, ∀ΔGm ∈ G¯m , ∀m (31)
(32)
Proof: It is noted that the first constraint of (31) is equivalent to maxΔH∈H¯ 2 σr2 Tr(H† So H) + σd2 νo ≤ 1. We prove Lemma 4 by reductio ad absurdum. From the constraints of (31), if (So , νo ) is the optimal solution to problem (31), (So , νo ) should not be (0, 0). Suppose that (So , νo ) satisfies maxΔH∈H¯ 2 σr2 Tr(H† So H) + σd2 νo < 1. We can find a scaler a0 (a0 > 1) which satisfies maxΔH∈H¯ 2 σr2 Tr(H† (a0 So )H)+ σd2 a0 νo = 1. It is noted that (a0 So , a0 νo ) is a feasible point of problem (31). Since So = 0, we have minΔh∈H¯ 1 σx2 h† (a0 So )h > minΔh∈H¯ 1 σx2 h† So h. It is contradictory with the assumption that (So , νo ) is the optimal solution. Using Lemma 4, we will prove that the SDP (31) is equivalent to the problem (15). Denote the optimal objective values of (15) and (31) as f1o and f2o , respectively. We need to prove f1o = f2o . First, we will show that f1o ≤ f2o . Suppose that Wo is the optimal solution to problem (15). It can be verified that the pair (S , ν ) = (Wo /(maxΔH∈H¯ 2 σr2 Tr(H† Wo H) + σd2 ), 1/(maxΔH∈H¯ 2 σr2 Tr(H† Wo H) + σd2 )) is feasible for the problem (31). Therefore, we have f1o ≤ f2o . Next, we will show that f1o ≥ f2o . Since (So , νo ) is the optimal solution to problem (31), it can be verified that W = So /νo is feasible for problem (15). Using Lemma 4, we have f2o = minΔh∈H¯ 1 σx2 h† So h = (minΔh∈H¯ 1 σx2 h† So /νo h)/(maxΔH∈H¯ 2 σr2 Tr(H† So /νo H)+ σd2 ) = minΔh∈H¯ 1 ,ΔH∈H¯ 2 σx2 h† W h/(σr2 Tr(H† W H) + σd2 ), which indicates f1o ≥ f2o . Thus, we prove that the SDP (31) is equivalent to problem (15). Using the epigraph reformulation for (31), we obtain the SDP (16). A PPENDIX B P ROOF OF P ROPOSITION 2 Suppose that Wo and So are the optimal solutions to the problems (15) and (23), respectively. It is found that Wo is a feasible point of the problem (23). We have σx2 Tr(H†1 So H1 ) + σr2 Tr(So )
¯1 S0,ν≥0 Δh∈H
¯2 ΔH∈H σx2 Tr(H†1 SH1 ) + σr2 Tr(S) Tr(G†m SGm ) − νIm ≤ 0,
must satisfy
≤ σx2 Tr(H†1 Wo H1 ) + σr2 Tr(Wo ) ≤ PR
(33)
where the last inequality comes from that fact that Wo is the optimal solution of the problem (15). From (33), we find that So is a feasible point of the problem (15), which implies that σx2 h† So h ≤ γ. σr2 Tr(H† So H) + σd2
(34)
In addition, since So is the optimal solution of the problem (23), we have σx2 h† So h 2 σr Tr(H† So H)
+ σd2
≥ γ − Δγ.
(35)
From (34) and (35), we obtain γ − Δγ ≤
σx2 h† So h ≤γ σr2 Tr(H† So H) + σd2
(36)
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which indicates that So can achieve at least (1 − Δγ γ )optimal to the problem (15). When Δγ = 0, we have 2 † σx h So h = γ, i.e., So is also the optimal solution σr2 Tr(H† So H)+σd2 of the problem (15). A PPENDIX C P ROOF OF L EMMA 3
and necessary conditions for a primal-dual point to be optimal [24]. We can use the KKT conditions to prove Proposition 3. The Lagrangian function of the SDP (24) is given by L(X ) = σx2 Tr(H†1 SH1 ) + σr2 Tr(S) − Tr(Υ0 (S, λ1 , λ2 )Q0 ) M ˆ 3m (S, μm )Qm ) − Tr(SR) Tr(Υ (41) − m=1
We rewrite the first constraint in the problem (23) as ¯ 1 , ΔH ∈ H ¯2. ≥ + ∀Δh ∈ H (37) Then we apply the S-Procedure to (37) with respect to the ¯ 1 . Thus, (37) can be equivalently uncertainty set Δh ∈ H expressed as
¯ + σ2 S ˘ σx2 Sh λ1 Q x ˘† S ˘ † Sh ˘ − γˆ σr2 Tr(H† SH) 0, σx2 h ς + σx2 h ¯2 (38) ∀ΔH ∈ H σx2 h† Sh
γˆ σr2 Tr(H† SH)
−η1 λ1 − η2 λ2 −
γˆ σd2 ,
for some λ1 ≥ 0 and ς = −λ1 θ1 − γˆ σd2 . To remove the ¯ 2 , we make use of the following uncertainty set ΔH ∈ H extension of the S-Procedure [28]. Lemma 5: The data matrices (A, B, C, D, F, G, H) satisfy
H F + GX 0, (F + GX)† C + X† B + B† X + X† AX ∀ I − X† DX 0 if and only if there exists t ≥ 0 such that ⎡ ⎤ H F G ⎣ F† C − tI ⎦ 0. B† † G B A + tD ¯ 2 as We rewrite Tr(H† SH) and ∀ΔH ∈ H
M
ηm+2 μm
m=1
in which X = {S, R, Q, μ, η, λ} consists of all the primal and dual variables with Q = {Qm 0}M m=0 , R 0, λ = {λi ≥ 0}2i=1 and η = {ηm ≥ 0}M+2 . For ease of exposition, m=1 we rewrite Υ0 (S, λ1 , λ2 ) as Υ0 (S, λ1 , λ2 ) = Ω0 (λ1 , λ2 ) + P†1 SP1 − P†2 (I ⊗ S)P2 (42) where
⎡
¯ λ1 Q 0 0 0 −λ1 θ1 − λ2 − γˆ σd2 Ω0 (λ1 , λ2 ) = ⎣ 0 λ2 ¯ 0 0 I ⊗Q θ 2 ˘ 0 , P2 = γˆ σr2 0 h ˆ I P1 = σx I h
where
ˆ C = Substituting A = −ˆ γ σr2 I ⊗ S, B = −ˆ γ σr2 (I ⊗ S)h, ˆ and D = 1 I ⊗ Q. ¯ ˆ † (I ⊗ S)h, ˘ † Sh ˘ − γˆ σr2 h −λ1 θ1 − γˆσd2 + σx2 h θ2 into Lemma 5, we can express (38) as the following LMI ⎤ ⎡ ¯ + σ2 S ˘ σx2 Sh 0 λ1 Q x ⎥ ⎢ ˆ † (I ⊗ S) ˘†S σx2 h ρ −ˆ γ σr2 h ⎦ ⎣ ˆ λ2 I ⊗ Q ¯ − γˆ σ 2 I ⊗ S 0 −ˆ γ σr2 (I ⊗ S)h r θ2 (40)
Similar to (21), we can obtain (26) by using the S-Procedure. A PPENDIX D P ROOF OF P ROPOSITION 3 The problem (24) is also a convex SDP with a linear objective function and LMI constraints, which satisfies Slater’s constraint qualification condition: there exists a strictly feasible point for the problem (24). For example, the point S = (1 − Δγ 2γ )So is strictly feasible. It is noted that if S = So , i.e., Δγ = 0, the point S may not be strictly feasible. Thus, the strong duality holds and the KKT conditions are the sufficient
⎤ ⎦
.
ˆ 3m (S, μm ) as Similarly, we can rewrite Υ ˆ 3m (S, μm ) = Ω3m (μm ) − P† (I ⊗ S)P3m Υ 3m
¯ † (I ⊗ S)Δh ¯ + 2Re{((I ⊗ S)h) ˆ † Δh} ¯ Tr(H† SH) = Δh † ˆ ˆ +h (I ⊗ S)h; (39) 1 ¯ : Δh ¯† ¯ ≤ 1. ¯ Δh ¯ 2 ⇔ ∀Δh I⊗Q ∀ΔH ∈ H θ2
0.
485
¯m μm I ⊗ T Ω3m (μm ) = 0 ˆm . P3m = I g
0 Im − μm θ3
(43)
,
Substituting (42)-(43) into (41), we can express the Lagrangian function as L(X ) =σx2 Tr(SH1 H†1 ) + σr2 Tr(S) − Tr(SR) N − Tr(SP1 Q0 P†1 ) + Tr SU(k,k) k=1
+
M N
(44)
(k,k) + φ(μ, η, λ) Tr SVm
m=1 k=1 (k,k)
where U(k,k) 0 and Vm 0 are block submatrices located in the diagonal of P2 Q0 P†2 and P3m Q3m P†3m with appropriate size, respectively, i.e., ⎤ ⎡ U(1,1) . . . U(1,N ) ⎥ ⎢ .. .. .. P2 Q0 P†2 = ⎣ ⎦, . . . ⎡
P3m Q3m P†3m
U(N,1) (1,1)
Vm ⎢ .. =⎢ . ⎣ (N,1) Vm
. . . U(N,N ) ... .. .
(1,N )
Vm .. .
(N,N )
⎤ ⎥ ⎥. ⎦
. . . Vm
In (44), the function φ(μ, η, λ) consists of the terms unrelated to S and R, which are not relevant to the proof. We only
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consider the KKT conditions relevant to the proof, given by ∂L(X ) = 0, ∂S Υ0 (S, λ1 , λ2 )Q0 = 0,
(45a) (45b)
SR = 0, Υ0 (S, λ1 , λ2 ) 0, S 0, R 0, λ1 ≥ 0, λ2 ≥ 0.
(45c) (45d) (45e)
From (44) and (45a), we have σx2 H1 H†1 + σr2 I − R − P1 Q0 P†1 +
N
M N
U(k,k) +
(k,k) Vm = 0.
(46)
m=1 k=1
k=1
Multiplying both sides of the above equation by S and using (45c), we obtain ⎛ ⎞ ⎜ ⎟ N M N ⎜ ⎟ ⎜ 2 (k,k) ⎟ S ⎜σx H1 H†1 + σr2 I + U(k,k) + Vm ⎟ ⎜ ⎟ (47) m=1 k=1 k=1 ⎝ !" #⎠ Θ
=
SP1 Q0 P†1 .
It is found that Θ is a positive-definite matrix which has full rank. Thus, rank(S) = rank (SΘ) = rank SP1 Q0 P†1 ≤ rank P1 Q0 P†1 .
(48)
The last inequality is from a basic rank inequality for the product [29]. From (48), if we can prove that of two matrices rank P1 Q0 P†1 = 1, we have rank(S) ≤ 1. Therefore, in
the following, we will focus on the rank of P1 Q0 P†1 . Applying (42) to (45b), we have
Ω0 (λ1 , λ2 )Q0 + P†1 SP1 Q0 − P†2 (I ⊗ S)P2 Q0 = 0. (49) Multiplying both sides of the above equation by P†1 , we obtain Ω0 (λ1 , λ2 )Q0 P†1 + P†1 SP1 Q0 P†1 − P†2 (I ⊗ S)P2 Q0 P†1 = 0. (50) Using the following equations, [ I
λ1 ¯ ˘ 0 ]) Q(P1 − [ 0 h σx (51) 0 ]P†1 = σx I, [ I 0 0 ]P†2 = 0
0 0 ]Ω0 (λ1 , λ2 ) = [ I 0
and multiplying both sides of (50) by [ I λ1 ¯ Q(P1 − [ 0 σx i.e.,
˘ h
0 0 ], we obtain
† † 0 ])Q0 P1 + σx SP1 Q0 P1 = 0, (52)
λ1 ¯ λ1 ¯ Q + σx S P1 Q0 P†1 = Q[ 0 σx σx
˘ h
† 0 ]Q0 P1 . (53)
¯ = I ⊗ Q 0 since Q 0. If λ1 > 0, It is found that Q λ1 ¯ we have σx Q + σx S 0 due to S 0. Suppose that
λ1 = 0, we obtain SP1 Q0 P†1 = 0 by (52), which leads to S = 0 since Θ 0 in (47). However, S = 0 means that Υ0 (S, λ1 , λ2 ) = Ω0 (λ1 , λ2 ) is a nondefinite matrix, which violates the assumption that the SDP (24) is feasible. ¯ + σx S 0. From (53), Therefore, we have λ1 > 0 and σλx1 Q we can obtain λ1 ¯ rank P1 Q0 P†1 = rank Q + σx S P1 Q0 P†1 σ x λ1 ¯ † ˘ = rank Q[ 0 h 0 ]Q0 P1 σx ' ( ˘ 0 ] ≤ rank [ 0 h = 1.
(54)
Combining (48) and (54), we obtain rank(S) ≤ 1.
(55)
Because S = 0, the optimal S to the SDP (24) is rank-one. R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum-sharing in fading environments,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 649–658, Feb. 2007. [3] G. Zheng, S. Ma, K. K. Wong, and T. S. Ng, “Robust beamforming in cognitive radio,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 570–575, Feb. 2010. [4] 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. [5] J. Wang, G. Scutari, and D. P. Palomar, “Robust MIMO cognitive radio via game theory,” IEEE Trans. Signal Process., vol. 59, no. 3, pp. 1183– 1201, Mar. 2011. [6] 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. [7] Y. Huang, Q. Li, W. Ma, and S. Zhang, “Robust multicast beamforming for spectrum sharing-based cognitive radios,” IEEE Trans. Signal Process., vol. 60, no. 1, pp. 527–533, Jan. 2012. [8] E. A. Gharavol, Y. C. Liang, and K. Mouthaan, “Robust downlink beamforming in multiuser MISO cognitive radio networks with imperfect channel-state information,” IEEE Trans. Veh. Technol., vol. 59, no. 6, pp. 2852–2860, Jul. 2010. [9] 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. [10] X. Zhang, D. P. Palomar, and B. Ottersten, “Statistically robust design of linear MIMO transceivers,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3678–3689, Aug. 2008. [11] J. Wang and D. P. Palomar, “Worst-case robust MIMO transmission with imperfect channel knowledge,” IEEE Trans. Signal Process., vol. 57, no. 8, pp. 3086–3100, Aug. 2009. [12] M. Botros and T. N. Davidson, “Convex conic formulations of robust downlink precoder designs with quality of service constraints,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 4, pp. 714–724, Dec. 2007. [13] L. Li, X. Zhao, H. Xu, G. Y. Li, D. Wang, and A. Soong, “Simplified relay selection and power allocation in cooperative cognitive radio systems,” IEEE Trans. Wireless Commun., vol. 10, no. 1, pp. 33–36, Jan. 2011. [14] M. Xia and S. Aissa, “Cooperative AF relaying in spectrum-sharing systems: performance analysis under average interference power constraints and Nakagami-m fading,” IEEE Trans. Commun., vol. 60, no. 6, pp. 1523–1533, Jun. 2012. [15] L. Luo, P. Zhang, G. Zhang, and J. Qin, “Outage performance for cognitive relay networks with underlay spectrum sharing,” IEEE Commun. Lett., vol. 15, no. 7, pp. 710–712, Jul. 2011. [16] R. Zhang and Y. C. Liang, “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 88–102, Feb. 2008.
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[17] R. Zhang, Y. C. Liang, and S. Cui, “Dynamic resource allocation in cognitive radio networks: a convex optimization perspective,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 102–114, May 2010. [18] P. Ubaidulla and S. Aissa, “Robust distributed cognitive relay beamforming,” in Proc. 2012 IEEE VTC – Spring, pp. 1–5. [19] L. Sanguinetti, A. A. D’Amico, and Y. Rong, “A tutorial on the optimization of amplify-and-forward MIMO relay systems,” IEEE J. Sel. Areas Commun., vol. 30, no. 8, 2012, pp. 1331–1346. Sept. 2012. [20] G. Zheng, K. K. Wong, A. Paulraj, and B. Ottersten, “Robust collaborative-relay beamforming,” IEEE Trans. Signal Process., vol. 57, no. 8, pp. 3130–3143, Aug. 2009. [21] R. Mo, J. Lin, Y. H. Chew, and W. H. Chin, “Relay precoder design for non-regenerative MIMO relay networks with imperfect channel state information,” in Proc. 2010 IEEE ICC, pp. 1–5. [22] P. Janis, C.-H. Yu, K. Doppler, C. Ribeiro, C. Wijting, K. Hugl, O. Tirkkonen, and V. Koivunen, “Device-to-device communication underlaying cellular communications systems,” Int. J. Commun., Network and System Sciences, doi:10.4236/ijcns.2009.23019, pp. 169–247, 2009. [23] G. Zheng, K. K. Wong, and T. S. Ng, “Robust MIMO in the downlink: a worst-case optimization with ellipsoidal uncertainty regions,” EURASIP J. Adv. Signal Process., vol. 2008, Article ID 609028, 15 pages. [24] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [25] A. Beck and Y. Eldar, “Strong duality in nonconvex quadratic optimization with two quadratic constraints,” SIAM J. Opt., vol. 17, no. 3, pp. 844–860, Jul. 2006. [26] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming. Available: http://cvxr.com/cvx. [27] I. Polik and T. Terlaky, “Interior point methods for nonlinear optimization,” in Nonlinear Optimization, 1st ed., G. Di Pillo, F. Schoen, Eds. Springer, 2010. [28] Z.-Q. Luo, J. F. Sturm, and S. Zhang, “Multivariate nonnegative quadratic mappings,” SIAM J. Optimiz., vol. 14, no. 4, pp. 1140–1162, 2004. [29] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1985.
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Quanzhong Li received his B.S. degree from Sun Yat-Sen University (SYSU), Guangzhou, China, in 2009. He is currently working toward the Ph.D. degree at the School of Information Science and Technology, SYSU. His research interests are in wireless communications powered by energy harvesting, cognitive radio, cooperative communications, and multiple-input-multiple-output (MIMO) communications. Qi Zhang (S’04–M’11) received the B.Eng. (Hons.) and M.S. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, Sichuan, China, in 1999 and 2002, respectively. He received the Ph.D. degree in Electrical and Computer Engineering from the National University of Singapore (NUS), Singapore, in 2007. He is currently an Associate Professor with the School of Information Science and Technology, Sun Yat-Sen University, China. From 2007 to 2008, he was a Research Fellow in the Communications Lab, Department of Electrical and Computer Engineering, NUS. From 2008 to 2011, he was at the Center for Integrated Electronics, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences and The Chinese University of Hong Kong. His research interests are in wireless communications powered by energy harvesting, cooperative communications, ultra-wideband (UWB) communications. Jiayin Qin received the M.S. degree in radio physics from Huazhong Normal University, China, in 1992 and the Ph.D. degree in Electronics from Sun Yat-Sen University (SYSU), Guangzhou, China, in 1997. Since 1999, he has been a professor with the School of Information Science and Technology, SYSU, China. From 2002 to 2004, he was the Head of the Department of Electronics and Communication Engineering, SYSU, China. From 2003 to 2008, he was the Vice Dean of the School of Information Science and Technology, SYSU, China. Dr. Qin is the recipient of the New Century Excellent Talent, Ministry of Education (MOE), China in 1999, the Second Young Teacher Award of Higher Education Institutions, MOE, China in 2001, and the Seventh Science and Technology Award for Chinese Youth in 2001. His research areas include wireless communication and submillimeter wave technology.
