# An Improved Robust Beamforming Design for Cognitive Multiantenna Relay Networks

Authors: Lulu Zhao; Guang Liang; Huijie Liu
Journal: Mobile Information Systems
 (2017)

---

## Abstract

This paper investigates the robust relay beamforming design for the multiantenna nonregenerative cognitive relay networks (CRNs). Firstly, it is proved that the optimal beamforming matrix could be simplified as the product of a variable vector and the conjugate transposition of a known channel response vector. Then, by exploiting the optimal beamforming matrix with simplified structure, an improved robust beamforming design is proposed. Analysis and simulation results show that, compared with the existing suboptimal scheme, the proposed method can achieve higher worst-case channel capacity with lower computational complexity.


---

## Body

## 1. Introduction

Cognitive radio [1] has been proposed as a potential technology to improve the spectrum utilization and alleviate the spectrum shortage problem in wireless communication system. In underlay cognitive radio network, the secondary users (SUs) are allowed to share the spectrum licensed to the primary users (PUs) only when the interference power from SUs to PUs remains below a predefined threshold. The threshold is defined as maximum tolerable interference level below which the PUs could maintain reliable communication [2, 3]. On the other hand, the interference constraint would require very low transmit power for SUs [3], which restricts the achievable channel capacity for SUs. In order to increase the channel capacity, the cognitive relay networks (CRNs) have been actively investigated to help improve the communication of SUs [4–6]. The relaying technique is proved to be an effective way to extend the coverage and enhance the performance of wireless communication systems [7]. Generally, there are mainly two schemes to implement relaying: regenerative scheme and nonregenerative scheme [8, 9]. In regenerative relaying scheme, the relay decodes the received signal and then retransmits it to the destination receiver after appropriate reencoding, whereas, in nonregenerative relaying scheme, the relay simply scales the received signal and forwards it to the destination receiver. Compared with regenerative scheme, the nonregenerative relaying scheme is relatively simple and more attractive. In addition, beamforming is an efficient and popular approach to exploit the spatial diversity offered by multiple antennas, which has been commonly reported for nonregenerative CRNs with single multiantenna relay [10, 11] or multiple relays [12, 13].Most of the beamforming designs mentioned above assume the availability of perfect channel state information (CSI). However, the practical available CSI would be imperfect due to various factors, for example, estimation and feedback errors. In CRNs, more CSI errors may be caused due to the limited cooperation between PUs and SUs. The performance of relay beamforming design, which is based on the assumption of perfect CSI, would degrade in the case with CSI errors. Hence, it is of critical importance to develop robust relay beamforming designs which take into consideration the CSI errors to guarantee the quality of service (QoS) requirement. Robust relay beamforming designs for CRNs have been studied in [13–16]. According to the way that the CSI errors are modeled, the design approaches are mainly divided into chance constrained methods [14, 15] and the worst-case constrained methods [13, 16]. The method of chance constrained robust design assumes that the CSI errors are random variables following the known statistical distribution, and the robustness can be achieved in a probabilistic sense; for example, the QoS requirements are met with a high probability, while the method of worst-case constrained robust design assumes that the CSI errors lie in bounded uncertainty regions, and the QoS requirements are guaranteed for all possible errors within the uncertainty regions, thereby, achieving absolute robustness. Based on the latter method, [17] studied the robust beamforming design problem for the multiantenna nonregenerative CRN where a pair of SUs communicate through a multiantenna relay with imperfect CSIs. The SUs and relay share the spectrum licensed to M PUs. The CSI errors are bounded by elliptically uncertainty regions. The objective of the robust beamforming design is to maximize the worst-case achievable channel capacity for SUs by optimizing the relay beamforming matrix, subject to the transmit power constraint of relay and the worst-case interference constraints of PUs. However, the original optimization problem of the robust beamforming design is nonconvex and thus difficult to be solved. Then, as an alternative solution to the original problem, an approximation problem was solved by neglecting the dependence existed between the received signal component and noise component at SU due to the same CSI error. However, because the approximation problem is not equivalent to the original one, the existing method is a suboptimal scheme which often yields suboptimal solutions for the original problem.In order to avoid the performance loss due to suboptimal solutions, an improved robust relay beamforming scheme for CRN is proposed in this paper. Firstly, it has been proved that the optimal beamforming matrix could be simplified as the product of a variable vector and the conjugate transposition of a known channel response vector. Then, by utilizing optimal beamforming matrix with simplified structure, the original problem is converted into an equivalent problem instead of the approximation problem solved in [17]. An optimal or near-optimal solution of the original problem can be obtained by solving the equivalent problem. It is proved that the performance loss of the near-optimal solution could be negligible. Therefore, the proposed method can obtain better performance than the suboptimal scheme. In addition, the computational complexity of proposed method is much lower than that of the suboptimal scheme. This is mainly because the variable in the equivalent problem in an N-dimension vector by assuming the number of antennas in the relay is N, while that in the approximation one is an N-order matrix.The rest of the paper is organized as follows. In Section2, the system model of CRN is presented. In Section 3, the improved robust relay beamforming design for CRN is proposed. In Section 4, numerical simulations are presented to illustrate the performance of the proposed method. Finally, the conclusions are drawn in Section 5.Notations. Vectors are written in lower case boldface letters, while matrices are denoted by upper case boldface letters. In is the n×n identity matrix and 0 is a zero vector or matrix. CN×M denotes the space of N×M matrix with complex entries. The superscript (·)H stands for the Hermitian transposition of a complex vector or matrix. · and · denote the absolute value of a complex scalar and Frobenius norm of a vector or matrix, respectively. tr(X) and rank(X) represent the trace and rank of matrix X, respectively. Furthermore, X⪰0 and X≻0 mean X is Hermitian positive semidefinite and positive definite matrix, respectively. x~CN(μ,Σ) represents that the random vector x follows the circular symmetric complex Gaussian distribution with mean vector μ and covariance matrix Σ.


## 2. System Model and Problem Formulation

### 2.1. System Model

A two-hop nonregenerative CRN is considered which consists of an SU-transmitter (SU-Tx), an SU receiver (SU-Rx), a cognitive relay, andM PUs. The SUs and the relay are allowed to share the same spectrum with M PUs. The relay is equipped with N antennas while other nodes are equipped with single antenna. The same assumption as in [17] applies for the system model; that is, reliable communication link is established by the relay with no direct link between the SU-Tx and SU-Rx. The scenario is typical for relay-assisted device-to-device (D2D) communications where two D2D users in an underlay cellular network communicate with the help of a femtocell [18, 19]. The configuration is illustrated in Figure 1.Figure 1
The system model for the CRN.The CRN operates in a half-duplex mode and the communication based on relay takes two time slots. In the first time slot, the SU-Tx transmits signal to the relay. The signal received at the relay can be expressed as(1)r=hs+nr,where h denotes the channel response from the SU-Tx to relay, s is the transmit symbol at the SU-Tx with E[s2]=σs2, and nr is the additive Gaussian noise vector at the relay with nr~CN(0,σr2I).In the second time slot, the relay multiplies the received signalr with a beamforming matrix R∈CN×N and forwards the processed signal to SU-Rx. The signal forwarded by the relay is(2)x=Rr=Rhs+Rnr.Then, the transmit power of the relay is(3)σs2Rh2+σr2R2.The received signal at the SU-Rx is expressed as(4)y=gHx+nd=gHRhs+gHRnr+nd,where gH denotes the channel response from the relay to SU-Rx and nd is the additive Gaussian noise at the SU-Rx with nd~CN(0,σd2). Then, the received signal-to-noise ratio (SNR) at the SU-Rx can be expressed as(5)SNR=σs2gHRh2σr2gHR2+σd2.The interference power from the relay to the mth PU can be expressed as(6)σs2gmHRh2+σr2gmHR2,where gmH denotes the channel response from the relay to mth PU.


### 2.2. CSI Errors

As in [17], it is assumed that the practical available CSIs of relay-to-PU links and relay-to-SU-Rx link at relay are imperfect and the actual CSI is within the neighborhood of the imperfect CSI which is obtained from estimation (for relay-to-PU CSI) or feedback information (for relay-to-SU-Rx CSI). Specifically, the actual CSIs g and gm can be represented as(7)g=g^+Δg,gm=g^m+Δgm,m=1,2,…,M,where g^ and g^m denote the imperfect practical available CSIs, respectively, and Δg and Δgm denote the CSI errors for g and gm, respectively. The CSI errors Δg and Δgm are bounded by the ellipsoidal uncertainty regions(8)G=Δg∣ΔgHQΔg≤1,(9)Gm=Δgm∣ΔgmHQmΔgm≤1,m=1,2,…,M,respectively, where the matrices Q≻0 and {Qm≻0}m=1M are used to determine the qualities of CSIs and assumed to be known [17, 20, 21].


### 2.3. Problem Formulation

The objective of the robust relay beamforming design is to maximize the worst-case achievable channel capacity for SUs by optimizing relay beamforming matrix, subject to worst-case interference constraints at PUs and transmit power constraint at relay. The robust beamforming problem can be expressed as(10a)P1:maxRminΔg∈G12log21+σs2gHRh2σr2gHR2+σd2(10b)s.t.σs2Rh2+σr2R2≤Pr,(10c)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,m=1,2,…,M,where Pr is the transmit power budget at the relay. Constraint (10c) shows that, to ensure the communication of PUs, the interference power from relay to mth PU should be below a threshold, denoted as Im. It is noted that only the relay beamforming optimization is considered in problem P1.


### 2.4. Optimal Transmit Power of SU-Tx

In CRN, the interference power from SU-Tx tomth PU should also be below Im. Therefore, the transmit power of SU-Tx is limited by(11)σs2hm2≤Im,m=1,2,…,M,where hm denotes the actual channel response from SU-Tx to the mth PU. It is noted that the SU-Tx also has imperfect SU-Tx-to-PU CSI. Then, the actual CSI hm can be expressed as(12)hm=h^m+Δhm,where h^m denotes imperfect practical available CSI and Δhm is the CSI error bounded by |Δhm|2≤ϵm. For the imperfect CSI case, constraint (11) should be rewritten as(13)maxΔhm2≤ϵmσs2h^m+Δhm2≤Im,∀m.From (13), we can get(14)σs2≤min⁡Ps,Imh^m+ϵm2,∀m,where Ps denotes the transmit power budget of SU-Tx.Defineσ¯s2≜min(Ps,Im/(|h^m|+ϵm)2,∀m). In [17], σ¯s2 is adopted as the transmit power of SU-Tx without illustrating the reasons for adoption of σ¯s2 (note that the expression of σ¯s2 in [17] is given with minor error as σ¯s2≜min(Ps,Im/(|h^m|2+ϵm),∀m)). In fact, the optimal σs2 can be obtained by solving the following problem:(15a)P2:maxσs2,RminΔg∈G12log21+σs2gHRh2σr2gHR2+σd2(15b)s.t.σs2≤σ-s2,(15c)σs2Rh2+σr2R2≤Pr,(15d)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,m=1,2,…,M.In problem P2, the worst-case achievable channel capacity for SUs is maximized by jointly optimizing σs2 and R. It can be proved below that σ¯s2 is just the optimal σs2 for this problem.Proof.
Assume that the pairkσ¯s2,R is optimal solution of problem P2, where k satisfies 0<k≤1. If 0<k<1, it can be verified that (σ¯s2,kR) is also feasible and(16)kσ¯s2gHRh2σr2gHR2+σd2<σ¯s2gHkRh2σr2gHkR2+σd2always holds. Therefore, (σ¯s2,kR) is a better solution than (kσ¯s2,R), which is contradictory with the assumption that (kσ¯s2,R) is the optimal solution. Therefore, if and only if k=1, the pair (kσ¯s2,R) is optimal; that is, σ¯s2 is the optimal σs2 for problem P2.After the optimalσs2 is determined, problem P2 is equivalent to problem P1 with σs2=σ¯s2. Therefore, solving the former problem has been converted to solving the latter one with given σs2 as in [17].


## 2.1. System Model

A two-hop nonregenerative CRN is considered which consists of an SU-transmitter (SU-Tx), an SU receiver (SU-Rx), a cognitive relay, andM PUs. The SUs and the relay are allowed to share the same spectrum with M PUs. The relay is equipped with N antennas while other nodes are equipped with single antenna. The same assumption as in [17] applies for the system model; that is, reliable communication link is established by the relay with no direct link between the SU-Tx and SU-Rx. The scenario is typical for relay-assisted device-to-device (D2D) communications where two D2D users in an underlay cellular network communicate with the help of a femtocell [18, 19]. The configuration is illustrated in Figure 1.Figure 1
The system model for the CRN.The CRN operates in a half-duplex mode and the communication based on relay takes two time slots. In the first time slot, the SU-Tx transmits signal to the relay. The signal received at the relay can be expressed as(1)r=hs+nr,where h denotes the channel response from the SU-Tx to relay, s is the transmit symbol at the SU-Tx with E[s2]=σs2, and nr is the additive Gaussian noise vector at the relay with nr~CN(0,σr2I).In the second time slot, the relay multiplies the received signalr with a beamforming matrix R∈CN×N and forwards the processed signal to SU-Rx. The signal forwarded by the relay is(2)x=Rr=Rhs+Rnr.Then, the transmit power of the relay is(3)σs2Rh2+σr2R2.The received signal at the SU-Rx is expressed as(4)y=gHx+nd=gHRhs+gHRnr+nd,where gH denotes the channel response from the relay to SU-Rx and nd is the additive Gaussian noise at the SU-Rx with nd~CN(0,σd2). Then, the received signal-to-noise ratio (SNR) at the SU-Rx can be expressed as(5)SNR=σs2gHRh2σr2gHR2+σd2.The interference power from the relay to the mth PU can be expressed as(6)σs2gmHRh2+σr2gmHR2,where gmH denotes the channel response from the relay to mth PU.


## 2.2. CSI Errors

As in [17], it is assumed that the practical available CSIs of relay-to-PU links and relay-to-SU-Rx link at relay are imperfect and the actual CSI is within the neighborhood of the imperfect CSI which is obtained from estimation (for relay-to-PU CSI) or feedback information (for relay-to-SU-Rx CSI). Specifically, the actual CSIs g and gm can be represented as(7)g=g^+Δg,gm=g^m+Δgm,m=1,2,…,M,where g^ and g^m denote the imperfect practical available CSIs, respectively, and Δg and Δgm denote the CSI errors for g and gm, respectively. The CSI errors Δg and Δgm are bounded by the ellipsoidal uncertainty regions(8)G=Δg∣ΔgHQΔg≤1,(9)Gm=Δgm∣ΔgmHQmΔgm≤1,m=1,2,…,M,respectively, where the matrices Q≻0 and {Qm≻0}m=1M are used to determine the qualities of CSIs and assumed to be known [17, 20, 21].


## 2.3. Problem Formulation

The objective of the robust relay beamforming design is to maximize the worst-case achievable channel capacity for SUs by optimizing relay beamforming matrix, subject to worst-case interference constraints at PUs and transmit power constraint at relay. The robust beamforming problem can be expressed as(10a)P1:maxRminΔg∈G12log21+σs2gHRh2σr2gHR2+σd2(10b)s.t.σs2Rh2+σr2R2≤Pr,(10c)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,m=1,2,…,M,where Pr is the transmit power budget at the relay. Constraint (10c) shows that, to ensure the communication of PUs, the interference power from relay to mth PU should be below a threshold, denoted as Im. It is noted that only the relay beamforming optimization is considered in problem P1.


## 2.4. Optimal Transmit Power of SU-Tx

In CRN, the interference power from SU-Tx tomth PU should also be below Im. Therefore, the transmit power of SU-Tx is limited by(11)σs2hm2≤Im,m=1,2,…,M,where hm denotes the actual channel response from SU-Tx to the mth PU. It is noted that the SU-Tx also has imperfect SU-Tx-to-PU CSI. Then, the actual CSI hm can be expressed as(12)hm=h^m+Δhm,where h^m denotes imperfect practical available CSI and Δhm is the CSI error bounded by |Δhm|2≤ϵm. For the imperfect CSI case, constraint (11) should be rewritten as(13)maxΔhm2≤ϵmσs2h^m+Δhm2≤Im,∀m.From (13), we can get(14)σs2≤min⁡Ps,Imh^m+ϵm2,∀m,where Ps denotes the transmit power budget of SU-Tx.Defineσ¯s2≜min(Ps,Im/(|h^m|+ϵm)2,∀m). In [17], σ¯s2 is adopted as the transmit power of SU-Tx without illustrating the reasons for adoption of σ¯s2 (note that the expression of σ¯s2 in [17] is given with minor error as σ¯s2≜min(Ps,Im/(|h^m|2+ϵm),∀m)). In fact, the optimal σs2 can be obtained by solving the following problem:(15a)P2:maxσs2,RminΔg∈G12log21+σs2gHRh2σr2gHR2+σd2(15b)s.t.σs2≤σ-s2,(15c)σs2Rh2+σr2R2≤Pr,(15d)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,m=1,2,…,M.In problem P2, the worst-case achievable channel capacity for SUs is maximized by jointly optimizing σs2 and R. It can be proved below that σ¯s2 is just the optimal σs2 for this problem.Proof.
Assume that the pairkσ¯s2,R is optimal solution of problem P2, where k satisfies 0<k≤1. If 0<k<1, it can be verified that (σ¯s2,kR) is also feasible and(16)kσ¯s2gHRh2σr2gHR2+σd2<σ¯s2gHkRh2σr2gHkR2+σd2always holds. Therefore, (σ¯s2,kR) is a better solution than (kσ¯s2,R), which is contradictory with the assumption that (kσ¯s2,R) is the optimal solution. Therefore, if and only if k=1, the pair (kσ¯s2,R) is optimal; that is, σ¯s2 is the optimal σs2 for problem P2.After the optimalσs2 is determined, problem P2 is equivalent to problem P1 with σs2=σ¯s2. Therefore, solving the former problem has been converted to solving the latter one with given σs2 as in [17].


## 3. Robust Relay Beamforming Design For CRN

Using the monotonicity of the logarithmic function, the optimization problemP1 can be equivalently expressed as(17a)P3:maxRminΔg∈Gσs2gHRh2σr2gHR2+σd2(17b)s.t.σs2Rh2+σr2R2≤Pr,(17c)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,∀m.


### 3.1. Suboptimal Scheme

In [17], a suboptimal solution to the problem P3 is obtained by solving the following approximation problem P4:(18a)P4:maxRminΔg∈Gσs2gHRh2maxΔg∈Gσr2gHR2+σd2(18b)s.t.σs2Rh2+σr2R2≤Pr,(18c)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,∀m.However, the objective function (18a) is only the lower bound of (17a) since(19)minΔg∈Gσs2gHRh2maxΔg∈Gσr2gHR2+σd2≤minΔg∈Gσs2gHRh2σr2gHR2+σd2always holds for any matrix R, which means that problem P4 is not equivalent to problem P3. As a result, the existing method is not the optimal scheme for problem P3.


### 3.2. Proposed Method

To simplify the beamforming problemP3, we introduce the following Lemma 1.Lemma 1.
AssumeRo is the optimal solution of problem P3; decompose Ro=whH+Th⊥H, where w∈CN×1, T∈CN×N-1, and h⊥ is the orthonormal basis for the null space of h. Then, whH is also the optimal solution of problem P3.Proof.
See AppendixA.Lemma1 indicates that the optimal beamforming matrix R could be a rank-one matrix whH with some w∈CN×1. Then, by replacing R with whH, the problem P3 can be equivalently reformulated as(20a)P5:maxwminΔg∈Gσs2h4gHw2σr2h2gHw2+σd2(20b)s.t.σs2h4+σr2h2w2≤Pr,(20c)σs2h4+σr2h2gmHw2≤Im,∀Δgm∈Gm,∀m.Defineu(w)≜minΔg∈Gσs2h4gHw2/σr2h2gHw2+σd2 and v(w)≜minΔg∈GgHw2. Since σs2h4gHw2/σr2h2gHw2+σd2 is an increasing function with respect to gHw2, u(w) can be represented by v(w) as(21)uw=σs2h4vwσr2h2vw+σd2.Since u(w) is also an increasing function with respect to v(w), an alternative method for maximizing u(w) is to maximize v(w). Then, the optimal solution of problem P5 can also be obtained by solving the following problem P6:(22a)P6:maxwvw(22b)s.t.σs2h4+σr2h2w2≤Pr,(22c)σs2h4+σr2h2gmHw2≤Im,∀Δgm∈Gm,∀m.Different from P4, problem P6 is equivalent to problem P3. Define W≜wwH and α=1/(σs2h4+σr2h2) and introduce an auxiliary variable τ; the problem P6 can be equivalently converted into(23a)P7:maxW⪰0,ττ(23b)s.t.trW≤αPr,(23c)gmHWgm≤αIm,∀Δgm∈Gm,∀m,(23d)gHWg≥τ,∀Δg∈G,(23e)rankW=1.Denote the optimal solution of problemP6 as wo. Then, the pair wowoH,v(wo) is the optimal solution of problem P7. The problem P7 has semi-infinite constraints (23c) and (23d), which are intractable. To make the problem P7 tractable, the S-Procedure [22] is employed to convert the constraints (23c) and (23d) into linear matrix inequalities (LMIs) [23].By applying theS-Procedure, constraint (23c) can be reformulated as(24)-g^mHWg^m+αIm-sm-g^mHW-Wg^m-W+smQm⪰0,∀mfor some sm≥0 and constraint (23d) can be reformulated as(25)g^HWg^-τ-s0g^HWWg^W+s0Q⪰0for some s0≥0. Using (24)-(25) and relaxing the rank-one constraint (23e), problem P7 can be converted into a convex semidefinite programming (SDP) problem P8 as follows:(26a)P8:maxW⪰0,ττ(26b)s.t.trW≤αPr,(26c)-g^mHWg^m+αIm-sm-g^mHW-Wg^m-W+smQm⪰0,sm≥0,∀m,(26d)g^HWg^-τ-s0g^HWWg^W+s0Q⪰0,s0≥0.The convex SDP problem P8 can be solved by standard inner point method [23]. Denote the optimal solution of problem P8 as Wo,τo with τo=minΔg∈GgHWog. It is obviously that τo is the upper bound of v(wo); that is, v(wo)≤τo always holds. Moreover, if Wo is rank-one, the pair Wo,τo is also the optimal solution of problem P7. Then, wo can be obtained by decomposing Wo=wowoH and v(wo) achieves its upper bound; that is, v(wo)=τo. Otherwise, consider the following problem P9:(27a)P9:minW⪰0trW(27b)s.t.gmHWgm≤αIm,∀Δgm∈Gm,∀m,(27c)gHWg≥1-δτo,∀Δg∈G,where 0<δ<1. Note that minΔg∈GgHWog=τo>(1-δ)τo; therefore, Wo is a feasible solution of problem P9. Denote the optimal solution of problem P9 as W^o; the objective function value tr(W^o) satisfies tr(W^o)≤tr(Wo)≤αPr.By applying theS-Procedure, constraint (27b) can be reformulated as(28)g^HWg^-1-δτo-s0g^HWWg^W+s0Q⪰0for some s0≥0. Using (24) and (28), problem P9 can be equivalently transformed into a convex SDP problem P10 as(29a)P10:minW⪰0trW(29b)s.t.-g^mHWg^m+αIm-sm-g^mHW-Wg^m-W+smQm⪰0,sm≥0,∀m,(29c)g^HWg^-1-δτo-s0g^HWWg^W+s0Q⪰0,s0≥0.The problem P10 can be also solved by standard inner point method. In addition, a lemma about the optimal solution of problem P10 is presented as follows.Lemma 2.
When0<δ<1, the optimal solution W^o must be rank-one.Proof.
See AppendixB.Lemma2 indicates that W^o is rank-one with any small positive δ. Decompose W^o=w^ow^oH; then, w^o is a feasible solution of problem P6. This is because w^o satisfies(30)w^o2=trW^o≤αPr,gmHw^o2=gmHW^ogm≤αIm,∀Δgm∈Gm,∀m.Then, the objective function value v(w^o) for problem P6 satisfies(31)vw^o=minΔg∈GgHw^o2=minΔg∈GgHW^og≥1-δτo.Substituting v(wo)≤τo and v(w^o)≥(1-δ)τo into (21) will yield(32)uwo-uw^o=σs2h4vwoσr2h2vwo+σd2-σs2h4vw^oσr2h2vw^o+σd2≤σs2h4τoσr2h2τo+σd2-σs2h41-δτoσr2h21-δτo+σd2=σs2h4σd2·δτoσr2h2τo+σd2σr2h21-δτo+σd2<σs2h4·δτoσd2.That is, the performance gap between u(wo) and u(w^o) is less than (σs2h4τo/σd2)δ. Therefore, by setting δ to be a positive number small enough, for example, δ=10-6, the performance loss of near-optimal solution w^o would be negligible.The proposed algorithm for robust beamforming problemP1 is shown in Algorithm 1.Algorithm 1:The proposed algorithm.
(1) solve the SDP problemP8 to obtain the optimal solution Wo,τo; (2)if  rank(Wo)=1  then (3)  DecomposeWo=wowoH; (4)else (5)  Setδ=10-6, solve the SDP problem P10 to obtain the optimal solution W^o; (6)  DecomposedW^o=w^ow^oH, set wo=w^o; (7)end if (8) Calculate the optimal/near-optimal beamforming matrixRo=w0hH.


### 3.3. Computational Complexity

The computational complexity of Algorithm1 mainly comes from the computational complexity of the SDP problems P8 and P10. From [24], the computational complexity for solving an SDP problem within a tolerance ϵ is O((mn3.5+m2n2.5+m3n0.5)·log⁡(1/ϵ)), where n is the dimension of the semidefinite cone and m is the number of linear constraints. Thus, the computational complexity for solving the SDP problems P8 and P10 is O(((M+1)N7+(M+1)2N5+(M+1)3N)·log⁡(1/ϵ)). Note that the computational complexity of suboptimal scheme is O(((M+1)N14+(M+1)2N10+(M+1)3N2)·log⁡(1/ϵ)) [17]; the computational complexity of our proposed method is much lower than that of the suboptimal scheme, especially at large N.


## 3.1. Suboptimal Scheme

In [17], a suboptimal solution to the problem P3 is obtained by solving the following approximation problem P4:(18a)P4:maxRminΔg∈Gσs2gHRh2maxΔg∈Gσr2gHR2+σd2(18b)s.t.σs2Rh2+σr2R2≤Pr,(18c)σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,∀m.However, the objective function (18a) is only the lower bound of (17a) since(19)minΔg∈Gσs2gHRh2maxΔg∈Gσr2gHR2+σd2≤minΔg∈Gσs2gHRh2σr2gHR2+σd2always holds for any matrix R, which means that problem P4 is not equivalent to problem P3. As a result, the existing method is not the optimal scheme for problem P3.


## 3.2. Proposed Method

To simplify the beamforming problemP3, we introduce the following Lemma 1.Lemma 1.
AssumeRo is the optimal solution of problem P3; decompose Ro=whH+Th⊥H, where w∈CN×1, T∈CN×N-1, and h⊥ is the orthonormal basis for the null space of h. Then, whH is also the optimal solution of problem P3.Proof.
See AppendixA.Lemma1 indicates that the optimal beamforming matrix R could be a rank-one matrix whH with some w∈CN×1. Then, by replacing R with whH, the problem P3 can be equivalently reformulated as(20a)P5:maxwminΔg∈Gσs2h4gHw2σr2h2gHw2+σd2(20b)s.t.σs2h4+σr2h2w2≤Pr,(20c)σs2h4+σr2h2gmHw2≤Im,∀Δgm∈Gm,∀m.Defineu(w)≜minΔg∈Gσs2h4gHw2/σr2h2gHw2+σd2 and v(w)≜minΔg∈GgHw2. Since σs2h4gHw2/σr2h2gHw2+σd2 is an increasing function with respect to gHw2, u(w) can be represented by v(w) as(21)uw=σs2h4vwσr2h2vw+σd2.Since u(w) is also an increasing function with respect to v(w), an alternative method for maximizing u(w) is to maximize v(w). Then, the optimal solution of problem P5 can also be obtained by solving the following problem P6:(22a)P6:maxwvw(22b)s.t.σs2h4+σr2h2w2≤Pr,(22c)σs2h4+σr2h2gmHw2≤Im,∀Δgm∈Gm,∀m.Different from P4, problem P6 is equivalent to problem P3. Define W≜wwH and α=1/(σs2h4+σr2h2) and introduce an auxiliary variable τ; the problem P6 can be equivalently converted into(23a)P7:maxW⪰0,ττ(23b)s.t.trW≤αPr,(23c)gmHWgm≤αIm,∀Δgm∈Gm,∀m,(23d)gHWg≥τ,∀Δg∈G,(23e)rankW=1.Denote the optimal solution of problemP6 as wo. Then, the pair wowoH,v(wo) is the optimal solution of problem P7. The problem P7 has semi-infinite constraints (23c) and (23d), which are intractable. To make the problem P7 tractable, the S-Procedure [22] is employed to convert the constraints (23c) and (23d) into linear matrix inequalities (LMIs) [23].By applying theS-Procedure, constraint (23c) can be reformulated as(24)-g^mHWg^m+αIm-sm-g^mHW-Wg^m-W+smQm⪰0,∀mfor some sm≥0 and constraint (23d) can be reformulated as(25)g^HWg^-τ-s0g^HWWg^W+s0Q⪰0for some s0≥0. Using (24)-(25) and relaxing the rank-one constraint (23e), problem P7 can be converted into a convex semidefinite programming (SDP) problem P8 as follows:(26a)P8:maxW⪰0,ττ(26b)s.t.trW≤αPr,(26c)-g^mHWg^m+αIm-sm-g^mHW-Wg^m-W+smQm⪰0,sm≥0,∀m,(26d)g^HWg^-τ-s0g^HWWg^W+s0Q⪰0,s0≥0.The convex SDP problem P8 can be solved by standard inner point method [23]. Denote the optimal solution of problem P8 as Wo,τo with τo=minΔg∈GgHWog. It is obviously that τo is the upper bound of v(wo); that is, v(wo)≤τo always holds. Moreover, if Wo is rank-one, the pair Wo,τo is also the optimal solution of problem P7. Then, wo can be obtained by decomposing Wo=wowoH and v(wo) achieves its upper bound; that is, v(wo)=τo. Otherwise, consider the following problem P9:(27a)P9:minW⪰0trW(27b)s.t.gmHWgm≤αIm,∀Δgm∈Gm,∀m,(27c)gHWg≥1-δτo,∀Δg∈G,where 0<δ<1. Note that minΔg∈GgHWog=τo>(1-δ)τo; therefore, Wo is a feasible solution of problem P9. Denote the optimal solution of problem P9 as W^o; the objective function value tr(W^o) satisfies tr(W^o)≤tr(Wo)≤αPr.By applying theS-Procedure, constraint (27b) can be reformulated as(28)g^HWg^-1-δτo-s0g^HWWg^W+s0Q⪰0for some s0≥0. Using (24) and (28), problem P9 can be equivalently transformed into a convex SDP problem P10 as(29a)P10:minW⪰0trW(29b)s.t.-g^mHWg^m+αIm-sm-g^mHW-Wg^m-W+smQm⪰0,sm≥0,∀m,(29c)g^HWg^-1-δτo-s0g^HWWg^W+s0Q⪰0,s0≥0.The problem P10 can be also solved by standard inner point method. In addition, a lemma about the optimal solution of problem P10 is presented as follows.Lemma 2.
When0<δ<1, the optimal solution W^o must be rank-one.Proof.
See AppendixB.Lemma2 indicates that W^o is rank-one with any small positive δ. Decompose W^o=w^ow^oH; then, w^o is a feasible solution of problem P6. This is because w^o satisfies(30)w^o2=trW^o≤αPr,gmHw^o2=gmHW^ogm≤αIm,∀Δgm∈Gm,∀m.Then, the objective function value v(w^o) for problem P6 satisfies(31)vw^o=minΔg∈GgHw^o2=minΔg∈GgHW^og≥1-δτo.Substituting v(wo)≤τo and v(w^o)≥(1-δ)τo into (21) will yield(32)uwo-uw^o=σs2h4vwoσr2h2vwo+σd2-σs2h4vw^oσr2h2vw^o+σd2≤σs2h4τoσr2h2τo+σd2-σs2h41-δτoσr2h21-δτo+σd2=σs2h4σd2·δτoσr2h2τo+σd2σr2h21-δτo+σd2<σs2h4·δτoσd2.That is, the performance gap between u(wo) and u(w^o) is less than (σs2h4τo/σd2)δ. Therefore, by setting δ to be a positive number small enough, for example, δ=10-6, the performance loss of near-optimal solution w^o would be negligible.The proposed algorithm for robust beamforming problemP1 is shown in Algorithm 1.Algorithm 1:The proposed algorithm.
(1) solve the SDP problemP8 to obtain the optimal solution Wo,τo; (2)if  rank(Wo)=1  then (3)  DecomposeWo=wowoH; (4)else (5)  Setδ=10-6, solve the SDP problem P10 to obtain the optimal solution W^o; (6)  DecomposedW^o=w^ow^oH, set wo=w^o; (7)end if (8) Calculate the optimal/near-optimal beamforming matrixRo=w0hH.


## 3.3. Computational Complexity

The computational complexity of Algorithm1 mainly comes from the computational complexity of the SDP problems P8 and P10. From [24], the computational complexity for solving an SDP problem within a tolerance ϵ is O((mn3.5+m2n2.5+m3n0.5)·log⁡(1/ϵ)), where n is the dimension of the semidefinite cone and m is the number of linear constraints. Thus, the computational complexity for solving the SDP problems P8 and P10 is O(((M+1)N7+(M+1)2N5+(M+1)3N)·log⁡(1/ϵ)). Note that the computational complexity of suboptimal scheme is O(((M+1)N14+(M+1)2N10+(M+1)3N2)·log⁡(1/ϵ)) [17]; the computational complexity of our proposed method is much lower than that of the suboptimal scheme, especially at large N.


## 4. Simulations

In this section, the performance of proposed method is demonstrated through numerical simulations. The channel vectors are assumed ash, g^, and g^mH, ∀m~CN(0,I), and the noise power is assumed as σr2=σd2=σ2. The uncertainty regions are assumed to be norm-bounded; that is, Q=(1/ωs)I and Qm=(1/ωp)I, ∀m, where (ωs,ωp) determines the quality of the CSIs. The maximum interference power thresholds are assumed as Im=I,∀m. Three different CSI errors (ωs,ωp)=(0.01,0.01), (ωs,ωp)=(0.01,0.1), and (ωs,ωp)=(0.1,0.1) and two different maximum interference power thresholds I/σ2=0dB and I/σ2=-10dB are considered in the simulations. The number of the antennas in the relay N is assumed to be 4 and the number of PUs M is assumed to be 2. CVX toolbox [25] is used to solve the SDP problems numerically. In all simulations, the worst-case achievable channel capacity for SUs, expressed as(33)minΔg∈G12log21+σs2gHRh2σr2gHR2+σd2,is evaluated with 1000 randomly generated CSI errors satisfying the model (8) for each channel realization. The simulation results are averaged over 1000 randomly generated channel realizations.Figure2 shows the relationship between average worst-case achievable channel capacity of proposed method (denoted as “Pro” in the legend) and ST transmit power (σs2/σ2) at Pr/σ2=10dB, together with the worst-case capacity of suboptimal scheme (denoted as “Sub”). I/σ2 is set to be 0 dB and −10 dB in Figures 2(a) and 2(b), respectively. It can be observed from Figure 2 that the channel capacity increases with the increasing of σs2 as illustrated in Section 2.4. We can find that the proposed method outperforms the suboptimal scheme for all σs2. Specially, it can be seen from Figure 2(a) that when I/σ2=0dB, σs2/σ2=0dB, and (ωs,ωp)=(0.1,0.1), the worst-case capacity of proposed method and the suboptimal scheme is about 0.8 bps/Hz and 0.53 bps/Hz, respectively; that is, the proposed method improves the worst-case capacity of 50% compared with the suboptimal scheme.Figure 2
Average worst-case achievable channel capacity versus ST transmit powerσs2/σ2. (a) I/σ2=0dB; (b) I/σ2=-10dB.
(a) (b)Denote the average worst-case achievable channel capacity of proposed method and the suboptimal scheme asγPro and γSub, respectively. The capacity improvement, defined as γPro-γSub, is shown in Figure 3 for Pr/σ2=10dB. From Figure 3, we can observe that the capacity improvement becomes smaller as σs2 increases. This is because the increase of σs2 would cause both of the relay transmit constraint (17b) and interference power constraint (17c) becoming tighter, reducing the feasible region of problem P3. The reduction of feasible region may make the suboptimal solution tend to approach the optimal one, which makes γSub closer to γPro and leads to a smaller capacity improvement. Moreover, it is shown that the capacity improvement increases with the increase of ωs. The reason is that as ωs increases, the difference between the objective function (17a) and its lower bound (18a) becomes larger by neglecting the correlation of Δg in suboptimal scheme, leading to the increase of the capacity improvement. Furthermore, the capacity improvement decreases with the increase of ωp due to the fact that the increasing of ωp tightens the interference power constraint (17c), reducing the feasible region of problem P3. In addition, we can also see that, as I increases, the capacity improvement becomes larger. This is because the increase of I loosens the interference constraint (17c), leading to an expansion of the feasible region of problem P3.Figure 3
Average worst-case achievable channel capacity improvement versus ST transmit powerσs2/σ2.Figure4 shows the average worst-case achievable channel capacity versus relay transmit power budget (Pr/σ2) at σs2/σ2=5dB. I/σ2 is set to be 0 dB and −10 dB in Figures 4(a) and 4(b), respectively. It also can be found from Figure 4 that the proposed method outperforms the suboptimal scheme for all Pr. The capacity improvement at σs2/σ2=5dB is shown in Figure 5. From Figures 4 and 5, we can observe that only when Pr is below some threshold, both γPro and the capacity improvement increase as Pr increases. The reason would be analysis as follows. Consider the following problem:(34a)P11:maxRσs2Rh2+σr2R2(34b)s.t.σs2gmHRh2+σr2gmHR2≤Im,∀Δgm∈Gm,∀m.Denote the optimal objective function value of problem P11 as P¯r. Then, when Pr<P¯r, increasing Pr expands the feasible region of problem P3; γPro as well as the capacity improvement could become larger. On the other hand, when Pr≥P¯r, the feasible region of both problem P3 and problem P4 keeps invariant as Pr increases, indicating that both γPro and γSub keep almost constant, which leads the capacity improvement to also keep invariant.Figure 4
Average worst-case achievable channel capacity versus relay transmit power budgetPr/σ2. (a) I/σ2=0dB; (b) I/σ2=-10dB.
(a) (b)Figure 5
Average worst-case achievable channel capacity improvement versus relay transmit power budgetPr/σ2.


## 5. Conclusion

An improved robust relay beamforming design for multiantenna nonregenerative CRN is developed in this paper. The objective is to maximize the worst-case achievable channel capacity for SUs subject to worst-case interference constraint at PUs and transmit power constraint at relay. By employing the optimal beamforming matrix with simplified structure, the original robust beamforming problem is converted into an equivalent problem. Then, an optimal or near-optimal solution of the original problem can be obtained by solving the equivalent one. Analysis and simulation results show that the proposed method can achieve higher capacity for SUs and require lower computational complexity than suboptimal scheme.



---

*Source: 2719543-2017-02-14.xml*