An Improved Robust Beamforming Design for Cognitive Multiantenna Relay Networks

Zhao, Lulu; Liang, Guang; Liu, Huijie

doi:https://doi.org/10.1155/2017/2719543

Mobile Information Systems

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Abstract Introduction System Model Conclusion Appendix Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 2719543 | https://doi.org/10.1155/2017/2719543

An Improved Robust Beamforming Design for Cognitive Multiantenna Relay Networks

Lulu Zhao,^1,2Guang Liang,²and Huijie Liu²

Academic Editor: Alessandro Bazzi

Received24 Oct 2016

Accepted17 Jan 2017

Published14 Feb 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.

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 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 -dimension vector by assuming the number of antennas in the relay is , while that in the approximation one is an -order matrix.

The rest of the paper is organized as follows. In Section 2, 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. is the identity matrix and is a zero vector or matrix. denotes the space of matrix with complex entries. The superscript 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. and represent the trace and rank of matrix , respectively. Furthermore, and mean is Hermitian positive semidefinite and positive definite matrix, respectively. represents that the random vector follows the circular symmetric complex Gaussian distribution with mean vector and covariance matrix .

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, and PUs. The SUs and the relay are allowed to share the same spectrum with PUs. The relay is equipped with 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.

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 aswhere denotes the channel response from the SU-Tx to relay, is the transmit symbol at the SU-Tx with , and is the additive Gaussian noise vector at the relay with .

In the second time slot, the relay multiplies the received signal with a beamforming matrix and forwards the processed signal to SU-Rx. The signal forwarded by the relay isThen, the transmit power of the relay isThe received signal at the SU-Rx is expressed aswhere denotes the channel response from the relay to SU-Rx and is the additive Gaussian noise at the SU-Rx with . Then, the received signal-to-noise ratio (SNR) at the SU-Rx can be expressed asThe interference power from the relay to the th PU can be expressed aswhere denotes the channel response from the relay to th 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 and can be represented aswhere and denote the imperfect practical available CSIs, respectively, and and denote the CSI errors for and , respectively. The CSI errors and are bounded by the ellipsoidal uncertainty regionsrespectively, where the matrices and 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 aswhere is the transmit power budget at the relay. Constraint (10c) shows that, to ensure the communication of PUs, the interference power from relay to th PU should be below a threshold, denoted as . It is noted that only the relay beamforming optimization is considered in problem .

2.4. Optimal Transmit Power of SU-Tx

In CRN, the interference power from SU-Tx to th PU should also be below . Therefore, the transmit power of SU-Tx is limited bywhere denotes the actual channel response from SU-Tx to the th PU. It is noted that the SU-Tx also has imperfect SU-Tx-to-PU CSI. Then, the actual CSI can be expressed aswhere denotes imperfect practical available CSI and is the CSI error bounded by . For the imperfect CSI case, constraint (11) should be rewritten asFrom (13), we can getwhere denotes the transmit power budget of SU-Tx.

Define . In [17], is adopted as the transmit power of SU-Tx without illustrating the reasons for adoption of (note that the expression of in [17] is given with minor error as ). In fact, the optimal can be obtained by solving the following problem:In problem , the worst-case achievable channel capacity for SUs is maximized by jointly optimizing and . It can be proved below that is just the optimal for this problem.

Proof. Assume that the pair is optimal solution of problem , where satisfies . If , it can be verified that is also feasible andalways holds. Therefore, is a better solution than , which is contradictory with the assumption that is the optimal solution. Therefore, if and only if , the pair is optimal; that is, is the optimal for problem .

After the optimal is determined, problem is equivalent to problem with . Therefore, solving the former problem has been converted to solving the latter one with given as in [17].

3. Robust Relay Beamforming Design For CRN

Using the monotonicity of the logarithmic function, the optimization problem can be equivalently expressed as

3.1. Suboptimal Scheme

In [17], a suboptimal solution to the problem is obtained by solving the following approximation problem :However, the objective function (18a) is only the lower bound of (17a) sincealways holds for any matrix , which means that problem is not equivalent to problem . As a result, the existing method is not the optimal scheme for problem .

3.2. Proposed Method

To simplify the beamforming problem , we introduce the following Lemma 1.

Lemma 1. Assume is the optimal solution of problem ; decompose , where , , and is the orthonormal basis for the null space of . Then, is also the optimal solution of problem .

Proof. See Appendix A.

Lemma 1 indicates that the optimal beamforming matrix could be a rank-one matrix with some . Then, by replacing with , the problem can be equivalently reformulated as

Define and . Since is an increasing function with respect to , can be represented by asSince is also an increasing function with respect to , an alternative method for maximizing is to maximize . Then, the optimal solution of problem can also be obtained by solving the following problem :Different from , problem is equivalent to problem . Define and and introduce an auxiliary variable ; the problem can be equivalently converted into

Denote the optimal solution of problem as . Then, the pair is the optimal solution of problem . The problem has semi-infinite constraints (23c) and (23d), which are intractable. To make the problem tractable, the -Procedure [22] is employed to convert the constraints (23c) and (23d) into linear matrix inequalities (LMIs) [23].

By applying the -Procedure, constraint (23c) can be reformulated asfor some and constraint (23d) can be reformulated asfor some . Using (24)-(25) and relaxing the rank-one constraint (23e), problem can be converted into a convex semidefinite programming (SDP) problem as follows:The convex SDP problem can be solved by standard inner point method [23]. Denote the optimal solution of problem as with . It is obviously that is the upper bound of ; that is, always holds. Moreover, if is rank-one, the pair is also the optimal solution of problem . Then, can be obtained by decomposing and achieves its upper bound; that is, . Otherwise, consider the following problem :where . Note that ; therefore, is a feasible solution of problem . Denote the optimal solution of problem as ; the objective function value satisfies .

By applying the -Procedure, constraint (27b) can be reformulated asfor some . Using (24) and (28), problem can be equivalently transformed into a convex SDP problem asThe problem can be also solved by standard inner point method. In addition, a lemma about the optimal solution of problem is presented as follows.

Lemma 2. When , the optimal solution must be rank-one.

Proof. See Appendix B.

Lemma 2 indicates that is rank-one with any small positive . Decompose ; then, is a feasible solution of problem . This is because satisfiesThen, the objective function value for problem satisfiesSubstituting and into (21) will yieldThat is, the performance gap between and is less than . Therefore, by setting to be a positive number small enough, for example, , the performance loss of near-optimal solution would be negligible.

The proposed algorithm for robust beamforming problem is shown in Algorithm 1.

(1) solve the SDP problem to obtain the optimal solution ;
(2) if then
(3) Decompose ;
(4) else
(5) Set , solve the SDP problem to obtain the optimal solution ;
(6) Decomposed , set ;
(7) end if
(8) Calculate the optimal/near-optimal beamforming matrix .

3.3. Computational Complexity

The computational complexity of Algorithm 1 mainly comes from the computational complexity of the SDP problems and . From [24], the computational complexity for solving an SDP problem within a tolerance is , where is the dimension of the semidefinite cone and is the number of linear constraints. Thus, the computational complexity for solving the SDP problems and is . Note that the computational complexity of suboptimal scheme is [17]; the computational complexity of our proposed method is much lower than that of the suboptimal scheme, especially at large .

4. Simulations

In this section, the performance of proposed method is demonstrated through numerical simulations. The channel vectors are assumed as , , and , , and the noise power is assumed as . The uncertainty regions are assumed to be norm-bounded; that is, and , , where determines the quality of the CSIs. The maximum interference power thresholds are assumed as . Three different CSI errors , , and and two different maximum interference power thresholds dB and dB are considered in the simulations. The number of the antennas in the relay is assumed to be 4 and the number of PUs 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 asis 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.

Figure 2 shows the relationship between average worst-case achievable channel capacity of proposed method (denoted as “Pro” in the legend) and ST transmit power () at dB, together with the worst-case capacity of suboptimal scheme (denoted as “Sub”). 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 as illustrated in Section 2.4. We can find that the proposed method outperforms the suboptimal scheme for all . Specially, it can be seen from Figure 2(a) that when dB, dB, and , 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.

(a)

(b)

Denote the average worst-case achievable channel capacity of proposed method and the suboptimal scheme as and , respectively. The capacity improvement, defined as , is shown in Figure 3 for dB. From Figure 3, we can observe that the capacity improvement becomes smaller as increases. This is because the increase of would cause both of the relay transmit constraint (17b) and interference power constraint (17c) becoming tighter, reducing the feasible region of problem . The reduction of feasible region may make the suboptimal solution tend to approach the optimal one, which makes closer to and leads to a smaller capacity improvement. Moreover, it is shown that the capacity improvement increases with the increase of . The reason is that as increases, the difference between the objective function (17a) and its lower bound (18a) becomes larger by neglecting the correlation of in suboptimal scheme, leading to the increase of the capacity improvement. Furthermore, the capacity improvement decreases with the increase of due to the fact that the increasing of tightens the interference power constraint (17c), reducing the feasible region of problem . In addition, we can also see that, as increases, the capacity improvement becomes larger. This is because the increase of loosens the interference constraint (17c), leading to an expansion of the feasible region of problem .

Figure 4 shows the average worst-case achievable channel capacity versus relay transmit power budget () at dB. 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 . The capacity improvement at dB is shown in Figure 5. From Figures 4 and 5, we can observe that only when is below some threshold, both and the capacity improvement increase as increases. The reason would be analysis as follows. Consider the following problem:Denote the optimal objective function value of problem as . Then, when , increasing expands the feasible region of problem ; as well as the capacity improvement could become larger. On the other hand, when , the feasible region of both problem and problem keeps invariant as increases, indicating that both and keep almost constant, which leads the capacity improvement to also keep invariant.

(a)

(b)

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.

Appendix

A. Proof of Lemma 1

Proof. Using and , we can getInequalities (A.1) imply that is a feasible solution of problem . Since is the optimal solution of problem , we can obtainOn the other hand, we haveCombining (A.2) and (A.3), we can obtainTherefore, is also the optimal solution of problem .

B. Proof of Lemma 2

Proof. Assume the optimal solution of problem is ; we can verify that is a strictly feasible solution of problem sincealways holds. Therefore, the problem satisfies Slater’s constraint qualification condition. Thus, the strong duality holds and the KKT conditions are the sufficient and necessary conditions for a primal-dual point to be optimal [23].
DefineThe Lagrangian function of problem can be written aswhere consists of all the primal and dual variables with , , , , and .
The KKT conditions relevant to the proof are given bySubstituting (B.5) into (B.6a), we haveFrom (B.7), we can getMultiplying both sides of (B.8) by and using (B.6c), we can obtainPremultiplying and postmultiplying equation (B.6b) by and , respectively, we haveSubstituting (B.3) into (B.10) will yieldFrom (B.11), we can obtainAdding to both sides of (B.12), we can getIf , from (B.13), we haveCombining (B.9) and (B.14), we can obtainSince always holds, from (B.15), we have . Then, substituting into (B.3), we can getHowever, cannot be semidefinite positive matrix, which violates the assumption that the SDP problem is feasible. Therefore, and should satisfy and , respectively.
Using , , and , we haveBecause is full rank, we can obtainSubstituting (B.13) into (B.18) will yieldThe last inequality is from a basic rank inequality for the product of two matrices [26].
Because is full rank, we haveSubstituting (B.9) into (B.20), we can getCombing (B.19) and (B.21), we can conclude thatOn the other hand, since holds, we have ; that is, the optimal solution of problem is rank-one.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the Natural Science Foundation of China (no. 61401278).

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Copyright © 2017 Lulu Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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