Optimization of a MIMO amplify-and- forward relay system with channel state information estimation error and feedback delay
- Ying Zhang^{1}Email author,
- Danni Zhao^{1},
- Huapeng Zhao^{1},
- Yinjiang Chen^{1} and
- Ning Kang^{1}
https://doi.org/10.1186/s13634-016-0400-9
© The Author(s) 2016
Received: 25 June 2015
Accepted: 13 September 2016
Published: 10 November 2016
Abstract
This paper addresses the robust design of a multiple-input multiple-output amplify-and-forward relay system against channel state information (CSI) mismatch due to estimation error and feedback delay. The estimation error and feedback delay are expressed using appropriate models, from which we derive the conditional mean square error (MSE) between the desired and the received signals upon the estimated CSI. The conditional MSE is then minimized to optimize the relay beamforming matrix with relay transmission power constraint. It is shown that the proposed optimization problem reduces to the conventional minimum MSE problem when CSI mismatch vanishes. By analyzing the structure of the optimal beamforming matrix, the optimization problem is simplified so that it can be directly solved using the genetic algorithm (GA). To further reduce the computational load, we develop a relaxed version of the optimization problem. It is found that the relaxation enables us to efficiently solve the problem using water filling strategy. Computer simulations show that both GA and water filling solutions are superior to conventional solutions without CSI mismatch consideration, while the water filling is 1000 times faster than the GA.
Keywords
Channel state information mismatch Conditional expectation Multiple-input multiple-output Minimum mean square error Relay network1 Introduction
Relaying technique is capable of extending communication range and coverage by providing link to shadowed users via relay nodes. From the perspective of signal processing, the cooperative relays can be viewed as a virtual antenna array which provides spatial diversity to combat frequency/time fading of channels. Attracted by its obvious merits, relaying technique has received extensive study in recent years [1–3].
The advantages of multiple-input multiple-output (MIMO) systems can be exploited in relay communications by accommodating multiple antennas at the nodes [4]. Recently, many works concentrate on designing non-regenerative MIMO relay systems. With perfect channel state information (CSI) assumption of all hops, optimal designs of MIMO amplify-and-forward (AF) relay networks were proposed [5–9]. To take CSI mismatch into account, robust design of a three-node MIMO relay system against CSI mismatch for linear non-regenerative MIMO relays was established in [10], where quality of service was attained by minimizing the averaged transmission power subject to mean square error (MSE) constraints at each data stream. Assuming the CSI uncertainty lies in a norm bounded region, two performance metrics, mutual information and MSE, were adopted to design the MIMO AF relay precoders in [11]. With the goal to minimize the MSE between the transmitted and the estimated symbol, Zhang proposed a robust precoder for MIMO AF relay systems against channel estimation error [12]. In [13], MSE minimization criterion was proposed to deal with CSI mismatch, and a closed-form solution was derived. In [14], two schemes aiming to maximize the signal-to-interference noise ratio were proposed to deal with CSI mismatch. In [15], the channel quantization error was considered and robust precoding schemes were proposed based on zero forcing and minimum MSE criterion. Also, a novel precoding scheme for MIMO relay systems in the presence of imperfect CSI was introduced in [16], where a base station precoding matrix and a relay station precoding matrix were created.
All the aforementioned algorithms focus on enhancing robustness against CSI estimation error. Besides CSI estimation error, other factors can also cause mismatch. In [17], quatized CSI feedback was considered and a strategy of scaling quantization quality of both two-hop links was proposed. Besides, channel feedback delay also has a significant influence on the performance of an AF relay system [18]. Therefore, it is highly desirable to consider both CSI estimation error and feedback delay when designing the MIMO AF relay system.
This paper presents a robust design of a MIMO AF relay system against CSI feedback delay and estimation error. The conditional MSE between the desired and received signals upon the estimated CSI is derived based on models of CSI feedback delay and estimation error. The conditional MSE is minimized subject to relay power constraint to optimize the beamforming matrix. Two solutions, one global and one relaxed, are proposed. Computer simulations show that both solutions outperform the MMSE strategy in terms of bit error rate (BER). It is also shown that ignoring the feedback delay results in higher BER. Although the relaxed solution is not as good as the global solution, it requires 1000 times less CPU time than the global solution. Hence, the relaxed solution provides a fast option for practical use.
2 Relay system mode
where s is the transmitted signal, W denotes the equalization matrix at the destination node, Q denotes the relay beamforming matrix and n=H _{2} Q n _{1}+n _{2} is the additive noise at the destination node. n _{1} and n _{2} are the additive complex Gaussian noises with zero mean and covariance \({\sigma _{1}^{2}}\mathbf {I}_{N_{r}}\) and \({\sigma _{2}^{2}}\mathbf {I}_{N_{s}}\), respectively.
In the conventional MMSE strategy, Q and W are optimized by minimizing \(E\left (\parallel \mathbf {s}-\mathbf {\widehat {s}}{\parallel _{2}^{2}}\right)_{\mathbf {s},\mathbf {n}}\) subject to relay transmission power constraint [19]. This criterion is optimal if the estimated channel matrices are the same as the real ones. However, due to estimation error and feedback delay, there is usually certain mismatch between the estimated and the real channel matrices in practice. Therefore, a more realistic algorithm should take the CSI mismatch into account when designing Q and W.
3 Models of CSI feedback delay and estimation error
3.1 Source to relay CSI
where \(\epsilon _{1}^{(i,j)}\) denotes the estimation error which is modeled by a complex Gaussian variable with zero mean and variance \(\sigma _{\epsilon _{1}}^{2}={{\sigma _{1}^{2}}}/{(N_{1}-1)}\). N _{1} is the number of training samples to obtain \(\widehat {h}_{1}^{(i,j)}\), and \({\sigma _{1}^{2}}\) is the noise level at relay node.
where \(\sigma ^{2}_{h_{1}|\widehat {h}_{1}}=\sigma _{h_{1}}^{2}-{\sigma _{h_{1}}^{4}}{/\left (\sigma _{h_{1}}^{2}+\sigma _{\epsilon _{1}}^{2}\right)}\).
3.2 Relay to destination CSI
where ρ is the normalized correlation coefficient calculated as ρ=J _{0}(2π f _{ d } τ). J _{0} denotes the first kind Bessel function of order zero, and f _{ d } is the maximum Doppler frequency. ζ ^{(i,j)} in (5) is a complex Gaussian variable with zero mean and variance \(\left (1-\rho ^{2}\right)\sigma _{h_{2}}^{2}\).
where \(\epsilon _{2}^{(i,j)}\) is a complex Gaussian variable with zero mean and variance \(\sigma ^{2}_{\epsilon _{2}}={{\sigma _{2}^{2}}}/{(N_{2}-1)}\). N _{2} and and \({\sigma _{2}^{2}}\) are the number of training samples and the noise level in estimating \(\widehat {h}_{2}^{\tau (i,j)}\), respectively.
where \(\sigma ^{2}_{h_{2}|\widehat {h}_{2}}=\sigma _{h_{2}}^{2}-{\rho ^{2}\sigma _{h_{2}}^{4}}/{\left (\sigma _{h_{2}}^{2}+\sigma _{\epsilon _{2}}^{2}\right)}\).
4 Robust design of a MIMO AF relay system
4.1 Minimum conditional MSE criterion
where P _{ r } is the upper bound of relay transmission power.
where W _{opt} denotes the optimal solution of W, and \(\mathbf {R}_{n}={\sigma _{s}^{2}}E\left [\mathbf {HH}^{H}|\mathbf {\widehat {H}}_{1},\mathbf {\widehat {H}}_{2}\right ] +{\sigma _{1}^{2}}E\left [\mathbf {H}_{2}\mathbf {QQ}^{H}\mathbf {H}_{2}^{H}|\mathbf {\widehat {H}}_{2}\right ] +{\sigma _{2}^{2}}\mathbf {I}_{N_{s}}\).
and λ _{1,i } and λ _{2,i } denote the i ^{th} singular values of \(\overline {\mathbf {H}}_{1}\) and \(\overline {\mathbf {H}}_{2}\), respectively. Details of transformation from (11) to (13) are presented in Appendix B. The optimization problem (13) is easier to solve than (11). Once the optimal Φ _{1} is derived, Q _{opt} and W _{opt} are computed using (12) and (10), respectively.
which are equivalent to (24) and (25) of [19]. Therefore, the minimum conditional MSE criterion reduces to the MSE criterion when the CSI mismatch vanishes.
4.2 Global solution by genetic algorithm
Substituting α ϕ _{ i } into (13a) yields the value of the fitness function of GA.
4.3 Relaxed solution by water filling strategy
It is observed from (13a) that the terms of b and c contain ϕ _{ i },i=1,…,M, which prevent deriving an analytical solution to (13). To avoid high computational loads of using the global searching algorithm, a relaxed version of (13) is proposed here.
and (x)^{+}=max(x,0), μ is the Lagrange constant which should be chosen such that (13b) is satisfied.
5 Computer simulations
To demonstrate validity and advantages of the proposed strategy, the following simulation scenarios are devised. The channel coefficients are assumed to be complex Gaussian variables with zero mean and variance \(\sigma _{h_{1}}^{2}\) and \(\sigma _{h_{2}}^{2}\). The signal-to-noise ratio (SNR) of the backward and forward channels are defined as \({\mathrm {SNR_{1}}}={\sigma _{s}^{2}}\sigma _{h_{1}}^{2}/{\sigma _{1}^{2}}\) and \({\mathrm {SNR_{2}}}=P_{r}\sigma _{h_{2}}^{2}/{\sigma _{2}^{2}}L\), respectively. The source is generated from a QPSK constellation. It is assumed that the number of data samples is 120, containing N _{ tr }=20 training samples. The number of M is 4. The relay transmission power P _{ r } is 0 dB. The BER is derived from 100 independent trials for all the plots. The proposed algorithm is compared with the conventional MMSE strategy and the robust method which only considers CSI estimation error.
Computational time (second) comparison of 100 independent trials with M=L
Methods | L=4 | L=8 | L=16 | L=32 |
---|---|---|---|---|
GA | 140.78 | 176.75 | 396.56 | 2739.67 |
Relaxed solution | 0.103 | 0.164 | 0.214 | 0.360 |
6 Conclusions
In this paper, we consider the robust design of an AF relay system against CSI mismatch. A relay system equipped with multiple antennas is considered. The conditional expectation of the mean square error is minimized with respect to precoding matrix and equalization coefficient. Computer simulations show that when the estimated CSI is different from the real CSI, the proposed strategy outperforms the MMSE strategies with perfect CSI assumption. The advantages include lower BER with different values of ρ, SNR _{1}, SNR _{2}, and L. Also, the proposed strategy is more likely to give small BER compared to the conventional MMSE strategy. Therefore, the proposed strategy is more reliable to be used in real applications.
7 Appendix A
Proof of the Theorem
Assume \(\mathbf {Q}=\mathbf {V}_{2}\boldsymbol {\Phi }\mathbf {U}_{1}^{H}\), where \(\boldsymbol {\Phi }=\left (\begin {array}{cc} \boldsymbol {\Phi }_{1} & \boldsymbol {\Phi }_{2} \\ \boldsymbol {\Phi }_{3} & \boldsymbol {\Phi }_{4} \end {array} \right)\) and \(\boldsymbol {\Phi }_{1}\in \mathcal {C}^{N_{s}\times N_{s}}\).
where \(\boldsymbol {\Phi }_{i}^{T}\) denotes the i ^{th} row of Φ.
where \(\mathbf {z}\in \mathcal {C}^{N_{s}\times 1}\).
Also, it is easy to show that Φ=Φ _{0} helps to save relay transmission power. Therefore, (11) has a solution as \(\mathbf {Q}=\mathbf {V}_{2}\boldsymbol {\Phi }_{0}\mathbf {U}_{1}^{H}\). □
8 Appendix B
Proof of Lemma 1
where \(b=\sigma _{h_{2}|\widehat {h}_{2}}^{2}\sum _{i=1}^{N_{s}}\left (\lambda _{1,i}^{2}+\sigma _{h_{1}|\widehat {h}_{1}}^{2}\right)\mid \phi _{i}\mid ^{2}\), and u _{2,i } denotes the i ^{th} column of U _{2}.
where \(c=\sigma _{h_{2}|\widehat {h}_{2}}^{2}\sum _{i=1}^{N_{s}}\mid \phi _{i}\mid ^{2}\).
Declarations
Acknowledgements
This work was supported by grants from the Central University Research Founding (ZYGX2014J018), National Science Foundation of China (61671121), and UESTC Research Start-up funding (ZYGX2016KYQD106).
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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