Robust Blind Separation for MIMO Systems against Channel Mismatch Using Second-Order Cone Programming

Zhongqiang Luo ＊, Chengjie Li , Lidong Zhu

1 School of Automation & Information Engineering, Sichuan University of Science and Engineering, China, Zigong 643000, China

2 National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China,Chengdu 611731, China

* The corresponding author, email: luozhongqiang@126.com

I. INTRODUCTION

Multiple-Input Multiple-Output (MIMO) systems have received considerable attention due to its dramatic capacity gain and high quality of communication performance [1-3]. The ideal capacity gain and the performance of source recovery are obtained based on the premise that the channel state information (CSI) is acquired perfectly. However, to acquire perfect CSI is difficult on account of an increased number of channel parameters to be estimated at the receiver. Moreover, the channel environments are complex, which always lead to time-varying channel and channel distortion.Therefore, in practice, the channel state condition is always unmatched which will give rise to the performance loss of MIMO systems[2, 3]. This problem motivated us to develop effective methods to improve the system performance of MIMO communications.

It is an undeniable fact that the CSI ac-quisition is crucial to guarantee system performance in MIMO systems. In a general way, the channel estimation methods can be classified into two types: pilot sequence assisted non-blind method and blind estimation method [3]. The pilot sequence based methods need occupy extra frequency resource which will reduce the spectrum effectiveness of the system. Moreover, the real-time of nonblind channel estimation method is not fit in time-varying channel condition due to requirements of frequent pilot sequence. For all the above reasons, it is attractive to use the blind methods which depend on the statistical property of the source to carry out estimation work.For the past few years, the direct blind source recovery is implemented by blind source separation (BSS) based on independent component analysis (ICA) which arouses much attention in wireless communication. The goal of BSS/ICA aims at the recovery of a set of unknown signals from their mixtures, with the only knowledge of their general statistical/independent properties [4-6]. BSS has important applications in wireless communications, such as CDMA [7, 8, 9, 15, 17], OFDM [10-12] and MIMO [13, 14], and so on. With the help of BSS, the effect of nuisance channel estimation can be formulated as mixing matrix in a BSS framework. The source signals can be recovered from only received mixed signals [7-13].

The ICA-based blind extraction for MIMO system with the channel mismatch problem is proposed in this paper based on approximate negentropy maximization using SOC constraint.

Fig. 1 Scene graph of system model

This paper focuses on the problem of blind separation for MIMO system under the condition of channel mismatch. There are few papers which report the research results about this problem [9, 14]. We consider a channel model with a bounded fluctuant error in the channel mismatch state. A robust blind separation/extraction method is developed,which is based on the approximate negentropy maximization subject to the SOC constraint[15-17]. The proposed blind source recovery method can be self-adaptive adjust to be fit for a bounded fluctuant error in the channel mismatch state. The proposed method can effectively improve the system performance of MIMO system under the condition of channel mismatch. Theoretical analysis and simulation results illustrate that the proposed algorithm has low computational complexity and the effective performance.

The remaining of this paper is organized as follows. In Section II, the system model and problem formulation are described. Section III illustrates the proposed robust blind separation/extraction method. Section IV shows the computational complexity and performance analysis. Section V demonstates the simulation results and discussions. Conclusions and remarks are given in Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Assume that a multiuser MIMO (MU-MIMO)downlink system in a single cell, the number of base station antennas is. The scene graph of the system model is shown in Fig. 1. The total number of mobile user in a cell is, and theactive users communicate simultaneously. In Fig.1, the solid lines denote active users, and the dashed lines show unactive users.

Namely,

Usually, it is assumed that the receiver has perfect knowledge of channel state conditionof the user used to recover the symbols of user. However, this may not be ture in real systems. Because the channel conditionmay include channel distortion or channel estimation error, which lead to channel mismatch. Under unknown channel distortion and channel estimation errors, the actual received signal channel model may be described as

where the channel mismatch termis modeled as a vector of complex Gaussian random variables with zero mean and varianceFurthermore, the channel mismatch in matrix form is described as

In a channel mismatch state, the nonblind detectors, such as ZF (zero-forcing) and MMSE (minimum mean square error) can not attain ideal performance gain. As shown in Fig.2 and Fig.3 (simulation parameters is in Section V), the degraded capacity and BER performance gain is obtained under the condition of channel mismatch in MIMO system.Moreover, the conventional BSS algorithm is lack of robustness for separation or extraction of the source signals because of the fluctuant channel matrix (which is also known as mixing matrix in BSS model). In next section, the robust algorithm is proposed to overcome the above-mentioned problem to improve system receiving performance.

Fig. 2 Capacity gain performance under two diffierent channel conditions

Fig. 3 BER performance under two diffierent channel conditions

III. RUBST BLIND SEPARATION USING SECOND-ORDER CONE PROGRAMMING

3.1 The principle of ICA-based blind separation method

From the statistical independence point of view, BSS is also known as ICA, which is based on an assumption of the independence and non-Gaussianity of the sources. ICA method always includes two steps: first, the independence principle based cost function is constructed; second, the optimation operation of the cost function is carried out. Based on the non-Gaussian maximization principle,negentropy maximization is proposed by Hyv?rinen and Karhunen as the cost function for blind separation [4, 5].

Negentropy is obtained from differential entropy and defined as [5, 6]

For complex-valued sources, find a certain signal tap and makewhereis a certain signal tap-weight vector and the superscriptdenotes the complex conjugate transposition. According to the principle of independent component analysis (ICA) based blind separation, the latent source of the received mixed-up signals can be extracted by maximizing their approximate negentropies.This inference is extended to complex-valued sources in [6], making the cost function

Gis core function which is chosen for non-Gaussianity maximization in. Due to that communication signals are always sub-Gaussian signal, a suitablefunction with fast computation is chosen, as follows [4, 5],

3.2 Blind separation subject to second-order cone constraint

Taking into accout the channel mismatch condition, the channel of desired user is modeled as

Without loss of generality, suppose that user 1 is the desired user. Assume that channel mismatch term can be bounded by some constantthat is,The actual channel condition can be described as a vector in the set

Combined with the model nd the actual channel condition, then yielding

In cost function, the additional second-order cone (SOC) programming condition is introduced for ICA formulation. This constraint can be rewritten as

Furthermore, the problem of an be formulated as

When the cost function (15) ismaximizing,the equality constraintis satisfied. Omit an extra constraint imaginary part ofbecause the equality constraintguarantees that the value ofis real-valued and positive. Using the equality constraint, the weight vectoris obtained by rewriting the convex optimization problem of (15) as

The optimization problem of worst-case performance by maximizing the approximate negentropy subject to the SOC constraint is solved. The constrained optimization problem in ill be derived by the quasi-Newton iteration.Newton’s method is based on the Lagrangian function

The Hessian matrix of the cost function s

The algorithm can be further simplified by multiplying both sides of (20) byWhile choosing a sufficiently smallit can be easily shown that the left-hand side of the equality in (20)wmultiplyingcan be set aswhere

Step 1.After centeringtake a small initial vectoris estimated in advance or known, ut channel condition is channel mismatch due to channel distortion or time-varying or estimation error) Set interation number

Step 2.Update

(21) shown in the bottom at this page.

Step 3.Check the convergence of.If the error measure iswhereis the terminating error value, letand go back to Step 2. Otherwise, output the vectorAccording to ICA method, the step that projectthroughcould enforce the unit variance constraintafter each step.

Remarks: For MIMO blind detection, a new modification of the classical ICA has been proposed that omits the complex pre-whitening step and directly incorporates the SOC constraint onto the correct solution. In MIMO system, the proposed ICA-SOC detection is developed to combat small-to-medium,norm-bounded channel mismatch. It is worth noting that the proposed detector can be implemented by choosing a proper value forwithout estimating. Due to the algorithmic parameter(including to the Lagrange multiplier), the parameter, and the unknown channel mismatch), the proposed ICASOC possesses the capability of self-adjusting computation against channel mismatch.Self-adjusting computation refers to a model of computing where computations can automatically respond to changes in varying channel mismatch, which means that parametercan be kept a proper constant under varying circumstances by using self-adjustedandfor ICA iterations. The proposed algorithm is nonsensitive to channel mismatch.

IV. COMPUTATIONAL COMPLEXITY AND PERFORMANCE ANALYSIS

4.1 Computation complexity analysis

The computational complexities of the proposed detector in including projection) are aboutin total. The desired weight/filter vector of the ICA-SOC detector can be obtained by calculating. Assumed that the number of symbols iswhich is used for each batching processing to form the data matrixFirst, the computational complexity of computing the autocorrelation matrix

The computational complexity of the proposed detectors will then be compared with the non-blind detector, ZF and MMSE detector, and conventional blind ICA detector. In the subspace-based MMSE detector,the EVD (eigenvalue decomposition) of autocorrelation matrix has the complexity ofand the complexity of the projection of the desired signature waveform onto signal subspace isThus, the final complexity of the subspace-based MMSE is of orderBesides, in the ICA, the computational complexities of autocorrelation matrix, prior subspace estimation, pre-whitening of the received data, and each unit-gainbased ICA iteration areandrespectively. Thus, the final complexity of the ICA is of order

4.2 Performance analysis

The conventional source recovery scheme needs CSI, which is always obtained by using channel estimation operation through the pilot sequence. Even if the CSI is acquired perfectly, the estimation of the source signals in a noise model is not a trivial task. Although the maximum likelihood sequence estimator is the optimal detector, its computational load can be prohibitive in scenarios involving a large number of users and highly dispersive channels. Trading off complexity for performance,linear receivers are based on estimation of a linear transformation fulfilling certain suboptimal criterion, and data are then detected asThe zero forcing (ZF) detectors aim at the joint minimization of ISI and ICI in the absence of noise, and can thus be expressed as the least-squares problem,

The solution to s readily computed as

In practice, the CSI is not perfect, which is mismatch due to channel estimation errors. Namely, the actual channel matrix iswhich will lead to the damaged receving performance. Moreover, the ZF detector can lead to severe noise amplification in noisy scenarios.

The minimum mean square error (MMSE)detector can alleviate noise amplification drawback in ZF detector. The MMSE detector can be decribed as

Considering the constrained condition, we can use extra information to help blind separation assignment. Thus the ICA-based blind separation scheme can be strengthened for more performance refinement. The rationale behind constrained ICA-aided is that the channel mismatch condition is integrated into the cost function of blind separation to construct robust SOC programming optimization problem. Thus the constrained blind separation can lead to the receiving performance improvement.

In the following section, the simulation experiments are implemented to verify the performance of the proposed algorithm against the channel mismatch problem.

Fig. 4 Capacity as a function of SNR for different methods

V. SIMULATIONS AND DISCUSSIONS

To demonstrate the effectiveness of the proposed ICA-SOC algorithm, we conduct simulation experiments to evaluate the performance of MIMO systems compared with those of the existing non-blind detectors which include zero-forcing (ZF) and minimum mean square error (MMSE), and ICA-based blind detector.The simulation parameter is thatwithout user scheduling policy. Whenthe random selection method is used to determine the active user set. Theusers are allocated the same transmit power. The QPSK symbols are transmitted for source signals with the same frequency. The data sample length is 2000. The number of base station and the mobile station is 4. The channel vector of desired user 1 is estimated ashowever the actual channel condition is with a bounded fluctuateis channel mismatch parameter) due to channel distortion or channel estimation error. The channel mismatch termis a vector of complex Gaussian random variables with zero mean and variance.

Fig. 4 shows the system capacity and the average capacity of the single user for the different methods. When the SNR is lower, the influence of noise will be a main factor. ZF method can eliminate multiple user inference(MUI) but has a noise amplification negative effect. Therefore, ICA-based method (ICA and ICA-SOC) obtains higher average capacity than the ZF method. The MMSE method takes into account the influence of noise, thus has the better system capacity than ZF method and ICA based method. With the increase of SNR,MUI gradually becomes the main factors influencing the capacity. Inperfect channel estimation is helpful for eliminating MUI,thus the non-blind method has high capacity in high SNR. However, inthe channel mismatch leads to performance deterioration of non-blind methods. The system capacity tends to an asymptotic valuewith the increase of SNR. As shown in Fig. 4, inthe ICA-based method acquires better average capacity than ZF method in whole SNR range,and is superior to MMSE method when SNR＞15dB. We can see that the proposed ICASOC is better than non-blind method and ICA method under channel mismatch conditions,which can improve the impaired capacity gain of MIMO.

Fig. 5 gives a BER performance of the different methods. In a lower SNR, the noise is main factor which affects the BER performance. Inthe BER performance of MMSE is best, and the ICA-based method(ICA and ICA-SOC) is better than ZF method.At the condition ofand low SNR, the channel mismatch has a small effect on nonblind method, thus the BER performance of MMSE method is better than ICA-based blind method. With the increase of SNR, ICA-based blind method gets better separation performance. However, MUI gradually becomes the main factors influencing the BER. Therefore,inand high SNR, the BER performance of MMSE method tends to saturate. As shown in Fig. 5, at the condition ofand SNR＞14dB, the BER of MMSE method is worse than the ICA-SOC blind method. Due to the noise amplification effect of ZF method, the BER performance is worst in the whole SNR range at the same condition of. The performance of the ICA-SOC method is near to the performance of the MMSE in perfect channel condition. All in all, the ICA-SOC is robust to the influence of a channel mismatch from the previous analysis.

Fig. 6 illustrates the robustness of different method against channel mismatch. The SNR is set at 25 dB. With the increase of, the non-blind methods become worse gradually.And the ICA-based is also affected by the increasing. However, the proposed ICA-SOC almost has no influence, which can adjust to overcome the influence of channel mismatch.

VI. CONCLUSIONS

As a powerful technique, BSS can offer additional interference suppression capability,since also the independence of the source signals is utilized. BSS can mitigate the performance drops due to erroneous timing and channel estimation problem. As a motivation,in this paper, the ICA-based blind extraction for MIMO system with the channel mismatch problem is proposed based on approximate negentropy maximization using SOC constraint.The proposed blind detector can provide robustness against channel mismatch to achieve better performance than the existing blind detectors. Simulation results have demonstrated that the effective performance of the proposed method is obtained. The proposed blind separation/extraction method is appealing and attractive for MIMO system subject to time-varying channel in the future work.

Fig. 5 BER performance as a function of SNR for different methods

Fig. 6 BER performance as a function of d for different methods

ACKNOWLEDGEMENTS

The authors would like to thank the reviewers for their detailed reviews and constructive comments, which have helped improve the quality of this paper. This work is fully supported by Sichuan Youth Science and Technology Innovation Research Team Project(No. 2015TD0022), and the Talents Project of Sichuan University of Science and Engineering (No. 2017RCL11 and No. 2017RCL10),and the first batch of science and technology plan key R&D project of Sichuan province(No.2017GZ0068).

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