A second-order blind equalization method robust to ill-conditioned SIMO FIR channels
Introduction
One of the major challenges in wireless communications is channel fading caused by multipath propagation. Channel fading can generate considerable inter-symbol interference (ISI) and thus result in severe signal distortion at the receiver. For single-input–multiple-output (SIMO) finite-impulse-response (FIR) channels, the effects of channel fading can be greatly reduced by applying a proper equalizer to the received signals. This process is called channel equalization. Traditionally, a known training sequence of symbols is transmitted to the receiver to estimate the SIMO FIR channel and then the obtained channel information is used to derive the equalizer. In spite of its simplicity, this approach is inherently inefficient since it wastes the bandwidth and limits system throughput. Moreover, the training session would be unrealistic in certain applications such as asynchronous wireless networks. Because of these reasons, methods which do not use a training sequence are preferred, which refers to the so-called ‘blind’ (channel) equalization.
As will be shown in Section 2, the SIMO FIR channel system is equivalent to a virtual multiple-input–multiple-output (MIMO) instantaneous mixing system. In this virtual MIMO mixing system, the ‘source’ signals include the real source signal and its delayed versions , where L is the order of the SIMO FIR channel. So the blind equalization of an SIMO FIR channel can be cast as a blind source separation (BSS) problem, where the dimension of the virtual instantaneous mixing matrix is dependent on the channel order L and the number of channel outputs I. If , the virtual mixing matrix will have more columns than rows, which corresponds to the so-called underdetermined BSS. Although various BSS methods for instantaneous mixing systems have been reported in the literature [1], [2], [3], [4], these methods cannot be directly used to recover the source signal due to several reasons: i) the ‘source’ signals in the virtual MIMO mixing system are mutually identical up to certain time delay and can be non-sparse as well, ii) the samples of the ‘source’ signals (resp. the entries of the virtual mixing matrix) might not have the same sign, and iii) there exists inherent permutation ambiguity in BSS, which would lead to unknown time delay in source signal recovery. To perform blind equalization for an SIMO FIR channel, both the special structure of the channel and the property of the source signal should be exploited.
Most of the existing blind equalization methods employ either the higher-order statistics (HOS) or second-order statistics (SOS) of the source signal. The constant modulus algorithm (CMA) is the most popular HOS-based blind equalization algorithm, which aims at minimizing the deviation of the amplitude of the equalizer output from a constant. Based on the concept of CMA, various variations of CMA have been proposed [5], [6], [7], [8], [9], [10]. The method in [5] uses a dual-mode scheme, i.e., the CMA is used to open the ‘eye’ of the channel outputs and the decision-directed (DD) algorithm follows for further equalization. However, the decision on switching from the CMA to the DD algorithm is made in an ad hoc manner, which may result in the ill convergence of the DD algorithm. In [6], a single-mode scheme is presented to equalize the channel system. However, as shown in [11], the method in [6] cannot deal with communication signals which have more than four constellation points. The restriction on the number of constellation points is removed for M-ary phase-shift keying (MPSK) signals in [7] and time-varying channels with a special structure are considered in [8]. The method in [9] and [10] can equalize both SIMO and MIMO FIR channels. A drawback of the CMA-based methods is the delay ambiguity, i.e., the recovered source signal might have an unknown time delay. In addition, it is known that HOS-based methods normally need a large number of data samples to obtain satisfactory performance [12]. The usage of huge data samples not only increases the computational complexity but also is prohibitive in some applications. Therefore, the blind equalization methods based on SOS are preferable [13].
With regard to blind equalization using SOS, major efforts have been devoted to the blind identification of SIMO FIR channel [14], [15], [16], [17], [18], [19], [20]. Once the SIMO FIR channel is estimated, the corresponding equalizer can be easily obtained by computing the pseudo-inverse of the block-Toeplitz matrix associated with the SIMO FIR channel. Among the existing SOS-based blind identification techniques, the cross-relation (CR) methods [14], [15] are based on noise-free channel models and thus sensitive to noise added to the channel outputs. The linear prediction (LP) methods [16], [17], [18] have low computational costs but they are sensitive to observation noise as well. In contrast, the subspace (SS) methods [19], [20] show better performance when the channel outputs are contaminated by additive noise. The method in [20], called truncated transfer matrix (TTM) method, only utilizes the covariance matrices of the stacked channel outputs at time lags 0 and 1. So it is easy to implement. However, we shall show in Section 3 that the TTM method will fail in some situations.
We would like to note that the SIMO FIR channels encountered in many practical applications could be ill-conditioned [21], [22], [23], i.e., the block-Toeplitz matrix associated with the channel is of near column rank deficiency. This can occur when there is a lot of receiver diversity, or when channel impulse response contains small tail or precursor coefficients. A common shortcoming of most SOS-based blind identification techniques is that their performance degrades significantly if the SIMO FIR channel is ill-conditioned. Although some iterative algorithms have been proposed for estimating ill-conditioned channels [24], [25], [26], these algorithms suffer from local minimum problem and thus could yield incorrect channel estimation result. Consequently, the equalizer cannot be accurately obtained as its derivation depends on the precise estimation of the channel impulse responses. Some SOS-based methods have also been proposed to perform direct channel equalization [29], [30]. The method in [29] requires the SIMO channel lengths to be known (or estimated) a priori. This requirement usually cannot be satisfied in practical applications and channel length estimation is a challenging task. Regarding the method in [30], it requires the source signal to be colored and thus is not applicable to blind equalization of SIMO channels driven by i.i.d. signal.
In this paper, we propose a new SOS-based blind equalization method, with particular focus on ill-conditioned SIMO FIR channels driven by i.i.d. signals. Similar to the TTM method, the proposed method only employs the covariance matrices of the stacked channel outputs at time lags 0 and 1, making it easy to implement. Furthermore, since our method exploits the i.i.d. assumption on the source signals, it is robust to ill-conditioned channels. Besides, whilst the equalization methods in [5], [6], [7], [8], [27] and [28] suffer from time delay ambiguity, the signal recovered by our method does not have any time delay.
The remainder of this paper is organized as follows. Section 2 formulates the problem of blind equalization of SIMO FIR channels. A brief analysis is conducted in Section 3 to show that the TTM method in [20] does not guarantee to work under the given assumptions. The new blind equalization method is presented in Section 4. Section 5 illustrates the performance of the proposed method, in comparison with three existing benchmark methods. Finally, some conclusions are drawn in Section 6.
Section snippets
Problem formulation
We consider an SIMO FIR channel system whose ith channel output is where is the source signal, is the ith additive noise, and is the ith channel impulse response of order L. The channel order L can be estimated using the methods in [31], [32], [33], [34]. The vector form of (1) can be expressed as where and the superscript stands for
Analysis of TTM method
Let be the identity matrix and stand for the Jordan matrix with the following form: The TTM method in [20] is based on the following assumptions:
- A1)
The complex input symbols are i.i.d. with unit variance.
- A2)
The complex noise vector sequence is zero-mean, temporally white, and its covariance matrix is equal to .
- A3)
The random variables and are independent for every .
- A4)
It holds that
- A5)
The block-Toeplitz matrix related to H has
Proposed blind equalization method
In this section, we shall propose a novel blind equalization method, which adopts all of the assumptions required by the TTM method except for the assumption A4). The proposed method can directly obtain a zero-forcing (ZF) equalization vector b from the stacked channel outputs such that , where α is a nonzero constant. As a result, the source signal can be recovered to a very great extent(up to a scalar) by the ZF equalizer in the absence of additive noise. Since
Simulations
In this section, we use four different SIMO channel systems driven by quadrature phase-shift keying (QPSK) signal to illustrate the validity of the proposed approach. The performance of our method is also compared with the HOS-based blind equalization method in [9] and [10], labeled as CM, and three classical SOS-based blind identification methods: the CR method [15], the SS method [19], and the TTM method [20]. Whilst our method and the CM method in [9] and [10] can directly estimate the ZF
Conclusion
In this paper, we present a new SOS-based blind equalization method for SIMO FIR channels. The proposed method directly estimates the desired ZF equalizer by using the covariance matrices of the stacked channel outputs at time lags 0 and 1. We show that if noise is absent at the channel outputs, the obtained ZF equalizer can recover the source signal to a very great extent (up to a scalar). Then the obtained ZF equalizer is transformed to a linear MMSE equalizer to reduce the negative impact of
Acknowledgements
This work was supported by the Australian Research Council under grants DP110102076 and LP120100239, the Xinmiao Foundation of Guangzhou University, and the National Basic Research Program of China (973 Program) under Grant 2011CB302201, the Program for New Century Excellent Talents in University of China under Grant NCET-12-0384, and the National Natural Science Foundation of China under Grant 61172180.
Yong Xiang received the B.E. and M.E. degrees from the University of Electronic Science and Technology of China, Chengdu, China, in 1983 and 1989, respectively. In 2003, he received the Ph.D. degree from The University of Melbourne, Melbourne, Australia.
He was with the Southwest Institute of Electronic Equipment of China, Chengdu, from 1983 to 1986. In 1989, he joined the University of Electronic Science and Technology of China, where he was a Lecturer from 1989 to 1992 and an Associate
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Yong Xiang received the B.E. and M.E. degrees from the University of Electronic Science and Technology of China, Chengdu, China, in 1983 and 1989, respectively. In 2003, he received the Ph.D. degree from The University of Melbourne, Melbourne, Australia.
He was with the Southwest Institute of Electronic Equipment of China, Chengdu, from 1983 to 1986. In 1989, he joined the University of Electronic Science and Technology of China, where he was a Lecturer from 1989 to 1992 and an Associate Professor from 1992 to 1997. He was a Senior Communications Engineer with Bandspeed Inc., Melbourne, Australia, from 2000 to 2002. He is currently an Associate Professor with the School of Information Technology at Deakin University, Australia. His research interests include signal and system estimation, information and network security, wireless sensor networks, multimedia (speech/image/video) processing, compressed sensing, and biomedical signal processing.
Liu Yang received the B.S. degree from Jiangsu University, Zhenjiang, China and the Ph.D. degree from South China University of Technology, Guangzhou, China, in 2007 and 2013, respectively.
She is a Lecturer at the School of Computer Science and Educational Software, Guangzhou University, Guangzhou, China. Her current research interests include intelligent information processing, blind signal processing, and parallel factor analysis.
Dezhong Peng received the B.S. degree in applied mathematics and the M.S. and Ph.D. degrees in computer software and theory from the University of Electronic Science and Technology of China, Chengdu, China, in 1998, 2001, and 2006, respectively.
He was a Postdoctoral Research Fellow at the School of Engineering, Deakin University, Australia, from 2007 to 2009. Currently, he is a Professor at the Machine Intelligence Laboratory, College of Computer Science, Sichuan University, Chengdu, China. His current research interests include blind signal processing and neural networks.
Shengli Xie received the M.S. degree in mathematics from Central China Normal University, Wuhan, China, and the Ph.D. degree in control theory and applications from the South China University of Technology, Guangzhou, China, in 1992 and 1997, respectively.
He is the Director of the Laboratory for Intelligent Information Processing and a Full Professor with the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. He has authored or co-authored two monographs, a dozen of patents and more than 80 scientific papers in journals and conference proceedings. His current research interests include automatic controls, signal processing, and image processing.