1998 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 6, JUNE 2006 Blind Equalization of SIMO FIR Channels Driven by Colored Signals With Unknown Statistics Jun Fang, Member, IEEE, A. Rahim Leyman, Member, IEEE, and Yong Huat Chew, Member, IEEE Abstract—We consider the blind equalization of single-input multiple-output finite impulse response channel driven by colored signals. The statistics of the input colored signals are unknown a priori. By exploiting the inherent structural relationship be- tween the source autocorrelation matrices of different delay lags, the closed-form zero-forcing and minimum mean-square-error equalizers of desired delays are estimated from the second-order statistics of the received data. The blind equalizability conditions of the proposed method are investigated. Numerical simulation results are presented to illustrate the performance of the proposed algorithm. Index Terms—Blind equalization, colored source, single-input multiple-output system. I. INTRODUCTION I N most practical wireless communication environments, intersymbol interference (ISI) exists and has a severe distor- tion effect on the transmitted signals. ISI arises from channel dispersion due to multipath propagation and limited channel bandwidth. Traditionally, to accurately recover the transmitted symbols, we achieve channel identification or equalization by resorting to training sequences. However, these periodically transmitted training sequences result in reduced effective channel data rate. In contrast, the so-called blind identifica- tion/equalization methods do not require training sequences but rely only on the observations of system outputs and some a priori statistical information of transmitted signals. Early blind techniques exploited higher order statistics of the received data to estimate the channel and compute the equalizer [1], [2]. More recently, in the pioneering work [3], it is shown that the single-input multiple-output (SIMO) finite impulse response (FIR) channel can be perfectly identified/equalized from the second-order statistics of the received data under quite general assumptions. Following [3], numerous second-order statistics (SOS)-based blind identification/equalization methods [4]–[8] have been proposed. However, most existing SOS-based methods are applied to the independent identically distributed (i.i.d.) input signals. The works that considers the case of Manuscript received December 9, 2004; revised July 18, 2005. This work was supported by the Agency for Science, Technology and Research (A STAR), Singapore, under Research Grant 022-106-0041. The associate editor coordi- nating the review of this manuscript and approving it for publication was Dr. Franz Hlawatsch. J. Fang is with the Department of Electrical and Computer Engi- neering, National University of Singapore, 119260, Singapore (e-mail: g0202082@nus.edu.sg). A. R. Leyman and Y. H. Chew are with the Institute for Infocomm Re- search, A STAR, 119613 Singapore (e-mail: larahim@i2r.a-star.edu.sg; chewyh@i2r.a-star.edu.sg). Digital Object Identifier 10.1109/TSP.2006.874406 correlated input signals are much less; see [9]–[11]. In fact, colored sources indeed occur in practice. For example, colored sources arise as a result of channel encoding [12]. Specifically, [9]–[11] treated the case where the input signal statistics are colored and known. As for the case where the input statistics are colored but unknown, it seems that it is much more difficult to devise an SOS-based algorithm since no prior statistical information of the transmitted signals can be utilized. One solution to this problem is given in [4], which proposed a subspace-based method by exploiting the block Toeplitz struc- ture of the channel convolution matrix, and thus required no knowledge of input statistics whatsoever. The extension of [4] to multiple-input multiple-output (MIMO) systems was studied in [13]. There are some other works [14], [15] that studied blind identification/equalization of an MIMO FIR channel driven by colored signals with unknown statistics. However, both [14] and [15] constitute a two-step approach that is based on [4]. They first determine source separating vectors or decorrelators to separate the sources. Once the sources are separated, the second step utilizes the subspace method [4] to estimate the resulted SIMO systems and the original MIMO systems. Some deterministic approaches that can handle arbitrarily correlated source signals have been discussed in [16]–[19] for blind SIMO channel identification/equalization. They are most effective at high signal-to-noise ratio (SNR) and for small data sample sce- narios. Note that the deterministic method [16] has its statistical version whose performance is similar as the subspace method [4]. Besides the above-mentioned methods, the mutually ref- erenced equalizers (MREs) method proposed in [20] and the constrained minimum output energy algorithm presented in [21] are also important work in blind SIMO channel identifica- tion/equalization. Reference [20] was developed on the concept of mutually referenced equalizers, i.e., the outputs of the set of filters (equalizers) act as training signals for each other. The method does not rely on the specific assumptions concerning the input statistics, and several variations of the MRE criterion including a stochastic criterion using the second-order statistics have been derived. Another interesting work [21] explored the popular constrained minimum output energy approach to derive the optimal blind equalizers. As indicated in [21], the method is also insensitive to the color of the input signals. In this paper, we study the blind equalization of SIMO FIR channel when the input signals are colored but the source sta- tistics are unknown. It is shown that although the statistical in- formation of the transmitted signals is not available, we can still estimate the equalizers of desired delays from the second-order statistics of the received data by exploiting the inherent struc- tural relationship between source autocorrelation matrices of 1053-587X/$20.00 © 2006 IEEE