406 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 2, FEBRUARY 2004 Blind Identification of MIMO FIR Systems Driven by Quasistationary Sources Using Second-Order Statistics: A Frequency Domain Approach Kamran Rahbar, James P. Reilly, Member, IEEE, and Jonathan H. Manton Abstract—This paper discusses a frequency domain method for blind identification of multiple-input multiple-output (MIMO) convolutive channels driven by white quasistationary sources. The sources can assume arbitrary probability distributions, and in some cases, they can even be all Gaussian distributed. We also show that under slightly more restrictive assumptions, the algorithm can be applied to the case when the sources are colored, nonsta- tionary signals. We demonstrate that by using the second-order statistics of the channel outputs, under mild conditions on the nonstationarity of sources, and under the condition that channel is column-wise coprime, the impulse response of the MIMO channel can be identified up to an inherent scaling and permutation ambiguity. We prove that by using the new algorithm, under the stated assumptions, a uniform permutation across all frequency bins is guaranteed, and the inherent frequency-dependent scaling ambiguities can be resolved. Hence, no post processing is required, as is the case with previous frequency domain algorithms. We further present an efficient, two-step frequency domain algorithm for identifying the channel. Numerical simulations are presented to demonstrate the performance of the new algorithm Index Terms—Blind identification, blind signal separation, MIMO systems, nonstationarity. I. INTRODUCTION B LIND identification of a multiple-input, multiple-output (MIMO) finite impulse response (FIR) system deals with identifying the impulse response of a unknown system using only the system output data and, in particular, without any (or the least amount of) knowledge about the inputs. Multichannel blind identification has been of great interest to both the communications and signal processing communities, and there have been numerous publications in both societies on this subject (see [1] for a review of recent blind channel estimation and identification techniques). Extensive literature has been Manuscript received March 27, 2002; revised April 3, 2003. The work of K. Rabhar and J. P. Reilly was supported by Mitel Corporation, Canada; The Centre for Information Technology Ontario (CITO); and the Natural Sciences and En- gineering Research Council of Canada (NSERC). The work of J. H., Manton was supported by the Australian Research Council and the Special Research Center for Ultra-Broadband Information Networks. The associate editor coor- dinating the review of this paper and approving it for publication was Dr. Inbar Fijalkow. K. Rahbar and J. P. Reilly are with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 (e-mail: kamran.rahbar@zarlink.com; reillyj@mcmaster.ca). J. H. Manton is with the Department of Electrical and Electronic Engi- neering, The University of Melbourne, Parkville, Victoria 3010, Australia (e-mail: jon@ee.mu.oz.au). Digital Object Identifier 10.1109/TSP.2003.820988 dedicated to a special instance of the MIMO channel identifi- cation problem, namely, the single-input single-output (SISO) case [2], [3]. When there is more than one output derived from a common source, the problem is referred to as single-input, multiple-output (SIMO) blind identification [4], [5] 1 . In the literature, there are fewer works related to the more general MIMO problem. An exception is in the memoryless channel case, where MIMO blind identification is closely related to blind source separation (BSS) for instantaneous mixtures. This latter problem has recently been discussed extensively in both the signal processing and neural network literature [6]. In addition, see [7] for indeterminacy and identification conditions in this case. Blind identification of a MIMO channel can be used for blind signal separation and blind multichannel equalization in a con- volutive environment. In most methods for BSS of convolutive mixtures, the sources are only separated up to a distorted (fil- tered) version of the original sources [8], [9]. Identification of the complete MIMO channel not only permits blind signal sepa- ration but, in addition, gives the potential for equalization of the outputs so that the original sources can be fully recovered. This by itself has many applications, e.g., in data communication for eliminating the ISI without knowledge of the input signal or in speech processing for reverberation cancellation and speech en- hancement and in biomedical instrumentation for suppressing cardio or other interfering signals from desired signals, such as electromyogram or electrogastrogram signals. In this paper, we consider the problem of blind identifica- tion of MIMO channels with finite memory. Previous work in this area can be divided into two groups. The first group uses higher order statistical (HOS) methods that exploit the higher order moments (or higher order spectra) of the output signals to identify the channel, e.g., [10] and [11]. For HOS methods, a non-Gaussian condition on the inputs is necessary. In addition, the main limitation of the HOS methods is their slow conver- gence due to large estimation variance of higher order moments. As a result of this, they usually require large sample sizes for good time-averaged estimates of the higher order statistics [12]. The second group are the second-order statistical (SOS) methods that rely only on the second-order moments of the output signals to identify the channel [8], [13], [14]. SOS methods have the following advantages: They usually have a simple implementation, and in some cases, a closed-form 1 In fact, for some cases, SISO can often be converted into a SIMO problem by oversampling the outputs of the channel. 1053-587X/04$20.00 © 2004 IEEE