1506 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 8, OCTOBER 1998 Closed-Form Blind Channel Identification and Source Separation in SDMA Systems Through Correlative Coding Jo˜ ao M. F. Xavier, Student Member, IEEE, Victor A. N. Barroso, Member, IEEE, and Jos´ e M. F. Moura, Fellow, IEEE Abstract— We address the problem of blind identification of multiuser multiple-input multiple-output (MIMO) finite-impulse response (FIR) digital systems. This problem arises in spatial division multiple access (SDMA) architectures for wireless com- munications. We present a closed-form, i.e., noniterative, consis- tent estimator for the MIMO channel based only on second- order statistics. To obtain this closed form we introduce spec- tral/correlation asymmetry between the sources by filtering each source output with adequate correlative filters. Our algorithm uses the closed form MIMO channel estimate to cancel the intersymbol interference (ISI) due to multipath propagation and to discriminate between the sources at the wireless base station receiver. Simulation results show that, for single-user channels, this technique yields better channel estimates in terms of mean- square error (MSE) and better probability of error than a well- known alternative method. Finally, we illustrate its performance for MIMO channels in the context of the global system for mobile communications (GSM) system. Index Terms— Blind channel identification, global system for mobile communications (GSM), intersymbol interference, MIMO, multipath propagation, SIMO, wireless digital communications. I. INTRODUCTION W E study blind channel identification in the context of digital multiple-input/multiple-output (MIMO) sys- tems. This problem arises naturally in spatial division multiple access (SDMA) architectures for wireless communications. In SDMA, multiple users transmit simultaneously in time using the same frequency narrowband channel, thus increasing the cellular capacity without the need for additional RF spectrum [1]. In code division multiple access (CDMA) systems, the users also transmit simultaneously in time but each user’s signal is spread over a larger frequency region. Tong et al. [2] presented a major breakthrough for the blind identification of digital single-input/multiple-output (SIMO) systems, i.e., multichannel filters driven by a single digital Manuscript received August 26, 1997; revised April 13, 1998. This work was supported in part by the NATO Collaborative Research Grants Pro- gramme, SA. 5-02-05 (CRG.971184) 1202/97/JARC-501. The work of the J. M. F. Xavier and V. A. N. Barroso was also supported by FEDER and PRAXIS XXI, under contract PRAXIS/3/3.1/TPR/23/94. J. M. F. Xavier and V. A. N. Barroso are with the Instituto Superior T´ ecnico–Instituto de Sistemas e Rob´ otica, 1096 Lisboa Codex, Portugal (e- mail: jxavier@isr.ist.utl.pt; vab.@isr.ist.utl.pt). J. M. F. Moura is with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail: moura@ece.cmu.edu). Publisher Item Identifier S 0733-8716(98)07898-6. source. They assume that the array at the base station cell site receives an oversampled weighted linear superposition of the emitted signal. Their algorithm exploits the cyclostationary property associated with virtual channels created by temporal and/or spatial oversampling of digital communications signals. These authors derived a closed-form (noniterative) asymp- totically exact estimator, i.e., a consistent estimator, for the SIMO channel which relies only on second-order statistics of the received signals. When compared to other well-known blind equalization methods, their approach exhibits two main advantages, though at the expense of added computational complexity. • Because it is a closed-form algorithm, it does not suffer from the irregular convergence properties [5] of most adaptive iterative methods, such as the constant modulus (CM) algorithms [3], [4]. This drawback is due to the existence of many local minima attractors in the cost functions. Also, the overall performance of gradient- based optimization techniques strongly depends on the learning rate parameter, usually chosen by a trial-and- error procedure. • Since only second-order statistics are involved, the ap- proach in [2] is feasible with short data packets. This con- trasts with high-order cumulants-based methods [6]–[8] which require a significant amount of data in order to attain equivalent results in terms of mean-square error (MSE). In wireless radio communications, this is an im- portant issue since only a few data samples are available for processing during the time interval along which the channel can be assumed time invariant. After [2], many other closed-form approaches have been proposed for the blind identification of digital SIMO-finite impulse response (FIR) systems. Xu et al. [9] model the input process as a deterministic signal and exploit special algebraic properties of oversampled systems. Since no statistical model is assumed, the algorithm works with a very small number of data samples. Moulines et al. [10] take advantage of the orthogonality between the signal and noise subspaces spanned by the data covariance matrix. They exploit the structure of the filtering matrix connecting the transmitted digital sequence to the oversampled array outputs in order to significantly reduce the number of estimated coefficients. The linear prediction approach, introduced by Slock [11] and generalized by Abed- Meraim et al. [12] and Gesbert et al. [13], exhibits robustness 0733–8716/98$10.00  1998 IEEE