1814 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 9, SEPTEMBER 2004 Blind Source Extraction From Convolutive Mixtures in Ill-Conditioned Multi-Input Multi-Output Channels Yuanqing Li, Jun Wang, Senior Member, IEEE, and Andrzej Cichocki Abstract—This paper presents a new approach to blind source extraction from convolutive mixtures in multi-input multi-output (MIMO) channels. Two ill-conditioned cases are considered: the number of sensors is less than the number of sources and the number of sensors is greater than or equal to the number of sources but the system is noninvertible. Although there exist several works related to ill-conditioned dynamic MIMO channels, especially on blind channel identification, how to obtain a true source only from observable convolutive mixtures is still an open problem. In this paper, beginning with introducing two blind ex- traction models for blind deconvolution in ill-conditioned MIMO channels, we discuss the extractability issue. Results from our extractability analysis (a necessary and sufficient condition) show that it is possible to extract individual sources from the outputs. Furthermore, all potentially separable sources (at most equal to the number of sensors) can be extracted sequentially based on these extraction models. A cost function based on cross cumulant is discussed along with the Gauss–Newton algorithm. Finally, a simulation example is presented for illustration. Index Terms—Blind extraction, convolutive mixtures, ex- tractability, multichannel deconvolution. I. INTRODUCTION B LIND deconvolution recovers unknown sources from convolutive mixtures without the information about the signal channels. It has significant potential for applications in numerous technical areas such as array processing, speech and image enhancement, digital communications [1]–[6], to name a few. Blind deconvolution of multi-input multi-output (MIMO) dynamic channels has received considerable attention in recent years. Many studies on blind deconvolution of MIMO systems have been reported; see the references herein. For example, in [5], [7]–[13], various algorithms (e.g., cumulant-based super-exponential algorithm, adaptive algorithms, Godard cost Manuscript received August 15, 2002; revised February 1, 2004. This work was supported by the National Natural Science Foundation of China under Grant 60004004 including E5303220, the Excellent Young Teachers Program of MOE, China, and by the Hong Kong Research Grants Council under Grant CUHK4203/04E. Y. Li is with the the Institute of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, China, with the Labo- ratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Saitama 3510198, Japan, and also with the Institute for Infocom Research, Sin- gapore 119613. J. Wang is with the Automation and Computer-Aided Engineering Depart- ment, The Chinese University of Hong Kong, Shatin, Hong Kong. A. Cichocki is with the Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Saitama 3510198, Japan, and also with the De- partment of Electrical Engineering, Warsaw University of Technology, Warsaw PL 00 661, Poland. Digital Object Identifier 10.1109/TCSI.2004.832723 function, etc.) are developed to recover sources sequentially or simultaneously from convolutive mixtures. [14], [15] explore the geometrical structures of the manifold of finite-impulse response (FIR) filters and develop a natural gradient algorithm for blind deconvolution. Independent component analysis (ICA) techniques has also been used in blind deconvolution. For instance, in [17], several blind source separation (BSS) techniques are extended to blind deconvolution. Most of these studies assume that the number of sources is less than or equal to the number of sensors, and the MIMO system is invertible. From the discussions in [18], we also can see that this assump- tion plays important roles. Incidentally, several papers discuss related issues for MIMO static systems with more sources than senors; e.g., [19] and [20]. In [19], by resorting to higher order statistics and mul- tiway array decomposition, it was proved that static MIMO sys- tems with fewer outputs than inputs can be identified; an ef- fective algorithm was proposed to extract three digital sources [e.g., binary phase-shift keying (BPSK) or quadrature phase shift keying (OPSK) sources] from the two observations using the discrete distribution of the sources. Since separability (or extractability) conditions are weaker for a system with digital sources than for analog sources in general [23], the algorithms different from that in [19] should be developed for blind sepa- ration or extraction of analog sources from their ill-conditioned mixtures. In [20], it is discussed that the identification of static systems with sources and two sensors using a joint character- istic function of the random variables (observations); but they did not discuss how to obtain these sources after the systems are identified. In [22], blind identification of multichannel moving average parameter matrices was discussed using higher-order statistics. Five or six assumptions regarding the channels are presented related to channel identification. Similar to [20], there was no discussion about how to estimate the sources. There exist classical methods for blind separation for ill-con- ditioned systems with sources and sensors (here, ). That is, only sources are extracted or separated using general approaches (e.g., in [24]–[26]); the other sources are con- sidered to be noise and are not extracted. In [26], if the system is ill-conditioned (undermodeled case), a local extremum of the criterion only results in a new mixture of several sources. In fact, if an ill-conditioned system is assumed to be an invertible one, and sources are assumed to be noise, then the estimated signals are, theoretically, new mixtures of sources instead of separated sources, of which the signal-to-noise ratios may be very low. Blind source extraction is an effective method for recovering sources from instantaneous mixtures in ill-conditioned cases 1057-7122/04$20.00 © 2004 IEEE