An Analytical Solution for 2nd Order Statistics Based Blind Mimo Channel Identification

We address the problem of blind identification of multiple-input multiple-output (MIMO) channels, based only on second-order statistics. We complement the theoretical study of a previously introduced closed-form solution derived within a correlative coding framework. This consists in pre-filtering the stationary random data processes, in order to attain diversity in the respective spectral power densities. The induced structure permits the receiver to fully recover the unknown MIMO channel from the correlation matrices of the observed channel outputs. 1. PROBLEM STATEMENT Motivation. Blind channel identification finds direct application in the growing field of mobile wireless communications. In these scenarios, the data links must oftenly be re-established after sudden breakdowns due to severe signal fading (incoherent combination of the multiple signal arrivals). Blind methods permit to restart the communication link without requiring the sender to transmit reference (training) signals, hence attaining a higher spectral efficiency. SIMO channels (single-user). In the context of blind equalization of single-input multiple-output (SIMO) channels, the work by Tong et al. [1] has attracted much attention. They derived an analytical solution (i.e., non-iterative) for the unknown channel coefficients, by exploiting only the second-order statistics (SOS) of the system outputs. See also the subspace method [2] for another blind SOS-based channel identification technique. Notice that SOS-based approaches are almost mandatory in fast changing environments, e.g., multipath wireless channels, since the relatively short latency time of the propagation space-time channel may jeopardize the accuracy of higher-order statistics (HOS) (i.e., cumulant-based) approaches, which tipically require larger data sets than SOS approaches for the same channel precision. MIMO channels (multiple users). In this paper, we address the problem of blind channel identification in the context of multiple users. The problem of blind identification of multiple-input multiple-output (MIMO) channels occurs frequently in many fields of interest, e.g., in space division multiple access (SDMA) networks for wireless communication [3, 4, 5, 6]. In these architectures, the spatial dimension is efficiently exploited for user parallelism. Several sources within the same cell are allowed to share This work was partially funded by FEDER/FCT Programmatic Funding/101. the same time/frequency slot, in order to boost the overall network capacity. At the base station, signal separation is still possible thanks to the so-called spatial diversity principle: each mobile source induces a distinct spatial signature at the antenna array receiver, which makes user discrimination a theoretically feasible problem. When the uplink space-time signatures activated by the sources are unknown at the receiver, blind methods play a key role in filling in this missing information for the signal separation algorithms. Partial blind MIMO channel identification. Generalization of the single-user approaches in [1, 2] for the multiple user situation has been studied in [7, 8]. Therein, it is shown that they cannot completely solve for the unknown MIMO filtering matrix. At best, those methods can only simplify the convolutive mixture problem up to several static mixtures of the input signals. Each static mixture contains the users subject to the same channel memory order (degree of intersymbol interference). Thus, complete signal separation is only achieved for the special case when each user experiences a distinct delay spread (a condition also known as full memory diversity). The whitening approaches in [9, 10] are also SOS techniques which permit to reduce convolutive mixtures into static ones, under relaxed channel assumptions: infinite-impulse response (IIR) channels are allowed, as well as minimum phase common zeros among the subchannels; moreover, they can circumvent the usual column-reduced system condition. Residual static mixtures. To resolve the residual instantaneous mixtures left by the techniques in [7, 8, 9, 10], it is usually assumed further that the sources driving the space-time channel belong to one of these three classes: (i) non-Gaussian, (ii) constant-modulus or (iii) finite-alphabet sources. Then, popular blind source separation (BSS) algorithms can be applied, e.g., (i) the joint diagonalization procedure in [11], (ii) the analytical constant modulus algorithm (ACMA) [12] or (iii) the iterative tehcniques in [3, 4, 5, 6], respectively. Complete blind MIMO channel identification. In contrast with the methods in [7, 8, 9, 10], the recently introduced transmitter induced conjugate cyclostationarity (TICC) approach by Chevreuil and Loubaton [13] provides a complete solution (i.e., no extra BSS algorithm required) for the blind MIMO channel identification problem. The basic idea consists in assigning a distinct conjugate cyclic frequency (baseband rotation) per user, which permits the receiver to easily decouple the users, hence reducing the original MIMO problem into several SIMO ones. Those can be solved by the single-user methods in [1, 2]. The main drawback of the TICC approach is lack of robustness with respect to carrier frequency misadjustements. Previous work. In [14, 15], we introduced the closed-form correlative coding (CFC2) approach for blind MIMO channel identification. As the TICC methodology, it provides complete channel identification based only on second-order statistics. However, it is less vulnerable to local oscillator synchronization, since it relies on a distinct concept for signal separation [16]. Namely, we exploit spectral diversity in the SOS of the emitted data sequences. This is obtained by correlative filters which judiciously color the statistics of the white information sequences prior to transmission. This inserted structure permits the receiver to blindly recover the MIMO channel from the SOS of the channel outputs. Contribution. Here, we extend the theoretical study of [15] in two directions. (i) We prove that the MIMO channel orders (assumed known in [15]) may in fact also be obtained in the integrated framework of correlative coding. In equivalent terms, these unknown system parameters are also implicitly contained in the SOS of the channel outputs. A straightforward methodology to estimate them is sketched. (ii) We prove that the required SOS conditions on the MIMO input sources stated in [15] can be attained by minimum phase correlative filters with arbitrary nonzero degree. The minimum-phase property is attractive for (suboptimal) direct inversion of the correlative filters after channel estimation, thus avoiding the computationally intensive Viterbi detection algorithm. Paper organization. Section 2 briefly reviews the data model and the main assumptions. In section 3, we prove that the users’ delay spreads are uniquely identifiable from the correlation matrices of the system outputs. Section 4 is devoted to the correlative filters. It is shown that minimum phase FIR filters of arbitrary non-zero degree can fulfill the previously stated assumptions. Notation. All signals are in discrete-time. Matrices (uppercase) and vectors are in boldface type. C m and C denote the set of n m matrices and the set of n-dimensional column vectors with complex entries. The notations ( )T , ( ) and ( ) stand for the transpose, Moore-Penrose pseudo-inverse, and the Hermitean operator, respectively. The symbol In stands for the n n identity, and jjAjj is the Frobenius norm of A. When the dimensions are clear from the context, the subscripts are dropped. Diagonal concatenation of matrices is given by diag (A1;A2; ;Am). For A 2 C n n , (A) = f 1; 2; ; n g denotes its eigenvalues (including multiplicities). The set of polynomials with coefficients in C and indeterminate z 1 is denoted by C [z]. The polynomial f(z) = P d k=0 f(k)z k 2 C [z] is said to have degree d, written deg f(z) = d, if f(d) 6= 0. The subsets of all polynomials with degree d and degree at most d are denoted by Cn(d) [z] and C n d [z], respectively. Similar definitions hold for C n [z] and C m [z], the set of n 1 polynomial vectors and n m polynomial matrices, respectively. We shall identify Cnd [z] and C n(d+1) by associating to f(z) = P d k=0 f(k)z k the vector f = f(0) ;f(1) ; : : : ;f(d) T . For f(z) = P d