Blind MIMO channel identification using cumulant tensor decomposition

Abstract

In this paper, we exploit the symmetry properties of fourthorder cumulants to develop a new blind identification algorithm for multiple-input multiple-output (MIMO) instantaneous channels. The proposed algorithm utilizes the Parallel Factor (Parafac) decomposition of the 4th-order cumulant tensor by solving a single-step (SS) least squares (LS) problem. This approach is shown to hold for channels with more sources than sensors. A simplified approach using a reduced-order tensor is also discussed. Computer simulations are provided to illustrate the performance of the proposed identification algorithms. I. THE BLIND IDENTIFICATION PROBLEMCumulants of order higher than two can be viewed as tensors with a highly symmetrical structure. For about two decades, exploiting the cumulant symmetries with a tensor formalism has been an important research topic. The Parallel Factor (Parafac) decomposition of a Pth-order tensor with rank Q consists in decomposing it into a sum of Q rank-one tensors [1]. The keypoint in the use of the Parafac decomposition is its uniqueness property, which can be assured under simple conditions, stated by the Kruskal Theorem [2]. Parafac does not induce neither rotational ambiguities nor orthogonality constraints, as it is the case with matrix singular value decomposition. For that reason, the use of cumulant tensor factorization allows for avoiding the pre-whitening step, a time-consuming operation responsible for increased estimation errors [3].The alternating least squares (ALS) algorithm consists in fitting a Pth-order Parafac model by iteratively minimizing, in an alternate way, P least squares (LS) cost functions. Our focus in this paper is to exploit the redundancies of the parallel factors of the 4th-order cumulant tensor in the minimization problem in order to develop a new blind channel identification (BCI) algorithm. We consider the problem of blind multiple-input multiple-output (MIMO) channel (mixture) identification in the context of a multiuser system characterized by instantaneous complex-valued channels. We introduce new algorithms based on the Parafac decomposition of cumulant tensors, as an extension of the algorithms proposed in [4] for the case of single-input single-output FIR channels. Our main contribution consists in exploiting the redundancies in the Parafac components to estimate the channel matrix by solving a single LS minimization problem. This approach greatly simplifies the estimation problem and allows us to introduce a new blind MIMO channel estimation algorithm based on a single-step (SS) LS optimization procedure.During the two last decades, several BCI methods making use of the redundancies in the 4th-order cumulants have been proposed [5]. Such approaches include, for instance, the popular joint approximate diagonalization of eigenmatrices (JADE) algorithm [6], which is based on second and fourth-order statistics. It is now well-known that in the case of linear mixtures the BCI problem is closely related to the (simultaneous) diagonalization of symmetric cum...

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