SUPER-SYMMETRIC DECOMPOSITION of the FOURTH-ORDER CUMULANT TENSOR. BLIND IDENTIFICATION of MORE SOURCES THAN SENSORS

New ideas for Higher-Order Array Processing are introduced. The paper focuses on fourth-order cumulant statistics. They are expressed in an index-free formalism (that is believed to be of general interest) allowing the exploitation of all their symmetry properties. We show that, when dealing with 4-index quantities, symmetries are related to rank properties. The rich symmetry structure yields a whole class of new identification algorithms. Main features are : • Use of fourth-order cumulant statistics only : yielding asymptotically unbiased estimates in presence of Gaussian noise regardless of its spatial structure. • Blind Identification : directional vectors are estimated without a priori knowledge of the array manifold. • More source than sensors can be identified. The paper is mainly theoretical but, as a first application, a new algebraic technique for blind identification is included and a few others are briefly indicated. INTRODUCTION This paper deals with narrow-band array processing using higer-order statistics. We consider the problem of identifying directional vectors without knowing the array manifold (blind identification) by resorting only to 4th-order cumulant statistics. Various cumulant based methods for blind identification have recently been reported [1-3]. They all exploit both secondand fourth-order information. In contrast, this paper considers the Fourth-Order Only Blind Identification (FOOBI) problem. Restriction to 4th-order offers two advantages. Firstly, while 2nd-order cumulants are affected by additive Gaussian noise, 4th-order cumulants are not, so that no assumption about noise spatial structure is necessary. Secondly, more sources than sensors can be identified. Reported techniques [1-3] assume at most as many sources as sensors since they rely (explicitely or not) on a prior directional vector orthogonalization based on 2nd-order information. In contrast, FOOBI approaches can identify more sources than sensors by turning the orthogonality constraint into a less stringent symmetry constraint. It is important to take into account the symmetric structure of cumulants. In [3], two hermitian-like symmetries of 4thorder statistics were used. By taking into account an additional symmetry, the whole group of symmetries satisfied by 4thorder statistics of complex variables can be generated. This is why it is sufficient to consider a third symmetry, because it is, in some sense, the "last" symmetry. The paper is organized as follows. The 4th-order identification equation is established for independent sources in Gaussian noise. An index-free formalism is then introduced to handle 4th-order quantities and deal with all their symmetries. Symmetries are then shown to be related to rank properties. We give an exemple of a FOOBI algorithm and allude to other possible algorithms. FOURTH-ORDER ONLY BLIND IDENTIFICATION The standard linear model for a N-sensor array listening to S narrow-band point sources is most commonly written as : (1) X(t) = A α(t) + N (t) where X (t) and N (t) are N× 1 complex vectors representing measurements and additive noise. The source signals are stacked up in the S× 1 complex vector α(t) and the S columns of A are the N× 1 directional vectors. In the following, we use a slightly different notation for eq. (1) : we drop explicit dependence on t, we denote by αp the p-th element of vector α(t) and by P the p-th column of matrix A (hence P appears as a "self-indexed" symbol) so that (1) is rewritten as :