DOA Estimation Using Coherent Signal - Subspace Method Based On Fourth - Order Cumulants

In this paper the advantage provided by the fourth order cumulant domain is exploited, that is the suppression of additive Gaussian noise sources. Thus we examine the effect of combining the transformation matrices of the Coherent Signal Subspace Method with spatial fourth order cumulant matrices for the estimation of the direction of arrival of non Gaussian wideband signals in spatially correlated, Gaussian noise of unknown covariance. Instead of spatial covariance matrices which are used by the CSS Method, the transformation matices are used here in order to align the Signal Subspaces of the fourth order cumulant matrices at the temporal frequencies in which each snapshot of the array outputs is decomposed, with the Signal Subspace at center frequency. It is shown with simulations that the new method can suppress noise and resolve signals in cases where the spatial covariance based methods do not but the estimates present higher bias and strong fluctuation. INTRODUCTION The problem of estimating the angles of arrival of non Gaussian wideband signals in spatially correlated Gaussian noises of unknown correlation matrix, using an array of !.ensorS. is addressed. This is a situation that is often encountered in practice where a deviation of the signals as stochastic processes from being Gaussian is observed and the noise correlation matrix is not spatially white as is required by the signal subspace based methods in order to give reasonable estimates. The Coherent Signal Subspace (CSS) method [ 13 has been developed for the estimation of diredon of arrival of wideband signals received by an anay of sensors. Here, the effect of combining the Transformation matrices of the CSS methods with fourth order cumulant matrices is examined. This is motivated by the known property of Gaussian processes that all cumulant spectra of order greater than two are identical to ;en, [2]. So, by using the fourth order cumulants of the array data, any additive Gaussian noises corrupting non Gaussian signals will (in principle) be suppressed. Array processing methods for narrowband signals in spatially correlated Gaussian noise have been introduced in [3]. There, the processing is performed in the time domain ithile, all the processing here is performed in the 'rcquency domain. In the following, the CSS method is reviewed briefly and the new method is explained in detail. Its performance is assessed with simulations and a short discussion with comments and observations is provided. PROBLEM FORMULATION A wavefield generated by M wideband sources in the presence of noise is sampled temporally and spatially by a passive array of N (N>M) hydrophones with a known arbitrary geometry. The source signals are characterized as zero mean, non Gaussian stationary stochastic processes over the observation interval To, bandlimited to a common frequency band with bandwidth B which may be of the same order of magnitude as the center frequency fo. The source signal vector s(t) is defined as where T denotes transpose of a vector or a matrix. Note that throughout this paper, small letters in bold print will indicate vectors while capital letters in bold print will indicate matrices. The signal xi(t), received at the i-th hydrophone, can be expressed as where aim is the amplitude response of the i-th hydrophone to the m-th source, Ti, is the propagation time difference between the i-th hydrophone and the reference hydrophone and ni(t) is the additive noise at the i-th hydrophone. The observation interval To is divided into K nonoverlapping snapshot intervals T, and for each of these intervals the array output signals Xi(t) are decomposed into J frequency components xi(fj), j=1, ... J. via Fast Fourier Transform 0. So, essentially. we sample K times each frequency component of the output signals, thus obtaining the data set xk(fj), j=1, ... J; k=l, ..., K. From Eqn. (2), xi(fj) will be given by