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U6.15 Maximum Likelihood Direction Finding o StochasticSources: A Separable Solution f Amin G. Jaffer Hughes Aircraft Company Fullerton, CA 92634 approaches, e.g.[4], have been limited to b r u t e - f o r c e maximizationprocedures over all unknown parameters (including the full unknown sources covariance matrix). A novel method is developed in the present paper for maximum likelihood direction finding using a stochastic (complex Gaussian) signal model. It is shown that the problem reduces to a separable one, with the ML estimates of the angle parameters being obtained by maximizing a function over only the angle parameters. The ML estimate of the unknown sources covariance matrix is obtained by an explicit closed-form formula. The method results in a substantial reduction of the parameter set required for numerical optimization compared to procedures based on direct maximization over all parameters; in fact, a reduction from (M2+ M) real parameters to M real parameters for M sources is effected. The method developed here also reveals a close relationship between the ML techniques based on deterministic and stochastic signal models. Some computer simulation results on the ML technique developed here are also presented and compared with the ML technique based on a deterministic model 131, 151 as well as an eigenvector decomposition method[I]. 2. MODEL DEFINITION AND PROBLEM STATEMENT

ABSTRACT

A novel method is presented for maxir,,m likelihood direction finding of stochastic sources which may be correlated. It is shown that the maximum likelihood estimates of the angle parameters and unknown source covariance matrix may be obtained in a separable form, i.e. the angle parameters are obtained by maximizing a function of only the angle parameters. The source covariance matrix estimate is then obtained by an explicit formula. This results in a significant reduction of the dimensionality of the optimization problem to be solved compared to previous approaches based on direct maximization over all parameters. Computer simulation results are presented to demonstrate the performance of the proposed method.

1. INTRODUCTION

Various high resolution techniques for estimating the directions-of-arrival of multiple planewave signals incident on an array of sensors have been reported in the literature, including eigenstructure based methods and maximum likelihood (ML) techniques[I]-[3]. These techniques are of wide applicability since they tend to result in fractional beamwidth resolution of signals and enhanced adaptive array performance in suppressing directional interferences. Most of the reported ML techniques e.g.[2],[3] have been based on a signal model where the complex amplitudes of the signal sources are regarded as unknown but nonrandom parameters to be estimated, together with the angle parameters, from the array data. It is known that this results in a separable least-squares problem with attendant reduction of the dimensionality of the opt

imization problem needed to solve for obtaining the angle estimates[2],[3]. However, in many situations the appropriate signal model corresponds to that of multiple random sources which may be statistically correlated due to, for example, multipath conditions or being intentionally introduced (smart jamming). Although the maximum likelihood direction finding formulation using a stochastic signal model has been presented before, the previous

Consider M stochastic narrowband signal sources located at angles 01, 0 2, ..., O M whose emitted signals a r e simultaneously incident on an array of N sensors, N M. The received (N x 1) array data vector x k for the k*"snapshot" can be modeled as X k= D (k%Bk+Yk

, k= 1, 2,..., K

(1 1

The M-dimensional amplitude vector a k is assumed to be a zero-mean complex Gaussian random vector with (unknown) source covariance matrix P= E[ia k a kt] where E and t denote the expectation and conjugate transpose operations a I( is assumed statistically independent from respectively. snapshot to snapshot. y.k is additive zero-mean complex Gaussian noise vector (N x 1) with covariance matrix Rv= 02 IN where I N is the identity matrix of order N. yk is assumed independent snapshot to snapshot as well as independent of the is an N x M matrix of sources complex amplitude vector a k. D direction vectors of the M sources. The ith column of D (a), denoted by di (e,), represents the vector response of the sensor array to the plane-wave signal received from the i* signal source and has the general form

(a)

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I'

2893 CH2561-9/88/0000-2893$1.00 0 1988 IEEE

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