On Maximum-Likelihood Localization of Coherent Signals - IEEE Xplore

IEEE: TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO IO, OCTOBER 1996

2475

On Maximum-Likelihood Localization of Coherent Signals Jacob Sheinvald, Mati Wax, Fellow, IEEE, and Anthony J. Weiss, Senior Member, IEEE

A hskact-We consider the problem of localizing multiple signal sources in the special case where all the signals are known a priori to be coherent. A maximum-likelihood estimator (MLE) is constructed for this special case, and its asymptotical performance is analyzed via the Cramer-Rao hound (CRB). It is proved that the CRH for this case is identical to the CRB for the case that no prior knowledge on coherency is exploited, thus establishing a quite surprising result that, asymptotically, the localization errors are not reduced by exploiting this prior knowledge. Also, we prove that in the coherent case the deterministic signals model and the stochastic signals model yield MLE’s that are asymptotically identical. Simulation results confirming these theoretical results are included.

I. INTRODUCTION

T

1 WE problem of localizing multiple narrowband sources by

a passive sensor array is common to diverse areas such as radar, communication, sonar, seismology, and radioastronomy. This paper is focused on estimating the directions-of-arrival (DOA’s) of the sources in the special case where the signals are fully correlated. This case, referred to as the coherent signals case, appears in specular multipath propagation and is, therefore, of great practical importance. As is well known, some asymptotically efficient generalpuq ose localization techniques, such as the maximumlikelihood estimator (MLE) in [ 11, designed for arbitrarily correlated signals, are capable of coping also with coherent signals. Naturally, these estimators take no advantage of the prior knowledge about the coherency of the signals. Exploitation of such a prior knowledge usually reduces the estimation errors. For example, it is shown in [4] that exploitation of a prior knowledge that the signals are unccirrelated indeed reduces the estimation errors. An MLE specialized for the coherent case and based on a deterministic model for the signals was derived in [2]. In this model, the signal samples are considered as unknown parameters, and as a result the overall number of parameter; grows linearly with the number of observations. In this paper, we develop an MLE specially tailored for the coherent case that, unlike [2], employs a stochastic model for Manuscript received November 21, 1994; revised April 15, 1996. The assocate editor coordinating the review of this paper and approving it for publkation was Prof. Fu Li. J. Sheinvald is with RAFAEL 83, P.O.B. 2250, Haifa, Israel 31021 and with he Electrical Engineering Department, Tel-Aviv University, Tel-Aviv, Israel 69978 (e-mail: [email protected]). M. Wax is with RAFAEL 83, Haifa, Israel 31021. A. J. Weiss is with the Electrical Engineering Department, Tel-Aviv University, Tel-Aviv, Israel 69978. Publisher Item Identifier S 10.53-587X(96)07119-X.

the signals, with the number of parameters to be estimated being fixed. Consequently, the resulting MLE is guaranteed to be asymptotically efficient. We then investigate its performance in comparison with the general-purpose stochastic signals MLE [l], and with the deterministic coherent signals MLE [2]. Surprisingly enough, it turns out that, although the number of parameters in the latter estimator grows with the number of observations which usually leads to asymptotic inefficiency [6], the performance of all these estimators is identical asymptotically (i.e., for a large number of samples), thus establishing the following facts: 1) exploitation of the prior knowledge on coherency is not advantageous (asymptntically). 2) the deterministic coherent signal MLE asymptotically attains the the Cramer-Rao bound (CRB) corresponding to the stochastic signal model. The paper is organized as follows. We formulate the problem in Section 11. In Section 111, we present our specially tailored MLE, and in Section IV, we compare its corresponding CRB with the CRB corresponding to the general-purpose MLE. In Section V, we compare it with the deterministic coherent signal MLE [ 2 ] .In Section VI, we present simulation results that demonstrate the estimator performance, and finally, in Section VII, we present some concluding remarks.

11. PROBLEMFORMULATION Consider q wavefronts impinging on an array consisting of p sensors from sources located at Q1 . . . ! 8,. For simplicity, assume that the sensors and the sources are all located on the same plane, and that the sources are in the far field of the array, so that the wavefronts are plane, with { H i } representing their DOA’s. In addition, assume that the sources emit narrowband signals (i.e., each source bandwith is much smaller than the reciprocal of the time delay across the array), all centered around a common frequency. Let s k ( t ) denote the complex envelope of the kth source signal. Let xi(t) denote the complex envelope of the signal received at the ith sensor, and ~ ( t=) ( x l ( t ) x, ~ ( t .).,. ~ ~ ( denote t ) ) a ~vector formed from these complex envelopes, with T denoting transposition. In the presence of additive noise, this received vector can be expressed as ~

4

z(t)

U(Q,)Sk(t)

+ n(t)

(1)

k=l

where u(8) is the steering vector of the array expressing its complex response to a planar wavefront arriving from direction

1053-587W96$05.00 0 1996 IEEE

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. IO, OCTOBER 1996

2476

8,and n(t) is a vector formed from the complex envelopes of the sensor noises. This expression can be written more compactly as

+

~ ( t=)A(B)s(t) n ( t )

6

def

(0, . .

A(8 = [ u ( H ~ ) ,

*

p ( X 14) =

3

(T-”

IR(4) 1

CXP[ -tr(R-’

(d)k)])”

(5)

where / . I denotes determinant, tr( ) denotes trace, ( ) H denotes complex-conjugate transposition, k=ilefXXH/mis the sample-covariance matrix, R(q5) is the array covariance matrix, and represents a vector formed from the unknown parameters:

,~ ( e , ) ]

andl

We shall assume that the isteering vectors {.(e)} are known for all 0 E 8, where 8 denotes the field-of-view. Now, assuming that the signals are all coherent, so that { s z ( t )are } all some complex multiples of a common signal, say s o ( t ) , then s ( f ) can be expressed as s(t) = Dso(t)

and A2 it follows that the probability From assumptions density function (pdf) for the data is given by

(2)

where dcf

111. THE MLE FOR COHERENT SIGNALS

Using (2)-(4), the array covariance is given by

R(d,) = A ( 8 ) / l / j H A H (+n21 8 ) = b(0, /3)bH(8,/”) +oLI ( 6 ) where

b( 0. /I)