ML ESTIMATION AND CRB FOR NARROWBAND AR SIGNALS ON A SENSOR ARRAY Langford B White School of Electrical and Electronic Engineering The University of Adelaide, Australia Lang.White@adelaide.edu.au Peter J Sherman Departments of Aero. Eng. and Statistics Iowa State University, Ames IA, USA shermanp@iastate.edu ABSTRACT This paper considers the exploitation of temporal correlation in inci- dent sources in a narrowband array processing scenario. The MLE and CRB are derived for parameter estimation of spatially uncorre- lated first order Gaussian autoregressive source signals with additive Gaussian spatially and temporally uncorrelated sensor noise. These are compared to the MLE and CRB for the usual uncorrelated (WN) sources model. The paper deals with the case where the number of data snapshots is small. Numerical simulations show that (i) there is no significant performance gain in the correlated signal case, and sig- nificantly, (ii) the WN MLE performance does degrade in the pres- ence of source correlation, which appears to be in contrast to some recently published work. Index Terms— array signal processing, direction-of-arrival es- timation, autoregressive models, maximum likelihood 1. INTRODUCTION Conventional likelihood based narrowband sensor array signal es- timation generally assumes that the incident signals are either (i) deterministic and unknown, or (ii) realisations of zero-mean tem- porally uncorrelated wide-sense stationary Gaussian random pro- cesses with known (spatial) covariance. Maximum Likelihood es- timation (MLE) for the signals’ angles-of-arrival (AoA) for these models is a conventional approach, and its performance has been analysed in detail (see e.g. [2]). In particular, there is a well-known “threshold” phenomenon whereby the performance of the MLE de- viates markedly from the associated Cramer-Rao bound (CRB) be- low a certain Signal-to-Noise ratio and/or number of independent data “snapshots”. Although the MLE offers superior threshold per- formance, its computational requirements have led to many other approaches, such as signal subspace (e.g. ESPRIT [3]) or noise sub- space (e.g. MUSIC [4]) based methods. These methods offer similar performance to MLE for large number of data snapshots, but their thresholding behaviour is significantly worse. Thus in investigating the performance of various AoA estimators, the criteria used are the CRB associated with the particular signal model (the best attainable “above-threshold” performance), and the SNR, or number of snap- shots when thresholding occurs. 1.1. Background and Motivation In practical scenarios, the signals incident on the array may not be well-modelled at baseband as independent (white noise) samples. Typically, the data collection system selects a passband filter and ap- propriate sample rate to yield Nyquist sampling for the largest band- width signal incident on the array at the specified carrier frequency. Other signals, which may possess smaller bandwidths, will thus gen- erally yield correlated samples at baseband. It is therefore natural to ask whether this correlation can be exploited in the design of an esti- mator for all incident signals’ AoAs. Studies such as [7] have shown that the MLE designed for independent Gaussian data samples is ro- bust to the presence of temporal correlation in the signals’ samples, however there are no detailed studies concerning the performance of the MLE designed specifically for correlated signals. Some results are available concerning the CRB for correlated signals, when the spatio-temporal source correlation matrix is known [5], and these re- sults show that the CRB for correlated sources is lower than that for uncorrelated sources. However for large number of data snapshots, the difference between these CRBs becomes smaller. The threshold- ing behaviour of the correlated sources MLE has not been studied. We are specifically interested in those processing scenarios where computational complexity is not a limiting factor and that compu- tationally intensive techniques such as the correlated sources MLE can be justified if better performance can be obtained. It should be pointed out at this stage, that a signal state-space based approach us- ing the Expectation-Maximisation (EM) algorithm for ML AoA esti- mation of general linear Gauss-Markov sources with known models, has been proposed in [8]. This algorithm iteratively applies a fixed- interval Kalman smoother to perform estimation of the signals, and a likelihood based method using the resulting signal estimates to find the AoAs. 1.2. Existing Work Since we are interested in potential exploitation of source correla- tion, we will focus on this issue in this paper. Source correlation was addressed specifically as its main focus in [6] where it was concluded that covariance based estimators designed for uncorrelated sources (including the MLE) are robust to temporal source correlation when the additive sensor noise is itself temporally uncorrelated (the case considered in this paper). Methods designed to use temporal correla- tion to advantage are described in [10] and [5], however these papers do not consider the performance attained by an MLE specifically de- signed for the temporally correlated sources case, but rather an ap- proximation method which intrinsically estimates temporal source correlation lags. However, these papers do present the CRB for the temporally correlated source case, and show that it lower than the corresponding CRB for the temporally uncorrelated (white) source case. In [7], simulation results which include numerical studies of the behaviour of the white sources based MLE when temporal cor- relation is present, show that this MLE is indeed robust to the pres- ence of source correlation. The paper claims that the performance so attained is similar to that obtained “for techniques specifically de- signed for the circumstance”. In this respect, they refer to [11] which does not address the MLE for the temporally correlated source case,