590 IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL 40. NO 3. MARCH 1992 Analysis of Subspace Fitting and ML Techniques for Parameter Estimation from Sensor Array Data Bjom Ottersten, Member, IEEE, Mats Viberg, Member, IEEE, and Thomas Kailath, Fellow, IEEE Abstract-Signal parameter estimation from sensor array to be asymptotically e@cient, i.e., it achieves the Cra- data is a problem that is encountered in many engineering ap- m&-Rao bound (CRB) on the estimation error variance. plications. Under the assumption of Gaussian distributed In this sense, ML has the best possible asymptotic prop- emitter signals, the so-called stochastic maximum likelihood (ML) technique is known to be statistically efficient, i.e., the erties. estimation error covariance attains the Cramer-Rao bound The sensor noise is often regarded a superposition of (CRB) asymptotically. Herein, it is shown that also the multi- several “error sources.” Due to the central limit theorem, dimensional signal subspace method, termed weighted sub- it is therefore natural to model the noise as a Gaussian space fitting (WSF), is asymptotically efficient. This also results random process. For the signal waveforms, two main in a novel, compact matrix expression for the CRB on the es- timation error variance. The asymptotic analysis of the ML and models have appeared in the literature. One approach as- WSF methods is extended to deterministic emitter signals. The sumes that also the signal waveforms are Gaussian. The asymptotic properties of the estimates for this case are shown corresponding ML method and CRB have been formu- to be identical to the Gaussian emitter signal case, i.e., inde- lated and studied in several papers, see, e.g., [I], [4]- pendent of the actual signal waveforms. Conclusions, concern- ing the modeling aspect of the sensor array problem are drawn. [8], and are referred to as the stochastic ML and CRB, respectively. It is easily checked that the stochastic like- lihood function is sufficiently regular, resulting in an I. INTRODUCTION asymptotically efficient ML method. ENSOR array processing has been an active research In many applications, the signal waveforms are not well Sarea for several years. The problems under consider- approximated by Gaussian random processes. It has then ation concern information extraction from measurements been proposed to model the signals as arbitrary determin- using spatially distributed sensors. The measured outputs istic sequences. The corresponding deterministic (or con- are assumed to be noise-corrupted superpositions of nar- ditional) ML method is studied in, for instance, [3], [6], row-band plane waves. Given observations of the sensor [9], [ 101. The deterministic likelihood function does not outputs, the objective is to estimate unknown parameters meet the required regularity conditions, and the determin- associated with the wavefronts. These parameters can in- istic ML estimate does not achieve the corresponding clude bearings and/or elevation angles, signal wave- CRB. This unusual fact was noted in [lo], where a com- forms, center frequencies, etc. Areas such as radar arrays, pact expression for the deterministic CRB was derived. radio and microwave communication, acoustic sensor ar- The weighted subspace fitting (WSF) approach to the rays in underwater applications and the seismic explora- estimation problem 191, [ 1 11 is a multidimensional signal tion industry, are all concerned with estimating parame- subspace method and belongs to the same general class of ters from observations of a sensor array output. subspace fitting methods as deterministic ML, conven- A vast number of methods have been proposed in the tional beamforming, MUSIC [ 11, [2], and ESPRIT [ 121. literature for solving the estimation problem, see for ex- The asymptotic properties of the estimates have been de- ample, [1]-[3]. When formulated in an appropriate statis- rived and WSF has been shown to yield the lowest esti- tical framework, the maximum likelihood (ML) principle mation error variance in the class of subspace fitting provides a systematic way to obtain an estimator. Under methods [13]. Herein, it is shown that the WSF method certain regularity conditions, the ML estimator is known has the same asymptotic properties as the stochastic ML technique. Consequently, WSF is asymptotically efficient Manuscript received September 18, 1989; revised January 25, 1991. This for Gaussian signal waveforms. This in turn results in a work was supported in part by the SDIiIST Program managed by the Office of Naval Research under Contract N00014-85-K-0550 and by the Joint Ser- compact matrix expression for the stochastic CRB, which vices Program at Stanford University (U.S. Army, U.S. Navy, U.S. Air has not been available previously. Force) under contract DAAL03-88-C-0011. Although the analysis of the deterministic ML method B. Ottersten is with the Department of Telecommunication Theory, Royal Institute of Technology, S-10044 Stockholm, Sweden. [9], [14] does not assume Gaussian signal waveforms, M. Viberg is with the Department of Electrical Engineering. Linkoping most other results do. For instance, the asymptotic prop- University, S-581 83 Linkoping, Sweden. erties of the stochastic ML method are known only for the T. Kailath is with the Information Systems Laboratory, Stanford Uni- versity, Stanford, CA 94305. Gaussian case. Also, the asymptotic analysis of eigenvec- IEEE Log Number 9105667. tor-based methods as reported in, for example, [ 131, [ 151, 3053-S87X/92$03.00 0 1992 lEEE 59 I OTTERSTEN PI al.: ANALYSIS OF SUBSPACE FITTING AND ML TECHNIQUES [ 161, relies on the assumption of Gaussian signal wave- where O0 is a parameter vector corresponding to the true forms. This is because the asymptotic analysis of the sam- signal parameters. The vector a(t9,) = [a,(O,) . am(8,)] ple covariance eigenvectors, e.g., [17],is based on in- contains the sensor responses to a unit wavefront having dependent, zero-mean, Gaussian observations. Herein, parameters e,. The collection of these vectors over the pa- the analysis is extended to deterministic signal wave- rameter space of interest forms. The asymptotic distribution of the signal eigen- vectors is derived for a general “deterministic signal in Q. = {ace,) I e, E e> (3) additive Gaussian noise,” model, and applied to the sen- is called the array manifold. It is assumed that the array sor array estimation problem. It is found that the stochas- manifold vectors hpve bounded third derivatives with re- tic ML technique and all multidimensional subspace fit- spect to the parameters, and that for any collection of m ting methods mentioned above preserve their asymptotic distinct e,, the matrix A(8) has full rank. In general, each properties regardless of the actual signal waveforms. wavefront is parameterized by several signal parameters, During the review process, it has come to the authors’ such as bearing and elevation angle, polarization, range, attention that many of the results presented herein parallel center frequency, etc. The results presented here apply to those of [18].The compact expression for the stochastic a general parameterization. However, to avoid unneces- CRB appears also in [18],and the asymptotic properties sary notational complexity, we restrict the discussion to of the stochastic ML method are derived. Furthermore, a the one-parameter problem. Thus, 0, is a real scalar, re- subspace-based estimation procedure termed method of ferred to as the direction of arrival (DOA), and 8, = [e,, direction estimation (MODE) is analyzed. The latter tech- ..., e,] is a real d-dimensional vector of unknown pa- nique is asymptotically identical to the WSF method when rameters. Collect N independent observations, x( l), the emitter covariance matrix has full rank. However, for ... ,x(N), of the array output. Given these observations, coherent sources only WSF is efficient. An important dif- the main interest for our purposes is in estimating the un- ference between the analysis presented in [18] and in the known DOA’s. However, the parameter space usually in- present paper is the assumption of full rank emitter co- cludes other unknowns as well, see Section 111-A. variance matrix made in [18].Our analysis is valid for The vector of signal waveforms s(t) is assumed to be a arbitrary signal correlation, including full coherence. stationary, temporally white, I zero-mean complex Gauss- Also, the papers differ in the methods of derivation. As a ian random process with second moments consequence we present results not found in [ 181, for ex- ample, the asymptotic distribution of the signal eigenvec- E[s(t)s*(l)] = SI/ (4) tors is derived under quite general conditions, the CRB E[s(t)sT(Z)]= 0 (5) on the noise variance is given, and a result on separable likelihood functions is provided in Appendix A. where 6,, is the Kronecker delta. In the analysis of Section V, this assumption is replaced by a much weaker require- 11. ASSUMPTIONSAND NOTATION ment on ~(t). Assume that d narrow-band plane waves impinge on an The additive noise n(t)is modeled as a stationary, tem- array of m sensors, where m > d. The measured array porally white, zero-mean complex Gaussian random pro- output is a weighted superposition of the wavefronts, cor- cess. For simplicity, we will also require n(t) to be spa- rupted by additive noise. The narrow-band signal as- tially white, i.e., E[n(t)n*(t)] = a2Z. The assumption of sumption allows us to model the time delays of the wave- spatially white noise is no restriction if the noise covari- fronts at different sensors as phase shifts. The measured ance is known (up to an unknown scalar a*).The noise is sensor outputs are also affected by the individual sensor assumed to be uncorrelated with the signal waveforms. gain and phase responses, modeled as a complex weight- The covariance matrix of the array output has the follow- ing of the wavefronts. The output of the ith sensor is rep- ing familiar structure: resented by R = E[x(t)x*(t)]= A(8,)SA*(O0) + a2Z.