APPLICATION OF SUPERRESOLUTION TECHNIQUES

5.1

INTRODUCTION

Superresolution schemes for enhanced spatial estimation are considered whenever conventional beamforming fails to deliver the required bearing accuracy o f a sensor array under a constrained aperture size. These are spectral estimation algorithms which adaptively use information provided by the received signals, usually through second-order statistics of their temporal Fourier transforms, with the aim of surpassing the Rayleigh resolution limit of Fourier-based direction-finding methods. According to the classical Rayleigh resolution criterion, two uncorrelated equi-power far-field sources are just resolvable when their angular separation equals half the null- to-null delay-and-sum beamwidth of the receiving array.^ If the angular separation of the two sources is decreased below this limit, the two (mainlobe) pattern peaks in the spectral response o f a conventional beamformer will merge into a single maximum representing the two sources. In practice the beamforming receiver may also be adversely affected by the presence of additional noise which can mask the received signals altogether, let alone allow their resolution. High-resolution estimation o f directions of arrival (DOA) from noisy data has therefore attracted intensive research interest for over 30 years and a variety of different spatial spectral estimation algorithms have been developed or adapted from the closely related context of time-ffequency signal analysis.

In the simplest scenario often investigated, spatial and temporal samples of the received signals are assumed to originate from a number of narrowband sources all having the same centre frequency. A somewhat more realistic model consists of narrowband sources centred at known but different frequencies. In both cases, as well as in the more general case of wideband sources o f known spectral densities, the problem is effectively that of a one-dimensional spectral estimation in the spatial domain.2

^ For a uniformly illuminated line source or linear array of length DM wavelengths, this angular separation is approximately equal to A/D radians.

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The spatial distribution of the array sensors may or may not be restricted by the specific algorithm used while the sensors themselves may take a variety of forms such as antenna elements or acoustic hydrophones in radar and sonar systems respectively. In any event, the digitised array outputs are assumed to be available, together with a precise knowledge of the array geometry (in some cases, the arrangement of the array sensors need only be partially known, subject to some structural constraint) and (for most estimators) a calibrated record o f the array element patterns over the relevant frequency band and angular sector.

Two different types o f data models are generally used to map the observation space of received signals onto the parameter space of estimated directions of arrivals [Sto 90b]. Under the stochastic (or unconditional) model the signals emanating from the distant sources are regarded as random processes, and are ordinarily assumed to be stationary zero-mean jointly complex Gaussian, uncorrelated with the additive noises. The deterministic (conditional) model, on the other hand, assumes the received signals to be non-random. In both cases the additive noises received or generated at the array channels are considered to be random, and commonly taken as stationary zero-mean uncorrelated (“spatially white”) complex Gaussian processes.

In section 5.2 we review and compare several superresolution schemes pertinent to arbitrarily shaped arrays, as well as methods apposite to arrays of some constrained geometry. The algorithms considered may involve a scalar (‘one-dimensional’) search in which the DOA parameters are sought one at a time, a vector parameter search for the simultaneous (‘multi-dimensional’) estimation of all directions of arrivals, or they may employ no parameter search at all. Section 5.3 then considers the application of such techniques to circular arrays and in particular to their phase-mode outputs. It is shown that this particular kind of ‘beam-space’ formulation allows superresolution algorithms which are specific to linear arrays to be also applicable to circular arrays. The (pre-processing) transformation from ‘element space’ to ‘mode space’ also allows the decorrelation of coherent sources or multipathed signals received by a circular array, through spatial and omni-directional frequency smoothing; these two techniques, which have hitherto been restricted to linear arrays, are discussed in the context of circular arrays in section 5.4. Section 5.5 concludes this chapter with a simulated study of DOA estimation procedures under various multiple-source scenarios, in which the performances of a representative superresolution algorithm using linear and circular arrays are examined and compared.

Strictly speaking a spatial spectra estimator is one-dimensional only when the estimated (DOA) parameters are evaluated one by one.

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5.2

GENERAL OVERVIEW

52.1 CRAMÉR‘RAO LOWER BOUND

The effectiveness of any method for the estimation of a K y \ vector <j> o f K DOA parameters from a random observation vector u, is ordinarily evaluated in accordance with the following quality criteria:

z. R e s o lv a b ility - its ability to reveal the presence of two equal-power sources of nearly equal bearings

ii. B ia s - the average error in estimating the location of a source. An estimate is unbiased when its expected value equals the parameter itself.

in. V a r ia b ility - the variance o f the estimation bearing (which is also the variance of the estimation error)

Since the direct computation of the bias and variance of the estimated DOA parameters is in general difficult to achieve, the usual procedure is to derive a lower bound on the variance of each estimated parameter. A popular lower bound on the variance var Q>k) of any unbiased estimate of a parameter is provided by the Cramér~Rao lower bound (CRLB), the determination of which is closely linked to the Fisher information matrix. The ^'À:"th element of the latter Ky^K matrix J is defined as:

j , . , . = . . .^^2.1)

provided the derivatives exist and are absolutely integrable. In (52.1) U is a random vector of the received array data, represented by the observed sample vector m, /u(mI^) is the conditional joint probability density function (PDF) for the random vector £/, viewed as a function of the DOA parameter vector 0 , and S is the expectation operator. It can be shown [Van 68], [Rag 73] that for an unbiased vector estimate ^

cov (ÿ) > j ' . . . (522)

where J '^ is the inverse o f J , cov (ÿ) is the covariance matrix for the estimated parameter vector (or in other words the estimation error covariance matrix) and the matrix inequality is in the sense that the difference matrix [cov (^) - J '^ ] is positive