The multiple hypothesis testing problem of the detection-estimation of an unknown number of independent Gaussian point sources is adequately addressed by likelihood ratio (LR) maximization over the ...set of admissible covariance matrix models. We introduce nonasymptotic lower and upper bounds for the maximum LR. Since LR optimization is generally a nonconvex multiextremal problem, any practical solution could now be tested against these bounds, enabling a high probability of recognizing nonoptimal solutions. We demonstrate that in many applications, the lower bound is quite tight, with approximate maximum likelihood (ML) techniques often unable to approach this bound. The introduced lower bound analysis is shown to be very efficient in determining whether or not performance breakdown has occurred for subspace-based direction-of-arrival (DOA) estimation techniques. We also demonstrate that by proper LR maximization, we can extend the range of signal-to-noise ratio (SNR) values and/or number of data samples wherein accurate parameter estimates are produced. Yet, when the SNR and/or sample size falls below a certain limit for a given scenario, we show that ML estimation suffers from a discontinuity in the parameter estimates: a phenomenon that cannot be eliminated within the ML paradigm.
The problem of estimating the number of independent Gaussian sources and their parameters impinging upon an antenna array is addressed for scenarios that are problematic for standard techniques, ...namely, under "threshold conditions" (where subspace techniques such as MUSIC experience an abrupt and dramatic performance breakdown). We propose an antenna geometry-invariant method that adopts the generalized-likelihood-ratio test (GLRT) methodology, supported by a maximum-likelihood-ratio lower-bound analysis that allows erroneous solutions ("outliers") to be found and rectified. Detection-estimation performance in both uniform circular and linear antenna arrays is shown to be significantly improved compared with conventional techniques but limited by the performance-breakdown phenomenon that is intrinsic to all such maximum-likelihood (ML) techniques
The well-known general problem of signal detection in background interference is addressed for situations where a certain statistical description of the interference is unavailable, but is replaced ...by the observation of some secondary (training) data that contains only the interference. For the broad class of interferences that have a large separation between signal-and noise-subspace eigenvalues, we demonstrate that adaptive detectors which use a diagonally loaded sample covariance matrix or a fast maximum likelihood (FML) estimate have significantly better detection performance than the traditional generalized likelihood ratio test (GLRT) and adaptive matched filter (AMI') detection techniques, which use a maximum likelihood (ML) covariance matrix estimate. To devise a theoretical framework that can generate similarly efficient detectors, two major modifications are proposed for Kelly's traditional GLRT and AMF detection techniques. First, a two-set GLRT decision rule takes advantage of an a priori assignment of different functions to the primary and secondary data, unlike the Kelly rule that was derived without this. Second, instead of ML estimates of the missing parameters in both GLRT and AMF detectors, we adopt expected likelihood (EL) estimates that have a likelihood within the range of most probable values generated by the actual interference covariance matrix. A Gaussian model of fluctuating target signal and interference is used in this study. We demonstrate that, even under the most favorable loaded sample-matrix inversion (LSMI) conditions, the theoretically derived EL-GLRT and FL-AMF techniques (where the loading factor is chosen from the training data using the EL matching principle) gives the same detection performance as the loaded AMF technique with a proper a priori data-invariant loading factor. For the least favorable conditions, our EL-AMF method is still superior to the standard AMF detector, and may be interpreted as an intelligent (data-dependent) method for selecting the loading factor.
This paper considers the problem of direction-of-arrival (DOA) estimation for multiple uncorrelated plane waves incident on 'partially augmentable' antenna arrays, whose difference set of ...interelement spacings is not complete. The DOA estimation problem for the case when the number of sources exceeds the number of contiguous covariance lags gives rise to the covariance matrix completion problem. Maximum-entropy (ME) positive-definite (p.d.) completion for partially specified Toeplitz covariance matrices is developed using convex programming techniques. By this approach, the classical Burg ME extension problem for the given set of covariance lags is generalized for the situation when some lags are missing. For DOA estimation purposes, we find the p.d. Toeplitz matrix with fixed eigensubspace dimension that is the closest approximation of the ME-completed matrix. Computer simulation results are presented to demonstrate the high DOA estimation accuracy of the proposed technique compared with the corresponding Cramer-Rao bound.
This paper addresses the problem of ambiguities in direction of arrival (DOA) estimation for nonuniform (sparse) linear arrays. Usually, DOA estimation ambiguities are associated with linear ...dependence among the points on the antenna array manifold, that is, the steering vectors degenerate so that each may be expressed as a linear combination of the others. Most nonuniform array geometries, including the so-called "minimum redundancy" arrays, admit such manifold ambiguities. While the standard subspace algorithms such as MUSIC fail to provide unambiguous DOA estimates under these conditions, we demonstrate that this failure does not necessarily imply that consistent and asymptotically effective DOA estimates do not exist. We demonstrate that in most cases involving uncorrelated Gaussian sources, manifold ambiguity does not necessarily imply nonidentifiability; most importantly, we introduce algorithms designed to resolve manifold ambiguity. We also show that for situations where the number of sources exceeds the number of array sensors, a new class of locally nonidentifiable scenario exists.
Previous studies dealing with direction-of-arrival (DOA) estimation for uncorrelated planes waves incident on nonuniform M-sensor arrays assumed that the number of signal sources m was known or had ...already been estimated. In the "conventional" case (m < M), traditional detection techniques such as Akaike's information criterion (AIC) and minimum description length (MDL) that are based on the equality of several smallest eigenvalues in the covariance matrix may be applied, although we demonstrate that these results can be misleading for nonuniform arrays. In the "superior" case (m greater than or equal to M), these standard techniques are not applicable. We introduce a new approach to the detection problem for "fully augmentable" arrays (whose set of intersensor differences is complete). We show that the well-known direct augmentation approach applied to the sample covariance matrix is not a solution by itself since the resulting Toeplitz matrix is generally not positive definite for realistic sample volumes. We propose a transformation of this augmented matrix into a positive definite Toeplitz matrix T sub( mu ) with the proper number of equal minimum eigenvalues that are appropriate for the candidate number of sources mu . Comparison of the results of these best-fit transformations over the permissible range of candidates then allows us to select the most likely number of sources m using traditional criteria and yields uniquely defined DOA's. Simulation results demonstrate the high performance of this method. Since detection techniques for superior scenarios have not been previously described in the literature, we compare our method with the standard AIC and MDL techniques in a conventional case with similar Cramer-Rao bound (CRB) and find that it has a similar detection performance.
The maximum likelihood ratio (LR) lower bound analysis introduced in our previous papers is applied to support the detection-estimation of multiple Gaussian spread (distributed, scattered) sources. ...Since angular spreading eliminates any "noise eigensubspace" from the spatial covariance matrix, traditional detection techniques based on the equality of noise-subspace eigenvalues are not applicable here. Brute-force "focusing", which is based on the Schur-Hadamard inverse, is shown to be inefficient. Our technique is based on generalized likelihood-ratio test (GLRT) principles and involves LR maximization over the set of admissible covariance matrix models. The introduced technique yields results that statistically exceed the LR generated by the exact covariance matrix, which is used as the lower bound. High optimization efficiency drives high detection-estimation performance that, nevertheless, breaks down under certain threshold conditions. It is demonstrated that this breakdown phenomenon is not curable within the maximum likelihood (ML) paradigm since these highly erroneous solutions are still "better" than the true covariance matrix (as measured by the LR).
This paper considers the problem of direction-of-arrival (DOA) estimation for multiple uncorrelated plane waves incident on so-called `fully augmentable' sparse linear arrays. In situations where a ...decision is made on the number of existing signal sources (m) prior to the estimation stage, we investigate the conditions under which DOA estimation accuracy is effective (in the maximum-likelihood sense). In the case where m is less than the number of antenna sensors (M), a new approach called `MUSIC-maximum-entropy equalization' is proposed to improve DOA estimation performance in the `preasymptotic region' of finite sample size (N) and signal-to-noise ratio. A full-sized positive definite (p.d.) Toeplitz matrix is constructed from the MxM direct data covariance matrix, and then, alternating projections are applied to find a p.d. Toeplitz matrix with m-variate signal eigensubspace (`signal subspace truncation'). When m greater than or equal to M, Cramer-Rao bound analysis suggests that the minimal useful sample size N is rather large, even for arbitrarily strong signals. It is demonstrated that the well-known direct augmentation approach (DAA) cannot approach the accuracy of the corresponding Cramer-Rao bound, even asymptotically (as N arrow right infinity ) and, therefore, needs to he improved. We present a new estimation method whereby signal subspace truncation of the DAA augmented matrix is used for initialization and is followed by a local maximum-likelihood optimization routine. The accuracy of this method is demonstrated to be asymptotically optimal for the various superior scenarios (m greater than or equal to M) presented.
For pt.II see ibid., vol.51, no.6, p.1492-507 (2003). We investigate nonidentifiability conditions for the detection-estimation problem with multiple uncorrelated plane waves incident upon a ...nonuniform (sparse) linear antenna array. Specifically, we define conditions under which a given Hermitian covariance matrix has a nonunique (multiple) decomposition into an admissible number of dyads weighted by the source powers and a white-noise identity matrix. Our method is based on the Proukakis-Manikas technique of generating ambiguity generator sets (AGSs) and allows ambiguous sets of sources associated with a given sparse antenna geometry to be determined.
We investigate direction-of-arrival (DOA) estimation involving nonuniform linear arrays, where the sensor positions may be noninteger values expressed in half-wavelength units, with some number of ...uncorrelated Gaussian sources that is greater than or equal to the number of sensors. We introduce an approach whereby the (noninteger) co-array is treated as the most appropriate virtual array when considering an augmented covariance matrix. Since such virtual arrays have an incomplete set of covariance lags, we discuss various completion philosophies to fill in the missing elements of the associated partially specified Hermitian covariance matrix. This process is followed by the application of an algorithm that searches for a specific number of plane wavefronts, yielding the minimum fitting error for the specified covariance lags in the neighborhood of the completion-initialized DOA estimates. In this way, we are able to resolve possible ambiguity and to achieve asymptotically optimal estimation accuracy. Numerical simulations of DOA estimation demonstrate a close proximity to the Cramer-Rao bound.
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