Date of Award


Degree Type


Degree Name

Doctor of Philosophy in Electrical Engineering


Electrical, Computer, and Biomedical Engineering

First Advisor

Kaushallya Adhikari


The expected value and covariance function of a product processor for a spatially uncorrelated white guassian process has been derived for known sparse geometries such as nested or coprime with uniform taper. When a non-uniform taper is applied, the expected value of the product processor becomes the convolution of the true spatial power spectral density and the spatial Fourier transform of the difference coarray. A windowed estimate of the true autocorrelation function can be obtained by taking the inverse Fourier transform of the output. By extending that function to an estimate of the sample covariance matrix, it can be used in subspace-based algorithms such as MUSIC or ESPRIT. These results are extended to colored processes and multidimensional arrays.

Subspace-based algorithms make use of the eigenvectors of the sample covariance matrix or the singular vectors of the received array data. The subspaces spanned by the singular vectors or eigenvectors are perturbed away from the ensemble due to additive noise causing performance losses. This can be improved by using the optimal signal or noise subspace basis estimate instead. A statistically optimal estimate of the unperturbed subspace basis in terms of perturbed signal and orthogonal bases has been derived and is accurate up to the first-order terms in the additive noise matrix. A methodology to find these optimal estimates in sparse arrays that possess a shift-invariant structure is derived here and can be additionally interpolated to match the basis of a fully populated array with equal aperture. The second-order optimal approximation for the unperturbed subspace is derived. The results are applied to a uniform linear array (ULA) processing model for both full and sparse geometries. In the case of a fully sampled array, the first-order terms carry the bulk of information and an extension to second-order does little to reduce estimation error. However, for a sparse array, the performance is improved more clearly.

The variance on DOA estimation using an optimal subspace basis estimate on shift-invariant arrays reaches the Cramer-Rao lower bound. However, unlike other sparse geometries such as nested or coprime arrays, SISAs are not able to estimate more sources than the number of sensors. Presented is the addition of spatial matrix filtering to reduce the rank of the input passed into the estimation algorithm. Matrix filters allow spatial frequencies within its pass band to remain nearly distortionless while simultaneously attenuating ambient noise in the stop band. The reduction in rank does not affect the arrays resolution capability or dimension of the coarray. Therefore we can enable a SISA to estimate more sources than the number of sensors. Derived here is a more generalized optimal subspace estimation algorithm designed to handle the correlation introduced by the matrix filtering pre-processor. By doing so, the SISA achieves unprecedented estimation performance for SNR regions typically considered unusable.



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