Product Processing for Tapered Sparse Arrays

TL;DR

This paper extends the analysis of product array processing by deriving the expected value and covariance when using non-uniform tapers, applicable to multidimensional arrays, enhancing understanding of spatial spectral density estimation.

Contribution

It introduces the derivation of expected value and covariance functions for product array processing with non-uniform tapers, including multidimensional array configurations, which was not previously addressed.

Findings

Expected value is the convolution of true PSD with the Fourier transform of the difference coarray.

Covariance function of the product processor output is derived for non-uniform tapers.

Results are extended to multidimensional array geometries.

Abstract

The product processor output has recently been introduced as a spatial power spectral density estimate, unifying product arrays such as coprime arrays, nested arrays, and standard uniform line arrays. The expected value and covariance function of this estimate for a white Gaussian process was derived in previous work over these various array configurations. However, this prior work used a uniform taper in all cases. In this paper, we show that when product arrays are windowed with non-uniform tapers, the expected value of the product processor output is the convolution of the true spatial power spectral density with the spatial Fourier transform of the difference coarray. This expected value makes a Fourier transform pair with a spatial autocorrelation estimate obtained by windowing the true autocorrelation function. We also derive the covariance function of the product processor output with non-uniform tapers, and compare these derived statistics for the aforementioned array geometries. Also, in prior work, the moments were provided only for linear arrays; this paper extends the estimation results to multidimensional arrays.