Spatial Spectral Estimation with Product Processing of a Pair of Colinear Arrays

The product of the scanned responses of two colinear interleaved arrays estimates the spatial power spectral density of the received signal. This estimate has mathematical structure similar to the classical periodogram and hence is called the generalized periodogram. The generalized periodogram includes sparse arrays such as coprime arrays and two level nested arrays. This paper derives the analytical expected value of the generalized periodogram output for any signal. The generalized periodogram expected value is the convolution of the true spectral density with a windowing function. This result reveals that the expected product spectrum is not guaranteed to be positive definite. The paper also derives the covariance function of the generalized periodogram for a white Gaussian process. Finally, the paper illustrates the implicit equivalence between the product spectrum and estimation of the correlation function from the products of appropriately spaced pairs of sensors.