Characterization of orthogonal subspaces for alias-free reconstruction of damped complex exponential modes in sparse arrays

We consider the problem of estimating p damped complex exponentials from spatial samples of their weighted sum, taken by a sparse sensor array. Our focus is on a particular sparse array geometry, where the array can be thought of as a subsampled version of a dense (with half-wavelength spacings) uniform line array, plus an extra sensor that is posited at a location on the array that allows us to resolve aliasing ambiguities. This array geometry is a special, but canonical, example of a co-prime sensor array. Our main result is a 2p-parameter characterization of the so-called orthogonal subspace. This is the subspace that is orthogonal to the subspace spanned by the columns of the generalized Vandermonde matrix of modes in the sparse array. This characterization allows us to extend methods of linear prediction and approximate least squares, such as iterative quadratic maximum likelihood (IQML), for estimating mode parameters.