Let a measurement consist of a linear combination of damped complex exponential modes, plus noise. The problem is to estimate the parameters of these modes, as in line spectrum estimation, vibration analysis, speech processing, system identification, and direction of arrival estimation. Our results differ from standard results of modal analysis to the extent that we consider sparse and co-prime samplings in space, or equivalently sparse and co-prime samplings in time. Our main result is a characterization of the orthogonal subspace. This is the subspace that is orthogonal to the signal subspace spanned by the columns of the generalized Vandermonde matrix of modes in sparse or co-prime arrays. This characterization is derived in a form that allows us to adapt modern methods of linear prediction and approximate least squares, such as iterative quadratic maximum likelihood (IQML), for estimating mode parameters. Several numerical examples are presented to demonstrate the validity of the proposed modal estimation methods, and to compare the fidelity of modal estimation with sparse and co-prime arrays, versus SNR. Our calculations of Cram{e}r-Rao bounds allow us to analyze the loss in performance sustained by sparse and co-prime arrays that are compressions of uniform linear arrays.
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