Modal Analysis Using Co-Prime Arrays

We address the problem of estimating mode parameters from noisy observations of a linear combination of the corresponding modes. This problem arises 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 co-prime samplings in space, or equivalently co-prime samplings in time. Our main result is a characterization of the orthogonal subspace for this problem. This is the subspace that is orthogonal to the signal subspace spanned by the columns of the generalized Vandermonde matrix of modes in co-prime samplings. This characterization is derived in a form that allows us to adapt modern methods of subspace signal processing to co-prime sampled signals. Several numerical examples are presented to demonstrate the application of the proposed modal estimation method. We state and prove theorems on identifiability of the modes and calculate a Cramér-Rao bound that allows us to analyze the performance of co-prime arrays that are subsamplings of uniform linear arrays of the same apertures.