Modal Analysis Using Sparse and Co-prime Arrays

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, This work is supported in part by NSF under grants CCF-1018472 and CCF-1422658. A preliminary version of a subset of the results reported here was presented at the 2014 Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2-5, 2014, in a paper entitled “Characterization of orthogonal subspaces for alias-free reconstruction of damped complex exponential modes in sparse arrays,” by P. Pakrooh, A. Pezeshki, and L. L. Scharf. P. Pakrooh is with the Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA (e-mail: [email protected]). L. L. Scharf is with the Department of Mathematics and the Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA (e-mail: [email protected]). A. Pezeshki is with the Department of Electrical and Computer Engineering and the Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA (e-mail: [email protected]). April 7, 2015 DRAFT ar X iv :1 50 4. 01 25 8v 1 [ cs .I T ] 6 A pr 2 01 5 MODAL ANALYSIS USING SPARSE AND CO-PRIME ARRAYS 2 versus SNR. Our calculations of Cramér-Rao bounds allow us to analyze the loss in performance sustained by sparse and co-prime arrays that are compressions of uniform linear arrays. Index Terms Co-pime array, IQML, modal analysis, orthogonal subspaces, sparse array