Uncertainty principle and sparse reconstruction in pairs of orthonormal rational function bases

Abstract

This paper presents theoretical results on the uncertainty principle and the sparse reconstruction of rational transfer functions in a dictionary of two orthonormal rational function (ORF) bases. The uncertainty principle concerning pairs of representations of rational transfer functions in different ORF bases is established. It is shown that a rational transfer function cannot have a sparse representation simultaneously in two different mutually incoherent ORF bases. The uniqueness for the sparse representation is derived as a direct consequence of this uncertainty principle. A reconstruction method for a rational transfer function in a pair of ORF bases is proposed. The sparse reconstruction result shows that, given a rational transfer function with a sufficiently sparse representation in a given dictionary of two ORF bases, the sparse representation can be recovered by solving a linear programming problem. A lower bound is provided on the number of frequency response measurements required to recover the sparse representation with high probability.