Uncertainty Principle and Sparse Reconstruction in Pairs of Orthonormal Rational Function Bases

Abstract

Most rational systems can be described in terms of orthonormal basis functions. This paper considers the reconstruction of a sparse coefficient vector for a rational transfer function under a pair of orthonormal rational function bases and from a limited number of linear frequency-domain measurements. We prove the uncertainty principle concerning pairs of compressible representation of orthonormal rational functions in the infinite dimensional function space. The uniqueness of compressible representation using such pairs is provided as a direct consequence of uncertainty principle. The bound of the number of measurements which guarantees the replacement of 1_0 optimization searching for the unique sparse reconstruction by 1_1 optimization using random sampling on the unit circle with high probability is provided as well.