Sparse System Identification in Pairs of Pulse and Takenaka--Malmquist Bases

Abstract

This paper considers the identification of a rational transfer function with sparse coefficients, under a pair of pulse and Takenaka--Malmquist (TM) bases and from a limited number of linear frequency domain measurements. We propose to concatenate pulse and TM basis functions in the representation of the transfer function, and prove the uniqueness of the sparse representation. We provide the sufficient condition under which the sparse system can be identified by $\ell_1$ optimization from random samples with high probability, and also provide the computation algorithm. Numerical examples show that the concatenated pulse and TM bases give a much sparser representation, with much lower reconstruction order compared to using only pulse basis functions and less dependency on the knowledge of the true system poles compared to using only TM basis functions.