Matching Pursuits may yield superior results to Orthogonal Matching Pursuits when secondary information is estimated from the signal model

In this paper, we compare the Orthogonal Matching Pursuits and the Matching Pursuits algorithm as used with the same dictionary in a stationary spectral estimation problem. Specifically, we develop an over-complete dictionary using the Fourier and Hirschman Optimal Transforms. Then, we apply a periodogram spectral estimation algorithm using this dictionary to a signal consisting of closely-spaced-in-frequency sinusoids in additive white noise. It is well-known that the matching pursuit algorithm is not guaranteed to converge as more dictionary elements are used to represent the signal, the orthogonal version must. What is not well-known, at least to the authors, is that the frequency estimation (i.e. the secondary inference that is desired from the signal model) is not highly connected to the modeling performance. In fact, we will show that while the energy in the residual is lower for the orthogonal matching pursuits, the frequency estimation error is lower for the spectral estimated derived using matching pursuits.