Sparse and cross-term free time-frequency distribution based on Hermite functions

Abstract

Hermite functions are an effective tool for improving the resolution of the single-window spectrogram. In this paper, we analyze the Hermite functions in the ambiguity domain and show that the higher order terms can introduce undesirable cross-terms in the multiwindow spectrogram. The optimal number of Hermite functions depends on the location and spread of signal auto-terms in the ambiguity domain. We apply and compare several sparsity measures, namely ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm, the Gini index and the time-frequency concentration measure, for determining the optimal number of Hermite functions, leading to the most desirable time-frequency representation. Among the employed measures, the Gini index provides the sparsest solution. This solution corresponds to a well-concentrated and cross-term reduced time-frequency signature.