Spectral estimation with the Hirschman optimal transform filter bank and compressive sensing

Abstract

The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. Unlike the Heisenberg-Weyl measure, the Hirschman notion of joint uncertainty is based on the entropy rather than the energy [1]. Furthermore, as we noted in [2], the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to resolve two limiting cases of localization in frequency, viz pure tones and additive white noise. We found in [3] that the HOT has a superior resolution to the DFT when two pure tones are close in frequency. In this paper, we improve on that method to present a more complete spectral analysis tool. Here, we implement a stationary spectral estimation method using compressive sensing (in particular, Iterative Hard Thresholding) on HOT filterbanks. We compare its frequency resolution to that of a DFT filterbank using compressive sensing. In particular, we compare the performance of the HF with that of the DFT in resolving two close frequency components in additive white Gaussian noise (AWGN). We find the HF method to be superior to the DFT method in frequency estimation, and ascribe the difference to the HOT's relationship to entropy.