Using entropy to improve the resolution in non-parametric spectral estimation

Detection and estimation of frequencies in composite signal is a very important topic. The resolution issue is the most fundamental. The higher the resolution, the more precise information we can get from the signal.;The traditional frequency detection and estimation is evaluated by conventional Discrete Fourier Transform (DFT) based periodogram. The basis of the DFT is the Heisenberg-Weyl measure, which 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. When the two frequency components are very close to each other in frequency, the two peaks in the periodogram will emerge, and it is hard for the DFT to distinguish between them. However the new proposed Hirschman Optimal Transform (HOT) based periodogram has the ability to resolve them. Unlike the Heisenberg-Weyl measure, the Hirschman notion of joint uncertainty is based on entropy rather than energy. Furthermore, its definition extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals. The HOT is superior to the DFT and Discrete Cosine Transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz pure tones and additive white noise.;In this dissertation I implement a stationary spectral estimation using three methods: (1) matching pursuit method whose dictionary members are constructed from the combination of HOT-based and DFT atoms (elements). (2) filter bank method whose filter banks are constructed from HOT and DFT matrices. (3) compressive sensing method i.e Iterative Hard Thresholding (IHT) combined with matching pursuit and filter bank methods. I call the resulting algorithm the HOT-DFT (HF) periodogram. I compare its performance (in terms of frequency resolution) with a standard DFT-based periodogram. I find the HF to be superior to the DFT in frequency estimation, and ascribe the difference to the HOT's relationship to entropy.