Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data

TL;DR

This paper analyzes the properties of the Gabor and Morlet transforms for spectral analysis of discretely sampled data, proposing a fixed window approach that satisfies energy and reconstruction theorems and can be tailored for specific resolution needs.

Contribution

It introduces a method to construct a single, fixed window from arbitrary windows that ensures energy and reconstruction theorems are satisfied in spectral analysis.

Findings

Fixed window with uniform frequency sampling satisfies energy and reconstruction theorems.

The proposed approach allows tailoring window shape for specific time/frequency resolution.

Method extends naturally to nonuniform sampling without major modifications.

Abstract

The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy and reconstruction theorems; however, transform pairs based on a variable window or nonuniform frequency sampling in general do not. Instead of selecting the shape of the window as some function of the central frequency, we propose constructing a single window with unit energy from an arbitrary set of windows which is applied over the entire frequency axis. By virtue of using a fixed window with uniform frequency sampling, such a transform satisfies the energy and reconstruction theorems. The shape of the window can be tailored to meet the requirements of the investigator in terms of time/frequency resolution. The algorithm extends naturally to the case of nonuniform signal sampling without modification beyond identification of the Nyquist interval.