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

• Main Author: Robert W. Johnson
• Format: Article
• Language: English
• Published: MDPI AG 2013-06-01
• Series: Axioms
• Subjects: Fourier transform; Gabor transform; Morlet transform; multiresolution analysis
• Online Access: http://www.mdpi.com/2075-1680/2/3/286

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 that 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.