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Axioms 2013, 2(3), 286-310; doi:10.3390/axioms2030286
Article

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

Alphawave Research, 29 Stanebrook Court, Jonesboro, GA 30238, USA
Received: 24 April 2013 / Revised: 20 May 2013 / Accepted: 21 May 2013 / Published: 24 June 2013
(This article belongs to the Special Issue Wavelets and Applications)
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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 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.
Keywords: Fourier transform; Gabor transform; Morlet transform; multiresolution analysis Fourier transform; Gabor transform; Morlet transform; multiresolution analysis
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Johnson, R.W. Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data. Axioms 2013, 2, 286-310.

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