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

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Received: 24 April 2013; in revised form: 20 May 2013 / Accepted: 21 May 2013 / Published: 24 June 2013
(This article belongs to the Special Issue Wavelets and Applications)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
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MDPI and ACS Style

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.

AMA Style

Johnson RW. Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data. Axioms. 2013; 2(3):286-310.

Chicago/Turabian Style

Johnson, Robert W. 2013. "Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data." Axioms 2, no. 3: 286-310.

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