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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; in revised form: 20 May 2013 / Accepted: 21 May 2013 / Published: 24 June 2013
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.
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.
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.