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

Axioms

关键词: Fourier transform;    Gabor transform;    Morlet transform;    multiresolution analysis;   

DOI  :  10.3390/axioms2030286

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2013 by the authors; licensee MDPI, Basel, Switzerland.