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

摘要

The properties of the Gabor and Morlet transforms are examined with respectto the Fourier analysis of discretely sampled data. Forward and inversetransform pairs based on a fixed window with uniform sampling of the frequencyaxis can satisfy numerically the energy and reconstruction theorems; however,transform pairs based on a variable window or nonuniform frequency sampling ingeneral do not. Instead of selecting the shape of the window as some functionof the central frequency, we propose constructing a single window with unitenergy from an arbitrary set of windows which is applied over the entirefrequency axis. By virtue of using a fixed window with uniform frequencysampling, such a transform satisfies the energy and reconstruction theorems.The shape of the window can be tailored to meet the requirements of theinvestigator in terms of time/frequency resolution. The algorithm extendsnaturally to the case of nonuniform signal sampling without modification beyondidentification of the Nyquist interval.