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.
Highlights
The primary criticism leveled at the use of the continuous wavelet transform for the spectral analysis of discretely sampled data is that it fails to give quantitatively meaningful results
There is nothing to be gained by working on the scale axis rather than frequency other than a headache in dealing with the effect of aliasing, as one requires the same set of basis functions Θ and minimal order Nmin to satisfy the fundamental theorems of spectral analysis in either case
The power spectrum given by the layered window in (c) does incorporate features of either Gabor transform shown in (a) and (b); in a sense, it has combined the Gabor transforms over its range of scale in a way that preserves the energy and reconstruction theorems
Summary
The primary criticism leveled at the use of the continuous wavelet transform for the spectral analysis of discretely sampled data is that it fails to give quantitatively meaningful results. The discrete implementation of the continuous wavelet transform and its inverse do not Axioms 2013, 2 satisfy the energy and reconstruction theorems of spectral analysis. The goal of this investigation is to devise a multiresolution analysis that does satisfy those theorems. The review by Torrence and Compo [1] remains a popular resource for practitioners of wavelet analysis It relies on the method by Farge [2] for the reconstruction of the data signal. We will propose an algorithm for a layered window transform whose spectral density is similar to that of the Morlet transform yet which satisfies the energy and reconstruction theorems. The programs used for this analysis are available online [20]
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