Symmetrization and enhancement of the continuous Morlet transform

TL;DR

This paper introduces symmetrization and renormalization techniques for the continuous Morlet wavelet transform, improving spectral resolution and power estimation in signal analysis.

Contribution

It proposes methods to symmetrize and renormalize the Morlet wavelet transform, enhancing spectral analysis accuracy and resolution.

Findings

Symmetrization improves the transform's response symmetry.

Renormalization based on power reduces wavelet truncation effects.

Deconvolution enhances spectral resolution and reduces leakage.

Abstract

The forward and inverse wavelet transform using the continuous Morlet basis may be symmetrized by using an appropriate normalization factor. The loss of response due to wavelet truncation is addressed through a renormalization of the wavelet based on power. The spectral density has physical units which may be related to the squared amplitude of the signal, as do its margins the mean wavelet power and the integrated instant power, giving a quantitative estimate of the power density with temporal resolution. Deconvolution with the wavelet response matrix reduces the spectral leakage and produces an enhanced wavelet spectrum providing maximum resolution of the harmonic content of a signal. Applications to data analysis are discussed.