Generalized Morse wavelets

• Published in: IEEE transactions on signal processing Vol. 50; no. 11; pp. 2661 - 2670
• Main Authors: Olhede, S.C., Walden, A.T.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-11-2002; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Computation; Continuous wavelet transforms; Discrete Fourier transforms; Discrete wavelet transforms; Eigenfunctions; Eigenvalues; Eigenvalues and eigenfunctions; Exact sciences and technology; Fourier transforms; Information, signal and communications theory; Mathematical analysis; Proving; Signal analysis; Signal and communications theory; Signal processing; Signal representation. Spectral analysis; Signal, noise; Spectral analysis; Spectrogram; Stochastic processes; Studies; Telecommunications and information theory; Wavelet; Wavelet analysis; Wavelet transforms; Signal processing; Time frequency domain method; Wavelet transformation; Signal analysis; Spectral analysis
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2002.804066

This paper examines the class of generalized Morse wavelets, which are eigenfunction wavelets suitable for use in time-varying spectrum estimation via averaging of time-scale eigenscalograms. Generalized Morse wavelets of order k (the corresponding eigenvalue order) depend on a doublet of parameters (/spl beta/, /spl gamma/); we extend results derived for the special case /spl beta/ = /spl gamma/ = 1 and include a proof of "the resolution of identity." The wavelets are easy to compute using the discrete Fourier transform (DFT) and, for (/spl beta/, /spl gamma/) = (2m, 2), can be computed exactly. A correction of a previously published eigenvalue formula is given. This shows that for /spl gamma/ > 1, generalized Morse wavelets can outperform the Hermites in energy concentration, contrary to a conclusion based on the /spl gamma/ = 1 case. For complex signals, scalogram analyses must be carried out using both the analytic and anti-analytic complex wavelets or odd and even real wavelets, whereas for real signals, the analytic complex wavelet is sufficient.