Wavelet Transform With Tunable Q-Factor

• Published in: IEEE transactions on signal processing Vol. 59; no. 8; pp. 3560 - 3575
• Main Author: Selesnick, I. W.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Constant-Q transform; Continuous wavelet transforms; Detection, estimation, filtering, equalization, prediction; Discrete Fourier transforms; Discrete wavelet transforms; Exact sciences and technology; filter bank; Filter banks; Information, signal and communications theory; Oversampling; Q factor; Reconstruction; Redundancy; Sampling; Sampling, quantization; Signal and communications theory; Signal processing; Signal, noise; Synthesis; Telecommunications and information theory; Transforms; wavelet transform; Wavelet transforms; Q-factor; Q factor; Fast Fourier transformation; Redundancy; wavelet transform; Oversampling; Discrete wavelet transforms; Implementation; Wavelet transformation; Discrete time; Constant-Q transform; Signal processing; Filter bank; Discrete signal; Sampling; Localization
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2011.2143711

This paper describes a discrete-time wavelet transform for which the Q-factor is easily specified. Hence, the transform can be tuned according to the oscillatory behavior of the signal to which it is applied. The transform is based on a real-valued scaling factor (dilation-factor) and is implemented using a perfect reconstruction over-sampled filter bank with real-valued sampling factors. Two forms of the transform are presented. The first form is defined for discrete-time signals defined on all of Z. The second form is defined for discrete-time signals of finite-length and can be implemented efficiently with FFTs. The transform is parameterized by its Q-factor and its oversampling rate (redundancy), with modest oversampling rates (e.g., three to four times overcomplete) being sufficient for the analysis/synthesis functions to be well localized.