The complex - valued continuous wavelet transform as a preprocessorfor auditory

In this paper we draw links between the widely used gammatone lter auditory model and wavelet theory. From the viewpoint of wavelet theory the beneet from linking these research elds is a fast method for the computation of a timescale representation. From the viewpoint of auditory ltering the beneets are the existence of methods for the detection of signal singularities and for resynthesis. Our method has proved to be useful for the analysis of music pieces with a limited spectral overlap of the diierent signal components. It has been implemented for further research in automated music transcription and auditory source separation, but might also be of interest for sound synthesis systems based on the analysis and transformation of acoustic signals. 1 Linear Time-Frequency Distributions Given a one-dimensional acoustic time signal s(t) a time-frequency distribution (TFD) is a two-dimensionalrepre-sentation of s(t) with time and frequency as its parameters. Basically a TFD tells us, which frequencies occur at which times in the input signal. There is, however, no unique TFD of a given signal s(t). For a general overview of some diierent time-frequency representations the reading of Hlawatsch and Boudreaux-Bartels, 1992] is recommended. 1.1 Why Wavelets? The short time Fourier transform (STFT) of a signal s(t) is given by F s (t; f) = Z 1 ?1 s() g (? t) e ?j2f dd; (1) where the asterisk denotes complex conjugation. This is technically a windowed version of the Fourier transform, where a single window of constant shape is sliding along the time axis. The continuous wavelet transform (CWT) is given by W s (b; a) = 1 p a Z 1 ?1 s() g ? b a dd; a > 0; (2) or equivalently in the frequency domain using Parseval's identity W s (b; a) = p a Z 1 ?1 S(f) G (af)e j2fb df; a > 0: (3) As a is scaling the mother-wavelet g(t), it is called scale parameter, as b shifts it in time, we call it shift parameter. Given the Fourier transform G(f) of the mother wavelet g(t) the reconstruction of the time signal can be computed by a dadb a 2 ; (4) where c g = Z 1 ?1 jG(f)j 2 jfj df < 1: (5) The latter equation is a necessary and suucient condition for admissibility of a time function g(t) as a mother wavelet. Eq.5 implies that Z 1 ?1 jg(t)j 2 …