Auditory-motivated Gammatone wavelet transform

• We propose the construction of a continuous wavelet transform motivated by the human auditory system.
• A family of wavelets are derived from the popular Gammatone function model for the auditory impulse response, via repeated differentiation.
• The wavelets have a time-bandwidth product close to 0.5, in addition to being causal.
• We investigate the vanishing moments property of the proposed Gammatone wavelets.
• We show the application of the proposed wavelets to singularity and transient detection.

The ability of the continuous wavelet transform (CWT) to provide good time and frequency localization has made it a popular tool in time–frequency analysis of signals. Wavelets exhibit constant-Q property, which is also possessed by the basilar membrane filters in the peripheral auditory system. The basilar membrane filters or auditory filters are often modeled by a Gammatone function, which provides a good approximation to experimentally determined responses. The filterbank derived from these filters is referred to as a Gammatone filterbank. In general, wavelet analysis can be likened to a filterbank analysis and hence the interesting link between standard wavelet analysis and Gammatone filterbank. However, the Gammatone function does not exactly qualify as a wavelet because its time average is not zero. We show how bona fide wavelets can be constructed out of Gammatone functions. We analyze properties such as admissibility, time-bandwidth product, vanishing moments, which are particularly relevant in the context of wavelets. We also show how the proposed auditory wavelets are produced as the impulse response of a linear, shift-invariant system governed by a linear differential equation with constant coefficients. We propose analog circuit implementations of the proposed CWT. We also show how the Gammatone-derived wavelets can be used for singularity detection and time–frequency analysis of transient signals.