A 'gammachirp' function as an optimal auditory filter with the Mellin transform

A 'gammachirp' function has been derived as an optimal auditory filter function in terms of minimal uncertainty in a joint time and modified-scale representation if the scale transform defined by Cohen (1989) is used in the auditory system. The gammatone function, which is widely used as the impulse response of a linear auditory filter, is a first-order approximation of the 'gammachirp' function consisting of a chirp carrier with an envelope that is a gamma distribution function. The optimality of the 'gammachirp' function is argued for the general Mellin transform since Cohen's scale transform is a specific example of the Mellin transform. A sample speech signal is analyzed to demonstrate the properties of a joint time and scale distribution derived with a short-time Mellin transform in comparison with a short-time Fourier spectrum.