The Fourier–Mellin transform and mammalian hearing

Abstract

A combined Fourier–Mellin transform yields a representation of a signal that is independent of delay and scale change. Such a representation should be useful for speech analysis, where delay and scale differences degrade the performance of correlation operations or other similarity measures. At least two different versions of a combined Fourier–Mellin transform can be implemented. The simplest version (the ‖F‖2−‖M‖2transform) completely eliminates spectral phase information, while a slightly more complicated version (the ?−? transform) preserves some phase information. Both versions can be synthesized with a Fourier transform and an exponential‐sampling algorithm. Exponential sampling produces a frequency scale distortion that is similar to the effect of the cochlea. The ‖F‖2−‖M‖2transform can also be implemented with a bank of proportional bandwidth filters. If the relative phase between spectral components is preserved, then a Fourier–Mellin transformer can perform compression of linear‐period modulated signals. Such signals are used for echolocation by bats and cetaceans. The same approach that gives scale and delay invariance can be used to obtain other transform conbinations that provide insensitivity to a variety of distortions. The combined transforms can also be used for analyzing these distortions.