Shear Madness: New Orthonormal Bases and Frames Using Chirp Functions

The proportional-bandwidth and constant-bandwidth time-frequency signal decompositions of the wavelet, Gabor, and Wilson orthonormal bases have attracted substantial interest for representing nonstationary signals. However, these representations are limited in that they are based on rectangular tessellations of the time-frequency plane. While much effort has gone into methods for designing nice wavelet and window functions for these frameworks, little consideration has been given to methods for constructing orthonormal bases employing nonrectangular time-frequency tilings. In this note, we take a first step in this direction by deriving two new families of orthonormal bases and frames employing elements that shear, or chirp, in the time-frequency plane, in addition to translate and scale. The new scale-shear fan bases and shift-shear chevron bases are obtained by operating on an existing wavelet, Gabor, or Wilson basis set with two special unitary warping transformations. In addition to the theoretical benefit of broadening the class of valid time-frequency plane tilings, these new bases could possibly also be useful for representing certain types of signals, such as chirping and dispersed signals.