Shear madness: new orthonormal bases and frames using chirp functions

The proportional-bandwidth and constant-bandwidth timefrequency signal decompositions of the wavelet, Gabor, and Wilson Manuscript received August 21, 1992; revised May 26, 1993. The Guest Editor coordinating the review of this paper and approving it for publication was Dr. Patrick Flandrin. This work was supported in part by the Sound Group of the computer-based Education Research Laboratory, University of Illinois, in part by an NSERC-NATO postdoctoral fellowship, in part by the Joint Services Electronics Program under Grant N00014-90J-1270, and in part by the National Science Foundation under Grant MIP 9012747. R. G . Baraniuk is with the Department of Electrical and Computer Engineering, Rice University, Houston, TX 77251-1892. D. L. Jones is with the Coordinated Science Laboratory. University of Illinois, Urbana, IL 61801. IEEE Log Number 92 12 184. 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 timefrequency plane, in addition to translate and scale. The new scale-shear fun 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.