Wide-band ambiguity functions and affine Wigner distributions

Abstract

A symmetric wideband ambiguity function is proposed, by recognizing the relationship between ambiguity functions and the theory of group representations. By slightly modifying the 2D Fourier transform of the symmetric wideband ambiguity function, we recover the affine Wigner distribution, a Mellin-frequency/time-scale phase space function with properties that parallel those of the Wigner distribution. The affine Wigner distribution was first proposed by the Bertrands: they defined a class of time-frequency distributions satisfying covariance with respect to time-shifts and dilations, and singled out this function by imposing certain axiomatic requirements, just as the usual Wigner distribution uniquely satisfies certain axioms among the class of time-frequency distributions that are covariant with respect to time-shifts and frequency-shifts. Our derivation, which is different, is based on recognizing that the duality between the narrowband ambiguity function and the usual Wigner distribution is an example of a general framework that has been studied in the subject of Fourier analysis on groups. Our description of this group-theoretic framework is new in the signal processing literature. The classical Fourier and Mellin transforms, Wigner distribution, and affine Wigner distribution result from applications of this theory to different groups, and it therefore provides a unifying perspective.