Bilinear time-frequency representations of signals: the shift-scale invariant class

The authors consider the class of bilinear time-frequency representations (BTFR´s) that are invariant (or covariant) to time shifts, frequency shifts, and time-frequency scalings. This “shift-scale invariant” class is the intersection of the classical shift-invariant (Cohen) class and the recently defined affine class. The mathematical description of shift-scale invariant BTFR´s is in terms of a 1-D kernel and is thus particularly simple. The paper concentrates on the time-frequency localization properties of shift-scale invariant BTFR´s. Since any shift-scale invariant BTFR is a superposition of generalized Wigner distributions, the time-frequency localization of the family of generalized Wigner distributions is studied first. For those shift-scale invariant BTFR´s that may be interpreted as smoothed versions of the Wigner distribution (e.g., the Choi-Williams distribution), an analysis in the Fourier transform domain shows interesting peculiarities regarding time-frequency concentration and interference geometry properties