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

Abstruct-We 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 tbat may be interpreted as smoothed versions of the Wigner distribution (e.g., the ChoLWilliams distribution), an analysis in the Fourier transform domain shows interesting peculiarities regarding time-frequency concentration and interference geometry properties.