Distributions for time-frequency analysis: a generalization of Choi-Williams and the Butterworth distribution

The authors generalize the Choi-Williams (1989) time-frequency exponential distribution (ED) and propose the Butterworth distribution (BUD). The kernels of both act as 2-D lowpass filters in the ambiguity function plane with variable filter characteristics. Increasing the order parameters results in flatter passbands and narrower transition regions, approaching ideal lowpass filters. The scaling parameters can be selected to scale the kernel´s passband edge or stopband edge. It is shown that the BUD and the GED satisfy all the desirable properties of the ED, and optimum design equations for the BUD kernel parameters are derived. An optional order parameter quantization is discussed, and examples that demonstrate the superior nature of the GED and the BUD over the ED in removing cross-terms while retaining desirable auto-terms are given