Design of time-frequency representations using a multiform, tiltable exponential kernel

A novel Cohen´s (1981) class time-frequency representation with a tiltable, generalized exponential kernel capable of attaining a wide diversity of shapes in the ambiguity function plane is proposed for improving the time-frequency analysis of multicomponent signals. The first advantage of the proposed kernel is its ability to generate a wider variety of passband shapes, e.g., rotated ellipses, generalized hyperbolas, diamonds, rectangles, parallel strips at arbitrary angles, crosses, snowflakes, etc., and narrower transition regions than conventional Cohen´s class kernels; this versatility enables the new kernel to suppress undesirable cross terms in a broader variety of time-frequency scenarios. The second advantage of the new kernel is that closed form design equations can now be easily derived to select kernel parameters that meet or exceed a given set of user specified passband and stopband design criteria in the ambiguity function plane. Thirdly, it is shown that simple constraints on the parameters of the new kernel can be used to guarantee many desirable properties of time-frequency representations. The well known Choi-Williams (1989) exponential kernel, the generalized exponential kernel, and Nuttall´s (1990) tilted Gaussian kernel are special cases of the proposed kernel