Design of High-Resolution Quadratic TFDs with Separable Kernels

Abstract

A separable kernel gives separate control of the frequency-smoothing and time- smoothing of the WVD: the lag-dependent factor causes a convolution in the fre- quency direction in the (t, f) plane, while the Doppler-dependent factor causes a convolution in the time direction. A Doppler-independent (DI) kernel smoothes the WVD in the frequency direction only, reducing the inner artifacts and preserving the time marginal. A lag-independent (LI) kernel smoothes the WVD in the time direction only, reducing the cross-terms and preserving the frequency marginal. For an LI kernel, slower variations in component frequencies allow easier sup- pression of cross-terms and higher frequency resolution in the TFD; such kernels should therefore be considered for representing multicomponent signals with slowly- varying instantaneous frequencies. For a multicomponent signal with at least one highly nonlinear IF law, a lag-dependent factor is also needed to suppress the inner artifacts. The separable-kernel approach allows a complete understanding and appraisal of the properties and behavior of the smoothed WVD [2], the B-distribution [5] and the modified B-distribution [3,4]. It also enables the construction of high-resolution quadratic TFDs using the classical smoothing functions commonly encountered in digital signal processing and spectral analysis.