On alias-free formulations of discrete-time Cohen's class of distributions

The transition of the Cohen's (1989) class of distributions from the continuous-time case to the discrete-time case is not straightforward because of aliasing problems. We classify the aliasing problems, which occur for joint time-frequency representations (TFRs), into two categories: type-I and type-II aliasings. Type-I aliasing can be avoided by properly defined discrete-time versions of some members of Cohen's class (in particular, properly defined kernels), whereas type-II aliasing can be reduced and/or eliminated by increasing the sampling rate. A type-I alias-free formulation of the discrete-time Cohen's class (AF-DTCC), which is equivalent to the AF-GDTFT of Joeng and Williams (see ibid., vol.40, no.2, p.1084, 1992) is then introduced based on the fact that the Cohen's class can be expressed as the 2-D Fourier transform of the generalized ambiguity function (AF). Based on this definition, two discretization schemes for kernel functions are presented in both the AF domain and the time-lag domain, and are shown to be equivalent under certain conditions. We also do the following: (1) we show that a discrete-time Wigner-Ville distribution (DWVD) and discrete-time spectrogram (DSPG) are type-I alias-free and members of AF-DTCC; (2) we use all the available correlation information from a given data sequence by using the Woodward AF instead of the Sussman AF; (3) we give kernel constraints in the AF domain for various distribution properties; and (4) we provide a type-I and type-II alias-free formulation for those distributions whose kernel functions satisfy the finite frequency-support constraint.