A deductive construction of third-order time-frequency distributions

Abstract

Abstract In 1988, a third-order Wigner-Ville distribution was introduced by Gerr. This distribution was defined by analogy with the second-order Wigner-Ville representation. Later, a class of generalized higher-order time-frequency distributions was presented by Fonollosa and Nikias using the notion of local multicorrelation functions. In this paper we address the problem of deriving such generalized representations deductively, as done by Flandrin for the second-order case. For physical reasons, we show that this derivation must be done on a third-order inter-representation (i.e. a representation for three different signals). Thus, if we impose invariance under a time shift and under some frequency shifts to such a third-order time-frequency distribution, we can define an extension to the third order of the so-called Cohen's class. In this class, the distributions are parametrized by an arbitrary function, and a desired property on the representation is equivalent to a constraint on this function. The construction described below is concerned with deterministic signals. In order to work with random signals, we introduce a third-order time-frequency spectrum, namely, the Wigner-Ville bispectrum, which is shown to be valid for a certain class of random processes. We then propose as an application of it the study of the detection of quadratic phase coupling in a non-stationary context.