Logarithmic Uncertainty Relations for Odd or Even Signals Associate with Wigner–Ville Distribution

Heisenberg’s uncertainty relation is a basic principle in the applied mathematics and signal processing community. The logarithmic uncertainty relation, which is a more general form of Heisenberg’s uncertainty relation, describes the relationship between a function and its Fourier transform. In this paper, we consider several logarithmic uncertainty relations for a odd or even signal f(t) related to the Wigner–Ville distribution and the linear canonical transform. First, the logarithmic uncertainty relations associated with the Wigner–Ville distribution of a signal f(t) based on the Fourier transform are obtained. We then generalize the logarithmic uncertainty relation to the linear canonical transform domain and derive a number of theorems relating to the Wigner–Ville distribution and the ambiguity function; finally, the logarithmic uncertainty relations are obtained for the Wigner–Ville distribution associated with the linear canonical transform. We present an example in which the theorems derived in this paper can be used to provide an estimation for a practical signal.