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.