Relationships Between $\Psi _{ {\tt B}}$-Energy Operator and Some Time-Frequency Representations

Abstract

Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator is an energy operator that measures the interactions between two complex signals. In this letter, new properties of Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator are presented. Connections between Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator and some time-frequency representations (cross-ambiguity function, short-time Fourier transform, Zak transform, and Gabor coefficients) are established. Link between Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator of two input signals and their cross-spectrum is also derived. For two equal input signals, we find that Fourier transform of Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator is proportional to the second derivative of the ambiguity function. The established links show the ability of Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator to analyze nonstationary signals. A numerical example is provided for illustrating how to estimate the second order moment, of a FM signal, using Ψ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> operator. We compare the result to the moment given by the Wigner Ville distribution.