Joint spatial-frequency distribution of Gaussian signals and prospects of its applications for description of optical systems

We consider a joint spatial-frequency distribution as a result of superposition of two known Wigner and Ville distributions. The joint distribution is characterized by a certain parameter t that plays a part of measure of superposition: the joint frequency-spatial distribution passes into the Ville one at t=0 and coincides with the Wigner distribution at t=1 . For the values 0< t<1 there exists a row of ‘mixed’ distributions, which represent superposition of the Wigner and Ville distributions. Explicit expression for the joint distribution of Gaussian signal is obtained by means of analytical calculations. We present graphic illustrations for the shape of joint distribution of Gaussian signal at different values of the joint parameter t. It is demonstrated that the Wigner distribution is formed as a rotation of Ville distribution on the information diagram by the angle proportional to the joint parameter t .