Wigner distribution function (WDF) obtained for some signals is due to the auto-correlation function inherent in its formulation; however, the latter also causes the cross-term problem. Therefore, if we want to analyze a single-term signal, using the WDF may be the best approach; if the signal is composed of multiple components, some other methods like the Gabor transform, Gabor-Wigner distribution or Modified B-Distribution functions may be better choices. As an illustration, magnitudes from non-localized Fourier analysis cannot distinguish the signals: But time–frequency analysis can. TF analysis and Random Process For a random process x(t), we cannot find the explicit value of x(t). The value of x(t) is expressed as a probability function. Auto-covariance function In usual, we suppose that for any t, (alternative definition of the auto-covariance function) Power spectral density (PSD) Relation between the WDF (Wigner Distribution Function) and the random process Relation between the ambiguity function and the random process Stationary random process: the statistical properties do not change with t. Its auto-covariance function: for any , Therefore, PSD, White noise: , where is some constant. When x(t) is stationary, , (invariant with ) , (nonzero only when ) For white noise, Filter Design for White noise : energy of the signal : area of the time frequency distribution of the signal The PSD of the white noise is If varies with and is nonzero when , then is a non-stationary random process. If 's have zero mean for all 's 's are mutually independent for all 's and 's if , then Random process for STFT (Short Time Fourier Transform) should be satisfied. Otherwise, for zero-mean random process, Decompose by the AF and the FRFT Any non-stationary random process can be expressed as a summation of the fractional Fourier transform (or chirp multiplication) of stationary random process. Applications The following applications need not only the time–frequency distribution functions but also some operations to the signal. The Linear canonical transform (LCT) is really helpful. By LCTs, the shape and location on the time–frequency plane of a signal can be in the arbitrary form that we want it to be. For example, the LCTs can shift the time–frequency distribution to any location, dilate it in the horizontal and vertical direction without changing its area on the plane, shear (or twist) it, and rotate it (Fractional Fourier transform). This powerful operation, LCT, make it more flexible to analyze and apply