The Fourier transform of formula_61 is  The following proposition gives necessary and sufficient conditions to ensure that formula_63 satisfies marginal energy properties like those of the Wigner–Ville distribution. Aforementioned, the Wigner distribution function is a member of the class of quadratic time-frequency distributions (QTFDs) with the kernel function formula_68. The definition of Wigner distribution is as follows:  We can design time-frequency energy distributions that satisfy the scaling property  as does the Wigner–Ville distribution. If   then  This is equivalent to imposing that  and hence  The Rihaczek and Choi–Williams distributions are examples of affine invariant Cohen's class distributions. The kernel of Choi–Williams distribution is defined as follows:  where "α" is an adjustable parameter. The kernel of Rihaczek distribution is defined as follows:  With this particular kernel a simple calculation proves that  The kernel of cone-shape distribution function is defined as follows:  where "α" is an adjustable parameter. See Transformation between distributions in time-frequency analysis. More such QTFDs and a full list can be found in, e.g., Cohen's text cited. A time-varying spectrum for non-stationary processes is defined from the expected Wigner–Ville distribution. Locally stationary processes appear in many physical systems where random fluctuations are produced by a mechanism that changes slowly in time. Such processes can be approximated locally by a stationary process. Let formula_79 be a real valued zero-mean process with covariance  The covariance operator "K" is defined for any deterministic signal formula_66 by  For locally stationary processes, the eigenvectors of "K" are well approximated by the Wigner–Ville spectrum. The properties of the covariance formula_83 are studied as a function of formula_84 and formula_85:  The process is "wide-sense stationary" if the covariance depends only on formula_84:  The eigenvectors are the complex exponentials formula_89 and the corresponding eigenvalues are given by the power spectrum  For non-stationary processes, Martin and Flandrin have introduced a "time-varying spectrum"  To avoid convergence issues we suppose that "X" has compact support so that formula_92 has compact support in formula_93. From above we can write  which proves that the time varying spectrum is the expected value of the Wigner–Ville transform of the process "X".