Bilinear time–frequency distribution

Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time;[1] this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems.

A mixed approach is required in time–frequency analysis techniques which are especially effective in analyzing non-stationary signals, whose frequency distribution and magnitude vary with time.

This distribution function is mathematically similar to a generalized time–frequency representation which utilizes bilinear transformations.

Compared with other time–frequency analysis techniques, such as short-time Fourier transform (STFT), the bilinear-transformation (or quadratic time–frequency distributions) may not have higher clarity for most practical signals, but it provides an alternative framework to investigate new definitions and new methods.

While it does suffer from an inherent cross-term contamination when analyzing multi-component signals, by using a carefully chosen window function(s), the interference can be significantly mitigated, at the expense of resolution.

The Wigner–Ville distribution is a quadratic form that measures a local time-frequency energy given by: The Wigner–Ville distribution remains real as it is the fourier transform of f(u + τ/2)·f*(u − τ/2), which has Hermitian symmetry in τ.

It can also be written as a frequency integration by applying the Parseval formula: Let

The interference term is a real function that creates non-zero values at unexpected locations (close to the origin) in the

Interference terms present in a real signal can be avoided by computing the analytic part

The interference terms are oscillatory since the marginal integrals vanish and can be partially removed by smoothing

Since the interferences take negative values, one can guarantee that all interferences are removed by imposing that The spectrogram and scalogram are examples of positive time-frequency energy distributions.

The resulting time-frequency energy density is From the Moyal formula, which is the time frequency averaging of a Wigner–Ville distribution.

The smoothing kernel thus can be written as The loss of time-frequency resolution depends on the spread of the distribution

with a sufficiently wide Gaussian defines positive energy density.

The general class of time-frequency distributions obtained by convolving

with an arbitrary kernel θ is called a Cohen's class, discussed below.

There is no positive quadratic energy distribution Pf that satisfies the following time and frequency marginal integrals: The definition of Cohen's class of bilinear (or quadratic) time–frequency distributions is as follows: where

The relationship between the two kernels is the same as the one between the WD and the AF, namely two successive Fourier transforms (cf.

i.e. or equivalently The class of bilinear (or quadratic) time–frequency distributions can be most easily understood in terms of the ambiguity function, an explanation of which follows.

, these relations can be generalized using a time-dependent power spectral density or equivalently the famous Wigner distribution function of

For multi-component signals in general, the distribution of its auto-term and cross-term within its Wigner distribution function is generally not predictable, and hence the cross-term cannot be removed easily.

However, as shown in the figure, for the ambiguity function, the auto-term of the multi-component signal will inherently tend to close the origin in the ητ-plane, and the cross-term will tend to be away from the origin.

With this property, the cross-term in can be filtered out effortlessly if a proper low-pass kernel function is applied in ητ-domain.

The kernel of Choi–Williams distribution is defined as follows: where α is an adjustable parameter.

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.

by For locally stationary processes, the eigenvectors of K are well approximated by the Wigner–Ville spectrum.

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

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.