Editing Time–frequency analysis (section)

===TF filtering and signal decomposition===

The goal of filter design is to remove the undesired component of a signal. Conventionally, we can just filter in the time  domain or in the frequency domain individually as shown below.

[[Image:filter tf.jpg]]

The filtering methods mentioned above can’t work well for every signal which may overlap in the time domain or in the frequency domain. By using the time–frequency distribution function, we can filter in the Euclidean time–frequency domain or in the fractional domain by employing the [[fractional Fourier transform]]. An example is shown below.

[[Image:filter fractional.jpg]]

Filter design in time–frequency analysis always deals with signals composed of multiple components, so one cannot use WDF due to cross-term. The Gabor transform, Gabor–Wigner distribution function, or Cohen's class distribution function may be better choices.

The concept of signal decomposition relates to the need to separate one component from the others in a signal; this can be achieved through a filtering operation which require a filter design stage. Such filtering is traditionally done in the time domain or in the frequency domain; however, this may not be possible in the case of non-stationary signals that are multicomponent as such components could overlap in both the time domain and also in the frequency domain; as a consequence, the only possible way to achieve component separation and therefore a signal decomposition is to implement a time–frequency filter.