Editing Time–frequency representation (section)

== Formulation of TFRs and TFDs ==

One form of TFR (or TFD) can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such representations and formulations are known as [[quadratic function|quadratic]] or "bilinear" TFRs or TFDs (QTFRs or QTFDs) because the representation is quadratic in the signal (see [[Bilinear time–frequency distribution]]). This formulation was first described by [[Eugene Wigner]] in 1932 in the context of [[quantum mechanics]] and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the [[Wigner–Ville distribution]], as it was shown in <ref>B. Boashash, "Note on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech. and Signal Processing, vol. 36, issue 9, pp 1518–1521, Sept. 1988. {{doi|10.1109/29.90380}}</ref> that Wigner's formula needed to use the [[analytic signal]] defined in Ville's paper to be useful as a representation and for a practical analysis. Today, QTFRs include the [[spectrogram]] (squared magnitude of [[short-time Fourier transform]]), the [[scaleogram]] (squared magnitude of Wavelet transform) and the smoothed pseudo-Wigner distribution. 

Although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms, also called "interferences". The cross-terms caused by the bilinear structure of TFDs and TFRs may be useful in some applications such as classification as the cross-terms provide extra detail for the recognition algorithm. However, in some other applications, these cross-terms may plague certain quadratic TFRs and they would need to be reduced. One way to do this is obtained by comparing the signal with a different function. Such resulting representations are known as linear TFRs because the representation is linear in the signal. An example of such a representation is the ''windowed Fourier transform'' (also known as the [[short-time Fourier transform]]) which localises the signal by modulating it with a [[window function]], before performing the Fourier transform to obtain the frequency content of the signal in the region of the window.