Time-frequency representation

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A time-frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time-frequency analysis means analysis of a TFR.

A signal, as a function of time, may be considered as a representation with perfect temporal resolution. The magnitude of the Fourier transform (FT) of the signal may be considered as a representation with perfect spectral resolution but with no temporal information (although the magnitude of the FT conveys frequency content, it fails to convey where in time different events occur in the signal). TFRs provide some temporal information and some spectral information, simultaneously. Thus, TFRs are used for the analysis of signals containing multiple time-varying frequencies.

One form of TFR can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such formulations are known as quadratic TFRs because the representation is quadratic in the signal. 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. Unfortunately, although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms whenever multiple frequencies are superimposed. This was partly addressed by the development of the Choi-Williams distribution in 1989 but most recent applications of TFRs have turned to linear methods.

The cross-terms which plague quadratic TFRs may be evaded by comparing the signal with a different function. Such representations are known as linear TFRs because the representation is linear in the signal. The windowed Fourier transform (also known as the short-time Fourier transform) 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. Wavelet transforms, in particular the continuous wavelet transform, expand the signal in terms of wavelet functions which are localised in both time and frequency. Thus the wavelet transform of a signal may be represented in terms of both time and frequency. Before 1991, the notions of time, frequency and amplitude used to generate a TFR from a wavelet transform were derived intuitively. In 1991, Nathalie Delprat gave the first quantitative derivation of these relationships, based upon a stationary phase approximation.

TFRs are often complex-valued fields over time and frequency, where the modulus of the field represents "energy density" (the concentration of the root mean square over time and frequency) or amplitude, and the argument of the field represents phase.

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