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 into a TFR.
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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Explanation
A signal, as a function of time, may be considered as a representation with perfect time resolution. In contrast, the magnitude of the Fourier transform (FT) of the signal may be considered as a representation with perfect spectral resolution but with no time information because the magnitude of the FT conveys frequency content but it fails to convey when, in time, different events occur in the signal.
TFRs provide a bridge between these two representations in that they provide some temporal information and some spectral information simultaneously. Thus, TFRs are useful for the analysis of signals containing multiple time-varying frequencies.
Formulation of TFRs
Quadratic forms
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 (QTFRs) 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. Today, various QTFRs include but not limited to spectrogram (squared magnitude of short-time Fourier transform), scaleogram (squared magnitude of Wavelet transform) and the smoothed pseudo-Wigner distribution. In fact, a whole class of representation known as the Cohen's class distribution function fall in this category. A detailed discussion on QTFRs with their property can be found in [1].
Although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms. The following can be used to estimate which QTFRs contain cross terms.
Given a QTFR E(t,f) defined on 
, define a constant 
and a set ![\mathcal{C} = \{(t,f)\in \mathbb{R}^2 : |E(t,f)|>T,\,\forall T\in[0,E_0]\}](http://wpcontent.answers.com/math/c/5/3/c539ec009ee0db491d3b75901b5b6ead.png)
. The QTFR, E(t,f) is cross-term free if 
is a convex set.
Linear forms
The cross-terms which plague certain 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
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[2] gave the first quantitative derivation of these relationships, based upon a stationary phase approximation.
Linear canonical transformation
Linear canonical transformations are the linear transforms of the time–frequency representation that preserve the symplectic form. These include and generalize the Fourier transform, fractional Fourier transform, and others, thus providing a unified view of these transforms in terms of their action on the time–frequency domain.
See also
References
- ^ Antonia Papandreou-Suppappola, Franz Hlawatsch, and G. Faye Boudreaux-Bartels, Quadratic Time-Frequency Representations with Scale Covariance and Generalized Time-Shift Covariance: A Unified Framework for the Affine, Hyperbolic, and Power Classes, DIGITAL SIGNAL PROCESSING 8, 3--48 (1998), ARTICLE NO. SP970303
- ^ http://www.limsi.fr/Individu/delprat/
External links
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