A time–frequency representation (TFR) is a view of a

signal A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...

(taken to be a function of time) represented over both time and

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved by using a formulation often called "Time–Frequency Distribution", abbreviated as TFD. TFRs are often complex-valued fields over time and frequency, where the modulus of the field represents either amplitude or "energy density" (the concentration of the

root mean square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...

over time and frequency), and the

argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...

of the field represents phase.

Background and motivation

, as a function of time, may be considered as a representation with perfect ''time resolution''. In contrast, the magnitude of the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

(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 representation and analysis of signals containing multiple time-varying frequencies.

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 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 Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...

in 1932 in the context of

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

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 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 A spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time. When applied to an audio signal, spectrograms are sometimes called sonographs, voiceprints, or voicegrams. When the data are represen ...

(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.

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. Continuous wavelet transform analysis is very useful for identifying non-stationary signals in

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...

, such as those related to climate or landslides. The notions of time, frequency, and amplitude used to generate a TFR from a wavelet transform were originally developed intuitively. In 1992, a quantitative derivation of these relationships was published, 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

fractional Fourier transform In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n' ...

, and others, thus providing a unified view of these transforms in terms of their action on the time–frequency domain.

References

External links