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

signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...

(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. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

Time–frequency analysis In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains ''simultaneously,'' using various time–frequency representations. Rather than viewing a 1-dimensional signal (a ...

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 and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...

over time and frequency), and the

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

of the field represents phase.

Background and motivation

, as a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

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

magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...

of the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

(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 Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. S ...

). This formulation was first described by

Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; 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 con ...

in 1932 in the context of

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the

Wigner–Ville distribution The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quan ...

, 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 short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divid ...

), the

scaleogram 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 represe ...

(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

) 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

, 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. 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 In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as k \to \infty . This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin. It is closel ...

Linear canonical transformation

Linear canonical transformation In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the ac ...

s are the

linear transform In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

s of the time–frequency representation that preserve the

symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument s ...

. 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'' n ...

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

References

External links

DiscreteTFDs — software for computing time–frequency distributions

TFTB — Time–Frequency ToolBox

Time stretched short time Fourier transform for time-frequency analysis of ultra wideband signals
{{DEFAULTSORT:Time-frequency representation

representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...