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signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various
time–frequency representation 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 the time–frequency domain provided by a TFR. This is achieved ...
s. Rather than viewing a 1-dimensional signal (a function, real or complex-valued, whose domain is the real line) and some transform (another function whose domain is the real line, obtained from the original via some transform), time–frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time–frequency transform. The mathematical motivation for this study is that functions and their transform representation are tightly connected, and they can be understood better by studying them jointly, as a two-dimensional object, rather than separately. A simple example is that the 4-fold periodicity 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 ...
– and the fact that two-fold Fourier transform reverses direction – can be interpreted by considering the Fourier transform as a 90° rotation in the associated time–frequency plane: 4 such rotations yield the identity, and 2 such rotations simply reverse direction ( reflection through the origin). The practical motivation for time–frequency analysis is that classical
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
assumes that signals are infinite in time or periodic, while many signals in practice are of short duration, and change substantially over their duration. For example, traditional musical instruments do not produce infinite duration sinusoids, but instead begin with an attack, then gradually decay. This is poorly represented by traditional methods, which motivates time–frequency analysis. One of the most basic forms of time–frequency analysis is the
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 divide ...
(STFT), but more sophisticated techniques have been developed, notably
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s and
least-squares spectral analysis Least-squares spectral analysis (LSSA) is a method of estimating a Spectral density estimation#Overview, frequency spectrum based on a least-squares fit of Sine wave, sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the ...
methods for unevenly spaced data.


Motivation

In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, time–frequency analysis is a body of techniques and methods used for characterizing and manipulating signals whose statistics vary in time, such as
transient Transience or transient may refer to: Music * ''Transient'' (album), a 2004 album by Gaelle * ''Transience'' (Steven Wilson album), 2015 * Transience (Wreckless Eric album) Science and engineering * Transient state, when a process variable or ...
signals. It is a generalization and refinement of
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
, for the case when the signal frequency characteristics are varying with time. Since many signals of interest – such as speech, music, images, and medical signals – have changing frequency characteristics, time–frequency analysis has broad scope of applications. Whereas the technique 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 ...
can be extended to obtain the frequency spectrum of any slowly growing
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
signal, this approach requires a complete description of the signal's behavior over all time. Indeed, one can think of points in the (spectral) frequency domain as smearing together information from across the entire time domain. While mathematically elegant, such a technique is not appropriate for analyzing a signal with indeterminate future behavior. For instance, one must presuppose some degree of indeterminate future behavior in any telecommunications systems to achieve non-zero entropy (if one already knows what the other person will say one cannot learn anything). To harness the power of a frequency representation without the need of a complete characterization in the time domain, one first obtains a time–frequency distribution of the signal, which represents the signal in both the time and frequency domains simultaneously. In such a representation the frequency domain will only reflect the behavior of a temporally localized version of the signal. This enables one to talk sensibly about signals whose component frequencies vary in time. For instance rather than using tempered distributions to globally transform the following function into the frequency domain one could instead use these methods to describe it as a signal with a time varying frequency. : x(t)=\begin \cos( \pi t); & t <10 \\ \cos(3 \pi t); & 10 \le t < 20 \\ \cos(2 \pi t); & t > 20 \end Once such a representation has been generated other techniques in time–frequency analysis may then be applied to the signal in order to extract information from the signal, to separate the signal from noise or interfering signals, etc.


Time–frequency distribution functions


Formulations

There are several different ways to formulate a valid time–frequency distribution function, resulting in several well-known time–frequency distributions, such as: *
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 divide ...
(including the
Gabor transform The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the Sine wave, sinusoidal frequency and phase (waves), phase content of local sections of a signal as it changes over time ...
), * Wavelet transform, *
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. ...
function (
Wigner distribution function The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, ...
, or WDF), * Modified Wigner distribution function, Gabor–Wigner distribution function, and so on (see Gabor–Wigner transform). * Hilbert–Huang transform More information about the history and the motivation of development of time–frequency distribution can be found in the entry
Time–frequency representation 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 the time–frequency domain provided by a TFR. This is achieved ...
.


Ideal TF distribution function

A time–frequency distribution function ideally has the following properties: #High resolution in both time and frequency, to make it easier to be analyzed and interpreted. #No cross-term to avoid confusing real components from artifacts or noise. #A list of desirable mathematical properties to ensure such methods benefit real-life application. #Lower computational complexity to ensure the time needed to represent and process a signal on a time–frequency plane allows real-time implementations. Below is a brief comparison of some selected time–frequency distribution functions. To analyze the signals well, choosing an appropriate time–frequency distribution function is important. Which time–frequency distribution function should be used depends on the application being considered, as shown by reviewing a list of applications. The high clarity of the Wigner distribution function (WDF) obtained for some signals is due to the auto-correlation function inherent in its formulation; however, the latter also causes the cross-term problem. Therefore, if we want to analyze a single-term signal, using the WDF may be the best approach; if the signal is composed of multiple components, some other methods like the Gabor transform, Gabor-Wigner distribution or Modified B-Distribution functions may be better choices. As an illustration, magnitudes from non-localized Fourier analysis cannot distinguish the signals: : x_1 (t)=\begin \cos( \pi t); & t <10 \\ \cos(3 \pi t); & 10 \le t < 20 \\ \cos(2 \pi t); & t > 20 \end : x_2 (t)=\begin \cos( \pi t); & t <10 \\ \cos(2 \pi t); & 10 \le t < 20 \\ \cos(3 \pi t); & t > 20 \end But time–frequency analysis can.


TF analysis and random processes

For a random process x(t), we cannot find the explicit value of x(t). The value of x(t) is expressed as a probability function.


General random processes

* Auto-covariance function (ACF) R_x(t,\tau) :R_x(t,\tau) = E (t+\tau/2)x^*(t-\tau/2)/math> :In usual, we suppose that E (t)= 0 for any t, :E (t+\tau/2)x^*(t-\tau/2)/math> :=\iint x(t+\tau/2,\xi_1)x^*(t-\tau/2,\xi_2)P(\xi_1,\xi_2)d\xi_1d\xi_2 :(alternative definition of the auto-covariance function) :\overset(t,\tau)=E (t)x(t+\tau)/math> * Power spectral density (PSD) S_x(t,f) :S_x(t,f) = \int_^ R_x(t,\tau)e^d\tau * Relation between the WDF (Wigner Distribution Function) and the PSD :E _x(t,f)= \int_^ E (t+\tau/2)x^*(t-\tau/2)cdot e^\cdot d\tau :::::= \int_^ R_x(t,\tau)\cdot e^\cdot d\tau= S_x(t,f) * Relation between the ambiguity function and the ACF :E _X(\eta,\tau)= \int_^ E (t+\tau/2)x^*(t-\tau/2)^dt :::::= \int_^ R_x(t,\tau)e^dt


Stationary random processes

* Stationary random process: the statistical properties do not change with t. Its auto-covariance function: R_x(t_1,\tau) = R_x(t_2,\tau) = R_x(\tau) for any t, Therefore, R_x(\tau) = E (\tau/2)x^*(-\tau/2)/math> =\iint x(\tau/2,\xi_1)x^*(-\tau/2,\xi_2)P(\xi_1,\xi_2)d\xi_1d\xi_2PSD, S_x(f) = \int_^ R_x(\tau)e^d\tau White noise: S_x(f) = \sigma , where \sigma is some constant. * When x(t) is stationary, E _x(t,f)= S_x(f) , (invariant with t) E _x(\eta,\tau)= \int_^ R_x(\tau)\cdot e^\cdot dt = R_x(\tau)\int_^ e^\cdot dt= R_x(\tau)\delta(\eta) , (nonzero only when \eta = 0)


Additive white noise

* For additive white noise (AWN), :E _g(t,f)= \sigma :E _x(\eta,\tau)= \sigma\delta(\tau)\delta(\eta) * Filter Design for a signal in additive white noise E_x: energy of the signal A : area of the time frequency distribution of the signal The PSD of the white noise is S_n(f) = \sigma SNR \approx 10\log_\frac SNR \approx 10\log_\frac


Non-stationary random processes

* If E _x(t,f)/math> varies with t and E _x(\eta,\tau)/math> is nonzero when \eta = 0, then x(t) is a non-stationary random process. * If *# h(t) = x_1(t)+x_2(t)+x_3(t)+......+x_k(t) *# x_n(t)'s have zero mean for all t's *# x_n(t)'s are mutually independent for all t's and \tau's :then: ::E _m(t+\tau/2)x_n^*(t-\tau/2)= E _m(t+\tau/2) _n^*(t-\tau/2)= 0 * if m \neq n, then ::E _h(t,f)= \sum_^k E _(t,f)/math> ::E _h(\eta,\tau)= \sum_^k E _(\eta,\tau)/math>


Short-time Fourier transform

* Random process for STFT (Short Time Fourier Transform) E (t)neq 0 should be satisfied. Otherwise, E (t,f)= E int_^ x(\tau)w(t-\tau)e^d\tau/math> =\int_^ E (\tau)(t-\tau)e^d\taufor zero-mean random process, E (t,f)= 0 * Decompose by the AF and the FRFT. Any non-stationary random process can be expressed as a summation of the fractional Fourier transform (or chirp multiplication) of stationary random process.


Applications

The following applications need not only the time–frequency distribution functions but also some operations to the signal. The
Linear canonical transform 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 ...
(LCT) is really helpful. By LCTs, the shape and location on the time–frequency plane of a signal can be in the arbitrary form that we want it to be. For example, the LCTs can shift the time–frequency distribution to any location, dilate it in the horizontal and vertical direction without changing its area on the plane, shear (or twist) it, and rotate it (
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' ...
). This powerful operation, LCT, make it more flexible to analyze and apply the time–frequency distributions. The time-frequency analysis have been applied in various applications like, disease detection from biomedical signals and images, vital sign extraction from physiological signals, brain-computer interface from brain signals, machinery fault diagnosis from vibration signals, interference mitigation in spread spectrum communication systems.


Instantaneous frequency estimation

The definition of
instantaneous frequency Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''compl ...
is the time rate of change of phase, or : \frac \frac \phi (t), where \phi (t) is the
instantaneous phase Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''compl ...
of a signal. We can know the instantaneous frequency from the time–frequency plane directly if the image is clear enough. Because the high clarity is critical, we often use WDF to analyze it.


TF filtering and signal decomposition

The goal of filter design is to remove the undesired component of a signal. Conventionally, we can just filter in the time domain or in the frequency domain individually as shown below. The filtering methods mentioned above can’t work well for every signal which may overlap in the time domain or in the frequency domain. By using the time–frequency distribution function, we can filter in the Euclidean time–frequency domain or in the fractional domain by employing 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' ...
. An example is shown below. Filter design in time–frequency analysis always deals with signals composed of multiple components, so one cannot use WDF due to cross-term. The Gabor transform, Gabor–Wigner distribution function, or Cohen's class distribution function may be better choices. The concept of signal decomposition relates to the need to separate one component from the others in a signal; this can be achieved through a filtering operation which require a filter design stage. Such filtering is traditionally done in the time domain or in the frequency domain; however, this may not be possible in the case of non-stationary signals that are multicomponent as such components could overlap in both the time domain and also in the frequency domain; as a consequence, the only possible way to achieve component separation and therefore a signal decomposition is to implement a time–frequency filter.


Sampling theory

By the
Nyquist–Shannon sampling theorem The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample r ...
, we can conclude that the minimum number of sampling points without
aliasing In signal processing and related disciplines, aliasing is a phenomenon that a reconstructed signal from samples of the original signal contains low frequency components that are not present in the original one. This is caused when, in the ori ...
is equivalent to the area of the time–frequency distribution of a signal. (This is actually just an approximation, because the TF area of any signal is infinite.) Below is an example before and after we combine the sampling theory with the time–frequency distribution: It is noticeable that the number of sampling points decreases after we apply the time–frequency distribution. When we use the WDF, there might be the cross-term problem (also called interference). On the other hand, using
Gabor transform The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the Sine wave, sinusoidal frequency and phase (waves), phase content of local sections of a signal as it changes over time ...
causes an improvement in the clarity and readability of the representation, therefore improving its interpretation and application to practical problems. Consequently, when the signal we tend to sample is composed of single component, we use the WDF; however, if the signal consists of more than one component, using the Gabor transform, Gabor-Wigner distribution function, or other reduced interference TFDs may achieve better results. The Balian–Low theorem formalizes this, and provides a bound on the minimum number of time–frequency samples needed.


Modulation and multiplexing

Conventionally, the operation of
modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...
and
multiplexing In telecommunications and computer networking, multiplexing (sometimes contracted to muxing) is a method by which multiple analog or digital signals are combined into one signal over a shared medium. The aim is to share a scarce resource� ...
concentrates in time or in frequency, separately. By taking advantage of the time–frequency distribution, we can make it more efficient to modulate and multiplex. All we have to do is to fill up the time–frequency plane. We present an example as below.
As illustrated in the upper example, using the WDF is not smart since the serious cross-term problem make it difficult to multiplex and modulate.


Electromagnetic wave propagation

We can represent an electromagnetic wave in the form of a 2 by 1 matrix : \begin x \\ y \end, which is similar to the time–frequency plane. When electromagnetic wave propagates through free-space, the
Fresnel diffraction In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff's diffraction formula, Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near and far field, near fi ...
occurs. We can operate with the 2 by 1 matrix : \begin x \\ y \end by LCT with parameter matrix : \begin a & b \\ c & d \end = \begin 1 & \lambda z \\ 0 & 1 \end, where ''z'' is the propagation distance and \lambda is the wavelength. When electromagnetic wave pass through a spherical lens or be reflected by a disk, the parameter matrix should be : \begin a & b \\ c & d \end = \begin 1 & 0 \\ -\frac & 1 \end and : \begin a & b \\ c & d \end = \begin 1 & 0 \\ \frac & 1 \end respectively, where ƒ is the focal length of the lens and ''R'' is the radius of the disk. These corresponding results can be obtained from : \begin a & b \\ c & d \end \begin x \\ y \end.


Optics, acoustics, and biomedicine

Light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
is an electromagnetic wave, so time–frequency analysis applies to optics in the same way as for general electromagnetic wave propagation. Similarly, it is a characteristic of acoustic signals, that their frequency components undergo abrupt variations in time and would hence be not well represented by a single frequency component analysis covering their entire durations. As acoustic signals are used as speech in communication between the human-sender and -receiver, their undelayedly transmission in technical communication systems is crucial, which makes the use of simpler TFDs, such as the Gabor transform, suitable to analyze these signals in real-time by reducing computational complexity. If frequency analysis speed is not a limitation, a detailed feature comparison with well defined criteria should be made before selecting a particular TFD. Another approach is to define a signal dependent TFD that is adapted to the data. In biomedicine, one can use time–frequency distribution to analyze the
electromyography Electromyography (EMG) is a technique for evaluating and recording the electrical activity produced by skeletal muscles. EMG is performed using an instrument called an electromyograph to produce a record called an electromyogram. An electromyo ...
(EMG),
electroencephalography Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignal, bio signals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in ...
(EEG), electrocardiogram (ECG) or
otoacoustic emissions An otoacoustic emission (OAE) is a sound that is generated from within the inner ear. Having been predicted by Austrian astrophysicist Thomas Gold in 1948, its existence was first demonstrated experimentally by British physicist David Kemp in 19 ...
(OAEs).


History

Early work in time–frequency analysis can be seen in the
Haar wavelet In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repr ...
s (1909) of
Alfréd Haar Alfréd Haar (; 11 October 1885, Budapest – 16 March 1933, Szeged) was a Kingdom of Hungary, Hungarian mathematician. In 1904 he began to study at the University of Göttingen. His doctorate was supervised by David Hilbert. The Haar me ...
, though these were not significantly applied to signal processing. More substantial work was undertaken by
Dennis Gabor Dennis Gabor ( ; ; 5 June 1900 – 9 February 1979) was a Hungarian-British physicist who received the Nobel Prize in Physics in 1971 for his invention of holography. He obtained British citizenship in 1946 and spent most of his life in Engla ...
, such as
Gabor atom In applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function. Overview In 1946, Denn ...
s (1947), an early form of
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s, and the
Gabor transform The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the Sine wave, sinusoidal frequency and phase (waves), phase content of local sections of a signal as it changes over time ...
, a modified
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 divide ...
. The Wigner–Ville distribution (Ville 1948, in a signal processing context) was another foundational step. Particularly in the 1930s and 1940s, early time–frequency analysis developed in concert with
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 ...
(Wigner developed the Wigner–Ville distribution in 1932 in quantum mechanics, and Gabor was influenced by quantum mechanics – see
Gabor atom In applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function. Overview In 1946, Denn ...
); this is reflected in the shared mathematics of the position-momentum plane and the time–frequency plane – as in the
Heisenberg uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
(quantum mechanics) and the
Gabor limit The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
(time–frequency analysis), ultimately both reflecting a symplectic structure. An early practical motivation for time–frequency analysis was the development of radar – see ambiguity function.


See also

* Motions in the time-frequency distribution *
Multiresolution analysis A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was int ...
*
Spectral density estimation In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal ...
* Time–frequency analysis for music signals *
Wavelet analysis A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...


References

{{DEFAULTSORT:Time-Frequency Analysis Signal processing