Time–frequency analysis: Difference between revisions

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{{Short description|Techniques and methods in signal processing}}

{{merge|Time–frequency representation|target=Time–frequency analysis, distribution, and representation|date=June 2019}}

{{See also|Time–frequency representation}}

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 representation]]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.<ref>L. Cohen, "Time–Frequency Analysis," ''Prentice-Hall'', New York, 1995. {{isbn|978-0135945322}}</ref><ref>E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009.</ref>

The mathematical motivation for this study is that functions and their transform representation are often 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]] – 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]] 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]] (STFT), but more sophisticated techniques have been developed, notably [[wavelet]]s and [[least-squares spectral analysis]] methods for unevenly spaced data.

==Motivation==

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[[File:X1(t).jpg|thumb]]

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.

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#'''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.<ref>{{Cite journal|lastlast1=Shafi|firstfirst1=Imran|last2=Ahmad|first2=Jamil|last3=Shah|first3=Syed Ismail|last4=Kashif|first4=F. M.|date=2009-06-09|title=Techniques to Obtain Good Resolution and Concentrated Time-Frequency Distributions: A Review|journal=EURASIP Journal on Advances in Signal Processing|language=en|volume=2009|issue=1|pages=673539|doi=10.1155/2009/673539|bibcode=2009EJASP2009..109S |issn=1687-6180|doi-access=free|hdl=1721.1/50243|hdl-access=free}}</ref>

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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.<ref>A. Papandreou-Suppappola, Applications in Time–Frequency Signal Processing (CRC Press, Boca Raton, Fla., 2002)</ref> 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:

: <math>x_1 (t)=\begin{cases}

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\cos(3 \pi t); & t > 20

\end{cases}</math>

[[File:X1-x2.jpg|thumb]]

But time–frequency analysis can.

== TF analysis and random processes<ref>{{Cite book |last=Ding |first=Jian-Jiun |title=Time frequency analysis and wavelet transform class notes |publisher=Graduate Institute of Communication Engineering, National Taiwan University (NTU) |year=2022 |___location=Taipei, Taiwan}}</ref> ==

==Applications==

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.

The following applications need not only the time–frequency distribution functions but also some operations to the signal. The [[Linear canonical transform]] (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]]). This powerful operation, LCT, make it more flexible to analyze and apply the time–frequency distributions.

=== General random processes ===

===Instantaneous frequency estimation===

* Auto-covariance function (ACF) <math>R_x(t,\tau)</math>

The definition of [[instantaneous frequency]] is the time rate of change of phase, or

:<math>R_x(t,\tau) = E[x(t+\tau/2)x^*(t-\tau/2)]</math>

:In usual, we suppose that <math>E[x(t)] = 0 </math> for any t,

: <math>E[x(t+\frac{1}{tau/2 \pi} \frac{d}{dt} \phi )x^*(t-\tau/2), ]</math>

:<math>=\iint x(t+\tau/2,\xi_1)x^*(t-\tau/2,\xi_2)P(\xi_1,\xi_2)d\xi_1d\xi_2</math>

:(alternative definition of the auto-covariance function)

:<math>\overset{\land}{R_x}(t,\tau)=E[x(t)x(t+\tau)]</math>

* Power spectral density (PSD) <math>S_x(t,f)</math>

:<math>S_x(t,f) = \int_{-\infty}^{\infty} R_x(t,\tau)e^{-j2\pi f\tau}d\tau</math>

* Relation between the [[Wigner distribution function|WDF (Wigner Distribution Function)]] and the PSD

:<math>E[W_x(t,f)] = \int_{-\infty}^{\infty} E[x(t+\tau/2)x^*(t-\tau/2)]\cdot e^{-j2\pi f\tau}\cdot d\tau</math>

:::::<math>= \int_{-\infty}^{\infty} R_x(t,\tau)\cdot e^{-j2\pi f\tau}\cdot d\tau</math><math>= S_x(t,f)</math>

* Relation between the [[ambiguity function]] and the ACF

:<math>E[A_X(\eta,\tau)] = \int_{-\infty}^{\infty} E[x(t+\tau/2)x^*(t-\tau/2)]e^{-j2\pi t\eta}dt</math>

:::::<math>= \int_{-\infty}^{\infty} R_x(t,\tau)e^{-j2\pi t\eta}dt</math>

=== Stationary random processes ===

where <math>\phi (t)</math> is the [[instantaneous phase]] 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.

* [[Stationary process|Stationary random process]]: the statistical properties do not change with t. Its auto-covariance function:

<math>R_x(t_1,\tau) = R_x(t_2,\tau) = R_x(\tau)</math> for any <math>t</math>, Therefore,

<math>R_x(\tau) = E[x(\tau/2)x^*(-\tau/2)]</math>

<math>=\iint x(\tau/2,\xi_1)x^*(-\tau/2,\xi_2)P(\xi_1,\xi_2)d\xi_1d\xi_2</math>PSD,

<math>S_x(f) = \int_{-\infty}^{\infty} R_x(\tau)e^{-j2\pi f\tau}d\tau</math> White noise:

<math>S_x(f) = \sigma</math> , where <math>\sigma</math> is some constant.[[File:Stationary random process's WDF and AF.jpg|right|frameless|440x440px]]

* When x(t) is stationary,

<math>E[W_x(t,f)] = S_x(f)</math> , (invariant with <math>t</math>)

<math>E[A_x(\eta,\tau)] = \int_{-\infty}^{\infty} R_x(\tau)\cdot e^{-j2\pi t\eta}\cdot dt</math>

===TF filtering and signal decomposition===

<math>= R_x(\tau)\int_{-\infty}^{\infty} e^{-j2\pi t\eta}\cdot dt</math><math>= R_x(\tau)\delta(\eta)</math> , (nonzero only when <math>\eta = 0</math>)

=== Additive white noise ===

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.

* For additive white noise (AWN),

:<math>E[W_g(t,f)] = \sigma</math>

:<math>E[A_x(\eta,\tau)] = \sigma\delta(\tau)\delta(\eta)</math>

* Filter Design for a signal in additive white noise

[[Image:filter tf.jpg]]

[[File:Filter design for white noise.jpg|left|thumb|440x440px]]

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]]. An example is shown below.

[[Image:filter fractional.jpg]]

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.

<math>E_x</math>: energy of the signal

===Sampling theory===

<math>A</math> : area of the time frequency distribution of the signal

By the [[Nyquist–Shannon sampling theorem]], we can conclude that the minimum number of sampling points without [[aliasing]] 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:

The PSD of the white noise is <math>S_n(f) = \sigma</math>

[[Image:sampling.jpg]]

It is noticeable that the number of sampling points decreases after we apply the time–frequency distribution.

<math>SNR \approx 10\log_{10}\frac{E_x}{\iint\limits_{(t,f)\in\text{signal part}} S_x(t,f)dtdf}</math>

When we use the WDF, there might be the cross-term problem (also called interference). On the other hand, using [[Gabor transform]] causes an improvement in the clarity and readability of the representation, therefore improving its interpretation and application to practical problems.

<math>SNR \approx 10\log_{10}\frac{E_x}{\sigma\Alpha}</math>

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.

=== Non-stationary random processes ===

The [[Balian–Low theorem]] formalizes this, and provides a bound on the minimum number of time–frequency samples needed.

* If <math>E[W_x(t,f)]</math> varies with <math>t</math> and <math>E[A_x(\eta,\tau)]</math> is nonzero when <math>\eta = 0</math>, then <math>x(t)</math> is a non-stationary random process.

* If

*# <math>h(t) = x_1(t)+x_2(t)+x_3(t)+......+x_k(t)</math>

*# <math>x_n(t)</math>'s have zero mean for all <math>t</math>'s

*# <math>x_n(t)</math>'s are mutually independent for all <math>t</math>'s and <math>\tau</math>'s

:then:

::<math>E[x_m(t+\tau/2)x_n^*(t-\tau/2)] = E[x_m(t+\tau/2)]E[x_n^*(t-\tau/2)] = 0</math>

* if <math>m \neq n</math>, then

'''Methods'''

::<math>E[W_h(t,f)] = \sum_{n=1}^k E[W_{x_n}(t,f)]</math>

Number of sampling points = sum of areas of time frequency distribution + the number of extra parameters

::<math>E[A_h(\eta,\tau)] = \sum_{n=1}^k E[A_{x_n}(\eta,\tau)]</math>

=== Short-time Fourier transform ===

* How to make the area of time-frequency smaller?

* Random process for [[Short-time Fourier transform|STFT (Short Time Fourier Transform)]]

<math>E[x(t)]\neq 0</math> should be satisfied. Otherwise,

<math>E[X(t,f)] = E[\int_{t-B}^{t+B} x(\tau)w(t-\tau)e^{-j2\pi f\tau}d\tau]</math>

<math>=\int_{t-B}^{t+B} E[x(\tau)]w(t-\tau)e^{-j2\pi f\tau}d\tau</math>for zero-mean random process, <math>E[X(t,f)] = 0</math>

* 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.

# Divide into several components.

# Use chirp multiplications, chirp convolutions, fractional Fourier transforms, or linear canonical transforms to reduce the area.

==Applications==

'''Step 1. Analytic Signal Conversion'''

The following applications need not only the time–frequency distribution functions but also some operations to the signal. The [[Linear canonical transform]] (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]]). 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.<ref>{{cite book |last1=Pachori |first1=Ram Bilas |title=Time-Frequency Analysis Techniques and Their Applications |publisher=CRC Press|url=https://www.routledge.com/Time-Frequency-Analysis-Techniques-and-their-Applications/Pachori/p/book/9781032435763?srsltid=AfmBOorjwRC4cJ-ABXieBsYLfFSmwQdGQ3GHNvL_O5pGnBMchjM8x7S8}}</ref><ref>{{cite book |last1=Boashash |first1=Boualem |title=Time-Frequency Signal Analysis and Processing: A Comprehensive Reference |publisher=Elsevier|url=https://www.sciencedirect.com/book/9780123984999/time-frequency-signal-analysis-and-processing}}</ref>

convert to signal side band signals (for real signals)

===Instantaneous frequency estimation===

<math>x(t)\rightarrow \begin{alignat}{2} x_a(t) & = x(t)+j x_H (t) \\ x_a(f) & = x(f)+jH(f)x(f)\\ \end{alignat}, \ x_H (t): \text{Hilbert transform of } x(t)

</math>

The definition of [[instantaneous frequency]] is the time rate of change of phase, or

<math>H(f) = \begin{cases} -j, & \text{for }f > 0 \\ j, & \text{for }f<0 \end{cases},x_a(f) = \begin{cases} 2x(f), & \text{for }f > 0 \\ 0, & \text{for }f<0 \end{cases}</math>

[[File:Analytic_signal_conversion_first_step.png|thumb|x(f) v.s. x_a(f) analytic signal conversion first step]]

'''Step 2. Separate the components'''

: <math>\frac{1}{2 \pi} \frac{d}{dt} \phi (t), </math>

Use STFT (due to multiple components) to separate the components

[[File:Separate_the_components.png|thumb|separate the frequency components]]

'''Step 3. Use shearing or rotation to minimize the "area" to each component'''

where <math>\phi (t)</math> is the [[instantaneous phase]] 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.

Use WDF (Wigner distribution function, due to single component and random process) to do this

===TF filtering and signal decomposition===

'''Step 4. Use the conventional sampling theory to sample each component'''

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.

* conventional sampling method

[[Image:filter tf.jpg]]

<math>x_d[n]=x(n\vartriangle_t), \vartriangle_t<1/F </math>

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]]. An example is shown below.

* reconstruction

[[Image:filter fractional.jpg]]

<math>x(t)=\sum_{n}x_d[n]sinc(\frac{t}{\vartriangle_t}-n)</math>

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.

* new sampling methods

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.

# <math>x(t)\rightarrow x_a(t) = x(t)+j x_H (t), \ x_H (t): \text{Hilbert transform of } x(t)</math>

# <math>x(t)\rightarrow x_a(t) = x_1(t)+x_2(t)+...+x_K(t)

</math>

# <math>y_k(t)=exp(j2\pi a_k t^2)x_k(t), k=1,2,...,K</math>

# <math>\begin{align} x_{d,k}[n] & = y_k(n\vartriangle_{t,k}) \\ & = exp(j2\pi a_k n^2 \vartriangle_{t,k}^2)x_k(n\vartriangle_{t,k}), k=1,2,...,K \\ \end{align}</math>

===Sampling theory===

* reconstruction

By the [[Nyquist–Shannon sampling theorem]], we can conclude that the minimum number of sampling points without [[aliasing]] 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:

# <math>y_k(t)=\sum_{n}x_{d,k}[n]sinc(\frac{t}{\vartriangle_{t,k}}-n)</math>

# <math>x_k(t)=exp(-j2\pi a_k t^2)y_k(t)</math>

# <math>x_a(t) = x_1(t)+x_2(t)+...+x_K(t)

</math>

# <math>x(t)=\real[x_a(t)]</math>

[[Image:sampling.jpg]]

Actually, no area of any signal time frequency distribution is limited. But we could choose a "threshold" Δ, then time-frequency analysis <math>\left\vert X(t,f) \right\vert >\bigtriangleup</math> or the area of distribution is limited. It sacrifices some accuracy to decide sampling points on "area".

It is noticeable that the number of sampling points decreases after we apply the time–frequency distribution.

* Theorem:

When we use the WDF, there might be the cross-term problem (also called interference). On the other hand, using [[Gabor transform]] causes an improvement in the clarity and readability of the representation, therefore improving its interpretation and application to practical problems.

If <math>x(t)</math> is time limited <math>(x(t)=0 \text{ for } t<t_1 \text{ and } t>t_2)</math>, then it is impossible to be frequency limited.

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.

If <math>x(t)</math> is frequency limited <math>(X(f)=0 \text{ for } f<f_1 \text{ and } f>f_2)</math>, then it is impossible to be time limited.

<math>err=\frac{\int_{-\infty}^{t_1} |x(t)|^2dt+\int_{t_2}^{\infty} |x(t)|^2dt+\int_{-\infty}^{f_1} |X_1(f)|^2df+\int_{f_2}^{\infty} |X_1(f)|^2df}{\int_{-\infty}^{\infty} |x(t)|^2dt}</math>

<math>X_1(f)=FT[x_1(t)], x_1(t)=x(t) \text{ for } t\in[t_1,t_2], x_1(t)=0 \text{ otherwise }</math>

* For the Wigner distribution function|Wigner distribution function (WDF)

<math>|x(t)|^2=\int_{-\infty}^{\infty} W_x(t,f)df,|X(f)|^2=\int_{-\infty}^{\infty} W_x(t,f)dt</math>

<math>\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} W_x(t,f)dfdt=\int_{-\infty}^{\infty}|x(t)|^2dt= \text{ energy of } x(t)</math>

<math>\begin{align} & \int_{-\infty}^{t_1} |x(t)|^2dt+\int_{t_2}^{\infty} |x(t)|^2dt+\int_{-\infty}^{f_1} |X_1(f)|^2df+\int_{f_2}^{\infty} |X_1(f)|^2df

\\ & = \int_{-\infty}^{t_1}\int_{-\infty}^{\infty} W_x(t,f)dfdt

+\int_{t_2}^{\infty}\int_{-\infty}^{\infty} W_x(t,f)dfdt

+\int_{-\infty}^{\infty}\int_{-\infty}^{f_1} W_{x_1}(t,f)dfdt

+\int_{-\infty}^{\infty}\int_{f_2}^{\infty} W_{x_1}(t,f)dfdt

\\ & = \int_{-\infty}^{t_1}\int_{-\infty}^{\infty} W_x(t,f)dfdt

+\int_{t_2}^{\infty}\int_{-\infty}^{\infty} W_x(t,f)dfdt

+\int_{t_1}^{t_2}\int_{-\infty}^{f_1} W_{x_1}(t,f)dfdt

+\int_{t_1}^{t_2}\int_{f_2}^{\infty} W_{x_1}(t,f)dfdt

\\ & \cong \int_{-\infty}^{t_1}\int_{-\infty}^{\infty} W_x(t,f)dfdt \text{ (A) }

+\int_{t_2}^{\infty}\int_{-\infty}^{\infty} W_x(t,f)dfdt \text{ (B) }

+\int_{t_1}^{t_2}\int_{-\infty}^{f_1} W_x(t,f)dfdt \text{ (C) }

+\int_{t_1}^{t_2}\int_{f_2}^{\infty} W_x(t,f)dfdt\text{ (D) }

\end{align} </math>

<math>err\cong1-\frac{\int_{t_1}^{t_2}\int_{f_1}^{f_2} W_x(t,f)dfdt}{\int_{-\infty}{\infty}|x(t)|^2dt}</math>

[[File:Summation_of_area_of_Wigner_distribution_function_on_time-frequency_analysis.png|thumb|summation of area of Wigner distribution function on time-frequency analysis]]

The [[Balian–Low theorem]] formalizes this, and provides a bound on the minimum number of time–frequency samples needed.

===Modulation and multiplexing===

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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.

'''Conventional Modulation Theory'''

The signals <math>x_1(t), x_2(t), x_3(t),......,x_K(t)</math> can be transmitted successfully if

<math>\begin{align} & \text{Allowed Bandwidth}\geq\sum_{k=1}^K B_k \\ & B_k\text{:the bandwidth (including the negative frequency part) of }x_k(t) \\ \end{align}</math>

'''Modulation Theory Based on Time-Frequency Analysis'''

The signals <math>x_1(t), x_2(t), x_3(t),......,x_K(t)</math> can be transmitted successfully if

<math>\begin{align} & \text{Allowed Time duration } \times \text{ Allowed Bandwidth }\geq\sum_{k=1}^K A_k \\ & A_k\text{:the area of the time-frequency distribution of }x_k(t) \\ \end{align}</math>

<br />

===Electromagnetic wave propagation===

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===Optics, acoustics, and biomedicine===

[[Light]] is an electromagnetic wave, so time–frequency analysis applies to optics in the same way as for general electromagnetic wave propagation.

[[Light]] is a kind of electromagnetic wave, so we apply the time–frequency analysis to optics in the same way as to electromagnetic wave propagation. In the same way, a characteristic of acoustic signals is that, often, its frequency varies really severely with time. Because the acoustic signals usually contain a lot of data, it is suitable to use simpler TFDs such as the Gabor transform to analyze the acoustic signals due to the lower computational complexity. If speed is not an issue, then a detailed 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.

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]] (EMG), [[electroencephalography]] (EEG), [[electrocardiogram]] (ECG) or [[otoacoustic emissions]] (OAEs).

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== See also ==

* [[ConeMotions in the time-shapefrequency distribution function]]

* [[Multiresolution analysis]]

* [[Spectral density estimation]]

* [[Time–frequency analysis for music signalsignals]]

* [[Wavelet analysis]]

== References ==