Time–frequency analysis: Difference between revisions

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

{{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 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]]).

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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|last1=Shafi|first1=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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\cos(3 \pi t); & t > 20

\end{cases}</math>

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But time–frequency analysis can.

== TF analysis and Randomrandom Processprocesses<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> ==

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 <math>R_x(t,\tau)</math>

<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,

* Auto-covariance function (ACF) <math>E[xR_x(t+\tau/2)x^*(t-,\tau/2)]</math>

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

:<math>E[W_gx(t,f+\tau/2)] = x^*(t-\sigmatau/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 random processPSD

:<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 random processACF

:<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 ===

* [[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,

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

<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>)

* For white noise,

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

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

Filter=== DesignAdditive for Whitewhite noise ===

* 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

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

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

=== Non-stationary random processes ===

* 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

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*# <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

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

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

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

#=== Short-time Fourier transform ===
* 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.

==Applications==

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>

===Instantaneous frequency estimation===