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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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===