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 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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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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\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> ==

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) <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,

:<math>E[x(t+\tau/2)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 ===

* [[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>

<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 ===

* 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]]

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

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

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

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

<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

*# <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

::<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===

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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 ==