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><ref>{{cite book |last=Pachori |first=Ram Bilas |title=Time-frequency analysis techniques and their applications |year=2023 |publisher=CRC Press |isbn=978-1-00-336798-7 |doi=10.1201/9781003367987}}</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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==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===