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{{Short description|Field of electrical engineering}}
{{Redirect-distinguish|Signal theory|Signalling theory|Signalling (economics)}}

[[File:Signal processing system.png|thumb|400px|Signal transmission using electronic signal processing. [[Transducer]]s convert signals from other physical [[waveform]]s to electric [[Electric current|current]] or [[voltage]] waveforms, which then are processed, transmitted as [[electromagnetic wave]]s, received and converted by another transducer to their final form.]]
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'''Signal processing''' is an [[electrical engineering]] subfield that focuses on analyzing, modifying and synthesizing ''[[signal]]s'', such as [[audio signal processing|sound]], [[image processing|images]], [[Scalar potential|potential fields]], [[Seismic tomography|seismic signals]], [[Altimeter|altimetry processing]], and [[scientific measurements]].<ref>{{cite journal|last=Sengupta|first=Nandini|author2=Sahidullah, Md|author3=Saha, Goutam|date=August 2016|title=Lung sound classification using cepstral-based statistical features|journal=Computers in Biology and Medicine|volume=75|issue=1|pages=118–129|doi=10.1016/j.compbiomed.2016.05.013|pmid=27286184}}</ref> Signal processing techniques are used to optimize transmissions, [[digital storage]] efficiency, correcting distorted signals, improve [[subjective video quality]], and to detect or pinpoint components of interest in a measured signal.<ref>{{cite book|title=Discrete-Time Signal Processing|author=Alan V. Oppenheim and Ronald W. Schafer|publisher=Prentice Hall|year=1989|isbn=0-13-216771-9|page=1}}</ref>

==History==
According to [[Alan V. Oppenheim]] and [[Ronald W. Schafer]], the principles of signal processing can be found in the classical [[numerical analysis]] techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital [[control system]]s of the 1940s and 1950s.<ref>{{cite book |title=Digital Signal Processing |year=1975 |publisher=[[Prentice Hall]] |isbn=0-13-214635-5 |author=Oppenheim, Alan V. |author2=Schafer, Ronald W. |page= 5}}</ref>

In 1948, [[Claude Shannon]] wrote the influential paper "[[A Mathematical Theory of Communication]]" which was published in the ''[[Bell System Technical Journal]]''.<ref>{{cite web |url=https://www.computerhistory.org/revolution/digital-logic/12/269/1331 |title=A Mathematical Theory of Communication – CHM Revolution |website=Computer History |access-date=2019-05-13}}</ref> The paper laid the groundwork for later development of information communication systems and the processing of signals for transmission.<ref name="fifty">{{cite book |title=Fifty Years of Signal Processing: The IEEE Signal Processing Society and its Technologies, 1948–1998 |publisher=The IEEE Signal Processing Society |year=1998 |url=https://signalprocessingsociety.org/uploads/history/history.pdf|archive-date=2016-03-04|archive-url=http://web.archive.org/web/20160304031943/https://signalprocessingsociety.org/uploads/history/history.pdf}}</ref>

Signal processing matured and flourished in the 1960s and 1970s, and [[digital signal processing]] became widely used with specialized [[digital signal processor]] chips in the 1980s.<ref name=fifty/>

== Definition of a signal ==
In signal processing, a signal is represented as a [[Function (mathematics)|function]] of time: <math>x(t)</math>, where this function is either<ref>Berber, S. (2021). Discrete Communication Systems. United Kingdom: Oxford University Press., page 9, https://books.google.com/books?id=CCs0EAAAQBAJ&pg=PA9</ref>
* deterministic (then one speaks of a deterministic signal) or
* a path <math>(x_t)_{t \in T}</math>, a realization of a [[stochastic process]] <math>(X_t)_{t \in T}</math>

==Categories==

===Analog===
{{main|Analog signal processing}}

Analog signal processing is for signals that have not been digitized, as in most 20th-century [[radio]], telephone, and television systems. This involves linear electronic circuits as well as nonlinear ones. The former are, for instance, [[passive filter]]s, [[active filter]]s, [[Electronic mixer|additive mixers]], [[integrator]]s, and [[Analog delay line|delay line]]s. Nonlinear circuits include [[compandor]]s, multipliers ([[frequency mixer]]s, [[voltage-controlled amplifier]]s), [[voltage-controlled filter]]s, [[voltage-controlled oscillator]]s, and [[phase-locked loop]]s.

===Continuous time===
[[Continuous-time signal]] processing is for signals that vary continuously in time and are not broken into individual interrupted points, i.e., [[Sampling (signal processing)|samples]].

The methods of signal processing include [[time domain]], [[frequency domain]], and [[complex frequency|complex frequency domain]]. This technology mainly discusses the modeling of a [[linear time-invariant]] continuous system, integral of the system's zero-state response, setting up system function and the continuous time filtering of deterministic signals. For example, in time domain, a continuous-time signal <math>x(t)</math> passing through a [[linear time-invariant]] filter/system denoted as <math>h(t)</math>, can be expressed at the output as

<math>
y(t) = \int_{-\infty}^\infty h(\tau) x(t - \tau) \, d\tau
</math>

In some contexts, <math>h(t)</math> is referred to as the impulse response of the system. The above [[convolution]] operation is conducted between the input and the system.

===Discrete time===
[[Discrete-time signal]] processing is for sampled signals, defined only at discrete points in time, and as such are quantized in time, but not in magnitude.

''Analog discrete-time signal processing'' is a technology based on electronic devices such as [[sample and hold]] circuits, analog time-division [[multiplexer]]s, [[analog delay line]]s and [[analog feedback shift register]]s. This technology was a predecessor of digital signal processing (see below), and is still used in advanced processing of gigahertz signals.<ref>{{cite web |url=https://microwavelab.nd.edu/research/analog-signal-processing/ |title=Microwave & Millimeter-wave Circuits and Systems |access-date=2024-10-20|archive-url=http://web.archive.org/web/20201029083454/https://microwavelab.nd.edu/research/analog-signal-processing/|archive-date=2020-10-29}}</ref>

The concept of discrete-time signal processing also refers to a theoretical discipline that establishes a mathematical basis for digital signal processing, without taking [[quantization error]] into consideration.

===Digital===
{{main|Digital signal processing}}

Digital signal processing is the processing of digitized discrete-time sampled signals. Processing is done by general-purpose [[computer]]s or by digital circuits such as [[ASIC]]s, [[field-programmable gate array]]s or specialized [[digital signal processor]]s. Typical arithmetical operations include [[Fixed-point arithmetic|fixed-point]] and [[floating-point]], real-valued and complex-valued, multiplication and addition. Other typical operations supported by the hardware are [[circular buffer]]s and [[lookup table]]s. Examples of algorithms are the [[fast Fourier transform]] (FFT), [[finite impulse response]] (FIR) filter, [[Infinite impulse response]] (IIR) filter, and [[adaptive filter]]s such as the [[Wiener filter|Wiener]] and [[Kalman filter]]s.

===Nonlinear===
Nonlinear signal processing involves the analysis and processing of signals produced from [[nonlinear system]]s and can be in the time, [[frequency]], or spatiotemporal domains.<ref name="Billings">{{cite book |last=Billings |first=S. A. |title=Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains |publisher=Wiley |year=2013 |isbn=978-1-119-94359-4 }}</ref><ref name="VSA">{{cite book |author=Slawinska, J. |author2=Ourmazd, A. |author3=Giannakis, D. |title=2018 IEEE Statistical Signal Processing Workshop (SSP) |chapter=A New Approach to Signal Processing of Spatiotemporal Data |pages=338–342 |publisher=IEEE Xplore |year=2018 |doi=10.1109/SSP.2018.8450704|isbn=978-1-5386-1571-3 |s2cid=52153144 }}</ref> Nonlinear systems can produce highly complex behaviors including [[bifurcation theory|bifurcations]], [[chaos theory|chaos]], [[harmonics]], and [[subharmonics]] which cannot be produced or analyzed using linear methods.

Polynomial signal processing is a type of non-linear signal processing, where [[polynomial]] systems may be interpreted as conceptually straightforward extensions of linear systems to the nonlinear case.<ref>{{cite book |author1=V. John Mathews |author2=Giovanni L. Sicuranza |title=Polynomial Signal Processing |date=May 2000 |isbn=978-0-471-03414-8 |publisher=Wiley}}</ref>

===Statistical ===
'''Statistical signal processing''' is an approach which treats signals as [[stochastic process]]es, utilizing their [[statistical]] properties to perform signal processing tasks.<ref name ="Scharf">{{cite book |first=Louis L. |last=Scharf |title=Statistical signal processing: detection, estimation, and time series analysis |publisher=[[Addison–Wesley]] |location=[[Boston]] |year=1991 |isbn=0-201-19038-9 |oclc=61160161}}</ref> Statistical techniques are widely used in signal processing applications. For example, one can model the [[probability distribution]] of noise incurred when photographing an image, and construct techniques based on this model to [[noise reduction|reduce the noise]] in the resulting image.

===Graph ===
'''Graph signal processing''' generalizes signal processing tasks to signals living on non-Euclidean domains whose structure can be captured by a weighted graph.<ref name ="Ortega">{{cite book |first=A. |last=Ortega |title=Introduction to Graph Signal Processing |publisher=[[Cambridge University Press]] |location=[[Cambridge]] |year=2022 |isbn=9781108552349}}</ref> Graph signal processing presents several key points such as sampling signal techniques,<ref name="Tanaka">{{cite journal|title=Generalized Sampling on Graphs with Subspace and Smoothness Prior|journal=IEEE Transactions on Signal Processing|date=2020|last1=Tanaka|first1=Y.|last2=Eldar|first2=Y.|volume=68 |pages=2272–2286 |doi=10.1109/TSP.2020.2982325 |arxiv=1905.04441 |bibcode=2020ITSP...68.2272T }}</ref> recovery techniques <ref name="Fascista">{{cite journal|title=Graph Signal Reconstruction under Heterogeneous Noise via Adaptive Uncertainty-Aware Sampling and Soft Classification|journal=IEEE Transactions on Signal and Information Processing over Networks|date=2024|last1=Fascista|first1=A.|last2=Coluccia|first2=A.|last3=Ravazzi|first3=C.|volume=10 |pages=277–293 |doi=10.1109/TSIPN.2024.3375593 |bibcode=2024ITSIP..10..277F }}</ref> and time-varying techiques.<ref name="Giraldo">{{cite journal|title=Reconstruction of Time-varying Graph Signals via Sobolev Smoothness|journal=IEEE Transactions on Signal and Information Processing over Networks|date=March 2022|last1=Giraldo|first1=J.|last2=Mahmood|first2=A. |last3=Garcia-Garcia|first3=B.|last4=Thanou|first4=D.|last5=Bouwmans|first5=T.|volume=8 |pages=201–214 |doi=10.1109/TSIPN.2022.3156886 |arxiv=2207.06439 |bibcode=2022ITSIP...8..201G }}</ref> Graph signal processing has been applied with success in the field of image processing, computer vision <ref name="Giraldo1">{{cite book|title=2020 IEEE International Conference on Image Processing (ICIP)|date=October 2020|last1=Giraldo|first1=J.|last2=Bouwmans|first2=T.|chapter= Semi-Supervised Background Subtraction of Unseen Videos: Minimization of the Total Variation of Graph Signals|pages= 3224–3228|doi= 10.1109/ICIP40778.2020.9190887|isbn= 978-1-7281-6395-6}}</ref>
<ref name="Giraldo2">{{cite book|title=2020 25th International Conference on Pattern Recognition (ICPR)|date=2020|last1=Giraldo|first1=J.|last2=Bouwmans|first2=T.|chapter=GraphBGS: Background Subtraction via Recovery of Graph Signals |pages=6881–6888 |doi=10.1109/ICPR48806.2021.9412999 |arxiv=2001.06404 |isbn=978-1-7281-8808-9 }}</ref>
<ref name="Giraldo3">{{cite book|title=Frontiers of Computer Vision|date=February 2021|chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-81638-4_3|last1=Giraldo|first1=J.|last2=Javed|first2=S.|last3=Sultana|first3=M.|last4=Jung|first4=S.|last5=Bouwmans|first5=T.|chapter=The Emerging Field of Graph Signal Processing for Moving Object Segmentation |series=Communications in Computer and Information Science |volume=1405 |pages=31–45 |doi=10.1007/978-3-030-81638-4_3 |isbn=978-3-030-81637-7 }}</ref> and sound anomaly detection.<ref name="Bouwmans1">{{cite book |last1=Mnasri |first1=Zied |last2=Giraldo |first2=Jhony H. |last3=Bouwmans |first3=Thierry |title=2024 32nd European Signal Processing Conference (EUSIPCO) |chapter=Anomalous Sound Detection for Road Surveillance Based on Graph Signal Processing |date=2024 |pages=161–165 |doi=10.23919/EUSIPCO63174.2024.10715291 |isbn=978-9-4645-9361-7 |chapter-url=https://hal.science/hal-04756448 }}</ref>

==Application fields==
[[File:Seismic Data Processing.jpg|thumb|Seismic signal processing]]
* [[Audio signal processing]]{{spaced ndash}} for electrical signals representing sound, such as [[Speech signal processing|speech]] or music<ref>{{cite journal
|last=Sarangi|first=Susanta |author2=Sahidullah, Md |author3=Saha, Goutam
|title=Optimization of data-driven filterbank for automatic speaker verification
|journal=Digital Signal Processing |date=September 2020 |volume=104
|article-number=102795 |doi= 10.1016/j.dsp.2020.102795|arxiv=2007.10729|bibcode=2020DSP...10402795S |s2cid=220665533 }}</ref>
* [[Image processing]]{{spaced ndash}} in digital cameras, computers and various imaging systems
* [[Video processing]]{{spaced ndash}} for interpreting moving pictures
* [[Wireless communication]]{{spaced ndash}} waveform generations, demodulation, filtering, equalization
* [[Control systems]]
* [[Array processing]]{{spaced ndash}} for processing signals from arrays of sensors
* [[Process control]]{{spaced ndash}} a variety of signals are used, including the industry standard [[4-20 mA current loop]]
* [[Seismology]]
* [[Feature extraction]], such as [[image understanding]], [[semantic audio]] and [[speech recognition]].
* Quality improvement, such as [[noise reduction]], [[image enhancement]], and [[echo cancellation]].
* Source coding including [[audio compression (data)|audio compression]], [[image compression]], and [[video compression]].
* [[Genomic]] signal processing<ref>{{cite journal|first1=D.|last1=Anastassiou|title=Genomic signal processing|journal=IEEE Signal Processing Magazine|volume=18|issue=4|pages=8–20|year=2001|publisher=IEEE|doi=10.1109/79.939833|bibcode=2001ISPM...18....8A }}</ref>
* In [[geophysics]], signal processing is used to amplify the signal vs the noise within [[time-series]] measurements of geophysical data. Processing is conducted within the [[time domain]] or [[frequency domain]], or both.<ref>{{cite book |last1=Telford |first1=William Murray |last2=Geldart |first2=L. P. |first3=Robert E. |last3= Sheriff |title=Applied geophysics |year=1990 |publisher=[[Cambridge University Press]] |isbn=978-0-521-33938-4}}</ref><ref>{{cite book|last1=Reynolds |first1=John M. |title=An Introduction to Applied and Environmental Geophysics |year=2011 |publisher=[[Wiley-Blackwell]] |isbn=978-0-471-48535-3}}</ref>

In communication systems, signal processing may occur at:{{cn|date=March 2025}}
* OSI layer 1 in the seven-layer [[OSI model]], the [[physical layer]] ([[modulation]], [[Equalization (communications)|equalization]], [[multiplexing]], etc.);
* OSI layer 2, the [[data link layer]] ([[forward error correction]]);
* OSI layer 6, the [[presentation layer]] (source coding, including [[analog-to-digital conversion]] and [[data compression]]).

==Typical devices ==
* [[Filter (signal processing)|Filters]]{{spaced ndash}} for example analog (passive or active) or digital ([[FIR filter|FIR]], [[IIR filter|IIR]], frequency domain or [[stochastic filter]]s, etc.)
* [[Sampling (signal processing)|Samplers]] and [[analog-to-digital converter]]s for [[signal acquisition]] and reconstruction, which involves measuring a physical signal, storing or transferring it as a digital signal, and possibly later rebuilding the original signal or an approximation thereof.
* [[Digital signal processor]]s (DSPs)

==Mathematical methods applied==
* [[Differential equations]]<ref name="Gaydecki2004">{{cite book|author=Patrick Gaydecki|title=Foundations of Digital Signal Processing: Theory, Algorithms and Hardware Design|url=https://books.google.com/books?id=6Qo7NvX3vz4C&q=%22differential+equation%22+OR+%22differential+equations%22&pg=PA40|year=2004|publisher=IET|isbn=978-0-85296-431-6|pages=40–}}</ref>{{spaced ndash}} for modeling system behavior, connecting input and output relations in linear time-invariant systems. For instance, a low-pass filter such as an [[RC circuit]] can be modeled as a differential equation in signal processing, which allows one to compute the continuous output signal as a function of the input or initial conditions.
* [[Recurrence relation]]s<ref name="Engelberg2008">{{cite book|author=Shlomo Engelberg|title=Digital Signal Processing: An Experimental Approach|url=https://books.google.com/books?id=z3CpcCHbtgIC|date=8 January 2008|publisher=Springer Science & Business Media|isbn=978-1-84800-119-0}}</ref>
* [[Transform theory]]
* [[Time-frequency analysis]]{{spaced ndash}} for processing non-stationary signals<ref>{{cite book|title=Time frequency signal analysis and processing a comprehensive reference|year=2003|publisher=Elsevier|location=Amsterdam|isbn=0-08-044335-4|edition=1|editor=Boashash, Boualem}}</ref>
* [[Linear canonical transformation]]
* [[Spectral estimation]]{{spaced ndash}} for determining the spectral content (i.e., the distribution of power over frequency) of a set of [[time series]] data points<ref>{{cite book|first1=Petre|last1=Stoica|first2=Randolph|last2=Moses|title=Spectral Analysis of Signals|year=2005|publisher=Prentice Hall|location=NJ|url=http://user.it.uu.se/%7Eps/SAS-new.pdf}}</ref>
*[[Statistical signal processing]]{{spaced ndash}} analyzing and extracting information from signals and noise based on their stochastic properties
*[[Linear time-invariant system]] theory, and [[transform theory]]
*[[Polynomial signal processing]]{{spaced ndash}} analysis of systems which relate input and output using polynomials
*[[System identification]]<ref name="Billings"/> and classification
*[[Calculus]]
*[[Coding theory]]
*[[Complex analysis]]<ref name="SchreierScharf2010">{{cite book|author1=Peter J. Schreier|author2=Louis L. Scharf|title=Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals|url=https://books.google.com/books?id=HBaxLfDsAHoC&q=%22complex+analysis%22|date=4 February 2010|publisher=Cambridge University Press|isbn=978-1-139-48762-7}}</ref>
*[[Vector spaces]] and [[Linear algebra]]<ref name="Little2019">{{cite book|author=Max A. Little|title=Machine Learning for Signal Processing: Data Science, Algorithms, and Computational Statistics|url=https://books.google.com/books?id=ejGoDwAAQBAJ&q=%22vector+space%22|date=13 August 2019|publisher=OUP Oxford|isbn=978-0-19-102431-3}}</ref>
*[[Functional analysis]]<ref name="DamelinJr2012">{{cite book|author1=Steven B. Damelin|author2=Willard Miller, Jr|title=The Mathematics of Signal Processing|url=https://books.google.com/books?id=MtPLYXQ9d9MC&q=%22functional+analysis%22|year=2012|publisher=Cambridge University Press|isbn=978-1-107-01322-3}}</ref>
*[[Probability]] and [[stochastic processes]]<ref name="Scharf"/>
*[[Detection theory]]
*[[Estimation theory]]
*[[Optimization]]<ref name="PalomarEldar2010">{{cite book|author1=Daniel P. Palomar|author2=Yonina C. Eldar|title=Convex Optimization in Signal Processing and Communications|url=https://books.google.com/books?id=UOpnvPJ151gC|year=2010|publisher=Cambridge University Press|isbn=978-0-521-76222-9}}</ref>
*[[Numerical methods]]
*[[Data mining]]{{spaced ndash}} for statistical analysis of relations between large quantities of variables (in this context representing many physical signals), to extract previously unknown interesting patterns

==See also==
* [[Algebraic signal processing]]
* [[Audio filter]]
* [[Bounded variation]]
* [[Dynamic range compression]]
* [[Information theory]]
* [[Least-squares spectral analysis]]
* [[Non-local means]]
* [[Reverberation]]
* [[Sensitivity (electronics)]]
* [[Similarity (signal processing)]]
* [[Wiener filter]]

==References==
{{reflist}}

==Further reading==
* {{cite book |first=Charles |last=Byrne |title=Signal Processing: A Mathematical Approach |publisher=[[Taylor & Francis]] |year=2014 |doi=10.1201/b17672 |isbn=9780429158711 |url=https://www.taylorfrancis.com/books/oa-mono/10.1201/b17672/signal-processing-charles-byrne}}
* {{cite book|last=P Stoica|first=R Moses|title=Spectral Analysis of Signals|year=2005|publisher=Prentice Hall|location=NJ|url=https://user.it.uu.se/%7Eps/SAS-new.pdf}}
* {{cite book |first=Athanasios |last=Papoulis |title=Probability, Random Variables, and Stochastic Processes |year=1991 |edition=third |publisher=McGraw-Hill |isbn=0-07-100870-5}}
* [[Ali H. Sayed]], Adaptive Filters, Wiley, NJ, 2008, {{isbn|978-0-470-25388-5}}.
* [[Thomas Kailath]], [[Ali H. Sayed]], and [[Babak Hassibi]], Linear Estimation, Prentice-Hall, NJ, 2000, {{isbn|978-0-13-022464-4}}.
* [https://sp4comm.org Signal Processing for Communications] – free online textbook by Paolo Prandoni and Martin Vetterli (2008)
* [https://dspguide.com Scientists and Engineers Guide to Digital Signal Processing] – free online textbook by Stephen Smith

==External links==
* [https://www.dsprelated.com/freebooks/sasp/ Julius O. Smith III: Spectral Audio Signal Processing] – free online textbook
* [https://sites.google.com/view/gsp-website/graph-signal-processing Graph Signal Processing Website] – free online website by Thierry Bouwmans (2025)

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[[Category:Signal processing| ]]
[[Category:Mass media technology]]
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Signal processing - Revision history

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