Signal processing

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 systems of the 1940s and 1950s.{{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}}

In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was published in the Bell System Technical Journal.{{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}} The paper laid the groundwork for later development of information communication systems and the processing of signals for transmission.{{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}}

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.

A signal is a function $x(t)$, where this function is eitherBerber, S. (2021). Discrete Communication Systems. United Kingdom: Oxford University Press., page 9, https://books.google.com/books?id=CCs0EAAAQBAJ&pg=PA9

• deterministic (then one speaks of a deterministic signal) or
• a path $(x\_t)\_\{t\; \backslash in\; T\}$, a realization of a stochastic process $(X\_t)\_\{t\; \backslash in\; T\}$

{{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 filters, active filters, additive mixers, integrators, and delay lines. Nonlinear circuits include compandors, multipliers (frequency mixers, voltage-controlled amplifiers), voltage-controlled filters, voltage-controlled oscillators, and phase-locked loops.

Continuous-time signal processing is for signals that vary with the change of continuous domain (without considering some individual interrupted points).

The methods of signal processing include time domain, frequency domain, and 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 $x(t)$ passing through a linear time-invariant filter/system denoted as $h(t)$, can be expressed at the output as

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

In some contexts, $h(t)$ is referred to as the impulse response of the system. The above convolution operation is conducted between the input and the system.

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 multiplexers, analog delay lines and analog feedback shift registers. This technology was a predecessor of digital signal processing (see below), and is still used in advanced processing of gigahertz signals.{{cite web |url=https://microwavelab.nd.edu/research/analog-signal-processing/ |title=Microwave & Millimeter-wave Circuits and Systems |access-date=2024-10-20}}

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.

{{main|Digital signal processing}}

Digital signal processing is the processing of digitized discrete-time sampled signals. Processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, multiplication and addition. Other typical operations supported by the hardware are circular buffers and lookup tables. Examples of algorithms are the fast Fourier transform (FFT), finite impulse response (FIR) filter, Infinite impulse response (IIR) filter, and adaptive filters such as the Wiener and Kalman filters.

Nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatiotemporal domains.{{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 }}{{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 }} Nonlinear systems can produce highly complex behaviors including bifurcations, 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.{{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}}

Statistical signal processing is an approach which treats signals as stochastic processes, utilizing their statistical properties to perform signal processing tasks.{{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}} 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 reduce the noise in the resulting image.

Graph signal processing generalizes signal processing tasks to signals living on non-Euclidean domains whose structure can be captured by a weighted graph.{{cite book |first=A. |last=Ortega |title=Introduction to Graph Signal Processing |publisher=Cambridge University Press |location=Cambridge |year=2022 |isbn=9781108552349}} Graph signal processing presents several key points such as sampling signal techniques,{{cite journal|title=Generalized Sampling on Graphs with Subspace and Smoothness Prior|journal=IEEE Transactions on Signal Processing|date=2020|url=https://ieeexplore.ieee.org/document/9043719|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 }} recovery techniques {{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|url=https://ieeexplore.ieee.org/document/10465260|last1=Fascista|first1=A.|last2=Coluccia|first2=A.|last3=Ravazzi|first3=C.|volume=10 |pages=277–293 |doi=10.1109/TSIPN.2024.3375593 |url-access=subscription}} and time-varying techiques.{{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|url=https://ieeexplore.ieee.org/document/9730033|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 }} Graph signal processing has been applied with success in the field of image processing, computer vision {{cite book|title=2020 IEEE International Conference on Image Processing (ICIP)|date=October 2020|chapter-url=https://ieeexplore.ieee.org/document/9190887|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}}

{{cite book|title=2020 25th International Conference on Pattern Recognition (ICPR)|date=2020|chapter-url=https://ieeexplore.ieee.org/document/9412999|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 }}

{{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 }} and sound anomaly detection.{{cite journal|title=Anomalous Sound Detection for Road Surveillance based on Graph Signal Processing|journal=European Conference on Signal Processing, EUSIPCO 2024|date=2024|url=https://ieeexplore.ieee.org/document/10715291|last1=Mnasri|first1=Z.|last2=Giraldo|first2=H. |last3=Bouwmans|first3=T.|pages=161–165 |doi=10.23919/EUSIPCO63174.2024.10715291 |isbn=978-9-4645-9361-7 }}

File:Seismic Data Processing.jpg

• Audio signal processing{{spaced ndash}} for electrical signals representing sound, such as speech or music{{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

|page=102795 |doi= 10.1016/j.dsp.2020.102795|arxiv=2007.10729|bibcode=2020DSP...10402795S |s2cid=220665533 }}

• 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, image compression, and video compression.
• Genomic signal processing{{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 }}
• 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.{{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}}{{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}}

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, 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).

• Filters{{spaced ndash}} for example analog (passive or active) or digital (FIR, IIR, frequency domain or stochastic filters, etc.)
• Samplers and analog-to-digital converters for signal acquisition and reconstruction, which involves measuring a physical signal, storing or transferring it as digital signal, and possibly later rebuilding the original signal or an approximation thereof.
• Digital signal processors (DSPs)

• Differential equations{{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–}}{{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 relations{{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}}
• Transform theory
• Time-frequency analysis{{spaced ndash}} for processing non-stationary signals{{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}}
• 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{{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}}
• 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 and classification
• Calculus
• Coding theory
• Complex analysis{{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}}
• Vector spaces and Linear algebra{{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}}
• Functional analysis{{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}}
• Probability and stochastic processes
• Detection theory
• Estimation theory
• Optimization{{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}}
• 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

• Algebraic signal processing
• Audio filter
• Bounded variation
• Digital image processing
• Dynamic range compression, companding, limiting, and noise gating
• Fourier transform
• Information theory
• Least-squares spectral analysis
• Non-local means
• Reverberation
• Sensitivity (electronics)
• Similarity (signal processing)

{{reflist}}

• {{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}}
• Kainam Thomas Wong [http://www.eie.polyu.edu.hk/~enktwong/]: Statistical Signal Processing lecture notes at the University of Waterloo, Canada.
• 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://www.sp4comm.org/ Signal Processing for Communications] – free online textbook by Paolo Prandoni and Martin Vetterli (2008)
• [http://www.dspguide.com Scientists and Engineers Guide to Digital Signal Processing] – free online textbook by Stephen Smith
• [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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