# Signal processing

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Field of electrical engineering

"Signal theory" redirects here; not to be confused with [Signalling theory](/source/Signalling_theory) or [Signalling (economics)](/source/Signalling_(economics)).

Signal transmission using electronic signal processing. [Transducers](/source/Transducer) convert signals from other physical [waveforms](/source/Waveform) to electric [current](/source/Electric_current) or [voltage](/source/Voltage) waveforms, which then are processed, transmitted as [electromagnetic waves](/source/Electromagnetic_wave), received and converted by another transducer to their final form.

The signal on the left looks like noise, but the signal processing technique known as [spectral density estimation](/source/Spectral_density_estimation) (right) shows that it contains five well-defined frequency components.

**Signal processing** is an [electrical engineering](/source/Electrical_engineering) subfield that focuses on analyzing, modifying and synthesizing *[signals](/source/Signal)*, such as [sound](/source/Audio_signal_processing), [images](/source/Image_processing), [potential fields](/source/Scalar_potential), [seismic signals](/source/Seismic_tomography), [altimetry processing](/source/Altimeter), and [scientific measurements](/source/Scientific_measurements).[1] Signal processing techniques are used to optimize transmissions, [digital storage](/source/Digital_storage) efficiency, correcting distorted signals, improve [subjective video quality](/source/Subjective_video_quality), and to detect or pinpoint components of interest in a measured signal.[2]

## History

According to [Alan V. Oppenheim](/source/Alan_V._Oppenheim) and [Ronald W. Schafer](/source/Ronald_W._Schafer), the principles of signal processing can be found in the classical [numerical analysis](/source/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](/source/Control_system) of the 1940s and 1950s.[3]

In 1948, [Claude Shannon](/source/Claude_Shannon) wrote the influential paper "[A Mathematical Theory of Communication](/source/A_Mathematical_Theory_of_Communication)" which was published in the *[Bell System Technical Journal](/source/Bell_System_Technical_Journal)*.[4] The paper laid the groundwork for later development of information communication systems and the processing of signals for transmission.[5]

Signal processing matured and flourished in the 1960s and 1970s, and [digital signal processing](/source/Digital_signal_processing) became widely used with specialized [digital signal processor](/source/Digital_signal_processor) chips in the 1980s.[5]

## Definition of a signal

In signal processing, a signal is represented as a [function](/source/Function_(mathematics)) of time: x ( t ) {\displaystyle x(t)} , where this function is either[6]

- deterministic (then one speaks of a deterministic signal) or

- a path ( x t ) t ∈ T {\displaystyle (x_{t})_{t\in T}} , a realization of a [stochastic process](/source/Stochastic_process) ( X t ) t ∈ T {\displaystyle (X_{t})_{t\in T}}

## Categories

### Analog

Main article: [Analog signal processing](/source/Analog_signal_processing)

Analog signal processing is for signals that have not been digitized, as in most 20th-century [radio](/source/Radio), telephone, and television systems. This involves linear electronic circuits as well as nonlinear ones. The former are, for instance, [passive filters](/source/Passive_filter), [active filters](/source/Active_filter), [additive mixers](/source/Electronic_mixer), [integrators](/source/Integrator), and [delay lines](/source/Analog_delay_line). Nonlinear circuits include [compandors](/source/Compandor), multipliers ([frequency mixers](/source/Frequency_mixer), [voltage-controlled amplifiers](/source/Voltage-controlled_amplifier)), [voltage-controlled filters](/source/Voltage-controlled_filter), [voltage-controlled oscillators](/source/Voltage-controlled_oscillator), and [phase-locked loops](/source/Phase-locked_loop).

### Continuous time

[Continuous-time signal](/source/Continuous-time_signal) processing is for signals that vary continuously in time and are not broken into individual interrupted points, i.e., [samples](/source/Sampling_(signal_processing)).

The methods of signal processing include [time domain](/source/Time_domain), [frequency domain](/source/Frequency_domain), and [complex frequency domain](/source/Complex_frequency). This technology mainly discusses the modeling of a [linear time-invariant](/source/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 ) {\displaystyle x(t)} passing through a [linear time-invariant](/source/Linear_time-invariant) filter/system denoted as h ( t ) {\displaystyle h(t)} , can be expressed at the output as

y ( t ) = ∫ − ∞ ∞ h ( τ ) x ( t − τ ) d τ {\displaystyle y(t)=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )\,d\tau }

In some contexts, h ( t ) {\displaystyle h(t)} is referred to as the impulse response of the system. The above [convolution](/source/Convolution) operation is conducted between the input and the system.

### Discrete time

[Discrete-time signal](/source/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](/source/Sample_and_hold) circuits, analog time-division [multiplexers](/source/Multiplexer), [analog delay lines](/source/Analog_delay_line) and [analog feedback shift registers](/source/Analog_feedback_shift_register). This technology was a predecessor of digital signal processing (see below), and is still used in advanced processing of gigahertz signals.[7]

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](/source/Quantization_error) into consideration.

### Digital

Main article: [Digital signal processing](/source/Digital_signal_processing)

Digital signal processing is the processing of digitized discrete-time sampled signals. Processing is done by general-purpose [computers](/source/Computer) or by digital circuits such as [ASICs](/source/ASIC), [field-programmable gate arrays](/source/Field-programmable_gate_array) or specialized [digital signal processors](/source/Digital_signal_processor). Typical arithmetical operations include [fixed-point](/source/Fixed-point_arithmetic) and [floating-point](/source/Floating-point), real-valued and complex-valued, multiplication and addition. Other typical operations supported by the hardware are [circular buffers](/source/Circular_buffer) and [lookup tables](/source/Lookup_table). Examples of algorithms are the [fast Fourier transform](/source/Fast_Fourier_transform) (FFT), [finite impulse response](/source/Finite_impulse_response) (FIR) filter, [Infinite impulse response](/source/Infinite_impulse_response) (IIR) filter, and [adaptive filters](/source/Adaptive_filter) such as the [Wiener](/source/Wiener_filter) and [Kalman filters](/source/Kalman_filter).

### Nonlinear

Nonlinear signal processing involves the analysis and processing of signals produced from [nonlinear systems](/source/Nonlinear_system) and can be in the time, [frequency](/source/Frequency), or spatiotemporal domains.[8][9] Nonlinear systems can produce highly complex behaviors including [bifurcations](/source/Bifurcation_theory), [chaos](/source/Chaos_theory), [harmonics](/source/Harmonics), and [subharmonics](/source/Subharmonics) which cannot be produced or analyzed using linear methods.

Polynomial signal processing is a type of non-linear signal processing, where [polynomial](/source/Polynomial) systems may be interpreted as conceptually straightforward extensions of linear systems to the nonlinear case.[10]

### Statistical

**Statistical signal processing** is an approach which treats signals as [stochastic processes](/source/Stochastic_process), utilizing their [statistical](/source/Statistical) properties to perform signal processing tasks.[11] Statistical techniques are widely used in signal processing applications. For example, one can model the [probability distribution](/source/Probability_distribution) of noise incurred when photographing an image, and construct techniques based on this model to [reduce the noise](/source/Noise_reduction) 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.[12] Graph signal processing presents several key points such as sampling signal techniques,[13] recovery techniques [14] and time-varying techiques.[15] Graph signal processing has been applied with success in the field of image processing, computer vision [16] [17] [18] and sound anomaly detection.[19]

## Application fields

Seismic signal processing

- [Audio signal processing](/source/Audio_signal_processing) – for electrical signals representing sound, such as [speech](/source/Speech_signal_processing) or music[20]

- [Image processing](/source/Image_processing) – in digital cameras, computers and various imaging systems

- [Video processing](/source/Video_processing) – for interpreting moving pictures

- [Wireless communication](/source/Wireless_communication) – waveform generations, demodulation, filtering, equalization

- [Control systems](/source/Control_systems)

- [Array processing](/source/Array_processing) – for processing signals from arrays of sensors

- [Process control](/source/Process_control) – a variety of signals are used, including the industry standard [4-20 mA current loop](/source/4-20_mA_current_loop)

- [Seismology](/source/Seismology)

- [Feature extraction](/source/Feature_extraction), such as [image understanding](/source/Image_understanding), [semantic audio](/source/Semantic_audio) and [speech recognition](/source/Speech_recognition).

- Quality improvement, such as [noise reduction](/source/Noise_reduction), [image enhancement](/source/Image_enhancement), and [echo cancellation](/source/Echo_cancellation).

- Source coding including [audio compression](/source/Audio_compression_(data)), [image compression](/source/Image_compression), and [video compression](/source/Video_compression).

- [Genomic](/source/Genomic) signal processing[21]

- In [geophysics](/source/Geophysics), signal processing is used to amplify the signal vs the noise within [time-series](/source/Time-series) measurements of geophysical data. Processing is conducted within the [time domain](/source/Time_domain) or [frequency domain](/source/Frequency_domain), or both.[22][23]

In communication systems, signal processing may occur at:[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

- OSI layer 1 in the seven-layer [OSI model](/source/OSI_model), the [physical layer](/source/Physical_layer) ([modulation](/source/Modulation), [equalization](/source/Equalization_(communications)), [multiplexing](/source/Multiplexing), etc.);

- OSI layer 2, the [data link layer](/source/Data_link_layer) ([forward error correction](/source/Forward_error_correction));

- OSI layer 6, the [presentation layer](/source/Presentation_layer) (source coding, including [analog-to-digital conversion](/source/Analog-to-digital_conversion) and [data compression](/source/Data_compression)).

## Typical devices

- [Filters](/source/Filter_(signal_processing)) – for example analog (passive or active) or digital ([FIR](/source/FIR_filter), [IIR](/source/IIR_filter), frequency domain or [stochastic filters](/source/Stochastic_filter), etc.)

- [Samplers](/source/Sampling_(signal_processing)) and [analog-to-digital converters](/source/Analog-to-digital_converter) for [signal acquisition](/source/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 processors](/source/Digital_signal_processor) (DSPs)

## Mathematical methods applied

- [Differential equations](/source/Differential_equations)[24] – 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](/source/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](/source/Recurrence_relation)[25]

- [Transform theory](/source/Transform_theory)

- [Time-frequency analysis](/source/Time-frequency_analysis) – for processing non-stationary signals[26]

- [Linear canonical transformation](/source/Linear_canonical_transformation)

- [Spectral estimation](/source/Spectral_estimation) – for determining the spectral content (i.e., the distribution of power over frequency) of a set of [time series](/source/Time_series) data points[27]

- [Statistical signal processing](/source/Statistical_signal_processing) – analyzing and extracting information from signals and noise based on their stochastic properties

- [Linear time-invariant system](/source/Linear_time-invariant_system) theory, and [transform theory](/source/Transform_theory)

- [Polynomial signal processing](/source/Polynomial_signal_processing) – analysis of systems which relate input and output using polynomials

- [System identification](/source/System_identification)[8] and classification

- [Calculus](/source/Calculus)

- [Coding theory](/source/Coding_theory)

- [Complex analysis](/source/Complex_analysis)[28]

- [Vector spaces](/source/Vector_spaces) and [Linear algebra](/source/Linear_algebra)[29]

- [Functional analysis](/source/Functional_analysis)[30]

- [Probability](/source/Probability) and [stochastic processes](/source/Stochastic_processes)[11]

- [Detection theory](/source/Detection_theory)

- [Estimation theory](/source/Estimation_theory)

- [Optimization](/source/Optimization)[31]

- [Numerical methods](/source/Numerical_methods)

- [Data mining](/source/Data_mining) – 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](/source/Algebraic_signal_processing)

- [Audio filter](/source/Audio_filter)

- [Bounded variation](/source/Bounded_variation)

- [Dynamic range compression](/source/Dynamic_range_compression)

- [Information theory](/source/Information_theory)

- [Least-squares spectral analysis](/source/Least-squares_spectral_analysis)

- [Non-local means](/source/Non-local_means)

- [Reverberation](/source/Reverberation)

- [Sensitivity (electronics)](/source/Sensitivity_(electronics))

- [Similarity (signal processing)](/source/Similarity_(signal_processing))

- [Wiener filter](/source/Wiener_filter)

## References

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1. **[^](#cite_ref-2)** Alan V. Oppenheim and Ronald W. Schafer (1989). *Discrete-Time Signal Processing*. Prentice Hall. p. 1. [ISBN](/source/ISBN_(identifier)) [0-13-216771-9](https://en.wikipedia.org/wiki/Special:BookSources/0-13-216771-9).

1. **[^](#cite_ref-3)** Oppenheim, Alan V.; Schafer, Ronald W. (1975). *Digital Signal Processing*. [Prentice Hall](/source/Prentice_Hall). p. 5. [ISBN](/source/ISBN_(identifier)) [0-13-214635-5](https://en.wikipedia.org/wiki/Special:BookSources/0-13-214635-5).

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1. ^ [***a***](#cite_ref-Billings_8-0) [***b***](#cite_ref-Billings_8-1) Billings, S. A. (2013). *Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains*. Wiley. [ISBN](/source/ISBN_(identifier)) [978-1-119-94359-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-119-94359-4).

1. **[^](#cite_ref-VSA_9-0)** Slawinska, J.; Ourmazd, A.; Giannakis, D. (2018). "A New Approach to Signal Processing of Spatiotemporal Data". *2018 IEEE Statistical Signal Processing Workshop (SSP)*. IEEE Xplore. pp. 338–342. [doi](/source/Doi_(identifier)):[10.1109/SSP.2018.8450704](https://doi.org/10.1109%2FSSP.2018.8450704). [ISBN](/source/ISBN_(identifier)) [978-1-5386-1571-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-5386-1571-3). [S2CID](/source/S2CID_(identifier)) [52153144](https://api.semanticscholar.org/CorpusID:52153144).

1. **[^](#cite_ref-10)** V. John Mathews; Giovanni L. Sicuranza (May 2000). *Polynomial Signal Processing*. Wiley. [ISBN](/source/ISBN_(identifier)) [978-0-471-03414-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-03414-8).

1. ^ [***a***](#cite_ref-Scharf_11-0) [***b***](#cite_ref-Scharf_11-1) Scharf, Louis L. (1991). *Statistical signal processing: detection, estimation, and time series analysis*. [Boston](/source/Boston): [Addison–Wesley](/source/Addison%E2%80%93Wesley). [ISBN](/source/ISBN_(identifier)) [0-201-19038-9](https://en.wikipedia.org/wiki/Special:BookSources/0-201-19038-9). [OCLC](/source/OCLC_(identifier)) [61160161](https://search.worldcat.org/oclc/61160161).

1. **[^](#cite_ref-Ortega_12-0)** Ortega, A. (2022). *Introduction to Graph Signal Processing*. [Cambridge](/source/Cambridge): [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [9781108552349](https://en.wikipedia.org/wiki/Special:BookSources/9781108552349).

1. **[^](#cite_ref-Tanaka_13-0)** Tanaka, Y.; Eldar, Y. (2020). "Generalized Sampling on Graphs with Subspace and Smoothness Prior". *IEEE Transactions on Signal Processing*. **68**: 2272–2286. [arXiv](/source/ArXiv_(identifier)):[1905.04441](https://arxiv.org/abs/1905.04441). [Bibcode](/source/Bibcode_(identifier)):[2020ITSP...68.2272T](https://ui.adsabs.harvard.edu/abs/2020ITSP...68.2272T). [doi](/source/Doi_(identifier)):[10.1109/TSP.2020.2982325](https://doi.org/10.1109%2FTSP.2020.2982325).

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1. **[^](#cite_ref-Giraldo1_16-0)** Giraldo, J.; Bouwmans, T. (October 2020). "Semi-Supervised Background Subtraction of Unseen Videos: Minimization of the Total Variation of Graph Signals". *2020 IEEE International Conference on Image Processing (ICIP)*. pp. 3224–3228. [doi](/source/Doi_(identifier)):[10.1109/ICIP40778.2020.9190887](https://doi.org/10.1109%2FICIP40778.2020.9190887). [ISBN](/source/ISBN_(identifier)) [978-1-7281-6395-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-7281-6395-6).

1. **[^](#cite_ref-Giraldo2_17-0)** Giraldo, J.; Bouwmans, T. (2020). "GraphBGS: Background Subtraction via Recovery of Graph Signals". *2020 25th International Conference on Pattern Recognition (ICPR)*. pp. 6881–6888. [arXiv](/source/ArXiv_(identifier)):[2001.06404](https://arxiv.org/abs/2001.06404). [doi](/source/Doi_(identifier)):[10.1109/ICPR48806.2021.9412999](https://doi.org/10.1109%2FICPR48806.2021.9412999). [ISBN](/source/ISBN_(identifier)) [978-1-7281-8808-9](https://en.wikipedia.org/wiki/Special:BookSources/978-1-7281-8808-9).

1. **[^](#cite_ref-Giraldo3_18-0)** Giraldo, J.; Javed, S.; Sultana, M.; Jung, S.; Bouwmans, T. (February 2021). ["The Emerging Field of Graph Signal Processing for Moving Object Segmentation"](https://link.springer.com/chapter/10.1007/978-3-030-81638-4_3). *Frontiers of Computer Vision*. Communications in Computer and Information Science. Vol. 1405. pp. 31–45. [doi](/source/Doi_(identifier)):[10.1007/978-3-030-81638-4_3](https://doi.org/10.1007%2F978-3-030-81638-4_3). [ISBN](/source/ISBN_(identifier)) [978-3-030-81637-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-81637-7).

1. **[^](#cite_ref-Bouwmans1_19-0)** Mnasri, Zied; Giraldo, Jhony H.; Bouwmans, Thierry (2024). ["Anomalous Sound Detection for Road Surveillance Based on Graph Signal Processing"](https://hal.science/hal-04756448). *2024 32nd European Signal Processing Conference (EUSIPCO)*. pp. 161–165. [doi](/source/Doi_(identifier)):[10.23919/EUSIPCO63174.2024.10715291](https://doi.org/10.23919%2FEUSIPCO63174.2024.10715291). [ISBN](/source/ISBN_(identifier)) [978-9-4645-9361-7](https://en.wikipedia.org/wiki/Special:BookSources/978-9-4645-9361-7).

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1. **[^](#cite_ref-23)** Reynolds, John M. (2011). *An Introduction to Applied and Environmental Geophysics*. [Wiley-Blackwell](/source/Wiley-Blackwell). [ISBN](/source/ISBN_(identifier)) [978-0-471-48535-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-48535-3).

1. **[^](#cite_ref-Gaydecki2004_24-0)** Patrick Gaydecki (2004). [*Foundations of Digital Signal Processing: Theory, Algorithms and Hardware Design*](https://books.google.com/books?id=6Qo7NvX3vz4C&q=%22differential+equation%22+OR+%22differential+equations%22&pg=PA40). IET. pp. 40–. [ISBN](/source/ISBN_(identifier)) [978-0-85296-431-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-85296-431-6).

1. **[^](#cite_ref-Engelberg2008_25-0)** Shlomo Engelberg (8 January 2008). [*Digital Signal Processing: An Experimental Approach*](https://books.google.com/books?id=z3CpcCHbtgIC). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-84800-119-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84800-119-0).

1. **[^](#cite_ref-26)** Boashash, Boualem, ed. (2003). *Time frequency signal analysis and processing a comprehensive reference* (1 ed.). Amsterdam: Elsevier. [ISBN](/source/ISBN_(identifier)) [0-08-044335-4](https://en.wikipedia.org/wiki/Special:BookSources/0-08-044335-4).

1. **[^](#cite_ref-27)** Stoica, Petre; Moses, Randolph (2005). [*Spectral Analysis of Signals*](http://user.it.uu.se/%7Eps/SAS-new.pdf) (PDF). NJ: Prentice Hall.

1. **[^](#cite_ref-SchreierScharf2010_28-0)** Peter J. Schreier; Louis L. Scharf (4 February 2010). [*Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals*](https://books.google.com/books?id=HBaxLfDsAHoC&q=%22complex+analysis%22). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-1-139-48762-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-139-48762-7).

1. **[^](#cite_ref-Little2019_29-0)** Max A. Little (13 August 2019). [*Machine Learning for Signal Processing: Data Science, Algorithms, and Computational Statistics*](https://books.google.com/books?id=ejGoDwAAQBAJ&q=%22vector+space%22). OUP Oxford. [ISBN](/source/ISBN_(identifier)) [978-0-19-102431-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-102431-3).

1. **[^](#cite_ref-DamelinJr2012_30-0)** Steven B. Damelin; Willard Miller, Jr (2012). [*The Mathematics of Signal Processing*](https://books.google.com/books?id=MtPLYXQ9d9MC&q=%22functional+analysis%22). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-1-107-01322-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-01322-3).

1. **[^](#cite_ref-PalomarEldar2010_31-0)** Daniel P. Palomar; Yonina C. Eldar (2010). [*Convex Optimization in Signal Processing and Communications*](https://books.google.com/books?id=UOpnvPJ151gC). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-0-521-76222-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-76222-9).

## Further reading

- Byrne, Charles (2014). [*Signal Processing: A Mathematical Approach*](https://www.taylorfrancis.com/books/oa-mono/10.1201/b17672/signal-processing-charles-byrne). [Taylor & Francis](/source/Taylor_%26_Francis). [doi](/source/Doi_(identifier)):[10.1201/b17672](https://doi.org/10.1201%2Fb17672). [ISBN](/source/ISBN_(identifier)) [9780429158711](https://en.wikipedia.org/wiki/Special:BookSources/9780429158711).

- P Stoica, R Moses (2005). [*Spectral Analysis of Signals*](https://user.it.uu.se/%7Eps/SAS-new.pdf) (PDF). NJ: Prentice Hall.

- Papoulis, Athanasios (1991). *Probability, Random Variables, and Stochastic Processes* (third ed.). McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [0-07-100870-5](https://en.wikipedia.org/wiki/Special:BookSources/0-07-100870-5).

- [Ali H. Sayed](/source/Ali_H._Sayed), Adaptive Filters, Wiley, NJ, 2008, [ISBN](/source/ISBN_(identifier)) [978-0-470-25388-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-470-25388-5).

- [Thomas Kailath](/source/Thomas_Kailath), [Ali H. Sayed](/source/Ali_H._Sayed), and [Babak Hassibi](/source/Babak_Hassibi), Linear Estimation, Prentice-Hall, NJ, 2000, [ISBN](/source/ISBN_(identifier)) [978-0-13-022464-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-022464-4).

- [Signal Processing for Communications](https://sp4comm.org) – free online textbook by Paolo Prandoni and Martin Vetterli (2008)

- [Scientists and Engineers Guide to Digital Signal Processing](https://dspguide.com) – free online textbook by Stephen Smith

## External links

- [Julius O. Smith III: Spectral Audio Signal Processing](https://www.dsprelated.com/freebooks/sasp/) – free online textbook

- [Graph Signal Processing Website](https://sites.google.com/view/gsp-website/graph-signal-processing) – free online website by Thierry Bouwmans (2025)

v t e Digital signal processing Theory Detection theory Discrete signal Estimation theory Nyquist–Shannon sampling theorem Sub-fields Audio signal processing Digital image processing Speech processing Statistical signal processing Techniques Z-transform Advanced z-transform Matched Z-transform method Bilinear transform Constant-Q transform Discrete cosine transform (DCT) Discrete Fourier transform (DFT) Discrete-time Fourier transform (DTFT) Impulse invariance Integral transform Laplace transform Post's inversion formula Starred transform Zak transform Sampling Aliasing Anti-aliasing filter Downsampling Nyquist rate / frequency Oversampling Quantization Sampling rate Undersampling Upsampling

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