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| '''Digital signal processing''' ('''DSP''') is the mathematical manipulation of an information signal to modify or improve it in some way. It is characterized by the representation of discrete time, discrete frequency, or other discrete domain [[signal (electronics)|signal]]s by a sequence of numbers or symbols and the processing of these signals.
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| The goal of DSP is usually to measure, filter and/or compress continuous real-world [[analog signals]]. The first step is usually to convert the signal from an analog to a digital form, by [[Sampling (signal processing)|sampling]] and then digitizing it using an [[analog-to-digital converter]] (ADC), which turns the analog signal into a stream of numbers. However, often, the required output signal is another analog output signal, which requires a [[digital-to-analog converter]] (DAC). Even if this process is more complex than analog processing and has a [[discrete signal|discrete value range]], the application of computational power to digital signal processing allows for many advantages over analog processing in many applications, such as [[error detection|error detection and correction]] in transmission as well as [[data compression]].<ref>{{cite book | title = Digital Signal Processing: Instant access | author = James D. Broesch, Dag Stranneby and William Walker | publisher = Butterworth-Heinemann | page = 3}}</ref>
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| Digital signal processing and [[analog signal processing]] are subfields of [[signal processing]]. DSP applications include: [[audio signal processing|audio]] and [[speech signal processing]], sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing, [[digital image processing]], signal processing for communications, control of systems, biomedical signal processing, seismic data processing, etc. DSP [[algorithm]]s have long been run on standard computers, as well as on specialized processors called [[digital signal processor]] and on purpose-built hardware such as [[application-specific integrated circuit]] (ASICs). Today there are additional technologies used for digital signal processing including more powerful general purpose [[microprocessor]]s, [[field-programmable gate array]]s (FPGAs), [[Digital Signal Controller|digital signal controllers]] (mostly for industrial apps such as motor control), and [[stream processing|stream processors]], among others.<ref>{{cite book | title = Digital Signal Processing and Applications | author = Dag Stranneby and William Walker | edition = 2nd ed. | publisher = Elsevier | year = 2004 | isbn = 0-7506-6344-8 | url = http://books.google.com/books?id=NKK1DdqcDVUC&pg=PA241&dq=called+digital+signal+processor+hardware+application-specific+integrated+circuit+general-purpose+microprocessors+field-programmable+gate+arrays+dsp+asic+fpga+stream}}</ref>
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| Digital signal processing can involve linear or nonlinear operations. Nonlinear signal processing is closely related to [[nonlinear system identification]] <ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> and can be implemented in the time, frequency, and spatio-temporal domains.
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| == Signal sampling ==
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| {{Main|Sampling (signal processing)}}
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| With the increasing use of [[computer]]s the usage of and need for digital signal processing has increased. To use an analog signal on a computer, it must be digitized with an analog-to-digital converter. Sampling is usually carried out in two stages, [[discretization]] and [[Quantization (signal processing)|quantization]]. In the discretization stage, the space of signals is partitioned into [[equivalence class]]es and quantization is carried out by replacing the signal with representative signal of the corresponding equivalence class. In the quantization stage the representative signal values are approximated by values from a finite set.
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| The [[Nyquist–Shannon sampling theorem]] states that a signal can be exactly reconstructed from its samples if the [[sampling frequency]] is greater than twice the highest frequency of the signal; but requires an infinite number of samples. In practice, the sampling frequency is often significantly more than twice that required by the signal's limited bandwidth.
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| Some (continuous-time) periodic signals become non-periodic after sampling, and some non-periodic signals become periodic after sampling. In general, for a periodic signal with period ''T'' to be periodic (with period ''N'') after sampling with [[sampling interval]] ''T<sub>s</sub>'', the following must be satisfied:
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| :<math>T_s N=Tk</math>
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| where ''k is an integer.<ref>{{Citation |last=Oppenheim |first=Alan V. |author1-link=Alan V. Oppenheim |first2=Ronald W. |last2=Schafer |author2-link=Ronald W. Schafer |first3=John R. |last3=Buck |title=Discrete-Time Signal Processing |edition=3rd |year= |publisher= |isbn= |page=15}}</ref>
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| == DSP domains ==
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| In DSP, engineers usually study digital signals in one of the following domains: [[time domain]] (one-dimensional signals), spatial domain (multidimensional signals), [[frequency domain]], and [[wavelet]] domains. They choose the domain in which to process a signal by making an informed guess (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal. A sequence of samples from a measuring device produces a time or spatial domain representation, whereas a [[discrete Fourier transform]] produces the frequency domain information, that is the [[frequency spectrum]]. Autocorrelation is defined as the [[cross-correlation]] of the signal with itself over varying intervals of time or space.
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| === Time and space domains ===
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| {{Main|Time domain}}
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| The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. [[Digital filter]]ing generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. There are various ways to characterize filters; for example:
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| * A "linear" filter is a [[linear transformation]] of input samples; other filters are "non-linear". Linear filters satisfy the superposition condition, i.e. if an input is a weighted linear combination of different signals, the output is an equally weighted linear combination of the corresponding output signals.
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| * A "causal" filter uses only previous samples of the input or output signals; while a "non-causal" filter uses future input samples. A non-causal filter can usually be changed into a causal filter by adding a delay to it.
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| * A "time-invariant" filter has constant properties over time; other filters such as [[adaptive filter]]s change in time.
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| * A "stable" filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An "unstable" filter can produce an output that grows without bounds, with bounded or even zero input.
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| * A "finite impulse response" ([[Finite impulse response|FIR]]) filter uses only the input signals, while an "infinite impulse response" filter ([[Infinite impulse response|IIR]]) uses both the input signal and previous samples of the output signal. FIR filters are always stable, while IIR filters may be unstable.
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| A filter can be represented by a [[block diagram]], which can then be used to derive a sample processing [[algorithm]] to implement the filter with hardware instructions. A filter may also be described as a [[difference equation]], a collection of [[Zero (complex analysis)|zeroes]] and [[pole (complex analysis)|pole]]s or, if it is an FIR filter, an [[impulse response]] or [[step response]].
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| The output of a linear digital filter to any given input may be calculated by [[convolution|convolving]] the input signal with the [[impulse response]].
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| === Frequency domain ===
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| {{Main|Frequency domain}}
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| Signals are converted from time or space domain to the frequency domain usually through the [[Fourier transform]]. The Fourier transform converts the signal information to a magnitude and phase component of each frequency. Often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.
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| The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to determine which frequencies are present in the input signal and which are missing.
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| In addition to frequency information, phase information is often needed. This can be obtained from the Fourier transform. With some applications, how the phase varies with frequency can be a significant consideration.
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| Filtering, particularly in non-realtime work can also be achieved by converting to the frequency domain, applying the filter and then converting back to the time domain. This is a fast, O(n log n) operation, and can give essentially any filter shape including excellent approximations to [[brickwall filter]]s.
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| There are some commonly used frequency domain transformations. For example, the [[cepstrum]] converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the harmonic structure of the original spectrum.
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| Frequency domain analysis is also called ''spectrum-'' or ''spectral analysis''.
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| ===Z-plane analysis===
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| {{main|Z-transform}}
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| Whereas analog filters are usually analyzed in terms of [[transfer function]]s in the [[s plane]] using [[Laplace transform]]s, digital filters are analyzed in the z plane in terms of Z-transforms. A digital filter may be described in the z plane by its characteristic collection of [[Zero (complex analysis)|zeroes]] and [[pole (complex analysis)|poles]]. The z plane provides a means for mapping digital frequency (samples/second) to real and imaginary z components, where <math>z=re^{j\Omega} </math> for continuous periodic signals and <math>\Omega = 2 \ F</math> (<math>F</math> is the digital frequency). This is useful for providing a visualization of the frequency response of a digital system or signal.
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| ===Wavelet===
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| {{main|Discrete wavelet transform}}
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| [[Image:Jpeg2000 2-level wavelet transform-lichtenstein.png|thumb|300px|An example of the 2D discrete wavelet transform that is used in [[JPEG2000]]. The original image is high-pass filtered, yielding the three large images, each describing local changes in brightness (details) in the original image. It is then low-pass filtered and downscaled, yielding an approximation image; this image is high-pass filtered to produce the three smaller detail images, and low-pass filtered to produce the final approximation image in the upper-left.]]
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| In [[numerical analysis]] and [[functional analysis]], a '''discrete wavelet transform''' (DWT) is any [[wavelet transform]] for which the [[wavelet]]s are discretely sampled. As with other wavelet transforms, a key advantage it has over [[Fourier transform]]s is temporal resolution: it captures both frequency ''and'' location information (location in time).
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| == Applications ==
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| The main applications of DSP are [[audio signal processing]], [[audio compression (data)|audio compression]], [[digital image processing]], [[video compression]], [[speech processing]], [[speech recognition]], [[digital communication]]s, [[RADAR]], [[SONAR]], [[seismology]] and [[biomedicine]]. Specific examples are [[speech encoding|speech compression]] and transmission in digital [[mobile phone]]s, [[digital room correction|room correction]] of sound in [[hi-fi]] and [[sound reinforcement]] applications, [[weather forecasting]], [[economic forecasting]], [[seismology|seismic]] data processing, analysis and control of [[industrial process]]es, [[medical imaging]] such as [[Computed axial tomography|CAT]] scans and [[MRI]], [[MP3]] compression, [[computer graphics]], [[image manipulation]], hi-fi [[loudspeaker]] [[audio crossover|crossovers]] and [[equalization]], and [[audio signal processing|audio effects]] for use with [[electric guitar]] [[amplifiers]].
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| == Implementation ==
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| Depending on the requirements of the application, digital signal processing tasks can be implemented on [[general purpose computer]]s (e.g. [[supercomputer]]s, [[mainframe computer]]s, or [[personal computer]]s) or with [[embedded processor]]s that may or may not include specialized [[microprocessor]]s called [[digital signal processor]]s.
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| Often when the processing requirement is not [[real-time computing|real-time]], processing is economically done with an existing general-purpose computer and the signal data (either input or output) exists in data files. This is essentially no different than any other data processing, except DSP mathematical techniques (such as the [[Fast Fourier transform|FFT]]) are used, and the sampled data is usually assumed to be uniformly sampled in time or space. For example: processing [[digital photograph]]s with software such as [[Photoshop]].
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| However, when the application requirement is [[real-time computing|real-time]], DSP is often implemented using [[Digital signal processor|specialized microprocessors]] such as the [[Motorola 56000|DSP56000]], the [[TMS320]], or the [[Super Harvard Architecture Single-Chip Computer|SHARC]]. These often process data using [[fixed-point arithmetic]], though some more powerful versions use [[floating point arithmetic]]. For faster applications [[FPGA]]s<ref>{{cite web
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| | last = JpFix
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| | title = FPGA-Based Image Processing Accelerator
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| | url = http://www.jpfix.com/About_Us/Articles/FPGA-Based_Image_Processing_Ac/fpga-based_image_processing_ac.html
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| | accessdate = 2008-05-10
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| }}</ref> might be used.
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| Beginning in 2007, multicore implementations of DSPs have started to emerge from companies including [[Freescale]] and [[Stream Processors, Inc]]. For faster applications with vast usage, [[Application-specific integrated circuit|ASIC]]s might be designed specifically. For slow applications, a traditional slower processor such as a microcontroller may be adequate. Also a growing number of DSP applications are now being implemented on [[Embedded Systems]] using powerful PCs with a [[Multi-core processor]].
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| == Techniques ==
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| <div style="-moz-column-count:2; column-count:2;">
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| * [[Bilinear transform]]
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| * [[Discrete Fourier transform]]
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| * [[Discrete-time Fourier transform]]
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| * [[Filter design]]
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| * [[LTI system theory]]
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| * [[Minimum phase]]
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| * [[Transfer function]]
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| * [[Z-transform]]
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| * [[Goertzel algorithm]]
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| * [[s-plane]]
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| </div>
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| == Related fields ==
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| * [[Analog signal processing]]
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| * [[Automatic control]]
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| * [[Computer Engineering]]
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| * [[Computer Science]]
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| * [[Data compression]]
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| * [[Dataflow programming]]
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| * [[Electrical engineering]]
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| * [[Fourier Analysis]]
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| * [[Information theory]]
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| * [[Machine Learning]]
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| * [[Real-time computing]]
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| * [[Stream processing]]
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| * [[Telecommunication]]
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| * [[Time series]]
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| * [[Wavelet]]
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| == References ==
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| {{Reflist}}
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| == Further reading ==
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| {{wikibooks|Digital Signal Processing}}
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| <div class="references-small">
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| *[[Alan V. Oppenheim]], [[Ronald W. Schafer]], John R. Buck : ''Discrete-Time Signal Processing'', Prentice Hall, ISBN 0-13-754920-2
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| *Boaz Porat: ''A Course in Digital Signal Processing'', Wiley, ISBN 0-471-14961-6
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| *Richard G. Lyons: ''Understanding Digital Signal Processing'', Prentice Hall, ISBN 0-13-108989-7
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| *Jonathan Yaakov Stein, ''Digital Signal Processing, a Computer Science Perspective'', Wiley, ISBN 0-471-29546-9
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| *Sen M. Kuo, Woon-Seng Gan: ''Digital Signal Processors: Architectures, Implementations, and Applications'', Prentice Hall, ISBN 0-13-035214-4
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| *Bernard Mulgrew, Peter Grant, John Thompson: ''Digital Signal Processing - Concepts and Applications'', Palgrave Macmillan, ISBN 0-333-96356-3
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| *{{cite book |author=Steven W. Smith |title=Digital Signal Processing: A Practical Guide for Engineers and Scientists |publisher=Newnes |date=2002 |isbn=0-7506-7444-X |url=http://www.dspguide.com}}
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| *Paul A. Lynn, Wolfgang Fuerst: ''Introductory Digital Signal Processing with Computer Applications'', John Wiley & Sons, ISBN 0-471-97984-8
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| *James D. Broesch: ''Digital Signal Processing Demystified'', Newnes, ISBN 1-878707-16-7
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| *John G. Proakis, Dimitris Manolakis: ''Digital Signal Processing: Principles, Algorithms and Applications'', 4th ed, Pearson, April 2006, ISBN 978-0131873742
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| *Hari Krishna Garg: ''Digital Signal Processing Algorithms'', CRC Press, ISBN 0-8493-7178-3
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| *P. Gaydecki: ''Foundations Of Digital Signal Processing: Theory, Algorithms And Hardware Design'', Institution of Electrical Engineers, ISBN 0-85296-431-5
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| *Paul M. Embree, Damon Danieli: ''C++ Algorithms for Digital Signal Processing'', Prentice Hall, ISBN 0-13-179144-3
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| *Vijay Madisetti, Douglas B. Williams: ''The Digital Signal Processing Handbook'', CRC Press, ISBN 0-8493-8572-5
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| *Stergios Stergiopoulos: ''Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging Real-Time Systems'', CRC Press, ISBN 0-8493-3691-0
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| *Joyce Van De Vegte: ''Fundamentals of Digital Signal Processing'', Prentice Hall, ISBN 0-13-016077-6
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| *Ashfaq Khan: ''Digital Signal Processing Fundamentals'', Charles River Media, ISBN 1-58450-281-9
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| *Jonathan M. Blackledge, Martin Turner: ''Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications'', Horwood Publishing, ISBN 1-898563-48-9
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| *Doug Smith: ''Digital Signal Processing Technology: Essentials of the Communications Revolution'', American Radio Relay League, ISBN 0-87259-819-5
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| *Charles A. Schuler: ''Digital Signal Processing: A Hands-On Approach'', McGraw-Hill, ISBN 0-07-829744-3
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| *[[James H. McClellan]], [[Ronald W. Schafer]], Mark A. Yoder: ''Signal Processing First'', Prentice Hall, ISBN 0-13-090999-8
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| *John G. Proakis: ''A Self-Study Guide for Digital Signal Processing'', Prentice Hall, ISBN 0-13-143239-7
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| *[[N. Ahmed]] and K.R. Rao (1975). Orthogonal Transforms for Digital Signal Processing. Springer-Verlag (Berlin – Heidelberg – New York), ISBN 3-540-06556-3.
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| </div>
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| {{Digital systems}}
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| {{DSP}}
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| {{DEFAULTSORT:Digital Signal Processing}}
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| [[Category:Digital electronics]]
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| [[Category:Digital signal processing|*]]
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| [[Category:Telecommunication theory]]
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| [[Category:Radar signal processing]]
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