Digital signal processing (DSP) is the numerical manipulation of signals, usually with the intention to measure, filter, produce or compress continuous analog signals. It is characterized by the use of digital signals to represent these signals as discrete time, discrete frequency, or other discrete domain signals in the form of a sequence of numbers or symbols to permit the digital processing of these signals.
Theoretical analyses and derivations are typically performed on discretetime signal models, created by the abstract process of sampling. Numerical methods require a digital signal, such as those produced by an analogtodigital converter (ADC). The processed result might be a frequency spectrum or a set of statistics. But often it is another digital signal that is converted back to analog form by a digitaltoanalog converter (DAC). Even if that whole sequence is more complex than analog processing and has a discrete value range, the application of computational power to signal processing allows for many advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression.^{[1]}
Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include 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, among others. DSP algorithms have long been run on standard computers, as well as on specialized processors called digital signal processors, and on purposebuilt hardware such as applicationspecific integrated circuit (ASICs). Currently, there are additional technologies used for digital signal processing including more powerful general purpose microprocessors, fieldprogrammable gate arrays (FPGAs), digital signal controllers (mostly for industrial applications such as motor control), and stream processors, among others.^{[2]}
Digital signal processing can involve linear or nonlinear operations. Nonlinear signal processing is closely related to nonlinear system identification^{[3]} and can be implemented in the time, frequency, and spatiotemporal domains.
Contents

Signal sampling 1

DSP domains 2

Time and space domains 2.1

Frequency domain 2.2

Zplane analysis 2.3

Wavelet 2.4

Applications 3

Implementation 4

Techniques 5

Related fields 6

References 7

Further reading 8
Signal sampling
The increasing use of computers has resulted in the increased use of, and need for, digital signal processing. To digitally analyze and manipulate an analog signal, it must be digitized with an analogtodigital converter. Sampling is usually carried out in two stages, discretization and quantization. In the discretization stage, the space of signals is partitioned into equivalence classes 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.
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 this requires an infinite number of samples. In practice, the sampling frequency is often significantly higher than twice that required by the signal's limited bandwidth.
Some (continuoustime) periodic signals become nonperiodic after sampling, and some nonperiodic 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_{s}, the following must be satisfied:

T_s N=Tk
where k is an integer.^{[4]}
DSP domains
In DSP, engineers usually study digital signals in one of the following domains: time domain (onedimensional signals), spatial domain (multidimensional signals), frequency domain, and wavelet domains. They choose the domain in which to process a signal by making an informed assumption (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 temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information, that is, the frequency spectrum. Autocorrelation is defined as the crosscorrelation of the signal with itself over varying intervals of time or space.
Time and space domains
The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Digital filtering 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:

A "linear" filter is a linear transformation of input samples; other filters are "nonlinear". Linear filters satisfy the superposition condition, i.e. if an input is a weighted linear combination of different signals, the output is a similarly weighted linear combination of the corresponding output signals.

A "causal" filter uses only previous samples of the input or output signals; while a "noncausal" filter uses future input samples. A noncausal filter can usually be changed into a causal filter by adding a delay to it.

A "timeinvariant" filter has constant properties over time; other filters such as adaptive filters change in time.

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.

A "finite impulse response" (FIR) filter uses only the input signals, while an "infinite impulse response" filter (IIR) uses both the input signal and previous samples of the output signal. FIR filters are always stable, while IIR filters may be unstable.
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 zeroes and poles or, if it is an FIR filter, an impulse response or step response.
The output of a linear digital filter to any given input may be calculated by convolving the input signal with the impulse response.
Frequency domain
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.
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.
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.
Filtering, particularly in nonrealtime 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 filters.
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.
Frequency domain analysis is also called spectrum or spectral analysis.
Zplane analysis
Whereas analog filters are usually analyzed in terms of transfer functions in the s plane using Laplace transforms, digital filters are analyzed in the z plane in terms of Ztransforms. A digital filter may be described in the z plane by its characteristic collection of zeroes and poles. The z plane provides a means for mapping digital frequency (samples/second) to real and imaginary z components, where z=re^{j\omega} for continuous periodic signals and \omega = 2 \pi F (F is the digital frequency). This is useful for providing a visualization of the frequency response of a digital system or signal.
Wavelet
An example of the 2D discrete wavelet transform that is used in
JPEG2000. The original image is highpass filtered, yielding the three large images, each describing local changes in brightness (details) in the original image. It is then lowpass filtered and downscaled, yielding an approximation image; this image is highpass filtered to produce the three smaller detail images, and lowpass filtered to produce the final approximation image in the upperleft.
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time).
Applications
The main applications of DSP are audio signal processing, audio compression, digital image processing, video compression, speech processing, speech recognition, digital communications, radar, sonar, financial signal processing, seismology and biomedicine. Specific examples are speech compression and transmission in digital mobile phones, room correction of sound in hifi and sound reinforcement applications, weather forecasting, economic forecasting, seismic data processing, analysis and control of industrial processes, medical imaging such as CAT scans and MRI, MP3 compression, computer graphics, image manipulation, hifi loudspeaker crossovers and equalization, and audio effects for use with electric guitar amplifiers.
Implementation
Depending on the requirements of the application, digital signal processing tasks can be implemented on general purpose computers.
Often when the processing requirement is not realtime, processing is economically done with an existing generalpurpose computer and the signal data (either input or output) exists in data files. This is essentially no different from any other data processing, except DSP mathematical techniques (such as the FFT) are used, and the sampled data is usually assumed to be uniformly sampled in time or space. For example: processing digital photographs with software such as Photoshop.
However, when the application requirement is realtime, DSP is often implemented using specialized microprocessors such as the DSP56000, the TMS320, or the SHARC. These often process data using fixedpoint arithmetic, though some more powerful versions use floating point. For faster applications FPGAs^{[5]} might be used. 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, ASICs 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 multicore processors.
Techniques
Related fields
References

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^ Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and SpatioTemporal Domains". Wiley, 2013

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Further reading

Alan V. Oppenheim, Ronald W. Schafer, John R. Buck : DiscreteTime Signal Processing, Prentice Hall, ISBN 0137549202

Boaz Porat: A Course in Digital Signal Processing, Wiley, ISBN 0471149616

Richard G. Lyons: Understanding Digital Signal Processing, Prentice Hall, ISBN 0131089897

Jonathan Yaakov Stein, Digital Signal Processing, a Computer Science Perspective, Wiley, ISBN 0471295469

Sen M. Kuo, WoonSeng Gan: Digital Signal Processors: Architectures, Implementations, and Applications, Prentice Hall, ISBN 0130352144

Bernard Mulgrew, Peter Grant, John Thompson: Digital Signal Processing  Concepts and Applications, Palgrave Macmillan, ISBN 0333963563


Paul A. Lynn, Wolfgang Fuerst: Introductory Digital Signal Processing with Computer Applications, John Wiley & Sons, ISBN 0471979848

James D. Broesch: Digital Signal Processing Demystified, Newnes, ISBN 1878707167

John G. Proakis, Dimitris Manolakis: Digital Signal Processing: Principles, Algorithms and Applications, 4th ed, Pearson, April 2006, ISBN 9780131873742

Hari Krishna Garg: Digital Signal Processing Algorithms, CRC Press, ISBN 0849371783

P. Gaydecki: Foundations Of Digital Signal Processing: Theory, Algorithms And Hardware Design, Institution of Electrical Engineers, ISBN 0852964315

Paul M. Embree, Damon Danieli: C++ Algorithms for Digital Signal Processing, Prentice Hall, ISBN 0131791443

Vijay Madisetti, Douglas B. Williams: The Digital Signal Processing Handbook, CRC Press, ISBN 0849385725

Stergios Stergiopoulos: Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging RealTime Systems, CRC Press, ISBN 0849336910

Joyce Van De Vegte: Fundamentals of Digital Signal Processing, Prentice Hall, ISBN 0130160776

Ashfaq Khan: Digital Signal Processing Fundamentals, Charles River Media, ISBN 1584502819

Jonathan M. Blackledge, Martin Turner: Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications, Horwood Publishing, ISBN 1898563489

Doug Smith: Digital Signal Processing Technology: Essentials of the Communications Revolution, American Radio Relay League, ISBN 0872598195

Charles A. Schuler: Digital Signal Processing: A HandsOn Approach, McGrawHill, ISBN 0078297443

James H. McClellan, Ronald W. Schafer, Mark A. Yoder: Signal Processing First, Prentice Hall, ISBN 0130909998

John G. Proakis: A SelfStudy Guide for Digital Signal Processing, Prentice Hall, ISBN 0131432397

N. Ahmed and K.R. Rao (1975). Orthogonal Transforms for Digital Signal Processing. SpringerVerlag (Berlin – Heidelberg – New York), ISBN 3540065563.
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