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**Digital signal processing** (**DSP**) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency.

A **digital signal processor** (**DSP**) is a specialized microprocessor, with its architecture optimized for the operational needs of digital signal processing.

**Signal processing** is a subfield of mathematics, information and electrical engineering that concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal.

In signal processing, **sampling** is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of samples.

- Signal sampling
- Domains
- Time and space domains
- Frequency domain
- Z-plane analysis
- Wavelet
- Applications
- Implementation
- Techniques
- Related fields
- References
- Further reading

Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, signal processing for telecommunications, control systems, biomedical engineering, seismology, among others.

**Analog signal processing** is a type of signal processing conducted on continuous analog signals by some analog means. "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal. Analog values are typically represented as a voltage, electric current, or electric charge around components in the electronic devices. An error or noise affecting such physical quantities will result in a corresponding error in the signals represented by such physical quantities.

**Audio signal processing** is a subfield of signal processing that is concerned with the electronic manipulation of audio signals. As audio signals may be represented in either digital or analog format, processing may occur in either domain. Analog processors operate directly on the electrical signal, while digital processors operate mathematically on the digital representation of that signal.

**Speech processing** is the study of speech signals and the processing methods of signals. The signals are usually processed in a digital representation, so speech processing can be regarded as a special case of digital signal processing, applied to speech signals. Aspects of speech processing includes the acquisition, manipulation, storage, transfer and output of speech signals. The input is called speech recognition and the output is called speech synthesis.

DSP can involve linear or nonlinear operations. Nonlinear signal processing is closely related to nonlinear system identification ^{ [1] } and can be implemented in the time, frequency, and spatio-temporal domains.

System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs. The applications of system identification include any system where the inputs and outputs can be measured and include industrial processes, control systems, economic data, biology and the life sciences, medicine, social systems and many more.

**Time domain** is the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various separate instants in the case of discrete time. An oscilloscope is a tool commonly used to visualize real-world signals in the time domain. A time-domain graph shows how a signal changes with time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.

In electronics, control systems engineering, and statistics, the **frequency domain** refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

The application of digital computation 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.^{ [2] } DSP is applicable to both streaming data and static (stored) data.

In information theory and coding theory with applications in computer science and telecommunication, **error detection and correction** or **error control** are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases.

In signal processing, **data compression**, **source coding**, or **bit-rate reduction** involves encoding information using fewer bits than the original representation. Compression can be either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information.

To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter (ADC).^{ [3] } Sampling is usually carried out in two stages, discretization and quantization. Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude. Quantization means each amplitude measurement is approximated by a value from a finite set. Rounding real numbers to integers is an example.

In electronics, an **analog-to-digital converter** is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an input analog voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.

In applied mathematics, **discretization** is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. **Dichotomization** is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable.

**Quantization**, in mathematics and digital signal processing, is the process of mapping input values from a large set to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms.

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 component in the signal. In practice, the sampling frequency is often significantly higher than twice the Nyquist frequency.^{ [4] }

In the field of digital signal processing, the **sampling theorem** is a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of *samples* to capture all the information from a continuous-time signal of finite bandwidth.

The **Nyquist frequency**, named after electronic engineer Harry Nyquist, is half of the sampling rate of a discrete signal processing system. It is sometimes known as the folding frequency of a sampling system. An example of folding is depicted in Figure 1, where f_{s} is the sampling rate and 0.5 f_{s} is the corresponding Nyquist frequency. The black dot plotted at 0.6 f_{s} represents the amplitude and frequency of a sinusoidal function whose frequency is 60% of the sample-rate (f_{s}). The other three dots indicate the frequencies and amplitudes of three other sinusoids that would produce the same set of samples as the actual sinusoid that was sampled. The symmetry about 0.5 f_{s} is referred to as *folding*.

Theoretical DSP analyses and derivations are typically performed on discrete-time signal models with no amplitude inaccuracies (quantization error), "created" by the abstract process of sampling. Numerical methods require a quantized signal, such as those produced by an ADC. The processed result might be a frequency spectrum or a set of statistics. But often it is another quantized signal that is converted back to analog form by a digital-to-analog converter (DAC).

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 assumption (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal and the processing to be applied to it. A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain representation.

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 principle, 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*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. - A
*time-invariant*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 (IIR) filter 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 zeros and poles or 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.

Signals are converted from time or space domain to the frequency domain usually through use of the Fourier transform. The Fourier transform converts the time or space information to a magnitude and phase component of each frequency. With some applications, how the phase varies with frequency can be a significant consideration. Where phase is unimportant, 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. Frequency domain analysis is also called *spectrum-* or *spectral analysis*.

Filtering, particularly in non-realtime work can also be achieved in the frequency domain, applying the filter and then converting back to the time domain. This can be an efficient implementation and can give essentially any filter response 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.

Digital filters come in both IIR and FIR types. FIR filters have many advantages, but are computationally more demanding. Whereas FIR filters are always stable, IIR filters have feedback loops that may become unstable and oscillate. The Z-transform provides a tool for analyzing stability issues of digital IIR filters. It is analogous to the Laplace transform, which is used to design and analyze analog IIR filters.

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.The accuracy of the joint time-frequency resolution is limited by the uncertainty principle of time-frequency.

Applications of DSP include audio signal processing, audio compression, digital image processing, video compression, speech processing, speech recognition, digital communications, digital synthesizers, radar, sonar, financial signal processing, seismology and biomedicine. Specific examples include speech coding and transmission in digital mobile phones, room correction of sound in hi-fi 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, audio crossovers and equalization, and audio effects units.^{ [5] }

DSP algorithms may be run on general-purpose computers and digital signal processors. DSP algorithms are also implemented on purpose-built hardware such as application-specific integrated circuit (ASICs). Additional technologies for digital signal processing include more powerful general purpose microprocessors, field-programmable gate arrays (FPGAs), digital signal controllers (mostly for industrial applications such as motor control), and stream processors.^{ [6] }

For systems that do not a have real-time computing requirement and the signal data (either input or output) exists in data files, processing may be done economically with a general-purpose computer. 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. An example of such an application is processing digital photographs with software such as Photoshop.

When the application requirement is real-time, DSP is often implemented using specialized or dedicated processors or microprocessors, sometimes using multiple processors or multiple processing cores. These may process data using fixed-point arithmetic or floating point. For more demanding applications FPGAs may be used.^{ [7] } For the most demanding applications or high-volume products, ASICs might be designed specifically for the application.

**Linear filters** process time-varying input signals to produce output signals, subject to the constraint of linearity. This results from systems composed solely of components classified as having a linear response. Most filters implemented in analog electronics, in digital signal processing, or in mechanical systems are classified as causal, time invariant, and linear signal processing filters.

In mathematics, **Fourier analysis** is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

In signal processing, a **digital filter** is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is an electronic circuit operating on continuous-time analog signals.

A **wavelet** is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing. Using a "reverse, shift, multiply and integrate" technique called convolution, wavelets can be combined with known portions of a damaged signal to extract information from the unknown portions.

**Filter design** is the process of designing a signal processing filter that satisfies a set of requirements, some of which are contradictory. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient degree to make it useful.

**Frequency response** is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output. In addition, if the system is time-invariant, then the frequency response also will not vary with time. Thus for LTI systems, the frequency response can be seen as applying the system's transfer function to a purely imaginary number argument representing the frequency of the sinusoidal excitation.

In signal processing, a **finite impulse response** (**FIR**) **filter** is a filter whose impulse response is of *finite* duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

**Infinite impulse response** (**IIR**) is a property applying to many linear time-invariant systems. Common examples of linear time-invariant systems are most electronic and digital filters. Systems with this property are known as *IIR systems* or *IIR filters*, and are distinguished by having an impulse response which does not become exactly zero past a certain point, but continues indefinitely. This is in contrast to a finite impulse response (FIR) in which the impulse response *h*(*t*) *does* become exactly zero at times *t* > *T* for some finite *T*, thus being of finite duration.

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.

**Direct digital synthesis** (**DDS**) is a method employed by frequency synthesizers used for creating arbitrary waveforms from a single, fixed-frequency reference clock. DDS is used in applications such as signal generation, local oscillators in communication systems, function generators, mixers, modulators, sound synthesizers and as part of a digital phase-locked loop.

In digital signal processing, **downsampling** and **decimation** are terms associated with the process of resampling in a multi-rate digital signal processing system. Both terms are used by various authors to describe the entire process, which includes lowpass filtering, or just the part of the process that does not include filtering. When downsampling (decimation) is performed on a sequence of samples of a *signal* or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate. The *decimation factor* is usually an integer or a rational fraction greater than one. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if compact disc audio at 44,100 samples/second is decimated by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called a *decimator*.

In a mixed-signal system, a **reconstruction filter** is used to construct a smooth analog signal from a digital input, as in the case of a digital to analog converter (DAC) or other sampled data output device.

The **Goertzel algorithm** is a technique in digital signal processing (DSP) that provides a means for efficient evaluation of individual terms of the discrete Fourier transform (DFT), thus making it useful in certain practical applications, such as recognition of DTMF tones produced by the buttons pushed on a telephone keypad. The algorithm was first described by Gerald Goertzel in 1958.

In signal processing, **multidimensional signal processing** covers all signal processing done using multidimensional signals and systems. While multidimensional signal processing is a subset of signal processing, it is unique in the sense that it deals specifically with data that can only be adequately detailed using more than one dimension. In m-D digital signal processing, useful data is sampled in more than one dimension. Examples of this are image processing and multi-sensor radar detection. Both of these examples use multiple sensors to sample signals and form images based on the manipulation of these multiple signals. Processing in multi-dimension (m-D) requires more complex algorithms, compared to the 1-D case, to handle calculations such as the Fast Fourier Transform due to more degrees of freedom. In some cases, m-D signals and systems can be simplified into single dimension signal processing methods, if the considered systems are separable.

- ↑ Billings, Stephen A. (Sep 2013).
*Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains*. UK: Wiley. ISBN 978-1-119-94359-4. - ↑ Broesch, James D.; Stranneby, Dag; Walker, William (2008-10-20).
*Digital Signal Processing: Instant access*(1 ed.). Butterworth-Heinemann-Newnes. p. 3. ISBN 9780750689762. - ↑ Walden, R. H. (1999). "Analog-to-digital converter survey and analysis".
*IEEE Journal on Selected Areas in Communications*.**17**(4): 539–550. doi:10.1109/49.761034. - ↑ Candes, E. J.; Wakin, M. B. (2008). "An Introduction To Compressive Sampling".
*IEEE Signal Processing Magazine*.**25**(2): 21–30. doi:10.1109/MSP.2007.914731. - ↑ Rabiner, Lawrence R.; Gold, Bernard (1975).
*Theory and application of digital signal processing*. Englewood Cliffs, NJ: Prentice-Hall, Inc. ISBN 978-0139141010. - ↑ Stranneby, Dag; Walker, William (2004).
*Digital Signal Processing and Applications*(2nd ed.). Elsevier. ISBN 0-7506-6344-8. - ↑ JPFix (2006). "FPGA-Based Image Processing Accelerator" . Retrieved 2008-05-10.

Wikibooks has a book on the topic of: Digital Signal Processing |

- N. Ahmed and K.R. Rao (1975). Orthogonal Transforms for Digital Signal Processing. Springer-Verlag (Berlin – Heidelberg – New York), ISBN 3-540-06556-3.
- Jonathan M. Blackledge, Martin Turner:
*Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications*, Horwood Publishing, ISBN 1-898563-48-9 - James D. Broesch:
*Digital Signal Processing Demystified*, Newnes, ISBN 1-878707-16-7 - Paul M. Embree, Damon Danieli:
*C++ Algorithms for Digital Signal Processing*, Prentice Hall, ISBN 0-13-179144-3 - Hari Krishna Garg:
*Digital Signal Processing Algorithms*, CRC Press, ISBN 0-8493-7178-3 - P. Gaydecki:
*Foundations Of Digital Signal Processing: Theory, Algorithms And Hardware Design*, Institution of Electrical Engineers, ISBN 0-85296-431-5 - Ashfaq Khan:
*Digital Signal Processing Fundamentals*, Charles River Media, ISBN 1-58450-281-9 - Sen M. Kuo, Woon-Seng Gan:
*Digital Signal Processors: Architectures, Implementations, and Applications*, Prentice Hall, ISBN 0-13-035214-4 - Paul A. Lynn, Wolfgang Fuerst:
*Introductory Digital Signal Processing with Computer Applications*, John Wiley & Sons, ISBN 0-471-97984-8 - Richard G. Lyons:
*Understanding Digital Signal Processing*, Prentice Hall, ISBN 0-13-108989-7 - Vijay Madisetti, Douglas B. Williams:
*The Digital Signal Processing Handbook*, CRC Press, ISBN 0-8493-8572-5 - James H. McClellan, Ronald W. Schafer, Mark A. Yoder:
*Signal Processing First*, Prentice Hall, ISBN 0-13-090999-8 - Bernard Mulgrew, Peter Grant, John Thompson:
*Digital Signal Processing - Concepts and Applications*, Palgrave Macmillan, ISBN 0-333-96356-3 - Boaz Porat:
*A Course in Digital Signal Processing*, Wiley, ISBN 0-471-14961-6 - John G. Proakis, Dimitris Manolakis:
*Digital Signal Processing: Principles, Algorithms and Applications*, 4th ed, Pearson, April 2006, ISBN 978-0131873742 - John G. Proakis:
*A Self-Study Guide for Digital Signal Processing*, Prentice Hall, ISBN 0-13-143239-7 - Charles A. Schuler:
*Digital Signal Processing: A Hands-On Approach*, McGraw-Hill, ISBN 0-07-829744-3 - Doug Smith:
*Digital Signal Processing Technology: Essentials of the Communications Revolution*, American Radio Relay League, ISBN 0-87259-819-5 - Smith, Steven W. (2002).
*Digital Signal Processing: A Practical Guide for Engineers and Scientists*. Newnes. ISBN 0-7506-7444-X. - Stein, Jonathan Yaakov (2000-10-09).
*Digital Signal Processing, a Computer Science Perspective*. Wiley. ISBN 0-471-29546-9. - Stergiopoulos, Stergios (2000).
*Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging Real-Time Systems*. CRC Press. ISBN 0-8493-3691-0. - Van De Vegte, Joyce (2001).
*Fundamentals of Digital Signal Processing*. Prentice Hall. ISBN 0-13-016077-6. - Oppenheim, Alan V.; Schafer, Ronald W. (2001).
*Discrete-Time Signal Processing*. Pearson. ISBN 1-292-02572-7. - Hayes, Monson H. Statistical digital signal processing and modeling. John Wiley & Sons, 2009. (with MATLAB scripts)

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