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**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",^{ [1] } such as sound, images, and biological measurements.^{ [2] } 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.^{ [3] }

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

**Information engineering** is the engineering discipline that deals with the generation, distribution, analysis, and use of information, data, and knowledge in systems. The field first became identifiable in the early 21st century.

**Electrical engineering** is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an identifiable occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone, and electric power distribution and use. Subsequently, broadcasting and recording media made electronics part of daily life. The invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object.

According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s.^{ [4] }

**Alan Victor Oppenheim** is a Professor of Engineering at MIT's Department of Electrical Engineering and Computer Science. He is also a principal investigator in MIT's Research Laboratory of Electronics (RLE), at the *Digital Signal Processing Group*. His research interests are in the general area of signal processing and its applications. He is coauthor of the widely used textbooks *Discrete-Time Signal Processing* and *Signals and Systems*. He is also editor of several advanced books on signal processing.

**Ronald W. Schafer** is an electrical engineer notable for his contributions to digital signal processing.

**Numerical analysis** is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Analog signal processing is for signals that have not been digitized, as in legacy radio, telephone, radar, and television systems. This involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators and delay lines. Non-linear circuits include compandors, multiplicators (frequency mixers and voltage-controlled amplifiers), voltage-controlled filters, voltage-controlled oscillators and phase-locked loops.

An **active filter** is a type of analog circuit implementing an electronic filter using active components, typically an amplifier. Amplifiers included in a filter design can be used to improve the cost, performance and predictability of a filter.

An **electronic mixer** is a device that combines two or more electrical or electronic signals into one or two composite output signals. There are two basic circuits that both use the term *mixer*, but they are very different types of circuits: additive mixers and multiplicative mixers. Additive mixers are also known as "**analog adders**" to distinguish from the related digital adder circuits.

An **integrator** in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output.

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

The methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology mainly discusses the modeling of 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

**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.

Discrete-time signal processing is for sampled signals, defined only at discrete points in time, and as such are quantized in time, but not in magnitude.

*Analog discrete-time signal processing* is a technology based on electronic devices such as sample and hold circuits, analog time-division multiplexers, analog delay lines and analog feedback shift registers. This technology was a predecessor of digital signal processing (see below), and is still used in advanced processing of gigahertz signals.

The concept of discrete-time signal processing also refers to a theoretical discipline that establishes a mathematical basis for digital signal processing, without taking quantization error into consideration.

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

Nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains.^{ [5] } Nonlinear systems can produce highly complex behaviors including bifurcations, chaos, harmonics, and subharmonics which cannot be produced or analyzed using linear methods.

Statistical signal processing is an approach which treats signals as stochastic processes, utilizing their statistical properties to perform signal processing tasks.^{ [6] } Statistical techniques are widely used in signal processing applications. For example, one can model the probability distribution of noise incurred when photographing an image, and construct techniques based on this model to reduce the noise in the resulting image.

- Audio signal processing – for electrical signals representing sound, such as speech or music
- Speech signal processing – for processing and interpreting spoken words
- Image processing – in digital cameras, computers and various imaging systems
- Video processing – for interpreting moving pictures
- Wireless communication – waveform generations, demodulation, filtering, equalization
- Control systems
- Array processing – for processing signals from arrays of sensors
- Process control – a variety of signals are used, including the industry standard 4-20 mA current loop
- Seismology
- Financial signal processing – analyzing financial data using signal processing techniques, especially for prediction purposes.
- Feature extraction, such as image understanding and speech recognition.
- Quality improvement, such as noise reduction, image enhancement, and echo cancellation.
- (Source coding), including audio compression, image compression, and video compression.
- Genomics, Genomic signal processing
^{ [7] }

In communication systems, signal processing may occur at:

- OSI layer 1 in the seven layer OSI model, the Physical Layer (modulation, equalization, multiplexing, etc.);
- OSI layer 2, the Data Link Layer (Forward Error Correction);
- OSI layer 6, the Presentation Layer (source coding, including analog-to-digital conversion and signal compression).

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

- Differential equations
- Recurrence relation
- Transform theory
- Time-frequency analysis – for processing non-stationary signals
^{ [8] } - Spectral estimation – for determining the spectral content (i.e., the distribution of power over frequency) of a time series
^{ [9] } - Statistical signal processing – analyzing and extracting information from signals and noise based on their stochastic properties
- Linear time-invariant system theory, and transform theory
- System identification and classification
- Calculus
- Vector spaces and Linear algebra
- Functional analysis
- Probability and stochastic processes
- Detection theory
- Estimation theory
- Optimization
- Numerical methods
- Time series
- 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

- ↑ Roland Priemer (1991).
*Introductory Signal Processing*. World Scientific. p. 1. ISBN 9971509199. - ↑ Sengupta, Nandini; Sahidullah, Md; Saha, Goutam (August 2016). "Lung sound classification using cepstral-based statistical features".
*Computers in Biology and Medicine*.**75**(1): 118–129. doi:10.1016/j.compbiomed.2016.05.013. - ↑ Alan V. Oppenheim and Ronald W. Schafer (1989).
*Discrete-Time Signal Processing*. Prentice Hall. p. 1. ISBN 0-13-216771-9. - ↑ Oppenheim, Alan V.; Schafer, Ronald W. (1975).
*Digital Signal Processing*. Prentice Hall. p. 5. ISBN 0-13-214635-5. - ↑ Billings, S. A. (2013).
*Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains*. Wiley. ISBN 1119943590. - ↑ Scharf, Louis L. (1991).
*Statistical signal processing: detection, estimation, and time series analysis*. Boston: Addison–Wesley. ISBN 0-201-19038-9. OCLC 61160161. - ↑ Anastassiou, D. (2001).
*Genomic signal processing*. IEEE. - ↑ Boashash, Boualem, ed. (2003).
*Time frequency signal analysis and processing a comprehensive reference*(1 ed.). Amsterdam: Elsevier. ISBN 0-08-044335-4. - ↑ Stoica, Petre; Moses, Randolph (2005).
*Spectral Analysis of Signals*(PDF). NJ: Prentice Hall.

- P Stoica, R Moses (2005).
*Spectral Analysis of Signals*(PDF). NJ: Prentice Hall. - Kay, Steven M. (1993).
*Fundamentals of Statistical Signal Processing*. Upper Saddle River, New Jersey: Prentice Hall. ISBN 0-13-345711-7. OCLC 26504848. - Papoulis, Athanasios (1991).
*Probability, Random Variables, and Stochastic Processes*(third ed.). McGraw-Hill. ISBN 0-07-100870-5. - Kainam Thomas Wong : Statistical Signal Processing lecture notes at the University of Waterloo, Canada.
- Ali H. Sayed, Adaptive Filters, Wiley, NJ, 2008, ISBN 978-0-470-25388-5.
- Thomas Kailath, Ali H. Sayed, and Babak Hassibi, Linear Estimation, Prentice-Hall, NJ, 2000, ISBN 978-0-13-022464-4.

- Signal Processing for Communications – free online textbook by Paolo Prandoni and Martin Vetterli (2008)
- Scientists and Engineers Guide to Digital Signal Processing – free online textbook by Stephen Smith
- Signal Processing Techniques for Determining Powerplant Characteristics
- The IEEE Signal Processing Society
- Bio-Medical Signal processing at a Glance

**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.

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 **cepstrum** is the result of taking the inverse Fourier transform (IFT) of the logarithm of the estimated spectrum of a signal. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with "kepstrum", which also exists. There is a *complex cepstrum*, a *real cepstrum*, a *power cepstrum*, and a *phase cepstrum*. The power cepstrum in particular has applications in the analysis of human speech.

**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.

In communication systems, signal processing, and electrical engineering, a **signal** is a function that "conveys information about the behavior or attributes of some phenomenon". In its most common usage, in electronics and telecommunication, this is a time varying voltage, current or electromagnetic wave used to carry information. A signal may also be defined as an "observable change in a quantifiable entity". In the physical world, any quantity exhibiting variation in time or variation in space is potentially a signal that might provide information on the status of a physical system, or convey a message between observers, among other possibilities. The *IEEE Transactions on Signal Processing* states that the term "signal" includes audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. In a later effort of redefining a signal, anything that is only a function of space, such as an image, is excluded from the category of signals. Also, it is stated that a signal may or may not contain any information.

**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.

A **time series** is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

**Welch's method**, named after P.D. Welch, is an approach for spectral density estimation. It is used in physics, engineering, and applied mathematics for estimating the power of a signal at different frequencies. The method is based on the concept of using periodogram spectrum estimates, which are the result of converting a signal from the time domain to the frequency domain. Welch's method is an improvement on the standard periodogram spectrum estimating method and on Bartlett's method, in that it reduces noise in the estimated power spectra in exchange for reducing the frequency resolution. Due to the noise caused by imperfect and finite data, the noise reduction from Welch's method is often desired.

**Homomorphic filtering** is a generalized technique for signal and image processing, involving a nonlinear mapping to a different domain in which linear filter techniques are applied, followed by mapping back to the original domain. This concept was developed in the 1960s by Thomas Stockham, Alan V. Oppenheim, and Ronald W. Schafer at MIT and independently by Bogert, Healy, and Tukey in their study of time series.

**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 signal processing, a **nonlinear****filter** is a filter whose output is not a linear function of its input. That is, if the filter outputs signals *R* and *S* for two input signals *r* and *s* separately, but does not always output α*R* + β*S* when the input is a linear combination α*r* + β*s*.

**Sample-rate conversion** is the process of changing the sampling rate of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application areas include image scaling and audio/visual systems, where different sampling rates may be used for engineering, economic, or historical reasons.

**Geophysical survey** is the systematic collection of geophysical data for spatial studies. Detection and analysis of the geophysical signals forms the core of Geophysical signal processing. The magnetic and gravitational fields emanating from the Earth's interior hold essential information concerning seismic activities and the internal structure. Hence, detection and analysis of the electric and Magnetic fields is very crucial. As the Electromagnetic and gravitational waves are multi-dimensional signals, all the 1-D transformation techniques can be extended for the analysis of these signals as well. Hence this article also discusses multi-dimensional signal processing techniques.

In statistical signal processing, the goal of **spectral density estimation** (**SDE**) is to estimate the spectral density of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

A **shift invariant system** is the discrete equivalent of a time-invariant system, defined such that if *y*(*n*) is the response of the system to *x*(*n*), then *y*(*n*–*k*) is the response of the system to *x*(*n*–*k*). That is, in a shift-invariant system the contemporaneous response of the output variable to a given value of the input variable does not depend on when the input occurs; time shifts are irrelevant in this regard.

In signal processing, a **filter** is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, and computer graphics.

**Simon Haykin** is an electrical engineer, noted for his pioneering work in Adaptive Signal Processing with emphasis on applications in radar and communications. He is currently a Distinguished University Professor at McMaster University in Hamilton, Ontario, Canada.

In signal processing, **discrete transforms** are mathematical transforms, often linear transforms, of signals between discrete domains, such as between discrete time and discrete frequency.

An **anamorphic stretch transform **(**AST**) also referred to as **warped stretch transform** is a physics-inspired signal transform that emerged from time stretch dispersive Fourier transform. The transform can be applied to analog temporal signals such as communication signals, or to digital spatial data such as images. The transform reshapes the data in such a way that its output has properties conducive for data compression and analytics. The reshaping consists of warped stretching in the Fourier domain. The name "Anamorphic" is used because of the metaphoric analogy between the warped stretch operation and warping of images in anamorphosis and surrealist artworks.

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