### Signal theory

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**Signal processing** is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of analog as well as digitized signals, representing time-varying or spatially varying physical quantities. Signals of interest can include sound, electromagnetic radiation, images, and sensor readings, for example biological measurements such as electrocardiograms, control system signals, telecommunication transmission signals, and many others.

## Contents

## Typical operations and applications

The goals of signal processing can roughly be divided into the following categories.

- Signal acquisition and reconstruction, which involves measuring a physical signal, storing it, and possibly later rebuilding the original signal or an approximation thereof. For digital systems, this typically includes sampling and quantization.
- Quality improvement, such as noise reduction, image enhancement, and echo cancellation.
- Signal compression (Source coding), including audio compression, image compression, and video compression.
- Feature extraction, such as image understanding and speech recognition.

In communication systems, signal processing may occur at OSI layer 1, the Physical Layer (modulation, equalization, multiplexing, etc.) in the seven layer OSI model, as well as at OSI layer 6, the Presentation Layer (source coding, including analog-to-digital conversion and signal compression).

## History

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 "digitalization" or digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s.^{[1]}

## Mathematical methods applied in signal processing

- Linear time-invariant system theory, and transform theory
- System identification and classification
- Calculus
- Differential equations
- Vector spaces and Linear algebra
- Functional analysis
- Probability and stochastic processes
- Detection theory
- Estimation theory
- Optimization
- Programming
- Numerical methods
- Iterative methods

## Categories of signal processing

### Analog signal processing

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.

### Discrete-time signal processing

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

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 look-up 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

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.^{[2]} Nonlinear systems can produce highly complex behaviors including bifurcations, chaos, harmonics, and subharmonics which cannot be produced or analyzed using linear methods.

## Fields of signal processing

- Statistical signal processing – analyzing and extracting information from signals and noise based on their stochastic properties
- Spectral estimation – for determining the spectral content (i.e., the distribution of power over frequency) of a time series
^{[3]} - 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
- Array processing – for processing signals from arrays of sensors
- Time-frequency analysis – for processing non-stationary signals
^{[4]} - Filtering – used in many fields to process signals
- Seismic signal processing
- Data mining
- Financial signal processing

## See also

- Audio filter
- Delay (audio effect)
- Dynamic range compression, companding, limiting, and noise gating
- Reverberation

## Notes and references

## External links

- 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