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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 signals, or measurements of time-varying or spatially varying physical quantities. Signals of interest can include sound, electromagnetic radiation, images, and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals, and many others.
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, 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 data compression).
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.
Mathematical methods applied in signal processing 
- Linear signals and systems, and transform theory
- System identification and classification
- Differential equations
- Vector spaces and Linear algebra
- Functional analysis
- Probability and stochastic processes
- Detection theory
- Estimation theory
- 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 that are considered as 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 digitised 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.
Fields of signal processing 
- Statistical signal processing – analyzing and extracting information from signals and noise based on their stochastic properties
- 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
- Filtering – used in many fields to process signals
- Seismic signal processing
- Data mining
- Financial signal processing
Notes and references 
- 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