In signal processing

Signal processing

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

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

Many common integral transforms used in signal processing have their discrete counterparts. For example, for the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 the counterpart is the discrete Fourier transform

Discrete Fourier transform

In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...

.

In addition to spectral analysis

Spectral analysis

Spectral analysis or Spectrum analysis may refer to:* Spectrum analysis in chemistry and physics, a method of analyzing the chemical properties of matter from bands in their visible spectrum...

 of signals, discrete transforms play important role in data compression

Data compression

In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....

, signal detection

Detection theory

Detection theory, or signal detection theory, is a means to quantify the ability to discern between information-bearing energy patterns and random energy patterns that distract from the information Detection theory, or signal detection theory, is a means to quantify the ability to discern between...

, digital filter

Digital filter

In electronics, computer science and mathematics, 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...

ing and correlation

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

 analysis.

Transforms between a discrete domain and a continuous domain are not discrete transforms. For example, the discrete-time Fourier transform

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function . But the DTFT requires an input function...

 and the Z-transform

Z-transform

In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

, from discrete time to continuous frequency, and the Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

, from continuous time to discrete frequency, are outside the class of discrete transforms.

Classical signal processing deals with one-dimensional discrete transforms. Other application areas, such as image processing

Image processing

In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

, computer vision

Computer vision

Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...

, high definition television, visual telephony, etc. make use of two-dimensional and in general, multidimensional discrete transforms.