In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, discrete transforms are
mathematical transforms, often
linear transform
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s, of signals between discrete domains, such as between discrete time and discrete frequency.
Many common
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...
s used in signal processing have their discrete counterparts. For example, for the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
the counterpart is the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...
.
In addition to
spectral analysis of signals, discrete transforms play important role in
data compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressi ...
,
signal detection
Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (c ...
,