# Discrete transform

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{{Short description|Mathematical transform on discrete signals}}
In [signal processing](/source/signal_processing), '''discrete transforms''' (or '''discrete integral transform''') are [mathematical transforms](/source/integral_transform), often [linear transform](/source/linear_transform)s, of signals between discrete domains, such as between discrete time and discrete frequency.<ref>
{{cite book
 | title = Television receivers
 | edition = 
 | author = Jerry C. Whitaker
 | publisher = McGraw-Hill Professional
 | year = 2001
 | isbn = 978-0-07-138042-3
 | page = 147
 | url = https://books.google.com/books?id=n3hxYt1u8vwC&dq=%22discrete+transforms+are%22&pg=PA147
 }}</ref>

Many common [integral transform](/source/integral_transform)s used in signal processing have their discrete counterparts. For example, for the [Fourier transform](/source/Fourier_transform) the counterpart is the [discrete Fourier transform](/source/discrete_Fourier_transform).

In addition to [spectral analysis](/source/frequency_spectrum) of signals,  discrete transforms play important role in [data compression](/source/data_compression), [signal detection](/source/detection_theory), [digital filter](/source/digital_filter)ing and [correlation](/source/correlation) analysis.<ref>
{{cite book
 | title = Signal coding and processing
 | edition = 2nd
 | author = Graham Wade
 | publisher = Cambridge University Press
 | year = 1994
 | isbn = 978-0-521-42336-6
 | page = 332
 | url = https://books.google.com/books?id=CJswCy7_W8YC&dq=%22discrete+transforms+are%22&pg=PA332
 }}</ref> The [discrete cosine transform](/source/discrete_cosine_transform) (DCT) is the most widely used [transform coding](/source/transform_coding) compression algorithm in [digital media](/source/digital_media), followed by the [discrete wavelet transform](/source/discrete_wavelet_transform) (DWT).

Transforms between a discrete domain and a continuous domain are not discrete transforms.  For example, the [discrete-time Fourier transform](/source/discrete-time_Fourier_transform) and the [Z-transform](/source/Z-transform), from discrete time to continuous frequency, and the [Fourier series](/source/Fourier_series), 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](/source/image_processing), [computer vision](/source/computer_vision), [high-definition television](/source/high-definition_television), visual telephony, etc. make use of two-dimensional and in general, multidimensional discrete transforms.

==See also==

*{{slink|List of transforms|Discrete transforms}}

==References==

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Adapted from the Wikipedia article [Discrete transform](https://en.wikipedia.org/wiki/Discrete_transform) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Discrete_transform?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.