Discrete cosine transform: Difference between revisions

Revision as of 04:51, 11 September 2024

Revision as of 04:56, 11 September 2024

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==Informal overview==

Like any Fourier-related transform, discrete [[Cosine transform|cosine transforms]] (DCTs) express a function or a signal in terms of a sum of [[sinusoid]]s with different [[frequencies]] and [[amplitude]]s. Like the [[discrete Fourier transform]] (DFT), a DCT operates on a function at a finite number of [[Discrete signal|discrete]] data points. The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of [[complex exponential]]s). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different [[boundary condition]]s from the DFT or other related transforms.

The Fourier-related transforms that operate on a function over a finite [[___domain of a function|___domain]], such as the DFT or DCT or a [[Fourier series]], can be thought of as implicitly defining an ''extension'' of that function outside the ___domain. That is, once you write a function <math>f(x)</math> as a sum of sinusoids, you can evaluate that sum at any <math>x</math>, even for <math>x</math> where the original <math>f(x)</math> was not specified. The DFT, like the Fourier series, implies a [[periodic function|periodic]] extension of the original function. A DCT, like a [[Sine and cosine transforms|cosine transform]], implies an [[even and odd functions|even]] extension of the original function.