Discrete cosine transform: Difference between revisions

Revision as of 23:38, 9 October 2024

Revision as of 07:39, 17 October 2024

Line 287:

\qquad \text{ for } ~ k = 0,\ \ldots\ N-1 ~.</math>

Some authors further multiply the <math>x_0 </math> and <math> x_{N-1} </math> terms by <math> \sqrt{2\,}\, ,</math> and correspondingly multiply the <math> X_0 </math> and <math> X_{N-1}</math> terms by <math> 1/\sqrt{2\,} \,,</math> which makes the DCT-I matrix [[orthogonal matrix|orthogonal]], if one further multiplies by an overall scale factor of <math> \sqrt{\tfrac{2}{N-1\,}\,} ,</math>, makes the DCT-I matrix [[orthogonal matrix|orthogonal]] but breaks the direct correspondence with a real-even [[Discrete Fourier transform|DFT]].

The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a [[Discrete Fourier transform|DFT]] of <math> 2(N-1) </math> real numbers with even symmetry. For example, a DCT-I of <math>N = 5 </math> real numbers <math> a\ b\ c\ d\ e </math> is exactly equivalent to a DFT of eight real numbers {{not a typo|<math> a\ b\ c\ d\ e\ d\ c\ b </math>}} (even symmetry), divided by two. (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.)