Editing Discrete cosine transform (section)

=== DCT-I ===
:<math>X_k
 = \frac{1}{2} (x_0 + (-1)^k x_{N-1}) 
 + \sum_{n=1}^{N-2} x_n \cos \left[\, \tfrac{\ \pi}{\,N-1\,} \, n \, k \,\right]
 \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, if one further multiplies by an overall scale factor of <math display="inline">\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 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.)

Note, however, that the DCT-I is not defined for <math>N</math> less than 2, while all other DCT types are defined for any positive <math>N</math>.

Thus, the DCT-I corresponds to the boundary conditions: <math>x_n</math> is even around <math>n = 0</math> and even around <math>n = N - 1</math>; similarly for <math>X_k</math>.