Editing Discrete cosine transform (section)

=== DCT V-VIII ===
DCTs of types I–IV treat both boundaries consistently regarding the point of symmetry: they are even/odd around either a data point for both boundaries or halfway between two data points for both boundaries. By contrast, DCTs of types V-VIII imply boundaries that are even/odd around a data point for one boundary and halfway between two data points for the other boundary.

In other words, DCT types I–IV are equivalent to real-even DFTs of even order (regardless of whether <math> N </math> is even or odd), since the corresponding DFT is of length <math> 2(N-1) </math> (for DCT-I) or <math> 4 N </math> (for DCT-II & III) or <math> 8 N </math> (for DCT-IV). The four additional types of discrete cosine transform<ref>{{harvnb|Martucci|1994}}</ref> correspond essentially to real-even DFTs of logically odd order, which have factors of <math> N \pm {1}/{2} </math> in the denominators of the cosine arguments.

However, these variants seem to be rarely used in practice. One reason, perhaps, is that [[Fast Fourier transform|FFT]] algorithms for odd-length DFTs are generally more complicated than [[Fast Fourier transform|FFT]] algorithms for even-length DFTs (e.g. the simplest radix-2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below.

(The trivial real-even array, a length-one DFT (odd length) of a single number {{mvar|a}}&nbsp;, corresponds to a DCT-V of length <math> N = 1 .</math>)