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Discrete Cosine Transform, Type II and III

The discrete cosine transform of type III is given by the matrix

DCT, type III

where ak = sqrt(1/2) for k = 0, and ak = 1 else. The DCT type II is the transpose of type III. The following structured, sparse matrix factorization represents a fast algorithm for a DCT, type III, size 8, and has been automatically generated (even in the presented Latex format) using the package AREP.

fast algorithm DCT, type III

By transposition we obtain a fast algorithm for the DCT(II):

fast algorithm DCT, type II
Note that these algorithms require 13 multiplications and 29 additions and are thus among the best ones known.

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