Type-II/III DCT/DST algorithms with reduced number of arithmetic operations

We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from 2Nlog2N+O(N) to 179Nlog2N+O(N) for a power-of-two transform size N. Furthermore, we show that an additional N   multiplications may be saved by a certain rescaling of the inputs or outputs, generalizing a well-known technique for N=8N=8 by Arai et al. These results are derived by considering the DCT to be a special case of a DFT of length 4N4N, with certain symmetries, and then pruning redundant operations from a recent improved fast Fourier transform algorithm (based on a recursive rescaling of the conjugate-pair split-radix algorithm). The improved algorithms for the DCT-III, DST-II, and DST-III follow immediately from the improved count for the DCT-II.