Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations

Authors:
Xuancheng Shao;Steven G. Johnson
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave. Rm. 2-388, Cambridge, MA 01239, USA;Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave. Rm. 2-388, Cambridge, MA 01239, USA
Venue:
Signal Processing
Year:
2008

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Abstract

We present algorithms for the type-IV discrete cosine transform (DCT-IV) and discrete sine transform (DST-IV), as well as for the modified discrete cosine transform (MDCT) and its inverse, 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 2Nlog"2N+O(N) to 179Nlog"2N+O(N) for a power-of-two transform size N, and the exact count is strictly lowered for all N=8. These results are derived by considering the DCT to be a special case of a DFT of length 8N, 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 DST-IV and MDCT follow immediately from the improved count for the DCT-IV.