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

TL;DR

This paper introduces new algorithms for type-IV DCT, DST, and MDCT that significantly reduce the number of arithmetic operations needed, improving efficiency without losing accuracy.

Contribution

It presents novel algorithms that lower the operation count for DCT-IV, DST-IV, and MDCT by leveraging symmetries and recursive rescaling of FFT algorithms.

Findings

Operation count reduced from ~2NlogN to ~(17/9)NlogN for power-of-two sizes

Exact count is lower for all N > 4

Algorithms maintain numerical accuracy

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 ~2NlogN to ~(17/9)NlogN for a power-of-two transform size N, and the exact count is strictly lowered for all N > 4. 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.