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

Xuancheng Shao, Steven G. Johnson

TL;DR

The paper addresses reducing the arithmetic complexity of the type-II/III DCT and DST for power-of-two sizes by deriving a new exact flop count through pruning a recently improved FFT based on recursive rescaling. It shows that the DCT-II can be realized as a DFT of length $4N$ with real-even inputs and zero interleaving, enabling a faster implementation; the new approach reduces the asymptotic cost from roughly $2N\log_{2}N$ to about $\frac{17}{9}N\log_{2}N$ real operations. A scaling technique saving $N$ multiplications extends Arai's eight-multiplication rule to all sizes. The improvements extend to DCT-III and DST-II/III via network transposition and symmetry, with normalization considerations and practical implications discussed.

Abstract

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 ~ 2N log_2 N to ~ (17/9) N log_2 N for a power-of-two transform size N. Furthermore, we show that a further N multiplications may be saved by a certain rescaling of the inputs or outputs, generalizing a well-known technique for N=8 by Arai et al. These results are derived by considering the DCT to be a special case of a DFT of length 4N, 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 DCT-III, DST-II, and DST-III follow immediately from the improved count for the DCT-II.

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

TL;DR

The paper addresses reducing the arithmetic complexity of the type-II/III DCT and DST for power-of-two sizes by deriving a new exact flop count through pruning a recently improved FFT based on recursive rescaling. It shows that the DCT-II can be realized as a DFT of length with real-even inputs and zero interleaving, enabling a faster implementation; the new approach reduces the asymptotic cost from roughly to about real operations. A scaling technique saving multiplications extends Arai's eight-multiplication rule to all sizes. The improvements extend to DCT-III and DST-II/III via network transposition and symmetry, with normalization considerations and practical implications discussed.

Abstract

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 ~ 2N log_2 N to ~ (17/9) N log_2 N for a power-of-two transform size N. Furthermore, we show that a further N multiplications may be saved by a certain rescaling of the inputs or outputs, generalizing a well-known technique for N=8 by Arai et al. These results are derived by considering the DCT to be a special case of a DFT of length 4N, 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 DCT-III, DST-II, and DST-III follow immediately from the improved count for the DCT-II.

Paper Structure

This paper contains 13 sections, 22 equations, 3 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. DCT-II in terms of DFT
  3. Conjugate-pair FFT and DCT-II
  4. Conjugate-pair split-radix FFT
  5. Fast DCT-II via split-radix FFT
  6. New FFT and DCT-II
  7. New FFT
  8. Fast DCT-II from new FFT
  9. Normalizations
  10. Fast DCT-III, DST-II, and DST-III
  11. DCT-III
  12. DST-II and DST-III
  13. Conclusion

Figures (3)

  • Figure 1: A DCT-II of size $N=8$ (open dots, $x_{0},\ldots,x_{7}$) is equivalent to a size-$4N$ DFT via interleaving with zeros (black dots) and extending in an even (squares) periodic (gray) sequence.
  • Figure 2: The DCT-II of $N=8$ points $x_{n}$ (open dots) is computed, in a conjugate-pair FFT of the $\tilde{N}=32$ extended data $\tilde{x}_{n}$ (squares) from figure \ref{['fig:dct-from-dft']}, via the DFT $Z_{k}$ of the circled points $\tilde{x}_{4n+1}$, corresponding to the even-indexed $x_{n}$ followed by the odd-indexed $x_{n}$.
  • Figure 3: Example of network transposition for a size-2 DCT-II (left). When the linear network is transposed (all edges are reversed), the resulting network computes the transposed-matrix operation: a size-2 DCT-III (right).