Type-II/III DCT/DST algorithms with reduced number of arithmetic operations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2007
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| _version_ | 1866915559932690432 |
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| author | Shao, Xuancheng Johnson, Steven G. |
| author_facet | Shao, Xuancheng Johnson, Steven G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_cs_0703150 |
| institution | arXiv |
| publishDate | 2007 |
| record_format | arxiv |
| spellingShingle | Type-II/III DCT/DST algorithms with reduced number of arithmetic operations Shao, Xuancheng Johnson, Steven G. Numerical Analysis Data Structures and Algorithms Mathematical Software F.2.1 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. |
| title | Type-II/III DCT/DST algorithms with reduced number of arithmetic operations |
| topic | Numerical Analysis Data Structures and Algorithms Mathematical Software F.2.1 |
| url | https://arxiv.org/abs/cs/0703150 |