| 引用本文: | 蒋增荣.一维变换与二维变换的等价以及二维DFT的向量算法.[J].国防科技大学学报,1987,(1):68-75.[点击复制] |
| Jiang Zengrong.Equivalence of One-Dimensional General Transforms and Two-Dimensional General Transforms and the Vector Algorithm of Two-Dimensional DFTs[J].Journal of National University of Defense Technology,1987,(1):68-75[点击复制] |
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| 一维变换与二维变换的等价以及二维DFT的向量算法 |
| 蒋增荣 |
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| 摘要: |
| 本文证明了当且仅当 [R]=[P]T [Q]时，一维变换 Y=[R]X与二维变换[Y]=[Q] [X] [P] 相互等价。此外，讨论了Hadamard 变换以及具有循环卷积特性的一维变换与二维变换的等价问题。最后，利用上述等价定理，导出了二维DFT的一种比行列算法更为有效的快速算法—向量算法。 |
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| 投稿日期：1985-02-18 |
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| Equivalence of One-Dimensional General Transforms and Two-Dimensional General Transforms and the Vector Algorithm of Two-Dimensional DFTs |
| Jiang Zengrong |
| Abstract: |
| In this paper，we discuss necessary and sufficient condition under which the one-dimensional general transforms equivalent to two-dimensional general transforms
Theorem：Let [X]，[Y] be the matrix representations of vectors X，Y respectively，and let [R]，[Q]，[P] be three matrices of order N M x N M，N x N，M x M respectively，[X]，[Y] are of order N x M，X ，Y are dimension N M. (1) If and only if [R]=[P]T [Q]，Y=[R]X and [Y]=[Q] [X] [Q] for any vector X，where denotes Kronecker Product of matrices;
(2) If Y=[R]X and [Y]=[Q] [X] [P] for any vector X，then [R] is symmetric，if and only if [Q]，[P] are symmetric or anti-symmetric matrices，simultaneously，and
Ri+jN,k+lN=±Ri+lN,k+jN
Ri+jN,k+ln=±Rk+jN,i+lN
(i，k=0，l，…，N-I;j,l=0，l，…，M-1)
Nextly，we have discussed the equivalence of Walsh-Hadamard transforms and that of any general transforms having the cyclic convolution property (CCP)(DFT，NTT，PT). Lastly，we have introduced a fast algorithm of 2D-DFT-Vector Algorithm by using above equivalence theorem，which is more efficient than the row-column method of 2D-DFTs. |
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