N
                 a power of 2

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 53-54 | 1 | 2 |

Section 1.3.3.2.3.3. N a power of 2

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.3.2.3.3. N a power of 2

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When $[N = 2^{\nu}]$ , the same method can be applied, except for a slight modification in the calculation of $[{\bf X}_{1}^{*}]$ . There is no primitive root modulo $[2^{\nu}]$ for $[\nu \gt 2]$ : the group $[U(2^{\nu})]$ is the direct product of two cyclic groups, the first (of order 2) generated by −1, the second (of order [N/4] ) generated by 3 or 5. One then uses a representation $[\eqalign{k &= (-1)^{m_{1}} 5^{m_{2}} \cr k^{*} &= (-1)^{m_{1}^{*}} 5^{m_{2}^{*}}}]$ and the reindexed core matrix gives rise to a two-dimensional convolution. The latter may be carried out by means of two 2D DFTs on $[2 \times (N/4)]$ points.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 53-54