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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 56   | 1 | 2 |

Section 1.3.3.2.3.3. N a power of 2

G. Bricognea

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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.








































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