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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 87-88   | 1 | 2 |

Section 1.3.4.3.5.3. Complex symmetric and antisymmetric transforms

G. Bricognea

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1.3.4.3.5.3. Complex symmetric and antisymmetric transforms

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The matrix is its own contragredient, and hence (Section 1.3.2.4.2.2 ) the transform of a symmetric (respectively antisymmetric) function is symmetric (respectively antisymmetric). In this case the group acts in both real and reciprocal space as . If with both factors diagonal, then acts by i.e. The symmetry or antisymmetry properties of X may be written with for symmetry and for antisymmetry.

The computation will be summarized as with the same indexing as that used for structure-factor calculation. In both cases it will be shown that a transform with and M diagonal can be computed using only partial transforms instead of .

 (i) Decimation in time Since we have and mod , so that the symmetry relations for each parity class of data read or equivalently Transforming by , this relation becomes Each parity class thus obeys a different symmetry relation, so that we may multiplex them in pairs by forming for each pair the vector Putting we then have the demultiplexing relations for each : which can be solved recursively. Transform values at the exceptional points where demultiplexing fails (i.e. where ) can be accumulated while forming Y. Only the unique half of the values of need to be considered at the demultiplexing stage and at the subsequent TW and F(2I) stages. (ii) Decimation in frequency The vectors of final results for each parity class obey the symmetry relations which are different for each . The vectors of intermediate results after the twiddle-factor stage may then be multiplexed in pairs as After transforming by , the results may be demultiplexed by using the relations which can be solved recursively as in Section 1.3.4.3.5.1 (b)(ii) .