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

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.3.5.3. Complex symmetric and antisymmetric transforms

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The matrix [-{\bf I}] is its own contragredient, and hence (Section 1.3.2.4.2.2[link]) the transform of a symmetric (respectively antisymmetric) function is symmetric (respectively antisymmetric). In this case the group [G = \{e, -e\}] acts in both real and reciprocal space as [\{{\bf I}, -{\bf I}\}]. If [{\bf N} = {\bf N}_{1} {\bf N}_{2}] with both factors diagonal, then [-e] acts by[\eqalign{ ({\bf m}_{1}, {\bf m}_{2}) \,&\longmapsto\, [{\bf N}_{1} {\boldzeta} ({\bf m}_{1}) - {\bf m}_{1}, {\bf N}_{2} {\boldzeta} ({\bf m}_{2}) - {\bf m}_{2} - {\boldzeta} ({\bf m}_{1})],\cr ({\bf h}_{2}, {\bf h}_{1}) \,&\longmapsto\, [{\bf N}_{2} {\boldzeta} ({\bf h}_{2}) - {\bf h}_{2}, {\bf N}_{1} {\boldzeta} ({\bf h}_{1}) - {\bf h}_{1} - {\boldzeta} ({\bf h}_{2})],}]i.e.[\eqalign{{\boldmu}_{2} (-e, {\bf m}_{1}) &= -{\boldzeta} ({\bf m}_{1}) \hbox{ mod } {\bf N}_{2} {\bb Z}^{n},\cr {\boldeta}_{1} (-e, {\bf h}_{2}) &= -{\boldzeta} ({\bf h}_{2}) \hbox{ mod } {\bf N}_{1} {\bb Z}^{n}.}]

The symmetry or antisymmetry properties of X may be written[X (-{\bf m}) = -\varepsilon X ({\bf m}) \hbox{ for all } {\bf m},]with [\varepsilon = + 1] for symmetry and [\varepsilon = -1] for antisymmetry.

The computation will be summarized as[{\bf X} {\buildrel{{\bf dec}({\bf N}_{1})} \over {\,\longmapsto\,}} {\bf Y} {\buildrel{\bar{F}({\bf N}_{2})} \over {\,\longmapsto\,}} {\bf Y}^{*} {\buildrel{\scriptstyle{\rm TW}} \over {\,\longmapsto\,}} {\bf Z} {\buildrel{\bar{F}({\bf N}_{1})} \over {\,\longmapsto\,}} {\bf Z}^{*} {\buildrel{{\bf rev}({\bf N}_{2})} \over {\,\longmapsto\,}} {\bf X}^{*}]with the same indexing as that used for structure-factor calculation. In both cases it will be shown that a transform [F({\bf N})] with [{\bf N} = {\bf 2M}] and M diagonal can be computed using only [2^{n-1}] partial transforms [F({\bf M})] instead of [2^{n}].

  • (i) Decimation in time [({\bf N}_{1} = 2{\bf I},{\bf N}_{2} = {\bf M})]

    Since [{\bf m}_{1} \in {\bb Z}^{n}/2{\bb Z}^{n}] we have [-{\bf m}_{1} = {\bf m}_{1}] and [{\boldzeta} ({\bf m}_{1}) = {\bf m}_{1}] mod [2{\bb Z}^{n}], so that the symmetry relations for each parity class of data [{\bf Y}_{{\bf m}_{1}}] read[Y_{{\bf m}_{1}} [{\bf M} {\boldzeta} ({\bf m}_{2}) - {\bf m}_{2} - {\bf m}_{1}] = \varepsilon Y_{{\bf m}_{1}} ({\bf m}_{2})]or equivalently[\tau_{{\bf m}_{1}} {\bf Y}_{{\bf m}_{1}} = \varepsilon \breve{{\bf Y}}_{{\bf m}_{1}}.]Transforming by [F({\bf M})], this relation becomes[e [-{\bf h}_{2} \cdot ({\bf M}^{-1} {\bf m}_{1})] {\bf Y}_{{\bf m}_{1}}^{*} = \varepsilon {\bf Y}_{{\bf m}_{1}}^{*}.]Each parity class thus obeys a different symmetry relation, so that we may multiplex them in pairs by forming for each pair [({\bf m}'_{1}, {\bf m}''_{1})] the vector[{\bf Y} = {\bf Y}_{{\bf m}'_{1}} + {\bf Y}_{{\bf m}''_{1}}.]Putting[\eqalign{e [-{\bf h}_{2} \cdot ({\bf M}^{-1} {\bf m}'_{1})] &= (c' + is') ({\bf h}_{2})\cr e [-{\bf h}_{2} \cdot ({\bf M}^{-1} {\bf m}''_{1})] &= (c'' + is'') ({\bf h}_{2})}]we then have the demultiplexing relations for each [{\bf h}_{2}]:[\displaylines{Y_{{\bf m}'_{1}}^{*} ({\bf h}_{2}) + Y_{{\bf m}''_{1}}^{*} ({\bf h}_{2})= Y^{*} ({\bf h}_{2})\cr (c' + is') ({\bf h}_{2}) Y_{{\bf m}'_{1}}^{*} ({\bf h}_{2}) + (c'' + is'') ({\bf h}_{2})Y_{{\bf m}''_{1}}^{*} ({\bf h}_{2})\cr \qquad\qquad\quad\, = \varepsilon Y^{*} [{\bf M} \boldzeta ({\bf h}_{2}) - {\bf h}_{2}]\hfill}]which can be solved recursively. Transform values at the exceptional points [{\bf h}_{2}] where demultiplexing fails (i.e. where [c' + is' = c'' + is'']) can be accumulated while forming Y.

    Only the unique half of the values of [{\bf h}_{2}] need to be considered at the demultiplexing stage and at the subsequent TW and F(2I) stages.

  • (ii) Decimation in frequency [({\bf N}_{1} = {\bf M},{\bf N}_{2} = 2{\bf I})]

    The vectors of final results [{\bf Z}_{{\bf h}_{2}}^{*}] for each parity class [{\bf h}_{2}] obey the symmetry relations[\tau_{{\bf h}_{2}} {\bf Z}_{{\bf h}_{2}}^{*} = \varepsilon \check{{\bf Z}}_{{\bf h}_{2}}^{*},]which are different for each [{\bf h}_{2}]. The vectors [{\bf Z}_{{\bf h}_{2}}] of intermediate results after the twiddle-factor stage may then be multiplexed in pairs as[{\bf Z} = {\bf Z}_{{\bf h}'_{2}} + {\bf Z}_{{\bf h}''_{2}}.]

    After transforming by [F({\bf M})], the results [{\bf Z}^{*}] may be demultiplexed by using the relations[\eqalign{Z_{{\bf h}'_{2}}^{*} ({\bf h}_{1})\quad &+ Z_{{\bf h}''_{2}}^{*} ({\bf h}_{1}) \phantom{- {\bf h}''_{2})} = Z^{*} ({\bf h}_{1})\cr Z_{{\bf h}'_{2}}^{*} ({\bf h}_{1} - {\bf h}'_{2}) &+ Z_{{\bf h}''_{2}}^{*} ({\bf h}_{1} - {\bf h}''_{2}) = \varepsilon Z^{*} [{\bf M} \boldzeta ({\bf h}_{1}) - {\bf h}_{1}]}]which can be solved recursively as in Section 1.3.4.3.5.1[link](b)(ii)[link].








































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