Generalized multiplexing

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, p. 82 | 1 | 2 |

Section 1.3.4.3.5.6. Generalized multiplexing

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

1.3.4.3.5.6. Generalized multiplexing

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So far the multiplexing technique has been applied to pairs of vectors with similar types of parity-related and/or conjugate symmetry properties, in particular the same value of ɛ.

It can be generalized so as to accommodate mixtures of vectors with different symmetry characteristics. For example if $[{\bf X}_{1}]$ is Hermitian-symmetric and $[{\bf X}_{2}]$ is Hermitian-antisymmetric, so that $[{\bf X}_{1}^{*}]$ is real-valued while $[{\bf X}_{2}^{*}]$ has purely imaginary values, the multiplexing process should obviously form $[{\bf X} = {\bf X}_{1} + {\bf X}_{2}]$ (instead of $[{\bf X} = {\bf X}_{1} + i{\bf X}_{2}]$ if both had the same type of symmetry), and demultiplexing consists in separating $[\eqalign{ {\bf X}_{1}^{*} &= {\scr Re}\; {\bf X}^{*}\cr {\bf X}_{2}^{*} &= i{\scr Im}\;{\bf X}^{*}.}]$

The general multiplexing formula for pairs of vectors may therefore be written $[{\bf X} = {\bf X}_{1} + \omega {\bf X}_{2},]$ where ω is a phase factor (e.g. 1 or i) chosen in such a way that all non-exceptional components of $[{\bf X}_{1}]$ and $[{\bf X}_{2}]$ (or $[{\bf X}_{1}^{*}]$ and $[{\bf X}_{2}^{*}]$ ) be embedded in the complex plane $[{\bb C}]$ along linearly independent directions, thus making multiplexing possible.

It is possible to develop a more general form of multiplexing/demultiplexing for more than two vectors, which can be used to deal with symmetry elements of order 3, 4 or 6. It is based on the theory of group characters (Ledermann, 1987).

References

Ledermann, W. (1987). Introduction to group characters, 2nd ed. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 82