Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 47   | 1 | 2 |

Section Relation between Fourier transforms

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Relation between Fourier transforms

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Finally, let us consider the relations between the Fourier transforms of these lattice distributions. Recalling the basic relation of Section[link], [\eqalign{{\scr F}[R_{\bf A}] &= {1 \over |\det {\bf A}|} R_{\bf A}^{*}\cr &= {1 \over |\det {\bf DB}|} T_{{\bf A}/{\bf B}}^{*} * R_{\bf B}^{*} \quad \quad \quad \quad \quad \quad \hbox{by (ii)}^{*}\cr &= \left({1 \over |\det {\bf D}|} T_{{\bf A}/{\bf B}}^{*}\right) * \left({1 \over |\det {\bf B}|} R_{\bf B}^{*}\right)}] i.e. [\displaylines{\quad (\hbox{iv})\hfill {\scr F}[R_{\bf A}] = S_{{\bf A}/{\bf B}}^{*} * {\scr F}[R_{\bf B}]\hfill}] and similarly: [\displaylines{\quad (\hbox{v})\hfill {\scr F}[R_{\bf B}^{*}] = S_{{\bf B}/{\bf A}} * {\scr F}[R_{\bf A}^{*}].\hfill}]

Thus [R_{\bf A}] (respectively [R_{\bf B}^{*}]), a decimated version of [R_{\bf B}] (respectively [R_{\bf A}^{*}]), is transformed by [{\scr F}] into a subdivided version of [{\scr F}[R_{\bf B}]] (respectively [{\scr F}[R_{\bf A}^{*}]]).

The converse is also true: [\eqalign{{\scr F}[R_{\bf B}] &= {1 \over |\det {\bf B}|} R_{\bf B}^{*}\cr &= {1 \over |\det {\bf B}|} {1 \over |\det {\bf D}|} ({\bf D}^{T})^{\#} R_{\bf A}^{*}\quad \quad \quad \quad \hbox{by (i)}^{*}\cr &= ({\bf D}^{T})^{\#} \left({1 \over |\det {\bf A}|} R_{\bf A}^{*}\right)}] i.e. [\displaylines{\quad (\hbox{iv}')\hfill {\scr F}[R_{\bf B}] = ({\bf D}^{T})^{\#} {\scr F}[R_{\bf A}]\hfill}] and similarly [\displaylines{\quad (\hbox{v}')\hfill {\scr F}[R_{\bf A}^{*}] = {\bf D}^{\#} {\scr F}[R_{\bf B}^{*}].\hfill}]

Thus [R_{\bf B}] (respectively [R_{\bf A}^{*}]), a subdivided version of [R_{\bf A}] (respectively [R_{\bf B}^{*}]) is transformed by [{\scr F}] into a decimated version of [{\scr F}[R_{\bf A}]] (respectively [{\scr F}[R_{\bf B}^{*}]]). Therefore, the Fourier transform exchanges subdivision and decimation of period lattices for lattice distributions.

Further insight into this phenomenon is provided by applying [\bar{\scr F}] to both sides of (iv) and (v) and invoking the convolution theorem: [\displaylines{\quad (\hbox{iv}'')\hfill \!\! R_{\bf A} = \bar{\scr F}[S_{{\bf A}/{\bf B}}^{*}] \times R_{\bf B} \;\hfill\cr \quad (\hbox{v}'')\hfill R_{\bf B}^{*} = \bar{\scr F}[S_{{\bf B}/{\bf A}}] \times R_{\bf A}^{*}. \hfill}] These identities show that multiplication by the transform of the period-subdividing distribution [S_{{\bf A}/{\bf B}}^{*}] (respectively [S_{{\bf B}/{\bf A}}]) has the effect of decimating [R_{\bf B}] to [R_{\bf A}] (respectively [R_{\bf A}^{*}] to [R_{\bf B}^{*}]). They clearly imply that, if [\boldell \in \Lambda_{\bf B}/\Lambda_{\bf A}] and [\boldell^{*} \in \Lambda_{\bf A}^{*}/\Lambda_{\bf B}^{*}], then [\eqalign{\bar{\scr F}[S_{{\bf A}/{\bf B}}^{*}] ({\boldell}) &= 1 \hbox{ if } {\boldell} = {\bf 0} \;\;\quad (i.e. \hbox{ if } {\boldell} \hbox{ belongs}\cr &{\hbox to 66pt{}}\hbox{to the class of } \Lambda_{\bf A}),\cr &= 0 \hbox{ if } {\boldell} \neq {\bf 0}\hbox{;}\cr \bar{\scr F}[S_{{\bf B}/{\bf A}}] ({\boldell}^{*}) &= 1 \hbox{ if } {\boldell}^{*} = {\bf 0} \quad (i.e. \hbox{ if } {\boldell}^{*} \hbox{ belongs}\cr &{\hbox to 60pt{}} \hbox{ to the class of } \Lambda_{\bf B}^{*}),\cr &= 0 \hbox{ if } {\boldell}^{*} \neq {\bf 0}.}] Therefore, the duality between subdivision and decimation may be viewed as another aspect of that between convolution and multiplication.

There is clearly a strong analogy between the sampling/periodization duality of Section[link] and the decimation/subdivision duality, which is viewed most naturally in terms of subgroup relationships: both sampling and decimation involve restricting a function to a discrete additive subgroup of the domain over which it is initially given.

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