Relation between Fourier transforms

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. 47 | 1 | 2 |

Section 1.3.2.7.2.4. Relation between Fourier transforms

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

1.3.2.7.2.4. 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 1.3.2.6.5, $[\eqalign{{\scr F}[R_{\bf A}] &= {1 \over |\det {\bf A}|} R_{\bf A}^{*}\cr &= {1 \over |\det {\bf DB}|} T_{{\bf A}/{\bf B}}^{*} * R_{\bf B}^{*} \quad \quad \quad \quad \quad \quad \hbox{by (ii)}^{*}\cr &= \left({1 \over |\det {\bf D}|} T_{{\bf A}/{\bf B}}^{*}\right) * \left({1 \over |\det {\bf B}|} R_{\bf B}^{*}\right)}]$ 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 1.3.2.6.6 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.

References

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