はじめに
他の記事から参照するための定義や事実を記載する。
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DFT基底
$d\in\naturalNumbers,\;N_l\in\naturalNumbers\;(l=1,2,\cdots,d),\;\bm{k},\bm{n}\in\integers^d$とする。次式で定義される、$\bm{n}$に関する離散座標信号を$d$次元DFTの第$\bm{k}$基底ベクトルという。
\[ W(\bm{k},\bm{n}) \coloneqq \left(\prod_{l=1}^d N_l\right)^{-1/2} \exp i\left(\sum_{l=1}^d \frac{k_l n_l}{N_l}2\pi\right) \]
DFTの定義
\[ \]
$d\in\naturalNumbers,\;N_l\in\naturalNumbers\;(l=1,2,\cdots,d),\;\bm{k}\in\integers^d$とする。$\Omega \coloneqq \{0,1,\cdots,N_1-1\}\times\{0,1,\cdots,N_2-1\}\times\cdots\times\{0,1,\cdots,N_d-1\}$とする。$f$を周期が$(N_1,N_2,\cdots,N_d)$であるような離散座標信号$f: \integers^d\to\complexNumbers;\;\bm{n} = [n_1,n_2,\cdots,n_d]^\top \mapsto f(\bm{n})$とするとき、次式で定義される、$\bm{k}$に関する離散座標信号を$f$の離散Fourier変換(Discrete Fourier Transform; DFT)という。
\[ \DFT\parens*{f}\parens*{\bm{k}} \coloneqq \sum_{\bm{n}\in\Omega} \conj{W(\bm{k},\bm{n})} f(\bm{n}) \]
定理: 巡回畳み込みのDFTはDFTの積に比例する
主張
$d,N_l,\bm{k},\Omega$の定義はDFTの定義と同じものとする。$f,g$を周期が$(N_1,N_2,\cdots,N_d)$であるような離散座標信号$f,g: \integers^d\to\complexNumbers;\;\bm{n} = [n_1,n_2,\cdots,n_d]^\top \mapsto f(\bm{n}),g(\bm{n})$とするとき、次が成り立つ。
\[ \DFT\parens*{f\circledast g}\parens*{\bm{k}} = \left(\prod_{l=1}^d N_l\right)^{1/2}\DFT\parens*{f}\parens*{\bm{k}}\DFT\parens*{g}\parens*{\bm{k}} \]
証明
$Proof$
$\bm{N} \coloneqq [N_1,\cdots,N_d]^\top$とする。
\begin{align*} \DFT\parens*{f\circledast g}\parens*{\bm{k}} &= \sum_{\bm{n}\in\Omega}\conj{W(\bm{k},\bm{n})} \left(f\circledast g\right)(\bm{n}) = \sum_{\bm{n}\in\Omega}\conj{W(\bm{k},\bm{n})} \sum_{\bm{m}\in\Omega} f(\bm{m})g((\bm{n}-\bm{m})\%\bm{N}) \\ &= \sum_{\bm{m}\in\Omega} f(\bm{m}) \sum_{\bm{n}\in\Omega} \conj{W(\bm{k},\bm{n})}g((\bm{n}-\bm{m})\%\bm{N}) \\ &= \sum_{\bm{m}\in\Omega} f(\bm{m}) \sum_{\bm{n}\in\Omega} \left(\prod_{l=1}^d N_l\right)^{1/2} \conj{W(\bm{k},\bm{m})} \conj{W(\bm{k},\bm{n}-\bm{m})} g((\bm{n}-\bm{m})\%\bm{N}) \\ &= \left(\prod_{l=1}^d N_l\right)^{1/2} \sum_{\bm{m}\in\Omega} \conj{W(\bm{k},\bm{m})}f(\bm{m}) \sum_{\bm{n}\in\Omega} \conj{W(\bm{k},(\bm{n}-\bm{m})\%\bm{N})} g((\bm{n}-\bm{m})\%\bm{N}) \\ &= \left(\prod_{l=1}^d N_l\right)^{1/2} \sum_{\bm{m}\in\Omega} \conj{W(\bm{k},\bm{m})}f(\bm{m}) \sum_{\bm{n}\in\Omega} \conj{W(\bm{k},\bm{n})} g(\bm{n}) \\ &= \left(\prod_{l=1}^d N_l\right)^{1/2}\DFT\parens*{f}\parens*{\bm{k}}\DFT\parens*{g}\parens*{\bm{k}} \end{align*}$\square$
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