畳み込みの性質

by nomura · 2025年2月12日

畳み込みの性質
関数\(f\left(x\right),g\left(x\right),h\left(x\right)\)、または数列\(f_{n},g_{n},h_{n}\)があるとき畳み込み\(f*g\)に関して次が成り立ちます。

(1)交換律

\[ f*g=g*f \]

(2)結合律

\[ \left(f*g\right)*h=f*\left(g*h\right) \]

(3)分配率

\[ f*\left(g+h\right)=f*g+f*h \]

(4)スカラー倍

\(c\in\mathbb{C}\)とする。
\begin{align*} c\left(f*g\right) & =\left(cf\right)*g\\ & =f*\left(cg\right) \end{align*}

(5)微分

\begin{align*} D\left(f*g\right)\left(x\right) & =\left(\left(Df\right)*g\right)\left(x\right)\\ & =\left(f*\left(Dg\right)\right)\left(x\right) \end{align*} \begin{align*} D\left(\left(f*g\right)_{n}\right) & =f_{n+1}g_{0}+\left(f*\left(Dg\right)\right)_{n}\\ & =f_{0}g_{n+1}+\left(\left(Df\right)*g\right)_{n} \end{align*}

(6)畳み込み定理

\[ \mathcal{F}\left(\left(f*g\right)\left(x\right)\right)=\mathcal{F}\left(\left(f\right)\left(x\right)\right)\mathcal{F}\left(\left(g\right)\left(x\right)\right) \] \[ \sum_{k=0}^{\infty}\left(f*g\right)_{k}=\sum_{j=0}^{\infty}f_{j}\sum_{k=0}^{\infty}g_{k} \]

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\(\mathcal{F}\left(f\right)\)はフーリエ変換

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(5)は\(f_{n},g_{n},n\in\mathbb{N}\)のときは、\(D\left(\left(f*g\right)_{n}\right)=f_{n+1}g_{1}+\left(f*\left(Dg\right)\right)_{n}=f_{1}g_{n+1}+\left(\left(Df\right)*g\right)_{n}\)になります。

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数列の微分は\(Da_{n}=a_{n+1}-a_{n}\)とします。

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フーリエ変換の他にラプラス変換・Z変化・メリン変換に対しても成り立ちます。

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フーリエ変換は
\[ \mathcal{F}_{x}\left[f\left(x\right)\right]\left(k\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{2\pi ikx}dx \] または、
\[ \mathcal{F}_{x}\left[f\left(x\right)\right]\left(\nu\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{i\nu x}dx \] とします。

(1)

\begin{align*} \left(f*g\right)\left(x\right) & =\int_{-\infty}^{\infty}f\left(t\right)g\left(x-t\right)dt\\ & =-\int_{\infty}^{-\infty}f\left(x-t\right)g\left(t\right)dt\cmt{t\rightarrow x-t}\\ & =\int_{-\infty}^{\infty}f\left(x-t\right)g\left(t\right)dt\\ & =\left(g*f\right)\left(x\right) \end{align*} \begin{align*} \left(f*g\right)_{n} & =\sum_{k=0}^{n}f_{k}g_{n-k}\\ & =\sum_{k=0}^{n}f_{n-k}g_{k}\cmt{k\rightarrow n-k}\\ & =\left(g*f\right)_{n} \end{align*}

(2)

\begin{align*} \left(\left(f*g\right)*h\right)\left(x\right) & =\int_{-\infty}^{\infty}\left(f*g\right)\left(t'\right)h\left(x-t'\right)dt'\\ & =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\left(t\right)g\left(t'-t\right)dt\cdot h\left(x-t'\right)dt'\\ & =\int_{-\infty}^{\infty}f\left(t\right)\int_{-\infty}^{\infty}g\left(t'-t\right)h\left(x-t'\right)dt'dt\\ & =\int_{-\infty}^{\infty}f\left(t\right)\int_{-\infty}^{\infty}g\left(t'\right)h\left(x-t-t'\right)dt'dt\cmt{t'\rightarrow t'+t}\\ & =\int_{-\infty}^{\infty}f\left(t\right)\left(g*h\right)\left(x-t\right)dt\\ & =\left(f*\left(g*h\right)\right)\left(x\right) \end{align*} \begin{align*} \left(\left(f*g\right)*h\right)_{n} & =\sum_{k'=0}^{n}\left(f*g\right)_{k'}h_{n-k'}\\ & =\sum_{k'=0}^{n}\sum_{k=0}^{k'}f_{k}g_{k'-k}h_{n-k'}\\ & =\sum_{k=0}^{n}f_{k}\sum_{k'=k}^{n}g_{k'-k}h_{n-k'}\\ & =\sum_{k=0}^{n}f_{k}\sum_{k'=0}^{n-k}g_{k'}h_{n-k-k'}\cmt{k'\rightarrow k'+k}\\ & =\sum_{k=0}^{n}f_{k}\left(g*h\right)_{n-k}\\ & =\left(f*\left(g*h\right)\right)_{n} \end{align*}

(3)

\begin{align*} \left(f*\left(g+h\right)\right)\left(x\right) & =\int_{-\infty}^{\infty}f\left(t\right)\left(g\left(x-t\right)+h\left(x-t\right)\right)dt\\ & =\int_{-\infty}^{\infty}f\left(t\right)g\left(x-t\right)dt+\int_{-\infty}^{\infty}f\left(t\right)h\left(x-t\right)dt\\ & =\left(f*g\right)\left(x\right)+\left(f*h\right)\left(x\right) \end{align*} \begin{align*} \left(f*\left(g+h\right)\right)_{n} & =\sum_{k=0}^{n}f_{k}\left(g+h\right)_{n-k}\\ & =\sum_{k=0}^{n}f_{k}g_{n-k}+\sum_{k=0}^{n}f_{k}h_{n-k}\\ & =\left(f*g\right)_{n}+\left(f*h\right)_{n} \end{align*}

(4)

\begin{align*} \left(c\left(f*g\right)\right)\left(x\right) & =c\int_{-\infty}^{\infty}f\left(t\right)g\left(x-t\right)dt\\ & =\int_{-\infty}^{\infty}cf\left(t\right)g\left(x-t\right)dt\\ & =\left(\left(cf\right)*g\right)\left(x\right) \end{align*} \begin{align*} \left(c\left(f*g\right)\right)\left(x\right) & =c\int_{-\infty}^{\infty}f\left(t\right)g\left(x-t\right)dt\\ & =\int_{-\infty}^{\infty}f\left(t\right)cg\left(x-t\right)dt\\ & =\left(f*\left(cg\right)\right)\left(x\right) \end{align*} \begin{align*} \left(c\left(f*g\right)\right)_{n} & =c\sum_{k=0}^{n}f_{k}g_{n-k}\\ & =\sum_{k=0}^{n}\left(cf\right)_{k}g_{n-k}\\ & =\left(\left(cf\right)*g\right)_{n} \end{align*} \begin{align*} \left(c\left(f*g\right)\right)_{n} & =c\sum_{k=0}^{n}f_{k}g_{n-k}\\ & =\sum_{k=0}^{n}f_{k}\left(cg\right)_{n-k}\\ & =\left(\left(cf\right)*g\right)_{n} \end{align*}

(5)

\begin{align*} D\left(f*g\right)\left(x\right) & =D\int_{-\infty}^{\infty}f\left(t\right)g\left(x-t\right)dt\\ & =\int_{-\infty}^{\infty}f\left(t\right)\left(Dg\left(x-t\right)\right)dt\\ & =\left(f*\left(Dg\right)\right)\left(x\right) \end{align*} \begin{align*} \left(D\left(f*g\right)\right)\left(x\right) & =D\left(g*f\right)\\ & =\left(g*\left(Df\right)\right)\left(x\right)\\ & =\left(\left(Df\right)*g\right)\left(x\right) \end{align*} \begin{align*} D\left(\left(f*g\right)_{n}\right) & =D_{n}\sum_{k=0}^{n}f_{k}g_{n-k}\\ & =\sum_{k=0}^{n+1}f_{k}g_{n+1-k}-\sum_{k=0}^{n}f_{k}g_{n-k}\\ & =f_{n+1}g_{0}+\sum_{k=0}^{n}f_{k}g_{n+1-k}-\sum_{k=0}^{n}f_{k}g_{n-k}\\ & =f_{n+1}g_{0}+\sum_{k=0}^{n}f_{k}\left(g_{n+1-k}-g_{n-k}\right)\\ & =f_{n+1}g_{0}+\sum_{k=0}^{n}f_{k}Dg_{n-k}\\ & =f_{n+1}g_{0}+\left(f*\left(Dg\right)\right)_{n} \end{align*} \begin{align*} D\left(\left(f*g\right)_{n}\right) & =\left(D\left(g*f\right)_{n}\right)\\ & =g_{n+1}f_{0}+\left(g*\left(Df\right)\right)_{n}\\ & =f_{0}g_{n+1}+\left(\left(Df\right)*g\right)_{n} \end{align*}

(6)

\begin{align*} \mathcal{F}_{x}\left[\left(f*g\right)\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\left(t\right)g\left(x-t\right)dte^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}f\left(t\right)\int_{-\infty}^{\infty}g\left(x-t\right)e^{-2\pi i\xi x}dxdt\\ & =\int_{-\infty}^{\infty}f\left(t\right)\mathcal{F}_{x}\left[g\left(x-t\right)\right]\left(\xi\right)dt\\ & =\int_{-\infty}^{\infty}f\left(t\right)e^{-2\pi i\xi t}\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)dt\\ & =\int_{-\infty}^{\infty}f\left(t\right)e^{-2\pi i\xi t}dt\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{t}\left[f\left(t\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \end{align*} \begin{align*} \sum_{k=0}^{\infty}\left(f*g\right)_{k} & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}f_{j}g_{k-j}\\ & =\sum_{j=0}^{\infty}f_{j}\sum_{k=j}^{\infty}g_{k-j}\\ & =\sum_{j=0}^{\infty}f_{j}\sum_{k=j}^{\infty}g_{k-j}\cmt{k\rightarrow k+j}\\ & =\sum_{j=0}^{\infty}f_{j}\sum_{k=0}^{\infty}g_{k} \end{align*}