9.2 Fourierova analýza

Příklady

Určete konvoluci funkcí $(f\ast g)(x)$:

9.23

$f(x)=\left\{ \begin{array}{r c l} 0 & &x\in(-\infty,0)\\[2pt] 2 & &x\in\langle 0,\infty) \end{array} \right.$

$g(x)=\left\{ \begin{array}{r c l} 0 & &x\in(-\infty,0)\\[2pt] 1 & &x\in\langle 0,\infty) \end{array} \right.$

$f\ast g=\left\{ \begin{array}{r c l} 0 & &x\in(-\infty,0)\\[2pt] 2x & &x\in\langle 0,\infty)\end{array}\right.$

9.24

$f(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \text{e}^{-2x} & & x\in\langle 0,\infty) \end{array}\right.$

$g(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \text{e}^{-x} & & x\in\langle 0,\infty) \end{array} \right.$

$f\ast g=\left\{ \begin{array}{r c l}0 & & x\in(-\infty,0)\\[2pt] \text{e}^{-x}\left(1-\text{e}^{-x}\right)& & x\in\langle 0,\infty)\end{array}\right.$

9.25

$f(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \text{e}^{-x} & & x\in\langle 0,\infty) \end{array} \right.$

$g(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \sin x & & x\in\langle 0,\infty) \end{array} \right.$

$f\ast g=\left\{ \begin{array}{r c l}0 & & x\in(-\infty,0)\\[2pt] \dfrac{1}{2}\left(\text{e}^{-x}+\sin x-\cos x\right) & & x\in\langle 0,\infty)\end{array}\right.$

9.26

$f(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \sin x & & x\in\langle 0,\infty) \end{array} \right.$

$g(x)=1$

$f\ast g=\left\{ \begin{array}{r c l}0& & x\in(-\infty,0)\\[4pt]1-\cos x & & x\in\langle 0,\infty)\end{array}\right.$

9.27

$f(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \sin x & & x\in\langle 0,\infty) \end{array} \right.$

$g(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[4pt] 1 & & x\in\left\langle 0,\dfrac{\pi}{2}\right\rangle\\[4pt] 0 & & x\in\left(\dfrac{\pi}{2},\infty\right)\\ \end{array}\right.$

$f\ast g=\left\{ \begin{array}{r c l}0 & & x\in(-\infty,0)\\[2pt] 1-\cos x & & x\in\left\langle 0,\dfrac{\pi}{2}\right\rangle\\[2pt] \sin x-\cos x & & x\in\left(\dfrac{\pi}{2},\infty\right)\end{array}\right.$

9.28

$f(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \sin x & & \in\langle 0,\infty) \end{array} \right.$

$g(x)=\sin(\text{a} x),\,\text{a}>0$

$f\ast g=\left\{ \begin{array}{r c l}0 & & x\in(-\infty,0)\\ \dfrac{1}{\text{a}^2–1}\left[\text{a}\sin x-\sin(\text{a} x)\right]& & x\in\langle 0,\infty)\end{array}\right.$

9.29

$f(x)=\left\{ \begin{array}{r c l} 0 & & x\in(-\infty,0)\\[2pt] \sin x & & x\in\langle 0,\infty) \end{array} \right.$

$g(x)=\left\{ \begin{array}{r c l} 0 & & x\in\left(-\infty,0\right)\\[2pt] 1-\dfrac{2}{\pi}x & & x\in\left\langle 0,\dfrac{\pi}{2}\right\rangle\\[4pt] 0 & & x\in\left(\dfrac{\pi}{2},\infty\right)\\ \end{array}\right.$

$f\ast g=\left\{ \begin{array}{r c l}0 & & x\in(-\infty,0) \\[2pt]1-\dfrac{2}{\pi}x+\dfrac{2}{\pi}\sin x-\cos x & & x\in\left\langle 0,\dfrac{\pi}{2}\right\rangle \\[2pt]\dfrac{2}{\pi}(\sin x+\cos x)-\cos x & & x\in\left(\dfrac{\pi}{2},\infty\right)\end{array}\right.$

9.30

Určete Fourierův obraz funkce:

$f(x)=1$ pro $x\in\left\langle-\dfrac{1}{2},\dfrac{1}{2}\right\rangle$, $f(x)=0$ pro $x\notin\left\langle -\dfrac{1}{2},\dfrac{1}{2}\right\rangle$,

$\widehat{f}(\omega)=\dfrac{2}{\omega}\sin\left(\dfrac{\omega}{2}\right)$
$f(x)=\text{e}^{-\text{b} x}$ pro $x\in\left\langle 0,\infty\right)$, $f(x)=0$ pro $x\in\left(-\infty,0\right)$, $\text{b}>0=\text{konst.}$,

$\widehat{f}(\omega)=\dfrac{1}{\text{b}+\text{i}\omega}$
$f(x)=\text{e}^{-\text{b}|x|}$ pro $x\in\left(-\infty,\infty\right)$,

$\widehat{f}(\omega)=\dfrac{2\text{b}}{\text{b}^2+\omega^2}$
$f(x)=\text{e}^{-\text{b} x^2}$ pro $x\in\left(-\infty,\infty\right)$,

$\widehat{f}(\omega)=\sqrt{\dfrac{\pi}{\text{b}}}\,\text{e}^{-\frac{\omega^2}{4\text{b}}}$
$f(x)=1+x$ pro $x\in\left\langle-1,0\right)$, $f(x)=1-x$ pro $x\in\left\langle 0,1\right)$, $f(x)=0$ pro $|x|>1$,

$\widehat{f}(\omega)=\dfrac{2}{\omega^2}\left(1-\cos\omega\right)$
$f(x)=x^2$ pro $x\in\left\langle-1,1\right\rangle$, $f(x)=0$ pro $|x|>1$.

$\widehat{f}(\omega)=\dfrac{2}{\omega^3}\left[(\omega^2–2)\sin\omega+2\omega\cos\omega\right]$

9.31

Určete Fourierův obraz funkce:

$f(x)=\sin x$ pro $x\in\left\langle 0,2\pi\right\rangle$, $f(x)=0$ pro $x\notin\left\langle 0,2\pi\right\rangle$,

$\widehat{f}(\omega)=\dfrac{\text{i}\omega\left(1-\text{e}^{-2\pi\text{i}\omega}\right)}{1-\omega^2}$
$f(x)=\cos x$ pro $x\in\left\langle 0,2\pi\right\rangle$, $f(x)=0$ pro $x\notin\left\langle 0,2\pi\right\rangle$,

$\widehat{f}(\omega)=-\dfrac{1-\text{e}^{-2\pi\text{i}\omega}}{\omega^2–1}$
$f(x)=\sin x\cos x$ pro $x\in\left\langle 0,2\pi\right\rangle$, $f(x)=0$ pro $x\notin\left\langle 0,2\pi\right\rangle$,

$\widehat{f}(\omega)=-\dfrac{1-\text{e}^{-2\pi\text{i}\omega}}{\omega^2–4}$
$f(x)=\sin^2x\cos^2x$ pro $x\in\left\langle 0,2\pi\right\rangle$, $f(x)=0$ pro $x\notin\left\langle 0,2\pi\right\rangle$,

$\widehat{f}(\omega)=-\dfrac{2\text{i}\left(1-\text{e}^{-2\pi\text{i}\omega}\right)}{\omega\left(\omega^2–16\right)}$
$f(x)=\text{A}^{\text{B} x}$ pro $x\in\left\langle 0,1\right\rangle$, $f(x)=0$ pro $x\notin\left\langle 0,1\right\rangle$.

$\widehat{f}(\omega)=\dfrac{1-\text{e}^{-\text{i}\omega}A^B}{\text{i}\omega-B\ln A}$

9.32

Ověřte platnost rovnice 9.10

na příkladu 9.23

$\widehat{f}(\omega)=-\dfrac{2\text{i}}{\omega}$,

$\widehat{g\,}(\omega)=-\dfrac{\text{i}}{\omega}$,

$\widehat{(f\ast g)}(\omega)=-\dfrac{2}{\omega^2}$
na příkladu 9.24

$\widehat{f}(\omega)=\dfrac{1}{2+\text{i}\omega}$,

$\widehat{g\,}(\omega)=\dfrac{1}{1+\text{i}\omega}$,

$\widehat{(f\ast g)}(\omega)=\dfrac{1}{2-\omega^2+3\text{i}\omega}$
na příkladu 9.25

$\widehat{f}(\omega)=\dfrac{1}{1+\text{i}\omega}$,

$\widehat{g\,}(\omega)=\dfrac{1}{1-\omega^2}$,

$\widehat{(f\ast g)}(\omega)=\dfrac{1}{(1+\text{i}\omega)(1-\omega^2)}$
na příkladu 9.26

$\widehat{f}(\omega)=\dfrac{1}{1-\omega^2}$,

$\widehat{g\,}(\omega)=\dfrac{1}{\text{i}\omega}$,

$\widehat{(f\ast g)}(\omega)=\dfrac{1}{\text{i}\omega(1-\omega^2)}$