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| optics:res2.2 [2020/02/07 17:14] – root | optics:res2.2 [2020/02/08 09:48] (текущий) – [д)] root | ||
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| Строка 17: | Строка 17: | ||
| Для функции $f(t)=\cos(\omega_{0}t)$ рассмотрим её образ: | Для функции $f(t)=\cos(\omega_{0}t)$ рассмотрим её образ: | ||
| - | $f_{\omega}=\frac{1}{2\pi}\intop\limits_{-\infty}^{\infty}f(t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\cos(\omega_{0}t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{e^{i\omega_{0}t}+e^{-i\omega_{0}t}}{2}e^{i\omega t}dt=$ | + | $$ |
| + | f_{\omega}=\frac{1}{2\pi}\intop\limits_{-\infty}^{\infty}f(t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\cos(\omega_{0}t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{e^{i\omega_{0}t}+e^{-i\omega_{0}t}}{2}e^{i\omega t}dt= | ||
| + | $$ | ||
| + | $$ | ||
| + | =\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{e^{i(\omega_{0}+\omega)t}+e^{i(\omega-\omega_{0})t}}{2}dt=\frac{1}{2}\left(\delta(\omega_{0}+\omega)+\delta(-\omega_{0}+\omega)\right), | ||
| + | $$ | ||
| + | где воспользовались определением | ||
| + | $$ | ||
| + | \delta(\omega_{0}+\omega)=\frac{1}{2\pi}\intop_{-\infty}^{\infty}e^{i(\omega_{0}+\omega)t}dt | ||
| + | $$ | ||
| + | дельта функции. | ||
| - | $=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\frac{e^{i(\omega_{0}+\omega)t}+e^{i(\omega-\omega_{0})t}}{2}dt=\frac{1}{2}\left(\delta(\omega_{0}+\omega)+\delta(-\omega_{0}+\omega)\right)$ | + | ===== б) ===== |
| - | $\delta(\omega_{0}+\omega)=\frac{1}{2\pi}\intop_{-\infty}^{\infty}e^{i(\omega_{0}+\omega)t}dt$ | + | Для функции распределения Гаусса: |
| - | \textemdash{} дельта функция. | + | $$f(t)=\exp(-\beta^{2}t^{2})$$ |
| + | для того что бы найти её образ | ||
| + | $$ | ||
| + | f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2})e^{i\omega t}dt | ||
| + | =\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2}+i\omega t)dt | ||
| + | $$ | ||
| + | вспомним значение интеграла: | ||
| + | $$ | ||
| + | \intop_{-\infty}^{\infty}\exp(-t^{2})dt=\sqrt{\pi}, | ||
| + | $$ | ||
| + | тогда приведём показатель | ||
| + | $$ | ||
| + | \beta^{2}t^{2}-i\omega t=\beta^{2}t^{2}-i\omega t\cdot\frac{2\beta}{2\beta}\pm\left(\frac{i\omega}{2\beta}\right)^{2}=\left(\beta t-\frac{i\omega}{2\beta}\right)^{2}-\left(\frac{i\omega}{2\beta}\right)^{2}=z^{2}+\left(\frac{\omega}{2\beta}\right)^{2}. | ||
| + | $$ | ||
| + | Теперь запишем и сам интеграл | ||
| + | $$ | ||
| + | f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2})e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp\left(-z^{2}-\left(\frac{\omega}{2\beta}\right)^{2}\right)d\left(\frac{z}{\beta}\right)= | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{\exp\left(-\left(\frac{\omega}{2\beta}\right)^{2}\right)}{2\pi\beta}\intop_{-\infty}^{\infty}\exp\left(-z^{2}\right)dz=\frac{\exp\left(-\left(\frac{\omega}{2\beta}\right)^{2}\right)}{2\sqrt{\pi}\beta} | ||
| + | $$ | ||
| - | \textemdash{} | + | ===== в) ===== |
| - | $f(t)=\exp(-\beta^{2}t^{2})$ | + | Образ ступенчатой функции: |
| - | $f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2})e^{i\omega t}dt$ | + | $$ |
| + | f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}f(t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}e^{i\omega t}dt= | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{i\omega2\pi}(e^{i\omega\frac{\tau}{2}}-e^{-i\omega\frac{\tau}{2}})=\frac{\sin\left(\omega\frac{\tau}{2}\right)}{\omega\pi}=\frac{\tau}{2\pi}\text{sinc}\left(\omega\frac{\tau}{2}\right) . | ||
| + | $$ | ||
| - | $\intop_{-\infty}^{\infty}\exp(-t^{2})dt=\sqrt{\pi}$ | + | ===== г) ===== |
| - | $\beta^{2}t^{2}-i\omega t=\beta^{2}t^{2}-i\omega t\cdot\frac{2\beta}{2\beta}\pm\left(\frac{i\omega}{2\beta}\right)^{2}=\left(\beta t-\frac{i\omega}{2\beta}\right)^{2}-\left(\frac{i\omega}{2\beta}\right)^{2}=z^{2}+\left(\frac{\omega}{2\beta}\right)^{2}$ | + | Образ полуволны в виде косинуса: |
| - | $f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2})e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp\left(-z^{2}-\left(\frac{\omega}{2\beta}\right)^{2}\right)d\left(\frac{z}{\beta}\right)=$ | + | $$ |
| + | f_{\omega}=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\cos(\frac{\pi t}{\tau})e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\frac{e^{i\frac{\pi | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{4\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\left(e^{it\left(\frac{\pi}{\tau}+\omega\right)}+e^{it\left(\omega-\frac{\pi}{\tau}\right)}\right)dt= | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{i4\pi}\left.\left(\frac{e^{it\left(\frac{\pi}{\tau}+\omega\right)}}{\frac{\pi}{\tau}+\omega}+\frac{e^{it\left(\omega-\frac{\pi}{\tau}\right)}}{\omega-\frac{\pi}{\tau}}\right)\right|_{-\frac{\tau}{2}}^{\frac{\tau}{2}}=\frac{1}{2\pi}\left(\frac{\sin\left(\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}+\frac{\sin\left(-\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)= | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{2\pi}\left(\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}-\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=\frac{\tau\cos\left(\frac{\tau}{2}\omega\right)}{\pi^{2}-\tau^{2}\omega^{2}} | ||
| + | .$$ | ||
| - | $\frac{\exp\left(-\left(\frac{\omega}{2\beta}\right)^{2}\right)}{2\pi\beta}\intop_{-\infty}^{\infty}\exp\left(-z^{2}\right)dz=\frac{\exp\left(-\left(\frac{\omega}{2\beta}\right)^{2}\right)}{2\sqrt{\pi}\beta}$ | + | ===== д) ===== |
| - | \textemdash{} | + | Образ равнобедренного треугольного сигнала: |
| - | $f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}f(t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}e^{i\omega t}dt=\frac{1}{i\omega2\pi}(e^{i\omega\frac{\tau}{2}}-e^{-i\omega\frac{\tau}{2}})=\frac{\sin\left(\omega\frac{\tau}{2}\right)}{\omega\pi}=\frac{\tau}{2\pi}sinc\left(\omega\frac{\tau}{2}\right)$ | + | $$ |
| - | + | f_{\omega}=\frac{1}{2\pi}\left(\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt+\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt\right). | |
| - | \textemdash{} | + | $$ |
| - | + | Вычислим для удобства отдельно неопределённый интеграл: | |
| - | $f_{\omega}=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\cos(\frac{\pi t}{\tau})e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\frac{e^{i\frac{\pi t}{\tau}}+e^{-i\frac{\pi t}{\tau}}}{2}e^{i\omega t}dt=\frac{1}{4\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\left(e^{it\left(\frac{\pi}{\tau}+\omega\right)}+e^{it\left(\omega-\frac{\pi}{\tau}\right)}\right)dt=$ | + | $$ |
| - | + | \intop te^{i\omega t}dt=\frac{1}{i\omega}\intop tde^{i\omega t}=\frac{1}{i\omega}\left(te^{i\omega t}-\intop e^{i\omega t}dt\right)=\frac{1}{i\omega}te^{i\omega t}+\frac{1}{\omega^{2}}e^{i\omega t} | |
| - | $\frac{1}{i4\pi}\left.\left(\frac{e^{it\left(\frac{\pi}{\tau}+\omega\right)}}{\frac{\pi}{\tau}+\omega}+\frac{e^{it\left(\omega-\frac{\pi}{\tau}\right)}}{\omega-\frac{\pi}{\tau}}\right)\right|_{-\frac{\tau}{2}}^{\frac{\tau}{2}}=\frac{1}{2\pi}\left(\frac{\sin\left(\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}+\frac{\sin\left(-\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=$ | + | $$ |
| - | + | и теперь, | |
| - | $\frac{1}{2\pi}\left(\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}-\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=\frac{\tau\cos\left(\frac{\tau}{2}\omega\right)}{\pi^{2}-\tau^{2}\omega^{2}}$ | + | $$ |
| - | + | \intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}+\frac{2}{i\omega\tau}te^{i\omega t}+\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{-\frac{\tau}{2}}^{0}= | |
| - | \textemdash{} | + | $$ |
| - | + | $$ | |
| - | $f_{\omega}=\frac{1}{2\pi}\left(\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt+\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt\right)$ | + | \frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}+\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}= |
| - | + | $$ | |
| - | $\intop te^{i\omega t}dt=\frac{1}{i\omega}\intop tde^{i\omega t}=\frac{1}{i\omega}\left(te^{i\omega t}-\intop e^{i\omega t}dt\right)=\frac{1}{i\omega}te^{i\omega t}+\frac{1}{\omega^{2}}e^{i\omega t}$ | + | $$ |
| - | + | \frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}. | |
| - | $\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}+\frac{2}{i\omega\tau}te^{i\omega t}+\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{-\frac{\tau}{2}}^{0}=$ | + | $$ |
| - | + | Теперь второй: | |
| - | $\frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}+\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}=\frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}$ | + | $$ |
| - | + | \intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}-\frac{2}{i\omega\tau}te^{i\omega t}-\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{0}^{\frac{\tau}{2}}= | |
| - | $\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}-\frac{2}{i\omega\tau}te^{i\omega t}-\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{0}^{\frac{\tau}{2}}=$ | + | $$ |
| - | + | $$ | |
| - | $-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}-\left(\frac{1}{i\omega}-\frac{2}{\omega^{2}\tau}\right)$ | + | -\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}-\left(\frac{1}{i\omega}-\frac{2}{\omega^{2}\tau}\right) |
| - | + | $$ | |
| - | $f_{\omega}=\frac{1}{2\pi}\left(\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}+\frac{2}{\omega^{2}\tau}\right)=\frac{2}{\pi\omega^{2}\tau}\left(1-e^{-i\omega\frac{\tau}{2}}\right)=\frac{i4e^{-i\omega\frac{\tau}{4}}\sin\left(\omega\frac{\tau}{4}\right)}{\pi\omega^{2}\tau}$ | + | Ну и окончательно, |
| + | $$ | ||
| + | f_{\omega}=\frac{1}{2\pi}\left(\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}+\frac{2}{\omega^{2}\tau}\right)= | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{1}{\pi\omega^{2}\tau}\left(2-e^{i\omega\frac{\tau}{2}}-e^{-i\omega\frac{\tau}{2}}\right)= | ||
| + | \frac{2}{\pi\omega^{2}\tau}\left(1-\cos (\omega\frac{\tau}{2})\right)= | ||
| + | $$ | ||
| + | $$ | ||
| + | \frac{4\sin^2\left(\omega\frac{\tau}{4}\right)}{\pi\omega^{2}\tau} | ||
| + | \frac{\tau }{4\pi} \text{sinc}^2\left(\omega\frac{\tau}{4}\right) | ||
| + | . | ||
| + | $$ | ||