Различия

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--- optics:res2.2 [2020/02/07 17:22] – [б)] root
+++ optics:res2.2 [2020/02/08 09:48] (текущий) – [д)] root
@@ Строка 36: / Строка 36: @@
 $$
 f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2})e^{i\omega t}dt
-=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2}-i\omega t)dt
+=\frac{1}{2\pi}\intop_{-\infty}^{\infty}\exp(-\beta^{2}t^{2}+i\omega t)dt
 $$
 вспомним значение интеграла:
@@ Строка 56: / Строка 56: @@
 ===== в) =====
-$f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}f(t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}e^{i\omega t}dt=\frac{1}{i\omega2\pi}(e^{i\omega\frac{\tau}{2}}-e^{-i\omega\frac{\tau}{2}})=\frac{\sin\left(\omega\frac{\tau}{2}\right)}{\omega\pi}=\frac{\tau}{2\pi}sinc\left(\omega\frac{\tau}{2}\right)$
+Образ ступенчатой функции:
-\textemdash{}
+$$
+f_{\omega}=\frac{1}{2\pi}\intop_{-\infty}^{\infty}f(t)e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}e^{i\omega t}dt=
+$$
+$$
+\frac{1}{i\omega2\pi}(e^{i\omega\frac{\tau}{2}}-e^{-i\omega\frac{\tau}{2}})=\frac{\sin\left(\omega\frac{\tau}{2}\right)}{\omega\pi}=\frac{\tau}{2\pi}\text{sinc}\left(\omega\frac{\tau}{2}\right) .
+$$
-$f_{\omega}=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\cos(\frac{\pi t}{\tau})e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\frac{e^{i\frac{\pi t}{\tau}}+e^{-i\frac{\pi t}{\tau}}}{2}e^{i\omega t}dt=\frac{1}{4\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\left(e^{it\left(\frac{\pi}{\tau}+\omega\right)}+e^{it\left(\omega-\frac{\pi}{\tau}\right)}\right)dt=$
+===== г) =====
-$\frac{1}{i4\pi}\left.\left(\frac{e^{it\left(\frac{\pi}{\tau}+\omega\right)}}{\frac{\pi}{\tau}+\omega}+\frac{e^{it\left(\omega-\frac{\pi}{\tau}\right)}}{\omega-\frac{\pi}{\tau}}\right)\right|_{-\frac{\tau}{2}}^{\frac{\tau}{2}}=\frac{1}{2\pi}\left(\frac{\sin\left(\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}+\frac{\sin\left(-\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=$
+Образ полуволны в виде косинуса:
-$\frac{1}{2\pi}\left(\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}-\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=\frac{\tau\cos\left(\frac{\tau}{2}\omega\right)}{\pi^{2}-\tau^{2}\omega^{2}}$
+$$
+f_{\omega}=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\cos(\frac{\pi t}{\tau})e^{i\omega t}dt=\frac{1}{2\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\frac{e^{i\frac{\pi t}{\tau}}+e^{-i\frac{\pi t}{\tau}}}{2}e^{i\omega t}dt=
+$$
+$$
+\frac{1}{4\pi}\intop_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\left(e^{it\left(\frac{\pi}{\tau}+\omega\right)}+e^{it\left(\omega-\frac{\pi}{\tau}\right)}\right)dt=
+$$
+$$
+\frac{1}{i4\pi}\left.\left(\frac{e^{it\left(\frac{\pi}{\tau}+\omega\right)}}{\frac{\pi}{\tau}+\omega}+\frac{e^{it\left(\omega-\frac{\pi}{\tau}\right)}}{\omega-\frac{\pi}{\tau}}\right)\right|_{-\frac{\tau}{2}}^{\frac{\tau}{2}}=\frac{1}{2\pi}\left(\frac{\sin\left(\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}+\frac{\sin\left(-\frac{\pi}{2}+\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=
+$$
+$$
+\frac{1}{2\pi}\left(\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\frac{\pi}{\tau}+\omega}-\frac{\cos\left(\frac{\tau}{2}\omega\right)}{\omega-\frac{\pi}{\tau}}\right)=\frac{\tau\cos\left(\frac{\tau}{2}\omega\right)}{\pi^{2}-\tau^{2}\omega^{2}}
+.$$
-\textemdash{}
+===== д) =====
-$f_{\omega}=\frac{1}{2\pi}\left(\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt+\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt\right)$
+Образ равнобедренного треугольного сигнала:
-$\intop te^{i\omega t}dt=\frac{1}{i\omega}\intop tde^{i\omega t}=\frac{1}{i\omega}\left(te^{i\omega t}-\intop e^{i\omega t}dt\right)=\frac{1}{i\omega}te^{i\omega t}+\frac{1}{\omega^{2}}e^{i\omega t}$
+$$
+f_{\omega}=\frac{1}{2\pi}\left(\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt+\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt\right).
-$\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}+\frac{2}{i\omega\tau}te^{i\omega t}+\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{-\frac{\tau}{2}}^{0}=$
+$$
+Вычислим для удобства отдельно неопределённый интеграл:
-$\frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}+\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}=\frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}$
+$$
+\intop te^{i\omega t}dt=\frac{1}{i\omega}\intop tde^{i\omega t}=\frac{1}{i\omega}\left(te^{i\omega t}-\intop e^{i\omega t}dt\right)=\frac{1}{i\omega}te^{i\omega t}+\frac{1}{\omega^{2}}e^{i\omega t}
-$\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}-\frac{2}{i\omega\tau}te^{i\omega t}-\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{0}^{\frac{\tau}{2}}=$
+$$
+и теперь, с его помощью, вычислим определённый. Сначала первый:
-$-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}-\left(\frac{1}{i\omega}-\frac{2}{\omega^{2}\tau}\right)$
+$$
+\intop_{-\frac{\tau}{2}}^{0}(1+2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}+\frac{2}{i\omega\tau}te^{i\omega t}+\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{-\frac{\tau}{2}}^{0}=
-$f_{\omega}=\frac{1}{2\pi}\left(\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}+\frac{2}{\omega^{2}\tau}\right)=\frac{2}{\pi\omega^{2}\tau}\left(1-e^{-i\omega\frac{\tau}{2}}\right)=\frac{i4e^{-i\omega\frac{\tau}{4}}\sin\left(\omega\frac{\tau}{4}\right)}{\pi\omega^{2}\tau}$
+$$
+$$
+\frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}+\frac{1}{i\omega}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}=
+$$
+$$
+\frac{1}{i\omega}+\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}.
+$$
+Теперь второй:
+$$
+\intop_{0}^{\frac{\tau}{2}}(1-2\frac{t}{\tau})e^{i\omega t}dt=\left.\left(\frac{1}{i\omega}e^{i\omega t}-\frac{2}{i\omega\tau}te^{i\omega t}-\frac{2}{\omega^{2}\tau}e^{i\omega t}\right)\right|_{0}^{\frac{\tau}{2}}=
+$$
+$$
+-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}-\left(\frac{1}{i\omega}-\frac{2}{\omega^{2}\tau}\right) .
+$$
+Ну и окончательно, используя сумму вычисленных интегралов:
+$$
+f_{\omega}=\frac{1}{2\pi}\left(\frac{2}{\omega^{2}\tau}-\frac{2}{\omega^{2}\tau}e^{-i\omega\frac{\tau}{2}}-\frac{2}{\omega^{2}\tau}e^{i\omega\frac{\tau}{2}}+\frac{2}{\omega^{2}\tau}\right)=
+$$
+$$
+\frac{1}{\pi\omega^{2}\tau}\left(2-e^{i\omega\frac{\tau}{2}}-e^{-i\omega\frac{\tau}{2}}\right)=
+\frac{2}{\pi\omega^{2}\tau}\left(1-\cos (\omega\frac{\tau}{2})\right)=
+$$
+$$
+\frac{4\sin^2\left(\omega\frac{\tau}{4}\right)}{\pi\omega^{2}\tau} =
+\frac{\tau }{4\pi} \text{sinc}^2\left(\omega\frac{\tau}{4}\right)
+.
+$$