From 542c71528960701b09cef9fa41e0d4235e574ef5 Mon Sep 17 00:00:00 2001 From: Anatoly Date: Sat, 8 May 2021 20:35:33 +0300 Subject: [PATCH] =?UTF-8?q?=D0=9F=D1=80=D0=B8=D0=B2=D0=B5=D0=BB=20=D0=B2?= =?UTF-8?q?=20=D0=BF=D0=BE=D1=80=D1=8F=D0=B4=D0=BE=D0=BA=20=D0=B3=D1=80?= =?UTF-8?q?=D0=B0=D1=84=D0=B8=D0=BA=D0=B8?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- term_paper.pdf | Bin 1170340 -> 1166475 bytes term_paper.tex | 301 ++++++++++++++++++++++++++++++++++++------------- 2 files changed, 225 insertions(+), 76 deletions(-) diff --git a/term_paper.pdf b/term_paper.pdf index 3a986a658d90f6f715fa29c79e3702cebe1493c3..0f23bbef62ed5e7e45239606306bfe863f2c6554 100644 GIT binary patch delta 155403 zcmV)eK&HQ>w?~VaM}UL@gaWh!Zy^IRHaM4&A`2*g#ac^?Ex8Te_phjBA9Bkh$?_{h z7764AW)TSK-R)U~TqY252?K%o`;nhz*KwU!pGg)?Ro8LJvMhc2Bo{3ISYY|n3+%3c zczgNov$hL5MQC_=`)NTSCpEBfMx?e^7cb`cwh$}=e*9qprGsC7NkhLaY*B$F 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$a_{макс}$. $w(а)$ мгновенных значений сообщения, функции распределения $F(a)$ и построить их графики (рис. \ref{fig:prob_plots}). - \[ w(a)=\frac{1}{a_{макс}-a_{мин}}=\frac1\Delta=\frac{1}{25,6+25,6}=0,02 \] - \[ F(a)=\int^a_{-\infty}w(a)da= - \int^a_{a_{мин}}\frac{1}{\Delta}da= - \begin{cases} - 1, & a > a_{макс}\\ - \frac{a-a_{мин}}{\Delta}, & a_{мин} \leq a \leq a_{макс}\\ - 0, & a < a_{мин} - \end{cases}\] + \begin{equation} + w(a)=\frac{1}{a_{макс}-a_{мин}}=\frac1\Delta=\frac{1}{25,6+25,6}=0,02 + \end{equation} + + \begin{equation} + F(a)=\int^a_{-\infty}w(a)da= + \int^a_{a_{мин}}\frac{1}{\Delta}da= + \begin{cases} + 1, & a > a_{макс}\\ + \frac{a-a_{мин}}{\Delta}, & a_{мин} \leq a \leq a_{макс}\\ + 0, & a < a_{мин} + \end{cases} + \end{equation} + где $\Delta = a_{макс}-a_{мин}=51,2\, В$. % Графики @@ -281,27 +291,36 @@ $a_{макс}$. \end{figure} \item Рассчитать математическое ожидание $\overline{A(t)}$ и дисперсию $D\{A(t)\}$ сообщения $A(t)$. - \[ \overline{A(t)}=\int^\infty_{-\infty}a\cdot w(a)da= - \int^{a_{макс}}_{a_{мин}}a \frac{1}{a_{макс}-a_{мин}} da= - \frac{a^2}{2\Delta} \Biggr|^{a_{макс}}_{a_{мин}}\! = - \frac{a_{макс}^2-a_{мин}^2}{2\Delta}=0 \] + \begin{equation} + \overline{A(t)}=\int^\infty_{-\infty}a\cdot w(a)da= + \int^{a_{макс}}_{a_{мин}}a \frac{1}{a_{макс}-a_{мин}} da= + \frac{a^2}{2\Delta} \Biggr|^{a_{макс}}_{a_{мин}}\! = + \frac{a_{макс}^2-a_{мин}^2}{2\Delta}=0 + \end{equation} - \begin{align*}\begin{split} + + \begin{align}\begin{split} D\{A(t)\}&=\int^\infty_{-\infty}(a-\overline{A(t)})^2 w(a)da= \int^{a_{макс}}_{a_{мин}}a^2w(a)da\\ &=\frac{a^3}{3\Delta}\Biggr|^{a_{макс}}_{a_{мин}}\! =\frac{a_\text{min}^2+a_\text{max}a_\text{min}+a_\text{max}^2}{3} =218,5 - \end{split}\end{align*} + \end{split}\end{align} \item Написать аналитическое выражение для спектральной плотности мощности $G_A(f)$ сообщения $A(t)$ и построить график (рис. \ref{fig:spectr_plot}). - \[ G_A(f)=\frac{D\{A(t)\}}{2f_В}=\frac{218,5}{2\cdot1,41\cdot 10^4} - =7,7 \,мВ^2/Гц \] - \[ G_A(f)=\begin{cases} - 7,7 \,мВ^2/Гц, & |f| \leq f_B\\ - 0, & |f| > f_B - \end{cases} \] + \begin{equation} + G_A(f)=\frac{D\{A(t)\}}{2f_В}=\frac{218,5}{2\cdot1,41\cdot 10^4} + =7,7 \,мВ^2/Гц + \end{equation} + + \begin{equation} + G_A(f)=\begin{cases} + 7,7 \,мВ^2/Гц, & |f| \leq f_B\\ + 0, & |f| > f_B + \end{cases} + \end{equation} + \begin{figure}[H] \centering \begin{tikzpicture} @@ -344,12 +363,13 @@ $a_{макс}$. является ли сообщение $A(t)$ эргодическим случайным процессом или не является таковым. - \begin{align*}\begin{split} + \begin{align}\begin{split} B_A(\tau)&=\int^\infty_{-\infty}\frac{G_A(f)}{2}e^{j2\pi f\tau}df =\int^{f_B}_{-f_B}\frac{G_A}{2}\cos{2\pi f\tau}df\\ &=\frac{G_A}2 \frac{\sin{2\pi f \tau}}{2\pi \tau}\Biggr|^{f_B}_{-f_B} =G_A\frac{\sin{2\pi f_B \tau}}{2\pi\tau} - \end{split}\end{align*} + \end{split}\end{align} + \begin{figure}[H] \centering \begin{tikzpicture} @@ -358,10 +378,16 @@ $a_{макс}$. \pgfmathsetmacro{\Ga}{0.0077} \begin{axis}[ width=10cm,height=6cm, - axis lines = left, + axis lines = middle, ylabel = {$B_A(\tau)$}, xlabel = {$\tau$}, + scaled x ticks = false, + xtick = {0.00205, 0.0041}, + xticklabels = {$\frac{1}{2f_B}$, $\frac{2}{2f_B}$}, + ytick = {0}, + yticklabels = {0}, ] + \graxisX; \addplot [ color=blue, samples=100, @@ -393,8 +419,11 @@ $a(t)$ сообщения $A(t)$ по времени. В моменты врем мгновенных значений реализации $a(t)$. Расстояние между отсчетами равно интервалу $\Delta t$, величина которого определяется в соответствии с теоремой Котельникова: -\[\Delta t \leq \frac{1}{2f_B};\, -f_d=\frac{1}{\Delta t}\geq2f_B\] +\begin{equation} + \Delta t \leq \frac{1}{2f_B};\, + f_d=\frac{1}{\Delta t}\geq2f_B +\end{equation} + где $f_d$ -- частота дискретизации. На втором этапе выполняется квантование точных отсчетов @@ -425,25 +454,52 @@ $L=2^k$, где $k$ -- целое число. \item Рассчитать интервал дискретизации $\Delta t$ для получения непрерывных отсчетов $a(t_i)$ реализации $a(t),\, t_i=i\cdot\Delta t,\, i=0,\pm1,\pm2,...$. - \[ \Delta t \leq \frac{1}{2f_B}=\frac1 {2\cdot 14100} = 3,546\cdot 10^{-5}\, с \] + \begin{equation} + \Delta t \leq \frac{1}{2f_B}=\frac1 {2\cdot 14100} = 3,546\cdot 10^{-5}\, с + \end{equation} + \item Рассчитать частоту дискретизации $f_d$. - \[ f_d=\frac{1}{\Delta t}\geq 2f_B=\frac{1}{3,546\cdot 10^{-5}}=28200 \] + \begin{equation} + f_d=\frac{1}{\Delta t}\geq 2f_B=\frac{1}{3,546\cdot 10^{-5}}=28200 + \end{equation} + \item Определить число уровней квантования $L$. - \[ k=9;\, L=2^9 = 512 \] + \begin{equation} + k=9;\, L=2^9 = 512 + \end{equation} + \item Рассчитать мощность шума квантования $P_{ШК}$ и сравнить ее с мощностью непрерывного сообщения $A(t)$. - \[ P_{ШК}=\Delta a^2/12 - =\frac{0,1^2}{12}=8,33\cdot10^{-4}\, В^2 \] - \[ P_{A(t)}=A^2(t)=1\, В^2\] - \[ P_{A(t)} >> P_{ШК} \] + \begin{equation} + P_{ШК}=\Delta a^2/12 + =\frac{0,1^2}{12}=8,33\cdot10^{-4}\, В^2 + \end{equation} + + \begin{equation} + P_{A(t)}=A^2(t)=1\, В^2 + \end{equation} + + \begin{equation} + P_{A(t)} >> P_{ШК} + \end{equation} + \item Найти минимальное число $k$ двоичных разрядов, требуемое для записи в двоичной форме любого номера $j$ из $L-1$ номеров уровней квантования. - \[ L-1=511_{10}=111111111_2 \] - \[ k_{люб}=9 \] + \begin{equation} + L-1=511_{10}=111111111_2 + \end{equation} + + \begin{equation} + k_{люб}=9 + \end{equation} + \item Записать $k$-разрядное двоичное число, соответствующее заданному уровню квантования $j$. - \[ j=377_{10}=101111001_2 \] + \begin{equation} + j=377_{10}=101111001_2 + \end{equation} + \item Начертить временную диаграмму отклика АЦП $b_{АЦП}(t)$ на заданный уровень квантования $j$ в виде последовательности импульсов, @@ -494,9 +550,11 @@ $L=2^k$, где $k$ -- целое число. \end{itemize} \item Структурная схема кодера. - \begin{center} + \begin{figure}[H] + \centering \includegraphics[scale=0.8]{coder2} - \end{center} + \caption{Структурная схема кодера} + \end{figure} \item Решетчатая диаграмма кодера. \input{coder_empty} @@ -515,7 +573,10 @@ $L=2^k$, где $k$ -- целое число. \hline \end{tabular} \end{center} - \[ \overline u = 11 10 00 01 10 10 01 11 11 01 11 00 00 \] + \begin{equation} + \overline u = 11 10 00 01 10 10 01 11 11 01 11 00 00 + \end{equation} + \item На решетчатой диаграмме кодера отметить путь, соответствующий полученным КС. @@ -584,12 +645,18 @@ $L=2^k$, где $k$ -- целое число. \caption{График реализации $c(t)$ с выхода сверточного кодера} \end{figure} - \[ C(t)=\sum^\infty_{n=-\infty}C_n\cdot g_1(t-nT_B) \] + \begin{equation} + C(t)=\sum^\infty_{n=-\infty}C_n\cdot g_1(t-nT_B) + \end{equation} + где $g_1(t)$ -- прямоугольный импульс длительностью $T_B$. - \[ g_1(t)=\begin{cases} - 1\,В, & 0\leq t \leq T_B;\\ - 0\,В, & t<0,\,t>T_B, - \end{cases} \] + \begin{equation} + g_1(t)=\begin{cases} + 1\,В, & 0\leq t \leq T_B;\\ + 0\,В, & t<0,\,t>T_B, + \end{cases} + \end{equation} + где $g_1(t-nT_B)$ -- прямоугольный импульс такой же формы, как и $g_1(t)$, но сдвинутый вправо относительно импульса $g_1(t)$ на величину $nT_B$, если $n>0$, или @@ -599,7 +666,9 @@ $L=2^k$, где $k$ -- целое число. случайной величины $C_n$ на $n$-интервале $T_B$. Величина $C_n$ принимает два дискретных значения $h(B)$ и $-h(B)$ с вероятностью $0,5$ каждое, \mbox{т. е.} - \[ P(h)=P(-h)=0,5. \] + \begin{equation} + P(h)=P(-h)=0,5. + \end{equation} Если в заданной реализации $c(t)$ на $n$-интервале передается информационный символ «1», то $c_n=h(B)$, @@ -611,16 +680,19 @@ $L=2^k$, где $k$ -- целое число. Написать аналитические выражения для случайных процессов $I(t)$ и $Q(t)$. \begin{equation} \label{eq:ItQt} - I(t)=\sum^\infty_{n=-\infty}I_n\cdot g_2(t-nT_S);\, - Q(t)=\sum^\infty_{n=-\infty}Q_n\cdot g_2(t-nT_S), + I(t)=\sum^\infty_{n=-\infty}I_n\cdot g_2(t-nT_S);\, + Q(t)=\sum^\infty_{n=-\infty}Q_n\cdot g_2(t-nT_S), \end{equation} где $g_(t)$ -- прямоугольный импульс длительностью $T_S=4T_B$. $T_S$ -- символьный интервал; $T_B$ -- бинарный интервал; - \[ g_2(t)=\begin{cases} - 1\,В, & 0\leq t \leq T_B;\\ - 0\,В, & t<0,\,t>T_B, - \end{cases} \] + \begin{equation} + g_2(t)=\begin{cases} + 1\,В, & 0\leq t \leq T_B;\\ + 0\,В, & t<0,\,t>T_B, + \end{cases} + \end{equation} + где $g_2(t-nT_S)$ -- прямоугольный импульс такой же формы, как и $g_2(t)$, но сдвинутый вправо относительно импульса $g_2(t)$ на величину $nT_S$, если $n>0$, или @@ -631,7 +703,10 @@ $L=2^k$, где $k$ -- целое число. которые согласно сигнальному созвездию (рис. \ref{fig:cam_16}) принимают четыре дискретных значения $-3h,\, -h,\, h,\, 3h$ с вероятностью 0,25 каждое, т. е. - \[ P(-3h)=P(-h)=P(h)=P(3h)=0,25. \] + \begin{equation} + P(-3h)=P(-h)=P(h)=P(3h)=0,25. + \end{equation} + \begin{figure}[H] \centering @@ -679,15 +754,21 @@ $L=2^k$, где $k$ -- целое число. и построить графики этих функций. Процесс $C(t)$ является случайным синхронным телеграфным сигналом. Его корреляционная функция имеет вид: - \[ B_C(\tau)=\begin{cases} - h^2(1-\frac{|\tau|}{T}),&|\tau|\leq T\\ - 0, & |\tau| > T - \end{cases}, \] + \begin{equation} + B_C(\tau)=\begin{cases} + h^2(1-\frac{|\tau|}{T}),&|\tau|\leq T\\ + 0, & |\tau| > T + \end{cases}, + \end{equation} + а спектральная плотность мощности - \[ G_C(\omega) - =\int^\infty_{-\infty}B_C(\tau)e^{-i\omega\tau}d\tau - =\int^\infty_{-\infty}B_C(\tau)\cos{\omega\tau}d\tau - =T\cdot h^2\cdot\frac{\sin^2(\frac{\omega T}{2})}{(\frac{\omega T}{2})^2}, \] + \begin{equation} + G_C(\omega) + =\int^\infty_{-\infty}B_C(\tau)e^{-i\omega\tau}d\tau + =\int^\infty_{-\infty}B_C(\tau)\cos{\omega\tau}d\tau + =T\cdot h^2\cdot\frac{\sin^2(\frac{\omega T}{2})}{(\frac{\omega T}{2})^2}, + \end{equation} + где $T=T_B$ -- длительность тактового интервала. \begin{figure}[H] \centering @@ -699,10 +780,15 @@ $L=2^k$, где $k$ -- целое число. axis lines = left, ylabel = {$B_C(\tau)$}, xlabel = {$\tau$}, + xtick={0}, + xticklabels={$0$}, + domain=-1.8:1.8, + ytick={0,0.98}, + yticklabels={0,0.25}, ] \addplot [ color=blue, - samples=100, + samples=5, ] {\h^2*(1-abs(x)/\T)}; \end{axis} @@ -717,9 +803,14 @@ $L=2^k$, где $k$ -- целое число. \pgfmathsetmacro{\h}{1} \begin{axis}[ width=10cm,height=6cm, - axis lines = left, + axis lines = middle, ylabel = {$G_C(\omega)$}, xlabel = {$\omega$}, + scaled y ticks = false, + ytick = {}, + yticklabels = {}, + xtick = {200, 400, -200, -400}, + xticklabels = {$\frac{1}{T_B}$, $\frac{2}{T_B}$, $-\frac{1}{T_B}$, $-\frac{2}{T_B}$}, ] \addplot [ color=blue, @@ -743,22 +834,34 @@ $L=2^k$, где $k$ -- целое число. мощности, поскольку они оба отличаются от процесса $C(t)$ лишь длительностью сигнального интервала $T_S=4T_B$. - \[ B_I(0)=B_Q(0)=D\{I(t)\}=D\{Q(t)\} \] - \[ G_I(0)=G_Q(0)=\frac{D\{I(t)\}}{T_S}=\frac{D\{Q(t)\}}{T_S} \] - \begin{align*}\begin{split} + \begin{equation} + B_I(0)=B_Q(0)=D\{I(t)\}=D\{Q(t)\} + \end{equation} + + \begin{equation} + G_I(0)=G_Q(0)=\frac{D\{I(t)\}}{T_S}=\frac{D\{Q(t)\}}{T_S} + \end{equation} + + \begin{align}\begin{split} D\{I(t)\}=D\{Q(t)\}& =\sum^4_{n=1}(i_n-\overline{I_n(t)})^2\cdot P(i_n)\\ &=0,25(-3h)^2+0,25(-h)^2+0,25h^2+0,25(3h)^2=5h^2 - \end{split}\end{align*} + \end{split}\end{align} Корреляционные функции: - \[ B_I(\tau)=B_Q(\tau)=\begin{cases} - 5h^2(1-\frac{|\tau|}{T_B}), & |\tau|\leq T_B\\ - 0, & |\tau| > T_B - \end{cases} \] + \begin{equation} + B_I(\tau)=B_Q(\tau)=\begin{cases} + 5h^2(1-\frac{|\tau|}{T_B}), & |\tau|\leq T_B\\ + 0, & |\tau| > T_B + \end{cases} + \end{equation} + Энергетический спектр: - \[ G_I(\omega)=G_Q(\omega) - =\int^\infty_{-\infty}B_C(\tau)e^{-i\omega\tau}d\tau - =T\cdot h^2\cdot\frac{\sin^2(\frac{\omega T}{2})}{(\frac{\omega T}{2})^2} \] + \begin{equation} + G_I(\omega)=G_Q(\omega) + =\int^\infty_{-\infty}B_C(\tau)e^{-i\omega\tau}d\tau + =T\cdot h^2\cdot\frac{\sin^2(\frac{\omega T}{2})}{(\frac{\omega T}{2})^2} + \end{equation} + \begin{figure}[H] \centering \begin{tikzpicture} @@ -769,10 +872,15 @@ $L=2^k$, где $k$ -- целое число. axis lines = left, ylabel = {$B(\tau)$}, xlabel = {$\tau$}, + xtick={0}, + xticklabels={$0$}, + domain=-1.8:1.8, + ytick={0,0.98}, + yticklabels={0,0.25}, ] \addplot [ color=red, - samples=100, + samples=5, ] {5*\h^2*(1-abs(x)/\T)}; \end{axis} @@ -791,6 +899,11 @@ $L=2^k$, где $k$ -- целое число. axis lines = left, ylabel = {$G(\omega)$}, xlabel = {$\omega$}, + scaled y ticks = false, + ytick = {}, + yticklabels = {}, + xtick = {50, 150, -50, -150}, + xticklabels = {$\frac{1}{T_S}$, $\frac{3}{T_S}$, $-\frac{1}{T_S}$, $-\frac{3}{T_S}$}, ] \addplot [ color=red, @@ -822,6 +935,10 @@ $L=2^k$, где $k$ -- целое число. axis lines = left, ylabel = {$B(\tau)$}, xlabel = {$\tau$}, + xtick={0}, + xticklabels={$0$}, + ytick={}, + yticklabels={}, ] \addplot [ color=blue, @@ -853,6 +970,11 @@ $L=2^k$, где $k$ -- целое число. axis lines = left, ylabel = {$G(\omega)$}, xlabel = {$\omega$}, + scaled y ticks = false, + ytick = {}, + yticklabels = {}, + xtick = {}, + xticklabels = {}, ] \addplot [ color=blue, @@ -954,7 +1076,11 @@ $L=2^k$, где $k$ -- целое число. axis lines = left, ylabel = {$S(\omega)$}, xlabel = {$\omega$}, - domain=-32:32, + domain = -32:32, + xtick = {0, 32, -32}, + xticklabels = {0, $\frac{2\pi}{T}$, $-\frac{2\pi}{T}$}, + ytick = {1.79, 1.35}, + yticklabels = {$T$, $\sqrt T$}, ] \addplot [ color=blue, @@ -968,6 +1094,8 @@ $L=2^k$, где $k$ -- целое число. ] {sqrt(\T/2*(1+cos(\PI*\T*abs(x))))}; \addlegendentry{$S_{x1}(\omega)$}; + \draw [dashed] (axis cs: \pgfkeysvalueof{/pgfplots/xmin},1.35) -- (axis cs: 0,1.35); + \draw [dashed] (axis cs: \pgfkeysvalueof{/pgfplots/xmin},1.79) -- (axis cs: 0,1.79); \end{axis} \end{tikzpicture} \caption{Графики спектральных плотностей @@ -989,6 +1117,10 @@ $L=2^k$, где $k$ -- целое число. ylabel = {$x(t)$}, xlabel = {$t$}, domain = -2:2, + xtick = {0, 1, -1}, + xticklabels = {0, T, -T}, + ytick = {0, 1, 1.6}, + yticklabels = {0, 1, $\frac{1.27}{\sqrt T}$} ] \addplot [no markers] gnuplot [ color=blue, @@ -1002,20 +1134,29 @@ $L=2^k$, где $k$ -- целое число. ] {sin(\PI*x*1.27/\T)/\PI/x*1.27*\T*cos(\PI*x*1.27/\T)/(1-4*(x*1.27)^2/\T^2)}; \addlegendentry{$x_1(t)$}; + + \draw [dashed] (axis cs: \pgfkeysvalueof{/pgfplots/xmin},1) -- (axis cs: 0,1); + \draw [dashed] (axis cs: \pgfkeysvalueof{/pgfplots/xmin},1.6) -- (axis cs: 0,1.6); \end{axis} \end{tikzpicture} \caption{Импульс Найквиста $x(t)$ и искомый импульс $x_1(t)$} \end{figure} \item Написать аналитические выражения для случайных процессов $I_ф(t)$ и $Q_ф(t)$. - \[ I_ф(t)=\sum^\infty_{n=-\infty}I_ng_3(t-nT), \] + \begin{equation} + I_ф(t)=\sum^\infty_{n=-\infty}I_ng_3(t-nT), + \end{equation} + где $i_n$ -- детерминированная величина, которая является реализацией случайной величины $I_n$. Величины $i_n$ в выражениях для $i(t)$ и $i_ф(t)$ принимают одинаковые значения на соответствующих символьных интервалах $T$. - \[ Q_ф(t)=\sum^\infty_{n=-\infty}Q_ng_3(t-nT), \] + \begin{equation} + Q_ф(t)=\sum^\infty_{n=-\infty}Q_ng_3(t-nT), + \end{equation} + где $I_n(t)$ и $Q_n(t)$ -- независимые случайные величины, принимающие известные дискретные значения с заданными вероятностями, какие они имеют в формулах (\ref{eq:ItQt}); @@ -1055,6 +1196,10 @@ $L=2^k$, где $k$ -- целое число. ylabel = {$B(\tau)$}, xlabel = {$\tau$}, domain = -2:2, + xtick = {0, 1, -1, 2, -2}, + xticklabels = {0, $T$, $-T$, $2T$, $-2T$}, + ytick = {0, 0.98}, + yticklabels = {0, $\frac{\overline{I^2_n}}{1.27^2}$}, ] \addplot [no markers] gnuplot [ color=blue, @@ -1086,6 +1231,10 @@ $L=2^k$, где $k$ -- целое число. ylabel = {$S(\omega)$}, xlabel = {$\omega$}, domain=-32:32, + xtick = {0, 32, -32}, + xticklabels = {0, $\frac{2\pi}{T}$, $-\frac{2\pi}{T}$}, + ytick = {1.79}, + yticklabels = {$\frac{\overline{I^2_n}\cdot T}{1.27^2}$}, ] \addplot [ color=blue,