From 3c280d4520933a54c0861ec5bc94b55a762fbba0 Mon Sep 17 00:00:00 2001 From: Anatoly Date: Thu, 6 May 2021 22:07:08 +0300 Subject: [PATCH] =?UTF-8?q?=D0=94=D0=BE=D0=B4=D0=B5=D0=BB=D0=B0=D0=BB=20?= =?UTF-8?q?=D0=A4=D0=9C=D0=A1?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- coder.tex | 75 +++--- decoder.tex | 16 +- images/cam_16.png | Bin 0 -> 33609 bytes images/coder2.png | Bin 0 -> 9386 bytes images/struct_scheme.png | Bin 0 -> 201707 bytes term_paper.pdf | Bin 586626 -> 598642 bytes term_paper.tex | 522 +++++++++++++++++++++++++++++++++++++-- 7 files changed, 547 insertions(+), 66 deletions(-) create mode 100644 images/cam_16.png create mode 100644 images/coder2.png create mode 100644 images/struct_scheme.png diff --git a/coder.tex b/coder.tex index 65aaac9..ce51248 100644 --- a/coder.tex +++ b/coder.tex @@ -1,42 +1,47 @@ -\begin{figure}[H] - \begin{tikzpicture}[x=1.2cm, y=-1cm] - \node at (-0.5,0) [left] {$s_1=00$}; - \node at (-0.5,1) [left] {$s_2=10$}; - \node at (-0.5,2) [left] {$s_3=01$}; - \node at (-0.5,3) [left] {$s_4=11$}; +%\begin{landscape} + \begin{figure}[H] + \begin{tikzpicture}[x=1cm, y=-1cm] + \node at (-0.5,0) [left] {$s_1=00$}; + \node at (-0.5,1) [left] {$s_2=10$}; + \node at (-0.5,2) [left] {$s_3=01$}; + \node at (-0.5,3) [left] {$s_4=11$}; - % Nodes - \foreach \x in {0,...,9} { - \node at (\x,-.7) {$\x$}; - \foreach \y in {0,...,3} { - \node (s\x\y) at (\x,\y) [circle,fill=black,scale=0.7] {}; + % Nodes + \foreach \x in {0,...,12} { + \node at (\x,-.7) {$\x$}; + \foreach \y in {0,...,3} { + \node (s\x\y) at (\x,\y) [circle,fill=black,scale=0.7] {}; + } } - } - % Edges - \trellisEdges{0}{0} - \trellisEdges{1}{0} - \trellisEdges{1}{1} - \foreach \x in {2,...,8} { - \foreach \y in {0,...,3} { - \trellisEdges{\x}{\y} + % Edges + \trellisEdges{0}{0} + \trellisEdges{1}{0} + \trellisEdges{1}{1} + \foreach \x in {2,...,11} { + \foreach \y in {0,...,3} { + \trellisEdges{\x}{\y} + } } - } - % Inputs and Outputs - \node at (-0.5,4) [left] {Входной бит}; - \node at (-0.5,5) [left] {Результат}; + % Inputs and Outputs + \node at (-0.5,4) [left] {ИС}; + \node at (-0.5,5) [left] {КС}; - \trellisInOut{0}{0.5}{1}{11} - \trellisInOut{1}{1.5}{0}{10} - \trellisInOut{2}{1.5}{1}{00} - \trellisInOut{3}{2}{1}{01} - \trellisInOut{4}{3}{1}{10} - \trellisInOut{5}{3}{1}{10} - \trellisInOut{6}{2.5}{0}{01} - \trellisInOut{7}{1}{0}{11} - \trellisInOut{8}{0.5}{1}{11} - \end{tikzpicture} + \trellisInOut{0}{0.5}{1}{11} + \trellisInOut{1}{1.5}{0}{10} + \trellisInOut{2}{1.5}{1}{00} + \trellisInOut{3}{2}{1}{01} + \trellisInOut{4}{3}{1}{10} + \trellisInOut{5}{3}{1}{10} + \trellisInOut{6}{2.5}{0}{01} + \trellisInOut{7}{1}{0}{11} + \trellisInOut{8}{0.5}{1}{11} + \trellisInOut{9}{1.5}{0}{01} + \trellisInOut{10}{1}{0}{11} + \trellisInOut{11}{0}{0}{00} + \end{tikzpicture} - \caption{Решетка кодера} \label{fig:coder} -\end{figure} \ No newline at end of file + \caption{Путь на решетке кодера} \label{fig:coder} + \end{figure} +%\end{landscape} \ No newline at end of file diff --git a/decoder.tex b/decoder.tex index 5bd1b74..0b1d92b 100644 --- a/decoder.tex +++ b/decoder.tex @@ -149,7 +149,7 @@ \trellisIn{2}{00} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=3$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=3$} \end{figure} \begin{figure} @@ -233,7 +233,7 @@ \trellisIn{3}{01} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=4$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=4$} \end{figure} \begin{figure} @@ -333,7 +333,7 @@ \trellisIn{4}{10} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=5$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=5$} \end{figure} \begin{figure} @@ -449,7 +449,7 @@ \trellisIn{5}{10} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=6$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=6$} \end{figure} \begin{figure} @@ -580,7 +580,7 @@ \trellisIn{6}{01} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=7$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=7$} \end{figure} \begin{figure} @@ -723,7 +723,7 @@ \trellisIn{7}{11} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=8$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=8$} \end{figure} @@ -883,7 +883,7 @@ \trellisIn{8}{11} \end{tikzpicture} - \caption{Сегмент решетки декодера от $t=0$, до $t=9$.} + \caption{Сегмент решетки декодера от $t=0$, до $t=9$} \end{figure} \begin{landscape} @@ -1089,6 +1089,6 @@ \trellisIn{11}{00} \end{tikzpicture} - \caption{Полная решетка декодера.} + \caption{Полная решетка декодера} \end{figure} \end{landscape} diff --git a/images/cam_16.png b/images/cam_16.png new file mode 100644 index 0000000000000000000000000000000000000000..d4a643d1cfd8feb78f688792934f6595b616dda5 GIT binary patch literal 33609 zcmd?QWmFwq(>4f%-~@Mff@?TOAru9 z<`$x&ic+GYq>2uNpk?#@uy#6*{YHf%UO#*gW4*!_veaeIX8>HctZ 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\end{axis} \end{tikzpicture} - \caption{График спектральной плотности мощности.} + \caption{График спектральной плотности мощности} \label{fig:spectr_plot} \end{figure} \item Найти аналитическое выражение для корреляционной функции @@ -363,43 +363,519 @@ $a_{макс}$. {\Ga*(sin(2*\PI*\fv*x))/(2*\PI*x)}; \end{axis} \end{tikzpicture} - \caption{График корреляционной функции $B_A(\tau)$.} + \caption{График корреляционной функции $B_A(\tau)$} \label{fig:coorel_plot} \end{figure} \end{enumerate} \subsection{Аналого-цифровой преобразователь} -\[ \Delta t \leq \frac{1}{2f_B}=\frac1 {2\cdot 14100} = 3,546\cdot 10^{-5}\, с \] -\[ f_d=\frac{1}{\Delta t}\geq 2f_B=\frac{1}{3,546\cdot 10^{-5}}=28200 \] -\[ 377_{10}=101111001_2 \] -\[ k=9;\, L=2^9 = 512 \] +Аналого-цифровой преобразователь (АЦП) преобразует реализации +аналогового (непрерывного) сообщения $A(t)$ в цифровую +форму, в поток двоичных символов: нулей и единиц, +т. е. в последовательность прямоугольных импульсов, +где «0» имеет нулевое напряжение, а «1» -- прямоугольный +импульс положительной полярности. +Амплитуда импульсов $U$ равна 1 В. + +Преобразование аналогового сигнала в цифровую форму +осуществляется в три этапа. + +На первом этапе производится дискретизация реализации +$a(t)$ сообщения $A(t)$ по времени. В моменты времени $t_i$ +берутся непрерывные по уровню отсчеты $a(t_i)$ +мгновенных значений реализации $a(t)$. Расстояние +между отсчетами равно интервалу $\Delta t$, величина которого +определяется в соответствии с теоремой Котельникова: +\[\Delta t \leq \frac{1}{2f_B};\, +f_d=\frac{1}{\Delta t}\geq2f_B\] +где $f_d$ -- частота дискретизации. + +На втором этапе выполняется квантование точных отсчетов +$a(t_i)$ по уровню. Для этого интервал $\Delta$, равный +разности $\Delta=a_{макс} - a_{мин}$, разбивается на уровни +квантования с постоянным шагом $\Delta a =0,1\, В$. +Уровни квантования нумеруются целыми числами +$0,1,2,3,...,L-1$. Нумерация уровней начинается с уровня, +которому соответствует значение $a_мин$, и заканчивается на +уровне, которому соответствует значение $a_макс$. Обычно +величина шага квантования $\Delta a$ выбирается так, чтобы +число уровней квантования $L$ можно было представить в виде +$L=2^k$, где $k$ -- целое число. + +Каждый аналоговый отсчет $a(t_i)$ заменяется значением +ближайшего к нему уровня квантования $j$ в виде целого числа, +удовлетворяющего неравенству $0\leq j \leq L-1$. +Получаем квантованный отсчет $j_{10}(t_i)$ в виде целого +числа в десятичной форме счисления. + +На третьем этапе число $j_{10}(t_i)$ в десятичной форме +переводится в двоичную форму счисления $j_2(t_i)$ в виде +последовательности $k$ двоичных +символов и на выходе АЦП появляется сигнал в виде двоичной цифровой последовательности из $k$ информационных символов. + +Требуется: +\begin{enumerate} + \item Рассчитать интервал дискретизации $\Delta t$ для + получения непрерывных отсчетов $a(t_i)$ реализации + $a(t),\, t_i=i\cdot\Delta t,\, i=0,\pm1,\pm2,...$. + \[ \Delta t \leq \frac{1}{2f_B}=\frac1 {2\cdot 14100} = 3,546\cdot 10^{-5}\, с \] + \item Рассчитать частоту дискретизации $f_d$. + \[ f_d=\frac{1}{\Delta t}\geq 2f_B=\frac{1}{3,546\cdot 10^{-5}}=28200 \] + \item Определить число уровней квантования $L$. + \[ k=9;\, L=2^9 = 512 \] + \item Рассчитать мощность шума квантования $P_{ШК}$ + и сравнить ее с мощностью непрерывного сообщения $A(t)$. + \[ P_{ШК}=\Delta a^2/12 + =\frac{0,1^2}{12}=8,33\cdot10^{-4}\, В^2 \] + \[ P_{A(t)}=A^2(t)=1\, В^2\] + \[ P_{A(t)} >> P_{ШК} \] + \item Найти минимальное число $k$ двоичных разрядов, + требуемое для записи в двоичной форме любого номера $j$ + из $L-1$ номеров уровней квантования. + \[ L-1=511_{10}=111111111_2 \] + \[ k_{люб}=9 \] + \item Записать $k$-разрядное двоичное число, + соответствующее заданному уровню квантования $j$. + \[ j=377_{10}=101111001_2 \] + \item Начертить временную диаграмму отклика АЦП + $b_{АЦП}(t)$ на заданный уровень квантования $j$ + в виде последовательности импульсов, + сопоставляя единичным символам прямоугольные импульсы + положительной полярности, а нулевым -- нулевые напряжения. + Амплитуда импульсов $U$ равна $2h$ B. Над импульсами + надписать значения соответствующих двоичных информационных + символов (ДИС). Длительность отклика АЦП на каждый отсчет + не должна превышать интервала дискретизации $\Delta t$. + \begin{figure}[H] + \centering + \begin{tikzpicture} + \draw[->, very thick] (0,1) -- (9.2,1); + \draw[->, very thick] (0,-0.2) -- (0,2.2); + + \draw (0,2) -- (1,2) -- (1,0); + \draw (1,0) -- (2,0) -- (2,2); + \draw (2,2) -- (6,2) -- (6,0); + \draw (6,0) -- (8,0) -- (8,2); + \draw (8,2) -- (9,2); + + \node at (0.5,2.5) {$1$}; + \node at (1.5,2.5) {$0$}; + \node at (2.5,2.5) {$1$}; + \node at (3.5,2.5) {$1$}; + \node at (4.5,2.5) {$1$}; + \node at (5.5,2.5) {$1$}; + \node at (6.5,2.5) {$0$}; + \node at (7.5,2.5) {$0$}; + \node at (8.5,2.5) {$1$}; + \end{tikzpicture} + \caption{Временная диаграмма отклика АЦП} + \end{figure} +\end{enumerate} \subsection{Кодер} -\begin{center} - \includegraphics[scale=0.8]{coder} +Используется помехоустойчивый сверточный код. - \begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | } - \hline - Входной сигнал &1&0&1&1&1&1&0&0&1\\ - \hline - Выходной сигнал &11&10&00&01&10&10&01&11&11\\ - \hline - \end{tabular} -\end{center} +\begin{enumerate} + \item Параметры сверточного кода. + \begin{itemize} + \item Степень кодирования $k/n=1/2$, + \item длина кодового ограничения $K=3$, + \item векторы связи $\overline g_1=111$ и + $\overline g_2=101$, + \item импульсная характеристика $h(k)=111011000...$, + \item кодовое расстояние $d=5$. + \end{itemize} -\subsubsection{Решетка кодера} + \item Структурная схема кодера. + \begin{center} + \includegraphics[scale=0.8]{coder2} + \end{center} -\input{coder} + \item Решетчатая диаграмма кодера. + \input{coder_empty} -Длительность двоичного символа \(T_В\) на выходе кодера: -\[T_В=\frac{\Delta t}{2k}=\frac{3,546\cdot 10^{-5}}{2\cdot 9}= -1,97\cdot 10^{-6}\,с\] + \item По решетчатой диаграмме сверточного кодера определить + последовательность кодовых символов (КС) $\overline u$ на выходе кодера + при условии, когда на вход кодера поступает 9-разрядная + двоичная последовательность информационных символов (ИС) + $\overline m$, соответствующая заданному уровню квантования $j$. + \begin{center} + \begin{tabular}{ |c|c|c|c|c|c|c|c|c|c|c|c|c|c| } + \hline + ИС &1&0&1&1&1&1&0&0&1&0&0&0&0\\ + \hline + КС &11&10&00&01&10&10&01&11&11&01&11&00&00\\ + \hline + \end{tabular} + \end{center} + \[ \overline u = 11 10 00 01 10 10 01 11 11 01 11 00 00 \] + + \item На решетчатой диаграмме кодера отметить путь, + соответствующий полученным КС. + \input{coder} +\end{enumerate} + +\subsection{Формирователь модулирующих символов} +Формирователь модулирующих символов служит для получения +модулирующих сигналов $I(t)$ и $Q(t)$, соответствующих заданному +виду модуляции. + +Требуется: +\begin{enumerate} + \item Изобразить сигнальное созвездие для заданного вида модуляции. + \begin{figure}[H] + \centering + \includegraphics[scale=0.6]{cam_16} + \caption{Сигнальное созвездие для КАМ-16} + \label{fig:cam_16} + \end{figure} + + \item Изобразить график реализации $c(t)$ случайного процесса + $C(t)$, формируемого с выхода блока сверточного кодера (К). + Реализация $с(t)$ поступает на вход блока ФМС на первых + 16 бинарных интервалах длительностью $T_B$. + Написать аналитическое выражение для + случайного процесса $C(t)$. + \begin{figure}[H] + \centering + \begin{tikzpicture}[x=0.8cm, y=-1cm] + \draw[->, very thick] (0,1) -- (16.2,1); + \draw[->, very thick] (0,2.2) -- (0,-0.2); + + \draw (0,0) -- (3,0) -- (3,2); + \draw (3,2) -- (7,2) -- (7,0); + \draw (7,0) -- (9,0) -- (9,2); + \draw (9,2) -- (10,2) -- (10,0); + \draw (10,0) -- (11,0) -- (11,2); + \draw (11,2) -- (13,2) -- (13,0); + \draw (13,0) -- (16,0); + + \node at (0.5,-0.5) {$1$}; + \node at (1.5,-0.5) {$1$}; + + \node at (2.5,-0.5) {$1$}; + \node at (3.5,-0.5) {$0$}; + + \node at (4.5,-0.5) {$0$}; + \node at (5.5,-0.5) {$0$}; + + \node at (6.5,-0.5) {$0$}; + \node at (7.5,-0.5) {$1$}; + + \node at (8.5,-0.5) {$1$}; + \node at (9.5,-0.5) {$0$}; + + \node at (10.5,-0.5) {$1$}; + \node at (11.5,-0.5) {$0$}; + + \node at (12.5,-0.5) {$0$}; + \node at (13.5,-0.5) {$1$}; + + \node at (14.5,-0.5) {$1$}; + \node at (15.5,-0.5) {$1$}; + \end{tikzpicture} + \caption{График реализации $c(t)$ с выхода сверточного кодера} + \end{figure} + + \[ C(t)=\sum^\infty_{n=-\infty}C_n\cdot g_1(t-nT_B) \] + где $g_1(t)$ -- прямоугольный импульс длительностью $T_B$. + \[ g_1(t)=\begin{cases} + 1\,В, & 0\leq t \leq T_B;\\ + 0\,В, & t<0,\,t>T_B, + \end{cases} \] + 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 + влево, если $n<0$; + + $C_n$ -- численный коэффициент, являющийся реализацией + 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.} + \[ P(h)=P(-h)=0,5. \] + + Если в заданной реализации $c(t)$ на $n$-интервале передается + информационный символ «1», то $c_n=h(B)$, + если передается символ «0», то $c_n=-h(B)$. + + \item В соответствии с сигнальным созвездием модулятора КАМ-16 + изобразить графики реализаций $i(t)$ и $q(t)$ на выходе + блока ФМС, соответствующие входной реализации $c(t)$. + Написать аналитические выражения для случайных процессов + $I(t)$ и $Q(t)$. + \[ I(t)=\sum^\infty_{n=-\infty}I_n\cdot g_2(t-nT_S);\, + Q(t)=\sum^\infty_{n=-\infty}Q_n\cdot g_2(t-nT_S), \] + где $g_(t)$ -- прямоугольный импульс длительностью + $T_S=4T_B$. $T_S$ -- символьный интервал; + $T_B$ -- бинарный интервал; + \[ g_2(t)=\begin{cases} + 1\,В, & 0\leq t \leq T_B;\\ + 0\,В, & t<0,\,t>T_B, + \end{cases} \] + 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 + влево, если $n<0$; + + $I_n$ и $Q_n$ -- независимые случайные величины, заданные на + символьном интервале с номером $n$, + 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{figure}[H] + \centering + \begin{tikzpicture}[x=0.8cm, y=-0.8cm] + \draw[->, very thick] (0,2) -- (16.2,2); + \draw[->, very thick] (0,4.2) -- (0,-0.2); + + \node at (-0.5,0) [left] {$3h$}; + \node at (-0.5,1) [left] {$h$}; + \node at (-0.5,2) [left] {$0$}; + \node at (-0.5,3) [left] {$-h$}; + \node at (-0.5,4) [left] {$-3h$}; + + \draw (0,4) -- (4,4) -- (4,1); + \draw (4,1) -- (8,1) -- (8,3); + \draw (8,3) -- (12,3) -- (12,0); + \draw (12,0) -- (16,0); + \end{tikzpicture} + \caption{График реализации $i(t)$} + \end{figure} + + \begin{figure}[H] + \centering + \begin{tikzpicture}[x=0.8cm, y=-0.8cm] + \draw[->, very thick] (0,2) -- (16.2,2); + \draw[->, very thick] (0,4.2) -- (0,-0.2); + + \node at (-0.5,0) [left] {$3h$}; + \node at (-0.5,1) [left] {$h$}; + \node at (-0.5,2) [left] {$0$}; + \node at (-0.5,3) [left] {$-h$}; + \node at (-0.5,4) [left] {$-3h$}; + + \draw (0,3) -- (4,3) -- (4,0); + \draw (4,0) -- (8,0) -- (8,3); + \draw (8,3) -- (12,3) -- (12,4); + \draw (12,4) -- (16,4); + \end{tikzpicture} + \caption{График реализации $q(t)$} + \end{figure} + + \item Написать аналитические выражения для корреляционной + функции $B_C(\tau)$ и спектральной плотности мощности + $G_C(\omega)$ входного случайного процесса $C(t)$ + и построить графики этих функций. + + Процесс $C(t)$ является случайным синхронным телеграфным сигналом. Его корреляционная функция имеет вид: + \[ B_C(\tau)=\begin{cases} + h^2(1-\frac{|\tau|}{T}),&|\tau|\leq T\\ + 0, & |\tau| > T + \end{cases}, \] + а спектральная плотность мощности + \[ G_C(\omega) + =\int^\infty_{-\infty}B_C(\tau)e^{-i\omega\tau}d\tau + =\int^\infty_{-\infty}B_C(\tau)\cos{\omega\tau}d\tau + =T\cdot h^2\cdot\frac{\sin^2(\frac{\omega T}{2})}{(\frac{\omega T}{2})^2}, \] + где $T=T_B$ -- длительность тактового интервала. + \begin{figure}[H] + \centering + \begin{tikzpicture} + \pgfmathsetmacro{\T}{1.8} + \pgfmathsetmacro{\h}{1} + \begin{axis}[ + width=10cm,height=6cm, + axis lines = left, + ylabel = {$B_C(\tau)$}, + xlabel = {$\tau$}, + ] + \addplot [ + color=blue, + samples=100, + ] + {\h^2*(1-abs(x)/\T)}; + \end{axis} + \end{tikzpicture} + \caption{График корреляционной функции $B_C(\tau)$} + \end{figure} + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \pgfmathsetmacro{\T}{1.8} + \pgfmathsetmacro{\h}{1} + \begin{axis}[ + width=10cm,height=6cm, + axis lines = left, + ylabel = {$G_C(\omega)$}, + xlabel = {$\omega$}, + ] + \addplot [ + color=blue, + samples=100, + domain=-400:400, + ] + {\h^2*\T*(sin(x*\T/2)^2)/(x*\T/2)^2}; + \end{axis} + \end{tikzpicture} + \caption{График спектральной плотности мощности + $G_C(\omega)$} + \end{figure} + + \item Написать аналитические выражения для + корреляционных функций $B_I(\tau)$ и $B_Q(\tau)$, + спектральных плотностей мощности $G_I(\omega)$ + и $G_Q(\omega)$ случайных процессов $I(t)$ и $Q(t)$. + Построить графики этих функций. + + Процессы $I(t)$ и $Q(t)$ будут иметь идентичные друг другу корреляционные функции и спектральные плотности + 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} + 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*} + Корреляционные функции: + \[ B_I(\tau)=B_Q(\tau)=\begin{cases} + 5h^2(1-\frac{|\tau|}{T_B}), & |\tau|\leq T_B\\ + 0, & |\tau| > T_B + \end{cases} \] + 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{figure}[H] + \centering + \begin{tikzpicture} + \pgfmathsetmacro{\T}{1.8*4} + \pgfmathsetmacro{\h}{1} + \begin{axis}[ + width=10cm,height=6cm, + axis lines = left, + ylabel = {$B(\tau)$}, + xlabel = {$\tau$}, + ] + \addplot [ + color=red, + samples=100, + ] + {5*\h^2*(1-abs(x)/\T)}; + \end{axis} + \end{tikzpicture} + \caption{График корреляционной функции + $B_I(\tau)$, $B_Q(\tau)$} + \end{figure} + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \pgfmathsetmacro{\T}{1.8*4} + \pgfmathsetmacro{\h}{1} + \begin{axis}[ + width=10cm,height=6cm, + axis lines = left, + ylabel = {$G(\omega)$}, + xlabel = {$\omega$}, + ] + \addplot [ + color=red, + samples=100, + domain=-200:200, + ] + {\h^2*\T*(sin(x*\T/2)^2)/(x*\T/2)^2}; + \end{axis} + \end{tikzpicture} + \caption{График спектральной плотности мощности + $G_I(\omega)$, $G_Q(\omega)$} + \end{figure} + + \item Сравнить графики корреляционных функций и спектральных + плотностей мощности сигналов на входе и выходе блока ФМС. + Привести краткое описание результатов сравнения и, + используя общие положения теории преобразования Фурье, + пояснить, почему спектр выходных сигналов уже спектра входного + сигнала. + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \pgfmathsetmacro{\T}{1.8} + \pgfmathsetmacro{\TB}{1.8*4} + \pgfmathsetmacro{\h}{1} + \begin{axis}[ + width=10cm,height=6cm, + axis lines = left, + ylabel = {$B(\tau)$}, + xlabel = {$\tau$}, + ] + \addplot [ + color=blue, + samples=100, + domain=-\T:\T, + ] + {\h^2*(1-abs(x)/\T)}; + \addlegendentry{$B_C(\tau)$}; + \addplot [ + color=red, + samples=100, + domain=-\TB:\TB, + ] + {5*\h^2*(1-abs(x)/\TB)}; + \addlegendentry{$B_I(\tau)$, $B_Q(\tau)$}; + \end{axis} + \end{tikzpicture} + \caption{Графики корреляционной функции $B_C(\tau)$ и $B_I(\tau)$} + \end{figure} + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \pgfmathsetmacro{\T}{1.8} + \pgfmathsetmacro{\TB}{1.8*4} + \pgfmathsetmacro{\h}{1} + \begin{axis}[ + width=10cm,height=6cm, + axis lines = left, + ylabel = {$G(\omega)$}, + xlabel = {$\omega$}, + ] + \addplot [ + color=blue, + samples=100, + domain=-300:300, + ] + {\h^2*\T*(sin(x*\T/2)^2)/(x*\T/2)^2}; + \addlegendentry{$G_C(\omega)$}; + \addplot [ + color=red, + samples=100, + domain=-300:300, + ] + {\h^2*\TB*(sin(x*\TB/2)^2)/(x*\TB/2)^2}; + \addlegendentry{$G_I(\omega)$, $G_Q(\omega)$}; + \end{axis} + \end{tikzpicture} + \caption{График спектральной плотности мощности + $G_C(\omega)$ и $G_I(\omega)$} + \end{figure} + + Выходной спектр уже, поскольку функция $G(\omega)$ равна + 0 при значениях $\omega = n/T$, а $T_S=4T_B$, поэтому + изгибы встречаются в 4 раза чаще. +\end{enumerate} \subsection{Декодер} -По каналу передавался код \(\overline{u}=11 10 00 01 10 10 01 11 11\). +По каналу передавался код +\(\overline{u}=11 10 00 01 10 10 01 11 11...\). Ошибка произошла на тактовом интервале \(q=3\). Таким образом, на вход декодера поступает последовательность -\(\overline{Z}=11 \overset{\times}{0} 0 00 01 10 10 01 11 11\). Крестиком обозначен ошибочно принятый символ. +\(\overline{Z}=11 \overset{\times}{0} 0 00 01 10 10 01 11 11...\). Крестиком обозначен ошибочно принятый символ. \subsubsection{Диаграмма декодера} \input{decoder}

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