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\cdot \Delta t \sim 2\pi \end{equation} -Для проверки соотношения неопределенностей работа разделена на три равноценные части, в каждой из которых сгенерирован сигнал определенной формы, обработан с помощью цифрового осциллографа, проверены соотношения неопределенности с помощью курсорных измерений. -1. Первая часть работы заключалась в исследовании спектра периодической последовательности прямоугольных импульсов (пример показан на рисунке \ref{прямоуг}). + \begin{figure}[h!] -\includegraphics[width=0.4\textwidth]{rect} -\caption{Пример периодической последовательности прямоугольных импульсов из \cite{labnik}} \label{rect} +\begin{center} +\includegraphics[width=\textwidth]{пример} +\caption{Примеры сигналов а) периодической последовательности прямоугольных импульсов, б) периодической последовательности цуг, в) модулированного по амплитуде сигнала из \cite{labnik}} \label{пример} +\end{center} \end{figure} +Для проверки соотношения неопределенностей работа разделена на три равноценные части, в каждой из которых сгенерирован сигнал определенной формы, обработан с помощью цифрового осциллографа, проверены соотношения неопределенности с помощью курсорных измерений. + +1. Первая часть работы заключалась в исследовании спектра периодической последовательности прямоугольных импульсов (пример показан на рисунке \ref{пример}). Теоретически рассчитано значение коэффициентов $c_n$ \cite{labnik}: \begin{equation} c_n = \dfrac{sin(\pi n \tau / T))}{\pi n} \end{equation} -2. Вторая часть работы состояла в исследовании спектра периодической последовательности цугов гармонических колебаний (пример показан на рисунке \ref{цуги}). -\begin{figure}[h!] -\begin{center} -\includegraphics[width=0.4\textwidth]{цуги} -\caption{Пример периодической последовательности цуг из \cite{labnik}} \label{цуги} -\end{center} -\end{figure} +2. Вторая часть работы состояла в исследовании спектра периодической последовательности цугов гармонических колебаний (пример показан на рисунке \ref{пример}). Теоретически известен спектр сигнала \cite{labnik}: \begin{equation} F(\omega) = \dfrac{\tau}{2T}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} @@ -90,13 +88,7 @@ F(\omega) = \dfrac{\tau}{2T}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} + \dfrac{\sin(\omega+\omega_0)\tau /2}{(\omega+\omega_0)\tau /2}\right] \end{equation} -3. Последняя часть заключалась в исследовании спектра гармонических сигналов, модулированных по амплитуде (пример показан на рисунке \ref{модулированный}). -\begin{figure}[h!] -\begin{center} -\includegraphics[width=0.4\textwidth]{Модулированный} -\caption{Пример модулированного по амплитуде сигнала из \cite{labnik}} \label{модулированный} -\end{center} -\end{figure} +3. Последняя часть заключалась в исследовании спектра гармонических сигналов, модулированных по амплитуде (пример показан на рисунке \ref{пример}). Теоретический вид сигнала \cite{labnik}: \begin{equation} f(t) = a_0 \cos (\omega_0 t) +\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t++\dfrac{ma_0}{2}\cos (\omega_0 -\Omega)t @@ -128,29 +120,18 @@ f(t) = a_0 \cos (\omega_0 t) +\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t++\dfrac{ma \label{прямоуг} \end{figure} При $\nu_{повт} = 700 Гц$ проведены измерения ширины спектра. Результаты -представлены в таблице \ref{dnu(tau)_tbl} и на рисунке \ref{dnu(tau)_img}. -\begin{table}[h!] -\caption{Зависимость ширины спектра от длительности спектра для последовательности прямоугольных импульсов при $\nu_{повт} = 700 Гц$} \label{dnu(tau)_tbl} -\begin{tabular}{|l|l|} -\hline -$\Delta\nu$, Hz & $\tau$, мкс \\ \hline -50200 & 20 \\ \hline -25200 & 40 \\ \hline -17200 & 60 \\ \hline -13000 & 80 \\ \hline -10200 & 100 \\ \hline -8600 & 120 \\ \hline -7400 & 140 \\ \hline -6600 & 160 \\ \hline -5800 & 180 \\ \hline -5000 & 200 \\ \hline -\end{tabular} -\end{table} +представлены на рисунке \ref{dnu(tau)_img}. \begin{figure}[h!] \begin{center} -\includegraphics[width=\textwidth]{dnu(tau)} -\caption{Зависимость ширины спектра от длительности спектра для последовательности прямоугольных импульсов} \label{dnu(tau)_img} +\includegraphics[width=0.85\textwidth]{dnu(tau)} +\caption{Зависимость ширины спектра от длительности спектра для последовательности прямоугольных импульсов при частоте повторения $\nu_{повт} = 700 Гц$} \label{dnu(tau)_img} +\end{center} +\end{figure} +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.85\textwidth]{a(n)} +\caption{Теоретический спектр прямоугольных импульсов при частоте повторения $\nu_{повт} = 1000 Гц$ и длительности импульса $\tau = 50 мкс$ из \cite{labnik}} \label{теор} \end{center} \end{figure} Рассчитан коэффициент наклона прямой: @@ -162,15 +143,9 @@ k = 0.9997 \pm 0.0039 Для сравнения экспериментальных и теоретических значений спектра для одного из сигналов (a) рассчитана теоретическую зависимость и изображена на графике \ref{теор}. Теоретический и экспериментальный спектр похожи. -\begin{figure}[h!] -\begin{center} -\includegraphics[width=\textwidth]{a(n)} -\caption{Теоретический спектр прямоугольных импульсов из \cite{labnik}} \label{теор} -\end{center} -\end{figure} -\subsection{Исследование спектра периодической последовательности цугов гармонических колебаний} -Для исследования спектра периодической последовательности цугов гармонических колебаний на генераторе создан сигнал последовательности синусоидальных цугов с разными параметрами, по которому на экране осциллографа получен спектр. (рис. \ref{спектр_цуги}) + +\subsection*{Исследование спектра периодической последовательности цугов гармонических колебаний} \begin{figure}[h!] \begin{minipage}[h!]{0.47\linewidth} @@ -192,76 +167,42 @@ k = 0.9997 \pm 0.0039 \begin{minipage}[h!]{0.47\linewidth} \center{\includegraphics[width=1\linewidth]{2_5}} \\e) $\nu = 70 кГц, T = 1 мс, N = 5$ \end{minipage} -\caption{Вид спектра для периодической последовательности цугов при разных частотах несущей $\nu_0$ = 50 кГц, периодах повторения $T$ = 1 мс, числах +\caption{Вид спектра для периодической последовательности цугов при разных частотах несущей $\nu$ = 50 кГц, периодах повторения $T$ = 1 мс, числах периодов в одном импульсе $N$ = 5} \label{спектр_цуги} \end{figure} +Для исследования спектра периодической последовательности цугов гармонических колебаний на генераторе создан сигнал последовательности синусоидальных цугов с разными параметрами, по которому на экране осциллографа получен спектр. (рис. \ref{спектр_цуги}) -Для проверки соотношения неопределенностей для данного сигнала при фиксированной длительности импульсов $\tau$ = 50 мкс измерены расстояния между соседними спектральными компонентами от периода повторения импульсов (табл. \ref{dnu(T)_tbl}, рис. \ref{dnu(T)_img}) - -\begin{table}[h!] -\caption{Зависимость расстояния между соседними спектральными компонентами от периода повторения импульсов при $\nu_0$ = 50 кГц и $N$ = 5} \label{dnu(T)_tbl} -\begin{tabular}{|l|l|} -\hline -T, ms & $\delta \nu$, Hz \\ \hline -0.2 & 6250 \\ \hline -1 & 2778 \\ \hline -1.5 & 4167 \\ \hline -2 & 1042 \\ \hline -2.5 & 1190 \\ \hline -3 & 735 \\ \hline -3.5 & 893 \\ \hline -4 & 1000 \\ \hline -4.5 & 1042 \\ \hline -5 & 1190 \\ \hline -\end{tabular} -\end{table} +Для проверки соотношения неопределенностей для данного сигнала при фиксированной длительности импульсов $\tau$ = 50 мкс измерены расстояния между соседними спектральными компонентами от периода повторения импульсов (рис. \ref{dnu(T)_img}) \begin{figure}[h!] \begin{center} \includegraphics[width=0.9\textwidth]{T(dnu)} -\caption{Зависимость расстояния между соседними спектральными компонентами от периода повторения импульсов} \label{dnu(T)_img} +\caption{Зависимость расстояния между соседними спектральными компонентами от периода повторения импульсов для периодической последовательности цугов при часоте несущей $\nu$ = 50 кГц и числе +периодов в одном импульсе $N$ = 5} \label{dnu(T)_img} \end{center} \end{figure} Теоретически известно (\cite{labnik}) точки должны хорошо ложиться на прямую, однако из графика видно, что это не так. Проблема заключается в снятии данных (был выбран неверный канал при курсорных измерениях). Поэтому подтвердить справедливость соотношения неопределенности невозможно. -\subsection{Исследование спектра гармонических сигналов, модулированных по амплитуде} +\subsection*{Исследование спектра гармонических сигналов, модулированных по амплитуде} Для исследования спектра гармонических сигналов, модулированных по амплитуде на генераторе создан сигнал, модулированный по амплитуде, по которому на экране осциллографа получается спектр (\ref{мод}). \begin{figure}[h!] \begin{center} -\includegraphics[width=\textwidth]{3} +\includegraphics[width=0.85\textwidth]{3} \caption{Спектр сигнала, модулированного по амплитуде, при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{мод} \end{center} \end{figure} -Измерена с помощью осциллографа глубину модуляции: +Измерена с помощью осциллографа глубина модуляции: \begin{equation} -m = \dfrac{A_{max}-A_{min}}{A_{max}+A_{min}} = \dfrac{1.54 - 0.04}{1.54 + 0.04} = 0.5, что сходится с установленным на генераторе +m = \dfrac{A_{max}-A_{min}}{A_{max}+A_{min}} = \dfrac{1.54 - 0.04}{1.54 + 0.04} = 0.5, что \hspace*{1mm} сходится \hspace*{1mm}с\hspace*{1mm} установленным \hspace*{1mm}на\hspace*{1mm} генераторе \end{equation} -Для проверки теоретической зависимости, изменяя глубину модуляции, измерена $\dfrac{a_{бок}}{а_{осн}}$ (табл. \ref{mod_tbl} и рис. \ref{mod_img}). +Для проверки теоретической зависимости, изменяя глубину модуляции, измерена $\dfrac{a_{бок}}{а_{осн}}$ (рис. \ref{mod_img}). -\begin{table}[h!] -\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$ при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} -\label{mod_tbl} -\begin{tabular}{|l|l|l|} -\hline -m & a\_бок & a\_центр \\ \hline -50 & 186 & 738 \\ \hline -10 & 38 & 738 \\ \hline -20 & 74 & 738 \\ \hline -30 & 110 & 738 \\ \hline -40 & 150 & 738 \\ \hline -60 & 222 & 738 \\ \hline -70 & 258 & 738 \\ \hline -80 & 298 & 738 \\ \hline -90 & 334 & 738 \\ \hline -100 & 370 & 738 \\ \hline -\end{tabular} -\end{table} \begin{figure}[h!] \begin{center} \includegraphics[width=0.9\textwidth]{a(m)} -\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$ при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{mod_img} +\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$ для сигнала, модулированного по амплитуде, при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{mod_img} \end{center} \end{figure} Определен коэффициент наклона прямой: @@ -279,7 +220,7 @@ k = 0.502 \pm 0.002 3. При обработке данных от спектра периодической последовательности цугов была обнаружена ошибка при снятии данных, что не позволило проверить соотношение неопределенностей. -4. Получен угол наклона графика зависимости $\dfrac{a_{бок}}{а_{осн}}$ от $m$ (0.5), подтверждено теоретическое значение этого угла (0.5). +4. Получен угол наклона графика зависимости $\dfrac{a_{бок}}{а_{осн}}$ от $m$ ($k$=0.5), подтверждено теоретическое значение этого угла ($k$=0.5). \begin{thebibliography}{}