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\section{Аннотация} -В работе исследуются спектры периодических сигналов: модулированный по амплитуде, прямоугольные импульсы и цуги. Проверяются теоретические зависимости параметров спектра на практике. +В работе исследованы спектры периодических сигналов: модулированный по амплитуде, прямоугольные импульсы и цуги. Проверены теоретические зависимости параметров спектра на практике. \section{Введение} -Задача описания поведения некоторой системы во времени зачастую -сводится к выяснению связи между "сигналом", подаваемым на "вход" -системы (обозначим его как $f(t)$), и её реакцией на "выходе" $g(t)$). Если произвольную функцию $f(t)$ удастся представить в виде -некоторой суммы ряда (конечного или бесконечного) гармонических -слагаемых, то при известной частотной характеристике $\lambda(\omega)$) задача -о связи воздействия и отклика системы будет решена. Такое разложение -называют спектральным. Оно имеет физический смысл: высоко добротный колебательный контур выделяет из подаваемого на него сигнала те спектральные компоненты, частоты которых близки к его собственной. Возможность разложить произвольную функцию $f(t)$ в ряд (или интеграл) -Фурье единственным и однозначным способом подразумевает и -возможность "собрать" сигнал любой формы, используя гармонические -колебания с подобранными амплитудами и фазами. В последнее время повсеместное распространение получила цифровая -обработка сигналов. Спектральный состав оцифрованного сигнала -может быть найден численно. Это и планируется проследить в работе. +Многие практические задачи описания поведения некоторой системы во времени зачастую сводятся к выяснению связи между "сигналом", подаваемым на "вход" +системы (обозначим его как $f(t)$), и её реакцией на "выходе" $g(t)$). Суть спектрального метода состоит в представлении произвольного воздействия в виде суперпозиции откликов на некоторые элементарные слагаемые. Данный метод используются для анализа многих сигналов, поэтому необходимо экспериментально ознакомиться с ним, сгенерировать и получить на осциллографе спектры различных периодических сигналов, проверить экспериментально соотношение неопределенности и отношения амплитуд гармоник при модулированных по амплитуде сигналах. - -\section{Поставка задачи} -Сгенерировать и получить на осциллографе спектры различных периодических сигналов. Проверить экспериментально соотношение неопределенности и отношения амплитуд гармоник при модулированных по амплитуде сигналах. - -\section{Теоретические сведения} -Задача описания поведения некоторой системы во времени зачастую -сводится к выяснению связи межу «сигналом», подаваемым на «вход» -системы (обозначим его как $f(t)$), и её реакцией на «выходе» ($g(t)$). -Для линейных стационарных фильтров: $g = \hat{\Lambda}[f]$, $\hat{\Lambda}$ - линейное преобразование. Из линейности системы: $f=\sum c_nf_n$, $g_n = c_n \cdot \hat{\Lambda}[f_n]$ -\begin{equation} -g(t) = \sum c_n \hat{\Lambda}[f_n] -\end{equation} -Выбор элементарных слагаемых - собственные векторы. -\begin{equation} -f(t) = \sum_n c_n e^{i\omega_n t} -\end{equation} -Такое представление - ряд Фурье. -\subsection*{Спект периодического процесса} -Периодический процесс - $f(t)=f(t+T)$ -\begin{equation} -f(t) = \sum_{n=-\inf}^{\inf} c_n e^{in\omega_0 t} -\end{equation} -Набор коэффициентов можно найти, домножив обе части прошлого равенства на $e^{-im\omega_0 t}$ и проинтегрировав по периоду: -\begin{equation} -c_n = \frac{1}{T}\int_0^T f(t)e^{-in\omega_0 t}dt -\end{equation} -\begin{figure}[h!] -\includegraphics[width=0.4\textwidth]{rect} -\caption{Периодическая последовательность прямоугольных импульсов} \label{rect} -\end{figure} -Рассмотрим спектр периодической последовательности прямоугольных импульсов (рис. \ref{rect}): -\begin{equation} -c_n = \frac{1}{T}\int_{-\tau / 2}^{\tau / 2} e^{-in\omega_0 t}dt = \dfrac{sin(\pi n \tau / T))}{\pi n} -\end{equation} - -\subsection*{Свойства спектров} -Справедливо для произвольного сигнала соотношение неопределенностей: +\section{Методика измерений} +В работе используются генератор сигналов произвольной формы, цифровой осциллограф с функцией быстрого преобразования Фурье. Для произвольного сигнала справедливо соотношение неопределенностей: \begin{equation} \Delta \omega \cdot \Delta t \sim 2\pi \end{equation} -Рассмотрим спектр обрывка синусоиды (цуг): +Работа разделена на три равноценные части, в каждой из которых генерируется сигнал определенной формы, обрабатывается с помощью цифрового осциллографа, проверяются 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} \label{rect} +\end{figure} +Теоретический расчет дает значение коэффициентов $c_n$: \begin{equation} -f(t) = f_0(t)\cos(\omega_0 t) +c_n = \dfrac{sin(\pi n \tau / T))}{\pi n} \end{equation} -Из спектра прямоугольного импульса: -\begin{equation} -F(\omega) = \dfrac{\tau}{2}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} -{(\omega-\omega_0)\tau /2} - + \dfrac{\sin(\omega+\omega_0)\tau /2}{\omega+\omega_0)\tau /2}\right] -\end{equation} -Получим спектр периодической последовательности цугов (рис. \ref{цуги}): + +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} \label{цуги} +\end{center} +\end{figure} +Теоретический расчет дает спектр сигнала: \begin{equation} F(\omega) = \dfrac{\tau}{2T}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} {(\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]{цуги} -\caption{Периодическая последовательность цуг} \label{цуги} +\includegraphics[width=0.4\textwidth]{Модулированный} +\caption{Пример модулированного сигнала} \label{модулированный} \end{center} \end{figure} -\subsection*{Модуляция} -Модулированные колебания: -\begin{equation} -f(t) = a(t) \cos (\omega_0 t+\varphi(t)) -\end{equation} -Простейшее амплитудно-модулированное колебание: -\begin{equation} -f(t)= a(t) \cos (\omega_0 t), \hspace{3mm} a(t) = a_0(1+m\cos(\Omega t)) -\end{equation} -В выражении $0 < m \leq 1$ - глубина модуляции, выражается: -\begin{equation} -m = \dfrac{a_{max}-a_{min}}{a_{max}+a_{min}} -\end{equation} -Из прошлой формулы можно получить: +Теоретический вид сигнала: \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 \end{equation} -\section{Методика измерений} -В работе используются генератор сигналов произвольной формы, цифровой осциллограф с функцией быстрого преобразования Фурье. + \section{Измерения и обработка данных} \subsection*{Исследования спектра периодической последовательности прямоугольных импульсов}