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diff --git a/exercise02/exercise02.tex b/exercise02/exercise02.tex new file mode 100644 index 0000000..d8ad77a --- /dev/null +++ b/exercise02/exercise02.tex @@ -0,0 +1,40 @@ +\phantomsection +\addcontentsline{toc}{section}{Exercise 2} +\section*{Exercise 2} + +\begin{question}[subtitle={Mono-chromatic Signals}] + Given is a mono-chromatic signal $u(t)$: + \begin{equation*} + u(t) = \SI{2}{V} \cdot \cos\left(2 \pi \cdot \SI{1}{MHz} \cdot t + \frac{\pi}{2} \right) + \end{equation*} + \begin{tasks} + \task + How much is the frequency and angular frequency? How much is the amplitude? How much is the phase? + \task + Give the phasor of the signal! + \task + An DC bias is added to the signal $u(t)$. + \begin{equation*} + u_2(t) = \SI{1}{V} + \SI{2}{V} \cdot \cos\left(2 \pi \cdot \SI{1}{MHz} \cdot t + \frac{\pi}{2} \right) + \end{equation*} + Is the resulting signal $u_2(t)$ still mono-chromatic? + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \task + \begin{itemize} + \item Frequency: \SI{1}{MHz} + \item Angular frequency: $2 \pi \cdot \SI{1}{MHz} = \SI{6283185.3}{s^{-1}}$ + \item Phase: $\SI{-\pi/2}{rad}$ or \SI{-90}{\degree} + \item Amplitude: \SI{2}{V} + \end{itemize} + \task + $\underline{U} = \SI{2}{V} \cdot e^{+j \frac{\pi}{2}}$ or $\underline{U} = \SI{2}{V} \angle +\frac{\pi}{2}$ + \task + No, the DC bias adds a mono-chromatic component with a frequency of $f = 0$. $u_2(t)$ is a Fourier series. + \end{tasks} +\end{solution} + +% Exercise: Is a sine wave with DC bias mono-chromatic -> no
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