\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