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Diffstat (limited to 'exercise02')
| -rw-r--r-- | exercise02/exercise02.tex | 299 |
1 files changed, 295 insertions, 4 deletions
diff --git a/exercise02/exercise02.tex b/exercise02/exercise02.tex index dcd06d2..51099f7 100644 --- a/exercise02/exercise02.tex +++ b/exercise02/exercise02.tex @@ -3,7 +3,7 @@ \section*{Exercise 2} \begin{question}[subtitle={Mono-chromatic Signals}] - Given is a mono-chromatic signal $u(t)$: + A mono-chromatic signal $u(t)$ is given: \begin{equation*} u(t) = \SI{2}{V} \cdot \cos\left(2 \pi \cdot \SI{1}{MHz} \cdot t + \frac{\pi}{2} \right) \end{equation*} @@ -37,17 +37,103 @@ \end{tasks} \end{solution} -\begin{question}[subtitle={Using the Fourier transform}] +\begin{question} + The following periodic signal is given. + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.8\linewidth, + scale only axis, + xlabel={$t \text{ in } \si{s}$}, + ylabel={$x(t) \text{ in } \si{V}$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-3.5, + xmax=3.5, + ymin=-0.5, + ymax=1.1, + xtick={-3.0, -2.5, ..., 3.0}, + %ytick={0, 0.5, ..., 1.5}, + %xticklabels={0, 1, $t_0$, 3, 4, ..., 10} + ] + \addplot[blue, thick] coordinates {(-3.0,-0.2) (-2.5,-0.2)}; + \addplot[blue, thick] coordinates {(-2.5,0.8) (-1.5,0.8)}; + \addplot[blue, thick] coordinates {(-1.5,-0.2) (-0.5,-0.2)}; + \addplot[blue, thick] coordinates {(-0.5,0.8) (0.5,0.8)}; + \addplot[blue, thick] coordinates {(0.5,-0.2) (1.5,-0.2)}; + \addplot[blue, thick] coordinates {(1.5,0.8) (2.5,0.8)}; + \addplot[blue, thick] coordinates {(2.5,-0.2) (3.0,-0.2)}; + + \addplot[blue, thick, dashed] coordinates {(-2.5,-0.2) (-2.5,0.8)}; + \addplot[blue, thick, dashed] coordinates {(-1.5,0.8) (-1.5,-0.2)}; + \addplot[blue, thick, dashed] coordinates {(-0.5,-0.2) (-0.5,0.8)}; + \addplot[blue, thick, dashed] coordinates {(0.5,0.8) (0.5,-0.2)}; + \addplot[blue, thick, dashed] coordinates {(1.5,-0.2) (1.5,0.8)}; + \addplot[blue, thick, dashed] coordinates {(2.5,0.8) (2.5,-0.2)}; + \end{axis} + \end{tikzpicture} + \end{figure} + + \begin{tasks} + \task + Find a functional expression for the above signal! + \task + What is the base frequency? + \task + Find the real-valued Fourier series coefficients $a_n$ and $b_m$! + \task + Find the complex-valued Fourier series coefficients $\underline{c}_n$! + \task + Plot the amplitude and phase spectra for $-5 \leq n \leq 5$! + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \task + \begin{equation*} + x(t) = \begin{cases} + \SI{-0.2}{V}, &\quad \text{ if} \; \left(\SI{-0.5}{s} + n \cdot \SI{2}{s}\right) \leq t < \left(\SI{0.5}{s} + n \cdot \SI{2}{s}\right) \\ + \SI{-0.2}{V}, &\quad \text{ if} \; \left(\SI{0.5}{s} + n \cdot \SI{2}{s}\right) \leq t < \left(\SI{1.5}{s} + n \cdot \SI{2}{s}\right) \\ + \end{cases} \qquad \forall \; n \in \mathbb{Z} + \end{equation*} + + \task + \begin{itemize} + \item Period: $T_0 = \SI{2}{s}$ + \item Base frequency: $f_0 = \SI{0,5}{Hz}$ + \item Base angular frequency: $\omega_0 = \SI{3.14}{s^{-1}}$ + \end{itemize} + + %TODO + \end{tasks} +\end{solution} + +\begin{question}[subtitle={Using the Fourier Transform}] Derive the Fourier transform, without using the duality, of \begin{tasks} \task - Derive the Fourier transform of the time shift, without using the duality! + the time shift \begin{equation*} \mathcal{F}\left\{\underline{f}(t - t_0)\right\} \end{equation*} \task - Derive the Fourier transform of the frequency shift, without using the duality! + the frequency shift \begin{equation*} \mathcal{F}\left\{e^{j \omega_0 t} \underline{f}(t)\right\} \end{equation*} @@ -119,3 +205,208 @@ % We obtain the same result as in b). The duality works. \acs{QED} \end{tasks} \end{solution} + +\begin{question}[subtitle={Fourier Transform of Signals}] + The following signal is given! + \begin{equation*} + \underline{x}(t) = j \cos\left(\omega_0 t\right) - \sin\left(\omega_0 t\right) + \end{equation*} + + \begin{tasks} + \task + What is the Fourier transform of the signal? + \task + Plot the amplitude and phase spectra! + \task + Why does the spectrum not fulfil the symmetry rules? + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \task + \begin{equation*} + \underline{X}\left(j \omega\right) = j 2 \pi \delta\left(\omega - \omega_0\right) + \end{equation*} + + \task + \begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega \text{ in } \si{Hz}$}, + ylabel={$\left|\underline{X}\left(j \omega\right)\right| \text{ in } \si{V/Hz}$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-2.5, + xmax=2.5, + ymin=0, + ymax=6.5, + xtick={-1, 0, 1}, + xticklabels={$-\omega_0$, 0, $\omega_0$}, + ytick={0, 6.28}, + yticklabels={0, 6.28}, + ] + \addplot[blue, thick] coordinates {(1,0) (1,6.28)}; + \addplot[blue, thick, only marks, mark=o] coordinates {(1,6.28)}; + \end{axis} + \end{tikzpicture} + \end{figure} + \end{minipage} + \hfill + \begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega \text{ in } \si{Hz}$}, + ylabel={$\arg\left(\underline{X}\left(j \omega\right)\right) \text{ in } \si{\degree}$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-2.5, + xmax=2.5, + ymin=-3.5, + ymax=3.5, + xtick={-1, 0, 1}, + xticklabels={$-\omega_0$, 0, $\omega_0$}, + ytick={-3.14, -1.57, 0, 1.57, 3.14}, + yticklabels={$-\pi$, $-\frac{\pi}{2}$, 0, $\frac{\pi}{2}$, $\pi$}, + ] + \addplot[blue, thick] coordinates {(1,0) (1,1.57)}; + \addplot[blue, thick, only marks, mark=o] coordinates {(1,1.57)}; + \end{axis} + \end{tikzpicture} + \end{figure} + \end{minipage} + + \task + The signal is complex-valued. The symmetry rules only apply for real-valued signals. + \end{tasks} +\end{solution} + +\begin{question} + The following circuit is given. + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[L, l=$L$, o-] ++(2,0) to[short, *-o] ++(2,0); + \draw (2, 0) to[C, l=$C$, -*] ++(0,-2); + \draw (0, -2) to[short, o-o] ++(4,0); + + \draw (0, 0) to[open, v=$u_i(t)$] (0, -2); + \draw (4, 0) to[open, v^=$u_o(t)$] (4, -2); + \end{circuitikz} + \end{figure} + \begin{tasks} + \task + Find a differential equation which connects $u_i(t)$ and $u_o(t)$! + \task + Determine the transfer function $\underline{H} \left(j \omega\right)$! + \task + Calculate the impulse response! + \task + Is the system causal? Why? + \task + What filter characteristic does the circuit have? Which order does the system have? + \end{tasks} +\end{question} + +\begin{question} + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[C, l=$C$, o-] ++(2,0) to[short, *-o] ++(2,0); + \draw (2, 0) to[R, l=$R$, -*] ++(0,-2); + \draw (0, -2) to[short, o-o] ++(4,0); + + \draw (0, 0) to[open, v=$u_i(t)$] (0, -2); + \draw (4, 0) to[open, v^=$u_o(t)$] (4, -2); + \end{circuitikz} + \end{figure} + + The high-pass filter has the following transfer function: + \begin{equation} + \underline{H}\left(j \omega\right) = \frac{j \omega RC}{j \omega RC + 1} + \end{equation} + with + \begin{itemize} + \item $R = \SI{100}{\ohm}$ + \item $C = \SI{470}{nF}$ + \end{itemize} + + \begin{tasks} + \task + Which order does the system have? + \task + What are the poles and zeroes of the system? Is the system stable? + \task + Determine and plot the amplitude response $\left|\underline{H}\left(j \omega\right)\right|$! + \task + Determine and plot the phase response $\arg\left(\underline{H}\left(j \omega\right)\right)$! + \task + The following signal is applied to the input of the system $u_i(t)$. + \begin{equation} + u_i(t) = \SI{2}{V} \cos\left(2 \pi \cdot \SI{25}{kHz} \cdot t\right) + \end{equation} + Calculate the output signal $u_o(t)$ as either a time domain function or a phasor! + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \task + 1st order + \begin{itemize} + \item Only one capacity as a memorizing component + \item Highest exponent is $1$. + \end{itemize} + + \task + Replace $j \omega$ by $\underline{s}$. + \begin{itemize} + \item Zero: $\underline{s}_0 = 0$ (Numerator of $\underline{H}\left(j \omega\right)$ must be zero) + \item Pole: Denominator of $\underline{H}\left(j \omega\right)$ must be zero + \begin{equation*} + \begin{split} + \underline{s}_{\infty} &= -\frac{1}{RC} \\ + &= -\frac{1}{\SI{100}{\ohm} \cdot \SI{470}{nF}} \\ + &= \SI{-21276.6}{s^{-1}} + \end{split} + \end{equation*} + \end{itemize} + The system is stable because the real part of its pole is negative. + + %TODO + \end{tasks} +\end{solution} |