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diff --git a/exercise04/exercise04.tex b/exercise04/exercise04.tex new file mode 100644 index 0000000..27551d4 --- /dev/null +++ b/exercise04/exercise04.tex @@ -0,0 +1,100 @@ +% SPDX-License-Identifier: CC-BY-SA-4.0 +% +% Copyright (c) 2020 Philipp Le +% +% Except where otherwise noted, this work is licensed under a +% Creative Commons Attribution-ShareAlike 4.0 License. +% +% Please find the full copy of the licence at: +% https://creativecommons.org/licenses/by-sa/4.0/legalcode + +\phantomsection +\addcontentsline{toc}{section}{Exercise 4} +\section*{Exercise 4} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{question}[subtitle={Sampling Periodic Signals}] + \begin{equation*} + u(t) = \SI{2}{V} \cos\left(2\pi \SI{2}{MHz} t + \SI{60}{\degree}\right) + \end{equation*} + The signal is sampled with a sampling period of $T_S = \SI{125}{\nano\second}$. The first sample taken is $u(t = 0)$. + + \begin{tasks} + \task + Plot the function from $t = 0$ to $t = \SI{1}{\micro\second}$! + + \task + Calculate the samples $n = 0 \dots 8$! + + \task + What is the DTFT of the signal? + + Hints: + \begin{equation*} + \begin{split} + x[n] = e^{-j a n} &= \underline{X}_{\frac{2\pi}{T_S}}\left(e^{-j T_S \omega}\right) = 2 \pi \cdot \delta \left(\omega + a\right) \\ + \cos\left(b\right) &= \frac{1}{2} \left(e^{j b} + e^{-j b}\right) + \end{split} + \end{equation*} + + \task + Can the DFT directly applied to the signal? If yes, determine the smallest $N$ and give the values of all $\underline{U}[k]$! + + \task + What is the longest possible sampling period? What must be considered at this sampling period? + + \task + Now, the sampling period is changed to $T_S = \SI{0.5}{\micro\second}$. There is no anti-aliasing filter. The reconstruction filter is an ideal low-pass filter with a cut-off frequency of \SI{50}{kHz}. Give the reconstructed output function in the time domain! Give an explanation in the frequency domain! + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \end{tasks} +\end{solution} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\begin{question}[subtitle={Sampling Non-Periodic Signals}] +% \begin{tasks} +% \end{tasks} +%\end{question} +% +%\begin{solution} +% \begin{tasks} +% \end{tasks} +%\end{solution} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{question}[subtitle={Quantization}] + The signal of task 1b) is now quantized. The quantizer has $8$ discrete values. These values are equally distributed between \SI{-2}{V} and \SI{2}{V}. Prior to sampling, the original time-continuous signal passed through an ideal low-pass filter with a cut-off frequency of \SI{4}{MHz}. + + \begin{tasks} + \task + Define a mapping from the value-continuous samples to the value-discrete samples! + + \task + The value-discrete samples are now pulse-code modulated. How many bits are required? + + \task + Determine the quantization error for each value-discrete sample! How much is the signal-to-noise ratio? + + \task + 3 bits are a very poor resolution. How many bits are appropriate for the quantizer to obtain the best signal-to-noise ratio? Effects of the window filter are neglected. Assume that the signal has passed through a processing chain with a total gain of \SI{25}{dB} and noise figure of \SI{12}{dB} prior to quantization. The input of the quantizer has an impedance of \SI{50}{\ohm}. % 14 bits + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \end{tasks} +\end{solution} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\begin{question}[subtitle={Decibel}] +% \begin{tasks} +% \end{tasks} +%\end{question} +% +%\begin{solution} +% \begin{tasks} +% \end{tasks} +%\end{solution} |