% SPDX-License-Identifier: CC-BY-SA-4.0
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% Copyright (c) 2020 Philipp Le
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\phantomsection
\addcontentsline{toc}{section}{Exercise 4}
\section*{Exercise 4}

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\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}

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%\begin{question}[subtitle={Sampling Non-Periodic Signals}]
%	\begin{tasks}
%	\end{tasks}
%\end{question}
%
%\begin{solution}
%	\begin{tasks}
%	\end{tasks}
%\end{solution}

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\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}

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%\begin{question}[subtitle={Decibel}]
%	\begin{tasks}
%	\end{tasks}
%\end{question}
%
%\begin{solution}
%	\begin{tasks}
%	\end{tasks}
%\end{solution}