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| -rw-r--r-- | exercise06/exercise06.tex | 126 |
1 files changed, 124 insertions, 2 deletions
diff --git a/exercise06/exercise06.tex b/exercise06/exercise06.tex index 954357e..9054ced 100644 --- a/exercise06/exercise06.tex +++ b/exercise06/exercise06.tex @@ -14,10 +14,38 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{question}[subtitle={Decibel}] - Proof mathematically that all poles of the FIR filter are $0$! +\begin{question}[subtitle={IIR Filter}] + The following IIR filter is given. + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw[o-] (-1,0) node[left, align=right]{$\underline{x}[n]$} -- (0,0); + \draw (0,-3) node[adder](Add1){}; + \draw (2,-3) node[adder](Add2){}; + \draw (0,0) to[amp,l=$\underline{b}_0$,>,-] (Add1.north) node[inputarrow,rotate=-90]{}; + \draw (0,0) to[short,*-] (2,0) to[twoport,t=$z^{-1}$,>,-] (Add2.north) node[inputarrow,rotate=-90]{}; + \draw (Add1.east) to[short] (Add2.west) node[inputarrow,rotate=0]{}; + \draw[-latex] (Add2.east) to[short] (4,-3) node[right, align=left]{$\underline{y}[n]$}; + \draw (3,-3) to[short,*-] (3,-6) to[twoport,t=$z^{-1}$,>,-] (0,-6) to[amp,l=$\underline{a}_0$,>,-] (Add1.south) node[inputarrow,rotate=90]{}; + \end{circuitikz} + \end{figure} + with: + \begin{itemize} + \item $\underline{a}_0 = 0.5$ + \item $\underline{b}_0 = 2$ + \end{itemize} \begin{tasks} + \task + Give the block diagram of the filter! + \task + Give the differential equation of the filter! + \task + How much is the filter order? + \task + Is the filter stable? + \task + Plot the amplitude and phase response between $0$ and $\pi$. \end{tasks} \end{question} @@ -28,6 +56,100 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{question}[subtitle={FIR Filter}] + An FIR filter with following coefficients is given. + \begin{itemize} + \item $b_0 = 1$. + \item $b_1 = 0.5 + j \cdot 1$. + \item $b_2 = 2$. + \end{itemize} + + The sampling rate of the digital system is \SI{2}{MHz}. + + \begin{tasks} + \task + Give the block diagram of the filter! + \task + Give the transfer function of the filter! + \task + Give the differential equation of the filter! + \task + How much is the filter order? + \task + Plot the amplitude and phase response between \SI{-1}{MHz} and \SI{1}{MHz}. + \task + Proof mathematically that all poles of the FIR filter are $0$! + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \end{tasks} +\end{solution} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{question}[subtitle={Down-sampling}] + An analogue signal $x(t)$ is digitized (sampled and quantized). + \begin{equation*} + x(t) = \sin\left(2 \pi \cdot \SI{96}{kHz} \cdot t\right) + \end{equation*} + The signal has been sampled by a \SI{8}{bit}-ADC at \SI{7.68}{MHz}. + + The signal $x[n]$ is decimated by $N = 40$. + + \begin{tasks} + \task + How much is the sampling rate of the decimated signal? + \task + Is the signal suitable to be decimated by $N = 40$? Explain why! What is the criterion? + \task + What is the optimal sampling phase? + \task + Explain the effect on the spectrum caused by down-sampling! + \task + How much is the processing gain? How much is the effective number of bits? + \end{tasks} +\end{question} + +\begin{solution} + \begin{tasks} + \end{tasks} +\end{solution} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{question}[subtitle={FFT}] + A series of the samples in the time-domain is given: + \begin{equation*} + x[n] = \left[2 \underline{-0.5} 1 -2 \right] + \end{equation*} + + \begin{remark} + The underline marks the sample at $n = 0$. + \end{remark} + + \begin{tasks} + \task + Calculate the DFT for $k = 0, \ldots, 3$! + \task + Calculate the FFT using the Cooley-Tuckey FFT algorithm! + \task + Compare the number of multiply-accumulate operations necessary for both methods in a) and b)! + \task + Draw the butterfly graph! + \task + Give th primitive roots of unity for each sub-FFT in the butterfly graph! + \end{tasks} +\end{question} + +\begin{solution} + %The signal is periodic with $N = 4$. + \begin{tasks} + \end{tasks} +\end{solution} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{question}[subtitle={Decibel}] % \begin{tasks} % \end{tasks} |