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\title{Analog Signal Processing \\ Midterms}

\begin{document}
\maketitle

\section{1. Midterm}
\begin{enumerate}
    \item Find $F(x)$ given  \[ f(x) =
        \begin{cases}
            t-1 &, (1;2) \\
            t &, [2;4] \\
            -\frac{1}{2} (t-6) &, (4;6] \\
        \end{cases}
        \]
    \item Define the imaginary part of the system given the real part is $ H_{Re} = 4-\omega^2 \quad ,\omega \in [-2;2]$
    \item Define the step response of a 2nd order Filter with Bessel approximation and $3.4$ kHz cutoff frequency.
        Determine the filter parameters.
    \item Define the transfer function of a 4th-order band-pass Chebyshev filter with central frequency $100$ MHz, $0.2$ dB passband ripple
        and bandwidth $5$ Mhz.
\end{enumerate}
\section{2. Midterm}
\begin{enumerate}
    \item Determine the component values and draw the implementation of a passive second-order high-pass filter in an RC configuration,
        with the component values indicated, a cutoff frequency of $250$ Hz and a Q factor of $0.35$. Choose decade values for the capacitances.
    \item Implement, using operational amplifiers, a fourth-order low-pass filter with the normalized transfer function
        \[H(s) = \frac{1}{s^2 + 1.8478 \cdot s + 1} \cdot \frac{1}{s^2 + 0.7654 \cdot s + 1}\]
        Perform element denormalization for a cutoff frequency of $12$ kHz. Choose decade values for the capacitances.
        Draw the implementatiopn with the denormalized component values.
    \item Implement, using transconductance amplifiers ($T= 40$\textdegree C, $U_{\text{NAP}} = \pm3.3V, U_\text{D} = 0.6$V)
        a second-order band-pass filter with a center frequency of $1$ kHz, a pole quality factor of $0.3$ and a gain of $6$ dB.
    \item A sporting event is being covered by a drone that produces an unwanted sound at a frequency of 366 Hz.
        Desing the simplest filter implementation that can remove this sound from the live broadcast, with a bandwidth of $\pm 5 $ Hz.
        Determine the transfer function and sketch the amplitude-frequency characteristic in detail.
        In the filter implementation, use all three types of passive elements, with the reactive elements connected in parallel,
        and the filter having no power supply.
\end{enumerate}
\end{document}