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\usepackage{tikz}
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% Ersetzt in den eckigen Klammern bitte die Ãœbungsnummer.
\title{Submission - Assignment [$10$]\\
\small{Introduction to Computer Graphics}\\
\large{Group 09}}

\author{[Steljana Lleshi, steljana.lleshi@rwth-aachen.de] \and [Iftekhar Ahmed, iftekhar.ahmed@rwth-aachen.de]}
\date{\today}
\maketitle

\section*{Theoretical exercise}
(a) The Fourier transform is \begin{equation*}
    \begin{split}
        F(\omega)&=\int_{-\infty}^{\infty}f(x)e^{-x}e^{-j\omega x}dx\\
        &=\int_{-\infty}^{0}f(x)e^{-x}e^{-j\omega x}dx+\int_{0}^{\infty}xe^{-x}e^{-j\omega x}dx\\
        &=0+\int_{0}^{\infty}xe^{-x}e^{-j\omega x}dx\\
        &=\lim_{0\to\infty}\bigg(\frac{xe^{-x(1+j\omega)}}{-(1+j\omega)}-\frac{e^{-x(1+j\omega)}}{(1+j\omega)^2}\bigg)\\
        &=\frac{1}{(1+j\omega)^2}
    \end{split}
\end{equation*}

\justify
(b) The Fourier transform is \begin{equation*}
    \begin{split}
        F(\omega)&=\int_{-\infty}^{\infty}e^{-a|x|}e^{-j\omega x}dx\\
        &=\int_{-\infty}^{0}e^{ax}e^{-j\omega x}dx+\int_{0}^{\infty}e^{-ax}e^{-j\omega x}dx\\
        &=\frac{1}{a-j\omega}+\frac{1}{a+j\omega}\\
        &=\frac{2a}{a^2+\omega^2}
    \end{split}
\end{equation*}

\justify
(c) The Nyquist-Shannon Sampling theorem is a fundamental one providing the condition on the sampling frequency of a band-width limited continuous-time signal in order to be able to reconstruct it perfectly from its discrete-time (sampled) version. It stated that the sampling frequency must be at least two times the highest frequency of the continuous-time signal spectrum. The sampling rate of an audio CD can be 44100 Hz because of Shannon’s sampling criterion which requires sampling to be two times of the maximum analogue frequency and that is nearly 22.5 KHZ.
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