67 \` table\\ |
67 \` table\\ |
68 \texttt{\footnotesize{09}} \> $2\cdot\frac{\sin\;\omega}{\omega}$ |
68 \texttt{\footnotesize{09}} \> $2\cdot\frac{\sin\;\omega}{\omega}$ |
69 \end{tabbing} |
69 \end{tabbing} |
70 } |
70 } |
71 |
71 |
72 \textbf{(b)} |
72 \noindent\textbf{(b)} \textsf{Subproblem 1} |
73 \onehalfspace{ |
73 \onehalfspace{ |
74 \begin{tabbing} |
74 \begin{tabbing} |
75 000\=\kill |
75 000\=\kill |
76 \textsf{Subproblem 1}\\ |
|
77 \texttt{\footnotesize{01}} \> Definition: $X(j\omega)=\int_\infty^\infty{x(t)\cdot e^{-j\omega t}}$\\ |
76 \texttt{\footnotesize{01}} \> Definition: $X(j\omega)=\int_\infty^\infty{x(t)\cdot e^{-j\omega t}}$\\ |
78 \`Insert Condition: $x(t) = 1\;$ for $\;\{1\leq t\;\land\;t\leq 3\}\;$ and $\;x(t)=0\;$ otherwise\\ |
77 \`Insert Condition: $x(t) = 1\;$ for $\;\{1\leq t\;\land\;t\leq 3\}\;$ and $\;x(t)=0\;$ otherwise\\ |
79 \texttt{\footnotesize{02}} \> $X(j\omega)=\int_{-1}^{1}{1\cdot e^{-j\omega t}}$\\ |
78 \texttt{\footnotesize{02}} \> $X(j\omega)=\int_{-1}^{1}{1\cdot e^{-j\omega t}}$\\ |
80 \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\ |
79 \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\ |
81 \texttt{\footnotesize{03}} \> $\int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{1}^3$\\ |
80 \texttt{\footnotesize{03}} \> $\int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{1}^3$\\ |
89 \texttt{\footnotesize{07}} \> $\frac{1}{j\omega}\cdot e^{j\omega2}\cdot(e^{j\omega} - e^{-j\omega})$\\ |
88 \texttt{\footnotesize{07}} \> $\frac{1}{j\omega}\cdot e^{j\omega2}\cdot(e^{j\omega} - e^{-j\omega})$\\ |
90 \`Simplification (trick)\\ |
89 \`Simplification (trick)\\ |
91 \texttt{\footnotesize{08}} \> $\frac{1}{\omega}\cdot e^{j\omega2}\cdot(\frac{e^{j\omega} - e^{-j\omega}}{j})$\\ |
90 \texttt{\footnotesize{08}} \> $\frac{1}{\omega}\cdot e^{j\omega2}\cdot(\frac{e^{j\omega} - e^{-j\omega}}{j})$\\ |
92 \` table\\ |
91 \` table\\ |
93 \texttt{\footnotesize{09}} \> $2\cdot e^{j\omega2}\cdot\frac{\sin\;\omega}{\omega}$\\ |
92 \texttt{\footnotesize{09}} \> $2\cdot e^{j\omega2}\cdot\frac{\sin\;\omega}{\omega}$\\ |
94 \textsf{Subproblem 2}\\ |
93 \noindent\textbf{(b)} \textsf{Subproblem 2}\\ |
95 \`Definition: $X(j\omega)=|X(j\omega)|\cdot e^{arg(X(j\omega))}$\\ |
94 \`Definition: $X(j\omega)=|X(j\omega)|\cdot e^{arg(X(j\omega))}$\\ |
96 \`$|X(j\omega)|$ is called \emph{Magnitude}\\ |
95 \`$|X(j\omega)|$ is called \emph{Magnitude}\\ |
97 \`$arg(X(j\omega))$ is called \emph{Phase}\\ |
96 \`$arg(X(j\omega))$ is called \emph{Phase}\\ |
98 \texttt{\footnotesize{10}} \> $|X(j\omega)|=\frac{2}{\omega}\cdot sin(\omega)$\\ |
97 \texttt{\footnotesize{10}} \> $|X(j\omega)|=\frac{2}{\omega}\cdot sin(\omega)$\\ |
99 \texttt{\footnotesize{11}} \> $arg(X(j\omega)=-2\omega$\\ |
98 \texttt{\footnotesize{11}} \> $arg(X(j\omega)=-2\omega$\\ |
100 \end{tabbing} |
99 \end{tabbing} |
101 } |
100 } |
102 |
|
103 %------------------------------------------------------------------------------ |
101 %------------------------------------------------------------------------------ |
104 %CONVOLUTION |
102 %CONVOLUTION |
105 |
103 |
106 \subsection{Convolution} |
104 \subsection{Convolution} |
107 \subsubsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the SPSC Problem Class 2, Summer term 2008}} |
105 \subsubsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the SPSC Problem Class 2, Summer term 2008}} |
150 } |
148 } |
151 |
149 |
152 %------------------------------------------------------------------------------ |
150 %------------------------------------------------------------------------------ |
153 %Z-Transformation |
151 %Z-Transformation |
154 |
152 |
155 \subsection{$\cal Z$-Transformation\label{sec:calc:ztrans}} |
153 \subsection{Z-Transformation\label{sec:calc:ztrans}} |
156 \subsubsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the signal processing problem class 5, summer term 2008.}} |
154 \subsubsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the signal processing problem class 5, summer term 2008.}} |
157 Determine the inverse $\cal{z}$ transform of the following expression. Hint: applay the partial fraction expansion. |
155 Determine the inverse $\cal{z}$ transform of the following expression. Hint: applay the partial fraction expansion. |
158 |
156 |
159 \begin{center} |
157 \begin{center} |
160 $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable |
158 $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable |