equal
deleted
inserted
replaced
19 |
19 |
20 |
20 |
21 %------------------------------------------------------------------------------ |
21 %------------------------------------------------------------------------------ |
22 %FOURIER |
22 %FOURIER |
23 |
23 |
24 \section*{Fourier Transformation} |
24 \subsection{Fourier Transformation} |
25 \subsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally fragment of the exam to \emph{Signaltransformationen VO} from 04.03.2011. Translated from German.}} |
25 \subsubsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally fragment of the exam to \emph{Signaltransformationen VO} from 04.03.2011. Translated from German.}} |
26 \textbf{(a)} Determine the fourier transform for the given rectangular impulse: |
26 \textbf{(a)} Determine the fourier transform for the given rectangular impulse: |
27 |
27 |
28 \begin{center} |
28 \begin{center} |
29 $x(t)= \left\{ |
29 $x(t)= \left\{ |
30 \begin{array}{lr} |
30 \begin{array}{lr} |
43 0 & else |
43 0 & else |
44 \end{array} |
44 \end{array} |
45 \right.$ |
45 \right.$ |
46 \end{center} |
46 \end{center} |
47 |
47 |
48 \subsection*{Solution} |
48 \subsubsection{Solution} |
49 \textbf{(a)} |
49 \textbf{(a)} |
50 \onehalfspace{ |
50 \onehalfspace{ |
51 \begin{tabbing} |
51 \begin{tabbing} |
52 000\=\kill |
52 000\=\kill |
53 \texttt{\footnotesize{01}} \> Definition: $X(j\omega)=\int_\infty^\infty{x(t)\cdot e^{-j\omega t}}$\\ |
53 \texttt{\footnotesize{01}} \> Definition: $X(j\omega)=\int_\infty^\infty{x(t)\cdot e^{-j\omega t}}$\\ |
101 } |
101 } |
102 |
102 |
103 %------------------------------------------------------------------------------ |
103 %------------------------------------------------------------------------------ |
104 %CONVOLUTION |
104 %CONVOLUTION |
105 |
105 |
106 \section*{Convolution} |
106 \subsection{Convolution} |
107 \subsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the SPSC Problem Class 2, Summer term 2008}} |
107 \subsubsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the SPSC Problem Class 2, Summer term 2008}} |
108 Consider the two discrete-time, linear and time-invariant (LTI) systems with the following impulse response: |
108 Consider the two discrete-time, linear and time-invariant (LTI) systems with the following impulse response: |
109 |
109 |
110 \begin{center} |
110 \begin{center} |
111 $h_1[n]=\left(\frac{3}{5}\right)^n\cdot u[n]$\\ |
111 $h_1[n]=\left(\frac{3}{5}\right)^n\cdot u[n]$\\ |
112 $h_1[n]=\left(-\frac{2}{3}\right)^n\cdot u[n]$ |
112 $h_1[n]=\left(-\frac{2}{3}\right)^n\cdot u[n]$ |
113 \end{center} |
113 \end{center} |
114 |
114 |
115 The two systems are cascaded seriell. Derive the impulse respinse of the overall system $h_c[n]$. |
115 The two systems are cascaded seriell. Derive the impulse respinse of the overall system $h_c[n]$. |
116 \subsection*{Solution} |
116 \subsubsection*{Solution} |
117 |
117 |
118 \doublespace{ |
118 \doublespace{ |
119 \begin{tabbing} |
119 \begin{tabbing} |
120 000\=\kill |
120 000\=\kill |
121 \texttt{\footnotesize{01}} \> $h_c[n]=h_1[n]*h_2[n]$\\ |
121 \texttt{\footnotesize{01}} \> $h_c[n]=h_1[n]*h_2[n]$\\ |
150 } |
150 } |
151 |
151 |
152 %------------------------------------------------------------------------------ |
152 %------------------------------------------------------------------------------ |
153 %Z-Transformation |
153 %Z-Transformation |
154 |
154 |
155 \section*{$\cal Z$-Transformation} |
155 \subsection{$\cal Z$-Transformation\label{sec:calc:ztrans}} |
156 \subsection*{Problem\endnote{Problem submitted by Bernhard Geiger. Originally part of the signal processing problem class 5, summer term 2008.}} |
156 \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. |
157 Determine the inverse $\cal{z}$ transform of the following expression. Hint: applay the partial fraction expansion. |
158 |
158 |
159 \begin{center} |
159 \begin{center} |
160 $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable |
160 $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable |
161 \end{center} |
161 \end{center} |
162 |
162 |
163 \subsection*{Solution} |
163 \subsubsection*{Solution} |
164 \onehalfspace{ |
164 \onehalfspace{ |
165 \begin{tabbing} |
165 \begin{tabbing} |
166 000\=\kill |
166 000\=\kill |
167 \textsf{Main Problem}\\ |
167 \textsf{Main Problem}\\ |
168 \texttt{\footnotesize{01}} \> $\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}}$ \\ |
168 \texttt{\footnotesize{01}} \> $\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}}$ \\ |