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Diffstat (limited to 'chapter02/content_ch02.tex')
| -rw-r--r-- | chapter02/content_ch02.tex | 62 |
1 files changed, 49 insertions, 13 deletions
diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index 994dfea..745c55d 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -94,7 +94,7 @@ A graphical view on the creation of a cosine signal is depicted in Figure \ref{f \centering \begin{tikzpicture} \begin{scope}[shift={(0, 0)}] - \draw[-latex] (0,0) -- (4.5,0) node[below, align=left]{$t$};
+ \draw[-latex] (0,0) -- (4.5,0) node[below, align=left]{$t$}; \draw[-latex] (0,-2.2) -- (0,2.2); \draw (1,0.1) -- (1,-0.1) node[below, align=center]{$\frac{T_0}{4}$}; \draw (2,0.1) -- (2,-0.1) node[below, align=center]{$\frac{T_0}{2}$}; @@ -107,7 +107,7 @@ A graphical view on the creation of a cosine signal is depicted in Figure \ref{f \end{scope} \begin{scope}[shift={(-4, 0)}] \draw[draw] (0:2) arc(0:360:2); - \draw[-latex] (0,0) -- (0,1) node[right, align=left]{$\Re$};
+ \draw[-latex] (0,0) -- (0,1) node[right, align=left]{$\Re$}; \draw[-latex] (0,0) -- (-1,0) node[below, align=center]{$\Im$}; \draw (180:1.9) -- (180:2.1) node[left, align=center]{$\frac{T_0}{4}$}; @@ -1690,6 +1690,15 @@ All ideal filters are non-causal and can therefore not be implemented in real. \subsubsection{Ideal Low Pass Filter} +\begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0,0) to[lowpass] (2,0); + \end{circuitikz} + \caption[Block symbol of a \acs{LPF}]{Block symbol of a \ac{LPF}} +\end{figure}% +\nomenclature[Bl]{\begin{circuitikz}[baseline={(current bounding box.center)}]\draw (0,0) to[lowpass] (2,0);\end{circuitikz}}{Low pass filter} + A \index{low pass filter} \textbf{\acf{LPF}} \begin{itemize} \item lets pass all signals below a \index{low pass filter!cut-off frequency} \textbf{cut-off frequency} $\omega_o$ (all signals within the \index{low pass filter!pass band} \textbf{pass band} $|\omega| < \omega_o$), @@ -1746,6 +1755,15 @@ A \index{low pass filter} \textbf{\acf{LPF}} \subsubsection{Ideal High Pass Filter} +\begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0,0) to[highpass] (2,0); + \end{circuitikz} + \caption[Block symbol of a \acs{HPF}]{Block symbol of a \ac{HPF}} +\end{figure}% +\nomenclature[Bh]{\begin{circuitikz}[baseline={(current bounding box.center)}]\draw (0,0) to[highpass] (2,0);\end{circuitikz}}{High pass filter} + A \index{high pass filter} \textbf{\acf{HPF}} \begin{itemize} \item blocks all signals below a \index{high pass filter!cut-off frequency} \textbf{cut-off frequency} $\omega_o$ (all signals within the \index{high pass filter!stopband} \textbf{stopband} $|\omega| < \omega_o$), @@ -1802,6 +1820,15 @@ A \index{high pass filter} \textbf{\acf{HPF}} \subsubsection{Ideal Band Pass Filter} +\begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0,0) to[bandpass] (2,0); + \end{circuitikz} + \caption[Block symbol of a \acs{BPF}]{Block symbol of a \ac{BPF}} +\end{figure}% +\nomenclature[Bl]{\begin{circuitikz}[baseline={(current bounding box.center)}]\draw (0,0) to[bandpass] (2,0);\end{circuitikz}}{Band pass filter} + A \index{band pass filter} \textbf{\acf{BPF}} \begin{itemize} \item lets pass all signals within a \index{band pass filter!pass band} \textbf{pass band} with the \index{band pass filter!bandwidth} \textbf{bandwidth} $\omega_b$ which is centred around the \index{band pass filter!centre frequency} \textbf{centre frequency} $\omega_c$: pass band $||\omega| - \omega_c| < \frac{\omega_b}{2}$ @@ -1867,16 +1894,25 @@ The \ac{BPF} can be seen as a \ac{LPF} frequency-shifted in both positive and ne \caption[Amplitude response of an ideal \acl{BPF}]{Amplitude response of an ideal \ac{BPF}} \end{figure} -\subsubsection{Ideal Band Elimination Filter} +\subsubsection{Ideal Band Stop Filter} + +\begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0,0) to[bandstop] (2,0); + \end{circuitikz} + \caption[Block symbol of a \acs{BSF}]{Block symbol of a \ac{BSF}} +\end{figure}% +\nomenclature[Bl]{\begin{circuitikz}[baseline={(current bounding box.center)}]\draw (0,0) to[bandstop] (2,0);\end{circuitikz}}{Band stop filter} -A \index{band elimination filter} \textbf{\acf{BEF}} +A \index{band stop filter} \textbf{\acf{BSF}} \begin{itemize} - \item blocks all signals within a \index{band eliminations filter!stopband} \textbf{stopband} with the \index{band elimination filter!bandwidth} \textbf{bandwidth} $\omega_b$ which is centred around the \index{band elimination filter!centre frequency} \textbf{centre frequency} $\omega_c$: stopband $||\omega| - \omega_c| < \frac{\omega_b}{2}$ - \item lets pass all signals outside the pass band: \index{band elimination filter!pass band} \textbf{pass band} is everything outside the stopband + \item blocks all signals within a \index{band stop filter!stopband} \textbf{stopband} with the \index{band stop filter!bandwidth} \textbf{bandwidth} $\omega_b$ which is centred around the \index{band stop filter!centre frequency} \textbf{centre frequency} $\omega_c$: stopband $||\omega| - \omega_c| < \frac{\omega_b}{2}$ + \item lets pass all signals outside the pass band: \index{band stop filter!pass band} \textbf{pass band} is everything outside the stopband \end{itemize} \begin{equation} - \underline{H}_{BEF}\left(j \omega\right) = 1 - \underline{H}_{BPF}\left(j \omega\right) = \begin{cases} + \underline{H}_{BSF}\left(j \omega\right) = 1 - \underline{H}_{BPF}\left(j \omega\right) = \begin{cases} 0 & \qquad \text{if } \; ||\omega| - \omega_c| < \frac{\omega_b}{2}, \\ 1 & \qquad \text{else} \end{cases} @@ -1890,7 +1926,7 @@ A \index{band elimination filter} \textbf{\acf{BEF}} width=0.8\linewidth, scale only axis, xlabel={$\omega$}, - ylabel={$A_{BEF}(\omega) = \left|\underline{H}_{BEF}(j \omega)\right|$}, + ylabel={$A_{BSF}(\omega) = \left|\underline{H}_{BSF}(j \omega)\right|$}, %grid style={line width=.6pt, color=lightgray}, %grid=both, grid=none, @@ -1924,7 +1960,7 @@ A \index{band elimination filter} \textbf{\acf{BEF}} \addplot[red, thick] coordinates {(45, 1) (50, 1)}; \end{axis} \end{tikzpicture} - \caption[Amplitude response of an ideal \acl{BEF}]{Amplitude response of an ideal \ac{BEF}} + \caption[Amplitude response of an ideal \acl{BSF}]{Amplitude response of an ideal \ac{BSF}} \end{figure} \subsection{Realizable Filters} @@ -2131,7 +2167,7 @@ The cut-off frequencies or bandwidth, respectively, is defined at those frequenc \end{figure} \end{minipage} -\subsubsection{Band Elimination Filter} +\subsubsection{Band Stop Filter} \begin{minipage}{0.45\linewidth} \begin{figure}[H] @@ -2144,7 +2180,7 @@ The cut-off frequencies or bandwidth, respectively, is defined at those frequenc \draw (0, 0) to[open, v=$u_i(t)$] (0, -4); \draw (4, 0) to[open, v^=$u_o(t)$] (4, -4); \end{circuitikz} - \caption{Real band elimination filter as an electrical network} + \caption{Real band stop filter as an electrical network} \end{figure} \end{minipage} \hfill @@ -2157,7 +2193,7 @@ The cut-off frequencies or bandwidth, respectively, is defined at those frequenc width=0.9\linewidth, scale only axis, xlabel={$\omega$}, - ylabel={$A_{BEF}(\omega) = \left|\underline{H}_{BEF}(j \omega)\right|$}, + ylabel={$A_{BSF}(\omega) = \left|\underline{H}_{BSF}(j \omega)\right|$}, %grid style={line width=.6pt, color=lightgray}, %grid=both, grid=none, @@ -2185,7 +2221,7 @@ The cut-off frequencies or bandwidth, respectively, is defined at those frequenc \addplot[red, dashed] coordinates {(15, 0) (15, 0.707) (25, 0.707) (25, 0)}; \end{axis} \end{tikzpicture} - \caption[Amplitude response of a real \acl{BEF}]{Amplitude response of a real \ac{BEF}. Negative $\omega$-axis omitted.} + \caption[Amplitude response of a real \acl{BSF}]{Amplitude response of a real \ac{BSF}. Negative $\omega$-axis omitted.} \end{figure} \end{minipage} |