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diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index f192067..02bc5d9 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -1058,11 +1058,11 @@ The Fourier transform of the Dirac delta function is the frequency-independent c \subsection{Fourier Transforms of Sinusoidal Functions} \begin{equation} - \mathcal{F} \left\{\cos\left(\omega_0 t\right)\right\} = \pi \left( \underbrace{\delta\left(\omega - \omega_0\right)}_{Poistive frequency} + \underbrace{\delta\left(\omega + \omega_0\right)}_{Negative frequency} \right) + \mathcal{F} \left\{\cos\left(\omega_0 t\right)\right\} = \pi \left( \underbrace{\delta\left(\omega - \omega_0\right)}_{\text{Poistive frequency}} + \underbrace{\delta\left(\omega + \omega_0\right)}_{\text{Negative frequency}} \right) \end{equation} \begin{equation} - \mathcal{F} \left\{\sin\left(\omega_0 t\right)\right\} = -j \pi \left( \underbrace{\delta\left(\omega - \omega_0\right)}_{Poistive frequency} - \underbrace{\delta\left(\omega + \omega_0\right)}_{Negative frequency} \right) + \mathcal{F} \left\{\sin\left(\omega_0 t\right)\right\} = -j \pi \left( \underbrace{\delta\left(\omega - \omega_0\right)}_{\text{Poistive frequency}} - \underbrace{\delta\left(\omega + \omega_0\right)}_{\text{Negative frequency}} \right) \end{equation} \begin{itemize} @@ -1355,7 +1355,7 @@ If $\underline{h}(t)$ is constrained to be zero for all $t < 0$ (red curve). It \begin{figure}[H] \centering \begin{tikzpicture} - \draw[->] (-2.2,0) -- (2.2,0) node[below, align=left]{$\Re\left\{\underline{s}\right\}$};
+ \draw[->] (-2.2,0) -- (2.2,0) node[below, align=left]{$\Re\left\{\underline{s}\right\}$}; \draw[->] (0,-2.2) -- (0,2.2) node[left, align=right]{$\Im\left\{\underline{s}\right\}$}; \draw[thick, red] (0,-2) -- (0,2); \draw[dashed, red] (0,1) -- (1,1.2) node[right, align=left, color=red]{$j \omega$}; @@ -1529,13 +1529,52 @@ A \index{low pass filter} \textbf{\acf{LPF}} \end{itemize} \begin{equation} - \underline{H}_{TPF}\left(j \omega\right) = \mathrm{rect}\left(\frac{1}{2} \cdot \frac{\omega}{\omega_o}\right) = \begin{cases} + \underline{H}_{LPF}\left(j \omega\right) = \mathrm{rect}\left(\frac{1}{2} \cdot \frac{\omega}{\omega_o}\right) = \begin{cases} 0 & \qquad \text{if } \; |\omega| > \omega_o, \\ 1 & \qquad \text{if } \; |\omega| < \omega_o \end{cases} \end{equation} -\todo{Amplitude response} +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{LPF}(\omega) = \left|\underline{H}_{LPF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-52, + xmax=52, + ymin=0, + ymax=1.5, + xtick={-20, 0, 20}, + xticklabels={$-\omega_o$, 0, $\omega_o$}, + ytick={0, 0.5, 1}, + ] + \addplot[red, thick] coordinates {(-50, 0) (-20, 0)}; + \addplot[red, dashed] coordinates {(-20, 0)(-20, 1)}; + \addplot[red, thick] coordinates {(-20, 1) (20, 1)}; + \addplot[red, dashed] coordinates {(20, 1) (20, 0)}; + \addplot[red, thick] coordinates {(20, 0) (50, 0)}; + \end{axis} + \end{tikzpicture} + \caption[Amplitude response of an ideal \acl{LPF}]{Amplitude response of an ideal \ac{LPF}} +\end{figure} \subsubsection{Ideal High Pass Filter} @@ -1546,13 +1585,52 @@ A \index{high pass filter} \textbf{\acf{HPF}} \end{itemize} \begin{equation} - \underline{H}_{HPF}\left(j \omega\right) = 1 - \underbrace{\mathrm{rect}\left(\frac{1}{2} \cdot \frac{\omega}{\omega_o}\right)}_{\text{Equals low pass filter}} = \begin{cases} + \underline{H}_{HPF}\left(j \omega\right) = 1 - \underline{H}_{LPF}\left(j \omega\right) = \begin{cases} 1 & \qquad \text{if } \; |\omega| > \omega_o, \\ 0 & \qquad \text{if } \; |\omega| < \omega_o \end{cases} \end{equation} -\todo{Amplitude response} +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{HPF}(\omega) = \left|\underline{H}_{HPF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-52, + xmax=52, + ymin=0, + ymax=1.5, + xtick={-20, 0, 20}, + xticklabels={$-\omega_o$, 0, $\omega_o$}, + ytick={0, 0.5, 1}, + ] + \addplot[red, thick] coordinates {(-50, 1) (-20, 1)}; + \addplot[red, dashed] coordinates {(-20, 1)(-20, 0)}; + \addplot[red, thick] coordinates {(-20, 0) (20, 0)}; + \addplot[red, dashed] coordinates {(20, 0) (20, 1)}; + \addplot[red, thick] coordinates {(20, 1) (50, 1)}; + \end{axis} + \end{tikzpicture} + \caption[Amplitude response of an ideal \acl{HPF}]{Amplitude response of an ideal \ac{HPF}} +\end{figure} \subsubsection{Ideal Band Pass Filter} @@ -1563,15 +1641,63 @@ A \index{band pass filter} \textbf{\acf{BPF}} \end{itemize} \begin{equation} - \underline{H}_{HPF}\left(j \omega\right) = \underbrace{\mathcal{F}\left\{\cos\left(\omega_c t\right)\right\} * \underbrace{\mathrm{rect}\left(\frac{1}{2} \cdot \frac{\omega}{\omega_c}\right)}_{\text{Equals low pass filter}}}_{\text{``Two-sided frequency shift''}} = \begin{cases} - 1 & \qquad \text{if } \; ||\omega| - \omega_c| < \frac{\omega_b}{2}, \\ - 0 & \qquad \text{else} - \end{cases} + \begin{split} + \underline{H}_{BPF}\left(j \omega\right) &= \underbrace{\mathcal{F}\left\{\frac{1}{\pi} \cos\left(\omega_c t\right)\right\} * \left.\underline{H}_{LPF}\left(j \omega\right)\right|_{\omega_o = \frac{1}{2} \omega_b}}_{\text{``Two-sided frequency shift''}} \\ + &= \left( \delta\left(\omega - \omega_c\right) + \delta\left(\omega + \omega_c\right) \right) * \mathrm{rect}\left(\frac{\omega}{\omega_b}\right) \\ + &= \mathrm{rect}\left(\frac{\omega - \omega_0}{\omega_b}\right) + \mathrm{rect}\left(\frac{\omega + \omega_0}{\omega_b}\right) \\ + &= \begin{cases} + 1 & \qquad \text{if } \; ||\omega| - \omega_c| < \frac{\omega_b}{2}, \\ + 0 & \qquad \text{else} + \end{cases} + \end{split} \end{equation} The \ac{BPF} can be seen as a \ac{LPF} frequency-shifted in both positive and negative direction by the centre frequency $\omega_c$. This special ``two-sided frequency shift'' will later be called \emph{modulation}. -\todo{Amplitude response} +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.8\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{BPF}(\omega) = \left|\underline{H}_{BPF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-52, + xmax=52, + ymin=0, + ymax=1.5, + xtick={-45, -32.5, -20, 0, 20, 32.5, 45}, + xticklabels={$-\omega_c - \frac{\omega_b}{2}$, $-\omega_c$, $-\omega_c + \frac{\omega_b}{2}$, 0, $\omega_c - \frac{\omega_b}{2}$, $\omega_c$, $\omega_c + \frac{\omega_b}{2}$}, + ytick={0, 0.5, 1}, + ] + \addplot[red, thick] coordinates {(-50, 0) (-45, 0)}; + \addplot[red, dashed] coordinates {(-45, 0)(-45, 1)}; + \addplot[red, thick] coordinates {(-45, 1) (-20, 1)}; + \addplot[red, dashed] coordinates {(-20, 1) (-20, 0)}; + \addplot[red, thick] coordinates {(-20, 0) (20, 0)}; + \addplot[red, dashed] coordinates {(20, 0) (20, 1)}; + \addplot[red, thick] coordinates {(20, 1) (45, 1)}; + \addplot[red, dashed] coordinates {(45, 1) (45, 0)}; + \addplot[red, thick] coordinates {(45, 0) (50, 0)}; + \end{axis} + \end{tikzpicture} + \caption[Amplitude response of an ideal \acl{BPF}]{Amplitude response of an ideal \ac{BPF}} +\end{figure} \subsubsection{Ideal Band Elimination Filter} @@ -1582,13 +1708,56 @@ A \index{band elimination filter} \textbf{\acf{BEF}} \end{itemize} \begin{equation} - \underline{H}_{HPF}\left(j \omega\right) = 1 - \underbrace{\left(\mathcal{F}\left\{\cos\left(\omega_c t\right)\right\} * \mathrm{rect}\left(\frac{1}{2} \cdot \frac{\omega}{\omega_c}\right)\right)}_{\text{Equals band pass filter}} = \begin{cases} + \underline{H}_{BEF}\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} \end{equation} -\todo{Amplitude response} +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.8\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{BEF}(\omega) = \left|\underline{H}_{BEF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-52, + xmax=52, + ymin=0, + ymax=1.5, + xtick={-45, -32.5, -20, 0, 20, 32.5, 45}, + xticklabels={$-\omega_c - \frac{\omega_b}{2}$, $-\omega_c$, $-\omega_c + \frac{\omega_b}{2}$, 0, $\omega_c - \frac{\omega_b}{2}$, $\omega_c$, $\omega_c + \frac{\omega_b}{2}$}, + ytick={0, 0.5, 1}, + ] + \addplot[red, thick] coordinates {(-50, 1) (-45, 1)}; + \addplot[red, dashed] coordinates {(-45, 1)(-45, 0)}; + \addplot[red, thick] coordinates {(-45, 0) (-20, 0)}; + \addplot[red, dashed] coordinates {(-20, 0) (-20, 1)}; + \addplot[red, thick] coordinates {(-20, 1) (20, 1)}; + \addplot[red, dashed] coordinates {(20, 1) (20, 0)}; + \addplot[red, thick] coordinates {(20, 0) (45, 0)}; + \addplot[red, dashed] coordinates {(45, 0) (45, 1)}; + \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}} +\end{figure} \subsection{Realizable Filters} @@ -1599,5 +1768,258 @@ Realizable filters \item Their phase response $\varphi(\omega)$ is not constantly zero. \end{itemize} +The cut-off frequencies or bandwidth, respectively, is defined at those frequencies, where the output signal power has dropped to $0.5$ in relation to the peak value. For voltages, this means $\sqrt{2} = 0.707$ of the peak voltage\footnote{$\sqrt{2} = 0.707$ is the crest factor for sinusoidal voltage signals.}. In decibel, this is \SI{-3}{dB}. + +\begin{itemize} + \item The order of the filter is the number of their memory components (capacitors, inductances, memory cells). The order corresponds to the highest exponent of $j \omega$ (or $\underline{s}$) in the transfer function of the filter. + \item A higher filter order yields a filter whose shape comes closer to the ideal filter (steeper slopes). + \item The filters depicted below are of low order and therefore very ``poor'' in performance. +\end{itemize} + +\subsubsection{Low Pass Filter} + +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[R, l=$R$, o-] ++(2,0) to[short, *-o] ++(2,0); + \draw (2, 0) to[C, l=$C$, -*] ++(0,-2); + \draw (0, -2) to[short, o-o] ++(4,0); + + \draw (0, 0) to[open, v=$u_i(t)$] (0, -2); + \draw (4, 0) to[open, v^=$u_o(t)$] (4, -2); + \end{circuitikz} + \caption{Real low pass filter as an electrical network} + \end{figure} + + Transfer function of this example: + \begin{equation} + \underline{H}\left(j \omega\right) = \frac{1}{j \omega RC + 1} + \end{equation} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{LPF}(\omega) = \left|\underline{H}_{LPF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-32, + xmax=32, + ymin=0, + ymax=1.2, + xtick={-5, 0, 5}, + xticklabels={$-\omega_o$, 0, $\omega_o$}, + ytick={0, 0.5, 0.707, 1}, + ] + % RC = 0.2 + \addplot[blue, thick, domain=-30:30, samples=100] plot (\x, {sqrt( 1 / ((0.2 * \x)^2 + 1) )}); + + \addplot[red, dashed] coordinates {(-5, 0) (-5, 0.707) (5, 0.707) (5, 0)}; + \end{axis} + \end{tikzpicture} + \caption[Amplitude response of a real \acl{LPF}]{Amplitude response of a real \ac{LPF}} + \end{figure} +\end{minipage} + +\subsubsection{High Pass Filter} + +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[C, l=$C$, o-] ++(2,0) to[short, *-o] ++(2,0); + \draw (2, 0) to[R, l=$R$, -*] ++(0,-2); + \draw (0, -2) to[short, o-o] ++(4,0); + + \draw (0, 0) to[open, v=$u_i(t)$] (0, -2); + \draw (4, 0) to[open, v^=$u_o(t)$] (4, -2); + \end{circuitikz} + \caption{Real high pass filter as an electrical network} + \end{figure} + + Transfer function of this example: + \begin{equation} + \underline{H}\left(j \omega\right) = \frac{j \omega RC}{j \omega RC + 1} + \end{equation} + + \textit{Remark:} The filter has one zero $\underline{s}_0 = 0 + j0$. The zero can be directly seen in the amplitude response: $A_{HPF}(\omega = 0) = 0$. The pole $\underline{s}_\infty = \frac{1}{RC} + j 0$ is not directly visible, because only the $j \omega$-axis of $\underline{s}$ is plotted. However, it indirectly determines the shape of the curve. +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{HPF}(\omega) = \left|\underline{H}_{HPF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-32, + xmax=32, + ymin=0, + ymax=1.2, + xtick={-5, 0, 5}, + xticklabels={$-\omega_o$, 0, $\omega_o$}, + ytick={0, 0.5, 0.707, 1}, + ] + % RC = 0.2 + \addplot[blue, thick, domain=-30:30, samples=100] plot (\x, {sqrt( (0.2 * \x)^2 / ((0.2 * \x)^2 + 1) )}); + + \addplot[red, dashed] coordinates {(-5, 0) (-5, 0.707) (5, 0.707) (5, 0)}; + \end{axis} + \end{tikzpicture} + \caption[Amplitude response of a real \acl{HPF}]{Amplitude response of a real \ac{HPF}} + \end{figure} +\end{minipage} + +\subsubsection{Band Pass Filter} + +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[C, l=$C$, o-] ++(2,0) to[L, l=$L$, -o] ++(2,0); + \draw (0, -2) to[short, o-o] ++(4,0); + + \draw (0, 0) to[open, v=$u_i(t)$] (0, -2); + \draw (4, 0) to[open, v^=$u_o(t)$] (4, -2); + \end{circuitikz} + \caption{Real band pass filter as an electrical network} + \end{figure} + \end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{BPF}(\omega) = \left|\underline{H}_{BPF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=west, + }, + xmin=0, + xmax=52, + ymin=0, + ymax=1.2, + xtick={0, 20}, + xticklabels={0, $\omega_c$}, + ytick={0, 0.5, 0.707, 1}, + ] + \addplot[blue, thick, domain=0:50, samples=100] plot (\x, {sqrt( 1 / ((0.2 * (\x - 20))^2 + 1) ) }); + + \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{BPF}]{Amplitude response of a real \ac{BPF}. Negative $\omega$-axis omitted.} + \end{figure} +\end{minipage} + +\subsubsection{Band Elimination Filter} + +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{circuitikz} + \draw (0, 0) to[short, o-o] ++(4,0); + \draw (2, 0) to[C, l=$C$, *-] ++(0,-2) to[L, l=$L$, -*] ++(0,-2); + \draw (0, -4) to[short, o-o] ++(4,0); + + \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} + \end{figure} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$A_{BEF}(\omega) = \left|\underline{H}_{BEF}(j \omega)\right|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=west, + }, + xmin=0, + xmax=52, + ymin=0, + ymax=1.2, + xtick={0, 20}, + xticklabels={0, $\omega_c$}, + ytick={0, 0.5, 0.707, 1}, + ] + \addplot[blue, thick, domain=0:50, samples=100] plot (\x, {sqrt( (0.2 * (\x - 20))^2 / ((0.2 * (\x - 20))^2 + 1) )}); + + \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.} + \end{figure} +\end{minipage} + \printbibliography[heading=subbibliography] \end{refsection} |