1 Proportional - Integral - Derivative
+
+
-
-
-1 Low Pass
-
-
-1.1 Proportional
-
-
-
-1.2 Integral
-
-
-1.3 Derivative
-
-
2 Low Pass
-
-
-
-
2.1 First Order
-
+
+
1.1 First Order Low Pass Filter
+\[ H(s) = \frac{1}{1 + s/\omega_0} \]
-
-
-w0 = 2*pi; % [rad/s] ++Parameters: +
+
-
+
- \(\omega_0\): cut-off frequency in [rad/s] +
+Characteristics: +
+-
+
- Low frequency gain of \(1\) +
- Roll-off equals to -20 dB/dec +
+Matlab code: +
+
+
-
w0 = 2*pi; % Cut-off Frequency [rad/s] + +H = 1/(1 + s/w0);
+
-
2.2 Second Order
-
+
+
1.2 Second Order
+\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
-
-
-
-w0 = 2*pi; % [rad/s] -xi = 0.3; ++Parameters: +
+
-
+
- \(\omega_0\): +
- \(\xi\): Damping ratio +
+Characteristics: +
+-
+
- Low frequency gain: 1 +
- High frequency roll off: - 40 dB/dec +
+Matlab code: +
+
+
-
w0 = 2*pi; % Cut-off frequency [rad/s] +xi = 0.3; % Damping Ratio + +H = 1/(1 + 2*xi/w0*s + s^2/w0^2);
+
-
2.3 Combine multiple filters
-
+
+
+1.3 Combine multiple first order filters
+
+
+
2 High Pass
+
+
+
+
2.1 First Order
+
+
+
++\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \] +
-
-
+
+2.4 Nice combination
-
+
++Parameters: +
+-
+
- \(\omega_0\): cut-off frequency in [rad/s] +
+Characteristics: +
+-
+
- High frequency gain of \(1\) +
- Low frequency slow of +20 dB/dec +
+Matlab code: +
+
+
+
+
+w0 = 2*pi; % Cut-off frequency [rad/s] + +H = (s/w0)/(1 + s/w0); ++
+
+
+
+
+
+2.2 Second Order
+
+
++\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] +
+ ++Parameters: +
+-
+
- \(\omega_0\): +
- \(\xi\): Damping ratio +
+Matlab code: +
+
+
+
+
+w0 = 2*pi; % [rad/s] +xi = 0.3; + +H = (s^2/w0^2)/(1 + 2*xi/w0*s + s^2/w0^2); ++
+
+
+
+
+2.3 Combine multiple filters
+
+
++\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \] +
+ ++Matlab code: +
+
+
+
+
+w0 = 2*pi; % [rad/s] +n = 3; + +H = ((s/w0)/(1 + s/w0))^n; ++
+
+
+
+
+
+3 Band Pass
+
+
+
+
+3.1 Second Order
+
+
+
+4 Notch
+
+
+
+
+4.1 Second Order
+
+\begin{equation}
+ \frac{s^2 + 2 g_c \xi \omega_n s + \omega_n^2}{s^2 + 2 \xi \omega_n s + \omega_n^2}
+\end{equation}
+
+
++Parameters: +
+-
+
- \(\omega_n\): frequency of the notch +
- \(g_c\): gain at the notch frequency +
- \(\xi\): damping ratio (notch width) +
+Matlab code: +
+
+
+
+
+gc = 0.02; +xi = 0.1; +wn = 2*pi; + +H = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); ++
+
+
+
+
+
+
+
+
+
+
+5 Chebyshev
+
+
+
+
+5.1 Chebyshev Type I
+
+
+
+
+n = 4; % Order of the filter +Rp = 3; % Maximum peak-to-peak ripple [dB] +Wp = 2*pi; % passband-edge frequency [rad/s] + +[A,B,C,D] = cheby1(n, Rp, Wp, 'high', 's'); +H = ss(A, B, C, D); ++
+
+
+6 Lead - Lag
+
+
+
+
+
+6.1 Lead
+
+\begin{equation}
+ H(s) = \frac{1 + \frac{s}{w_c/\sqrt{a}}}{1 + \frac{s}{w_c \sqrt{a}}}, \quad a > 1
+\end{equation}
+
+
++Parameters: +
+-
+
- \(\omega_c\): frequency at which the phase lead is maximum +
- \(a\): parameter to adjust the phase lead, also impacts the high frequency gain +
+Characteristics: +
+-
+
- the low frequency gain is \(1\) +
- the high frequency gain is \(a\) +
- the phase lead at \(\omega_c\) is equal to (Figure 10): +\[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \] +
+Matlab code: +
+
+
+
+
+a = 0.6; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi; % Frequency with the maximum phase lead [rad/s] + +H = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); ++
+
+
+
+
+
+
+
+
+
+6.2 Lag
+
+\begin{equation}
+ H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1
+\end{equation}
+
+
++Parameters: +
+-
+
- \(\omega_c\): frequency at which the phase lag is maximum +
- \(a\): parameter to adjust the phase lag, also impacts the low frequency gain +
+Characteristics: +
+-
+
- the low frequency gain is increased by a factor \(a\) +
- the high frequency gain is \(1\) (unchanged) +
- the phase lag at \(\omega_c\) is equal to (Figure 12): +\[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \] +
+Matlab code: +
+
+
+
+
+a = 0.6; % Amount of phase lag / width of the phase lag / high frequency gain +wc = 2*pi; % Frequency with the maximum phase lag [rad/s] + +H = (wc*sqrt(a) + s)/(wc/sqrt(a) + s); ++
+
+
+
+
+
+
+
+
+
+
+7 Complementary
+
+
-
-8 Performance Weight
+
+
+
+
-8.1 Nice combination
+
\begin{equation}
W(s) = G_c * \left(\frac{\frac{1}{\omega_0}\sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}}{\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{\left(\frac{G_\infty}{G_c}\right)^{\frac{2}{n}} - 1}} s + 1}\right)^n
\end{equation}
-
-
-n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; Gc = 1/2; -wL = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n; +n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; Gc = 1/2; +wL = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n; -n = 3; w0 = 2*pi*9; G0 = 10000; G1 = 0.1; Gc = 1/2; -wH = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n; --
-
-
-
-
-3 High Pass
-
-
-
-
-
-
-3.1 First Order
- -
-
-
-
-
-
-3.2 Second Order
- -
-
-
-
-
-4 Band Pass
-
-
-
-5 Notch
-
-
-6 Bump
-
-
-
n = 4; -w0 = 2*pi; -A = 10; - -a = sqrt(2*A^(2/n) - 1 + 2*A^(1/n)*sqrt(A^(2/n) - 1)); -G = ((1 + s/(w0/a))*(1 + s/(w0*a))/(1 + s/w0)^2)^n; -bodeFig({G}) +n = 3; w0 = 2*pi*9; G0 = 10000; G1 = 0.1; Gc = 1/2; +wH = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n;
-
-
-7 Chebyshev
-
-
-
-
-
-
8 Lead - Lag
-
-
-
-
-
-8.1 Lead
-
-
--\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z < \omega_p \] -
- --
-
[ ]Find a nice parametrisation to be able to specify the center frequency and the phase added
-[ ]Compute also the change in magnitude
-
-
-
-
-
-h = 2.0; -wz = 2*pi/h; % [rad/s] -wp = 2*pi*h; % [rad/s] - -H = (1 + s/wz)/(1 + s/wp); --
-
-
-
-8.2 Lag
+
+
-
-8.2 Alternative
-
--\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \] -
- --
-
[ ]Find a nice parametrisation to be able to specify the center frequency and the phase added
-[ ]Compute also the change in magnitude
-
-
-
-h = 2.0; -wz = 2*pi*h; % [rad/s] -wp = 2*pi/h; % [rad/s] +w0 = 2*pi; % [rad/s] +A = 1e-2; +M = 5; -H = (1 + s/wz)/(1 + s/wp); +H = (s/sqrt(M) + w0)^2/(s + w0*sqrt(A))^2;
-
-
-8.3 Lead Lag
-
-
--\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p} \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \] -
- -
-
-
-
-
-wz1 = 2*pi*1; % [rad/s] -wp1 = 2*pi*0.1; % [rad/s] -wz2 = 2*pi*10; % [rad/s] -wp2 = 2*pi*20; % [rad/s] - -H = (1 + s/wz1)/(1 + s/wp1)*(1 + s/wz2)/(1 + s/wp2); --
-
-
-9 Complementary
-
-
10 Performance Weight
-
-
-
-
-
-
-
-w0 = 2*pi; % [rad/s] -A = 1e-2; -M = 5; - -H = (s/sqrt(M) + w0)^2/(s + w0*sqrt(A))^2; --
+
-
-
11 Combine Filters
-
-
11.1 Additive
-
+
+>
#+end_src
-* Proportional - Integral - Derivative
-** Proportional
-
-** Integral
-
-** Derivative
-
* Low Pass
-** First Order
+** First Order Low Pass Filter
\[ H(s) = \frac{1}{1 + s/\omega_0} \]
+Parameters:
+- $\omega_0$: cut-off frequency in [rad/s]
+
+Characteristics:
+- Low frequency gain of $1$
+- Roll-off equals to -20 dB/dec
+
+Matlab code:
#+begin_src matlab
- w0 = 2*pi; % [rad/s]
+ w0 = 2*pi; % Cut-off Frequency [rad/s]
H = 1/(1 + s/w0);
#+end_src
#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
+ freqs = logspace(-2, 2, 1000);
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/lpf_first_order.csv');
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ plot(freqs, abs(squeeze(freqresp(H, freqs, 'Hz'))), 'k-');
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ hold off;
+ xline(1, '--', 'LineWidth', 2);
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ plot(freqs, 180/pi*angle(squeeze(freqresp(H, freqs, 'Hz'))), 'k-');
+ set(gca,'xscale','log');
+ yticks(-90:15:0); ylim([-90 0]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
#+end_src
-#+begin_src latex :file lpf_first_order.pdf :tangle figs/lpf_first_order.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- ymin=1e-2,
- ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_first_order.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_first_order.csv};
- \end{axis}
- \end{tikzpicture}
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_low_pass_first_order.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
-#+name: fig:lpf_first_order
-#+caption: First Order Low Pass Filter ([[./figs/lpf_first_order.png][png]], [[./figs/lpf_first_order.pdf][pdf]].
+#+name: fig:filter_low_pass_first_order
+#+caption:
#+RESULTS:
-[[file:figs/lpf_first_order.png]]
+[[file:figs/filter_low_pass_first_order.png]]
** Second Order
\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
+Parameters:
+- $\omega_0$:
+- $\xi$: Damping ratio
+
+Characteristics:
+- Low frequency gain: 1
+- High frequency roll off: - 40 dB/dec
+
+Matlab code:
#+begin_src matlab
- w0 = 2*pi; % [rad/s]
- xi = 0.3;
+ w0 = 2*pi; % Cut-off frequency [rad/s]
+ xi = 0.3; % Damping Ratio
H = 1/(1 + 2*xi/w0*s + s^2/w0^2);
#+end_src
#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
+ xis = [0.1, 0.3, 0.5, 0.7, 0.9];
+ Hs = {zeros(length(xis), 1)};
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/lpf_second_order.csv');
+ for xi_i = 1:length(xis)
+ Hs(xi_i) = {1/(1 + 2*xis(xi_i)/w0*s + s^2/w0^2)};
+ end
#+end_src
-#+begin_src latex :file lpf_second_order.pdf :tangle figs/lpf_second_order.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
+#+begin_src matlab :exports none
+ freqs = logspace(-2, 2, 1000);
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- ymin=1e-2,
- ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_second_order.csv};
- \end{axis}
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for xi_i = 1:length(xis)
+ plot(freqs, abs(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$\\xi = %.1f$', xis(xi_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'northeast');
+ hold off;
+ xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off');
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_second_order.csv};
- \end{axis}
- \end{tikzpicture}
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for xi_i = 1:length(xis)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(-180:30:0); ylim([-180 0]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
#+end_src
-#+name: fig:lpf_second_order
-#+caption: Second Order Low Pass Filter ([[./figs/lpf_second_order.png][png]], [[./figs/lpf_second_order.pdf][pdf]].
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_low_pass_second_order.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_low_pass_second_order
+#+caption:
#+RESULTS:
-[[file:figs/lpf_second_order.png]]
+[[file:figs/filter_low_pass_second_order.png]]
-** Combine multiple filters
+** Combine multiple first order filters
\[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \]
+Matlab code:
#+begin_src matlab
- w0 = 2*pi; % [rad/s]
- n = 3;
+ w0 = 2*pi; % Cut-off frequency [rad/s]
+ n = 3; % Filter order
H = (1/(1 + s/w0))^n;
#+end_src
#+begin_src matlab :exports none
+ ns = [1, 2, 3, 4];
+ Hs = {zeros(length(ns), 1)};
+
+ for n_i = 1:length(ns)
+ Hs(n_i) = {1/(1 + s/w0)^ns(n_i)};
+ end
+#+end_src
+
+#+begin_src matlab :exports none
+ freqs = logspace(-2, 2, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for n_i = 1:length(ns)
+ plot(freqs, abs(squeeze(freqresp(Hs{n_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$n = %i$', ns(n_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'northeast');
+ hold off;
+ xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off');
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for n_i = 1:length(ns)
+ plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hs{n_i}, freqs, 'Hz')))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(-360:45:0); ylim([-360 0]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_low_pass_first_order_add.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_low_pass_first_order_add
+#+caption:
+#+RESULTS:
+[[file:figs/filter_low_pass_first_order_add.png]]
+
+* High Pass
+** First Order
+\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \]
+
+Parameters:
+- $\omega_0$: cut-off frequency in [rad/s]
+
+Characteristics:
+- High frequency gain of $1$
+- Low frequency slow of +20 dB/dec
+
+Matlab code:
+#+begin_src matlab
+ w0 = 2*pi; % Cut-off frequency [rad/s]
+
+ H = (s/w0)/(1 + s/w0);
+#+end_src
+
+#+begin_src matlab :exports none
+ freqs = logspace(-2, 2, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ plot(freqs, abs(squeeze(freqresp(H, freqs, 'Hz'))), 'k-');
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ hold off;
+ xline(1, '--', 'LineWidth', 2);
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ plot(freqs, 180/pi*angle(squeeze(freqresp(H, freqs, 'Hz'))), 'k-');
+ set(gca,'xscale','log');
+ yticks(0:15:90); ylim([0, 90]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_high_pass_first_order.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_high_pass_first_order
+#+caption:
+#+RESULTS:
+[[file:figs/filter_high_pass_first_order.png]]
+
+** Second Order
+\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
+
+Parameters:
+- $\omega_0$:
+- $\xi$: Damping ratio
+
+Matlab code:
+#+begin_src matlab
+ w0 = 2*pi; % [rad/s]
+ xi = 0.3;
+
+ H = (s^2/w0^2)/(1 + 2*xi/w0*s + s^2/w0^2);
+#+end_src
+
+#+begin_src matlab :exports none
+ xis = [0.1, 0.3, 0.5, 0.7, 0.9];
+ Hs = {zeros(length(xis), 1)};
+
+ for xi_i = 1:length(xis)
+ Hs(xi_i) = {(s^2/w0^2)/(1 + 2*xis(xi_i)/w0*s + s^2/w0^2)};
+ end
+
+ freqs = logspace(-2, 2, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for xi_i = 1:length(xis)
+ plot(freqs, abs(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$\\xi = %.1f$', xis(xi_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'northeast');
+ hold off;
+ xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off');
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for xi_i = 1:length(xis)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(0:30:180); ylim([0 180]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_high_pass_second_order.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_high_pass_second_order
+#+caption:
+#+RESULTS:
+[[file:figs/filter_high_pass_second_order.png]]
+
+** Combine multiple filters
+\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \]
+
+Matlab code:
+#+begin_src matlab
+ w0 = 2*pi; % [rad/s]
+ n = 3;
+
+ H = ((s/w0)/(1 + s/w0))^n;
+#+end_src
+
+#+begin_src matlab :exports none
+ ns = [1, 2, 3, 4];
+ Hs = {zeros(length(ns), 1)};
+
+ for n_i = 1:length(ns)
+ Hs(n_i) = {((s/w0)/(1 + s/w0))^ns(n_i)};
+ end
+#+end_src
+
+#+begin_src matlab :exports none
+ freqs = logspace(-2, 2, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for n_i = 1:length(ns)
+ plot(freqs, abs(squeeze(freqresp(Hs{n_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$n = %i$', ns(n_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'northeast');
+ hold off;
+ xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off');
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for n_i = 1:length(ns)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{n_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(-180:45:180); ylim([-180 180]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_0'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_high_pass_first_order_add.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_high_pass_first_order_add
+#+caption:
+#+RESULTS:
+[[file:figs/filter_high_pass_first_order_add.png]]
+
+* TODO Band Pass
+** Second Order
+
+* Notch
+** Second Order
+\begin{equation}
+ \frac{s^2 + 2 g_c \xi \omega_n s + \omega_n^2}{s^2 + 2 \xi \omega_n s + \omega_n^2}
+\end{equation}
+
+Parameters:
+- $\omega_n$: frequency of the notch
+- $g_c$: gain at the notch frequency
+- $\xi$: damping ratio (notch width)
+
+Matlab code:
+#+begin_src matlab
+ gc = 0.02;
+ xi = 0.1;
+ wn = 2*pi;
+
+ H = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);
+#+end_src
+
+#+begin_src matlab :exports none
+ xi = 0.2;
+ gcs = [0.02, 0.1, 0.5];
+ Hs = {zeros(length(gcs), 1)};
+
+ for g_i = 1:length(gcs)
+ Hs(g_i) = {(s^2 + 2*gcs(g_i)*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2)};
+ end
+
freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/lpf_multiple_first_order.csv');
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for g_i = 1:length(gcs)
+ plot(freqs, abs(squeeze(freqresp(Hs{g_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$g_c = %.2f$', gm(g_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'southeast');
+ hold off;
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for g_i = 1:length(gcs)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{g_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(-90:30:90); ylim([-90 90]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
#+end_src
-#+begin_src latex :file lpf_multiple_first_order.pdf :tangle figs/lpf_multiple_first_order.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- ymin=1e-2,
- ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_multiple_first_order.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlpf_multiple_first_order.csv};
- \end{axis}
- \end{tikzpicture}
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_notch_xi.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
-#+name: fig:lpf_multiple_first_order
-#+caption: Combine Multiple First Order Low Pass Filter ([[./figs/lpf_multiple_first_order.png][png]], [[./figs/lpf_multiple_first_order.pdf][pdf]].
+#+name: fig:filter_notch_xi
+#+caption:
+#+RESULTS:
+[[file:figs/filter_notch_xi.png]]
+
+#+begin_src matlab :exports none
+ xis = [0.05, 0.1, 0.5];
+ gc = 0.1;
+ Hs = {zeros(length(xis), 1)};
+
+ for xi_i = 1:length(xis)
+ Hs(xi_i) = {(s^2 + 2*gc*xis(xi_i)*wn*s + wn^2)/(s^2 + 2*xis(xi_i)*wn*s + wn^2)};
+ end
+
+ freqs = logspace(-1, 1, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for xi_i = 1:length(xis)
+ plot(freqs, abs(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$\\xi = %.2f$', xis(xi_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'southeast');
+ hold off;
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for xi_i = 1:length(xis)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{xi_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(-90:30:90); ylim([-90 90]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_notch_gc.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_notch_gc
+#+caption:
+#+RESULTS:
+[[file:figs/filter_notch_gc.png]]
+
+* TODO Chebyshev
+** Chebyshev Type I
+
+#+begin_src matlab
+ n = 4; % Order of the filter
+ Rp = 3; % Maximum peak-to-peak ripple [dB]
+ Wp = 2*pi; % passband-edge frequency [rad/s]
+
+ [A,B,C,D] = cheby1(n, Rp, Wp, 'high', 's');
+ H = ss(A, B, C, D);
+#+end_src
+
+* Lead - Lag
+** Lead
+\begin{equation}
+ H(s) = \frac{1 + \frac{s}{w_c/\sqrt{a}}}{1 + \frac{s}{w_c \sqrt{a}}}, \quad a > 1
+\end{equation}
+
+Parameters:
+- $\omega_c$: frequency at which the phase lead is maximum
+- $a$: parameter to adjust the phase lead, also impacts the high frequency gain
+
+Characteristics:
+- the low frequency gain is $1$
+- the high frequency gain is $a$
+- the phase lead at $\omega_c$ is equal to (Figure [[fig:filter_lead_effect_a_phase]]):
+ \[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \]
+
+Matlab code:
+#+begin_src matlab
+ a = 0.6; % Amount of phase lead / width of the phase lead / high frequency gain
+ wc = 2*pi; % Frequency with the maximum phase lead [rad/s]
+
+ H = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+#+end_src
+
+#+begin_src matlab :exports none
+ wc = 2*pi;
+ as = [1.2, 2, 5];
+ Hs = {zeros(length(as), 1)};
+
+ for a_i = 1:length(as)
+ Hs(a_i) = {(1 + s/(wc/sqrt(as(a_i))))/(1 + s/(wc*sqrt(as(a_i))))};
+ end
+
+ freqs = logspace(-2, 2, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for a_i = 1:length(as)
+ plot(freqs, abs(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$a = %.1f$', as(a_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'northwest');
+ hold off;
+ xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off');
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for a_i = 1:length(as)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(0:15:60); ylim([0 60]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_c'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'top', 'LabelOrientation', 'horizontal');
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_lead.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_lead
+#+caption:
+#+RESULTS:
+[[file:figs/filter_lead.png]]
+
+#+begin_src matlab :exports none
+ as = logspace(0, 1, 100);
+
+ figure;
+ plot(as, 180/pi*(atan(sqrt(as)) - atan(1./sqrt(as))))
+ xlabel('$a$'); ylabel('Phase added at $\omega_c$ [deg]');
+ yticks(0:15:60); ylim([0, 60]);
+ xticks(1:1:10); xlim([as(1), as(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_lead_effect_a_phase.pdf', 'width', 'normal', 'height', 'normal');
+#+end_src
+
+#+name: fig:filter_lead_effect_a_phase
+#+caption:
+#+RESULTS:
+[[file:figs/filter_lead_effect_a_phase.png]]
+
+** Lag
+\begin{equation}
+ H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1
+\end{equation}
+
+Parameters:
+- $\omega_c$: frequency at which the phase lag is maximum
+- $a$: parameter to adjust the phase lag, also impacts the low frequency gain
+
+Characteristics:
+- the low frequency gain is increased by a factor $a$
+- the high frequency gain is $1$ (unchanged)
+- the phase lag at $\omega_c$ is equal to (Figure [[fig:filter_lag_effect_a_phase]]):
+ \[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \]
+
+Matlab code:
+#+begin_src matlab
+ a = 0.6; % Amount of phase lag / width of the phase lag / high frequency gain
+ wc = 2*pi; % Frequency with the maximum phase lag [rad/s]
+
+ H = (wc*sqrt(a) + s)/(wc/sqrt(a) + s);
+#+end_src
+
+#+begin_src matlab :exports none
+ wc = 2*pi;
+ as = [1.2, 2, 5];
+ Hs = {zeros(length(as), 1)};
+
+ for a_i = 1:length(as)
+ Hs(a_i) = {(wc*sqrt(as(a_i)) + s)/(wc/sqrt(as(a_i)) + s)};
+ end
+
+ freqs = logspace(-2, 2, 1000);
+
+ figure;
+ tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ % Magnitude
+ ax1 = nexttile;
+ hold on;
+ for a_i = 1:length(as)
+ plot(freqs, abs(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$a = %.1f$', as(a_i)));
+ end
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+ axis padded
+ legend('location', 'southwest');
+ hold off;
+ xline(1, '--', 'LineWidth', 2, 'HandleVisibility', 'off');
+
+ % Phase
+ ax2 = nexttile;
+ hold on;
+ for a_i = 1:length(as)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Hs{a_i}, freqs, 'Hz'))), '-');
+ end
+ set(gca,'xscale','log');
+ yticks(-60:15:0); ylim([-60 0]);
+ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+ hold off;
+ xline(1, '--', {'\omega_c'}, 'LineWidth', 2, 'LabelVerticalAlignment', 'bottom', 'LabelOrientation', 'horizontal');
+
+
+ linkaxes([ax1,ax2],'x');
+ xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_lag.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:filter_lag
+#+caption:
+#+RESULTS:
+[[file:figs/filter_lag.png]]
+
+#+begin_src matlab :exports none
+ as = logspace(0, 1, 100);
+
+ figure;
+ plot(as, 180/pi*(atan(1./sqrt(as)) - atan(sqrt(as))))
+ xlabel('$a$'); ylabel('Phase added at $\omega_c$ [deg]');
+ yticks(-60:15:0); ylim([-60, 0]);
+ xticks(1:1:10); xlim([as(1), as(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+ exportFig('figs/filter_lag_effect_a_phase.pdf', 'width', 'normal', 'height', 'normal');
+#+end_src
+
+#+name: fig:filter_lag_effect_a_phase
+#+caption:
+#+RESULTS:
+[[file:figs/filter_lag_effect_a_phase.png]]
+
+* TODO Complementary
+
+* TODO Performance Weight
** Nice combination
\begin{equation}
W(s) = G_c * \left(\frac{\frac{1}{\omega_0}\sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}}{\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{\left(\frac{G_\infty}{G_c}\right)^{\frac{2}{n}} - 1}} s + 1}\right)^n
@@ -364,735 +783,8 @@
wH = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n;
#+end_src
-* High Pass
-** First Order
-\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \]
-
-#+begin_src matlab
- w0 = 2*pi; % [rad/s]
-
- H = (s/w0)/(1 + s/w0);
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/hpf_first_order.csv');
-#+end_src
-
-#+begin_src latex :file hpf_first_order.pdf :tangle figs/hpf_first_order.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- ymin=1e-2,
- ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_first_order.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_first_order.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:hpf_first_order
-#+caption: First Order High Pass Filter ([[./figs/hpf_first_order.png][png]], [[./figs/hpf_first_order.pdf][pdf]].
-#+RESULTS:
-[[file:figs/hpf_first_order.png]]
-
-
-** Second Order
-\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
-
-#+begin_src matlab
- w0 = 2*pi; % [rad/s]
- xi = 0.3;
-
- H = (s^2/w0^2)/(1 + 2*xi/w0*s + s^2/w0^2);
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/hpf_second_order.csv');
-#+end_src
-
-#+begin_src latex :file hpf_second_order.pdf :tangle figs/hpf_second_order.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- ymin=1e-2,
- ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_second_order.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_second_order.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:hpf_second_order
-#+caption: Second Order High Pass Filter ([[./figs/hpf_second_order.png][png]], [[./figs/hpf_second_order.pdf][pdf]].
-#+RESULTS:
-[[file:figs/hpf_second_order.png]]
-
-
-** Combine multiple filters
-\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \]
-
-#+begin_src matlab
- w0 = 2*pi; % [rad/s]
- n = 3;
-
- H = ((s/w0)/(1 + s/w0))^n;
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/hpf_multiple_first_order.csv');
-#+end_src
-
-#+begin_src latex :file hpf_multiple_first_order.pdf :tangle figs/hpf_multiple_first_order.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- ymin=1e-2,
- ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_multiple_first_order.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/mathpf_multiple_first_order.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:hpf_multiple_first_order
-#+caption: Combine Multiple First Order High Pass Filter ([[./figs/hpf_multiple_first_order.png][png]], [[./figs/hpf_multiple_first_order.pdf][pdf]].
-#+RESULTS:
-[[file:figs/hpf_multiple_first_order.png]]
-
-
-
-* Band Pass
-
-* Notch
-
-* Bump
-#+begin_src matlab
- n = 4;
- w0 = 2*pi;
- A = 10;
-
- a = sqrt(2*A^(2/n) - 1 + 2*A^(1/n)*sqrt(A^(2/n) - 1));
- G = ((1 + s/(w0/a))*(1 + s/(w0*a))/(1 + s/w0)^2)^n;
- bodeFig({G})
-#+end_src
-
-
-* Chebyshev
-** Chebyshev Type I
-
-#+begin_src matlab
- n = 4; % Order of the filter
- Rp = 3; % Maximum peak-to-peak ripple [dB]
- Wp = 2*pi; % passband-edge frequency [rad/s]
-
- [A,B,C,D] = cheby1(n, Rp, Wp, 'high', 's');
- H = ss(A, B, C, D);
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/cheby1_hpf.csv');
-#+end_src
-
-#+begin_src latex :file cheby1_hpf.pdf :tangle figs/cheby1_hpf.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- % ymin=1e-2,
- % ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matcheby1_hpf.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-270,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matcheby1_hpf.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:cheby1_hpf
-#+caption: First Order Low Pass Filter ([[./figs/cheby1_hpf.png][png]], [[./figs/cheby1_hpf.pdf][pdf]].
-#+RESULTS:
-[[file:figs/cheby1_hpf.png]]
-
-* Lead - Lag
-** Lead
-\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z < \omega_p \]
-
-- [ ] Find a nice parametrisation to be able to specify the center frequency and the phase added
-- [ ] Compute also the change in magnitude
-
-#+begin_src matlab
- h = 2.0;
- wz = 2*pi/h; % [rad/s]
- wp = 2*pi*h; % [rad/s]
-
- H = (1 + s/wz)/(1 + s/wp);
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/lead_filter.csv');
-#+end_src
-
-#+begin_src latex :file lead_filter.pdf :tangle figs/lead_filter.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- % ymin=1e-2,
- % ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_filter.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-90,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_filter.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:lead_filter
-#+caption: Lead Filter ([[./figs/lead_filter.png][png]], [[./figs/lead_filter.pdf][pdf]].
-#+RESULTS:
-[[file:figs/lead_filter.png]]
-
-** Lag
-\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \]
-
-- [ ] Find a nice parametrisation to be able to specify the center frequency and the phase added
-- [ ] Compute also the change in magnitude
-
-#+begin_src matlab
- h = 2.0;
- wz = 2*pi*h; % [rad/s]
- wp = 2*pi/h; % [rad/s]
-
- H = (1 + s/wz)/(1 + s/wp);
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-1, 1, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/lag_filter.csv');
-#+end_src
-
-#+begin_src latex :file lag_filter.pdf :tangle figs/lag_filter.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- % ymin=1e-2,
- % ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlag_filter.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.1,
- xmax=10,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-90,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlag_filter.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:lag_filter
-#+caption: Lag Filter ([[./figs/lag_filter.png][png]], [[./figs/lag_filter.pdf][pdf]].
-#+RESULTS:
-[[file:figs/lag_filter.png]]
-
-** Lead Lag
-\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p} \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \]
-
-#+begin_src matlab
- wz1 = 2*pi*1; % [rad/s]
- wp1 = 2*pi*0.1; % [rad/s]
- wz2 = 2*pi*10; % [rad/s]
- wp2 = 2*pi*20; % [rad/s]
-
- H = (1 + s/wz1)/(1 + s/wp1)*(1 + s/wz2)/(1 + s/wp2);
-#+end_src
-
-#+begin_src matlab :exports none
- freqs = logspace(-2, 2, 1000);
- resp = squeeze(freqresp(H, freqs, 'Hz'));
- Ha = abs(resp);
- Hp = 180/pi*phase(resp);
-
- T = table(freqs', Ha, Hp, 'VariableNames', {'freqs', 'amplitude', 'phase'});
- writetable(T,'mat/lead_lag_filter.csv');
-#+end_src
-
-#+begin_src latex :file lead_lag_filter.pdf :tangle figs/lead_lag_filter.tex :exports results
- \setlength\fwidth{8cm}
- \setlength\fheight{6cm}
-
- \definecolor{mycolor1}{rgb}{0.00000,0.44700,0.74100}%
- \definecolor{mycolor2}{rgb}{0.85000,0.32500,0.09800}%
-
- \begin{tikzpicture}
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0.5\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.01,
- xmax=100,
- xticklabels={{}},
- xminorticks=true,
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymode=log,
- % ymin=1e-2,
- % ymax=1e1,
- % ytick={1e-14, 1e-12, 1e-10, 1e-8, 1e-6},
- % yticklabels={{1e-14}, {}, {1e-10}, {}, {1e-6}},
- yminorticks=true,
- ylabel={Amplitude},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids,
- yminorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=amplitude, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_lag_filter.csv};
- \end{axis}
-
- \begin{axis}[%
- width=\fwidth,
- height=0.45\fheight,
- at={(0\fwidth,0\fheight)},
- scale only axis,
- separate axis lines,
- every outer x axis line/.append style={black},
- every x tick label/.append style={font=\color{black}},
- every x tick/.append style={black},
- xmode=log,
- xmin=0.01,
- xmax=100,
- xminorticks=true,
- xlabel={Frequency [Hz]},
- every outer y axis line/.append style={black},
- every y tick label/.append style={font=\color{black}},
- every y tick/.append style={black},
- ymin=-90,
- ymax=90,
- ytick={-360, -270, -180, -90, 0, 90},
- ylabel={Phase [deg]},
- axis background/.style={fill=white},
- xmajorgrids,
- xminorgrids,
- ymajorgrids
- ]
- \addplot[color=black, mark=none]
- table [x=freqs, y=phase, col sep=comma] {/home/thomas/Cloud/thesis/matlab/filters/matlead_lag_filter.csv};
- \end{axis}
- \end{tikzpicture}
-#+end_src
-
-#+name: fig:lead_lag_filter
-#+caption: Lead Lag Filter ([[./figs/lead_lag_filter.png][png]], [[./figs/lead_lag_filter.pdf][pdf]].
-#+RESULTS:
-[[file:figs/lead_lag_filter.png]]
-
-* Complementary
-* Performance Weight
+** Alternative
#+begin_src matlab
w0 = 2*pi; % [rad/s]
A = 1e-2;
@@ -1160,12 +852,12 @@
#+RESULTS:
[[file:figs/weight_first_order.png]]
-* Combine Filters
+* TODO Combine Filters
** Additive
- [ ] Explain how phase and magnitude combine
** Multiplicative
-* Filters representing noise
+* TODO Filters representing noise
Let's consider a noise $n$ that is shaped from a white-noise $\tilde{n}$ with unitary PSD ($\Phi_\tilde{n}(\omega) = 1$) using a transfer function $G(s)$.
The PSD of $n$ is then:
\[ \Phi_n(\omega) = |G(j\omega)|^2 \Phi_{\tilde{n}}(\omega) = |G(j\omega)|^2 \]
+
diff --git a/index.org b/index.org
index 881e656..60393ca 100644
--- a/index.org
+++ b/index.org
@@ -46,310 +46,729 @@
<9 Combine Filters
+
+
+
+
-9.1 Additive
+[ ]Explain how phase and magnitude combine
-
+11.2 Multiplicative
+
+
+9.2 Multiplicative
+
+
10 Filters representing noise
+
+
+
++Let’s consider a noise \(n\) that is shaped from a white-noise \(\tilde{n}\) with unitary PSD (\(\Phi_\tilde{n}(\omega) = 1\)) using a transfer function \(G(s)\). +The PSD of \(n\) is then: +\[ \Phi_n(\omega) = |G(j\omega)|^2 \Phi_{\tilde{n}}(\omega) = |G(j\omega)|^2 \] +
+ ++The PSD \(\Phi_n(\omega)\) is expressed in \(\text{unit}^2/\text{Hz}\). +
+ ++And the root mean square (RMS) of \(n(t)\) is: +\[ \sigma_n = \sqrt{\int_{0}^{\infty} \Phi_n(\omega) d\omega} \] +
+
+
10.1 First Order Low Pass Filter
+
+
+\[ G(s) = \frac{g_0}{1 + \frac{s}{\omega_c}} \] +
+ +
+
+
+g0 = 1; % Noise Density in unit/sqrt(Hz) +wc = 1; % Cut-Off frequency [rad/s] + +G = g0/(1 + s/wc); + +% Frequency vector [Hz] +freqs = logspace(-3, 3, 1000); + +% PSD of n in [unit^2/Hz] +Phi_n = abs(squeeze(freqresp(G, freqs, 'Hz'))).^2; + +% RMS value of n in [unit, rms] +sigma_n = sqrt(trapz(freqs, Phi_n)) ++
+\[ \sigma = \frac{1}{2} g_0 \sqrt{\omega_c} \] +with: +
+-
+
- \(g_0\) the Noise Density of \(n\) in \(\text{unit}/\sqrt{Hz}\) +
- \(\omega_c\) the bandwidth over which the noise is located, in rad/s +
- \(\sigma\) the rms noise +
+If the cut-off frequency is to be expressed in Hz: +\[ \sigma = \frac{1}{2} g_0 \sqrt{2\pi f_c} = \sqrt{\frac{\pi}{2}} g_0 \sqrt{f_c} \] +
+ + ++Thus, if a sensor is said to have a RMS noise of \(\sigma = 10 nm\ rms\) over a bandwidth of \(\omega_c = 100 rad/s\), we can estimated the noise density of the sensor to be (supposing a first order low pass filter noise shape): +\[ g_0 = \frac{2 \sigma}{\sqrt{\omega_c}} \quad \left[ m/\sqrt{Hz} \right] \] +
+ +
+
+
+2*10e-9/sqrt(100) ++
+2e-09 ++ + +
+
+
+6*0.5*20e-12*sqrt(2*pi*100) ++
+1.504e-09 ++