diff --git a/index.html b/index.html index df70045..e3fcfe1 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + List of filters - Matlab Implementation @@ -26,78 +26,78 @@
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List of filters - Matlab Implementation

Table of Contents

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1 Low Pass

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1 Low Pass

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1.1 First Order Low Pass Filter

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1.1 First Order Low Pass Filter

\[ H(s) = \frac{1}{1 + s/\omega_0} \] @@ -129,7 +129,7 @@ H = 1/(1 + s -

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filter_low_pass_first_order.png

@@ -137,8 +137,8 @@ H = 1/(1 + s -
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1.2 Second Order

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1.2 Second Order

\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] @@ -172,15 +172,15 @@ H = 1/(1 + 2 -

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filter_low_pass_second_order.png

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1.3 Combine multiple first order filters

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1.3 Combine multiple first order filters

\[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \] @@ -198,7 +198,7 @@ H = (1/(1 + s -

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filter_low_pass_first_order_add.png

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2 High Pass

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2 High Pass

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2.1 First Order

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2.1 First Order

\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \] @@ -243,15 +243,15 @@ H = (s/w0)/(1 -

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filter_high_pass_first_order.png

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2.2 Second Order

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2.2 Second Order

\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \] @@ -277,15 +277,15 @@ H = (s^2/w0 -

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filter_high_pass_second_order.png

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2.3 Combine multiple filters

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2.3 Combine multiple filters

\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \] @@ -303,7 +303,7 @@ H = ((s/w0)/(1 -

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filter_high_pass_first_order_add.png

@@ -311,21 +311,21 @@ H = ((s/w0)/(1
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3 Band Pass

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3 Band Pass

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3.1 Second Order

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3.1 Second Order

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4 Notch

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4 Notch

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4.1 Second Order

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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} @@ -353,13 +353,13 @@ H = (s^2 + 2 -
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filter_notch_xi.png

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filter_notch_gc.png

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5 Chebyshev

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5 Chebyshev

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5.1 Chebyshev Type I

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5.1 Chebyshev Type I

n = 4; % Order of the filter
@@ -387,12 +387,12 @@ H = ss(A, B, C, D);
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6 Lead - Lag

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6 Lead - Lag

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6.1 Lead

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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 @@ -412,7 +412,7 @@ Characteristics:
  • the low frequency gain is \(1\)
  • the high frequency gain is \(a\)
    • -
  • the phase lead at \(\omega_c\) is equal to (Figure 10): +
  • the phase lead at \(\omega_c\) is equal to (Figure 10): \[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \]
@@ -428,21 +428,21 @@ H = (1 + s/(wc -
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filter_lead.png

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filter_lead_effect_a_phase.png

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6.2 Lag

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6.2 Lag

\begin{equation} H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1 @@ -462,7 +462,7 @@ Characteristics:
  • the low frequency gain is increased by a factor \(a\)
  • the high frequency gain is \(1\) (unchanged)
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  • the phase lag at \(\omega_c\) is equal to (Figure 12): +
  • the phase lag at \(\omega_c\) is equal to (Figure 12): \[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \]
@@ -478,13 +478,13 @@ H = (wc*sqrt(a) + s)
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filter_lag.png

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filter_lag_effect_a_phase.png

@@ -492,16 +492,16 @@ H = (wc*sqrt(a) + s)
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7 Complementary

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7 Complementary

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8 Performance Weight

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8 Performance Weight

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8.1 Nice combination

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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 @@ -520,8 +520,8 @@ wH = Gc*(((G1/Gc) -
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8.2 Alternative

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8.2 Alternative

w0 = 2*pi; % [rad/s]
@@ -533,7 +533,7 @@ H = (s/sqrt(M) + w0)
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weight_first_order.png

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9 Combine Filters

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9 Combine Filters

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9.1 Additive

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9.1 Additive

  • [ ] Explain how phase and magnitude combine
    • @@ -554,12 +554,12 @@ H = (s/sqrt(M) + w0)
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9.2 Multiplicative

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9.2 Multiplicative

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10 Filters representing noise

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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 \(G(s)\). @@ -577,8 +577,8 @@ And the root mean square (RMS) of \(n(t)\) is:

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10.1 First Order Low Pass Filter

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10.1 First Order Low Pass Filter

\[ G(s) = \frac{g_0}{1 + \frac{s}{\omega_c}} \] @@ -646,7 +646,7 @@ Thus, if a sensor is said to have a RMS noise of \(\sigma = 10 nm\ rms\) over a

Author: Dehaeze Thomas

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Created: 2020-10-26 lun. 14:03

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Created: 2020-10-29 jeu. 10:09

diff --git a/index.org b/index.org index 60393ca..4a3ef13 100644 --- a/index.org +++ b/index.org @@ -4,8 +4,8 @@ #+EMAIL: dehaeze.thomas@gmail.com #+AUTHOR: Dehaeze Thomas -#+HTML_LINK_HOME: ./index.html -#+HTML_LINK_UP: ./index.html +#+HTML_LINK_HOME: ../index.html +#+HTML_LINK_UP: ../index.html #+HTML_HEAD: #+HTML_HEAD: