718 lines
45 KiB
HTML
718 lines
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HTML
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">List of filters - Matlab Implementation</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orge8f7972">1. Proportional - Integral - Derivative</a>
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<ul>
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<li><a href="#org8aafed7">1.1. Proportional</a></li>
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<li><a href="#org37b43e8">1.2. Integral</a></li>
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<li><a href="#orgfb65936">1.3. Derivative</a></li>
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</ul>
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</li>
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<li><a href="#orgfed2e4e">2. Low Pass</a>
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<ul>
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<li><a href="#orga8faa42">2.1. First Order</a></li>
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<li><a href="#org170a9b0">2.2. Second Order</a></li>
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<li><a href="#orgf8f61f9">2.3. Combine multiple filters</a></li>
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<li><a href="#orge265051">2.4. Nice combination</a></li>
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</ul>
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</li>
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<li><a href="#org7e3ece6">3. High Pass</a>
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<ul>
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<li><a href="#org291d74d">3.1. First Order</a></li>
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<li><a href="#org5d4ea15">3.2. Second Order</a></li>
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<li><a href="#orge71c2dd">3.3. Combine multiple filters</a></li>
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</ul>
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</li>
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<li><a href="#org791f35b">4. Band Pass</a></li>
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<li><a href="#orge054b4d">5. Notch</a></li>
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<li><a href="#orgf94d78a">6. Bump</a></li>
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<li><a href="#org6255d82">7. Chebyshev</a>
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<ul>
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<li><a href="#org2bfe9f5">7.1. Chebyshev Type I</a></li>
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</ul>
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</li>
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<li><a href="#org127884a">8. Lead - Lag</a>
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<ul>
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<li><a href="#orgb58ebc8">8.1. Lead</a></li>
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<li><a href="#org700da69">8.2. Lag</a></li>
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<li><a href="#orgf2125da">8.3. Lead Lag</a></li>
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</ul>
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</li>
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<li><a href="#orga776beb">9. Complementary</a></li>
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<li><a href="#org18d2496">10. Performance Weight</a></li>
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<li><a href="#org2ee1e4e">11. Combine Filters</a>
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<ul>
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<li><a href="#org890b908">11.1. Additive</a></li>
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<li><a href="#org6f10b80">11.2. Multiplicative</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orge8f7972" class="outline-2">
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<h2 id="orge8f7972"><span class="section-number-2">1</span> Proportional - Integral - Derivative</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org8aafed7" class="outline-3">
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<h3 id="org8aafed7"><span class="section-number-3">1.1</span> Proportional</h3>
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</div>
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<div id="outline-container-org37b43e8" class="outline-3">
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<h3 id="org37b43e8"><span class="section-number-3">1.2</span> Integral</h3>
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</div>
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<div id="outline-container-orgfb65936" class="outline-3">
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<h3 id="orgfb65936"><span class="section-number-3">1.3</span> Derivative</h3>
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</div>
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</div>
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<div id="outline-container-orgfed2e4e" class="outline-2">
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<h2 id="orgfed2e4e"><span class="section-number-2">2</span> Low Pass</h2>
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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-orga8faa42" class="outline-3">
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<h3 id="orga8faa42"><span class="section-number-3">2.1</span> First Order</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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\[ H(s) = \frac{1}{1 + s/\omega_0} \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
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H = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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</div>
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<div id="org92da067" class="figure">
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<p><img src="figs/lpf_first_order.png" alt="lpf_first_order.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>First Order Low Pass Filter (<a href="./figs/lpf_first_order.png">png</a>, <a href="./figs/lpf_first_order.pdf">pdf</a>.</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org170a9b0" class="outline-3">
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<h3 id="org170a9b0"><span class="section-number-3">2.2</span> Second Order</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
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xi = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">3</span>;
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H = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>w0<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
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</pre>
|
|
</div>
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|
|
|
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|
<div id="orgd09d719" class="figure">
|
|
<p><img src="figs/lpf_second_order.png" alt="lpf_second_order.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Second Order Low Pass Filter (<a href="./figs/lpf_second_order.png">png</a>, <a href="./figs/lpf_second_order.pdf">pdf</a>.</p>
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</div>
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|
</div>
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</div>
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|
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|
<div id="outline-container-orgf8f61f9" class="outline-3">
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|
<h3 id="orgf8f61f9"><span class="section-number-3">2.3</span> Combine multiple filters</h3>
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|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
\[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \]
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</p>
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|
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|
<div class="org-src-container">
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|
<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
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|
n = <span class="org-highlight-numbers-number">3</span>;
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H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
|
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</pre>
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|
</div>
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|
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|
<div id="org05c14e7" class="figure">
|
|
<p><img src="figs/lpf_multiple_first_order.png" alt="lpf_multiple_first_order.png" />
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|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Combine Multiple First Order Low Pass Filter (<a href="./figs/lpf_multiple_first_order.png">png</a>, <a href="./figs/lpf_multiple_first_order.pdf">pdf</a>.</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orge265051" class="outline-3">
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<h3 id="orge265051"><span class="section-number-3">2.4</span> Nice combination</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
\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
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|
\end{equation}
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<div class="org-src-container">
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<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">2</span>; w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">11</span>; G0 = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">10</span>; G1 = <span class="org-highlight-numbers-number">1000</span>; Gc = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>;
|
|
wL = Gc<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
|
|
|
|
n = <span class="org-highlight-numbers-number">3</span>; w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">9</span>; G0 = <span class="org-highlight-numbers-number">10000</span>; G1 = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; Gc = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>;
|
|
wH = Gc<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>G1<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7e3ece6" class="outline-2">
|
|
<h2 id="org7e3ece6"><span class="section-number-2">3</span> High Pass</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
</div>
|
|
<div id="outline-container-org291d74d" class="outline-3">
|
|
<h3 id="org291d74d"><span class="section-number-3">3.1</span> First Order</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \]
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgc492a24" class="figure">
|
|
<p><img src="figs/hpf_first_order.png" alt="hpf_first_order.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>First Order High Pass Filter (<a href="./figs/hpf_first_order.png">png</a>, <a href="./figs/hpf_first_order.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org5d4ea15" class="outline-3">
|
|
<h3 id="org5d4ea15"><span class="section-number-3">3.2</span> Second Order</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
|
xi = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">3</span>;
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>w0<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>xi<span class="org-type">/</span>w0<span class="org-type">*</span>s <span class="org-type">+</span> s<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>w0<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orga8fa21b" class="figure">
|
|
<p><img src="figs/hpf_second_order.png" alt="hpf_second_order.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Second Order High Pass Filter (<a href="./figs/hpf_second_order.png">png</a>, <a href="./figs/hpf_second_order.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-orge71c2dd" class="outline-3">
|
|
<h3 id="orge71c2dd"><span class="section-number-3">3.3</span> Combine multiple filters</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \]
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
|
n = <span class="org-highlight-numbers-number">3</span>;
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orge98e2ed" class="figure">
|
|
<p><img src="figs/hpf_multiple_first_order.png" alt="hpf_multiple_first_order.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Combine Multiple First Order High Pass Filter (<a href="./figs/hpf_multiple_first_order.png">png</a>, <a href="./figs/hpf_multiple_first_order.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="outline-container-org791f35b" class="outline-2">
|
|
<h2 id="org791f35b"><span class="section-number-2">4</span> Band Pass</h2>
|
|
</div>
|
|
|
|
<div id="outline-container-orge054b4d" class="outline-2">
|
|
<h2 id="orge054b4d"><span class="section-number-2">5</span> Notch</h2>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf94d78a" class="outline-2">
|
|
<h2 id="orgf94d78a"><span class="section-number-2">6</span> Bump</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">4</span>;
|
|
w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>;
|
|
A = <span class="org-highlight-numbers-number">10</span>;
|
|
|
|
a = sqrt<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>A<span class="org-type">^</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span> <span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>A<span class="org-type">^</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>A<span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span> <span class="org-type">-</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
G = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-rainbow-delimiters-depth-3">(</span>w0<span class="org-type">/</span>a<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-rainbow-delimiters-depth-3">(</span>w0<span class="org-type">*</span>a<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
|
|
bodeFig<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">{</span>G<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org6255d82" class="outline-2">
|
|
<h2 id="org6255d82"><span class="section-number-2">7</span> Chebyshev</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
</div>
|
|
<div id="outline-container-org2bfe9f5" class="outline-3">
|
|
<h3 id="org2bfe9f5"><span class="section-number-3">7.1</span> Chebyshev Type I</h3>
|
|
<div class="outline-text-3" id="text-7-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">4</span>; <span class="org-comment">% Order of the filter</span>
|
|
Rp = <span class="org-highlight-numbers-number">3</span>; <span class="org-comment">% Maximum peak-to-peak ripple [dB]</span>
|
|
Wp = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% passband-edge frequency [rad/s]</span>
|
|
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>A,B,C,D<span class="org-rainbow-delimiters-depth-1">]</span> = cheby1<span class="org-rainbow-delimiters-depth-1">(</span>n, Rp, Wp, <span class="org-string">'high', 's'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
H = ss<span class="org-rainbow-delimiters-depth-1">(</span>A, B, C, D<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org175fae7" class="figure">
|
|
<p><img src="figs/cheby1_hpf.png" alt="cheby1_hpf.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>First Order Low Pass Filter (<a href="./figs/cheby1_hpf.png">png</a>, <a href="./figs/cheby1_hpf.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org127884a" class="outline-2">
|
|
<h2 id="org127884a"><span class="section-number-2">8</span> Lead - Lag</h2>
|
|
<div class="outline-text-2" id="text-8">
|
|
</div>
|
|
<div id="outline-container-orgb58ebc8" class="outline-3">
|
|
<h3 id="orgb58ebc8"><span class="section-number-3">8.1</span> Lead</h3>
|
|
<div class="outline-text-3" id="text-8-1">
|
|
<p>
|
|
\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z < \omega_p \]
|
|
</p>
|
|
|
|
<ul class="org-ul">
|
|
<li class="off"><code>[ ]</code> Find a nice parametrisation to be able to specify the center frequency and the phase added</li>
|
|
<li class="off"><code>[ ]</code> Compute also the change in magnitude</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = <span class="org-highlight-numbers-number">2</span>.<span class="org-highlight-numbers-number">0</span>;
|
|
wz = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
|
|
wp = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgbcca770" class="figure">
|
|
<p><img src="figs/lead_filter.png" alt="lead_filter.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Lead Filter (<a href="./figs/lead_filter.png">png</a>, <a href="./figs/lead_filter.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org700da69" class="outline-3">
|
|
<h3 id="org700da69"><span class="section-number-3">8.2</span> Lag</h3>
|
|
<div class="outline-text-3" id="text-8-2">
|
|
<p>
|
|
\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \]
|
|
</p>
|
|
|
|
<ul class="org-ul">
|
|
<li class="off"><code>[ ]</code> Find a nice parametrisation to be able to specify the center frequency and the phase added</li>
|
|
<li class="off"><code>[ ]</code> Compute also the change in magnitude</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = <span class="org-highlight-numbers-number">2</span>.<span class="org-highlight-numbers-number">0</span>;
|
|
wz = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>h; <span class="org-comment">% [rad/s]</span>
|
|
wp = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>h; <span class="org-comment">% [rad/s]</span>
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org4ddf70a" class="figure">
|
|
<p><img src="figs/lag_filter.png" alt="lag_filter.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Lag Filter (<a href="./figs/lag_filter.png">png</a>, <a href="./figs/lag_filter.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf2125da" class="outline-3">
|
|
<h3 id="orgf2125da"><span class="section-number-3">8.3</span> Lead Lag</h3>
|
|
<div class="outline-text-3" id="text-8-3">
|
|
<p>
|
|
\[ H(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p} \frac{1 + s/\omega_z}{1 + s/\omega_p}, \quad \omega_z > \omega_p \]
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wz1 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad/s]</span>
|
|
wp1 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad/s]</span>
|
|
wz2 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">10</span>; <span class="org-comment">% [rad/s]</span>
|
|
wp2 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">20</span>; <span class="org-comment">% [rad/s]</span>
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz1<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp1<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wz2<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span>wp2<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orge8b5da5" class="figure">
|
|
<p><img src="figs/lead_lag_filter.png" alt="lead_lag_filter.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Lead Lag Filter (<a href="./figs/lead_lag_filter.png">png</a>, <a href="./figs/lead_lag_filter.pdf">pdf</a>.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga776beb" class="outline-2">
|
|
<h2 id="orga776beb"><span class="section-number-2">9</span> Complementary</h2>
|
|
</div>
|
|
<div id="outline-container-org18d2496" class="outline-2">
|
|
<h2 id="org18d2496"><span class="section-number-2">10</span> Performance Weight</h2>
|
|
<div class="outline-text-2" id="text-10">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
|
A = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">2</span>;
|
|
M = <span class="org-highlight-numbers-number">5</span>;
|
|
|
|
H = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>M<span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">+</span> w0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span>s <span class="org-type">+</span> w0<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span>A<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="figs/weight_first_order.png" alt="weight_first_order.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2ee1e4e" class="outline-2">
|
|
<h2 id="org2ee1e4e"><span class="section-number-2">11</span> Combine Filters</h2>
|
|
<div class="outline-text-2" id="text-11">
|
|
</div>
|
|
<div id="outline-container-org890b908" class="outline-3">
|
|
<h3 id="org890b908"><span class="section-number-3">11.1</span> Additive</h3>
|
|
<div class="outline-text-3" id="text-11-1">
|
|
<ul class="org-ul">
|
|
<li class="off"><code>[ ]</code> Explain how phase and magnitude combine</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6f10b80" class="outline-3">
|
|
<h3 id="org6f10b80"><span class="section-number-3">11.2</span> Multiplicative</h3>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2019-10-08 mar. 13:27</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|