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< a accesskey = "h" href = "./index.html" > UP < / a >
|
< a accesskey = "H" href = "./index.html" > HOME < / a >
< / div > < div id = "content" >
< h1 class = "title" > List of filters - Matlab Implementation< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< 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 >
< li > < a href = "#org37b43e8" > 1.2. Integral< / a > < / li >
< li > < a href = "#orgfb65936" > 1.3. Derivative< / a > < / li >
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< / ul >
< / 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 >
< li > < a href = "#org170a9b0" > 2.2. Second Order< / a > < / li >
< li > < a href = "#orgf8f61f9" > 2.3. Combine multiple filters< / a > < / li >
< li > < a href = "#orge265051" > 2.4. Nice combination< / a > < / li >
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< / ul >
< / 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 >
< li > < a href = "#org5d4ea15" > 3.2. Second Order< / a > < / li >
< li > < a href = "#orge71c2dd" > 3.3. Combine multiple filters< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org791f35b" > 4. Band Pass< / a > < / li >
< li > < a href = "#orge054b4d" > 5. Notch< / a > < / li >
< li > < a href = "#orgf94d78a" > 6. Bump< / a > < / li >
< li > < a href = "#org6255d82" > 7. Chebyshev< / a >
< ul >
< li > < a href = "#org2bfe9f5" > 7.1. Chebyshev Type I< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org127884a" > 8. Lead - Lag< / a >
< ul >
< li > < a href = "#orgb58ebc8" > 8.1. Lead< / a > < / li >
< li > < a href = "#org700da69" > 8.2. Lag< / a > < / li >
< li > < a href = "#orgf2125da" > 8.3. Lead Lag< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orga776beb" > 9. Complementary< / a > < / li >
< li > < a href = "#org18d2496" > 10. Performance Weight< / a > < / li >
< li > < a href = "#org2ee1e4e" > 11. Combine Filters< / a >
< ul >
< li > < a href = "#org890b908" > 11.1. Additive< / a > < / li >
< li > < a href = "#org6f10b80" > 11.2. Multiplicative< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-orge8f7972" class = "outline-2" >
< 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" >
< / div >
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< div id = "outline-container-org8aafed7" class = "outline-3" >
< h3 id = "org8aafed7" > < span class = "section-number-3" > 1.1< / span > Proportional< / h3 >
< / div >
< div id = "outline-container-org37b43e8" class = "outline-3" >
< h3 id = "org37b43e8" > < span class = "section-number-3" > 1.2< / span > Integral< / h3 >
< / div >
< div id = "outline-container-orgfb65936" class = "outline-3" >
< h3 id = "orgfb65936" > < span class = "section-number-3" > 1.3< / span > Derivative< / h3 >
< / div >
< / div >
< div id = "outline-container-orgfed2e4e" class = "outline-2" >
< h2 id = "orgfed2e4e" > < span class = "section-number-2" > 2< / span > Low Pass< / h2 >
< div class = "outline-text-2" id = "text-2" >
< / div >
< div id = "outline-container-orga8faa42" class = "outline-3" >
< h3 id = "orga8faa42" > < span class = "section-number-3" > 2.1< / span > First Order< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< p >
\[ H(s) = \frac{1}{1 + s/\omega_0} \]
< / 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-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 > ;
< / pre >
< / 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" / >
< / p >
< 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 >
< / div >
< / div >
< / div >
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< div id = "outline-container-org170a9b0" class = "outline-3" >
< h3 id = "org170a9b0" > < span class = "section-number-3" > 2.2< / span > Second Order< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
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< p >
\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
< / 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-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 > ;
< / pre >
< / div >
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< div id = "orgd09d719" class = "figure" >
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< 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 >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgf8f61f9" class = "outline-3" >
< h3 id = "orgf8f61f9" > < span class = "section-number-3" > 2.3< / span > Combine multiple filters< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
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< p >
\[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \]
< / 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-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;
< / pre >
< / div >
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< div id = "org05c14e7" class = "figure" >
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< p > < img src = "figs/lpf_multiple_first_order.png" alt = "lpf_multiple_first_order.png" / >
< / 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 >
< / div >
< / div >
< / div >
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< div id = "outline-container-orge265051" class = "outline-3" >
< 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
\end{equation}
< div class = "org-src-container" >
< 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 >
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< / div >
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< / 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" >
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< 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 >
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< div id = "orgc492a24" class = "figure" >
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< 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 >
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< 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" >
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< 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 >
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< div id = "orga8fa21b" class = "figure" >
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< 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 >
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< 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" >
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< 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 >
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< div id = "orge98e2ed" class = "figure" >
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< 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 >
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< 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 >
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< / div >
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< 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 >
2019-08-17 10:50:06 +02:00
< / div >
2020-10-26 09:43:09 +01:00
< 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" >
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< / div >
2020-10-26 09:43:09 +01:00
< 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" >
2019-08-17 10:50:06 +02:00
< 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 >
2020-10-26 09:43:09 +01:00
< div id = "org175fae7" class = "figure" >
2019-08-17 10:50:06 +02:00
< 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 >
2020-10-26 09:43:09 +01:00
< 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 >
2019-08-17 10:50:06 +02:00
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-10-26 09:43:09 +01:00
< p class = "date" > Created: 2019-10-08 mar. 13:27< / p >
2019-08-17 10:50:06 +02:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
