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< title > Bank of Linear Filters - Matlab< / title >
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< meta name = "author" content = "Dehaeze Thomas" / >
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< h1 class = "title" > Bank of Linear Filters - Matlab< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org90266e3" > 1. Low Pass< / a >
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< ul >
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< li > < a href = "#org4291475" > 1.1. First Order Low Pass Filter< / a > < / li >
< li > < a href = "#org88f266d" > 1.2. Second Order< / a > < / li >
< li > < a href = "#org1ee2123" > 1.3. Combine multiple first order filters< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd7f4531" > 2. High Pass< / a >
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< ul >
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< li > < a href = "#org5859621" > 2.1. First Order< / a > < / li >
< li > < a href = "#org988fb36" > 2.2. Second Order< / a > < / li >
< li > < a href = "#orgd1ae949" > 2.3. Combine multiple filters< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org0871964" > 3. Band Pass< / a >
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< ul >
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< li > < a href = "#org020bd3f" > 3.1. Second Order< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org226333f" > 4. Notch< / a >
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< ul >
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< li > < a href = "#org5d6517b" > 4.1. Second Order< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgff2aff5" > 5. Chebyshev< / a >
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< ul >
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< li > < a href = "#orgb7b715f" > 5.1. Chebyshev Type I< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orge46a59c" > 6. Lead - Lag< / a >
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< ul >
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< li > < a href = "#org6336fc3" > 6.1. Lead< / a > < / li >
< li > < a href = "#org3797972" > 6.2. Lag< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org5e67a97" > 7. Complementary< / a > < / li >
< li > < a href = "#org74fba97" > 8. Performance Weight< / a >
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< ul >
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< li > < a href = "#org64d3111" > 8.1. Nice combination< / a > < / li >
< li > < a href = "#org978d37b" > 8.2. Alternative< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org043cd0b" > 9. Combine Filters< / a >
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< ul >
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< li > < a href = "#orgbb8d210" > 9.1. Additive< / a > < / li >
< li > < a href = "#orgc6eac8f" > 9.2. Multiplicative< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org2616858" > 10. Filters representing noise< / a >
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< ul >
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< li > < a href = "#orgb12ce21" > 10.1. First Order Low Pass Filter< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org90266e3" class = "outline-2" >
< h2 id = "org90266e3" > < span class = "section-number-2" > 1< / span > Low Pass< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< / div >
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< div id = "outline-container-org4291475" class = "outline-3" >
< h3 id = "org4291475" > < span class = "section-number-3" > 1.1< / span > First Order Low Pass Filter< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< p >
\[ H(s) = \frac{1}{1 + s/\omega_0} \]
< / p >
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< p >
Parameters:
< / p >
< ul class = "org-ul" >
< li > \(\omega_0\): cut-off frequency in [rad/s]< / li >
< / ul >
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< p >
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Characteristics:
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< / p >
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< ul class = "org-ul" >
< li > Low frequency gain of \(1\)< / li >
< li > Roll-off equals to -20 dB/dec< / li >
< / ul >
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< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % Cut-off Frequency [rad/s]< / span >
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H = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > w0);
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< / pre >
< / div >
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< div id = "org3a9bd55" class = "figure" >
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< p > < img src = "figs/filter_low_pass_first_order.png" alt = "filter_low_pass_first_order.png" / >
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org88f266d" class = "outline-3" >
< h3 id = "org88f266d" > < span class = "section-number-3" > 1.2< / span > Second Order< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< p >
\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
< / p >
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< p >
Parameters:
< / p >
< ul class = "org-ul" >
< li > \(\omega_0\):< / li >
< li > \(\xi\): Damping ratio< / li >
< / ul >
< p >
Characteristics:
< / p >
< ul class = "org-ul" >
< li > Low frequency gain: 1< / li >
< li > High frequency roll off: - 40 dB/dec< / li >
< / ul >
< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % Cut-off frequency [rad/s]< / span >
xi = 0.3; < span class = "org-comment" > % Damping Ratio< / span >
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H = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > 2< 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 > 2< span class = "org-type" > /< / span > w0< span class = "org-type" > ^< / span > 2);
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< / pre >
< / div >
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< div id = "org666b377" class = "figure" >
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< p > < img src = "figs/filter_low_pass_second_order.png" alt = "filter_low_pass_second_order.png" / >
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1ee2123" class = "outline-3" >
< h3 id = "org1ee2123" > < span class = "section-number-3" > 1.3< / span > Combine multiple first order filters< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< p >
\[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \]
< / p >
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< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % Cut-off frequency [rad/s]< / span >
n = 3; < span class = "org-comment" > % Filter order< / span >
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H = (1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > w0))< span class = "org-type" > ^< / span > n;
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< / pre >
< / div >
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< div id = "orgb9222b5" class = "figure" >
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< p > < img src = "figs/filter_low_pass_first_order_add.png" alt = "filter_low_pass_first_order_add.png" / >
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< / p >
< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd7f4531" class = "outline-2" >
< h2 id = "orgd7f4531" > < span class = "section-number-2" > 2< / span > High 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-org5859621" class = "outline-3" >
< h3 id = "org5859621" > < 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 >
\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \]
< / p >
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< p >
Parameters:
< / p >
< ul class = "org-ul" >
< li > \(\omega_0\): cut-off frequency in [rad/s]< / li >
< / ul >
< p >
Characteristics:
< / p >
< ul class = "org-ul" >
< li > High frequency gain of \(1\)< / li >
< li > Low frequency slow of +20 dB/dec< / li >
< / ul >
< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % Cut-off frequency [rad/s]< / span >
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H = (s< span class = "org-type" > /< / span > w0)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > w0);
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< / pre >
< / div >
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< div id = "org422aab2" class = "figure" >
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< p > < img src = "figs/filter_high_pass_first_order.png" alt = "filter_high_pass_first_order.png" / >
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org988fb36" class = "outline-3" >
< h3 id = "org988fb36" > < 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 >
\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]
< / p >
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< p >
Parameters:
< / p >
< ul class = "org-ul" >
< li > \(\omega_0\):< / li >
< li > \(\xi\): Damping ratio< / li >
< / ul >
< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % [rad/s]< / span >
xi = 0.3;
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H = (s< span class = "org-type" > ^< / span > 2< span class = "org-type" > /< / span > w0< span class = "org-type" > ^< / span > 2)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > 2< 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 > 2< span class = "org-type" > /< / span > w0< span class = "org-type" > ^< / span > 2);
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< / pre >
< / div >
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< div id = "org66dc7df" class = "figure" >
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< p > < img src = "figs/filter_high_pass_second_order.png" alt = "filter_high_pass_second_order.png" / >
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd1ae949" class = "outline-3" >
< h3 id = "orgd1ae949" > < 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" >
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< p >
\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \]
< / p >
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< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % [rad/s]< / span >
n = 3;
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H = ((s< span class = "org-type" > /< / span > w0)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > w0))< span class = "org-type" > ^< / span > n;
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< / pre >
< / div >
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< div id = "orgc70858b" class = "figure" >
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< p > < img src = "figs/filter_high_pass_first_order_add.png" alt = "filter_high_pass_first_order_add.png" / >
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< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0871964" class = "outline-2" >
< h2 id = "org0871964" > < span class = "section-number-2" > 3< / span > Band Pass< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< / div >
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< div id = "outline-container-org020bd3f" class = "outline-3" >
< h3 id = "org020bd3f" > < span class = "section-number-3" > 3.1< / span > Second Order< / h3 >
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< / div >
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< / div >
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< div id = "outline-container-org226333f" class = "outline-2" >
< h2 id = "org226333f" > < span class = "section-number-2" > 4< / span > Notch< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< / div >
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< div id = "outline-container-org5d6517b" class = "outline-3" >
< h3 id = "org5d6517b" > < span class = "section-number-3" > 4.1< / span > Second Order< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
\begin{equation}
\frac{s^2 + 2 g_c \xi \omega_n s + \omega_n^2}{s^2 + 2 \xi \omega_n s + \omega_n^2}
\end{equation}
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< p >
Parameters:
< / p >
< ul class = "org-ul" >
< li > \(\omega_n\): frequency of the notch< / li >
< li > \(g_c\): gain at the notch frequency< / li >
< li > \(\xi\): damping ratio (notch width)< / li >
< / ul >
< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > gc = 0.02;
xi = 0.1;
wn = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ;
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H = (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > gm< span class = "org-type" > *< / span > xi< span class = "org-type" > *< / span > wn< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > wn< span class = "org-type" > ^< / span > 2)< span class = "org-type" > /< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > xi< span class = "org-type" > *< / span > wn< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > wn< span class = "org-type" > ^< / span > 2);
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< / pre >
< / div >
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< div id = "orgbb197d6" class = "figure" >
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< p > < img src = "figs/filter_notch_xi.png" alt = "filter_notch_xi.png" / >
< / p >
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< / div >
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< div id = "org5b3ff85" class = "figure" >
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< p > < img src = "figs/filter_notch_gc.png" alt = "filter_notch_gc.png" / >
< / p >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgff2aff5" class = "outline-2" >
< h2 id = "orgff2aff5" > < span class = "section-number-2" > 5< / span > Chebyshev< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< / div >
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< div id = "outline-container-orgb7b715f" class = "outline-3" >
< h3 id = "orgb7b715f" > < span class = "section-number-3" > 5.1< / span > Chebyshev Type I< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > n = 4; < span class = "org-comment" > % Order of the filter< / span >
Rp = 3; < span class = "org-comment" > % Maximum peak-to-peak ripple [dB]< / span >
Wp = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % passband-edge frequency [rad/s]< / span >
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[A,B,C,D] = cheby1(n, Rp, Wp, < span class = "org-string" > 'high'< / span > , < span class = "org-string" > 's'< / span > );
H = ss(A, B, C, D);
< / pre >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orge46a59c" class = "outline-2" >
< h2 id = "orge46a59c" > < span class = "section-number-2" > 6< / span > Lead - Lag< / h2 >
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< div class = "outline-text-2" id = "text-6" >
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< / div >
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< div id = "outline-container-org6336fc3" class = "outline-3" >
< h3 id = "org6336fc3" > < span class = "section-number-3" > 6.1< / span > Lead< / h3 >
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< div class = "outline-text-3" id = "text-6-1" >
\begin{equation}
H(s) = \frac{1 + \frac{s}{w_c/\sqrt{a}}}{1 + \frac{s}{w_c \sqrt{a}}}, \quad a > 1
\end{equation}
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< p >
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Parameters:
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< / p >
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< ul class = "org-ul" >
< li > \(\omega_c\): frequency at which the phase lead is maximum< / li >
< li > \(a\): parameter to adjust the phase lead, also impacts the high frequency gain< / li >
< / ul >
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< p >
Characteristics:
< / p >
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< ul class = "org-ul" >
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< li > the low frequency gain is \(1\)< / li >
< li > the high frequency gain is \(a\)< / li >
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< li > the phase lead at \(\omega_c\) is equal to (Figure < a href = "#orgefcd37b" > 10< / a > ):
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\[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \]< / li >
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< / ul >
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< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > a = 0.6; < span class = "org-comment" > % Amount of phase lead / width of the phase lead / high frequency gain< / span >
wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % Frequency with the maximum phase lead [rad/s]< / span >
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H = (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > sqrt(a)))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > sqrt(a)));
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< / pre >
< / div >
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< div id = "orga3b4f13" class = "figure" >
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< p > < img src = "figs/filter_lead.png" alt = "filter_lead.png" / >
< / p >
< / div >
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< div id = "orgefcd37b" class = "figure" >
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< p > < img src = "figs/filter_lead_effect_a_phase.png" alt = "filter_lead_effect_a_phase.png" / >
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3797972" class = "outline-3" >
< h3 id = "org3797972" > < span class = "section-number-3" > 6.2< / span > Lag< / h3 >
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< div class = "outline-text-3" id = "text-6-2" >
\begin{equation}
H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1
\end{equation}
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< p >
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Parameters:
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< / p >
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< ul class = "org-ul" >
< li > \(\omega_c\): frequency at which the phase lag is maximum< / li >
< li > \(a\): parameter to adjust the phase lag, also impacts the low frequency gain< / li >
< / ul >
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< p >
Characteristics:
< / p >
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< ul class = "org-ul" >
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< li > the low frequency gain is increased by a factor \(a\)< / li >
< li > the high frequency gain is \(1\) (unchanged)< / li >
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< li > the phase lag at \(\omega_c\) is equal to (Figure < a href = "#orgbed77d1" > 12< / a > ):
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\[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \]< / li >
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< / ul >
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< p >
Matlab code:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > a = 0.6; < span class = "org-comment" > % Amount of phase lag / width of the phase lag / high frequency gain< / span >
wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % Frequency with the maximum phase lag [rad/s]< / span >
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H = (wc< span class = "org-type" > *< / span > sqrt(a) < span class = "org-type" > +< / span > s)< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > sqrt(a) < span class = "org-type" > +< / span > s);
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< / pre >
< / div >
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< div id = "orgce1450b" class = "figure" >
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< p > < img src = "figs/filter_lag.png" alt = "filter_lag.png" / >
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< / p >
< / div >
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< div id = "orgbed77d1" class = "figure" >
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< p > < img src = "figs/filter_lag_effect_a_phase.png" alt = "filter_lag_effect_a_phase.png" / >
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< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5e67a97" class = "outline-2" >
< h2 id = "org5e67a97" > < span class = "section-number-2" > 7< / span > Complementary< / h2 >
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< / div >
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< div id = "outline-container-org74fba97" class = "outline-2" >
< h2 id = "org74fba97" > < span class = "section-number-2" > 8< / span > Performance Weight< / h2 >
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< div class = "outline-text-2" id = "text-8" >
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< / div >
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< div id = "outline-container-org64d3111" class = "outline-3" >
< h3 id = "org64d3111" > < span class = "section-number-3" > 8.1< / span > Nice combination< / h3 >
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< div class = "outline-text-3" id = "text-8-1" >
\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}
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< div class = "org-src-container" >
< pre class = "src src-matlab" > n = 2; w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 11; G0 = 1< span class = "org-type" > /< / span > 10; G1 = 1000; Gc = 1< span class = "org-type" > /< / span > 2;
wL = Gc< span class = "org-type" > *< / span > (((G1< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (1< span class = "org-type" > /< / span > n)< span class = "org-type" > /< / span > w0< span class = "org-type" > *< / span > sqrt((1< span class = "org-type" > -< / span > (G0< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n))< span class = "org-type" > /< / span > ((G1< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n)< span class = "org-type" > -< / span > 1))< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > (G0< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (1< span class = "org-type" > /< / span > n))< span class = "org-type" > /< / span > (1< span class = "org-type" > /< / span > w0< span class = "org-type" > *< / span > sqrt((1< span class = "org-type" > -< / span > (G0< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n))< span class = "org-type" > /< / span > ((G1< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n)< span class = "org-type" > -< / span > 1))< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 1))< span class = "org-type" > ^< / span > n;
n = 3; w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 9; G0 = 10000; G1 = 0.1; Gc = 1< span class = "org-type" > /< / span > 2;
wH = Gc< span class = "org-type" > *< / span > (((G1< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (1< span class = "org-type" > /< / span > n)< span class = "org-type" > /< / span > w0< span class = "org-type" > *< / span > sqrt((1< span class = "org-type" > -< / span > (G0< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n))< span class = "org-type" > /< / span > ((G1< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n)< span class = "org-type" > -< / span > 1))< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > (G0< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (1< span class = "org-type" > /< / span > n))< span class = "org-type" > /< / span > (1< span class = "org-type" > /< / span > w0< span class = "org-type" > *< / span > sqrt((1< span class = "org-type" > -< / span > (G0< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n))< span class = "org-type" > /< / span > ((G1< span class = "org-type" > /< / span > Gc)< span class = "org-type" > ^< / span > (2< span class = "org-type" > /< / span > n)< span class = "org-type" > -< / span > 1))< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 1))< span class = "org-type" > ^< / span > n;
< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org978d37b" class = "outline-3" >
< h3 id = "org978d37b" > < span class = "section-number-3" > 8.2< / span > Alternative< / h3 >
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< div class = "outline-text-3" id = "text-8-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > ; < span class = "org-comment" > % [rad/s]< / span >
A = 1e< span class = "org-type" > -< / span > 2;
M = 5;
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H = (s< span class = "org-type" > /< / span > sqrt(M) < span class = "org-type" > +< / span > w0)< span class = "org-type" > ^< / span > 2< span class = "org-type" > /< / span > (s < span class = "org-type" > +< / span > w0< span class = "org-type" > *< / span > sqrt(A))< span class = "org-type" > ^< / span > 2;
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< / pre >
< / div >
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< div id = "orgb351bf7" class = "figure" >
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< p > < img src = "figs/weight_first_order.png" alt = "weight_first_order.png" / >
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org043cd0b" class = "outline-2" >
< h2 id = "org043cd0b" > < span class = "section-number-2" > 9< / span > Combine Filters< / h2 >
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< div class = "outline-text-2" id = "text-9" >
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< / div >
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< div id = "outline-container-orgbb8d210" class = "outline-3" >
< h3 id = "orgbb8d210" > < span class = "section-number-3" > 9.1< / span > Additive< / h3 >
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< div class = "outline-text-3" id = "text-9-1" >
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< ul class = "org-ul" >
< li class = "off" > < code > [  ]< / code > Explain how phase and magnitude combine< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-orgc6eac8f" class = "outline-3" >
< h3 id = "orgc6eac8f" > < span class = "section-number-3" > 9.2< / span > Multiplicative< / h3 >
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< / div >
< / div >
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< div id = "outline-container-org2616858" class = "outline-2" >
< h2 id = "org2616858" > < span class = "section-number-2" > 10< / span > Filters representing noise< / h2 >
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< div class = "outline-text-2" id = "text-10" >
< p >
Let’ s consider a noise \(n\) that is shaped from a white-noise \(\tilde{n}\) with unitary PSD (\(\Phi_\tilde{n}(\omega) = 1\)) using a transfer function \(G(s)\).
The PSD of \(n\) is then:
\[ \Phi_n(\omega) = |G(j\omega)|^2 \Phi_{\tilde{n}}(\omega) = |G(j\omega)|^2 \]
< / p >
< p >
The PSD \(\Phi_n(\omega)\) is expressed in \(\text{unit}^2/\text{Hz}\).
< / p >
< p >
And the root mean square (RMS) of \(n(t)\) is:
\[ \sigma_n = \sqrt{\int_{0}^{\infty} \Phi_n(\omega) d\omega} \]
< / p >
< / div >
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< div id = "outline-container-orgb12ce21" class = "outline-3" >
< h3 id = "orgb12ce21" > < span class = "section-number-3" > 10.1< / span > First Order Low Pass Filter< / h3 >
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< div class = "outline-text-3" id = "text-10-1" >
< p >
\[ G(s) = \frac{g_0}{1 + \frac{s}{\omega_c}} \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > g0 = 1; < span class = "org-comment" > % Noise Density in unit/sqrt(Hz)< / span >
wc = 1; < span class = "org-comment" > % Cut-Off frequency [rad/s]< / span >
G = g0< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wc);
< span class = "org-comment" > % Frequency vector [Hz]< / span >
freqs = logspace(< span class = "org-type" > -< / span > 3, 3, 1000);
< span class = "org-comment" > % < / span > < span class = "org-comment" > < span class = "org-constant" > PSD < / span > < / span > < span class = "org-comment" > of n in [unit^2/Hz]< / span >
Phi_n = abs(squeeze(freqresp(G, freqs, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2;
< span class = "org-comment" > % < / span > < span class = "org-comment" > < span class = "org-constant" > RMS < / span > < / span > < span class = "org-comment" > value of n in [unit, rms]< / span >
sigma_n = sqrt(trapz(freqs, Phi_n))
< / pre >
< / div >
< p >
\[ \sigma = \frac{1}{2} g_0 \sqrt{\omega_c} \]
with:
< / p >
< ul class = "org-ul" >
< li > \(g_0\) the Noise Density of \(n\) in \(\text{unit}/\sqrt{Hz}\)< / li >
< li > \(\omega_c\) the bandwidth over which the noise is located, in rad/s< / li >
< li > \(\sigma\) the rms noise< / li >
< / ul >
< p >
If the cut-off frequency is to be expressed in Hz:
\[ \sigma = \frac{1}{2} g_0 \sqrt{2\pi f_c} = \sqrt{\frac{\pi}{2}} g_0 \sqrt{f_c} \]
< / p >
< p >
Thus, if a sensor is said to have a RMS noise of \(\sigma = 10 nm\ rms\) over a bandwidth of \(\omega_c = 100 rad/s\), we can estimated the noise density of the sensor to be (supposing a first order low pass filter noise shape):
\[ g_0 = \frac{2 \sigma}{\sqrt{\omega_c}} \quad \left[ m/\sqrt{Hz} \right] \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > 2< span class = "org-type" > *< / span > 10e< span class = "org-type" > -< / span > 9< span class = "org-type" > /< / span > sqrt(100)
< / pre >
< / div >
< pre class = "example" >
2e-09
< / pre >
< div class = "org-src-container" >
< pre class = "src src-matlab" > 6< span class = "org-type" > *< / span > 0.5< span class = "org-type" > *< / span > 20e< span class = "org-type" > -< / span > 12< span class = "org-type" > *< / span > sqrt(2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100)
< / pre >
< / div >
< pre class = "example" >
1.504e-09
< / pre >
< / div >
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< / div >
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< / div >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-11-12 10:35:54 +01:00
< p class = "date" > Created: 2020-11-12 jeu. 10:35< / p >
2019-08-17 10:50:06 +02:00
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< / body >
< / html >
