Update Content - 2022-10-27

content/zettels/discrete_transfer_functions.md

@@ -35,6 +35,9 @@ There are several methods to go from the analog to the digital domain, `Tustin`
 
 ## Obtaining analytical formula of filter {#obtaining-analytical-formula-of-filter}
 
+
+### Procedure {#procedure}
+
 The Matlab [Symbolic Toolbox](https://fr.mathworks.com/help/symbolic/) can be used to obtain analytical formula for discrete transfer functions.
 
 Let's consider a notch filter:
@@ -56,13 +59,6 @@ First the symbolic variables are declared (`Ts` is the sampling time, `s` the La
 syms gc wn xi Ts s z
 ```
 
-The symbolic formula of the notch filter is defined:
-
-```matlab
-%% Notch Filter - Symbolic representation
-Ga = (s^2 + 2*xi*gc*s*wn + wn^2)/(s^2 + 2*xi*s*wn + wn^2);
-```
-
 Then the bi-linear transformation is performed to go from continuous to discrete:
 
 ```matlab
@@ -70,6 +66,13 @@ Then the bi-linear transformation is performed to go from continuous to discrete
 s = 2/Ts*(z - 1)/(z + 1);
 ```
 
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Notch Filter - Symbolic representation
+Ga = (s^2 + 2*xi*gc*s*wn + wn^2)/(s^2 + 2*xi*s*wn + wn^2);
+```
+
 Finally, the numerator and denominator coefficients can be extracted:
 
 ```matlab
@@ -90,6 +93,126 @@ den = (Ts^2*wn^2 - 4*Ts*wn*xi + 4) + (2*Ts^2*wn^2 - 8) * z + (Ts^2*wn^2 + 4*Ts*w
 ```
 
 
+### Second Order Low Pass Filter {#second-order-low-pass-filter}
+
+Let's consider a second order low pass filter:
+
+\begin{equation}
+  G(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_n} + \frac{s^2}{\omega\_n^2}}
+\end{equation}
+
+with:
+
+-   \\(\omega\_n\\): Cut off frequency
+-   \\(\xi\\): damping ratio
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms wn xi Ts s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Second Order Low Pass Filter - Symbolic representation
+Ga = 1/(1 + 2*xi*s/wn + s^2/wn^2);
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+gain = 1/(Ts^2*wn^2 + 4*Ts*wn*xi + 4)
+```
+
+```text
+num = (Ts^2*wn^2) + (2*Ts^2*wn^2) * z^-1 + (Ts^2*wn^2) * z^-2
+```
+
+```text
+den = 1 + (2*Ts^2*wn^2 - 8) * z^-1 + (Ts^2*wn^2 - 4*Ts*wn*xi + 4) * z^-2
+```
+
+And the transfer function is equal to `gain * num/den`.
+
+
+### Second Order High Pass Filter {#second-order-high-pass-filter}
+
+Let's consider a second order low pass filter:
+
+\begin{equation}
+  G(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_n} + \frac{s^2}{\omega\_n^2}}
+\end{equation}
+
+with:
+
+-   \\(\omega\_n\\): Cut off frequency
+-   \\(\xi\\): damping ratio
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms wn xi Ts s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Second Order Low Pass Filter - Symbolic representation
+Ga = (s^2/wn^2)/(1 + 2*xi*s/wn + s^2/wn^2);
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+gain = 1/(Ts^2*wn^2 + 4*Ts*wn*xi + 4)
+```
+
+```text
+num = (4) + (-8) * z^-1 + (4) * z^-2
+```
+
+```text
+den = 1 + (2*Ts^2*wn^2 - 8) * z^-1 + (Ts^2*wn^2 - 4*Ts*wn*xi + 4) * z^-2
+```
+
+And the transfer function is equal to `gain * num/den`.
+
+
 ## Variable Discrete Filter {#variable-discrete-filter}
 
 Once the analytical formula of a discrete transfer function is obtained, it is possible to vary some parameters in real time.