filters-matlab-bank/index.org

Bank of Linear Filters - Matlab

• Low Pass
  • First Order Low Pass Filter
  • Second Order
  • Combine multiple first order filters
• High Pass
  • First Order
  • Second Order
  • Combine multiple filters
• Band Pass
  • Second Order
• Notch
  • Second Order
• Chebyshev
  • Chebyshev Type I
• Lead - Lag
  • Lead
  • Lag
• Complementary
• Performance Weight
  • Nice combination
  • Alternative
• Combine Filters
  • Additive
  • Multiplicative
• Filters representing noise
  • First Order Low Pass Filter

Low Pass

First Order Low Pass Filter

\[ H(s) = \frac{1}{1 + s/\omega_0} \]

Parameters:

• $\omega_0$: cut-off frequency in [rad/s]

Characteristics:

• Low frequency gain of $1$
• Roll-off equals to -20 dB/dec

Matlab code:

  w0 = 2*pi; % Cut-off Frequency [rad/s]

  H = 1/(1 + s/w0);

Second Order

\[ H(s) = \frac{1}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]

Parameters:

• $\omega_0$:
• $\xi$: Damping ratio

Characteristics:

• Low frequency gain: 1
• High frequency roll off: - 40 dB/dec

Matlab code:

  w0 = 2*pi; % Cut-off frequency [rad/s]
  xi = 0.3; % Damping Ratio

  H = 1/(1 + 2*xi/w0*s + s^2/w0^2);

Combine multiple first order filters

\[ H(s) = \left( \frac{1}{1 + s/\omega_0} \right)^n \]

Matlab code:

  w0 = 2*pi; % Cut-off frequency [rad/s]
  n = 3; % Filter order

  H = (1/(1 + s/w0))^n;

High Pass

First Order

\[ H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \]

Parameters:

• $\omega_0$: cut-off frequency in [rad/s]

Characteristics:

• High frequency gain of $1$
• Low frequency slow of +20 dB/dec

Matlab code:

  w0 = 2*pi; % Cut-off frequency [rad/s]

  H = (s/w0)/(1 + s/w0);

Second Order

\[ H(s) = \frac{s^2/\omega_0^2}{1 + 2 \xi / \omega_0 s + s^2/\omega_0^2} \]

Parameters:

• $\omega_0$:
• $\xi$: Damping ratio

Matlab code:

  w0 = 2*pi; % [rad/s]
  xi = 0.3;

  H = (s^2/w0^2)/(1 + 2*xi/w0*s + s^2/w0^2);

Combine multiple filters

\[ H(s) = \left( \frac{s/\omega_0}{1 + s/\omega_0} \right)^n \]

Matlab code:

  w0 = 2*pi; % [rad/s]
  n = 3;

  H = ((s/w0)/(1 + s/w0))^n;

TODO Band Pass

Second Order

Notch

Second Order

\begin{equation} \frac{s^2 + 2 g_c \xi \omega_n s + \omega_n^2}{s^2 + 2 \xi \omega_n s + \omega_n^2} \end{equation}

Parameters:

• $\omega_n$: frequency of the notch
• $g_c$: gain at the notch frequency
• $\xi$: damping ratio (notch width)

Matlab code:

  gc = 0.02;
  xi = 0.1;
  wn = 2*pi;

  H = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);

TODO Chebyshev

Chebyshev Type I

  n = 4; % Order of the filter
  Rp = 3; % Maximum peak-to-peak ripple [dB]
  Wp = 2*pi; % passband-edge frequency [rad/s]

  [A,B,C,D] = cheby1(n, Rp, Wp, 'high', 's');
  H = ss(A, B, C, D);

Lead - Lag

Lead

\begin{equation} H(s) = \frac{1 + \frac{s}{w_c/\sqrt{a}}}{1 + \frac{s}{w_c \sqrt{a}}}, \quad a > 1 \end{equation}

Parameters:

• $\omega_c$: frequency at which the phase lead is maximum
• $a$: parameter to adjust the phase lead, also impacts the high frequency gain

Characteristics:

• the low frequency gain is $1$
• the high frequency gain is $a$
• the phase lead at $\omega_c$ is equal to (Figure fig:filter_lead_effect_a_phase): \[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \]

Matlab code:

  a  = 0.6;  % Amount of phase lead / width of the phase lead / high frequency gain
  wc = 2*pi; % Frequency with the maximum phase lead [rad/s]

  H = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));

Lag

\begin{equation} H(s) = \frac{w_c \sqrt{a} + s}{\frac{w_c}{\sqrt{a}} + s}, \quad a > 1 \end{equation}

Parameters:

• $\omega_c$: frequency at which the phase lag is maximum
• $a$: parameter to adjust the phase lag, also impacts the low frequency gain

Characteristics:

• the low frequency gain is increased by a factor $a$
• the high frequency gain is $1$ (unchanged)
• the phase lag at $\omega_c$ is equal to (Figure fig:filter_lag_effect_a_phase): \[ \angle H(j\omega_c) = \tan^{-1}(1/\sqrt{a}) - \tan^{-1}(\sqrt{a}) \]

Matlab code:

  a  = 0.6;  % Amount of phase lag / width of the phase lag / high frequency gain
  wc = 2*pi; % Frequency with the maximum phase lag [rad/s]

  H = (wc*sqrt(a) + s)/(wc/sqrt(a) + s);

TODO Complementary

TODO Performance Weight

Nice combination

\begin{equation} W(s) = G_c * \left(\frac{\frac{1}{\omega_0}\sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}}{\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{\left(\frac{G_\infty}{G_c}\right)^{\frac{2}{n}} - 1}} s + 1}\right)^n \end{equation}

  n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; Gc = 1/2;
  wL = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n;

  n = 3; w0 = 2*pi*9; G0 = 10000; G1 = 0.1; Gc = 1/2;
  wH = Gc*(((G1/Gc)^(1/n)/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + (G0/Gc)^(1/n))/(1/w0*sqrt((1-(G0/Gc)^(2/n))/((G1/Gc)^(2/n)-1))*s + 1))^n;

Alternative

  w0 = 2*pi; % [rad/s]
  A = 1e-2;
  M = 5;

  H = (s/sqrt(M) + w0)^2/(s + w0*sqrt(A))^2;

TODO Combine Filters

Additive

• Explain how phase and magnitude combine

Multiplicative

TODO Filters representing noise

Let's consider a noise $n$ that is shaped from a white-noise $\tilde{n}$ with unitary PSD ($\Phi_\tilde{n}(\omega) = 1$) using a transfer function $G(s)$. The PSD of $n$ is then: \[ \Phi_n(\omega) = |G(j\omega)|^2 \Phi_{\tilde{n}}(\omega) = |G(j\omega)|^2 \]

The PSD $\Phi_n(\omega)$ is expressed in $\text{unit}^2/\text{Hz}$.

And the root mean square (RMS) of $n(t)$ is: \[ \sigma_n = \sqrt{\int_{0}^{\infty} \Phi_n(\omega) d\omega} \]

First Order Low Pass Filter

\[ G(s) = \frac{g_0}{1 + \frac{s}{\omega_c}} \]

  g0 = 1; % Noise Density in unit/sqrt(Hz)
  wc = 1; % Cut-Off frequency [rad/s]

  G = g0/(1 + s/wc);

  % Frequency vector [Hz]
  freqs = logspace(-3, 3, 1000);

  % PSD of n in [unit^2/Hz]
  Phi_n = abs(squeeze(freqresp(G, freqs, 'Hz'))).^2;

  % RMS value of n in [unit, rms]
  sigma_n = sqrt(trapz(freqs, Phi_n))

\[ \sigma = \frac{1}{2} g_0 \sqrt{\omega_c} \] with:

• $g_0$ the Noise Density of $n$ in $\text{unit}/\sqrt{Hz}$
• $\omega_c$ the bandwidth over which the noise is located, in rad/s
• $\sigma$ the rms noise

If the cut-off frequency is to be expressed in Hz: \[ \sigma = \frac{1}{2} g_0 \sqrt{2\pi f_c} = \sqrt{\frac{\pi}{2}} g_0 \sqrt{f_c} \]

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): \[ g_0 = \frac{2 \sigma}{\sqrt{\omega_c}} \quad \left[ m/\sqrt{Hz} \right] \]

  2*10e-9/sqrt(100)

2e-09

  6*0.5*20e-12*sqrt(2*pi*100)

1.504e-09