Filter Design

Posted on Edited on In

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import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

# Parameters: 4th order filter with 1kHz cutoff
order = 4
w_cutoff = 2 * np.pi * 1000
w = np.logspace(2, 5, 1000) # Frequency range from 100Hz to 100kHz

# Generate Butterworth coefficients and frequency response
b_but, a_but = signal.butter(order, w_cutoff, analog=True)
w_but, h_but = signal.freqs(b_but, a_but, worN=w)

# Generate Chebyshev (1dB ripple) response
b_cheb, a_cheb = signal.cheby1(order, 1, w_cutoff, analog=True)
w_cheb, h_cheb = signal.freqs(b_cheb, a_cheb, worN=w)

# Generate Bessel response
b_bess, a_bess = signal.bessel(order, w_cutoff, analog=True)
w_bess, h_bess = signal.freqs(b_bess, a_bess, worN=w)

freqs = w / (2 * np.pi)

t = np.linspace(0, 0.005, 1000) # 5ms window

# Calculate Step Responses
t_but, y_but = signal.step((b_but, a_but), T=t)
t_cheb, y_cheb = signal.step((b_cheb, a_cheb), T=t)
t_bess, y_bess = signal.step((b_bess, a_bess), T=t)

Butterworth Filters

Butterworth Filters [https://people.eecs.ku.edu/~demarest/212/Butterworth%20Filters.pdf]

Stephen Roberts, Signal Processing & Filter Design B3 option: Lecture 2 - Frequency Selective Filters [https://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l2.pdf]

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% Parameters
wc = 100;               % Cutoff frequency in rad/s
orders = [1, 2, 6];     % Orders to compare
w = logspace(0, 3, 1000); % Frequency range (1 to 1000 rad/s)

for n = orders
    % 1. Design Analog Filter (the 's' flag is critical)
    [b, a] = butter(n, wc, 's');
    
    % 2. Calculate Complex Frequency Response
    h = freqs(b, a, w);
    
    % 3. Calculate Group Delay
    % For analog filters, we manually differentiate phase: gd = -d(phi)/dw
    phase = unwrap(angle(h));
    gd = -diff(phase) ./ diff(w);
    
    % --- Plot Magnitude (dB) ---
    subplot(3,1,1);
    semilogx(w, 20*log10(abs(h)), 'LineWidth', 2, 'DisplayName', ['n=' num2str(n)]);
    hold on;
    
    % --- Plot Phase (Degrees) ---
    subplot(3,1,2);
    semilogx(w, phase * 180/pi, 'LineWidth', 2);
    hold on;
    
    % --- Plot Group Delay ---
    subplot(3,1,3);
    semilogx(w(1:end-1), gd, 'LineWidth', 2);
    hold on;
end

% Formatting
subplot(3,1,1)
ylabel('Magnitude (dB)'); grid on; xline(wc, '--r', 'HandleVisibility','off'); legend('show');
ylim([-60 5]);

subplot(3,1,2); ylabel('Phase (deg)'); grid on; xline(wc, '--r', 'HandleVisibility','off');

subplot(3,1,3); ylabel('Group Delay (sec)'); xlabel('Frequency (rad/s)'); 
grid on; xline(wc, '--r', 'HandleVisibility','off');

Bessel Filters

Stephen Roberts, Signal Processing & Filter Design B3 option: Lecture 3 - Transient Response and Transforms [https://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l3.pdf]

Bessel filter is often called Bessel–Thomson filter or simply Thomson filter

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besself: Bessel analog filter design [b,a] = besself(n, Wo)

the transfer function coefficients of an \(n\)th-order lowpass analog Bessel filter, where Wo is the angular frequency up to which the filter's group delay is approximately constant. Larger values of n produce a group delay that better approximates a constant up to Wo.

scipy.signal.bessel(N, Wn, btype='low', analog=True, output='ba', norm='phase')

norm='phase' — The filter is normalized such that the phase response reaches its midpoint at angular (e.g. rad/s) frequency Wn

This is the default, and matches MATLAB's implementation.

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octave:8> [b,a] = besself(5,1)
b = 1
a =

   1.0000   3.8107   6.7767   6.8864   3.9363   1.0000
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b_bess, a_bess = signal.bessel(5, 1, btype='low', analog=True, norm='phase')
print(b_bess, a_bess)

# [1.] [1.         3.81070121 6.77667372 6.88636765 3.93628343 1.        ]

bessel_5th_wn1

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import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

order = 5
cutoff_rad = 1

b_bess, a_bess = signal.bessel(order, cutoff_rad, btype='low', analog=True, norm='phase')

# Frequency response computation (Linear scale)
w = np.linspace(0, 8 * cutoff_rad, 1000) # Frequency range 0 to 2*fc

# Frequency response for phase
w_bess, h_bess = signal.freqs(b_bess, a_bess, worN=w)

# Phase Response (unwrapped to avoid 360 degree jumps)
phase_bess = np.unwrap(np.angle(h_bess))

plt.figure(figsize=(10, 5))
plt.plot(w_bess, np.degrees(phase_bess), label='Bessel', linewidth=2, linestyle='-')
plt.axvline(cutoff_rad, color='red', linestyle=':', label='Cutoff')
plt.title('5th Order Bessel Filter Phase Response w/ Wn=1')
plt.ylabel('Phase [degrees]'); plt.xlabel('Frequency [rad/s]')
plt.grid(True); plt.legend()
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Wn = 1;

[bb, ab] = butter(2, Wn, 's');
[bs, as] = besself(2, Wn);

pb = roots(ab);
ps = roots(as);

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RC LPF

Kwantae Kim, Integrated Analog Systems D - Lecture 02 (Continuous-Time Filters) [https://youtu.be/B7-kr5zV3NA]

—, Integrated Analog Systems D - Lecture 03 (Continuous-Time Filters) [https://youtu.be/6GdDiwaKDZw]

—, Integrated Analog Systems D - Lecture 05 (Continuous-Time Filters) [https://youtu.be/LHhEK1RlC6w]

C. K. Singh and G. C. Temes, "R-C Ladder Design: A Tutorial [CAS Education]," in IEEE Circuits and Systems Magazine, vol. 24, no. 4, pp. 65-69, Fourthquarter 2024 [https://sci-net.xyz/10.1109/MCAS.2024.3462661]

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[Gist link]

2nd-Order RC LPF

Loading effect & limitation

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Phase Magin with damping Factor \(\zeta\)

image-20260524103924774 \[ \boxed{\phi_\text{PM}\approx 100\cdot \zeta} \]

General 2nd-Order RC LPF

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Laplace Transform

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Stability Analysis

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Bode's noise theorem

Bode's noise theorem holds only for linear, passive networks in thermal equilibrium, meaning every lossy element is at the same temperature T and the only noise is the thermal noise of that loss.

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A short circuit is the limit of a capacitor with infinite capacitance, since its impedance go to 0 only when capacitance go to infinity.

For 1st RC filter, \(C_\infty=C\), and \(C_0\) (the capacitance "seen" at the output) is \(C\) in parallel with a short, which is infinite, i.e. \(C_0=C+\infty=\infty\) \[ V_n^2=kT(\frac{1}{C} - \frac{1}{\infty})=\frac{kT}{C} \]

Active Filters

Kwantae Kim, Integrated Analog Systems D - Lecture 05 (Continuous-Time Filters) [https://youtu.be/LHhEK1RlC6w]

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[Gist link]

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TODO 📅

Single-Pole LPF Algorithms

Neil Robertson, Model a Sigma-Delta DAC Plus RC Filter [https://www.dsprelated.com/showarticle/1642.php]

Jason Sachs, Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter [https://www.embeddedrelated.com/showarticle/779.php]

—. Return of the Delta-Sigma Modulators, Part 1: Modulation [https://www.dsprelated.com/showarticle/1517/return-of-the-delta-sigma-modulators-part-1-modulation]

  • Derivatives Approximation (\(H_p(s)=\frac{1}{s\tau +1}\))

    \[\begin{align} H_p(z)&=\frac{\frac{T_s}{T_s+\tau}}{1+(\frac{T_s}{T_s+\tau}-1)z^{-1}}\tag{EQ-0}\\ H_p(z)&=\frac{\frac{T_s}{\tau}}{1+(\frac{T_s}{\tau}-1)z^{-1}}\tag{EQ-1} \end{align}\]

  • Matched z-Transform (Root Matching) \[ H_p(z)=\frac{1-e^{-T_s/\tau}}{1-e^{-T_s/\tau}z^{-1}}\tag{EQ-2} \] EQ-2 is connected with EQ-1 by \(1 - e^{-\Delta t/\tau} \approx \frac{\Delta t}{\tau}\)

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import numpy as np
import matplotlib.pyplot as plt
import scipy.signal


dt=0.002; tau=0.05

T=1
t = np.arange(0,T+1e-5,dt)
x = 1-np.abs(4*t-1)
x[4*t>2] = 0.95; x[t>0.75] = 0.02

alpha_derv = dt / (dt + tau)
alpha_derv_approx = dt / tau

yfilt_derv = scipy.signal.lfilter([alpha_derv], [1, alpha_derv - 1], x)
yfilt_derv_approx = scipy.signal.lfilter([alpha_derv_approx], [1, alpha_derv_approx - 1], x)

a1 = -np.exp(-dt/tau)
b0 = [1 + a1]
a = [1, a1]
y_filt_match = scipy.signal.lfilter(b0, a, x)

plt.figure(figsize=(20,10))
plt.plot(t, x, color='k', linewidth=3, label='x')
plt.plot(t, yfilt_derv, color='red', linewidth=3, label=r'$H_p(z)=\frac{\frac{T_s}{T_s+\tau}}{1+(\frac{T_s}{T_s+\tau}-1)z^{-1}}$')
plt.plot(t, yfilt_derv_approx, color='green', marker='D', linestyle='dashed',markersize=4, linewidth=3, label=r'$H_p(z)=\frac{\frac{T_s}{\tau}}{1+(\frac{T_s}{\tau}-1)z^{-1}}$')
plt.plot(t, y_filt_match, color='m', marker='x', linestyle='dashed',markersize=4, linewidth=3, label=r'$H_p(z)=\frac{1-e^{-T_s/\tau}}{1-e^{-T_s/\tau}z^{-1}}$')

plt.legend(loc='upper right', fontsize=20)
plt.grid(which='both');plt.xlabel('Time (s)');plt.show()
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x(t) ──┬── R ──┬── y(t)
       │       │
       │       C
       │       │
       └───────┴── gnd

Three discretizations of the same continuous prototype, all valid first-order LPFs but with different sample-domain behavior \(\alpha = \frac{T}{T+\tau}\):

Form Difference equation Transfer function Notes
Backward Euler (above) \(y_n = (1-\alpha) y_{n-1} + \alpha\, x_n\) \(\dfrac{\alpha}{1 - (1-\alpha) z^{-1}}\) Implicit; needs \(x_n\) before computing \(y_n\)
Delayed leaky integrator \(y_n = (1-\alpha) y_{n-1} + \alpha\, x_{n-1}\) \(\dfrac{\alpha z^{-1}}{1 - (1-\alpha) z^{-1}}\) One extra sample of delay; same magnitude response
Bilinear (Tustin) \(y_n = (1-\alpha)y_{n-1} + \tfrac{\alpha}{2}(x_n + x_{n-1})\) \(\dfrac{(\alpha/2)(1 + z^{-1})}{1 - (1-\alpha) z^{-1}}\) Adds zero at \(z = -1\); better frequency-response match
Forward Euler \(y_n = (1 - T/\tau)y_{n-1} + (T/\tau)\,x_{n-1}\) \(\dfrac{(T/\tau) z^{-1}}{1 - (1 - T/\tau) z^{-1}}\) Unstable when \(T > 2\tau\)

Pole magnitude \(|1-\alpha| < 1\) always — backward Euler is unconditionally stable, unlike forward Euler (\(\alpha = T/\tau\)), which goes unstable when \(T > 2\tau\)

All four collapse to the same continuous-time filter as \(T \to 0\), but they're not interchangeable at finite \(T\) — the delayed leaky integrator in particular adds one sample of group delay that the others don't.

Discrete-Time Integrators

Qasim Chaudhari. Discrete-Time Integrators [https://wirelesspi.com/discrete-time-integrators/]

David Johns (University of Toronto) "Oversampled Data Converters" Course (2019) [https://youtu.be/qIJ2LORYmyA]

Delaying Integrator

Delay-free Integrator

image-20250615124417691

Discrete-Time Differentiator

Qasim Chaudhari. Design of a Discrete-Time Differentiator [https://wirelesspi.com/design-of-a-discrete-time-differentiator/]

TODO 📅

reference

Boris Murmann. EE315A VLSI Signal Conditioning Circuits

Bill Redman-White, ISSCC 2009 Tutorial: T1 : Continuous-Time Filters

B. Nikolic, "Tutorial: Filtering in RF Transceivers," 2014 IEEE International Solid-State Circuits Conference Digest of Technical Papers (ISSCC), San Francisco, CA, USA, 2014

W. Sansen, "Short Course: Power Limits for Amplifiers and Filters," 2012 IEEE International Solid-State Circuits Conference, San Francisco, CA, USA, 2012

Antonio Liscidini, 2018 New Trends in Analog Filters

M. Babaie, "Tutorial: Role of Current-Mode Passive Mixers and N-Path Filters in RF Receivers," 2023 IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, CA, USA, 2023

Stephen Roberts, Signal Processing & Filter Design B3 option [https://www.robots.ox.ac.uk/~sjrob/Teaching/sp_course.html]

Butterworth, Chebyshev & Bessel filters [https://analogcircuitdesign.com/butterworth-and-chebyshev-filters/]

Qasim Chaudhari. FIR vs IIR Filters – A Practical Comparison [https://wirelesspi.com/fir-vs-iir-filters-a-practical-comparison/]

—. Finite Impulse Response (FIR) Filters [https://wirelesspi.com/finite-impulse-response-fir-filters/]

—. Why FIR Filters have Linear Phase [https://wirelesspi.com/why-fir-filters-have-linear-phase/]

—. Moving Average Filter [https://wirelesspi.com/moving-average-filter/]

—. Cascaded Integrator Comb (CIC) Filters – A Staircase of DSP. [https://wirelesspi.com/cascaded-integrator-comb-cic-filters-a-staircase-of-dsp/]

Hideo Okawara's Mixed Signal Lecture Series [https://tomverbeure.github.io/2024/01/06/Hideo-Okawara-Mixed-Signal-Lecture-Series.html]

How to generate complex poles without inductor? [https://a2d2ic.wordpress.com/2020/02/19/basics-on-active-rc-low-pass-filters/]