Analog & Digital Filters
Transfer functions, filter families (Butterworth, Chebyshev, Bessel), and digital filter design — FIR, IIR, windowed sinc, and the bilinear transform.
1. Filter Transfer Functions
A linear time-invariant (LTI) filter is characterized by its transfer function\(H(s)\) in the Laplace domain (analog) or \(H(z)\) in the Z-domain (digital). For a general N-th order system:
\[ H(s) = \frac{b_M s^M + \cdots + b_1 s + b_0}{a_N s^N + \cdots + a_1 s + a_0} \]
The frequency response is \(H(j\omega) = H(s)\big|_{s=j\omega}\). The four standard filter types are:
2. Butterworth Filter — Maximally Flat
The Butterworth filter has the flattest possible passband — no ripple. The magnitude response is:
\[ |H(j\omega)| = \frac{1}{\sqrt{1 + \left(\dfrac{\omega}{\omega_c}\right)^{2n}}} \]
At \(\omega = \omega_c\): \(|H| = 1/\sqrt{2} \approx -3\) dB for all orders \(n\)
Poles lie equally spaced on a circle of radius \(\omega_c\) in the left half s-plane. For order \(n\), the stopband rolls off at \(-20n\) dB/decade beyond \(\omega_c\). Higher order = steeper roll-off but slower transient response.
Determining the required order: given stopband attenuation \(A_s\) dB at frequency \(\omega_s\), \(n \geq \log(10^{A_s/10}-1) / (2\log(\omega_s/\omega_c))\).
3. Chebyshev & Bessel Filters
Chebyshev Type I — Equiripple Passband
Allows controlled ripple \(\epsilon\) in the passband to achieve steeper roll-off than Butterworth of the same order:
\[ |H|^2 = \frac{1}{1 + \varepsilon^2 T_n^2(\omega/\omega_c)} \]
\(T_n\) is the Chebyshev polynomial of degree \(n\). Type II has equiripple stopband instead.
Bessel Filter — Linear Phase
Maximally linear phase response (constant group delay) in the passband. Preserves waveshape — critical for pulse and data signals where phase distortion causes intersymbol interference.
Trade-off: slowest magnitude roll-off of the three families. Ideal for pulse shaping in communications.
Comparison Summary
4. Digital Filters: FIR vs IIR
FIR — Finite Impulse Response
Output depends only on current and past inputs:
\[ y[n] = \sum_{k=0}^{M} h[k]\,x[n-k] \]
- • Always stable (no feedback)
- • Linear phase achievable (symmetric coefficients)
- • Higher order needed for sharp cutoff
- • Windowed sinc design: multiply ideal sinc by Hamming/Blackman window
IIR — Infinite Impulse Response
Output also depends on past outputs (recursive):
\[ y[n] = \sum_k b_k x[n-k] - \sum_k a_k y[n-k] \]
- • Lower order for equivalent sharpness
- • Can be unstable if poles outside unit circle
- • Non-linear phase; may need all-pass correction
- • Bilinear transform maps analog prototype to digital
Bilinear Transform
Maps an analog prototype \(H(s)\) to a digital filter \(H(z)\) with no aliasing using the substitution:
\[ s = \frac{2}{T_s}\cdot\frac{z-1}{z+1} \]
Frequency warping occurs: analog frequency \(\Omega\) maps to \(\omega = 2\arctan(\Omega T_s/2)\). Pre-warp the critical frequency before design.
Python Simulation
Butterworth LP orders 1–5, Butterworth vs Chebyshev comparison, FIR windowed-sinc design, step responses, phase responses, and pole-zero plot.
Click Run to execute the Python code
Code will be executed with Python 3 on the server