Bessel Filter Design & Simulation Software

This software provides a comprehensive environment for designing and simulating Bessel filters of various types (Low-pass, High-pass, Band-pass, Band-stop) in both analog and digital domains. The application features graphical visualization of filter characteristics including magnitude/phase responses, pole-zero plots, and time-domain responses.

Overview

• Supports 4 filter types: Low-pass, High-pass, Band-pass, Band-stop
• Works in both analog and digital domains
• Automatic order determination or manual specification
• Visualizations:
  • Magnitude response (linear/dB scale)
  • Phase response and group delay
  • Pole-zero plots with unit circle for digital filters
  • Step and impulse responses
• Multiple coefficient representations:
  • Transfer function
  • State-space
  • Second-order sections (SOS)

Mathematical Models

Bessel Polynomials

The software uses Bessel polynomials to design the analog low-pass prototype. The recursive formula for Bessel polynomials is:

B₀(s) = 1
B₁(s) = s + 1
Bₙ(s) = (2n-1)Bₙ₋₁(s) + s²Bₙ₋₂(s)

Filter Transformations

1. Low-pass Prototype Normalization:
   • Poles are scaled to achieve unity cutoff frequency: s → s/ωc
2. Frequency Transformations:
   • Low-pass to High-pass: s → ωc/s
   • Low-pass to Band-pass: s → (s² + ω0²)/(Bs)
   • Low-pass to Band-stop: s → Bs/(s² + ω0²) where ω0 is center frequency and B is bandwidth
3. Bilinear Transform (Analog to Digital): s → (2/T)(z-1)/(z+1) where T is sampling period

Pole-Zero Analysis

• Poles are roots of the Bessel polynomial
• Zeros are determined by filter type:
  • Low-pass: No finite zeros
  • High-pass: Multiple zeros at origin
  • Band-pass: Complex conjugate zeros at center frequency
  • Band-stop: Zeros at origin and complex frequencies

Response Calculations

1. Magnitude Response: |H(jω)| = 1/|D(jω)| where D(jω) is denominator polynomial evaluated at s=jω
2. Phase Response: φ(ω) = -arg(D(jω)) (in degrees)
3. Group Delay: τ(ω) = -dφ/dω ≈ -Δφ/Δω (numerical differentiation)
4. Time Domain Responses:
   • Step response: Output when input u(t) = 1
   • Impulse response: Output when input δ(t) = 1 at t=0

Screenshots