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Home/Math Visualization/Butterworth / Chebyshev IIR

Butterworth / Chebyshev IIR

This interactive simulator explores Butterworth / Chebyshev IIR in Math Visualization. Design Butterworth, Chebyshev I/II LP/HP filters: |H(f)|, phase, impulse response, and z-plane pole–zero plot via bilinear transform. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Math Visualization.

Key terms

  • butterworth
  • chebyshev
  • iir
  • iir filter design
  • math
  • visualization

Filter design

Family

4
120Hz

Presets

Measured values

familyButterworth
order4
designed f_c120Hz
measured f_{−3 dB}119.9Hz
# poles / zeros4 / 4
stable?yes

How it works

IIR digital filter design: pick an analog prototype (Butterworth flat passband, Chebyshev I equiripple in passband, Chebyshev II equiripple in stopband), apply the bilinear transform with pre-warping and watch the magnitude response, optional phase, and the z-plane pole / zero map update live. Crank up the order to sharpen the roll-off (steeper transition); pull f_c through the spectrum to see how poles slide along the unit circle. Chebyshev I gains roll-off at the cost of in-band ripple R_p; Chebyshev II is flat in the passband but ripples in the stopband around R_s.

Key equations

Butterworth: |H(jΩ)|² = 1 / (1 + (Ω/Ωc)^{2N})
Chebyshev I: |H(jΩ)|² = 1 / (1 + ε² Tₙ²(Ω/Ωp)), ε = √(10^{R_p/10} − 1)
Bilinear: z = (1 + sT/2) / (1 − sT/2), pre-warp Ωc = (2/T) tan(ωc/2)