Why do Chebyshev filters have ripple but often need lower order?

Allowing equiripple error in the passband (Type I) or stopband (Type II) uses the approximation budget more efficiently than a maximally flat Butterworth design, so the same transition width is often reached with fewer poles.

What does the bilinear transform do?

It maps the stable left-half s-plane into the interior of the unit circle in z via z = (1+s)/(1−s) (up to sampling-rate scaling), preserving stability while warping analog frequencies according to a tangent law that designers pre-compensate (prewarping).

Why look at poles and zeros, not only |H(f)|?

Poles near the unit circle create sharp resonances and long impulse ringing; zeros carve notches. The z-plane explains stability margins and transient behavior that a magnitude plot alone hides.

Butterworth / Chebyshev IIR

Design Butterworth, Chebyshev I/II LP/HP filters: |H(f)|, phase, impulse response, and z-plane pole–zero plot via bilinear transform.

About this model

Classic analog prototypes — Butterworth (maximally flat passband) and Chebyshev Type I/II (equiripple passband or stopband) — are mapped to digital IIR filters with the bilinear transform. This simulator designs low-pass and high-pass instances and shows magnitude |H(f)|, phase, impulse response, and the z-plane pole–zero plot. Order, cutoff, and ripple (for Chebyshev) place poles inside the unit circle; bilinear warping maps the analog frequency axis via the tangent relation. Assumptions include linear time-invariant design, ideal coefficient arithmetic, and no fixed-point quantization or limit-cycle analysis. Compare Butterworth versus Chebyshev and LP versus HP while watching poles move and the impulse response ring.

Who it's for: DSP, communications, and electrical engineering courses on digital filter design.

Key terms

Butterworth filter
Chebyshev filter
IIR filter
Bilinear transform
Pole-zero plot
Frequency response

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)

Frequently asked questions

Why do Chebyshev filters have ripple but often need lower order?: Allowing equiripple error in the passband (Type I) or stopband (Type II) uses the approximation budget more efficiently than a maximally flat Butterworth design, so the same transition width is often reached with fewer poles.
What does the bilinear transform do?: It maps the stable left-half s-plane into the interior of the unit circle in z via z = (1+s)/(1−s) (up to sampling-rate scaling), preserving stability while warping analog frequencies according to a tangent law that designers pre-compensate (prewarping).
Why look at poles and zeros, not only |H(f)|?: Poles near the unit circle create sharp resonances and long impulse ringing; zeros carve notches. The z-plane explains stability margins and transient behavior that a magnitude plot alone hides.

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Key equations

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About this model

How it works

Key equations

Frequently asked questions