IIR Filter Design: Butterworth and Chebyshev For students.

An Infinite Impulse Response filter, or IIR filter, feeds past output samples back into its computation, so its response to a single input pulse theoretically rings forever. That feedback lets an IIR filter achieve a very sharp cutoff with only a handful of coefficients, far fewer than an FIR filter needs for the same sharpness.

This efficiency matters where processing power or battery is limited: hearing aids, embedded audio effects and real-time control loops all favour IIR filters. The two classic families are Butterworth, maximally flat in the passband, and Chebyshev, which accepts controlled ripple for an even sharper transition. Knowing when to pick each is a core DSP skill.

Key Concepts

• Poles and zeros: roots of the denominator (poles) and numerator (zeros) set the filter's behaviour. For stability every pole must lie inside the unit circle.
• Butterworth: maximally flat passband with no ripple; the response rolls off monotonically.
• Chebyshev Type I: allows equal ripple in the passband to get a steeper roll-off than Butterworth of the same order.
• Chebyshev Type II: flat passband with ripple in the stopband instead.
• Analog prototype: classical designs start from a continuous-time filter and convert it with the bilinear transform.
• Frequency warping: the bilinear transform squeezes the analog axis into the digital range, so cutoffs must be pre-warped to land correctly.

Formulas

• Butterworth magnitude squared: |H(jω)|² = 1 / (1 + (ω/ωc)^(2N)), with ωc the 3 dB cutoff and N the order.
• Chebyshev Type I magnitude squared: |H(jω)|² = 1 / (1 + ε²·T_N²(ω/ωc)), where ε sets the passband ripple.
• Butterworth attenuation: A = 10·log10(1 + (ω/ωc)^(2N)) dB.
• Bilinear transform: s = (2/T)·(1 − z^(−1))/(1 + z^(−1)), mapping the analog s-plane to the digital z-plane.
• Assumption: phase is nonlinear, unlike linear-phase FIR designs; poles must stay inside the unit circle.

Worked Example

Design a Butterworth lowpass with 3 dB point at 1000 Hz and at least 20 dB attenuation at 2000 Hz. From 10·log10(1 + (2000/1000)^(2N)) ≥ 20 we get 2^(2N) ≥ 99. Taking logs: 2N ≥ log10(99)/log10(2) ≈ 6.63, so N ≥ 3.31, and the order must be the integer N = 4.

Verify: attenuation at 2000 Hz is 10·log10(1 + 2^8) = 10·log10(257) ≈ 24.1 dB, above the requirement. A 1 dB-ripple Chebyshev Type I of order 3 would also meet the spec, showing the trade-off: Chebyshev saves one stage but leaves passband ripple, while Butterworth stays perfectly flat.

Common Misconceptions

• Chebyshev is always better because it is sharper. Sharpness costs ripple. Measurement instruments needing a flat passband should use Butterworth.
• Higher order is always better. Very high-order IIR filters grow sensitive: coefficient rounding can push poles outside the unit circle into instability.
• The bilinear transform preserves frequencies exactly. It warps them; pre-warp with ω_analog = (2/T)·tan(ω_digital·T/2) or the cutoff lands wrong.

Real-World Applications

• Audio crossovers split bass and treble with low-order Butterworth sections.
• Anti-aliasing filters in front of ADCs use IIR stages to keep analog circuitry small.
• Biomedical monitors filter ECG signals with IIR stages on tiny microcontrollers.

Butterworth gives flatness, Chebyshev gives sharpness per order, and the bilinear transform converts either prototype to digital. Master the order-versus-ripple trade-off and a few coefficients meet almost any specification.