How it works
Designing a digital IIR Butterworth low-pass filter starts in the analog domain: compute the prototype transfer function H_a(s) with poles at angles (2k+N-1)π/(2N) on a unit circle in the s-plane. Then apply the bilinear transform s = 2f_s·(z-1)/(z+1) to map H_a(s) to H(z). This transform compresses the entire analog frequency axis into 0 to π, causing frequency warping: a desired digital cutoff of ω_d requires pre-warping the analog cutoff to Ω_a = 2f_s·tan(ω_d/2). Impulse invariance is an alternative method where h[n] = T·h_a(nT), but it suffers from aliasing and is only used for low-pass designs.
Key points to remember
IIR filters can be unstable — poles must lie strictly inside the unit circle in the z-plane for BIBO stability. Unlike FIR, IIR filters generally do not have linear phase because of the feedback structure. Butterworth filters have maximally flat magnitude response with no ripple in passband or stopband. Chebyshev Type I allows passband ripple (say, 1 dB) to achieve sharper roll-off for the same filter order. Elliptic filters have equal ripple in both bands and give the sharpest transition of all IIR types. A 2nd-order IIR section is called a biquad; cascading biquads is preferred for numerical stability in fixed-point implementations.
Exam tip
The examiner always asks you to compare the bilinear transform and impulse invariance methods — focus on the fact that bilinear transform avoids aliasing completely while impulse invariance does not, and be ready to derive the pre-warping formula.