How it works
Bilinear transform substitution: s = (2/T)·(1−z⁻¹)/(1+z⁻¹), where T is the sampling period. This maps the left-half s-plane entirely into the interior of the unit circle in the z-plane, guaranteeing stability. The frequency relationship: Ω = (2/T)·tan(ω/2), where Ω is the analog frequency and ω is the digital frequency (0 to π). Prewarping: design the analog prototype at Ωpre = (2/T)·tan(ωc/2) instead of the desired digital cutoff frequency ωc, so that after transformation the digital filter has its −3 dB point exactly at ωc. For a 1st-order Butterworth H(s) = 1/(s+1): substituting s → (2/T)·(1−z⁻¹)/(1+z⁻¹) yields H(z) directly by algebraic simplification.
Key points to remember
Bilinear transform preserves filter order — an Nth-order analog prototype maps to an Nth-order digital filter. No aliasing occurs because the entire jΩ axis maps to the unit circle, unlike impulse invariance where only the strip −π/T to π/T is mapped. The frequency compression is most severe near ω = π (digital Nyquist), where a very large range of analog frequencies all map to π — this is why stopband edges warp more than passband edges for high-order filters. Prewarping is absolutely essential for band-pass and high-pass digital filters designed via bilinear transform; failure to prewarp shifts the actual cutoff from the specified value. The transformation H(z) = H_a(s)|_(s = (2/T)(1−z⁻¹)/(1+z⁻¹)) is applied by substituting and then rationalising to find the z-domain polynomial coefficients.
Exam tip
Every Anna University DSP paper asks you to design a first-order digital Butterworth LPF at a given cutoff frequency using the bilinear transform — show the prewarping step, substitute into H(s), simplify to H(z), and state that no aliasing occurs because the entire jΩ axis maps to the unit circle.