Questions: Bilinear Transform for Digital Filter Design
2 questions to test your understanding
Score: 0 / 2
Question 1 Short Answer
Why does the bilinear transform preserve stability while the forward Euler substitution does not?
Think about your answer, then reveal below.
Model answer: The bilinear transform maps the entire left half s-plane onto the interior of the unit circle in the z-plane. Any pole with Re(s) < 0 in the analog domain lands at |z| < 1 in the digital domain, preserving stability. Forward Euler maps the stable left half-plane to a small disk near z = 1, which does not encompass all stable digital poles — some stable analog poles map outside the unit circle.
The bilinear transform is a Möbius transformation that maps the imaginary jω axis to the unit circle. This bijection between the stability regions is exactly the property you need. Euler's method is a first-order approximation of the exponential map, which is globally accurate only at low frequencies and fails for lightly damped or near-Nyquist poles.
Question 2 Short Answer
You want a digital lowpass filter with −3 dB at 1000 Hz, sampled at 8000 Hz. What is the pre-warped analog frequency Ω_c to use in the analog prototype design?
Think about your answer, then reveal below.
Model answer: Digital frequency: ω_c = 2π × 1000/8000 = π/4 rad/sample. T = 1/8000 s. Pre-warped: Ω_c = (2/T)·tan(ω_c/2) = 16000·tan(π/8) ≈ 16000 × 0.4142 ≈ 6628 rad/s.
The pre-warp formula Ω_c = (2/T)·tan(ω_c/2) ensures that after the bilinear transform, the digital filter's −3 dB point lands exactly at ω_c = π/4 (1000 Hz). Without pre-warping, the bilinear transform's frequency compression would shift the −3 dB point lower than intended.