How it works
Design procedure: expand H_a(s) in partial fractions as Σ Ak/(s−sk). The digital filter H(z) = Σ Ak·T/(1−e^(sk·T)·z⁻¹), where T is the sampling period. Each s-plane pole at sk maps to a z-plane pole at e^(sk·T). Poles in the left-half s-plane (Re(sk) < 0) map inside the unit circle, preserving stability. The frequency response of the digital filter is H(e^(jω)) ≈ (1/T)·Σ Ha(j(ω−2πk)/T), showing that the digital response is the sum of shifted copies of the analog response — this aliasing is unavoidable and worsens as the analog filter's stopband attenuation is insufficient before the first alias appears at ω = π.
Key points to remember
Aliasing is the fundamental limitation of impulse invariance: the method cannot be used for high-pass or band-stop filters because their magnitude response does not decay to zero before fs/2, causing severe aliasing of the stopband into the passband. Impulse invariance preserves the shape of the impulse response and matches the analog frequency response well at frequencies well below π/T. The bilinear transform is preferred in modern design precisely because it eliminates aliasing at the cost of frequency warping. Both methods give the same filter order as the analog prototype. For a Butterworth analog prototype, the partial fraction expansion always has N distinct poles, each contributing one first-order section; pairing complex conjugate poles gives biquad (second-order section) implementations.
Exam tip
The examiner always asks you to compare impulse invariance and bilinear transform on four criteria — aliasing, frequency warping, suitable filter types, and stability preservation — write these four rows as a table and fill each cell concisely before writing any derivation.