Impulse Invariance

The impulse invariance method is a technique for designing IIR digital filters by converting an analog filter prototype directly into a digital filter such that the digital filter's impulse response is a sampled version of the analog filter's impulse response. It is one of the two main s-plane to z-plane mapping techniques taught in DSP courses, the other being the bilinear transformation. Understanding the mapping and its aliasing limitation is essential for GATE.

Core Concept Explanation

The impulse invariance method begins with an analog filter that has a known transfer function H_a(s). The analog impulse response h_a(t) is obtained from H_a(s) by inverse Laplace transform. The digital filter's impulse response is then defined as h[n] = T·h_a(n·T), where T is the sampling period (T = 1/Fs). This means the digital filter's response at each sample instant n exactly matches the analog filter's response at the corresponding time n·T, which is why the method is called impulse invariant.

In practice, the analog transfer function is expressed as a partial fraction expansion: H_a(s) = sum of A_k / (s - s_k), where s_k are the poles of the analog filter. Under impulse invariance mapping, each analog pole at s = s_k maps to a digital pole at z = e^(s_k·T). The corresponding digital transfer function is: H(z) = sum of T·A_k / (1 - e^(s_k·T)·z^(-1)). This direct pole mapping preserves stability because poles in the left half of the s-plane (σ < 0) map to locations inside the unit circle in the z-plane.

The most critical limitation of impulse invariance is frequency-domain aliasing. The digital frequency response H(e^jω) is related to the analog frequency response by: H(e^jω) = (1/T) · sum over k of H_a(j(ω/T + 2πk/T)). This sum shows that the digital spectrum is the periodic repetition of the analog spectrum with period 2π/T (or equivalently ωs = 2π·Fs). If the analog filter is not strictly bandlimited to half the sampling frequency, these periodic copies overlap and create aliasing distortion in the digital filter's frequency response.

Mathematical Expression

The mapping relationship between the s-plane and z-plane under impulse invariance is z = e^(sT). Substituting s = σ + jΩ, we get z = e^(σT)·e^(jΩT). The magnitude is |z| = e^(σT) and the angle is arg(z) = ΩT. Since e^(jΩT) is periodic in Ω with period 2π/T = Ωs (the sampling frequency), the entire vertical strip σ + j[-Ωs/2, Ωs/2] in the s-plane maps to the entire z-plane. All other strips map to the same z-plane values, which is the fundamental cause of aliasing. For the mapping to be accurate, the analog filter must have negligible energy above Ωs/2.

Practical Understanding

Because of aliasing, the impulse invariance method is only suitable for lowpass and bandpass filters where the analog frequency response decays sufficiently before Ωs/2. High-pass and band-stop filters have significant energy near ω = π in the digital domain, where the aliased copies cause severe distortion. For these cases, the bilinear transformation is always preferred because it has no aliasing.

The advantage of impulse invariance is that it preserves the time-domain shape of the response. For applications where the temporal behavior of the filter output must match the analog filter closely, such as in simulation of analog systems or in control applications, impulse invariance gives a physically meaningful digital equivalent. The bilinear transform, while alias-free, introduces frequency warping that changes the time-domain waveform.

Numerical Example

Given a first-order analog lowpass filter, apply the impulse invariance method to find the equivalent digital filter transfer function. The analog pole directly maps to a digital pole using z = e^(sT).

Given:
Analog prototype: H_a(s) = 1 / (s + 2) (first-order LPF, pole at s = -2)
Sampling frequency: Fs = 10 Hz, so T = 0.1 s

Why this formula applies:
Partial fraction gives A_1 = 1, pole s_1 = -2.
Impulse invariance maps each pole: z_k = e^(s_k × T)

Formula:
H(z) = T × A_k / (1 - e^(s_k·T)·z⁻¹)

Substitution:
z_1 = e^(-2 × 0.1) = e^(-0.2)
e^(-0.2) = 0.8187
H(z) = 0.1 / (1 - 0.8187·z⁻¹)

Calculation:
Numerator = T × A_1 = 0.1 × 1 = 0.1
Denominator pole at z = 0.8187, which is |0.8187| < 1 → stable

Final Answer:
H(z) = 0.1 / (1 - 0.8187·z⁻¹)
Digital pole at z = 0.8187 (inside unit circle, filter is stable)

Exam Tip: GATE frequently asks to identify the limitation of impulse invariance. The answer is always aliasing due to the periodic nature of the z = e^(sT) mapping. Also remember that impulse invariance is NEVER used for high-pass or band-stop filter design. The bilinear transformation is alias-free but introduces frequency warping, which requires pre-warping the analog prototype frequencies before design.

Mechanism of Aliasing in Impulse Invariance

• The mapping z = e^(sT) is many-to-one: the entire left half s-plane maps inside the unit circle, but each vertical strip of width 2π/T in the s-plane maps to the entire z-plane, causing spectral periodicity.
• For an analog lowpass filter with significant rolloff before Ωs/2, aliasing is negligible because the spectrum copies do not meaningfully overlap. This is the valid use case.
• For a highpass filter, the stopband is at low frequencies and the passband extends to Ωs/2 and beyond. The repeated copies in the digital domain completely distort the intended frequency response.
• Stability is preserved: a left half-plane analog pole (σ < 0) always maps to inside the unit circle (|z| = e^(σT) < 1), so a stable analog filter always produces a stable digital filter.
• The factor T in h[n] = T·h_a(nT) is a gain normalization that ensures the DC gain of the digital filter equals the DC gain of the analog filter when the sampling frequency is accounted for.

Quick Revision

• Impulse invariance sets h[n] = T·h_a(nT). Digital impulse response is a sampled version of the analog impulse response.
• Pole mapping: analog pole at s = s_k → digital pole at z = e^(s_k·T).
• Digital transfer function: H(z) = sum T·A_k / (1 - e^(s_k·T)·z^(-1)) from partial fraction of H_a(s).
• Aliasing occurs because z = e^(sT) is periodic in Ω with period Ωs = 2π/T.
• Valid only for lowpass and bandpass designs where analog spectrum is negligible above Ωs/2.
• Never use for highpass or bandstop: use bilinear transformation instead.
• Exam trap: Impulse invariance preserves stability (left half s-plane → inside unit circle) but does NOT preserve the frequency response shape exactly due to aliasing.