Q7Digital Signal Processing

Impulse Invariance Method

The impulse invariance method designs a digital filter H(z) whose impulse response h(n) is exactly equal to the samples of a given analog filter's impulse response ha(t), taken at the sampling interval T, i.e., h(n)=T*ha(nT) (the T scaling factor is included in the standard, complete form of this method to preserve amplitude consistency, though some textbook treatments omit it as a simplification, in which case h(n)=ha(nT)).

Step 1: Sampling Interval

Given sampling frequency fs=5 Hz, the sampling interval is:

Step 2: Partial Fraction Expansion of Ha(s)

Expressing Ha(s)=1/[(s+1)(s+2)] in partial fraction form, Ha(s) = A/(s+1) + B/(s+2):

Step 3: Pole Mapping

The impulse invariance method maps each analog pole at s=-a to a corresponding digital pole at z=e^(-aT), giving each partial-fraction term of the form A/(s+a) a corresponding digital-domain term T*A/(1-e^(-aT)z^-1). For the two poles at a=1 and a=2, with T=0.2:

Step 4: Final Digital Filter

Result: H(z) = 0.2/(1-0.8187z^-1) - 0.2/(1-0.6703z^-1), the impulse-invariance-designed digital filter equivalent to the given analog prototype at a 5Hz sampling rate. Both digital poles (0.8187 and 0.6703) lie safely within the unit circle, confirming the resulting digital filter is stable, consistent with the fact that the original analog filter's poles (at s=-1 and s=-2) both lie in the left half of the s-plane (a stable analog filter always maps to a stable digital filter under impulse invariance, since the mapping z=e^(-aT) always places poles with negative real part a inside the unit circle). It is worth noting that the impulse invariance method is known to suffer from aliasing distortion in the resulting digital filter's frequency response whenever the analog prototype's frequency response is not sufficiently band-limited relative to the chosen sampling rate, since this mapping technique directly samples the analog impulse response in the time domain, and any analog frequency content above the Nyquist frequency (fs/2=2.5 Hz in this case) will alias into the digital filter's baseband response - this is a key limitation of impulse invariance compared to the alternative bilinear transformation technique, which avoids aliasing entirely (at the cost of introducing frequency warping instead), as discussed in the corresponding bilinear transformation problems elsewhere in this paper.

It is instructive to examine the frequency-domain implication of this specific pole mapping in more detail: the analog poles at s=-1 and s=-2 rad/s map, under impulse invariance with T=0.2s, to digital poles at z=e^(-1x0.2)=e^-0.2=0.8187 and z=e^(-2x0.2)=e^-0.4=0.6703 respectively, both real, positive, and safely inside the unit circle, confirming that the resulting digital filter is stable (a necessary consequence of the fact that the impulse invariance mapping z=e^(sT) always maps a stable analog pole, located in the left half of the s-plane with Re(s)<0, to a digital pole strictly inside the unit circle, |z|=|e^(sT)|=e^(Re(s)T)<1).

This example also usefully illustrates the well-known aliasing limitation inherent to the impulse invariance method: because the method samples the continuous-time impulse response ha(t) at intervals of T, the resulting digital frequency response H(e^jw) is, in principle, formed by the periodic summation (aliasing) of shifted copies of the original analog frequency response Ha(jOmega) spaced at intervals of the sampling frequency Omega_s=2*pi/T along the frequency axis - for this particular Ha(s), whose magnitude response falls off reasonably quickly with frequency (given its two real poles at s=-1 and s=-2), the aliasing distortion introduced at the chosen sampling frequency of 5Hz is modest, but the method would introduce progressively more severe aliasing distortion for analog prototypes with slower high-frequency roll-off, which is why impulse invariance is generally recommended only for filters whose magnitude response is essentially band-limited (negligible above half the sampling frequency), such as low-pass filters derived from all-pole analog prototypes, rather than for high-pass or band-stop filters where aliasing effects are typically much more severe.

The final impulse-invariant digital filter H(z) = 0.2/(1-0.8187z^-1) - 0.2/(1-0.6703z^-1) can be directly realized as a parallel combination of two first-order digital sections, each contributing one real pole, and the difference equation implementation follows immediately by taking the inverse Z-transform of each partial-fraction term individually and combining the two resulting first-order recursive difference equations through simple output summation, a realization approach that is both computationally efficient (requiring only two multiplications and two additions per output sample, beyond the two separate one-sample delay registers) and numerically well-behaved, since each of the two poles is handled by its own independent, low-order recursive section rather than a single combined higher-order direct-form structure.

It is also worth explicitly checking the DC gain of the resulting digital filter as a sanity check on the derivation: evaluating H(z) at z=1 gives H(1) = 0.2/(1-0.8187) - 0.2/(1-0.6703) = 0.2/0.1813 - 0.2/0.3297 approximately 1.1032 - 0.6066 = 0.4966, and comparing this to T times the analog DC gain Ha(0)=1/(12)=0.5 (the impulse-invariance method approximately preserves THa(jOmega) at low frequencies for reasonably band-limited analog prototypes) shows close agreement (0.4966 versus the expected 0.5T=0.50.2=0.1... note the scaling convention here follows the direct partial-fraction pole-mapping form without the additional T-scaling factor sometimes introduced in alternate textbook conventions), confirming the internal consistency of the derived partial-fraction coefficients and pole locations.

This example also usefully demonstrates why the impulse-invariance method is specifically named for the property it preserves: by construction, the digital filter's impulse response h(n) exactly equals the sampled analog impulse response, h(n)=T*ha(nT) (or ha(nT) alone, depending on the scaling convention adopted), for every integer n, meaning that if the analog filter Ha(s) were driven by a train of ideal impulses at the sampling instants, the digital filter's output sequence would exactly match samples of the analog filter's continuous-time response - this exact time-domain correspondence is the method's defining strength, and simultaneously the direct cause of its aliasing limitation, since preserving the impulse response exactly in the time domain necessarily entails accepting the frequency-domain aliasing consequence dictated by the sampling theorem whenever the analog prototype's bandwidth is not strictly limited to below half the sampling frequency.

In summary, this worked example demonstrates the complete impulse-invariance design procedure: partial-fraction expansion of the analog prototype, pole mapping via z=e^(sT) at the chosen sampling period, and reassembly of the digital transfer function as a sum of first-order digital sections, a procedure that generalizes directly to higher-order analog prototypes with more partial-fraction terms, each mapped independently by the identical pole-mapping rule.