Sampling the Impulse Response

Sampling the impulse response of a system is of course quite elementary. The main thing to watch out for is aliasing, and the main disadvantage is high computational complexity when the impulse-response is long.

Since we have defined (in §7.2) the driving-point admittance $\Gamma (s)$ as the nominal transfer function of a system port, corresponding to defining the input as driving force and the output as resulting velocity (see Fig.7.3), we have that $\gamma(t)$ is defined as the system impulse response. Note, however, that the driving force and observed velocity need not be at the same physical point, and in general we may freely define any physical input and output points. Nevertheless, if the outputs are in velocity units and the inputs are in force units, then the transfer-function matrix will have units of admittance, and we will assume this for simplicity.

Sampling the impulse response can be expressed mathematically as $\gamma(t)\to T\gamma(nT) \to \gamma(n)$ .^9.2 In practice, we can only record a finite number of impulse-response samples. Usually a graceful taper (e.g., using the right half of an FFT window, such as the Hann window) yields better results than simple truncation. The system model is then implemented as a Finite Impulse Response (FIR) digital filter (§2.5.4). The next section describes the related impulse-invariant method for digital filter design which derives an infinite impulse response (IIR) digital filter that matches the analog filter impulse response exactly at the sampling times.

Sampling the impulse response has the advantage of preserving resonant frequencies (see next section), but its big disadvantage is aliasing of the frequency response. No ``system'' is truly bandlimited. For example, even a simple mass and dashpot with a nonzero initial condition produces a continuous decaying exponential response that is not bandlimited.

Before a continuous impulse response is sampled, a lowpass filter should be considered for eliminating all frequency components at half the sampling rate and above. In other words, the system itself should be ``lowpassed'' to avoid aliasing in many applications. (On the other hand, there are also many applications in which the frequency-response aliasing is not objectional to the ear.) If the system is linear and time invariant, and if we excite the system with input signals and initial conditions that are similarly bandlimited to less than half the sampling rate, no signal inside the system or appearing at the outputs will be aliased. In other words, these conditions yield an ideal bandlimited system simulation that remains exact (for the bandlimited signals) at the sampling instants.

Note, however, that time variation or nonlinearity (both common in practical instruments), together with feedback, will ``pump'' the signal spectrum higher and higher until aliasing is ultimately encountered (see §6.13). For this reason, feedback loops in the digital system may need additional lowpass filtering to attenuate newly generated high frequencies.

A sampled impulse response is an example of a nonparametric representation of a linear, time-invariant system. It is not usually regarded as a physical model, even when the impulse-response samples have a physical interpretation (such as when no anti-aliasing filter is used).