Sampling the Impulse Response

Sampling is elementary. Since we have defined the admittance $\Gamma (s)$ as the nominal transfer function, corresponding to defining the input as driving force and the output as resulting velocity (see Fig.K.3), we have that $\gamma(t)$ is defined as the system impulse response $\gamma(t)\to T\gamma(nT) \to \gamma(n)$.^Q.2 We are therefore digitizing a linear system by sampling its impulse response. The model is then implemented as a Finite Impulse Response (FIR) digital filter (§1.5.4). The next subsection describes the impulse-invariant method for digital filter design which derives an infinite impulse response (IIR) digital filter that matches the analog filter impulse response at the sampling points.

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 which 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 (crucial in all musical instruments) or nonlinearity (also quite common), together with feedback, will ``pump'' the signal spectrum higher and higher until aliasing is ultimately encountered (see Appendix S). 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).

Sampling the ideal impulse response results in an FIR filter, even when the original continuous-time model is represented by poles and zeros in the Laplace $s$ plane. The impulse invariant method of digitizing an analog filter, discussed in the next section, yields IIR filters that match the continuous-time prototype at the sampling instants.