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. J.3), we have that $\gamma(t)$ is defined as the system impulse response $\gamma(t)\to T\gamma(nT) \to \gamma(n)$.^P.2 We are therefore digitizing a linear system by sampling its impulse response. This is also known as the impulse invariant method in the context of digital filter design [323], because it preserves the impulse response exactly at the sampling points. The model is then implemented as a Finite Impulse Response (FIR) digital filter (see §1.5.4).

Sampling the impulse response has the advantage of preserving resonant frequency locations, but its big disadvantage is aliasing. 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, it should be lowpass filtered to eliminate all frequency components at half the sampling rate and above. In other words, the system itself should be ``lowpassed'' to avoid aliasing. Furthermore, we must agree to excite the system with input signals and initial conditions which are similarly bandlimited to less than half the sampling rate. If the system remains linear and time invariant, this will ensure no signal inside the system or appearing at the outputs will be aliased.

Note 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 R). For this reason, all feedback loops in the digital system must attenuate 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).