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 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 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
.^{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).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University