Practical Advice

In summary, the following pointers can be offered regarding nonlinear elements in a digital waveguide model:

• Verify that aliasing can be heard and sounds bad before working to get rid of it.

• Aliasing (bandwidth expansion) is reduced by smoothing ``corners'' in the nonlinearity.

• Consider an oversampling factor for nonlinear subsystems sufficient to accommodate the bandwidth expansion caused by the nonlinearity.

• Make sure there is adequate lowpass filtering in a feedback loop containing a nonlinearity.

As a specific example, consider the cubic nonlinearity used in a feedback loop (as in §9.1.6). This can be done with no aliasing at low levels (i.e., at levels below hard clipping) provided we use

• $3\times$ oversampling, and
• a lowpass filter with passband $\omega\in[-\pi/3,\pi/3]$ after the nonlinearity.

To avoid $3\times$ oversampling in the entire feedback loop, we may downsample by 3 after the lowpass filter and upsample by 3 just before the nonlinearity. If the lowpass filter is good, the downsampling by 3 is trivially accomplished by throwing away every 2 out of 3 samples. For upsampling, however, an additional third-band lowpass-filter is needed for the interpolation (§4.4).

A more agressive antialiasing scheme is to oversample by only two (or between 2 and 3) for cubic nonlinearities, in which case there is aliasing in the transition band, but it does not reach the passband. More generally, for an $N$ th-order nonlinearity, oversampling by $N/2$ suffices to keep aliasing out of the passband. This is a reasonable choice when the passband is the full audio band, or (using a bit more oversampling) when the lowpass filter is high order.