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

• oversampling, and
• a lowpass filter to after the nonlinearity.

To avoid 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).

Another variation is to oversample by two, in which case there is aliasing, but that aliasing does not reach the ``base band.'' Therefore, a half-band lowpass filter rejects both the second spectral image and the third, which is aliased onto the second.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.