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 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).

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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FDA in the Frequency Domain
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Stability of Nonlinear Feedback Loops