Next  |  Prev  |  Up  |  Top  |  REALSIMPLE Top

Cubic Nonlinear Distortion

To minimize aliasing, it is helpful to use nonlinearities that are approximated by polynomials of low order. An often-used cubic nonlinearity is given by [17]

f(x) = \left\{\begin{array}{ll}
-\frac{2}{3}, & x\leq -1 \\ [5pt]
x-\frac{x^3}{3}, & -1 < x < 1 \\ [5pt]
\frac{2}{3}, & x\geq 1. \\
\end{array} \right.
\end{displaymath} (6)

and diagrammed in Fig.11.13 An input gain may be used to set the desired degree of distortion. Analysis of spectral characteristics and associated aliasing due to nonlinearities appears in [14]. As discussed there, a non-saturating cubic nonlinearity does not alias at all when the input signal is oversampled by 2 or more and the nonlinearity is followed by a half-band lowpass filter, which eliminates aliasing since it is confined to the upper half-spectrum between $\pi/2$ and $\pi$ rad/sample. High quality commercial guitar distortion simulators are said to use oversampling factors of 4 to 8.

Figure 11: Soft-clipper defined by Eq.$\,$(6).
\resizebox{3in}{!}{\includegraphics{\figdir /cnl.eps}}

The cubic nonlinearity, being an odd function, produces only odd harmonics. To break the odd symmetry and bring in some even harmonics, a simple input offset can be used [10]. It was found empirically that a dc blocker [12] was needed to keep the signal properly centered in the output dynamic range. Since amplifier loudspeakers have a $+12$ dB/octave low-frequency response, at least two dc blockers are appropriate anyway.

While the cubic nonlinearity is the odd nonlinearity with the least aliasing (thereby minimizing oversampling and guard-filter requirements), it is sometimes criticized as overly weak as a nonlinearity, unless driven into the hard-clipping range where it is no longer bandlimited to three times the input signal bandwidth.

Next  |  Prev  |  Up  |  Top  |  REALSIMPLE Top

Download faust_strings.pdf

``Making Virtual Electric Guitars and Associated Effects Using Faust'', by Julius O. Smith III,
REALSIMPLE Project — work supported in part by the Wallenberg Global Learning Network .
Released 2013-08-22 under the Creative Commons License (Attribution 2.5), by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University