Nonlinear Distortion

While most musical vibrating strings are well approximated as linear, time-invariant systems, there are special cases in which nonlinear behavior is desired. This section summarizes a few examples.

One class of nonlinearity is purposely created in the design of some acoustic stringed instruments. For example, the Finnish Kantele exhibits nonlinear characteristics [215]. Perhaps a better known example is the Indian sitar, in which a curved ``jawari'' (functioning as a nonlinear bridge) serves to shorten the string gradually as it displaces toward the bridge. The Indian tambura also employs a thread perpendicular to the strings a short distance from the bridge, which serves to shorten the string whenever string displacement toward the bridge exceeds a certain distance. Finally, the slap bass playing technique for bass guitars involves hitting the string hard enough to cause it to beat against the neck during vibration [344]. In all of these cases, the string length is physically modulated in some manner each period, at least when the amplitude is sufficiently large.

Another widespread class of distortion, used in electric guitars, is clipping of the guitar waveform. While this more properly belongs in the category of ``virtual analog'' synthesis techniques [54,454,455,254,64], it is easy to add this effect to any string-simulation algorithm by passing the output signal through a nonlinear clipping function [464]. For example, a hard clipper has the characteristic (in normalized form)

$$f(x) = \left\{\begin{array}{ll} -1, & x\leq -1 \\ [5pt] x, & -1 \leq x \leq 1 \\ [5pt] 1, & x\geq 1 \\ \end{array} \right. \protect$$ (5.15)

where $x$ denotes the current input sample $x(n)$, and $f(x)$ denotes the output of the nonlinearity.

A soft clipper is similar to a hard clipper, but with the corners smoothed. A common choice of soft-clipper is the cubic nonlinearity, e.g. [464],

$$f(x) = \left\{\begin{array}{ll} -\frac{2}{3}, & x\leq -1 \\ [5pt]... ...1 \leq x \leq 1 \\ [5pt] \frac{2}{3}, & x\geq 1 \\ \end{array} \right. \protect$$ (5.16)

This particular soft-clipping characteristic is diagrammed in Fig. 4.22. An analysis of its spectral characteristics, with some discussion of aliasing it may cause, appears in Appendix R. An input gain may be used to set the desired degree of distortion.

Figure 4.22: Soft-clipper defined by Eq. (4.16).

A cubic nonlinearity, as well as any odd distortion law,^5.12 generates only odd-numbered harmonics (like in a square wave). For best results, and in particular for tube distortion simulation [29,367], it can be argued that some amount of even-numbered harmonics should also be present. Breaking the odd symmetry in any way will add even-numbered harmonics to the output as well. One simple way to accomplish this is to add an offset to the input signal, obtaining

$$y(n) = f[x(n) + c],$$

where $c$ is some small constant. (Signals $x(n)$ in practice are typically constrained to be zero mean by one means or another.)

Another method for breaking the odd symmetry is to add some square-law nonlinearity to obtain

$$f(x) = \alpha x^3 + \beta x^2 + \gamma x + \delta \protect$$ (5.17)

where $\beta$ controls the amount of square-law distortion in the more general third-order polynomial. The square-law is the most gentle nonlinear distortion in existence, adding only some second harmonic to a sinusoidal input signal. The constant $\delta$ can be set to zero the mean, on average; if the input signal $x(n)$ is zero-mean with variance is 1, then $\delta= - \beta$ will cancel the nonzero mean induced by the squaring term $\beta x^2$. Typically, the output of any audio effect is mixed with the original input signal to allow easy control over the amount of effect. The term $\gamma$ can be used for this, provided the constant gains for $x>1$ and $x<-1$ are modified accordingly, or $x$ is hard-clipped to the desired range at the input.