Enhancing Even Harmonics

A cubic nonlinearity, as well as any odd distortion law,^10.2 generates only odd-numbered harmonics (like in a square wave). For best results, and in particular for tube distortion simulation [31,398], it has been 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

$$f(x) = \alpha x^3 + \beta x^2 + \gamma x + \delta \protect$$ (10.6)

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.