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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

$\displaystyle 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

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


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA