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 [31,395], 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

where is some small constant. (Signals 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

where 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 can be set to zero the mean, on average; if the input signal is zero-mean with variance is 1, then will cancel the nonzero mean induced by the squaring term . 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 can be used for this, provided the constant gains for and are modified accordingly, or is hard-clipped to the desired range at the input.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.