[sdiy] Tanh and other distortion functions

[Richie Burnett]

Tanh is accurate for modelling lots of common saturation mechanisms like 
those that occur in the long-tailed pair differential amplifier.

However, two challenges presented by Tanh from a DSP viewpoint are that it 
results in a broad output spectrum (so needs adequate oversampling to avoid 
introducing aliasing,)  and the curve rounds off gradually over a wide input 
range so you either need a big look-up table with interpolation, or a 
high-order polynomial to model it accurately.

My favourite *cheap* saturation function for DSP audio work is:  y = 1.5 x - 
0.5 x^3

This gives an s-shape saturation curve for input values that lie between -1 
and 1.  The gradient (dy/dx) also goes to zero at x=-1 and x=1 meaning that 
you can clip larger input magnitudes to the range -1<x<1 without incurring a 
discontinuity in y or the first differential.  The output essentially tends 
smoothly to -1 or +1 as the input tends to -1 or +1 and then remains there 
for x<-1 or x>1.

This polynomial is easily implemented efficiently on a DSP with MAC 
operations by re-arranging into the Horner form:  y = ( 1.5 - 0.5 x^2 ) x
It's also easy to implement on a micro that doesn't have hardware-multiply 
support because the polynomial coefficients of 1.5 and 0.5 can easily be 
realised using a combination of binary right-shift and add.

This function can also be used as a cheap wave-shaping function to convert a 
triangle wave into a sinewave.  Only odd harmonics exist in the output, the 
strongest of which being the 3rd at about -38dB.

-Richie,