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With a hard knee you'd have an output as

output=Threshold+(Input-Threshold)/Ratio

Now, I found this pdf http://redmine.jamoma.org/attachments/download/175/JAES_V60_6_PG399.pdf

which lists equation

as a method to find the output given the knee, threshold, and input, so I tried this and here's what I got:

enter image description here

Which was supposed to be an example with a threshold of .5, knee of .1, and ratio of 3, yet the function isn't even continuous.

Is the equation from that pdf incorrect or am I interpreting this incorrectly?

2

After a bit more looking into it and experimenting I've found that the middle equation should be replaced with

xG - ((R - 1)/R)(xG - T + W/2)^2/(2W)

And it works correctly

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