On 2010-09-12 12:05, Mathieu Bouchard wrote:
On Sun, 12 Sep 2010, Martin Peach wrote:It's not the capacitors, it's the amplifier losing gain when it approaches the power supply.Yeah, but it seems to be a pattern similar to the one found in capacitors, because capacitor theory has exp(-x) all over it, and the only way that capacitors behave like [hip~] is when the signal is much below the capacity rating (µF)... otherwise they lose gain... when they don't, it's because exp(-x) can be well approximated by x.
I guess it's similar since capacitors charge at a rate proportional to a voltage difference, while transistors can supply charge carriers at a rate proportional to a voltage difference, so caps charge fastest when they are nearly empty and transistors have the best gain with small signal inputs.
The whole universe has exp written all over it in fact...
And then, exp is very close to tanh in several different ways, one of them being this (use gnuplot) : plot [-2:2] [-1:1] exp(x*sqrt(2))-1, 1-exp(-x*sqrt(2)), tanh(x), x I put the plain 'x' at the end to show what I mean above (though you already know that)
Of course, all the hyperbolic trig functions are made from exp, by definition.
http://en.wikipedia.org/wiki/Hyperbolic_functionAnother use of exp is the sigmoid function used in biology, that can be used to make a soft transition from one state to another as in 'fuzzy logic'.
Martin
sigmoid_neuron.pd
Description: application/puredata
sigmoid_neuron-help.pd
Description: application/puredata
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