Dan Slater's Article in CMJ, and X to the kth power - how do you do it?

[Martin Czech]

>> state-variable filter, but with the inverting stage replaced with an x^3
> stage (x cubed) and the signal input connected to the first integrator
> stage.  Slater's hypothetical module uses an x to the k function, where
> the circuit can be converted from a state variable topology to an Ueda
> Attractor by changing the value of k (for a state-variable filter, k=1;
> for the Ueda attractor, k=3).
> 
> How the heck would you do this?  Is there some common circuit that
> produces an x to the k function, where k can smoothly vary (i.e. by
> non-integer values) between 1 and 4?  x^1 is simple; x^2 could be a
> multiplier where the input is multiplied by itself; x^3 could be created
> by a second multiplier multiplying the input by the product of an x^2
> stage.  But how do you get, say, x^2.74?  And how do you vary these
> smoothly?  Could you use feedback from a multiplying stage, where a
> signal is multiplied by the output of the multiplier?  Or maybe feedback
> in a 2-multiplier configuration?
> 

I can't see how feedback will change the order of the circuit equation.

Stupid 1st order approach:

since log(x^k)=k*log(x)

x^k=exp{log(x^k)}=exp{k*log(x)}

a log,exp and multiplier circuit. And lots of problems: works only for x^k > 0.  A full-wave rectifier could enshure x^k >=0, the equal sign could
be avoided with some comparator logic, ie. for very small |x| ~ 0
another circuit could be switched in.  Ok, too complicated, not steady
arround 0. Next candidate , please.

m.c.

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