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Tue Jan 4 12:47:34 EST 2011

Practical d-plane formulations

As a continuation of [1].

Given a d-plane formulation, how to implement it?

It's simplest to start from the leaky discrete integrator and see what
it's d-plane formulation looks like.  The time domain formulation

   y[k+1] - y[k]  =  a y[k] + b x[k]
   y[k+1]        +=  ...

expresses the update of y in terms of the weighted magnitude of y and
x.  It corresponds to the frequency domain formulation

   d y  =  a y  + b x

   (d - a) y = b x

   y / x = b / d - a

The non-leaky integrator is then b / d.

Complex poles are handled in exaclty the same way: We take the complex
form of one of the complementary pairs, feed it a real input and
taking only the real part of the integrator as as output.

TODO:

  - Make the 2nd order implementation explicit.  Handwaving might be
    wrong.

  - What about d-zeros?




[1] entry://20110104-110954


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