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Tue Mar 19 16:49:07 EDT 2013

Partial fraction forms

http://www.wolframalpha.com/widgets/view.jsp?id=ec4a062bb304f88c2ba0b631d7acabbc

 LP(s)

 = pp' / (s - p) (s - p') 

      a          b
 = -------  + -------
    s - p      s - p'


   
  =>   as  + bs  = 0   => a = -b
     - ap' - pb' = pp' => a (p - p') = pp'

  or a = pp' / (p - p') = 2 |p|^2 / Im(p)


The wave form in the response is thus 90 degree shifted from a normal
exponential, of which the real part is a cosine and the imaginary part
a sine.  I.e. the response is a sine wave, gradually rising to follow
the pulse.  It doesn't jump immdiately, as would a high-pass filter.

 HP(s)

 = s^2 / (s - p) (s - p')

 = 1 + p^2 / 2 Im(p) (s-p)  + p'^2 / 2 Im(p) (s-p')

This has a direct path + a phase-shifted sinusoid.

 


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