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Tue Mar 19 15:17:14 EDT 2013

CT proto 2nd order filters.

Thinking about filters, re-building intuition..  These are all 2nd
order filters with p a complex pole and p' its complex conjugate.

* Band-pass filter (BPF).  Gain is zero at DC and infinity.  To have
  zero DC gain a zero at 0 is necessary.  To have have zero gain at
  infinity this should be the only zero.

          s
  ----------------
  (s - p) (s - p')


* High pass filter (HPF).  Derivative of BPF.  Limits to 1 when s->inf.

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

  High pass filters are closed under series composition (product).


* Low pass filter (LPF).  Integral of BPF.  Limits to 1 when s->0.

         pp'                   1
  ---------------- =  -------------------- 
  (s - p) (s - p')    (1 - s/p) (1 - s/p')

  A LPF is also an all-pole filter, meaning it has no (finite) zeros.
  Low pass filters are closed under series composition (product).


* All pass filter (APF).  Gain=1, only phase shift

  (s + p) (s + p')
  ----------------
  (s - p) (s - p')

  All pass filters are closed under series composition (product).


* Sum of one-pole filter (S1F).  Partial fraction expansion has no
  zeros.  Not sure if this is really so useful, but it does tend to
  show up from time to time, as they are the only ones closed under
  parallel composition (sum).

      s - Re(p)         1/2     1/2
  -----------------  = ----- + -----
  (s - p) (s - p')     s - p   s - p'




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