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Tue Mar 19 12:46:18 EDT 2013

## Standard filters from pole positions.

How to derive the LP/HP/BP/BS filters from a 2-pole discrete state
variable filter? There seem to be a couple of degrees of freedom here
that require a bit of extra information.
I'm picking the "orthogonal" SVF because it seems easiest to control.
Practically, I'm looking for B,C,D in
s_k+1 = A s_k + B i_k
o_k = C s_k + D i_k
where A is [ p_x p_y ] with (p_x, p_y) the pole location.
[ -p_y p_x ]
with i,o 1-dimensional, this boils down to 2+2+1 parameters.
Transfer frunction is:
o = (C (zI - A)^1 B + D) i
Maybe it's best to start from the relation HP = 1 - LP, and focus on
LP, then find out how to derive BP and BS. The latter 2 are not as
important as the LP, HP pair.
For LP, we need unit gain at z=1 or
C (I - A)^1 B + D = 1
Another way to go is to look at the 2D equation in terms as if it were
a 1D complex equation.
It's probably simpler to think in terms of complex numbers here.
Basically, a real 2-pole is the real part of a complex 1-pole.

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