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.