Tue Mar 19 22:43:19 EDT 2013
( EDITED 2013-5-7 - Original was wrong. It completely missed the
forward/backward thing and use a forward different transformation. )
According to , the direct digital approximation of the SVF is
stable for low frequencies, even at q=0 (Q=inf).
In Stilson's PhD  this is called the Chamberlin filter. Note that
this is _not_ a trivial transformation of the CT equations. It has
one forward and one backward difference to replace the 2 integrators.
See also . In this notation, the uptdate equations are g
_sequential_, where the update for s2 is used in the update for s1.
This in contrast to the usual formulation as simultaneous updates,
i.e. state space form.
s2' = s2 + f s1 : LP
s1' = s1 - f (q s1 + s2' + i) : HP
Interpreting these equations as simultaneous, i.e. using s2 instead of
s2' in the second equation, gives a forward difference approximation
and makes the filter unstable.
The parameters f and q are related to their continuous time
f = 2 sin ( pi F_c / F_s )
q = 1 / Q
Even in the presence of oversampling, this approach is a lot simpler
than the direct pole computation and interpolation approach from the
last couple of posts.