I'm attempting to calculate the steady-state state variables for a digital biquad filter direct form II (transposed). Illustration
For example, let assume the filter is fed a constant input of magnitude C. What would be the best method to calculate the internal state variables as the filter approaches steady state?
When I run a simulation, it appears that S1 and S2 are negatives of one another (S1 = -S2). Is there a way to calculate the exact values?
My best guess is that I need to set-up a system of equations and then find the steady-state output (given constant input C). However, my calculations seem to fall apart when I actually try to doing them.
Edit: So after working this problem a little longer, I was able to determine the following:
$$ S_{1}[n] = X[n]\cdot (Gain_{DC} - b_{0}) $$
The DC gain is then found by taking the Z transform of difference equations and setting Z = 1.