Is there a name for a single-pole recursive digital lowpass filter where the coefficients don't quite add up to 1?
For example: with the filter
y[n]=a0*x[n]+b1*y[n-1] instead of using coefficients
What if I used
Is there a name for this, or is it just functionally equivalent to something else like
Also, at what point will the filter become unstable? The example above where the coefficients add up to 1.01 looks stable to me but I think a0=0.009 and b1=1.001 would be unstable. Where's the line where it crosses from stable to unstable? b1=1?