In trying to answer another question, I got stuck applying what I thought would be a valid way to answer it: using the system function. Here's what I did:
First, I put the question
Y[i] = Y[i-1] + ALPHA * ( X[i] + Y[i-1] )
into DF I:
Y[i] = ALPHA * X[i] - (ALPHA-1) * Y[i-1]
This tells me that the system function is:
H(z) = ALPHA / ( 1 + (ALPHA-1) z^-1 )
Substituting:
z=e^(jw)
and
|H(z)| = .5 (or .707)
should allow me to calculate ALPHA, but I only get complex values for ALPHA. What am I doing wrong?