Determine if the filter with the following transfer function H(z) is stable. Show your work.
H(z) = (0.35z^2 + 0.70z + 0.35) / (z^2 - 1.41z +0.55)
To determine stability, we need to consider the poles of the transfer function H(z). If all the poles are inside the unit circle (i.e., have a magnitude less than 1), then the filter is stable.
In this case, we can find the poles by setting the denominator of H(z) equal to zero and solving for z:
z^2 - 1.41z + 0.55 = 0
Using the quadratic formula, we get:
z = (1.41 ± sqrt(1.41^2 - 4(1)(0.55))) / 2
z = 0.7044 or z = 0.7796
Both of these poles have a magnitude less than 1, so the filter is stable.