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 the stability of the filter with the given transfer function H(z), we need to analyze the poles of the transfer function.
The transfer function H(z) can be written as:
H(z) = (0.35z^2 + 0.70z + 0.35) / (z^2 - 1.41z + 0.55)
Setting the denominator equal to zero gives us the characteristic equation:
z^2 - 1.41z + 0.55 = 0
We can find the roots of this characteristic equation using the quadratic formula:
z = [1.41 ± sqrt((1.41)^2 - 4*0.55)] / 2
Calculating the values gives:
z = [1.41 ± sqrt(1.9881 - 2.2)] / 2
z = [1.41 ± sqrt(-0.2119)] / 2
z = [1.41 ± 0.46j] / 2
The poles of the transfer function are complex conjugate pairs, and the real parts are less than 1 in magnitude. Therefore, the filter is stable.
Conclusion: The filter with the transfer function H(z) = (0.35z^2 + 0.70z + 0.35) / (z^2 - 1.41z + 0.55) is stable.