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)

1 answer

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.