The output signal of an A/D converter is passed through a lowpass filter with transfer function is given by H(z)=(1-a)z/(z-a) for 0<a<1.Find the steady state output noise power due to quantization at the output of the digital filter.</p>

Solution:

$ \begin{aligned}\\ & \sigma_{\varepsilon}^2=\sigma_e^2 \frac{1}{2 \pi j} \oint_c H(z) H\left(z^{-1}\right) z^{-1} d z-(r) \\\\ H(z) &=\frac{(1-a) z}{z-a} ; H\left(z^{-1}\right)=\frac{(1-a) z^{-1}}{z^{-1}-a} \\\\ \sigma_{\varepsilon}^2 &=\sigma_c^2 \frac{1}{2 \pi j} \oint_c \frac{(1-a){ }^2\left(z^{-1}\right) d z}{(z-a)\left(z^{-1}-a\right)} \\\\ &=\sigma_e^2\left[\text { residue of } H(z) H\left(z^{-1}\right) z^{-1} \text { at } z=a\right.\\\\ +& \text { residue …