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Find the steady-state variance of the noise in the output due to quantization of input for the first order filter y(n)=ay(n-1)+x(n).
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Solution:

 Given y(n)=ay(n1)+x(n)Y(z)=az1Y(z)+X(z)Y(z)az1Y(z)=X(z)Y(z)[1az1]=X(z)H(z)=Y(z)X(z)=11az1=zzaH(z1)=z1z1a

$ \begin{aligned}\\ \sigma_{\varepsilon}^2 &=\sigma_e^2 \frac{1}{2 \pi j} \oint_C A(z) H\left(z^{-1}\right) z^{-1} d z \\\\ &=\sigma_e^2 \frac{1}{2 \pi j} \oint_c \frac{z}{z-a} \frac{z^{-1}}{z^{-1}-a} z^{-1} d …

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