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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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written 2.4 years ago by |
Solution:
Given y(n)=ay(n−1)+x(n)Y(z)=az−1Y(z)+X(z)Y(z)−az−1Y(z)=X(z)Y(z)[1−az−1]=X(z)H(z)=Y(z)X(z)=11−az−1=zz−aH(z−1)=z−1z−1−a
$ \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 …