Determine the frequency response and impulse response, $$ y(n)-\frac{1}{6} y(n-1)-\frac{1}{6} y(n-2)=x(n) $$

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written 3.0 years ago by

prajapatijaimin • 3.3k

Solution:

$ y(n)-\frac{1}{6} y(n-1)-\frac{1}{6} y(n-2)=x(n)\ $

Applying DTFT,

$ Y\left(e^{j \omega}\right)-\frac{1}{6} e^{-j \omega} Y\left(e^{j \omega}\right)-\frac{1}{6} e^{-2 j \omega} Y\left(e^{j \omega}\right)=X\left(e^{j \omega}\right)\ $

Frequency response,

$ H\left(e^{j \omega}\right)=\frac{Y\left(e^{j \omega}\right)}{X\left(e^{j \omega}\right)}=\frac{1}{1-\frac{1}{6} e^{-j \omega}-\frac{1}{6} e^{-2 j \omega}}\\ $

$ =\frac{e^{2 j \omega}}{e^{2 j \omega}-\frac{1}{6} e^{j \omega}-\frac{1}{6}}\\ $

$ \begin{gathered}\\ \frac{H\left(e^{j \omega}\right)}{e^{j \omega}}=\frac{e^{j \omega}}{e^{2 …

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