\begin{problem}
Graph the magnitude response and determine the first difference equation for the output signal $y(n)$ if the frequency response is given by:


$$
H(\omega) = \frac{ \left( e^{i\omega} - e^{i\frac{\pi}{4}}\right) \left(e^{i\omega} - e^{-i\frac{\pi}{4}}\right)}{ \left( e^{i\omega} - \alpha e^{i\frac{\pi}{4}}\right) \left(e^{i\omega} - \alpha e^{-i\frac{\pi}{4}}\right)   }
$$
\end{problem}

\begin{solution}
We can break this problem down to its four component vectors and find that it is a \emph{notch filter} that zeros out frequencies at $\pm \pi/4$ and is approximately 1 everywhere else.

To determine the difference equation, lets simplify the expression:

\begin{align*}
H(\omega) &= \frac{ \left( e^{i\omega} - e^{i\frac{\pi}{4}}\right) \left(e^{i\omega} - e^{-i\frac{\pi}{4}}\right)}{ \left( e^{i\omega} - \alpha e^{i\frac{\pi}{4}}\right) \left(e^{i\omega} - \alpha e^{-i\frac{\pi}{4}}\right)   } \\
H(\omega) &= \frac {e^{i2\omega} - e^{i\omega}e^{-i\pi/4}-e^{i\omega}e^{i\pi/4}+1}{e^{i2\omega} - \alpha e^{i\omega}e^{-i\pi/4}-\alpha e^{i\omega}e^{i\pi/4}+\alpha^2} \\
H(\omega) &= \frac {e^{i2\omega} - e^{i\omega} \left( e^{-i\pi/4}+e^{i\pi/4} \right) +1}{e^{i2\omega} - \alpha e^{i\omega} \left( e^{-i\pi/4}+e^{i\pi/4}\right)+\alpha^2} \\
H(\omega) &= \frac {e^{i2\omega} - e^{i\omega} \left( 2\cos(\frac{\pi}{4}) \right) +1}{e^{i2\omega} - \alpha e^{i\omega} \left( 2\cos(\frac{\pi}{4})\right)+\alpha^2} \\
H(\omega) &= \frac {e^{i2\omega} - e^{i\omega} \left( \sqrt{2} \right) +1}{e^{i2\omega} - \alpha e^{i\omega} \left( \sqrt{2}\right)+\alpha^2} \\
H(\omega) &= \frac {1 - \sqrt{2}e^{-i\omega} + e^{-i2\omega}}{1 - \alpha \sqrt{2}e^{-i\omega}+\alpha^2e^{-i2\omega}} \mbox{ multiply by } e^{-2\omega} \mbox{ on top and bottom}\\
\end{align*}

Now we have a much simpler form:
$$H(\omega) = \frac {1 - \sqrt{2}e^{-i\omega} + e^{-i2\omega}}{1 - \alpha \sqrt{2}e^{-i\omega}+\alpha^2e^{-i2\omega}} $$

Lets cross multiply (imagine having a 1 beneath $H(\omega)$:

\begin{align*}
H(\omega) - \alpha\sqrt{2}H(\omega)e^{-i\omega} + \alpha^2H(\omega)e^{-i2\omega} &= 1 - \sqrt{2}e^{-i\omega} + e^{-i2\omega} \\
H(\omega)e^{i\omega n} - \alpha\sqrt{2}H(\omega)e^{i\omega(n-1)} + \alpha^2H(\omega)e^{i\omega(n-2)} &= e^{i\omega n}  - \sqrt{2}e^{i\omega(n-1)} + e^{i\omega(n-2)} \\
y(n) - \alpha\sqrt{2}y(n-1) + \alpha^2y(n-2) &= x(n)  - \sqrt{2}x(n-1) + x(n-2)
\end{align*}

\end{solution}

Notch Filter