Problem $3)(50$ Points$)$

Consider an LTI DT system with frequency response given by:

$H(e^{jw})=\frac{1-e^{-j2w}}{1+0\mathrm{.}25e^{-j4w}}$

a$)$ Determine $H(z),$ plot its poles and zeros on a pole$/$zero plot, and use that plot to sketch magnitude response of the system for $-w.$

b$)$ If the system input is $x[n]=3+cos(\frac{n}{4})+cos(n),$ determine the steady$-$state output $y[n].$

c$)$ Use the z$-$transform to determine the impulse response of this system.

d$)$ Determine the difference equation for this system by inspection.

e$)$ Repeat part $($a$)$ if the numerator and denominator were switched. Does this agree with your knowledge of system stability?

Note: In part $3($a$),$ sketch the magnitude response approximately using the graphical approach.