a.
$ \begin{align} n &= ...\text{ -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7} ...\\ x\left[n\right] &= ... \text{ 0, 0, 0, 1, -1, 1, -1, 1, 0, 0, 0} ...\\ y\left[n\right] &= ...\text{ 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2 } ...\\ y\left[n\right] &= u\left[n\right] + u\left[n - 5\right] \end{align} $
b.
(i) Linear - Yes
(ii) Time invariant - Yes
(iii) Memoryless - No
(iv) Causal - Yes
(v) BIBO Stable - No (ex. when x[n] = 1 for all n)
c.
$ \begin{align} Y(\omega) &= X(\omega) + e^{-j\omega}X(\omega) + e^{-j\omega}Y(\omega) \\ Y(\omega)(1 - e^{-j\omega}) &= X(\omega)(1 + e^{-j\omega}) \\ H(\omega) &= \frac{Y(\omega)}{X(\omega)} = \frac{1 + e^{-j\omega}}{1 - e^{-j\omega}} \\ H(\omega) &= \frac{e^{-j\omega /2}(e^{j\omega /2} + e^{-j\omega /2})}{e^{-j\omega /2}(e^{j\omega /2} - e^{-j\omega /2})} \\ H(\omega) &= \frac{cos(\omega /2)}{jsin(\omega /2)} \end{align} $
d.
$ \begin{align} x[n] &= sin[\pi n/4] \\ x[n] &= \frac{e^{j \pi n/4} - e^{-j \pi n/4}}{2j} \end{align} $
In an LTI System, when
$ \begin{align} x[n] &= e^{j \omega_0 n}, \\ y[n] &= H(\omega_0)e^{j \omega_0 n} \end{align} $
Using H($ \omega $) from part c,
$ \begin{align} y[n] &= \frac{H(\pi /4)e^{j \pi n/4} - H(-\pi /4)e^{-j \pi n/4}}{2j} \\ y[n] &= \frac{1}{2j} \left( \frac{cos(\pi /8)e^{j \pi n/4}}{jsin(\pi /8)} - \frac{cos(-\pi /8)e^{-j \pi n/4}}{jsin(-\pi /8)} \right) \\ y[n] &= \frac{-1}{2} \left( \frac{cos(\pi /8)e^{j \pi n/4}}{sin(\pi /8)} + \frac{cos(\pi /8)e^{-j \pi n/4}}{sin(\pi /8)} \right) \\ y[n] &= \frac{-1}{2} \frac{cos(\pi /8)}{sin(\pi /8)} \left( e^{j \pi n/4} + e^{-j \pi n/4} \right) \\ y[n] &= \frac{-1}{2} \frac{cos(\pi /8)}{sin(\pi /8)} 2 cos(\pi n/4) \\ y[n] &= - \frac{cos(\pi /8)}{sin(\pi /8)}cos(\pi n/4) \\ \end{align} $
e.
$ \begin{align} y[n] &= x[n] + x[n-1] + y[n-1] \\ h[n] &= \delta[n] + \delta[n-1] + h[n-1] \\ n &= ...\text{ -3, -2, -1, 0, 1, 2, 3, 4} ...\\ h\left[n\right] &= ... \text{ 0, 0, 0, 1, 2, 2, 2, 2} ...\\ h[n] &= u[n] + u [n - 1] \end{align} $
Credit: Prof. Jan Allebach
