a) Find the z-transform of the following signals:
I) \( (u[n] - u[n - 6]) \cos[3n] + 0.3^n u[n] \sin[5n] \)
II) \( n^2 0.25^n u[n] \cos[2n] \)
b) Find the inverse z-transform of the following:
I) \( X(z) = \frac{z^2}{z^2 + 1} \)
II) \( X(z) = \frac{3z^2 + 1}{z^2 + 0.5z + 0.3125} \)
c) For \( x[n] = u[n] \), find \( y[4] \) of the following discrete-time system by iteration, then
find the closed-form solution \( y[n] \) using z-transforms. Show that the solution \( y[4] \)
from iteration equal \( y[4] \) from the closed-form solution.
\( 0.13y[n - 2] + 0.4y[n - 1] + y[n] = x[n] \), \( y[n] = 0 \) for all \( n < 0 \)
Finally, determine the stability the system.
