Find the first three terms of the causal signal - unit

Question

Q1 Find the first three terms of the causal signal f [k] (that is, find f [0], f [1] and f [2]) if

F[z] = 3x³ + 13z² + z / z³ + 7z² + 2z + 1

Find your answer by expanding F[x] as a power series in z^-1.

Q2 Using the 2-transform, solve

y[k + 1] + 2y[k] = f[k + 1]

with y[0] = 1 and f [k] = 3^-(k-1) u[k].

Q3 Using the Z-transform, solve

y[k + 2] + 2y[k + 1] + y[k] = f [k]

with y[-1] = 1, y[-2] = 0, and f [k] = u[k]

Note:

kγ^kU[k] <=> γz/ (z - γ)²

Q4 Using the z-transform, find h[k], the unit impulse response of the system described by the following equation:

y[k+ 2] + 2y [k + 1] + y[k] = 2f [k + 2] - f [k + 1]