Obtain the z transform and find the z transform - solve the

Question

Problem 1: Obtain the z transform of k³

Problem 2: find the z transform of the following signals

x₁(k) = Σa^h ,                                                                                            where a is a constant

x₂(k) = K²a^k

x₃(k) = kAcos(Ω₀k) ,                          k=0,1,2..., where Ω₀ and A are constants

Problem 3: obtain the inverse z transform of _{X(z) =} 1+ 2z +3z² + 4z³ + 5z⁴ /z⁴

Problem 4: by using the inversion method, obtain the inverse z transform of

X₁= z³+6z+1/(z -1)(z - 0.2)

X₂(z) = z² +z/(z- 0 .5)(Z^{2 -}1A 3: +0.64)

Problem 5:

Solve the following difference equations:

y(k +2)- y(k +1)+ 0.25y(k) = u(k -2),	y(0)	=1,	y(1)	= 2
y(k +2)- y(k +1) + 0.25y(k) = u(k +2),	y(0)	=1,	y(1)	= 2

Solve this problem for y(k), k=3,1,2... both analytically and numerically using MATLAB.

For the numerical solution, solve the problem for the first 30 samples. Plot both analytical and the numerical results in the same figure.