EE422

Solution to Homework #21

1.a) Solve the scalar state variable equation, x((k+1)T)=1/2x(kT)+4w(kT), x(0)=6 and w(kT)=discrete unit step (please identify the zero-input and zero-state portion of your solution).

Solution: Taking Z-transform of both sides:

b) If the sampling time is T=1/2 sec, use your answer to part a) solve for x(3.5 sec).

c) Use the property of linearity and shift invariance to solve x((k+1)T)=1/2x(kT)+4w(kT), with the new initial condition x(2T)=3 and w(kT)=2u((k-3)T).

e) Use the delay property to find the Z-transform of the discrete pulse:

f) Finally, find the Z-transform of the following discrete signal which is periodic for k ³ 0