Signals and Systems
Sharif University of Technology
Dr. Hamid Reza Rabiee
December 23, 2012
CE 40-242
Date Due:
Homework 8 (Chapter 10)
Problems
1. Using only fact that γ k u[k] ⇐⇒
z
z−γ
and the properties of z-transform find the z-transform of:
a. k 2 γ 2 u[k]
b. k 3 u[k]
c. ak [u[k] − u[k − m]]
d. ke−2k u[k − m]
2. Find the inverse z-transform of :
F [z] =
(e−2 −2)z
(z−e−2 )(z−2)
when the region of convergence is
(a)|z| > 2
(b) e−2 < |z| < 2
(c) |z| < e−2
3. Consider a left-sided sequence x[n] with Z-transform:
X(z) =
1
(1−0.5z −1 )(1−z −1 )
a. Write X(z) as a ratio of polynomials in z instead of z −1 .
b. Using a partial fraction expression, express X(z) as a sum of terms,where each term represents
a pole from your answer in part(a)
c. Determine x[n].
4. we are given the following five facts about a discrete-time signal x[n] with Z-transform X(z)
a. x[n] is real and right-sided.
b. X(z) has exactly two poles.
c. X(z) has to zeros at the origin.
π
d. X(z) has poles at z = 0.5ej 3
e. X(1) =
8
3
Determine X(z) and specify its region of convergence.
5. find the impulse response h[n] of the system with the following transfer function and verify that
the system is non-causal, even though h[n] is right-sided.
1
H(z) =
z2
z−1 , |z|
>1
6. Assume that X(z) is the Z-transform of the x[n], then find the Z-transform of the
∞
P
k=n+1
its ROC.
7. Problem 10-34.
2
x[k] and