CE 40763
DIGITAL SIGNAL PROCESSING
Homework 3 (Z Transform)
1- When the input to an LTI system is:
𝑥𝑛 =
1 𝑛
3
𝑢 𝑛 + 2𝑛 𝑢[−𝑛 − 1]
The corresponding output is:
𝑦 𝑛 =5
1 𝑛
3
𝑢 𝑛 +5
2 𝑛
3
𝑢[𝑛]
(a) Find the system function H(z) of the system. Plot the pole(s) and zero(s) of H(z) and
indicate the ROC.
(b) Find the impulse response h[n] of the system.
(c) Write a difference equation that is satisfied by the given input and output.
(d) Is the system stable? Is it causal?
2- Consider the z-transform X(z) whose pole-zero plot is as shown in figure 1.
(a) Determine the ROC of X(z) if it is known that the Fourier transform exists. For this case,
determine whether the corresponding sequence x[n] is right sided, left sided, or two
sided.
(b) How many possible two sided sequences have the pole zero plot shown in figure 1?
(c) Is it possible two-sided sequences have the pole-zero plot in figure 1 to be associated
with a sequence that is both stable and causal? If so, give the appropriate ROC.
Figure1
3- Determine the z-transform for each of the following sequences. Sketch the pole-zero plot
and indicate the ROC. Also indicate whether or not the discrete-time Fourier transform
exists.
a)
1 𝑛
−3
𝑢 −𝑛 − 2
b) 2𝑛 𝑢 −𝑛 +
c)
𝑛
1
1 𝑛
4
𝑢 𝑛−1
𝑛
2
4- Suppose we are given the following five facts about a particular LTI system S with
impulse response h[n] and z-transform H(z):
I.
II.
h[n] is real.
h[n] is right sided.
III.
limz→∞ H(z)=1.
IV.
H(z) has two zeros.
3
H(z) has one of its poles at a nonreal location on the circle defined by z = 4.
V.
Answer the following two questions:
a) Is S casual?
b) Is S stable?
5- Consider an even sequence x[n] (i.e., x[n] = x[-n]) with rational z-transform X(z).
a) From the definition of the z-transform show that X z = X
1
z
b) From your result in part (a), show that if a pole (or a zero) of X(z) occurs at
1
z = z0 , then a pole (or a zero) must also occur at z = z
0
6- Suppose we are given the following information for a discrete signal x[n] with ztransform X(z):
I.
x[n] is real and right sided.
II.
X(z) has exactly two poles.
III.
X(z) has two zeros at the origin.
IV.
X[z] has a pole at z = 0.5ej 3
π
V.
8
X[1] = 3
Determine X(z) and specify its region of convergence.
7- A causal LTI system is described by the difference equation:
y n −y n−1 −y n−2 =x n−1
a) Find the system function H(z). Plot the poles and zeros of H(z) and indicate the
ROC.
b) Find the unit sample response of the system.
c) You should have found the system to be unstable. Find a stable(noncausal) unit sample
response that satisfies the difference equation.