Signals and Systems
Sharif University of Technology
Dr. Hamid Reza Rabiee
December 11, 2011
CE 40-242
Date Due: Dey 4, 1390
Homework 7 (Chapter 9)
Problems
1. Determine the Laplace transform and the associated region of convergence and pole-zero plot
for each of the following functions of time:
9-21. a
9-21. b
9-21. c
1. x(t) =
2. x(t) =
R ∞ −st2
√1 u(t), Hint:
dt
−∞ e
πt
√1 e−αt u(t), where α ∈ C
πt
=
pπ
s
if Real{s} > 0
q
3. x(t) = 2 πt u(t)
4.
5.
2. Determine the function of time, x(t), for each of the following Laplace transforms and their
associated regions of convergence:
9-22. a
9-22. b
9-22. c
9-22. e
−s
1. X(s) = ( 1−es
2
)
3. Problem 9-6
4. Problem 9-27
5. Problem 9-35
1
6. Consider three real, causal, continuous-time LTI systems F , G, and H, arranged in a serial
architecture with the transfer functions given below:
(s−2)(s−5)
(s−100)(s−6)
F (s) = (s−7)
(s+2) , G(s) = (s+1)(s+6) , F (s) = (s+7)(s+5)
Sketch |Q(ω)|, the magnitude of the frequency response of the cascade system Q. Is the overall
system low-pass, band-pass, or high-pass?
7. Consider the signal x(t) with the laplace transform region of convergence a < Real{s} < b where
a < 0 < b. Determine the laplace transform region of convergence for y(t) = x(t)u(t).
8. Determine which of the following pole-zero diagrams could represent Laplace transforms of even
functions of time. Determine expressions for the time functions of those that can represent even
functions of time. For those that cannot, explain why they cannot.
How can you determine if a signal is even or not by looking at its Laplace transform and region
of convergence?
Practical Assignment
1. Use partial fraction expansion to find the inverse Laplace transform of each of the following. For
each case, find all finite poles and zeros and plot their location in the s-plane. Use Matlab to
plot the time functions. Qualitatively, how does the zero location affect the time response?
1. X(s) =
8(s+9)
(s+2)(s+10)
2. X(s) =
1
s2 +2s+2
3. X(s) =
3s2 +3s+1
s(s+1)2
2