with MATLAB® Applications
Signals

and

Systems
Steven T. Karris
Orchard Publications
www.orchardpublications.com
Second Edition
Includes
step-by-step
procedures
for designing
analog and
digital filters
Xm[] xn[]e
j2π
mn
N
–
n0=
N1–
∑
=
Orchard Publications, Fremont, California
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Signals and Systems
with MATLAB® Applications

Steven T. Karris
Orchard Publications
www.orchardpublications.com
Signals and Systems with MATLAB Applications, Second Edition
Copyright © 2003 Orchard Publications. All rights reserved. Printed in the United States of America. No part of this
publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system,
without the prior written permission of the publisher.
Direct all inquiries to Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538
Product and corporate names are trademarks or registered trademarks of the Microsoft™ Corporation and The
MathWorks™ Inc. They are used only for identification and explanation, without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Library of Congress Control Number: 2003091595
ISBN 0-9709511-8-3
Copyright TX 5-471-562
Preface
This text contains a comprehensive discussion on continuous and discrete time signals and systems
with many MATLAB® examples. It is written for junior and senior electrical engineering students,
and for self-study by working professionals. The prerequisites are a basic course in differential and
integral calculus, and basic electric circuit theory.
This book can be used in a two-quarter, or one semester course. This author has taught the subject
material for many years at San Jose State University, San Jose, California, and was able to cover all
material in 16 weeks, with 2½ lecture hours per week.
To get the most out of this text, it is highly recommended that Appendix A is thoroughly reviewed.
This appendix serves as an introduction to MATLAB, and is intended for those who are not familiar
with it. The Student Edition of MATLAB is an inexpensive, and yet a very powerful software
package; it can be found in many college bookstores, or can be obtained directly from
The MathWorks™ Inc., 3 Apple Hill Drive , Natick, MA 01760-2098
Phone: 508 647-7000, Fax: 508 647-7001

e-mail:

Signals Described in Math Form 1-1
The Unit Step Function 1-2
The Unit Ramp Function 1-10
The Delta Function 1-12
Sampling
Property of the Delta Function 1-12
Sifting
Property of the Delta Function 1-13
Higher
Order Delta Functions 1-15
Summary
1-19
Exercises 1-20
Solutions
to Exercises 1-21
Chapter 2
The Laplace Transformation
Definition of the Laplace Transformation 2-1
Properties of the Laplace
Transform 2-2
The Laplace
Transform of Common Functions of Time 2-12
The Laplace
Transform of Common Waveforms 2-23
Summary 2-
29
Exercises
2-34
Solutions to Exercises 2-37
Chapter 3

Expressing Differential Equations in State Equation Form 5-1
Solution of Single State Equations 5-
7
The State Transition Matrix
5-9
Computation of the State
Transition Matrix 5-11
Eigenvectors 5-18
Circuit Analysis with State Variables 5-22
Relationship between State Equations and
Laplace Transform 5-28
Summary 5-35
Exercises
5-39
Solutions to
Exercises 5-41
Chapter 6
The Impulse Response and Convolution
The Impulse Response in Time Domain 6-1
Even and
Odd Functions of Time 6-5
Convolution 6-7
Graphical Evaluation of the Convolution
Integral 6-8
Circuit Analysis with the Convolution Integral 6-18
Summary
6-20
Zs()
Y
s()

Summary 7-48
Exercises 7-51
Solutions
to Exercises 7-53
Chapter 8
The Fourier Transform
Definition and Special Forms 8-1
Special Forms of the Fourier Transform 8-2
Properties and Theorems of the
Fourier Transform 8-9
Fourier Transform Pairs of Common
Functions 8-17
Finding the Fourier Transform from
Laplace Transform 8-25
Fourier Transforms of Common Waveforms
8-27
Using MATLAB to Compute the Fourier Transform 8-33
The System Function and Applications
to Circuit Analysis 8-34
Summary 8-41
Exercises 8-
47
Solutions
to Exercises 8-49

iv Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
Chapter 9
Discrete Time Systems and the Z Transform
Definition and Special Forms 9-1

Design of Type I Chebyshev Analog Low-Pass Filters 11-22
Other Low-Pass Filter Approximations 11-34
High-Pass, Band-Pass, and Band-Elimination Filters 11-39
sz
Signals and Systems with MATLAB Applications, Second Edition v
Orchard Publications
Digital Filters 11-49
Summary
11-69
Exercises 11-73
Solutions
to Exercises 11-79
Appendix A
Introduction
to MATLAB®
MATLAB® and Simulink® A-1
Command Window A-
1
Roots of
Polynomials A-3
Polynomial Construction from
Known Roots A-4
Evaluation of a
Polynomial at Specified Values A-6
Rational Polynomials
A-8
Using MATLAB to Make Plots A-10
Subplots A-18
Multiplication,
Division and Exponentiation A-18

Solution of Simultaneous Equations with Matrices C-23
Exercises C-30
Signals and Systems with MATLAB Applications, Second Edition 1-1
Orchard Publications
Chapter 1
Elementary Signals
his chapter begins with a discussion of elementary signals that may be applied to electric net-
works. The unit step, unit ramp, and delta functions are introduced. The sampling and sifting
properties of the delta function are defined and derived. Several examples for expressing a vari-
ety of waveforms in terms of these elementary signals are provided.
1.1 Signals Described in Math Form
Consider the network of Figure 1.1 where the switch is closed at time .
Figure 1.1. A switched network with open terminals.
We wish to describe in a math form for the time interval . To do this, it is conve-
nient to divide the time interval into two parts, , and .
For the time interval
, the switch is open and therefore, the output voltage is zero. In
other words,
(1.1)
For the time interval
, the switch is closed. Then, the input voltage appears at the output,
i.e.,
(1.2)
Combining (1.1) and (1.2) into a single relationship, we get
(1.3)
We can express (1.3) by the waveform shown in Figure 1.2.
T
t0=
+
−

0t∞<<
⎩
⎨
⎧
=
Chapter 1 Elementary Signals
1-2
Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
Figure 1.2. Waveform for as defined in relation (1.3)
The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be dis-
continuous
if it exhibits points of discontinuity, that is, the function jumps from one value to another
without taking on any intermediate values.
1.2 The Unit Step Function
A well-known discontinuous function is the unit step function
*
that is defined as
(1.4)
It is also represented by the waveform of Figure 1.3.
Figure 1.3. Waveform for
In the waveform of Figure 1.3, the unit step function changes abruptly from to at .
But if it changes at instead, it is denoted as . Its waveform and definition are as
shown in Figure 1.4 and relation (1.5).

t()
0
1
t
u
0
t()
u
0
t() 01t0=
tt
0
= u
0
tt
0
–()
1
t
0
0
u
0
tt
0
–()
t
u
0
tt

⎩
⎨
⎧
=
01t t
0
–= u
0
tt
0
+()
t
−t
0
0
1
u
0
tt
0
+()
u
0
tt
0
+()
u
0
tt
0

S
= tT>
v
out
v
S
u
0
tT–()=
Chapter 1 Elementary Signals
1-4
Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
Figure 1.7. Waveform for Example 1.1
Other forms of the unit step function are shown in Figure 1.8.
Figure 1.8. Other forms of the unit step function
Unit step functions can be used to represent other time-varying functions such as the rectangular
pulse shown in Figure 1.9.
Figure 1.9. A rectangular pulse expressed as the sum of two unit step functions
Thus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c) is
represented as .
T
t
0
v
S
u
0
tT–()
v

(f)
(g)
(h)
(i)
−A
−A
−A
−A
−A
−A
A
A
A
Au
0
t–()
A– u
0
t()
A– u
0
tT–()
A– u
0
tT+()
Au
0
t– T+()
Au
0

t() u
0
t1–()–
Signals and Systems with MATLAB Applications, Second Edition 1-5
Orchard Publications
The Unit Step Function
The unit step function offers a convenient method of describing the sudden application of a voltage
or current source. For example, a constant voltage source of applied at , can be denoted
as . Likewise, a sinusoidal voltage source that is applied to a circuit at
, can be described as . Also, if the excitation in a circuit is a rect-
angular, or triangular, or sawtooth, or any other recurring pulse, it can be represented as a sum (dif-
ference) of unit step functions.
Example 1.2
Express the square waveform of Figure 1.10 as a sum of unit step functions. The vertical dotted lines
indicate the discontinuities at and so on.
Figure 1.10. Square waveform for Example 1.2
Solution:
Line segment { has height , starts at , and terminates at . Then, as in Example 1.1, this
segment is expressed as
(1.8)
Line segment | has height ,
starts at and terminates at . This segment is expressed
as
(1.9)
Line segment } has height
, starts at and terminates at . This segment is expressed as
(1.10)
Line segment ~ has height ,
starts at , and terminates at . It is expressed as
(1.11)

v
1
t() Au
0
t() u
0
tT–()–[]=
A– tT= t2T=
v
2
t() A– u
0
tT–()u
0
t2T–()–[]=
At2T= t3T=
v
3
t() Au
0
t2T–()u
0
t3T–()–[]=
A– t3T= t4T=
v
4
t() A– u
0
t3T–()u
0

0
t() u
0
tT–()–[]A– u
0
tT–()u
0
t2T–()–[]=
+Au
0
t2T–()u
0
t3T–()–[]A– u
0
t3T–()u
0
t4T–()–[]
vt() Au
0
t() 2u
0
tT–()– 2u
0
t2T–()2u
0
t3T–()– …++[]=
t
A
T– 2⁄
T2⁄

T
2
–
⎝⎠
⎛⎞
–==
t
1
0
T2⁄–
vt()
T2⁄
Signals and Systems with MATLAB Applications, Second Edition 1-7
Orchard Publications
The Unit Step Function
Figure 1.13. Equations for the linear segments of Figure 1.12
For line segment {,
(1.15)
and for line segment |,
(1.16)
Combining (1.15) and (1.16), we get
(1.17)
Example 1.5
Express the waveform of Figure 1.14 as a sum of unit step functions.
Figure 1.14. Waveform for Example 1.5.
Solution:
As in the previous example, we first find the equations of the linear segments { and | shown in Fig-
ure 1.15.
t
1

u
0
t()–=
v
2
t()
2
T
– t1+
⎝⎠
⎛⎞
u
0
t() u
0
t
T
2
–
⎝⎠
⎛⎞
–=
vt() v
1
t() v
2
t()+=
2
T


1
2
3
12
3
0
t
vt()
Chapter 1 Elementary Signals
1-8
Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
Figure 1.15. Equations for the linear segments of Figure 1.14
Following the same procedure as in the previous examples, we get
Multiplying the values in parentheses by the values in the brackets, we get
or
and combining terms inside the brackets, we get
(1.18)
Two other functions of interest are the
unit ramp function, and the unit impulse or delta function. We
will introduce them with the examples that follow.
Example 1.6
In the network of Figure 1.16 is a constant current source and the switch is closed at time .
Figure 1.16. Network for Example 1.6
1
2
3
12
3
0

0
t2–() t– 3+()u
0
t3–()–+
vt() 2t 1+()u
0
t() 2t 1+()– 3+[]u
0
t1–()+=
+ 3– t– 3+()+[]u
0
t2–() t– 3+()u
0
t3–()–
vt() 2t 1+()u
0
t() 2t 1–()u
0
t1–()– t– u
0
t2–()t3–()u
0
t3–()+=
i
S
t0=
v
C
t()
t0=

i
C
t() i
S
cons ttan== v
C
t()
v
C
t()
1
C

i
C
τ()τd
∞–
t
∫
=
∞–
t0<
τ
t0=
i
C
t()
i
C
t() i

∫
0
i
S
C

u
0
τ()τd
0
t
∫
+==
⎧
⎪
⎪
⎨
⎪
⎪
⎩
v
C
t()
i
S
C

tu
0
t()=

Similarly,
(1.27)
and in general,
u
1
t()
u
1
t()
u
1
t() u
0
τ()τd
∞–
t
∫
=
τ
u
0
t()
∞ to t–
Area 1 τ×τt===
1
τ
t
∞ to t–
u
1

t() u
2
t()
u
2
t()
0t0<
t
2
t0≥
⎩
⎨
⎧
= or u
2
t() 2u
1
τ()τd
∞–
t
∫
=
u
3
t()
0t0<
t
3
t0≥
⎩

Since the derivative of any constant is zero, the derivative of the unit step has a non-zero value
only at . The derivative of the unit step function is defined in the next section.
u
n
t()
0t0<
t
n
t0≥
⎩
⎨
⎧
= or u
n
t() 3u
n1–
τ()τd
∞–
t
∫
=
u
n1–
t()
1
n

d
dt


=
t0=
i
L
t() i
S
u
0
t()=
v
L
t() Li
S
d
dt

u
0
t()=
u
0
t() 01 t0=
u
0
t()
t0=
Chapter 1 Elementary Signals
1-12
Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications

δ t() 0 for all t0≠=
δ t() u
0
t()
−ε
ε
1
2ε
Figure (a)
Figure (b)
Area =1
ε
−ε
1
t
t
0
0
ε 0→
ε 0→ 12⁄ε
1 δ t()
1
δ t()
ft()δta–()fa()δt()=
a0=
ft()δt() f0()δt()=
Signals and Systems with MATLAB Applications, Second Edition 1-13
Orchard Publications
Sifting Property of the Delta Function
that is, multiplication of any function by the delta function results in sampling the function

f τ()δτ()τd
∞–
t
∫
f0()δτ()τd
∞–
t
∫
f τ() f0()–[]δτ()τd
∞–
t
∫
+=
f
0()
f0()δτ()τd
∞–
t
∫
f0() δτ()τd
∞–
t
∫
=
δ t() 0 for t0 and t0><=
f τ() f0()–[]
τ 0=
f0() f0()– 0==
f τ()δτ()τd
∞–