Tài liệu Signals And Systems With Matlab Applications P1 pptx

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
Visit us on the Internet
www.orchardpublications.com
or email us:
Signals and Systems
with MATLAB® Applications

Second Edition
Steven T. Karris
Students and working professionals will find
Signals and Systems with MATLAB® Applications,
Second Edition, to be a concise and easy-to-learn
text. It provides complete, clear, and detailed expla-
nations of the principal analog and digital signal
processing concepts and analog and digital filter
design illustrated with numerous practical examples.
This text includes the following chapters and appendices:
• Elementary Signals • The Laplace Transformation • The Inverse Laplace Transformation
• Circuit Analysis with Laplace Transforms • State Variables and State Equations • The Impulse
Response and Convolution • Fourier Series • The Fourier Transform • Discrete Time Systems
and the Z Transform • The DFT and The FFT Algorithm • Analog and Digital Filters
• Introduction to MATLAB • Review of Complex Numbers • Review of Matrices and Determinants
Each chapter contains numerous practical applications supplemented with detailed instructions
for using MATLAB to obtain quick solutions.
Steven T. Karris is the president and founder of Orchard Publications. He earned a bachelors
degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a mas-
ters degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and
has done post-master work at the latter. He is a registered professional engineer in California
and Florida. He has over 30 years of professional engineering experience in industry. In addi-
tion, he has over 25 years of teaching experience that he acquired at several educational insti-
tutions as an adjunct professor. He is currently with UC Berkeley Extension
.
ISBN 0-9709511-8-3
$39.95 U.S.A.
Signals and Systems
with MATLAB® Applications
Second Edition

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:

The elementary signals are reviewed in Chapter 1 and several examples are presented. The intent of
this chapter is to enable the reader to express any waveform in terms of the unit step function, and
subsequently the derivation of the Laplace transform of it. Chapters 2 through 4 are devoted to
Laplace transformation and circuit analysis using this transform. Chapter 5 discusses the state
variable method, and Chapter 6 the impulse response. Chapters 7 and 8 are devoted to Fourier series
and transform respectively. Chapter 9 introduces discrete-time signals and the Z transform.
Considerable time was spent on Chapter 10 to present the Discrete Fourier transform and FFT with
the simplest possible explanations. Chapter 11 contains a thorough discussion to analog and digital
filters analysis and design procedures. As mentioned above, Appendix A is an introduction to
MATLAB. Appendix B contains a review of complex numbers, and Appendix C discusses matrices.
New to the Second Edition
This is an refined revision of the first edition. The most notable changes are chapter-end summaries,
and detailed solutions to all exercises. The latter is in response to many students and working
professionals who expressed a desire to obtain the author’s solutions for comparison with their own.
The author has prepared more exercises and they are available with their solutions to those
instructors who adopt this text for their class.
The chapter-end summaries will undoubtedly be a valuable aid to instructors for the preparation of
presentation material.
2
The last major change is the improvement of the plots generated by the latest revisions of the
MATLAB® Student Version, Release 13.
Orchard Publications
Fremont, California
www.orchardpublications.com

Signals and Systems with MATLAB Applications, Second Edition i
Orchard Publications
Table of Contents
Chapter 1
Elementary Signals

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

The Inverse Laplace
Transformation
The Inverse Laplace Transform Integral 3-1
Partial Fraction Expansion 3-1
Case
where is Improper Rational Function ( ) 3-13
Alternate Method of Partial Fraction
Expansion 3-15
Summary 3-18
Fs() mn≥

ii Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
Exercises 3-20
Solutions
to Exercises 3-22
Chapter 4
Circuit Analysis with Laplace Transforms
Circuit Transformation from Time to Complex Frequency 4-1
Complex Impedance 4-8
Complex Admittance 4-10
Transfer Functions 4-13
Summary
4-16
Exercises
4-18
Solutions to Exercises 4-21
Chapter 5
State Variables
and State Equations

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()

Signals and Systems with MATLAB Applications, Second Edition iii
Orchard Publications
Exercises 6-22
Solutions
to Exercises 6-24
Chapter 7
Fourier
Series
Wave Analysis 7-1
Evaluation of
the Coefficients 7-2
Symmetry 7-7
Waveforms
in Trigonometric Form of Fourier Series 7-11
Gibbs
Phenomenon 7-24
Alternate Forms
of the Trigonometric Fourier Series 7-25
Circuit Analysis
with Trigonometric Fourier Series 7-29
The Exponential
Form of the Fourier Series 7-31
Line Spectra
7-35
Computation of RMS
Values from Fourier Series 7-40
Computation of Average
Power from Fourier Series 7-42
Numerical Evaluation of
Fourier Coefficients 7-44

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

Properties and Theorems of the
Z Tranform 9-3
The Z Transform of Common Discrete Time Functions 9-11
Computation of the
Z transform with Contour Integration 9-20
Transformation Between and Domains 9-22
The Inverse
Z Transform 9-24
The Transfer Function of Discrete Time Systems 9-38
State Equations for Discrete Time Systems 9-43
Summary 9-47
Exercises 9-52
Solutions to Exercises 9-54
Chapter 10
The DFT and the FFT Algorithm
The Discrete Fourier Transform (DFT) 10-1
Even and Odd Properties of the DFT 10-8
Properties and Theorems of the DFT 10-10
The Sampling Theorem 10-13
Number of Operations Required to Compute the DFT 10-16
The Fast Fourier Transform (FFT) 10-17
Summary 10-28
Exercises 10-31
Solutions to Exercises 10-33
Chapter 11
Analog and Digital Filters
Filter Types and Classifications 11-1
Basic Analog Filters 11-2
Low-Pass Analog Filters 11-7
Design of Butterworth Analog Low-Pass Filters 11-11

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

Script and Function Files A-25
Display
Formats A-30
Appendix B
Review of Complex Numbers
Definition of a Complex Number B-1
Addition and Subtraction of Complex Numbers
B-2
Multiplication of Complex Numbers B-3
Division of Complex Numbers B-4
Exponential and Polar Forms of Complex Numbers B-4
Appendix C
Matrices and Determinants
Matrix Definition C-1
Matrix Operations C-2
Special Forms of
Matrices C-5
Determinants
C-9
Minors and Cofactors C-12

vi Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
Cramer’s Rule C-16
Gaussian
Elimination Method C-19
The
Adjoint of a Matrix C-20
Singular and Non-Singular Matrices C-21
The Inverse of a Matrix C-21

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=
+
−

+
−
v
out
v
S
t0=
R
open terminals
v
out
∞ t +∞<<–
∞ t0<<– 0t∞<<
∞ t0<<– v
out
v
out
0 for ∞ t0 <<–=
0t∞<< v
S
v
out
v
S
for 0t∞ <<=
v
out
0 ∞– t0<<
v
S

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).

Figure 1.4. Waveform for
* In some books, the unit step function is denoted as , that is, without the subscript 0. In this text, however, we
will reserve the designation for any input when we discuss state variables in a later chapter.
0
v
out
v
S
t
v
out
u
0
t()
u
0
t()
ut()
ut()
u
0
t()
0t0<
1t0>
⎩
⎨
⎧
=
u
0

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

0
–()
Signals and Systems with MATLAB Applications, Second Edition 1-3
Orchard Publications
The Unit Step Function
(1.5)
If the unit step function changes abruptly from to at , it is denoted as . Its
waveform and definition are as shown in Figure 1.5 and relation (1.6).
Figure 1.5. Waveform for
(1.6)
Example 1.1
Consider the network of Figure 1.6, where the switch is closed at time .
Figure 1.6. Network for Example 1.1
Express the output voltage as a function of the unit step function, and sketch the appropriate
waveform.
Solution:
For this example, the output voltage for , and for . Therefore,
(1.7)
and the waveform is shown in Figure 1.7.
u
0
tt
0
–()
0tt
0
<
1tt
0
>

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

+()
0tt
0
–<
1tt
0
–>
⎩
⎨
⎧
=
tT=
+
−
+
−
v
out
v
S
tT=
R
open terminals
v
out
v
out
0= tT< v
out
v

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

out
0
t
t
t
t
Τ
−Τ
0
0
0
0
Τ
0
0
t
t
t
0
0
t
t
−Τ
−Τ
Τ
(a)
(b)
(c)
(d)
(e)

(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– T–()
A– u
0
t–()
A– u
0
t– T+()
A– u
0
t– T–()
0
0
0
t
t
t
1
1
1
u
0
t()
u
0
t1–()–
a()
b()
c()
u
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)

Thus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through (1.11),
that is,
24 Vt0=
24u
0
t() Vvt() V
m
ωt Vcos=
tt
0
= vt() V
m
ωtcos()u
0
tt
0
–() V=
T2T3T,,
{
|
}
~
t
vt()
3T
A
0
A–
T2T
At0= tT=

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

t4T–()–[]=
Chapter 1 Elementary Signals
1-6
Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
(1.12)
Combining like terms, we get
(1.13)
Example 1.3
Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions.
Figure 1.11. Symmetric rectangular pulse for Example 1.3
Solution:
This pulse has height , starts at , and terminates at . Therefore, with reference to
Figures 1.5 and 1.8 (b), we get
(1.14)
Example 1.4
Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions.
Figure 1.12. Symmetric triangular waveform for Example 1.4
Solution:
We first derive the equations for the linear segments { and | shown in Figure 1.13.
vt() v
1
t() v
2
t() v
3
t() v
4
t()+++=
Au

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⁄

0
it()
AtT2⁄–= tT2⁄=
it() Au
0
t
T
2
+
⎝⎠
⎛⎞
Au
0
t
T
2
–
⎝⎠
⎛⎞
– Au
0
t
T
2
+
⎝⎠
⎛⎞
u
0
t

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

0
T2⁄–
vt()
T2⁄
|
2
T
– t1+
{
2
T

t1+
v
1
t()
2
T

t1+
⎝⎠
⎛⎞
u
0
t
T
2
+
⎝⎠
⎛⎞

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

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

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

{
|
2t 1+
vt()
t
t– 3+
vt() 2t 1+()u
0
t() u
0
t1–()–[]3u
0
t1–()u
0
t2–()–[]+=
+ t– 3+()u
0
t2–()u
0
t3–()–[]
vt() 2t 1+()u
0
t() 2t 1+()u
0
t1–()– 3u
0
t1–()+=
3u
0
t2–()– t– 3+()u

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
S
R
C
−
+
Signals and Systems with MATLAB Applications, Second Edition 1-9
Orchard Publications
The Unit Step Function
Express the capacitor voltage as a function of the unit step.
Solution:
The current through the capacitor is , and the capacitor voltage is
*
(1.19)
where is a dummy variable.
Since the switch closes at , we can express the current as
(1.20)
and assuming that for , we can write (1.19) as
(1.21)
or
(1.22)
Therefore, we see that when a capacitor is charged with a constant current, the voltage across it is a
linear function and forms a
ramp with slope as shown in Figure 1.17.
Figure 1.17. Voltage across a capacitor when charged with a constant current source.
* Since the initial condition for the capacitor voltage was not specified, we express this integral with at the
lower limit of integration so that any non-zero value prior to would be included in the integration.
v
C
t()

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

S
u
0
t()=
v
C
t() 0= t0<
v
C
t()
1
C

i
S
u
0
τ()τd
∞–
t
∫
i
S
C

u
0
τ()τd
∞–
0

∫
0
i
S
C

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

tu
0
t()=

i
S
C⁄
v
C
t()
0
slope i
S
C⁄=
t
Chapter 1 Elementary Signals
1-10
Signals and Systems with MATLAB Applications, Second Edition
Orchard Publications
1.3 The Unit Ramp Function
The unit ramp function, denoted as , is defined as
(1.23)
where is a dummy variable.
We can evaluate the integral of (1.23) by considering the area under the unit step function from
as shown in Figure 1.18.
Figure 1.18. Area under the unit step function from
Therefore, we define as
(1.24)
Since is the integral of , then must be the derivative of , i.e.,
(1.25)
Higher order functions of can be generated by repeated integration of the unit step function. For
example, integrating twice and multiplying by
2, we define as
(1.26)

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
1
t()
0t0<
tt0≥
⎩
⎨
⎧
=
u
1
t() u
0
t() u
0
t() u
1
t()
d
dt

u
1
t() u
0
t()=
t
u
0

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≥
⎩

⎨
⎧
= or u
3
t() 3u
2
τ()τd
∞–
t
∫
=
Signals and Systems with MATLAB Applications, Second Edition 1-11
Orchard Publications
The Unit Ramp Function
(1.28)
Also,
(1.29)
Example 1.7
In the network of Figure 1.19, the switch is closed at time and for .
Figure 1.19. Network for Example 1.7
Express the inductor current in terms of the unit step function.
Solution:
The voltage across the inductor is
(1.30)
and since the switch closes at ,
(1.31)
Therefore, we can write (1.30) as
(1.32)
But, as we know, is constant (
or ) for all time except at where it is discontinuous.

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

u
n
t()=
t0= i
L
t() 0= t0<
R
i
S
t0=
L
v
L
t()
i
L
t()
+
−
`
i
L
t()
v
L
t() L
di
L
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

1.4 The Delta Function
The unit impulse or delta function, denoted as , is the derivative of the unit step . It is also
defined as
(1.33)
and
(1.34)
To better understand the delta function , let us represent the unit step as shown in Figure
1.20 (a).
Figure 1.20. Representation of the unit step as a limit.
The function of Figure 1.20 (a) becomes the unit step as . Figure 1.20 (b) is the derivative of
Figure 1.20 (a), where we see that as , becomes unbounded, but the area of the rectangle
remains . Therefore, in the limit, we can think of as approaching a very large spike or impulse
at the origin, with unbounded amplitude, zero width, and area equal to .
Two useful properties of the delta function are the sampling property and the sifting property.
1.5 Sampling Property of the Delta Function
The sampling property of the delta function states that
(1.35)
or, when ,
(1.36)
δ t()
δ t() u
0
t()
δτ()τd
∞–
t
∫
u
0
t()=

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

at the time instants where the delta function is not zero. The study of discrete-time systems is based
on this property.
Proof:
Since then,
(1.37)
We rewrite as
(1.38)
Integrating (1.37) over the interval and using (1.38), we get
(1.39)
The first integral on the right side of (1.39) contains the constant term ; this can be written out-
side the integral, that is,
(1.40)
The second integral of the right side of (1.39) is always zero because
and
Therefore, (1.39) reduces to
(1.41)
Differentiating both sides of (1.41), and replacing with , we get
(1.42)
1.6 Sifting Property of the Delta Function
The sifting property of the delta function states that
f
t() δt()
δ t() 0 for t0 and t0><=
f
t()δt() 0 for t0 and t0><=
f
t()
f
t() f0() ft() f0()–[]+=
∞ to t–

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
∞–

t
∫
f0() δτ()τd
∞–
t
∫
=
τ t
ft()δt() f0()δt()=
Sampling Property of δ t()
δ t()