Essentials of Digital Signal Processing
This textbook offers a fresh approach to digital signal processing
(DSP) that combines heuristic reasoning and physical appreciation
with sound mathematical methods to illuminate DSP concepts and
practices. It uses metaphors, analogies, and creative explanations
along with carefully selected examples and exercises to provide
deep and intuitive insights into DSP concepts.
Practical DSP requires hybrid systems including both discrete- and
continuous-time compo- nents. This book follows a holistic approach
and presents discrete-time processing as a seamless continuation of
continuous-time signals and systems, beginning with a review of
continuous-time sig- nals and systems, frequency response, and
filtering. The synergistic combination of continuous-time and
discrete-time perspectives leads to a deeper appreciation and
understanding of DSP concepts and practices.
Notable Features
2. Provides an intuitive understanding and physical appreciation of
essential DSP concepts with- out sacrificing mathematical
rigor
3. Illustrates concepts with 500 high-quality figures, more than
170 fully worked examples, and hundreds of end-of-chapter
problems
4. Encourages student learning with more than 150 drill exercises,
including complete and detailed solutions
5. Maintains strong ties to continuous-time signals and systems
concepts, with immediate access to background material with a
notationally consistent format, helping readers build on their
previous knowledge
6. Seamlessly integrates MATLAB throughout the text to enhance
learning
7. Develops MATLAB code from a basic level to reinforce connections
to underlying theory and sound DSP practice
B. P. Lathi holds a PhD in Electrical Engineering from Stanford
University and was previously a Professor of Electrical Engineering
at California State University, Sacramento. He is the author
of eight books, including Signal Processing and Linear
Systems (second ed., 2004) and, with Zhi Ding, Modern Digital and
Analog Communications Systems (fourth ed., 2009).
Sacramento State University,
32 Avenue of the Americas, New York, NY 10013-2473, USA
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www.cambridge.org Information on this title:
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c Bhagawandas P. Lathi, Roger Green 2014
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permission of Cambridge University Press.
First published 2014
Printed in the United States of America
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Library of Congress Cataloging in Publication Data
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persistence or accuracy of URLs for external or third-party
Internet Web sites referred to in this publication
9.6 Goertzel’s Algorithm . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 600 9.7 The Fast Fourier
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 603
9.7.1 Decimation-in-Time Algorithm . . . . . . . . . . . . . . . .
. . . . . . . . . . 604 9.7.2 Decimation-in-Frequency
Algorithm . . . . . . . . . . . . . . . . . . . . . . .
609
9.8 The Discrete-Time Fourier Series . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 612 9.9 Summary . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
617
A MATLAB 625
Preface
Since its emergence as a field of importance in the 1970s, digital
signal processing (DSP) has grown in exponential lockstep with
advances in digital hardware. Today’s digital age requires that
under- graduate students master material that was, until recently,
taught primarily at the graduate level. Many DSP textbooks remain
rooted in this graduate-level foundation and cover an exhaustive
(and exhausting!) number of topics. This book provides an
alternative. Rather than cover the broadest range of topics
possible, we instead emphasize a narrower set of core digital
signal processing con- cepts. Rather than rely solely on
mathematics, derivations, and proofs, we instead balance necessary
mathematics with a physical appreciation of subjects through
heuristic reasoning, careful examples, metaphors, analogies, and
creative explanations. Throughout, our underlying goal is to make
digital signal processing as accessible as possible and to foster
an intuitive understanding of the material.
Practical DSP requires hybrid systems that include both
discrete-time and continuous-time com- ponents. Thus, it is
somewhat curious that most DSP textbooks focus almost exclusively
on discrete- time signals and systems. This book takes a more
holistic approach and begins with a review of continuous-time
signals and systems, frequency response, and filtering. This
material, while likely familiar to most readers, sets the stage for
sampling and reconstruction, digital filtering, and other aspects
of complete digital signal processing systems. The synergistic
combination of continuous- time and discrete-time perspectives
leads to a deeper and more complete understanding of digital signal
processing than is possible with a purely discrete-time viewpoint.
A strong foundation of continuous-time concepts naturally
leads to a stronger understanding of discrete-time concepts.
Notable Features
Some notable features of this book include the following:
1. This text is written for an upper-level undergraduate audience,
and topic treatment is appro- priately geared to the junior and
senior levels. This allows a sufficiently detailed mathematical
treatment to obtain a solid foundation and competence in DSP
without losing sight of the basics.
2. An underlying philosophy of this textbook is to provide a simple
and intuitive understanding of essential DSP concepts without
sacrificing mathematical rigor. Much attention has been paid to
provide clear, friendly, and enjoyable writing. A physical
appreciation of the topics is attained through a balance of
intuitive explanations and necessary mathematics. Concepts are
illustrated using nearly 500 high-quality figures and over 170
fully worked examples. Fur- ther reinforcement is provided through
over 150 drill exercises, complete detailed solutions of
which are provided as an appendix to the book. Hundreds of
end-of-chapter problems provide students with additional
opportunities to learn and practice.
3. Unlike most DSP textbooks, this book maintains strong ties to
continuous-time signals and systems concepts, which helps readers
to better understand complete DSP systems. Further, by leveraging
off a solid background of continuous-time concepts, discrete-time
concepts are more easily and completely understood. Since the
continuous-time background material is
vii
included, readers have immediate access to as much or little
background material as necessary, all in a notationally-consistent
format.
4. MATLAB is effectively utilized throughout the text to enhance
learning. This MATLAB ma- terial is tightly and seamlessly
integrated into the text so as to seem a natural part of the
material and problem solutions rather than an added afterthought.
Unlike many DSP texts, this book does not have specific “MATLAB
Examples” or “MATLAB Problems” any more than it has “Calculator
Examples” or “Calculator Problems.” Modern DSP has evolved to the
point that sophisticated computer packages (such as MATLAB) should
be used every bit as naturally as calculus and calculators, and it
is this philosophy that guides the manner that MATLAB is
incorporated into the book.
Many DSP books rely on canned MATLAB functions to solve various
digital signal processing problems. While this produces results
quickly and with little effort, students often miss how problem
solutions are coded or how theory is translated into practice. This
book specifically avoids high-level canned functions and develops
code from a more basic level; this approach reinforces connections
to the underlying theory and develops sound skills in the practice
of DSP. Every piece of MATLAB code precisely conforms with
book concepts, equations, and notations.
Book Organization and Use
Roughly speaking, this book is organized into five parts.
1. Review of continuous-time signals and systems (Ch. 1) and
continuous-time (analog) filtering (Ch. 2).
2. Sampling and reconstruction (Ch. 3).
3. Introduction to discrete-time signals and systems (Ch. 4)
and the time-domain analysis of discrete-time systems (Ch.
5).
4. Frequency-domain analysis of discrete-time systems using the
discrete-time Fourier transform (Ch. 6) and the z-transform
(Ch. 7).
5. Discrete-time (digital) filtering (Ch. 8) and the
discrete-Fourier transform (Ch. 9).
The first quarter of this book (Chs. 1 and 2, about 150 pages)
focuses on continuous-time con- cepts, and this material can be
scanned or skipped by those readers who possess a solid background
in these areas. The last three quarters of the book (Chs. 3 through
9, about 450 pages) cover tra- ditional discrete-time concepts that
form the backbone of digital signal processing. The majority
of the book can be covered over a semester in a typical 3 or
4 credit-hour undergraduate-level course, which corresponds to
around 45 to 60 lecture-hours of contact.
As with most text books, this book can be adapted to accommodate a
range of courses and student backgrounds. Students with solid
backgrounds in continuous-time signals and systems can scan or
perhaps altogether skip the first two chapters. Students with
knowledge in the time-domain analysis of discrete-time signals and
systems can scan or skip Chs. 4 and 5. Courses
that do not wish to emphasize filtering operations can eliminate
coverage of Chs. 2 and 8. Many other options
exist as well. For example, students enter the 3-credit Applied
Digital Signal Processing and Filtering course at North Dakota
State University having completed a 4-credit Signals and Systems
course that covers both continuous-time and discrete-time concepts,
including Laplace and z-transforms but not including
discrete-time Fourier analysis. Given this student background, the
NDSU DSP course covers Chs. 2, 3, 6, 8, and
9, which leaves enough extra time to introduce (and
use) digital signal processing hardware from Texas Instruments;
Chs. 1, 4, 5, and 7 are recommended for
reading, but not required.
viii
Acknowledgments
We would like to offer our sincere gratitude to the many people who
have generously given their time and talents to the creation,
improvement, and refinement of this book. Books, particularly
sizable ones such as this, involve a seemingly infinite number of
details, and it takes the combined efforts of a good number of good
people to successfully focus these details into a quality result.
During the six years spent preparing this book, we have been
fortunate to receive valuable feedback and recommendations from
numerous reviewers, colleagues, and students. We are grateful for
the reviews provided by Profs. Zekeriya Aliyazicioglu of California
State Polytechnic University-Pomona, Mehmet Celenk of Ohio
University, Liang Dong of Western Michigan University, Jake Gunther
of Utah State University, Joseph P. Hoffbeck of the
University of Portland, Jianhua Liu of Embry- Riddle Aeronautical
University, Peter Mathys of the University of Colorado, Phillip A.
Mlsna of Northern Arizona University, S. Hossein
Mousavinezhad of Idaho State University, Kalyan Mondal of Fairleigh
Dickinson University, Anant Sahai of UC Berkeley, Jose Sanchez of
Bradley University, and Xiaomu Song of Widener University. We also
offer our heartfelt thanks for the thoughtful comments and
suggestions provided by the many anonymous reviewers, who
outnumbered the other reviewers more than two-to-one. We wish that
we could offer a more direct form of recognition to these
reviewers. Some of the most thoughtful and useful comments came
from students taking the Applied Digital Signal Processing and
Filtering course at North Dakota State University. Two students in
particular – Kyle Kraning and Michael Boyko – went above the call
of duty, providing over one hundred corrections and comments. For
their creative contributions of cartoon ideas, we also give thanks
to NDSU students Stephanie Rosen (Chs. 1, 4, and
5) and Tanner Voss (Ch. 2). Book writing is a
time-consuming activity, and one that inevitably causes hardship to
those who are close to an author. Thus, we offer our final thanks
to our families for their sacrifice, support, and love.
B. P. Lathi R. A. Green
DSP is always on the future’s horizon!
ix
Review of Continuous-Time Signals and Systems
This chapter reviews the basics of continuous-time (CT) signals and
systems. Although the reader is expected to have studied this
background as a prerequisite for this course, a thorough yet
abbreviated review is both justified and wise since a solid
understanding of continuous-time concepts is crucial to the study
of digital signal processing.
Why Review Continuous-Time Concepts?
It is natural to question how continuous-time signals and systems
concepts are relevant to digital signal processing. To answer this
question, it is helpful to first consider elementary signals and
systems structures.
In the most simplistic sense, the study of signals and systems is
described by the block diagram shown in Fig. 1.1a. An input
signal is fed into a system to produce an output signal.
Understanding this block diagram in a completely general sense is
quite difficult, if not impossible. A few well-chosen and
reasonable restrictions, however, allow us to fully understand and
mathematically quantify the character and behavior of the input,
the system, and the output.
input system output
(c)
Figure 1.1: Elementary block diagrams of (a) general, (b)
continuous-time, and (c) discrete-time signals and systems.
Introductory textbooks on signals and systems often begin by
restricting the input, the system, and the output to be
continuous-time quantities, as shown in Fig. 1.1b. This
diagram captures the basic structure of continuous-time signals and
systems, the details of which are reviewed later in this chapter
and the next. Restricting the input, the system, and the output to
be discrete-time (DT) quantities, as shown in Fig. 1.1c,
leads to the topic of discrete-time signals and systems.
Typical digital signal processing (DSP) systems are hybrids of
continuous-time and discrete-time systems. Ordinarily, DSP systems
begin and end with continuous-time signals, but they process
1
2 Chapter 1. Review of Continuous-Time Signals and Systems
signals using a digital signal processor of some sort. Specialized
hardware is required to bridge the continuous-time and
discrete-time worlds. As the block diagram of Fig. 1.2
shows, general DSP systems are more complex than either Figs.
1.1b or 1.1c allow; both CT and DT concepts are needed
to understand complete DSP systems.
x(t) continuous to discrete
x[n] discrete-time system
y(t)
Figure 1.2: Block diagram of a typical digital signal processing
system.
A more detailed explanation of Fig. 1.2 helps further
justify why it is important for us to review continuous-time
concepts. The continuous-to-discrete block converts a
continuous-time input signal into a discrete-time signal, which is
then processed by a digital processor. The discrete-time output of
the processor is then converted back to a continuous-time signal.†
Only with knowledge of continuous-time signals and
systems is it possible to understand these components of a DSP
system. Sampling theory, which guides our understanding of the
CT-to-DT and DT-to-CT converters, can be readily mastered with a
thorough grasp of continuous-time signals and systems.
Additionally, the discrete-time algorithms implemented on the
digital signal processor are often synthesized from continuous-time
system models. All in all, continuous-time signals and systems
concepts are useful and necessary to understand the elements of a
DSP system.
Nearly all basic concepts in the study of continuous-time signals
and systems apply to the discrete- time world, with some
modifications. Hence, it is economical and very effective to build
on the pre- vious foundations of continuous-time concepts. Although
discrete-time math is inherently simpler than continuous-time math
(summation rather than integration, subtraction instead of
differentia- tion), students find it difficult, at first, to grasp
basic discrete-time concepts. The reasons are not hard to find. We
are all brought up on a steady diet of continuous-time physics and
math since high school, and we find it easier to identify with the
continuous-time world. It is much easier to grasp many concepts in
continuous-time than in discrete-time. Rather than fight this
reality, we might use it to our advantage.
1.1 Signals and Signal Categorizations
A signal is a set of data or information. Examples
include telephone and television signals, monthly sales of a
corporation, and the daily closing prices of a stock market (e.g.,
the Dow Jones averages). In all of these examples, the signals are
functions of the independent variable time . This is
not always the case, however. When an electrical charge is
distributed over a body, for instance, the signal is the charge
density, a function of space rather than
time. In this book we deal almost exclusively with signals that are
functions of time. The discussion, however, applies equally well to
other independent variables.
Signals are categorized as either continuous-time or discrete-time
and as either analog or digital. These fundamental signal
categories, to be described next, facilitate the systematic and
efficient analysis and design of signals and systems.
1.1.1 Continuous-Time and Discrete-Time Signals
A signal that is specified for every value of time t
is a continuous-time signal . Since the signal is known
for every value of time, precise event localization is possible.
The tidal height data displayed in Fig. 1.3a is an example of
a continuous-time signal, and signal features such as daily tides
as well as the effects of a massive tsunami are easy to
locate.
(a)
Dec. 26 Dec. 27 Dec. 28 Dec. 29 Dec. 30
1150
1350
tidal height [cm], Syowa Station, Antarctica, UTC+3, 2004
tsunami event
September 2001
Internet/dot-com bubble
Figure 1.3: Examples of (a) continuous-time and (b) discrete-time
signals.
A signal that is specified only at discrete values of time is a
discrete-time signal . Ordinarily, the independent
variable for discrete-time signals is denoted by the integer
n. For discrete-time signals, events are localized within the
sampling period. The technology-heavy NASDAQ composite index
displayed in Fig. 1.3b is an example of a discrete-time
signal, and features such as the Internet/dot- com bubble as well
as the impact of the September 11 terrorist attacks are visible
with a precision that is limited by the one month sampling
interval.
1.1.2 Analog and Digital Signals
The concept of continuous-time is often confused with that of
analog. The two are not the same. The same is true of the concepts
of discrete-time and digital. A signal whose amplitude can take on
any value in a continuous range is an analog signal .
This means that an analog signal amplitude can take on an infinite
number of values. A digital signal , on the other hand,
is one whose amplitude can take on only a finite number of values.
Signals associated with typical digital devices take on only two
values (binary signals). A digital signal whose amplitudes can take
on L values is an L-ary signal of which
binary (L = 2) is a special case.
x(t)x(t)
(a) (b)
(c) (d)
c o n
t i
n u o
u s -
t i
m e
d
i s c
r e
t e
- t
i m
e
analog digital
Figure 1.4: Examples of (a) analog, continuous-time, (b) digital,
continuous-time, (c) analog, discrete-time, and (d) digital,
discrete-time sinusoids.
Signals in the physical world tend to be analog and continuous-time
in nature (Fig. 1.4a). Digital, continuous-time signals
(Fig. 1.4b) are not common in typical engineering systems. As
a result, when we refer to a continuous-time signal, an analog
continuous-time signal is implied.
Computers operate almost exclusively with digital, discrete-time
data (Fig. 1.4d). Digital repre- sentations can be difficult
to mathematically analyze, so we often treat computer signals as if
they were analog rather than digital (Fig.
1.4c). Such approximations are mathematically tractable and
provide needed insights into the behavior of DSP systems and
signals.
1.2 Operations on the Independent CT Variable
We shall review three useful operations that act on the independent
variable of a CT signal: shifting, scaling, and reversal. Since
they act on the independent variable, these operations do not
change the shape of the underlying signal. Detailed derivations of
these operations can be found in [1]. Although the
independent variable in our signal description is time, the
discussion is valid for functions having continuous independent
variables other than time (e.g., frequency or distance).
1.2.1 CT Time Shifting
A signal x(t) (Fig. 1.5a) delayed by b
> 0 seconds (Fig. 1.5b) is represented by x(t −
b). Similarly, the signal x(t) advanced by b > 0
seconds (Fig. 1.5c) is represented by x(t + b). Thus,
to time shift a signal x(t) by b seconds, we
replace t with t− b everywhere in the
expression for x(t). If b is positive, the
shift represents a time delay; if b is negative,
the shift represents a time advance by |b|. This is consistent
with the fact that a time delay of b seconds can
be viewed as a time advance of −b seconds.
x(t)
T 1 + b
T 1 − b
Figure 1.5: Time shifting a CT signal: (a) original signal, (b)
delay by b, and (c) advance by b.
when its argument t − b equals T 1, or
t = T 1 + b. Similarly,
x(t + b) starts when t + b
= T 1, or t = T 1 − b.
1.2.2 CT Time Scaling
A signal x(t), when time compressed by factor a
> 1, is represented by x(at). Similarly, a signal
time expanded by factor a > 1 is represented by
x(t/a). Figure 1.6a shows a signal x(t). Its
factor-2 time- compressed version is x(2t) (Fig. 1.6b),
and its factor-2 time-expanded version is x(t/2) (Fig.
1.6c). In general, to time scale a signal x(t) by
factor a, we replace t with at
everywhere in the expression for x(t). If a
> 1, the scaling represents time compression (by factor
a), and if 0 < a < 1, the scaling represents
time expansion (by factor 1/a). This is consistent with the fact
that time compression by factor a can be viewed as time
expansion by factor 1 /a.
As in the case of time shifting, time scaling operates on the
independent variable and does not change the underlying function.
In Fig. 1.6, the function x(·) has a maximum
value when its argument equals T 1. Thus, x(2t) has
a maximum value when its argument 2t equals T 1,
or t = T 1/2. Similarly, x(t/2) has a
maximum when t/2 = T 1, or t =
2T 1.
Drill 1.1 (CT Time Scaling)
1.2.3 CT Time Reversal
x(t)
x(2t)
2T 1 2T 2
Figure 1.6: Time scaling a CT signal: (a) original signal, (b)
compress by 2, and (c) expand by 2.
x(t) about the vertical axis is x(−t). Notice that time
reversal is a special case of the time-scaling operation
x(at) where a = −1.
x(t) x(−t)
tt0 0
(a) (b)
T 1
T 2
−T 1
−T 2
Figure 1.7: Time reversing a CT signal: (a) original signal and (b)
its time reverse.
1.2.4 Combined CT Time Shifting and Scaling
Many circumstances require simultaneous use of more than one of the
previous operations. The most general case is x(at − b),
which is realized in two possible sequences of operations:
1. Time shift x(t) by b to obtain x(t − b).
Now time scale the shifted signal x(t − b) by a
(i.e., replace t with at) to obtain
x(at − b).
2. Time scale x(t) by a to obtain x(at). Now
time shift x(at) by b a (i.e., replace t
with [t− b
a ]) to
obtain x
a[t − b a ] = x(at − b).
1.3. CT Signal Models 7
When a is negative, x(at) involves time scaling as
well as time reversal. The procedure, however, remains the same.
Consider the case of a signal x(−2t + 3) where a
= −2. This signal can be generated by advancing the
signal x(t) by 3 seconds to obtain x(t + 3). Next,
compress and reverse this signal by replacing t
with −2t to obtain x(−2t + 3). Alternately,
we may compress and reverse x(t) to obtain x(−2t); next,
replace t with t − 3/2 to delay this signal by
3/2 and produce x(−2[t − 3/2]) = x(−2t + 3).
Drill 1.2 (Combined CT Operations)
Using the signal x(t) shown in Fig. 1.6a, sketch the
signal y(t) = x(−3t − 4). Verify that y(t) has a
maximum value at t = T 1+4
−3 .
1.3 CT Signal Models
In the area of signals and systems, the unit step, the unit gate,
the unit triangle, the unit impulse, the exponential, and the
interpolation functions are very useful. They not only serve as a
basis for representing other signals, but their use benefits many
aspects of our study of signals and systems. We shall briefly
review descriptions of these models.
1.3.1 CT Unit Step Function u(t)
In much of our discussion, signals and processes begin at t
= 0. Such signals can be conveniently described in terms of
unit step function u(t) shown in Fig. 1.8a. This
function is defined by
u(t) =
0 t < 0
11
Figure 1.8: (a) CT unit step u(t) and (b) cos(2πt)u(t).
If we want a signal to start at t = 0 and have a value
of zero for t < 0, we only need to multiply the
signal by u(t). For instance, the signal cos(2πt) represents
an everlasting sinusoid that starts at t = −∞. The causal
form of this sinusoid, illustrated in Fig. 1.8b, can be
described as cos(2πt)u(t). The unit step function and its shifts
also prove very useful in specifying functions with different
mathematical descriptions over different intervals (piecewise
functions).
A Meaningless Existence?
8 Chapter 1. Review of Continuous-Time Signals and Systems
its own advantages, u(0) = 1/2 is particularly appropriate
from a theoretical signals and systems perspective. For real-world
signals applications, however, it makes no practical difference how
the point u(0) is defined as long as the value is finite. A
single point, u(0) or otherwise, is just one among an
uncountably infinite set of peers. Lost in the masses, any single,
finite-valued point simply does not matter; its individual
existence is meaningless.
Further, notice that since it is everlasting, a true unit step
cannot be generated in practice. One might conclude, given that
u(t) is physically unrealizable and that individual points
are inconse- quential, that the whole of u(t) is
meaningless. This conclusion is false.
Collectively the points of u(t) are well behaved
and dutifully carry out the desired function, which is greatly
needed in the mathematics of signals and systems.
1.3.2 CT Unit Gate Function Π(t)
We define a unit gate function Π( x) as a gate pulse of unit height
and unit width, centered at the origin, as illustrated in Fig.
1.9a. Mathematically,†
Π(t) =
. (1.2)
The gate pulse in Fig. 1.9b is the unit gate pulse Π(t)
expanded by a factor τ and therefore can be
expressed as Π(t/τ ). Observe that τ , the
denominator of the argument of Π( t/τ ), indicates the width
of the pulse.
Π(t) Π(t/τ )
Figure 1.9: (a) CT unit gate Π(t) and (b) Π(t/τ ).
Drill 1.3 (CT Unit Gate Representations)
1.3.3 CT Unit Triangle Function Λ(t)
We define a unit triangle function Λ(t) as a triangular pulse of
unit height and unit width, centered at the origin, as shown in
Fig. 1.10a. Mathematically,
Λ(t) =
2
†At |t| = 1 2
1.3. CT Signal Models 9
The pulse in Fig. 1.10b is Λ(t/τ ). Observe that here,
as for the gate pulse, the denominator τ of
the argument of Λ(t/τ ) indicates the pulse width.
Λ(t) Λ(t/τ )
Figure 1.10: (a) CT unit triangle Λ(t) and (b) Λ(t/τ ).
1.3.4 CT Unit Impulse Function δ(t)
The CT unit impulse function δ (t) is one of the most
important functions in the study of signals and systems. Often
called the Dirac delta function, δ (t) was first defined
by P. A. M. Dirac as
δ (t) = 0 for t = 0
and ∞ −∞ δ (t) dt = 1.
(1.4)
We can visualize this impulse as a tall, narrow rectangular pulse
of unit area, as illustrated in Fig. 1.11b. The width of this
rectangular pulse is a very small value , and its height is a
very large value 1/. In the limit → 0, this
rectangular pulse has infinitesimally small width, infinitely large
height, and unit area, thereby conforming exactly to the definition
of δ (t) given in Eq. (1.4). Notice that
δ (t) = 0 everywhere except at t = 0, where
it is undefined. For this reason a unit impulse is represented by
the spear-like symbol in Fig. 1.11a.
δ(t)
tt
tt
(a)
00
00
1
1
− 2
2
(b)
→ 0
10 Chapter 1. Review of Continuous-Time Signals and Systems
is not its shape but the fact that its effective duration (pulse
width) approaches zero while its area remains at unity. Both the
triangle pulse (Fig. 1.11c) and the Gaussian pulse (Fig.
1.11d) become taller and narrower as becomes
smaller. In the limit as → 0, the pulse height → ∞,
and its width or duration → 0. Yet, the area under each
pulse is unity regardless of the value of .
From Eq. (1.4), it follows that the function kδ (t) = 0
for all t = 0, and its area is k . Thus,
kδ (t) is an impulse function whose area is k
(in contrast to the unit impulse function, whose area is 1).
Graphically, we represent kδ (t) by either scaling our
representation of δ (t) by k or by
placing a k next to the impulse.
Properties of the CT Impulse Function
Without going into the proofs, we shall enumerate properties of the
unit impulse function. The proofs may be found in the literature
(see, for example, [ 1]).
1. Multiplication by a CT Impulse: If a function
φ(t) is continuous at t = 0, then
φ(t)δ (t) = φ(0)δ (t).
φ(t)δ (t − b) = φ(b)δ (t − b). (1.5)
∞
∞
−∞ φ(t)δ (t − b) dt = φ(b). (1.6)
Equation (1.6) states that the area under the product of a
function with a unit impulse is equal to the value of that
function at the instant where the impulse is located. This
property is very important and useful and is known as the
sampling or sifting property of
the unit impulse.
3. Relationships between δ(t) and u(t):
Since the area of the impulse is concentrated at one point t
= 0, it follows that the area under δ (t)
from −∞ to 0− is zero, and the area is unity once
we pass t = 0. The symmetry of δ (t),
evident in Fig. 1.11, suggests the area is 1/2 at
t = 0. Hence,
t
1 t > 0
δ (t) = d
dt u(t). (1.8)
The Unit Impulse as a Generalized Function
1.3. CT Signal Models 11
impulse function is zero everywhere except at t = 0,
and at this only interesting part of its range it is undefined.
These difficulties are resolved by defining the impulse as a
generalized function rather than an ordinary function. A
generalized function is defined by its effect on
other functions instead of by its value at every instant of
time.
In this approach, the impulse function is defined by the sampling
property of Eq. ( 1.6). We say nothing about what the impulse
function is or what it looks like. Instead, the impulse function is
defined in terms of its effect on a test function φ(t). We
define a unit impulse as a function for which the area under its
product with a function φ(t) is equal to the value of the
function φ(t) at the instant where the impulse is located.
Thus, we can view the sampling property of Eq. (1.6) as a
consequence of the classical (Dirac) definition of the unit impulse
in Eq. (1.4) or as the definition of the impulse
function in the generalized function approach.
A House Made of Bricks
In addition to serving as a definition of the unit impulse, Eq. (
1.6) provides an insightful and useful way to view an arbitrary
function.† Just as a house can be made of straw, sticks, or
bricks, a function can be made of different building materials such
as polynomials, sinusoids, and, in the case of Eq. (1.6), Dirac
delta functions.
To begin, let us consider Fig. 1.12b, where an input
x(t) is shown as a sum of narrow rectangular strips. As shown
in Fig. 1.12a, let us define a basic strip of unit height and
width τ as p(t) = Π(t/τ ). The rectangular
pulse centered at nτ in Fig. 1.12b has a
height x(nτ ) and can be expressed as
x(nτ ) p(t − nτ ). As τ →
0 (and nτ → τ ), x(t) is
the sum of all such pulses. Hence,
x(t) = lim τ →0
τ →0
Figure 1.12: Signal representation in terms of impulse
components.
Consistent with Fig. 1.11b, as τ → 0,
p(t − nτ )/τ → δ (t − nτ ).
Therefore,
x(t) = lim τ →0
∞
−∞
dτ. (1.9)
Equation (1.9), known as the sifting property, tells us that an
arbitrary function x(t) can be represented as a sum
(integral) of scaled (by x(τ )) and shifted (by
τ ) delta functions. Recognizing
12 Chapter 1. Review of Continuous-Time Signals and Systems
that δ (t− τ ) = δ (τ − t), we also
see that Eq. (1.9) is obtained from Eq. (1.6) by substituting
τ for t, t for b, and x(·)
for φ(·). As we shall see in Sec. 1.5, Eq. (1.9)
is very much a house of bricks, more than able to withstand the big
bad wolf of linear, time-invariant systems.
Drill 1.4 (CT Unit Impulse Properties)
Show that
(a) (t3 + 2t2 + 3t + 4)δ (t) = 4δ (t) (b)
δ (t)sin
t2 − π 2
= −δ (t)
(c) e−2tδ (t + 1) = e2δ (t + 1) (d)
∞ −∞ δ (τ )e−jωτ dτ = 1
(e) ∞ −∞ δ (τ − 2) cos
1.3.5 CT Exponential Function est
One of the most important functions in the area of signals and
systems is the exponential signal est, where s is
complex in general and given by
s = σ + jω .
Therefore, est = e(σ+jω)t = eσtejωt = eσt [cos(ωt)
+ j sin(ωt)] . (1.10)
The final step in Eq. (1.10) is a substitution based on
Euler’s familiar formula,
ejωt = cos(ωt) + j sin(ωt). (1.11)
A comparison of Eq. (1.10) with Eq. (1.11) suggests that
est is a generalization of the function ejωt , where
the frequency variable jω is generalized to a complex
variable s = σ + jω . For this reason we
designate the variable s as the complex
frequency .
For all ω = 0, est is complex valued. Taking just
the real portion of Eq. (1.10) yields
Re
est = eσt cos(ωt). (1.12)
From Eqs. (1.10) and (1.12) it follows that the function
est encompasses a large class of functions. The following
functions are special cases of est:
1. a constant k = ke0t (where s = 0
+ j0),
2. a monotonic exponential eσt (where s = σ
+ j0),
3. a sinusoid cos(ωt) = Re
ejωt (where s = 0 + jω), and
4. an exponentially varying sinusoid eσt cos(ωt) = Re
e(σ+jω)t (where s = σ
+ jω).
Figure 1.13 shows these functions as well as the
corresponding restrictions on the complex frequency
variable s. The absolute value of the imaginary part
of s is |ω| (the radian frequency), which
indicates the frequency of oscillation of est; the real
part σ (the neper frequency) gives
information about the rate of increase or decrease of the amplitude
of est. For signals whose complex frequencies lie on the
real axis (σ-axis, where ω = 0), the frequency of
oscillation is zero. Consequently these signals are constants (σ
= 0), monotonically increasing exponentials (σ > 0),
or monotonically decreasing exponentials (σ < 0). For
signals whose frequencies lie on the imaginary axis (ω-axis,
where σ = 0), eσt = 1. Therefore, these signals are
conventional sinusoids with constant amplitude.
ttt
ttt
=
=
σ < 0 σ = 0 σ > 0
Figure 1.13: Various manifestations of e(σ+j0)t
= eσt and Re {est} = eσt cos(ωt).
0 σ real axis
ω
i m a
g i n
a r y
a x
i s
e x p
o n e
n
t i
a
l l y
d e
c r e
a s i
n g
s i g
n a
l s
e x p
o n e
n
t i
a
l l l y
i n c
r e a
s i n
g s i
g n a
l s
l e
f t
h a
l f -
p
l a
n e
( σ
< 0
)
r i g
h t
h a
l f -
p
l a
n e
( σ
> 0
)
Figure 1.14: Complex frequency plane.
1.3.6 CT Interpolation Function sinc(t)
The “sine over argument” function, or sinc function, plays an
important role in signal processing .†
It is also known as the filtering or
interpolating function. We define
sinc(t) = sin(πt)
πt . (1.13)
Inspection of Eq. (1.13) shows the following:
†sinc(t) is also denoted by Sa(t) in the literature. Some authors
define sinc( t) as
sinc(t) = sin(t)
14 Chapter 1. Review of Continuous-Time Signals and Systems
1. The sinc function is symmetric about the vertical axis (an even
function).
2. Except at t = 0 where it appears indeterminate,
sinc(t) = 0 when sin(πt) = 0. This means that sinc(t) = 0 for
t = ±1, ±2, ±3, . . ..
3. Using L’Hopital’s rule, we find sinc(0) = 1 .
4. Since it is the product of the oscillating signal sin(πt) and
the decreasing function 1/(πt), sinc(t) exhibits sinusoidal
oscillations with amplitude rapidly decreasing as 1 /(πt).
Figure 1.15a shows sinc(t). Observe that sinc(t) = 0 for
integer values of t. Figure 1.15b shows sinc
(2t/3). The argument 2t/3 = 1 when t = 3/2. Therefore,
the first zero of this function for t > 0 occurs
at t = 3/2.
(a)
t
1
1
Figure 1.15: The sinc function.
Example 1.1 (Plotting Combined Signals)
Defining x(t) = e−tu(t), accurately plot the signal
y(t) = x −t+3
3
− 3
(−1.5 ≤ t ≤ 4.5).
This problem involves several concepts, including exponential and
unit step functions and oper- ations on the independent variable
t. The causal decaying exponential x(t) = e−tu(t)
is itself easy to sketch by hand, and so too are the individual
components x
−t+3 3
. The compo-
is a left-sided signal with jump discontinuity at
−t+3
3 = 0 or t = 3. The component
− 3 4x (t − 1) is a right-sided signal with jump discontinuity at
t − 1 = 0 or t = 1. The combination
y(t) = x −t+3
3
− 3
4x (t − 1), due to the overlap region between
(1 ≤ t ≤ 3), is difficult to accurately plot by
hand. MATLAB, however, makes accurate plots easy to generate.
01 u = @(t) 1.0*(t>0)+0.5*(t==0);
02 x = @(t) exp(-t).*u(t); y = @(t) x((-t+3)/3)-3/4*x(t-1);
03 t = (-1.5:.0001:4.5); plot(t,y(t)); xlabel(’t’);
ylabel(’y(t)’);
1.4. CT Signal Classifications 15
time vector is created, and the plots are generated. It is
important that the time vector t is created with
sufficiently fine resolution to adequately represent the jump
discontinuities present in y(t). Figure 1.16
shows the result including the individual components
x
−t+3 3
y(t)
0
1
3
− 3
Example 1.1
Drill 1.5 (Plotting CT Signal Models)
Plot each of the following signals:
(a) xa(t) = 2u(t + 2) − u(3− 3t) (b) xb(t) =
Π(πt) (c) xc(t) = Λ(t/10)
(d) xd(t) = Re
2t π
There are many possible signal classifications that are useful to
better understand and properly analyze a signal. In addition to the
continuous-time/discrete-time and the analog/digital signal
classifications already discussed, we will also investigate the
following classifications, which are suit- able for the scope of
this book:
1. causal, noncausal, and anti-causal signals,
2. real and imaginary signals,
3. even and odd signals,
4. periodic and aperiodic signals,
5. energy and power signals, and
6. deterministic and probabilistic signals.
1.4.1 Causal, Noncausal, and Anti-Causal CT Signals
A causal signal x(t) extends to the right,
beginning no earlier than t = 0. Mathematically,
x(t) is causal if
x(t) = 0 for t < 0. (1.14)
16 Chapter 1. Review of Continuous-Time Signals and Systems
The signals shown in Fig. 1.8 are causal. Any signal
that is not causal is said to be noncausal . Examples of
noncausal signals are shown in Figs. 1.6 and 1.7.
An anti-causal signal x(t) extends to the
left of t = 0. Mathematically, x(t) is
anti-causal if
x(t) = 0 for t ≥ 0. (1.15)
Notice that any signal x(t) can be decomposed into a causal
component plus an anti-causal compo- nent.
A right-sided signal extends to the right,
beginning at some point T 1. In other words, x(t)
is right-sided if x(t) = 0 for t < T 1.
The signals shown in Fig. 1.5 are examples of
right-sided signals. Similarly, a left-sided signal
extends to the left of some point T 1. Mathematically,
x(t) is left-sided if x(t) = 0 for t ≥
T 1. If we time invert a right-sided signal, then we
obtain a left-sided signal. Conversely, the time reversal of a
left-sided signal produces a right-sided signal. A causal signal is
a right-sided signal with T 1 ≥ 0, and an
anti-causal signal is a left-sided signal with
T 1 ≤ 0. Notice, however, that right-sided
signals are not necessarily causal, and left-sided signals are not
necessarily anti-causal. Signals that stretch indefinitely in both
directions are termed two-sided or
everlasting signals. The signals in Figs. 1.13
and 1.15 provide examples of two-sided
signals.
Comment
We postulate and study everlasting signals despite the fact that,
for obvious reasons, a true everlast- ing signal cannot be
generated in practice. Still, as we show later, many two-sided
signal models, such as everlasting sinusoids, do serve
a very useful purpose in the study of signals and systems.
1.4.2 Real and Imaginary CT Signals
A signal x(t) is real if, for all time, it
equals its own complex conjugate,
x(t) = x∗(t). (1.16)
A signal x(t) is imaginary if, for all time,
it equals the negative of its own complex conjugate,
x(t) = −x∗(t). (1.17)
The real portion of a complex signal x(t) is found by
averaging the signal with its complex conjugate,
Re {x(t)} = x(t) + x∗(t)
2 . (1.18)
The imaginary portion of a complex signal x(t) is found in a
similar manner,
Im {x(t)} = x(t) − x∗(t)
2 j . (1.19)
Notice that Im {x(t)} is a real signal.
Further, notice that Eq. (1.18) obeys Eq. (1.16) and j
times Eq. (1.19) obeys Eq. (1.17), as
expected.
Adding Eq. (1.18) and j times Eq. (1.19), we see that
any complex signal x(t) can be decomposed into a real portion
plus ( j times) an imaginary portion,
Re {x(t)} + jIm {x(t)} = x(t) + x∗(t)
2 + j
This representation is the familiar rectangular form.
Drill 1.6 (Variations of Euler’s Formula)
Using Euler’s formula ejt = cos(t) + j sin(t) and
Eqs. (1.18) and (1.19), show that
(a) cos(t) = ejt+e−jt
2 (b) sin(t) = ejt−e−jt
2j
Real Comments about the Imaginary
There is an important and subtle distinction between an
imaginary signal and the imaginary
portion of a signal: an imaginary signal is a complex signal
whose real portion is zero and is thus represented as j
times a real quantity, while the imaginary portion of a
signal is real and has no j present. One way to
emphasize this difference is to write Eq. (1.17) in an alternate
but completely equivalent way. A signal x(t) is imaginary if,
for all time,
x(t) = jIm {x(t)} .
From this expression, it is clear that an imaginary signal is never
equal to its imaginary portion but rather is equal to j
times its imaginary portion. We can view the
j in Eq. (1.20) as simply a mechanism to keep the two
real-valued components of a complex signal separate.
Viewing complex signals as pairs of separate real quantities offers
tangible benefits. Such a perspective makes clear that
complex signal processing , which is just signal
processing on complex- valued signals, is easily accomplished
in the real world by processing pairs of real signals . There
is a frequent misconception that complex signal processing is not
possible with analog systems. Again this is simply untrue. Complex
signal processing is readily implemented with traditional analog
electronics by simply utilizing dual signal
paths.
It is worthwhile to comment that the historical choices of the
terms “complex” and “imaginary” are quite unfortunate, particularly
from a signal processing perspective. The terms are prejudicial;
“complex” suggests difficult, and “imaginary” suggests something
that cannot be realized. Neither case is true. More often than not,
complex signal processing is more simple than the alternative, and
complex signals, as we have just seen, are easily realized in the
analog world.
What’s in a name?
Drill 1.7 (The Imaginary Part Is Real)
1.4.3 Even and Odd CT Signals
A signal x(t) is even if, for all time, it
equals its own reflection,
x(t) = x(−t). (1.21)
A signal x(t) is odd if, for all time, it
equals the negative of its own reflection,
x(t) = −x(−t). (1.22)
As shown in Figs. 1.17a and 1.17b, respectively, an
even signal is symmetrical about the vertical axis while an odd
signal is antisymmetrical about the vertical axis.
(a) (b)
xe(t) xo(t)
−T 1
00 tt
Figure 1.17: Even and odd symmetries: (a) an even signal
xe(t) and (b) an odd signal xo(t).
The even portion of a signal x(t) is found by averaging the
signal with its reflection,
xe(t) = x(t) + x(−t)
2 . (1.23)
As required by Eq. (1.21) for evenness, notice that
xe(t) = xe(−t). The odd portion of a signal x(t)
is found in a similar manner,
xo(t) = x(t) − x(−t)
2 . (1.24)
As required by Eq. (1.22) for oddness, xo(t) = −xo(−t).
Adding Eqs. (1.23) and (1.24), we see that any signal
x(t) can be decomposed into an even
portion plus an odd portion,
xe(t) + xo(t) = x(t) + x(−t)
2 +
or just x(t) = xe(t) + xo(t). (1.25)
Notice that Eqs. (1.23), (1.24), and (1.25) are remarkably similar
to Eqs. (1.18), (1.19), and (1.20), respectively.
Because xe(t) is symmetrical about the vertical axis, it
follows from Fig. 1.17a that
T 1
T 1
It is also clear from Fig. 1.17b that
T 1
−T 1 xo(t) dt = 0.
These results can also be proved formally by using the definitions
in Eqs. (1.21) and (1.22).
Example 1.2 (Finding the Even and Odd Portions of a
Function)
Determine and plot the even and odd components of x(t)
= e−atu(t), where a is real and >
0.
Using Eq. (1.23), we compute the even portion
of x(t) to be
xe(t) = x(t) + x(−t)
Similarly, using Eq. (1.24), the odd portion of x(t)
is
xo(t) = x(t) − x(−t)
2 .
Figures 1.18a, 1.18b, and 1.18c show the resulting
plots of x(t), xe(t), and xo(t),
respectively.
(a)
(b)
(c)
t
t
t
x(t)
0
0
0
Figure 1.18: Finding the even and odd components of x(t)
= e−atu(t).
Setting a = 1 for convenience, these plots are easily
generated using MATLAB.
01 t = linspace(-2,2,4001); x = @(t) exp(-t).*(t>0) +
0.5*(t==0);
02 xe = (x(t)+x(-t))/2; xo = (x(t)-x(-t))/2;
03 subplot(311); plot(t,x(t)); xlabel(’t’); ylabel(’x(t)’);
04 subplot(312); plot(t,xe); xlabel(’t’); ylabel(’x_e(t)’);
05 subplot(313); plot(t,xo); xlabel(’t’); ylabel(’x_o(t)’);
Example 1.2
Drill 1.8 (Even and Odd Decompositions)
A Different Prescription for Complex CT Signals
While a complex signal can be viewed using an even and odd
decomposition, doing so is a bit like a far-sighted man wearing
glasses intended for the near-sighted. The poor prescription blurs
rather than sharpens the view. Glasses of a different type are
required. Rather than an even and odd decomposition, the preferred
prescription for complex signals is generally a conjugate-symmetric
and conjugate-antisymmetric decomposition.
A signal x(t) is conjugate symmetric , or
Hermitian , if
x(t) = x∗(−t). (1.26)
A conjugate-symmetric signal is even in its real portion and odd in
its imaginary portion. Thus, a signal that is both conjugate
symmetric and real is also an even signal.
A signal x(t) is conjugate antisymmetric , or
skew Hermitian , if
x(t) = −x∗(−t). (1.27)
A conjugate-antisymmetric signal is odd in its real portion and
even in its imaginary portion. Thus, a signal that is both
conjugate antisymmetric and real is also an odd signal.
The conjugate-symmetric portion of a signal x(t) is given
by
xcs(t) = x(t) + x∗(−t)
2 . (1.28)
As required by Eq. (1.26), we find that xcs(t)
= x∗cs(−t). The conjugate-antisymmetric portion of a signal
x(t) is given by
xca(t) = x(t) − x∗(−t)
2 . (1.29)
As required by Eq. (1.27), notice that xca(t) = −x∗ca(−t).
Adding Eqs. (1.28) and (1.29), we see that any signal
x(t) can be decomposed into a conjugate-
symmetric portion plus a conjugate-antisymmetric portion,
xcs(t) + xca(t) = x(t) + x∗(−t)
2 +
Determine the conjugate-symmetric and conjugate-antisymmetric
portions of the following signals:
(a) xa(t) = ejt (b) xb(t) = jejt (c)
xc(t) = √
2ej(t+π/4)
1.4.4 Periodic and Aperiodic CT Signals
A CT signal x(t) is said to be
T -periodic if, for some positive constant
T ,
x(t) = x(t − T ) for all t. (1.31)
The smallest value
of T that satisfies the periodicity condition
of Eq. ( 1.31) is the fundamental
period T 0 of x(t). The signal in
Fig. 1.19 is a T 0-periodic signal. A
signal is aperiodic if it is not
x(t)
t
· · ·· · ·
Figure 1.19: A T 0-periodic signal.
periodic. The signals in Figs. 1.5, 1.6, and 1.8
are examples of aperiodic waveforms. By definition, a
periodic signal x(t) remains unchanged when time shifted by
one period. For
this reason a periodic signal, by definition, must start
at t = −∞ and continue forever;
if it starts or ends at some finite instant, say
t = 0, then the time-shifted signal x(t−T )
will start or end at t = T , and Eq. (1.31)
cannot hold. Clearly, a periodic signal is an everlasting signal;
not all everlasting signals, however, are periodic, as
Fig. 1.5 demonstrates.
Periodic Signal Generation by Periodic Replication of One
Cycle
Another important property of a periodic signal x(t) is that
x(t) can be generated by periodic
replication of any segment of x(t) of
duration T 0 (the period). As a result, we can
generate x(t) from any segment of x(t) with a
duration of one period by placing this segment and the reproduction
thereof end to end ad infinitum on either side. Figure
1.20 shows a periodic signal x(t) with period
T 0 = 6 generated in two ways. In Fig. 1.20a, the
segment (−1 ≤ t < 5) is repeated forever in
either direction, resulting in signal x(t). Figure
1.20b achieves the same end result by repeating the segment
(0 ≤ t < 6). This construction is possible with
any segment of x(t) starting at any instant as long as
the segment duration is one period.
1.4.5 CT Energy and Power Signals
The size of any entity is a number that indicates the largeness or
strength of that entity. Generally speaking, signal amplitude
varies with time. How can a signal that exists over time with
varying amplitude be measured by a single number that will indicate
the signal size or signal strength? Such a measure must consider
not only the signal amplitude but also its duration.
Signal Energy
By defining signal size as the area under x2(t), which is
always positive for real x(t), both signal amplitude and
duration are properly acknowledged. We call this measure the
signal energy E x, defined (for a real signal)
as
E x =
This definition can be generalized to accommodate complex-valued
signals as
E x =
x(t)
x(t)
t
t
0
· · ·
· · ·
· · ·
· · ·
Figure 1.20: Generation of the (T 0 = 6)-periodic signal
x(t) by periodic replication using (a) the segment (−1
≤ t < 5) and (b) the segment (0 ≤ t
< 6).
There are also other possible measures of signal size, such as the
area under |x(t)|. The energy measure, however, is not only
more tractable mathematically but is also more meaningful (as shown
later) in the sense that it is indicative of the energy that can be
extracted from the signal.
Signal energy must be finite for it to be a meaningful measure of
the signal size. A necessary condition for the energy to be finite
is that the signal amplitude → 0 as |t| → ∞
(Fig. 1.21a). Otherwise the integral in Eq. (1.32) will
not converge. A signal with finite energy is classified as an
energy signal .
x(t)x(t)
t
t
0
0
Figure 1.21: Examples of (a) finite energy and (b) finite power
signals.
Example 1.3 (Computing Signal Energy)
Compute the energy of
(a) xa(t) = 2Π(t/2) (b) xb(t) = sinc(t)
(a) In this particular case, direct integration is simple. Using
Eq. (1.32), we find that
E xa =
(2)2dt = 4t|1−1 = 8.
1.4. CT Signal Classifications 23
discussed in Sec. 1.9.8, makes it easy to determine that the
energy is E xb = 1, it is instructive to try
and obtain this answer by estimating the integral in Eq. ( 1.32).
We begin by plotting x2 b(t) = sinc2(t)
since energy is simply the area under this curve. As shown in