SECOND EDITION
`
`SIGNALS
`&
`SYSTEMS
`
`ALAN V. OPPENHEIM
`. ALAN s. w ILLSl(Y
`
`
`
`
`
`MASSACHUSETTS INSTITUTE OF TECHNOLOGY
`
`WITH
`
`s.HAMID NAWAB
`
`BOSTON UNIVERSITY
`
`•
`
`PRENTICE HALL
`UPPER SADDLE RIVER, NEW JERSEY 07458
`
`Petitioner Apple Inc.
`Ex. 1014, Cover
`
`

`

`Library of Congress Cataloging-in-Publication Data
`
`Oppenheim, Alun V.
`Signals und systems I Alan V. Oppenheim, Alan S. Willsky, with
`S. Hamid Nawab. - 2nd ed ..
`p.
`em. - Prentice-Hall signal processing series
`Includes bibliographical references and index.
`ISBN 0· 13-814757·4
`I. System analysis.
`I. Willsky, Alan S.
`IV. Series.
`QA402.063 1996
`621.382 '23-dc20
`
`2. Signal theory (Telecommunication)
`II. Nawab, Syed Hamid.
`III. Title.
`
`96 ·19945
`CIP
`
`Acquisitions editor: Tom Robbins
`Production service: TKM Productions
`Editorial/production supervision: Sharyn VItrano
`Copy editor: Brian Baker
`Interior and cover design: Patrice Van Acker
`Art director: Amy Rosen
`Managing editor: Dayan! Mendoza DeLeon
`Editor-in-Chief: Marcia Horton
`Director of production and manufacturing: David W. Riccardi
`Manufacturing buyer: Donna Sullivan
`Editorial assistant: Phyllis Morgan
`
`© 1997 by Alan V. Oppenheim and Alan S. Willsky
`© 1983 by Alan V. Oppenheim, Alan S. Willsky, and Jan T. Young
`Pearson Education
`Upper Saddle River, New Jersey 07458
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without permission in writing from the publisher.
`
`The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
`development, research, and testing of the theories and programs to determine their effectiveness. The author and
`publisher make no warranty of any kind, expressed or implied, with regard to these progranl~ or the documentation
`contained in this book. The author and publi~her shall not be liable in any event for incidental or consequential damages
`in connection with, or arising out of, the furnishing, performance, or use of these programs.
`
`P!jnted in the United States of America
`
`30 31 32 33 34 35 36 37 38 39 40 V092 18 17 16 15 14
`
`ISBN 0-13-814757-4
`
`Prentice-Hall International (UK) Limited, London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada Inc., Toronto
`Prentice-Hall Hispanoamericana, S.A., Mexico
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of Japan, Inc., Tokyo
`Prentice-Hall (Singapore) Asia Pte. Ltd., Singapore
`Editora Prentice-Hall do Brasil. Ltda., Rio de Janeiro
`
`Petitioner Apple Inc.
`Ex. 1014, Cover-2
`
`

`

`1
`SIGNALS AND SYSTEMS
`
`eword
`
`these
`icular
`e that
`1ly of
`tg.
`g his-
`vhich
`:antly
`s. We
`>ossi-
`ques.
`1ding
`naly-
`ineer.
`esen-
`eader
`lin an
`mtals
`mdto
`tch to
`
`1.0 INTRODUCTION
`
`As described in the Foreword, the intuitive notions of signals and systems arise in a rich va-
`riety of contexts. Moreover, as we will see in this book, there is an analytical framework-
`that is, a language for describing signals and systems and an extremely powerful set of tools
`for analyzing them-that applies equally well to problems in many fields. In this chapter,
`we begin our development of the analytical framework for signals and systems by intro-
`ducing their mathematical description and representations. In the chapters that follow, we
`build on this foundation in order to develop and describe additional concepts and methods
`that add considerably both to our understanding of signals and systems and to our ability
`to analyze and solve problems involving signals and systems that arise in a broad array of
`applications.
`
`1.1 CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS
`
`1 . 1 . 1 Examples and Mathematical Representation
`Signals may describe a wide variety of physical phenomena. Although signals can be rep-
`resented in many ways, in all cases the information in a signal is contained in a pattern of
`variations of some form. For example, consider the simple circuit in Figure 1.1. In this case, ·
`the patterns of variation over time in the source and capacitor voltages, Vs and Vc, are exam-
`ples of signals. Similarly, as depicted in Figure 1.2, the variations over time of the applied
`force f and the resulting automobile velocity v are signals. As another example, consider
`the human vocal mechanism, which produces speech by creating fluctuations in acous-
`tic pressure. Figure 1.3 is an illustration of a recording of such a speech signal, obtained by
`1
`
`Petitioner Apple Inc.
`Ex. 1014, p. 1
`
`

`

`2
`
`Signals and Systems
`
`Chap. 1
`
`R
`
`c
`
`Figure 1. 1 A simple RC circuit with source
`voltage Vs and capacitor voltage Vc.
`
`~--+-
`
`-p v
`
`Figure 1.2 An automobile responding to an
`applied force f from the engine and to a re(cid:173)
`tarding frictional force pv proportional to the
`automobile's velocity v.
`
`,- - - - - I - - - - -,- - - - -
`I
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`I
`
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`
`~----! - ---~----- L------ -- -------1 - ----~----J
`j
`sh
`oul
`d
`1
`1
`1
`
`r----~-----r----, -- ---r--- - 1-----~----r----~
`I
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`
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`w
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`----------------------~----1-----~- - --~----~
`a
`ch
`
`I
`
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`
`,- -- - - I - -- - -,- - - - - I--- - -,-- - - -I - - - - I - - - - -,- - --- ~
`1
`I
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`
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`
`~----!----~ - - - - -~ ----J-- -- -L- - -- 1- ----~-- - - J
`a
`se
`1
`
`Figure 1.3 Example of a record(cid:173)
`ing of speech. [Adapted from Ap(cid:173)
`plications of Digital Signal Process(cid:173)
`ing, A.V. Oppenheim, ed. {Englewood
`Cliffs, N.J.: Prentice-Hall, Inc., 1978),
`p. 121 .] The signal represents acous(cid:173)
`tic pressure variations as a function
`of time for the spoken words "should
`we chase." The top line of the figure
`corresponds to the word "should,"
`the second line to the word "we,"
`and the last two lines to the word
`"chase." {We have indicated the ap(cid:173)
`proximate beginnings and endings
`of each successive sound in each
`word.)
`
`using a microphone to sense variations in acoustic pressure, which are then converted into
`an electrical signal. As can be seen in the figure, different sounds correspond to different
`patterns in the variations of acoustic pressure, and the human vocal system produces intel-
`ligible speech by generating particular sequences of these patterns. Alternatively, for the
`monochromatic picture, shown in Figure 1.4, it is the pattern of variations in brightness
`across the image that is important.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 2
`
`

`

`Sec. 1.1
`
`Continuous-Time and Discrete-Time Signals
`
`3
`
`Figure 1.4 A monochromatic
`picture.
`Signals ru;e represented mathematically as functions of one or more independent
`variables. For example, a speech signal can be represented mathematically by acoustic
`pressure as a function of time, and a picture can be represented by brightness as a func-
`tion of two spatial variables. In this book, we focus our attention on signals involving a
`single independent variable. For convenience, we will generally refer to the independent
`variable as time, although it may not in fact represent time in specific applications. For
`example, in geophysics, signals representing variations with depth of physical quantities
`such as density, porosity, and electrical resistivity are used to study the structure of the
`earth. Also, knowledge of the variations of air pressure, temperature, and wind speed with
`altitude are extremely important in meteorological investigations. Figure 1.5 depicts a typ-
`ical example of annual average vertical wind profile as a function of height. The measured
`variations of wind speed with height are used in examining weather patterns, as well as
`wind conditions that may affect an aircraft during final approach and landing.
`Throughout this book we will be considering two basic types of signals: continuous-
`time signals and discrete-time signals. In the case of continuous-time signals the inde-
`pendent variable is continuous, and thus these signals are defined for a continuum of values
`
`26
`24
`22
`20
`18
`2' 16
`0 g_ 14
`'lfi 12
`&w
`(/)
`8
`6
`4
`2
`
`0
`
`I • I ~.
`
`Height (feet)
`
`Figure 1.5 Typical annual vertical
`wind profile. (Adapted from Crawford
`and Hudson, National Severe Storms
`Laboratory Report, ESSA ERLTM-NSSL
`48, August 1970.)
`
`Petitioner Apple Inc.
`Ex. 1014, p. 3
`
`

`

`4
`
`Signals and Systems
`
`Chap. 1
`
`400
`
`350
`
`300
`
`250
`
`200
`
`150
`
`100
`
`50
`
`0
`
`Jan. 5,1929
`
`Jan. 4,1930
`
`Figure 1.6 An example of a discrete-time signal: The weekly Dow-Jones
`stock market index from January 5, 1929, to January 4, 1930.
`
`of the independent variable. On the other hand, discrete-time signals are defined only at
`discrete times, and consequently, for these signals, the independent variable takes on only
`a discrete set of values. A speech signal as a function of time and atmospheric pressure
`as a function of altitude are examples of continuous-time signals. The weekly Dow-Jones
`stock market index, as illustrated in Figure 1.6, is an example of a discrete-time signal.
`Other examples of discrete-time signals can be found in demographic studies in which
`various attributes, such as average budget, crime rate, or pounds of fish caught, are tab-
`ulated against such discrete variables as family size, total population, or type of fishing
`vessel, respectively.
`To distinguish between continuous-time and discrete-time signals, we will use the
`symbol t to denote the continuous-time independent variable and n to denote the discrete-
`time independent variable. In addition, for continuous-time signals we will enclose the
`independent variable in parentheses ( · ), whereas for discrete-time signals we will use
`brackets [ · ] to enclose the independent variable. We will also have frequent occasions
`when it will be useful to represent signals graphically. Illustrations of a continuous-time
`signal x(t) and a discrete-time signal x[n] are shown in Figure 1.7. It is important to note
`that the discrete-time signal x[n] is defined only, for integer values of the independent
`variable. Our choice of ~raphical representation for x[n] emphasizes this fact, and for
`further emphasis we will on occasion refer to x[n] as a discrete-time sequence.
`A discrete-time signal x[n] may represent a phenomenon for which the independent
`variable is inherently discrete. Signals such as demographic data are examples of this. On
`the other hand, a very important class of discrete-time signals arises from the sampling of
`continuous-time signals. In this case, the discrete-time signal x[n] represents successive
`samples of an underlying phenomenon for which the independent variable is continuous.
`Because of their speed, computational power, and flexibility, modem digital processors are
`used to implement many practical systems, ranging from digital autopilots to digital audio
`systems. Such systems require the use of discrete-time sequences representing sampled
`versions of continuous-time signals-e.g., aircraft position, velocity, and heading for an
`
`Petitioner Apple Inc.
`Ex. 1014, p. 4
`
`

`

`Sec. 1.1
`
`Continuous-Time and Discrete-Time Signals
`
`5
`
`x(t)
`
`0
`
`(a)
`
`x[n)
`
`x[O)
`
`t
`
`n
`
`Figure 1. 7 Graphical representations of (a) continuous-time and (b) discrete(cid:173)
`time signals.
`
`autopilot or speech and music for an audio system. Also, pictures in newspapers-or in this
`book, for that matter-actually consist of a very fine grid of points, and each of these points
`represents a sample of the brightness of the corresponding point in the original image. No
`matter what the source of the data, however, the signal x[n] is defined only for integer
`values of n. It makes no more sense to refer to the 3 ~th sample of a digital speech signal
`than it does to refer to the average budget for a family with 24 family members.
`Throughout most of this book we will treat discrete-time signals and continuous-time
`signals separately but in parallel, so that we can draw on insights developed in one setting
`to aid our understanding of another. In Chapter 7 we will return to the question of sampling,
`and in that context we will bring continuous-time and discrete-time concepts together in
`order to examine the relationship between a continuous-time signal and a discrete-time
`signal obtained from it by sampling.
`1.1.2 Signal Energy and Power
`From the range of examples provided so far, we see that signals may represent a broad
`variety of phenomena. In many, but not all, applications, the signals we consider are di-
`rectly related to physical quantities capturing power and energy in a physical system. For
`example, if v(t) and i(t) are, respectively, the voltage and current across a resistor with
`resistance R, then the instantaneous power is
`p(t) = v(t)i(t) = ~v2(t).
`
`(1.1)
`
`·-
`
`lU
`·n
`.l
`li
`
`-~
`
`n
`1
`
`-
`
`r
`
`I!
`
`tJ
`e
`
`[,
`l
`g
`~i
`01 -
`
`Petitioner Apple Inc.
`Ex. 1014, p. 5
`
`

`

`6
`
`Signals and Systems
`
`Chap. 1
`
`(1.2)
`
`:::;; t :::;; tz is
`The total energy expended over the time interval ft
`tz p(t)dt = Jtz ~v2(t)dt,
`J
`IJ
`/1
`and the average power over this time interval is
`1 J tz
`1 J tz 1
`- - p(t) dt = - - -v2(t) dt.
`11 R
`ft
`tz -
`tz -
`t1
`11
`Similarly, for the automobile depicted in Figure 1.2, the instantaneous power dissipated
`through friction is p(t) = bv2(t), and we can then define the total energy and average
`power over a time interval in the same way as in eqs. (1.2) and (1.3).
`With simple physical examples such as these as motivation, it is a common and
`worthwhile convention to use similar terminology for power and energy for any continuous-
`time signal x(t) or any discrete-time signal x[n]. Moreover, as we will see shortly, we will
`frequently find it convenient to consider signals that take on complex values. In this case,
`the total energy over the time interval t 1 :::;; t :::;; t2 in a continuous-time signal x(t) is
`defined as
`
`(1.3)
`
`(1.4)
`
`•
`
`(1.5)
`
`where lxl denotes the magnitude of the (possibly complex) number x. The time-averaged
`power is obtained by dividing eq. (1.4) by the length, t2 -
`t1, of the time interval. Simi-
`larly, the total energy in a discrete-time signal x[n] over the time interval n 1 :::;; n :::;; n2 is
`defined as
`nz L lx[nll2
`n=n 1
`and dividing by the number of points in the interval, n2 - n1 + 1, yields the average power
`over the interval. It is important to remember that the,terms "power" and "energy" are used
`here independently of whether the quantities in eqs. (1.4) and (1.5) actually are related to
`physical energy. 1 Nevertheless, we will find it convenient to use these terms in a general
`fashion.
`Furthermore, in many systems we will be interested in examining power and energy
`in signals over an infinite time interval, i.e., for -oo < t < +oo or for -oo < n < +oo. In
`these cases, we define the total energy as limits of eqs. (1.4) and (1.5) as the time interval
`increases without bound. That is, in continuous time,
`J +oo
`Eoo ~ )~ -T lx(t)l2 dt = -oo lx(t)l2 dt,
`
`1.:
`
`(1.6)
`
`(1.7)
`
`IT
`
`and in discrete time,
`
`+N
`
`+oo
`
`Eoo ~ lim L lx[nll2 = L lx[n]jZ.
`N ..... oo
`
`n=-N
`n= -«>
`1Even if such a relationship does exist, eqs. (1.4) and (1.5) may have the wrong dimensions and scalings.
`For example, comparing eqs. (1.2) and (1.4), we see that if x(t) represents the voltage across a resistor, then
`eq. (1.4) must be divided by the resistance (measured, for example, in ohms) to obtain units of physical energy.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 6
`
`

`

`:hap. 1
`
`(1.2)
`
`(1.3)
`
`ipated
`rerage
`
`nand
`lUOUS-
`re will
`; case,
`x:(t) is
`
`(1.4)
`
`~raged
`Simi-
`; n2 is
`
`(1.5)
`
`power
`eused
`tted to
`eneral
`
`mergy
`1-oo. In
`1terval
`
`(1.6)
`
`(1.7)
`
`eatings.
`:or, then
`energy.
`
`Sec. 1.2
`
`Transformations of the Independent Variable
`
`7
`
`Note that for some signals the integral in eq. (1.6) or sum in eq. (1.7) might not converge-
`e.g., if x(t) or x[n] equals a nonzero constant value for all time. Such signals have infinite
`energy, while signals with Eoo < oo have finite energy.
`In an analogous fashion, we can define the time-averaged power over an infinite
`interval as
`
`1 IT
`/:::,
`Poo = lim ZT
`
`T->oo
`
`lx(t)l2 dt
`-T
`
`and
`
`(1.8)
`
`(1.9)
`
`2
`
`1
`Poo ~ Ji~ ZN + l L lx[n]l
`
`+N
`
`n=-N
`in continuous time and discrete time, respectively. With these definitions, we can identify
`three important classes of signals. The first of these is the class of signals with finite total
`energy, i.e., those signals for which Eoo < oo. Such a signal must have zero average power,
`since in the continuous time case, for example, we see from eq. (1.8) that
`1.
`O
`Eoo
`Poo = 1m ZT =
`.
`T-.oo
`An example of a finite-energy signal is a signal that takes on the value 1 for 0 :5 t :5 1
`and 0 otherwise. In this case, Eoo = 1 and Poo = 0.
`A second class of signals are those with finite average power Poo. From what we
`have just seen, if Poo > 0, then, of necessity, Eoo = oo. This, of course, makes sense, since
`if there is a nonzero average energy per unit time (i.e., nonzero power), then integrating
`or summing this over an infinite time interval yields an infinite amount of energy. For
`example, the constant signal x[n] = 4 has infinite energy, but average power Poo = 16.
`There are also signals for which neither Poo nor Eoo are finite. A simple example is the
`signal x(t) = t. We will encounter other examples of signals in each of these classes in
`the remainder of this and the following chapters.
`
`(1.10)
`
`I .2 TRANSFORMATIONS OF THE INDEPENDENT VARIABLE
`
`A central concept in signal and system analysis is that of the transformation of a signal.
`For example, in an aircraft control system, signals corresponding to the actions of the pilot
`are transformed by electrical and mechanical systems into changes in aircraft thrust or
`the positions of aircraft control surfaces such as the rudder or ailerons, which in turn are
`transformed through the dynamics and kinematics of the vehicle into changes in aircraft
`velocity and heading. Also, in a high-fidelity audio system, an input signal representing
`music as recorded on a cassette or compact disc is modified in order to enhance desirable
`characteristics, to remove recording noise, or to balance the several components of the
`signal (e.g., treble and bass). In this section, we focus on a very limited but important class
`of elementary signal transformations that involve simple modification of the independent
`variable, i.e., the time axis. As we will see in this and subsequent sections of this chapter,
`these elementary transformations allow us to introduce several basic properties of signals
`and systems. In later chapters, we will find that they also play an important role in defining
`and characterizing far richer and important classes of systems.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 7
`
`

`

`8
`
`Signals and Systems
`
`Chap. 1
`
`1.2.1 Examples of Transformations of the Independent Variable
`A simple and very important example of transforming the independent variable of a signal
`is a time shift. A time shift in discrete time is illustrated in Figure 1.8, in which we have
`two signals x[n] and x[n- n0 ] that are identical in shape, but that are displaced or shifted
`relative to each other. We will also encounter time shifts in continuous time, as illustrated
`in Figure 1.9, in which x(t- to) represents a delayed (if to is positive) or advanced (if to
`is negative) version of x(t). Signals that are related in this fashion arise in applications
`such as radar, sonar, and seismic signal processing, in which several receivers at different
`locations observe a signal being transmitted through a medium (water, rock, air, etc.). In
`this case, the difference in propagation time from the point of origin of the transmitted
`signal to any two receivers results in a time shift between the signals at the two receivers.
`A second basic transformation of the time axis is that of time reversal. For example,
`as illustrated in Figure 1.1 0, the signal x[- n] is obtained from the signal x[ n] by a reflec-
`tion about n = 0 (i.e., by reversing the signal). Similarly, as depicted in Figure 1.11, the
`signal x(- t) is obtained from the signal x(t) by a reflection about t = 0. Thus, if x(t) rep-
`resents an audio tape recording, then x( -t) is the same tape recording played backward.
`Another transformation is that of time scaling. In Figure 1.12 we have illustrated three
`signals, x(t), x(2t), and x(t/2), that are related by linear scale changes in the independent
`variable. If we again think of the example of x(t) as a tape recording, then x(2t) is that
`recording played at twice the speed, and x(t/2) is the recording played at half-speed.
`It is often of interest to determine the effect of transforming the independent variable
`of a given signal x(t) to obtain a signal of the form x(at + /3), where a and f3 are given
`numbers. Such a transformation of the independent variable preserves the shape of x(t),
`except that the resulting signal may be linearly stretched if Ia I < 1, linearly compressed
`if Ia I > 1, reversed in time if a < 0, and shifted in time if f3 is nonzero. This is illustrated
`in the following set of examples.
`x[n]
`
`n
`
`x[n-nol
`
`II
`
`0
`
`Figure 1.8 Discrete-time signals
`related by a time shift. In this figure
`no > 0, so that x[n - n0] is a delayed
`verson of x[n] (I.e., each point in x[n]
`occurs later in x[n - no]).
`
`n
`
`Petitioner Apple Inc.
`Ex. 1014, p. 8
`
`

`

`al
`'e
`d
`d
`·o
`s
`;t
`
`1
`i
`
`Sec. 1.2
`
`Transformations of the Independent Variable
`
`9
`
`x[n)
`
`x[-n)
`
`(b)
`
`n
`
`n
`
`Figure 1.1 o
`(a) A discrete-time signal x[n]; (b) its reflec(cid:173)
`tion x[- n] about n = 0.
`
`Figure 1.9 Continuous-time signals related
`by a time shift. In this figure to < 0, so that
`to) Is an advanced version of x(t) (i.e.,
`x(t -
`each point In x( t) occurs at an earlier time in
`x(t- to)).
`
`x(t)
`
`(a)
`
`x(-t)
`
`(b)
`
`x(t) d\
`x(2t) &
`~
`
`x(V2)
`
`(a) A continuous-time signal x(t); (b) its
`Figure 1.11
`reflection x(- t) about t = 0.
`
`Figure 1. 12 Continuous-time signals
`related by time scaling.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 9
`
`

`

`10
`
`Signals and Systems
`
`Chap. 1
`
`Example 1.1
`·" Given the signal x(t) shown in Figure 1.13(a), the signal x(t + 1) corresponds to an
`advance (shift to the left) by one unit along the taxis as illustrated in Figure 1.13(b).
`Specifically, we note that the value of x(t) at t = to occurs in x(t + 1) at t = to - 1. For
`
`'j x(U
`
`0
`
`1
`(a)
`
`2
`
`
`
`-1
`
`0
`
`-----~-:~ :----1::-~-x(-t+-1)-~----=----- t
`
`1
`(b)
`
`2
`
`-1
`
`0
`
`1
`(c)
`
`0
`
`4/3
`
`2/3
`(d)
`
`-2/3
`
`0
`
`2/3
`
`(e)
`
`(a) The continuous-time signal x(t) used in Examples 1.1- 1.3
`Figure 1.13
`to Illustrate transformations of the Independent variable; (b) the time-shifted
`signal x(t + 1 ); (c) the signal x( - t + 1} obtained by a time shift and a time
`reversal; (d) the time-scaled signal x(~t); and (e) the signal x( ~ t + 1) obtained
`by time-shifting and scaling.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 10
`
`

`

`Sec. 1.2
`
`Transformations of the Independent Variable
`
`11
`
`example, the value of x(t) at t = 1 is found in x(t + 1) at t = 1 - 1 = 0. Also, since
`•. x(t) is zero fort < 0, we have x(t + 1) zero fort < -1. Similarly, since x(t) is zero for
`-~ t > 2, x(t + 1) is zero fort > 1.
`Let us also consider the signal x(- t + 1 ), which may be obtained by replacing t
`'i~ ... ~
`with-tin x(t + 1). That is, x(-t + 1) is the time reversed version of x(t + 1). Thus,
`x( -t + 1) may be obtained graphically by reflecting x(t + 1) about the taxis as shown
`w) in Figure 1.13(c).
`
`Example 1.2
`
`1 Given the signal x(t), shown in Figure l.l3(a), the signal x( ~ t) corresponds to a linear
`~¥ • compression of x(t) by a factor of ~ as illustrated in Figure 1.13( d). Specifically we note
`• that the value of x(t) at t = to occurs in x(~t) at t = j to. For example, the value of
`· x(t) at t = 1 is found in x(~t) at t = ~ (1) == j. Also, since x(t) is zero fort < 0, we
`· have x(~t) zero fort< 0. Similarly, since x(t) is zero fort> 2, x(~ t) is zero fort> ~·
`
`Example 1.3
`'
`Suppose that we would like to determine the effect of transforming the independent vari-
`able of a given signal, x(t), to obtain a signal of the form x(at + {3), where a and {3 are
`given numbers. A systematic approach to doing this is to first delay or advance x(t) in
`. "'' accordance with the value of {3, and then to perform time scaling and/or time reversal on
`· , , the resulting signal in accordance with the value of a. The delayed or advanced signal is
`linearly stretched if /a I < 1, linearly compressed if /a I > 1, and reversed in time if a < 0.
`'
`To illustrate this approach, let us show how x( ~t + 1) may be determined for the
`signal x(t) shown in Figure l.l3(a). Since {3 = 1, we first advance (shift to the left) x(t)
`by 1 as shown in Figure l.13(b). Since /a/ = ~.we may linearly compress the shifted
`signal of Figure 1.13(b) by a factor of~ to obtain the signal shown in Figure 1.13(e).
`In addition to their use in representing physical phenomena such as the time shift
`in a sonar signal and the speeding up or reversal of an audiotape, transformations of the
`independent variable are extremely useful in signal and system analysis. In Section 1.6
`and in Chapter 2, we will use transformations of the independent variable to introduce and
`analyze the properties of systems. These transformations are also important in defining
`and examining some important properties of signals.
`
`1.2.2 Periodic Signals
`An important class of signals that we will encounter frequently throughout this book is
`the class of periodic signals. A periodic continuous-time signal x(t) has the property that
`there is a positive value of T for which
`x(t) = x(t + T)
`for all values oft. In other words, a periodic signal has the property that it is unchanged by a
`time shift ofT. In this case, we say that x(t) is periodic with period T. Periodic continuous-
`time signals arise in a variety of contexts. For example, as illustrated in Problem 2.61,
`the natural response of systems in which energy is conserved, such as ideal LC circuits
`without resistive energy dissipation and ideal mechanical systems without frictional losses,
`are periodic and, in fact, are composed of some of the basic periodic signals that we will
`introduce in Section 1.3.
`
`(1.11)
`
`Petitioner Apple Inc.
`Ex. 1014, p. 11
`
`

`

`12
`
`Signals and Systems
`
`Chap. 1
`
`···/\ 1\! 1\ ~ ...
`
`-2T
`
`-T
`
`0
`
`T
`
`2T
`
`Figure 1 . 14 A continuous-time
`periodic signal.
`
`An example of a periodic continuous-time signal is given in Figure 1.14. From the
`figure or from eq. (1.11), we can readily deduce that if x(t) is periodic with period T, then
`x(t) = x(t + mT) for all t and for any integer m. Thus, x(t) is also periodic with period
`2T, 3T, 4T, .... The .fundamental period To of x(t) is the smallest positive value ofT for
`which eq. (1.11) holds. This definition of the fundamental period works, except if x(t) is
`a constant. In this case the fundamental period is undefined, since x(t) is periodic for any
`choice ofT (so there is no smallest positive value). A signal x(t) that is not periodic will
`be referred to as an aperiodic signal.
`Periodic signals are defined analogously in discrete time. Specifically, a discrete-
`time signal x[n] is periodic with period N, where N is a positive integer, if it is unchanged
`by a time shift of N, i.e., if
`
`x[n] == x[n + N]
`(1.12)
`for all values of n. If eq. (1.12) holds, then x[n] is also periodic with period 2N, 3N, ....
`The .fundamental period No is the smallest positive value of N for which eq. (1.12) holds.
`An example of a discrete-time periodic signal with fundamental period No = 3 is shown
`in Figure 1.15.
`
`x[n]
`
`n
`
`Figure 1.15 A discrete-time pe-
`riodic signal with fundamental period
`No= 3.
`
`Example 1.4
`Let us illustrate the type of problem solving that may be required in determining whether
`• or not a given signal is periodic. The signal whose periodicity we wish to check is given
`,'' by
`
`if t < 0
`( ) = { cos(t)
`0 .
`, ( )
`'f
`X t
`smt
`112::
`From o·igonometry, we know that cos{t + 2'71') = cos(t) and sin(t + 2'71') = sin(t). Thus
`conside.ring t > 0 and 1 < 0 separately, we see that x(t) does repeat itself over every
`interval of length 2'71'. However as illustrated in Figure 1.16, x(t) also has a discontinuity
`at tbe time origin that does not recur at any other time. Since every featuJ·e in the shape of
`a periodic signal must recur periodically, we conclude that ihe signal x(c) is not periodic.
`
`(1.13)
`
`.,
`
`Petitioner Apple Inc.
`Ex. 1014, p. 12
`
`

`

`Sec. 1.2
`
`Transformations of the Independent Variable
`
`13
`
`' '
`
`x(t)
`
`Figure 1. 16 The signal x(t) considered in Example 1.4.
`
`1.2.3 Even and Odd Signals
`Another set of useful properties of signals relates to their symmetry under time reversal.
`A signal x(t) or x[n] is referred to as an even signal if it is identical to its time-reversed
`counterpart, i.e., with its reflection about the origin. In continuous time a signal is even if
`x( -t) = x(t),
`(1.14)
`while a discrete-time signal is even if
`x[-n] = x[n].
`
`(1.15)
`
`A signal is referred to as odd if
`
`x(-t) = -x(t),
`(1.16)
`x[-n] = -x[n].
`(1.17)
`An odd signal must necessarily be 0 att = 0 or n = 0, since eqs. (1.16) and (1.17) require
`that x(O) = - x(O) and x[O] = - x[O]. Examples of even and odd continuous-time signals
`are shown in Figure 1.17.
`
`x(t)
`
`x(t)
`
`(a) An even con(cid:173)
`Figure 1.17
`tinuous-time signal; (b) an odd
`continuous-time signal.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 13
`
`

`

`14
`
`Signals and Systems
`
`Chap. 1
`
`x[n] = { 1, n ~ 0
`0, n < 0
`
`- 3- 2-1 0 1 2 3
`
`.. .'!III
`&t~{x!nl} = l !: ~: ~
`···trr'hrr· ..
`
`2, n >0
`
`-3-2-1 0 1 2 3
`
`n
`
`n
`
`.I
`
`1
`
`-3-2-1
`
`tld{x!nl} = 1-~·. ~: ~
`2, n >0
`2 T r r
`···1110123
`n Figure 1 . I s Example of the even(cid:173)
`odd decomposition of a discrete-time
`- ! 2
`signal.
`An important fact is that any signal can be brolf.en into a sum of two signals, one of
`which is even and one of which is odd. To see this, consider the signal
`1
`Sv{x(t)} == 2 [x(t) + x(-t)],
`which is referred to as the even part of x(t). Similarly, the odd part of x(t) is given by
`1
`0d{x(t)} = 2[x(t)- x(-t)].
`It is a simple exercise to check that the even part is in fact even, that the odd part is odd,
`and that x(t) is the sum of the two. Exactly analogous definitions hold in the discrete-
`time case. An example of the even -odd decomposition of a discrete-time signal is given
`in Figure 1.18.
`
`(1.18)
`
`(1.19)
`
`I .3 EXPONENTIAL AND SINUSOIDAL SIGNALS
`
`Iil this section and the next, we introduce several basic continuous-time and discrete-time
`signals. Not only do these signals occur frequently, but they also serve as basic building
`blocks from which we can construct many other signals.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 14
`
`

`

`Sec. 1.3
`
`Exponential and Sinusoidal Signals
`
`15
`
`1.3.1 Continuous-Time Complex Exponential
`and Sinusoidal Signals
`
`The continuous-time complex exponential signal is of the form
`x(t) = cea1
`where C and a are, in general, complex numbers. Depending upon the values of these
`parameters, the complex exponential can exhibit several different characteristics.
`
`(1.20)
`
`,
`
`RealExponennalSignah
`As illustrated in Figure 1.19, if C and a are real,[in which case x(t) is called a real
`exponential], there are basically two types of behavior. H a is positive, then as t in-
`creases x(t) is a growing exponential, a form that is used in describing many different
`physical proces~es, including chain reactions in atomic explosions and complex chemical
`reactions. If a is negative, then x(t) is a decaying exponential, a signal that is also used
`to describe a wide variety of phenomena, including the process of radioactive decay and
`the responses of RC circuits and damped mechanical systems. In particular, as shown
`in Problems 2.61 and 2.62, the natural responses of the circuit in Figure 1.1 and the
`automobile in Figure 1.2 are decaying exponentials. Also, we note that for a = 0, x(t)
`is constant.
`
`x(t)
`
`(a)
`
`x(t)
`
`(b)
`
`Figure 1.19 Continuous-time real
`exponential x(t) = Clf1: (a) a > 0;
`(b) a< 0.
`
`Petitioner Apple Inc.
`Ex. 1014, p. 15
`
`

`

`16
`
`Signals and Systems
`
`Chap. 1
`
`Periodic Complex Exponential and Sinusoidal Signals
`A second important class of complex exponentials is obtained by constraining a to be
`purely imaginary. Specifically, consider
`
`x(t) = ejwot.
`
`(1.21)
`
`