DIGITAL
`SIGNAL
`PROCE.SSING
`
`Alan V. Oppenheim
`Department of Electrical Engineering
`Massachusetts Institute of Technology
`
`Ronald W. Schafer
`Bell Telephone Laboratories
`Murray Hill, New Jersey
`
`PRENTICE-HALL, INC., Englewood Cliffs, New Jersey.
`
`I )
`
`,
`
`1
`
`Micro Motion 1039
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`

`

`Library of Congre8s Cataloging In Publlcalion Data
`
`OPPENHEIM, ALAN V.
`Digital Signal PtocesS!ng.
`
`Includes bibliographical references.
`1. Signal tbeOry (Telecommunication) 2. Digital electronics. I. SCHAFER, RONALD W.
`joint author. n. Title.
`TK5102.5.0245
`621.3819'58'2
`ISBN 0-13-214635-5
`
`74-17280
`
`@ 1975 by Alan V. Oppenheim
`and Bell Telephon~ Laboratories, Inc.
`
`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.
`
`10 9 8 7 6 5 4 3 2 1
`
`Printed in the United States of America
`
`PRENTICE-HALL INTERNATIONAL, INC., London
`PRENTICE-HALL OF AUSTRALIA, PTy. LTD., Sydney
`PRENTICE-HALL OF CANADA, LTD., Toronto
`PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
`PRENTICE-HALL OF JAPAN, INC., Tokyo
`
`2
`
`

`

`I
`
`Discrete-Time Signals and Systems
`
`1.0 Introduction
`
`A signal can be defined as a function that conveys information, generally
`about the state or behavior of a physical system. Although signals can be
`represented in many ways, in all cases the information is contained in a
`pattern of variations of some form. For example, the signal may take the
`form of a pattern of time variations or a spatially varying pattern. Signals
`are represented mathematically as functions of one or more independent
`variables. For example, a speech signal would be represented mathematic(cid:173)
`ally as a function of time and a picture would be represented as a brightness
`function of two spatial variables. It is a common convention, and one that
`will be followed in this book, to refer to the independent variable of the
`mathematical representation of a signal as time, although it may in fact not
`represent time.
`The independent variable of the mathematical representation of a signal
`may be either continuous or discrete. Continuous-time signals are signals
`that are defined at a continuum of times and thus are represented by contin(cid:173)
`uous variable functions. Discrete-time signals are defined at discrete times
`and thus the independent variable takes on only discrete values; i.e., discrete(cid:173)
`time signals are represented as sequences of numbers. As we will see,
`signals such as speech or pictures may have either a continuous or a discrete
`variable representation, and if certain conditions hold, these representations
`are entirely equivalent.
`
`6
`
`In addil
`continuous
`discrete. Di
`discrete. Cc
`called analo
`In almo~
`to facilitate
`processing t
`usually take
`is in some se
`to design tn
`combined ir
`meter of as
`signal. Sign
`signals. Tha
`input and 0
`those for w
`analog syste,
`and digital s~
`Digital sigm
`discrete in b
`of this book
`The effects
`Discrete·
`they may be
`origin of tb
`manyattrae
`general-purr
`hardware. 1
`moreimporl
`analog hard'
`when sophis
`In this c
`signals and
`then for tw
`the class of
`this chapter
`that we del
`invariant c(
`texts [1-3].
`systems by
`This appro~
`fact forms i
`systems (sec
`
`3
`
`

`

`1.0
`
`Introduction
`
`7
`
`In addition to the fact that the independent variables can be either
`continuous or discrete, the signal amplitude may be either continuous or
`discrete. Digital signals are those for which both time and amplitude are
`discrete. Continuous-time, continuous-amplitude signals are sometimes
`called analog signals.
`In almost every area of science and technology, signals must be processed
`to facilitate the extraction of information. Thus, the development of signal
`processing techniques and systems is of great importance. These techniques
`usually take the form of a transformation of a signal into another signal that
`is in some sense more desirable than the original. For example, we may wish
`to design transformations for separating two or more signals that have been
`combined in some way; we may wish to enhance some component or para(cid:173)
`meter of a signal; or we may wish to estimate one or more parameters of a
`signal. ~ignal processing systems may be classified along the same lines as
`signals. That is, continuous-time systems are systems for which both the
`input and output are continuous-time signals and discrete-time systems are
`those for which the input and output are discrete-time signals. Similarly
`analog systems are systems for which the input and output are analog signals
`and digital systems are those for which the input and output are digital signals.
`Digital signal processing, then, deals with transformations of signals that are
`discrete in both amplitude and time. This chapter, and in fact the major part
`of this book, deals with discrete-time rather than digital signals and systems.
`The effects of discrete amplitude are considered in detail in Chapter 9.
`Discrete-time signals may arise by sampling a continuous-time signal or
`they may be generated directly by some discrete-time process. Whatever the
`origin of the discrete-time signals, digital signal processing systems have
`many attractive features. They can be realized with great flexibility using
`general-purpose digital computers, or they can be realized with digital
`hardware. They can, if necessary, be used to simulate analog systems or,
`more importantly, to realize signal transformations impossible to realize with
`analog hardware. Thus, digital representations of signals are' often desirable
`when sophisticated, signal processing is required .
`In this chapter we consider the fundamental concepts of discrete-time
`signals and signal processing systems first for one-dimensional signals and
`then for two-dimensional signals. We shall place the most emphasis on
`the class of linear shift-invariant discrete-time systems. It will be true in
`this chapter and succeeding ones that many of the properties and results
`that we derive will be similar to properties and results for linear time(cid:173)
`invariant continuous-time systems as presented in a variety of excellent
`texts [1-3]. In fact, it is possible to approach the discussion of discrete-time
`systems by treating sequences as analog signals that are impulse trains.
`This approach, if implemented carefully, can lead to correct results and in
`fact forms the basis for much of the classical discussion of sampled data
`systems (see, for example, [4-6]). In many present digital signal processing
`
`'$
`
`lation, generally
`u signals can be
`. contained in a
`al may take the
`pattern. Signals
`)re independent
`~ed mathematic-
`1 as a brightness
`m, and one that
`. variable of the
`may in fact not
`
`:ition of a signal
`;nals are signals
`ented by contin-
`1t discrete times
`es; i.e., discrete(cid:173)
`\s we will see,
`DUS or a discrete
`; representations
`
`4
`
`

`

`8 Discrete-Time Signals and Systems
`
`applications, however, not all sequences arise from sampling a continuous(cid:173)
`time signal. Furthermore, many discrete-time systems are not simply
`approximations to corresponding analog systems. Therefore, rather than
`attempt to force results from analog system theory into a discrete framework,
`we shall derive similar results starting within a framework and with notation
`suitable to discrete-time systems. Discrete-time signals will be related to
`analog signals only when necessary.
`
`1.1 Discrete-Time Signals-Sequences
`
`In discrete-time system theory. we are concerned with processing signals that·
`are represented by sequences. A sequence of numbers x, in which the nth
`number in the sequence is denoted x(n), is formally written as
`-00 < n < 00
`
`x = {x(n)},
`
`(1.1)
`
`x(n)
`
`x(O)
`x(1)
`x(2J
`
`x (-1)
`x(-2)
`
`Fig. 1.1 Graphical representation of a discrete-time signal.
`
`Although sequences do not always arise from sampling analog waveforms,
`for convenience we shall refer to x(n) as the "nth sample" of the sequence.
`Also, although strictly speaking x(n) denotes the nth number in the sequence,
`the notation of Eq. (1.1) is often unnecessarily cumbersome, and it is con(cid:173)
`venient and unambiguous to refer to "the sequence x(n)." Discrete-time
`signals (i.e., sequences) are often depicted graphically as shown in Fig. 1.1.
`Although the abscissa is drawn as a continuous line, it is important to recog(cid:173)
`nize that x(n) is only defined for integer values of n. It is not correct to think
`of x(n) as being zero for n not an integer; x(n) is simply undefined for non(cid:173)
`integer values of n.
`Some examples of sequences are shown in Fig. 1.2. The unit-sample
`sequence, ben), is defined as the sequence with values
`
`ben) = {~:
`
`0
`n
`n=O
`
`Fi
`
`As we wi
`crete-tim
`tinuous-1
`is often 1
`importar
`matical
`definitiOJ
`
`5
`
`

`

`:1 continuous-
`not simply
`, rather than
`:e framework,
`with notation
`be related to
`
`19 signals that·
`which the nth
`
`(1.1).
`
`--n
`
`)g waveforms,
`, the sequence.
`1 the sequence,
`and it is con(cid:173)
`Discrete-time
`wn in Fig. 1.1.
`,rtant to recog(cid:173)
`orrect to think
`~fined for non-
`
`he unit-sample
`
`1.1 Discrete-Time Signals-Sequences
`
`9
`
`Uoil ",mpl.
`
`'I
`
`-----}~.~.~.~.~.-.~.~.~.~o~.~.~.~.~.~.~.~.~.~.~. __ ---n
`
`_ ..................... ~. J ~II III~II1I_n
`
`Unit step
`
`o
`
`Sinusoidal
`
`Fig. 1.2 S(jme examples of sequences. The sequences shown play an
`important role in the analysis and representation of discrete(cid:173)
`time signals and systems.
`
`As we will see shortly, the unit-sample sequence plays the same role for dis(cid:173)
`crete-time signals and systems that the unit impulse function does for con(cid:173)
`tinuous-time signals and systems. For convenience the unit-sample sequence
`is often referred to as a discrete-time impulse, or simply as an impulse. It is
`important to note that a discrete-time impulse does not suffer from the mathe(cid:173)
`matical complications that a continuous-time impulse suffers from. Its
`definition is simple and precise. The unit-step sequence, u(n), has values
`
`u(n) = {~:
`
`n;;:::O
`n<O
`
`6
`
`