`
`Communication
`
`
`
`
`Systems
`Engineering
`
`John G. Proakis
`
`Masoud Salehi
`
`Northeastern University
`
`PRENTICE HALL, Englewood Cliffs, New Jersey 07632
`
`REMBRANDT EXHIBIT 2202
`
`REMBRANDT EXHIBIT 2202
`
`

`

`Library of Congress Cataloging-iml’ublication Data
`Proakis, John G.
`Communication systems engineering / John G. Proakis, Masoud
`Salehi.
`cm.
`p.
`Includes bibliographical references and index.
`ISBN 0-13-158932-6
`l. Telecommunication.
`TK5101.P75 1994
`621.382—dc20
`
`I. Salehi, Masoud.
`
`II. Title.
`
`93-23109
`CIP
`
`Acquisitions editor: DON FOWLEY
`Production editor: IRWIN ZUCKER
`Production coordinator: DAVID DICKEY
`Supplements editor: ALICE DWORKIN
`Copy editors: JOHN COOK and ANNA HALASZ
`Cover design: DESIGN SOLUTIONS
`Editorial assistant: JENNIFER KLEIN
`
`
`
`© 1994 by Prentice-Hall, Inc.
`A Paramount Communications Company
`Englewood Cliffs, New Jersey 07632
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without pemtission in writing from the publisher.
`
`Printed in the United States of America
`
`10987654321
`
`ISBN 0-].3-1156'132-[3
`
`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
`Simon & Schuster Asia Pte. Ltd., Singapore
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`
`REMBRANDT EXHIBIT 2202
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`REMBRANDT EXHIBIT 2202
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`

`

`438
`
`Digital Transmission Through an AWGN Channel
`
`Chap. 7
`
`We begin by developing a geometric representation for several different types
`of pulse modulation signals. Optimum demodulation and detection of these signals
`is then described. Finally. we evaluate the probability of error as a performance mea-
`sure for the different types of modulation signals on an AWGN channel. The various
`modulation methods are compared on the basis of their performance characteristics.
`their bandwidth requirements. and their implementation complexity.
`Initially. we will not
`impose bandwidth constraints in the design of signals
`for digital modulation. However. because the channel bandwidth is an important
`parameter that
`influences the design of most communication systems.
`the design
`of the modulator and demodulator for bandlimited channels is treated in depth in
`Chapter 8.
`
`7.1 PULSE MODULATION SIGNALS AND THEIR GEOMETRIC
`REPRESENTATION
`
`types of pulse modulation signals that are
`In this section. we introduce several
`used for the transmission of digital information and develop a geometric represen-
`tation of such signals. The pulse modulation signals considered include (I) pulse
`amplitude modulated signals. (2) pulse position modulated (orthogonal) signals.
`(3) bionhogonal signals. (4) simplex signals. and (5) signals generated from binary
`code sequences. First we describe how the digital
`information is conveyed with
`these types of signals.
`
`7.1.1 Pulse Modulation Signals
`
`ln pulse amplitude modulation (PAM). the information is conveyed by the amplitude
`of the pulse. For example.
`in binary PAM. the information bit
`I
`is represented by
`a pulse of amplitude A and the information bit 0 is represented by a pulse of
`amplitude —A as shown in Figure 7.|. Pulses are transmitted at a bit rate R), = l/T.
`bits per second. where T), is called the bit interval. Although the pulses are shown
`as rectangular.
`in practical systems. the rise time and decay time are nonzero and
`the pulses are generally smoother. The effect of the pulse shape on the spectral
`characteristics of the transmitted signal is considered in Chapter 8.
`
`um
`
`A
`
`o
`
`no)
`
`0
`
`-4
`
`T’
`
`’
`
`T),
`
`I
`
`nouns: 1.1. Example of signals for
`binary PAM.
`
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`

`Sec. 7.1
`
`Pulse Modulation Signals and Their Geometric Representation
`
`439
`
`In pulse position modulation (PPM). the information is conveyed by the time
`interval
`in which the pulse is transmitted. For example.
`in binary PPM.
`the bit
`interval is divided into two time slots of Tb/Z seconds each. The information bit I
`is represented by a pulse of amplitude A in the first time slot and the information
`bit 0 is represented by a pulse of amplitude A in the second time slot. as shown in
`Figure 7.2. The pulse shape within a time slot need not be rectangular.
`Another pulse modulation method for transmitting binary information is called
`on-off keying (00K). The transmitted signal for transmitting a l
`is a pulse of
`duration Tb. If a 0 is to be transmitted. no pulse is transmitted in the signal interval
`of duration Tb.
`The generalization of PAM and PPM to nonbinary (M -ary) pulse transmission
`is relatively straightforward. Instead of transmitting one bit at a time. the binary
`information sequence is subdivided into blocks of k bits. called symbols. and each
`block. or symbol.
`is represented by one of M = 2‘ pulse amplitude values for
`PAM and pulse position values for PPM. Thus. with k = 2. we have M = 4 pulse
`amplitude values. or pulse position values. Figure 7.3 illustrates the PAM and PPM
`signals for k = 2. M = 4. Note that when the bit rate R. is fixed. the symbol interval
`IS
`
`It
`T= — =kT
`Rb
`a
`
`7.|.I
`
`(
`
`)
`
`as shown in Figure 7.4.
`It
`is interesting to characterize the PAM and PPM signals in terms of their
`basic properties. For example. the M-ary PAM signal waveforms may be expressed
`as
`
`5.0) = Misfit). m = l.2.....M
`
`Kl“)
`
`A
`
`0
`
`:20)
`
`A
`
`D
`2
`
`T.
`
`l
`
`0
`
`I:
`2
`
`7*
`
`‘
`
`noun 1:. Exunple of signals for
`binary PPM.
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`

`

`81(1)
`
`3A
`
`810)
`
`Digital Tmnsmission Through an AWGN Channel
`
`Chap. 7
`
`
`
`T
`
`I
`
`T
`
`I
`
`0
`
`34“)
`
`0
`
`—3A
`
`T
`
`T
`
`I
`
`I
`
`1
`
`2
`
`.
`
`3.7
`
`4
`
`(I) M I 4 PAM signals
`
`1
`
`4
`
`33(1)
`
`0‘
`
`.
`
`1
`
`2
`
`
`
`8‘0)
`
`3_T
`4
`
`T
`
`l
`
`(b) M I 4 PPM signals
`
`ncmu EumpleuofM=4PAMmdPPMsigmls.
`
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`

`

`Sec. 7.1
`
`Pulse Modulation Signals and Their Geometric Representation
`
`441
`
`|~————r———-I
`
`0
`
`T.
`
`2T.
`
`*Tb
`
`T.- bit interval
`.
`T. symbol interval
`
`FIGURE 7.4. Relationship between the
`symbol interval and the bit interval.
`
`0 S f S T
`
`(7.].2)
`
`where gr“) is a pulse of some arbitrary shape as shown in Figure 7.5(a). We
`observe that the distinguishing feature among the M signals is the signal amplitude.
`All the M signals have the same pulse shape. Another important feature of these
`signals is their energies. We note that the signals have different energies. i.e..
`T
`T
`1-... =f 33,0)4: = A2,] gium. m= l.2.....M
`(7.1.3)
`
`0
`
`0
`
`In the case of PPM. the signal waveforms may be expressed as
`
`5.0) = A310 - (m — l)T/M). m =1.2.....M
`
`(m —- l)T/M 5 r 5 mT/M
`
`(7.1.4)
`
`where gym is a pulse of duration T/M and of arbitrary shape. as shown for
`example in Figure 7.5(b). A major distinguishing characteristic of these waveforms
`is that they are nonoverlapping. Consequently.
`
`/ s..(t)s..(r)dr = 0. m # n
`
`0
`
`(7.].5)
`
` r
`
`31(1)
`
`0
`
`1
`M
`
`(b)
`
`r
`
`(l)
`
`FIGURE 7.5. Signal pulses for (I) PAM and (1’) PPM.
`
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`

`“2
`
`Digital Transmission Through an AWGN Channel
`
`Chap. 7
`
`Such signal waveforms are said to be orthogonal. A second distinguishing feature
`of PPM signal waveforms is that all the waveforms have equal energy. i.e..
`T
`nT/M
`
`f s:(r)dt = A2]
`
`0
`
`(n—liT/M
`T/M
`
`gzu — (m — l)T/M)dr
`
`= A2]
`
`0
`
`g2(t)dt = a:
`
`all m
`
`(7.l.6)
`
`A major difference between a set of M PAM signal waveforms and a set of
`M PPM signal waveforms is the channel bandwidth required for their transmission.
`We observe that the channel bandwidth required to transmit the PAM signals is
`determined by the frequency characteristics of the pulse g1“). This basic pulse
`has a duration T. On the other hand.
`the basic pulse gr(!) in a set of M PPM
`signal waveforms has a duration T/M. Whatever pulse shape we choose for PAM
`and PPM and whatever definition of bandwidth that we employ.
`it
`is clear that
`the spectrum of the PPM pulse is M times wider than the spectrum of a PAM
`pulse. Consequently. the PAM signals do not require an expansion of the channel
`bandwidth as M increases. On the other hand. the PPM signal waveforms require
`an increase in the channel bandwidth as M increases.
`types of signal
`PAM and PPM are two examples of a variety of different
`sets that can be constructed for transmission of digital information over baseband
`channels. For example. if we take a set of M/2 PPM signals and construct the M/Z
`negative signal pulses. the combined set of M signal waveforms constitute a set of
`M bionhogonal signals. An example for M = 4 is shown in Figure 7.6. It is easy
`to see that all the M signals have equal energy. Furthermore. the channel bandwidth
`required to transmit the M signals is just one-half of that required to transmit M
`PPM signals.
`As another example. we demonstrate that from any set of M orthogonal signal
`waveforms. we can construct another set of M signal waveforms that are known as
`simplex signal waveforms. From the M orthogonal signals we subtract the average
`of the M signals. Thus.
`
`,
`I
`"
`s_(r) =s,,(t)— H 25.“)
`k=l
`
`Then. it follows that (see Problem 7.7) the energy of these signals 5;,(1) is
`T
`
`2;]; [s_(i)] dr=(l— HP;
`
`l
`
`r
`
`I
`
`I
`
`j; s_(t)s.(l)dt — M _ l
`
`_
`
`I
`
`'l‘,.m ;é n
`
`REMBRANDT EXHIBIT 2202
`
`(7.1.7)
`
`(7.l.8)
`
`(7.l.9)
`
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`

`

`Sec. 7.1
`
`Pulse Modulation Signals and Their Geometric Representation
`
`“3
`
`
`
`FIGURE 7.6. A set of M = 4 biorthogoml signal waveform.
`
`is the energy of
`is the energy of each of the orthogonal signals and '5;
`where T.
`each of the signals in the simplex signal set. Note that the waveforms in the simplex
`set have smaller energy than the waveforms in the orthogonal signal set. Second.
`we note that simplex signal waveforms are not orthogonal. Instead.
`they have a
`negative correlation. which is equal for all pairs of signal waveforms. It has been
`conjectured for several decades that among all the possible M -ary signal waveforms
`of equal energy 1;. the simplex signal set results in the smallest probability of error
`when used to transmit infomtation on an additive white Gaussian noise channel.
`However. this conjecture has not been proved.
`As a final example of the construction of M signal wavefomis. let us consider
`a set of M binary code words of the form
`
`c..=(c...c..2 ..... c..~). m=l.2.....M
`
`(7.1.10)
`
`where c.., = 0 or I for all m and j. N is called the block length or dimension of
`the code words. Given M code words. we can construct M signal waveforms by
`mapping a code bit c.., = 1
`into a pulse gr(r) of duration T/N and a code bit
`c.., = 0 into the negative pulse —gr(r).
`
`Example 7.1.]
`Given the code words
`
`l
`c. =[l
`l
`c¢=[l
`0
`c3=[l
`c.=[0 I
`
`l
`0
`l
`0
`
`l]
`0]
`0]
`I]
`
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`

`444
`
`Digital Transmission Through an AWGN Channel
`
`Chap. 7
`
`construct a set of M = 4 signal waveforms. as described above. using a rectangular
`pulse arm.
`
`is mapped into the rectangular pulse 370) of
`I
`Soltrtion As indicated above. a code bit
`duration 1/4 and a code bit 0 is mapped into the rectangular pulse —gr(r). Thus. we
`construct the four waveforms shown in Figure 7.7 that correspond to the four code
`words. It is interesting to note that the first time signal waveforms in Figure 7.7 are
`mutually orthogonal. but the fourth waveform is the negative of the third.
`
`There are numerous binary block codes that have been constructed over the
`past several decades. Several of these codes are described in Chapter IO. Any of
`these codes can be used to construct signal waveforms for M -ary signal transmission
`over a baseband channel.
`
`In addition to the pulse modulated signals described above. in Section 7.1.4.
`we describe other types of baseband signals that are synthesized from a set of code
`words.
`
`7.1.2 Geometric Repreeentation of Signal Waveforms
`
`We recall from Section 2.2 that the Gram-Schmidt orthogonalization procedure may
`be used to construct an orthonormal basis for a set of signals.
`In this section.
`we develop a geometric representation of signal waveforms as points in a signal
`space. Such a representation provides a compact characterization of signal sets
`for transmitting information over a channel and simplifies the analysis of their
`performance.
`
`1:0)
`
`T 2 F
`
`IGURE 7.7. A set of M :4 signal waveforms constructed from the code words in
`Example 7.1.].
`
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