The
`George Washington
`University
`
`Gelman
`
`Library
`
`
`
`
`
`
`
`
`
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`
`#HJ
`1%
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`
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`Page 1 of 52
`
`SAMSUNG EXHIBIT 1019
`
`Page 1 of 52
`
`SAMSUNG EXHIBIT 1019
`
`

`

`DIGITAL
`COMMUNICATIONS
`
`Page 2 of 52
`
`

`

`
`
`DIGITAL
`
`COMMUNICATIONS
`
`Fundamentals and Applications
`
`BERNARD SKLAR
`
`The Aerospace Corporation, El Segundo, California
`and
`
`University of California, Los Angeles
`
`PRENTICE HALL
`
`Englewood Cliffs, New Jersey 07632
`
`Page 3 of 52
`
`

`

`SKLAR, BERNARD (date)
`Digital communications.
`
`Bibliography: p.
`Includes index,
`I. Title.
`1. Digital communications.
`TK5103.7.SSS 1988
`621.38’0413
`ISBN 0434119394)
`
`87-1316
`
`Editorial/production supervision and
`interior design: Reynold Rieger
`Cover design: Wanda Lubelska Design
`Manufacturing buyers: Gordon Osbourne and Paula Benevento
`
`© 1988 by Prentice Hall
`A Division of Simon & Schuster
`Englewood Cliffs, New Jersey 07632
`
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`
`Prentice-Hall International (UK) Limited, London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada lnc., 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
`
`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.
`
`Printed in the United States of America
`
`1098765432]
`
`ISBN U-lB-E’ll‘lfl‘i-D DES
`
`Page 4 of 52
`
`Page 4 of 52
`
`

`

`CHAPTER '7
`
`Modulation and Coding
`Trade- Offs
`
`
`
`
`
`
`
`l
`
`
`From other
`SOU FCBS
`
`,
`Information
`source
`
`l
`l
`1
`I
`
`Source
`bits
`
`_
`
`“
`
`_
`Channel
`bits
`
`,
`
`"
`
`_
`
`“
`
`
`
`_ " "I
`I
`1
`I
`
`i
`
`Synch-
`ronization
`
`Digital
`waveform
`
`‘ Channel
`decode
`
`Information
`sink
`
`i
`
`l
`l
`1
`Source
`Channel
`I
`
`1
`bits
`bits
`}
`
`w Optional
`1
`3%
` [:3 Essential
`To_oth_er
`destinations
`
`
`Page 5 of 52
`
`381
`
`Page 5 of 52
`
`

`

`
`
`
`
`@GOALS OF THE COMMUNICATIONS SYSTEM DESIGNER
`System trade—offs are fundamental to all digital communication designs. The goals
`of the designer are (1) to maximize transmission bit rate, R; (2) to minimize prob-
`ability of bit error, PB; (3) to minimize required power, or equivalently, to min-
`imize required bit energy to noise power spectral density, Eb/No; (4) to minimize
`requ1red system bandwidth, W; (5) to maximize system utilization, that is, to
`provide reliable service for a maximum number of users with minimum delay and
`with maximum resistance to interference; and (6) to minimize system complexity,
`computational load, and system cost. A good system designer seeks to achieve
`all these goals simultaneously. However, goals 1 and 2 are clearly in conflict with
`goals 3 and 4; they call for simultaneously maximizing R, while minimizing PB,
`Eb/NO, and W. There are several constraints and theoretical limitations that ne—
`cessitate the trading off of any one system requirement with each of the others.
`Some of the constraints are:
`The Nyquist theoretical minimum bandwidth requirement
`The Shannon—Hartley capacity theorem (and the Shannon limit)
`Government regulations (e.g., frequency allocations)
`Technological limitations (e.g., state-of—the art components)
`Other system requirements (e.g., Satellite orbits)
`Some of the realizable modulation and coding trade—offs can best be viewed
`
`l
`it
`
`Page 6 of 52'
`
`382
`
`_
`
`Modulation and Coding Trade-Offs
`
`Chap. 7
`
`Page 6 of 52
`
`

`

`
`
`as a change in operating point on one of two performance planes. These planes
`will be referred to as the error probability plane and the bandwidth efficiency
`plane; they are described in the following sections.
`
`
`
`f 7.‘ ERROR PROBABILITY PLANE
`
`Figure 7.1 illustrates the family of PB versus Eb/No curves for the coherent de-
`tection of orthogonal signaling (Figure 7.1a) and multiple phase signaling (Figure
`7.1b). For signaling schemes that process k bits at a time, the signaling is called
`M—ary (see Section 3.8). The modulator uses one of its M = 2" waveforms to
`represent each k—bit sequence, where M is the size of the symbol set. Figure 7.1a
`illustrates the potential bit error improvement with orthogonal signaling as k (or
`M) is increased. For orthogonal signal sets, such as frequency shift keying (FSK)
`modulation, increasing the size of the symbol set can provide an improvement in
`PB, or a reduction in the Eb/No required, at the cost of increased bandwidth.
`Figure 7.1b illustrates potential bit error degradation with nonorthogonal signaling
`as k (or M) increases. For nonorthogonal signal sets, such as multiple phase shift
`keying (MPSK) modulation, increasing the size of the symbol set can reduce the
`bandwidth requirement, but at the cost of a degraded P3, or an increased Eb/No
`requirement. We shall refer to these families of curves (Figure 7.1a or b) as error“
`probability performance curves, and to the plane on which they are plotted as an
`error probability plane. Such a plane describes the locus of operating points avail—
`able for a particular type of modulation and coding. For a given system information
`’ rate, each curve in the plane can be associated with a different fixed minimum
`required bandwidth; therefore, the set of curves can be termed equibandwidth
`curves. As the curves move in the direction of the ordinate, the required trans—
`miséion bandwidth increases; as the curves move in the opposite direction, the
`required bandwidth decreases. Once a modulation and coding scheme and an
`available Eb/NO are determined, system operation is characterized by a particular
`point in the error probability plane. Possible trade—offs can be Viewed as changes
`in the operating point on one of the curves or as changes in the operating point
`from one curve to another curve of the family. These trade—offs are seen in Figure
`7.1a and b as changes in the system operating point in the direction shown by the
`arrows. Movement of the operating point along line 1, between points a and b,
`can be viewed as trading off PB for Eb/No performance (with W fixed). Similarly,
`movement along line 2, between points 0 and d, is seen as trading PB for W
`performance (with Eb/No fixed). Finally, movement along line 3, between points
`e and f, illustrates trading W for Eb/No performance (with PB fixed). Movement
`along line 1 is effected by increasing or decreasing the available Eb/No. This can
`be achieved, for example, by increasing transmitter power, which means that the
`trade-off might be accomplished simply by “turning a knob,” even after the sys-
`tem is configured. However, the other trade—offs (movement along line 2 or line
`3) involve some change in the system modulation or coding scheme, and therefore
`need to be accomplished during the system design phase.
`
`
`
`ll
`l
`
`Sec. 7.2
`
`Error Probability Plane
`
`‘
`
`383
`
`Page 7 of 52
`
`Page 7 of 52
`
`

`

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`(W) 8d ’Alllichaqmd 10143 348
`
`384
`
`Page 8 of 52
`
`
`
`
`
`
`
`AS@53qu55.:@8082“.mwaouoaooSmozkm332,banmfioaSheH5#80.5me
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`
`Page 8 of 52
`
`
`
`

`

`I
`
`
`
`NYQUlST MlNlMUM BANDWIDTH
`
`Every realizable system having some nonideal filtering will suffer from intersym—
`bol interference (ISI)——the tail of one pulse spilling over into adjacent symbol
`intervals so as to interfere with correct detection. Nyquist [1] showed that, in
`theory, Rs symbols per second could be detected without 181 in an RS/2 hertz
`minimum bandwidth (Nyquist bandwidth); this is‘a basic theoretical constraint,
`limiting the designer’s goal to expend as little bandwidth as possible (see Section
`2.11). In practice, RS hertz is typically required for the transmission of Rs symbols
`per second. In other words, typical digital communication throughput, Without
`181, is limited to 1 symbol/s per hertz. The modulation or coding system assigns
`to each symbol, of its set of M symbols, a k—bit meaning, where M =. 2". For a
`signaling scheme with a fixed bandwidth, such as MPSK, as k increases, the
`allowable data rate, R, increases, and hence the bandwidth efficiency, R/ W, mea-
`sured in bits per second per hertz, also increases.‘ For example, movement along
`line 3, from point e to point f in Figure 7.1b represents trading Eb/No for a reduced
`bandwidth requirement. In other words, with the same system bandwidth one can
`transmit at an increased data rate, hence at an increased R/ W.
`
`
`1"“!7.4‘ HANNON—HARTLEY CAPACITY THEOREM
`
`
`Shannon [2] showed that the system capacity, C, of a channel perturbed by ad—
`ditive white Gaussian noise (AWGN) is a function of the average received signal
`power, S, the average noise power, N, and the bandwidth, W. The capacity re—
`lationship (Shannon—Hartley theorem) can be stated as
`
`C = Wiog2 (1 + N)
`
`S
`
`i
`
`(7.1)
`
`When W is in hertz and the logarithm is taken to the base 2, as shown, the capacity
`is given in bits/s. It is theoretically possible to transmit information over such a
`channel at any rate, R, where R S C, with an arbitrarily small error probability
`by using a sufficiently complicated coding scheme. For an information rate
`R > C, it is not possible to find a code that can achieve an arbitrarily small error
`probability. Shannon’s work showed that the values of S, N, and W set a limit
`on transmission rate, not on error probability. Shannon [3] used Equation (7.1)
`to graphically exhibit a bound for the achievable performance of practical systems.
`This plot, shown in Figure 7. 2, gives the normalized channel capacity CLBL1n
`b131HWct1on of the channel signal-to--noise ratioISNR). A related plot,
`shownin Figure 7. 3, indicates the normalized channel bandwidth W/ C1n Hz/bits
`as a function of SNR in the channel. Figure 7.3 is sometimes used to illustrate
`the power—bandwidth trade—off inherent in the ideal channel. However, it is not
`a pure trade-off [4] because the detected noise power is proportional to bandwidth.
`
`Sec. 7.4
`
`Shannon-Hartley Capacity Theorem
`
`N = NOW ,
`
`(7.2)
`
`385
`
`
`
`Page 9 of 52
`
`Page 9 of 52
`
`

`

`C/W (bits/s/Hz)
`
`16
`
`8
`
`4
`
`Unattainable
`region
`
`
`
`
`Practical
`systems
`
`
`
`50
`
`Figure 7.2 Normalized channel
`capacity versus channel SNR.
`
`Substituting Equation (7.2) into Equation (7.1) and rearranging terms yields
`
`S
`W
`ng <
`NOW)
`
`C
`— = l
`
`l +
`
`'
`
`'
`
`7.
`
`(
`
`3)
`
`For the case Where transmission bit rate is equal to channel capacity, R = C,
`we can use the identity presented in Equation (3.94) to write
`
`
`S
`'
`
`NoC
`
`E,
`= —-
`
`No
`
`Hence we can modify Equation (7.3) as follows:
`
`C
`—— = l
`
`E, C
`1 + —— —
`
`N0 (WM
`ng [
`W
`20W = 1 + — —N. (W)
`
`'
`
`E, C
`
`W
`E,
`— = -— ZC/W — 1
`No
`C <
`
`>
`
`‘
`
`7.4
`
`)
`
`(
`
`(7 )
`
`.5
`
`.6
`(7 >
`
`Figure 7.4 is a plot of W/C versus Eb/No in accordance with Equation (7.6).
`
`386
`
`Modulation and Coding Trade-Offs
`
`Chap. 7
`
`Page 10 of 52
`
`Page 10 of 52
`
`

`

`W/C (Hz/bits/s)
`
`Practical
`systems
`
`50
`
`
`
`Figure 7.3 Normalized channel
`bandwidth versus channel SNR.
`
`
`
`
`
`Unattainable
`region
`
`1/4
`
`1/8
`
`The asymptotic behavior of this curve as C/W—> 0 (or W/C —> 00) is‘discussed in
`the next section.
`'
`
`7.4.1 Shannon Limit
`
`There exists a limiting value of Eb/NO below which there can be no error-free
`communication at any information rate. Using the identity
`
`lim (1 + x)” = e
`x—>0
`
`we can calculate the limiting value of Eb/No as follows. Let
`
`( rag
`H
`No W
`
`Then from Equation (7.5) ,L
`
`g2 (
`
`1 + x “x
`)
`
`C—
`
`- = x lo
`W
`
`Eb
`1 = —- lo
`‘ No
`
`g2(
`
`1 + x “x
`)
`
`—
`
`Sec. 7.4
`
`Shannon-Hartley Capacity Theorem
`
`387
`
`Page 11 of 52
`
`Page 11 of 52
`
`

`

`W/C (Hz/bits/S)
`
`
`
`
`16
`
`8
`
`4
`
`l l
`
`Asymptote
`|
`loge =—1.59dB :
`l
`
`Practical
`systems
`
`Unattainable
`region
`
`
`
`1/18
`
`Figure 7 .4 Normalized channel bandwidth versus channel Eb/No.
`
`In the limit, as C/We 0, we get
`
`(77)
`
`H
`
`1
`
`——
`
`‘
`if}:
`logz e
`N0
`or, in decibels = — 1.59 dB
`
`0.693
`
`This value of Eb/No is called the Shannon limit. On Figure 7. la the Shannon limit
`is the PB versus Eb/No curve corresponding to k-—> 00. The curve is discontinuous,
`going from a value of PB = % to PB = 0 at Eb/No = —1‘.59 dB. It is not possible
`..in practice to reach the Shannon limit, because as k increases Without bound, the
`bandwidth requirement and the implementation complexity increase Without
`bound. Shannon’s work provided a theoretical proof for the existence of codes
`that could improve the PB performance, or reduce the Eb/No required, from the
`levels of the uncoded binary modulation schemes to levels approaching the limiting
`curve. For a bit error probability of 10‘s, binary phase shift keying (BPSK) mod--
`ulation' requires an Eb/No of 9.6 dB (the optimum uncoded binary modulation).
`Therefore, Shannon’s work promised the existence of a theoretical performance
`
`388
`
`Modulation and Coding Trade—Offs
`
`Chap. 7
`
`Page 12 of 52
`
`Page 12 of 52
`
`

`

`improvement of 11.2 dB over the performance of optimum uncoded binary mod-
`ulation, through the use of coding techniques. Today, most of that promised im—
`provement (approximately 7 dB) is realizable [5]. Optimum system design can
`best be described as a search for rational compromises or trade—offs among the
`various constraints and conflicting goals. The modulation and coding trade—off,
`that is, the selection of modulation and coding techniques to make the best use
`of transmitter power and channel bandwidth, is important, since there are strong
`incentives to reduce the cost of generating power and to conserve the radio
`spectrum.
`
`7.4.2 Entropy
`
`To design a communications system with a specified message handling capability,
`we need a metric for measuring the information content to be transmitted. Shan-
`non [2] developed such a metric, H, called the entropy of the message source
`(having n possible outputs). Entropy is defined as the average amount of infor-
`mation per source output and is expressed by
`
`' H = — Z pilogz p,-
`i=1
`
`bits/source output
`
`(7.8)
`
`where p,- is the probability of the ith output and 2 pi = 1. In the case of a binary
`message or a source having only two possible outputs, with probabilities p and
`q = (l — p), the entropy is written
`
`H = -(p10g2p + qlogz CI)
`
`(7.9)
`
`and is plotted versus p in Figure 7.5.
`The quantity H has a number of interesting properties,
`following:
`
`including the
`
`1. When the logarithm in Equation (7.8) is taken to the base 2, as shown, the
`unit for H is averagebits per event. The unit bit, here, is a measure of
`information content and is not to be confused with the term-“bit,” meaning
`“binary digit.”
`
`2. The term “entropy” has the same uncertainty connotation as it does in
`certain formulations of statistical mechanics. For the information source
`
`with two equally likely possibilities (e.g., the flipping of a fair coin), it can
`be seen from Figure 7.5 that the uncertainty in the event, and hence the
`average information content, is maximum. As the probabilities depart from
`the equally likely case, the average information content decreases. In the
`limit, when one of the probabilities goes to zero, H also goes to zero. We
`/know the result before the event happens, so the result conveys no additional
`information.
`
`3. To illustrate that information content is related to a priori probability (if the
`a priori message probability at the receiver is zero or one, we need not send
`the message), consider the following example: At the end of her nine-month
`
`
`
`
`
`..
`
`Sec. 7.4
`
`Shannon-Hartley Capacity Theoremi
`
`389
`
`Page 13 of 52
`
`Page 13 of 52
`
`

`

`1.0
`
`0.9
`
`.0.0O.0.0.0(A)J:-010'}\l00
`
`.0 l\)
`
`0.1
`
`Entropy,H(bits)
`
`0
`
`0.1
`
`0.2
`
`0.3
`
`0.4
`
`0.5
`
`0.6
`
`0.7
`
`0.8
`
`0.9
`
`1.0
`
`Probability, p
`
`Figure 7.5 Entropy versus probability (two events).
`
`pregnancy, a woman enters the deliVery room of a local hospital to give
`birth. Her husband waits anxiously in the waiting room. After some time,
`a physician approaches the husband and says: “Congratulations, you are
`the father of a child.” How much information has the physician given the
`father beyond the medical outcome? Almost none; the father has known
`with virtual certainty that a child was forthcoming. Had the physician said,
`“you are the father of a boy” or “you are the father of a gir ,” he would
`have transmitted 1 bit of information, since there was a 50% chance that the
`child could have been a boy or a girl.
`
`Example 7.1 Average Information Content in the English Language
`(a) Calculate the average information in bits/character for the English language, as-
`suming that each of the 26 characters in the alphabet occurs with equal likelihood.
`Neglect spaces and punctuation.
`.
`(b) Since the alphabetic characters do not appear with equal frequency in the English
`language (or any other language), the answer to part (a) will represent an upper
`bound on average information content per character. Repeat part (a) under the
`assumption that the alphabetic characters occur with the following probabilities:
`
`390
`
`Modulation and Coding Trade-Offs
`
`Chap. 7
`
`Page 14 of 52
`
`Page 14 of 52
`
`

`

`*U‘BWU’U
`
`0.10:
`
`for the letters a, e, o, t
`
`0.07:
`
`for the letters h, i, n, r, s
`
`0.02: \ for the letters 0, d, f, l, m, p, u, y
`
`0.01:
`
`for the letters b, g, j, k, q, V, W, x, z
`
`; Solution
`
`(a) H = — I; 561029 <26)
`
`2 4.7 bits/character:
`
`(b) H = —(4 X 0.110g20.1 + 5 X 0.07 logz 0.07
`+ 8 X 0.02 logz 0.02 + 9 X 0.0110g2 0.01)
`= 4.17 bits/character
`
`If we want to express the 26 letters of the alphabet with some binary-digit
`coding scheme, we generally need five binary digits for each character. Example
`7.1 demonstrates that there may be a way to encode the English language with a
`fewer number of binary digits per character, on the average, by exploiting the
`fact that the average amount of information contained within each character is
`less than 5 bits. The subject of source coding, which deals with this exploitation,
`is treated in Chapter 11.
`
`7.4.3 Equivooation and Effective Transmission Rate
`
`Suppose that we are transmitting information at a rate of 1000 binary symbols/s
`over a binary symmetric channel (defined in Section 5.3.1), and that the a priori
`probability of transmitting either a one or a zero is equally likely. Suppose also
`that the noise in the channel is so great that the probability of receiving a one is
`%, whatever was transmitted, and similarly for receiving a zero. In such a case,
`half the received symbols would be correct due to chance alone, and the system
`might appear to be providing 500 bits/s while actually no information is being
`received at all. Equally “good” reception could be obtained by dispensing with
`the channel entirely and “flipping a coin” within the receiver. The proper cor-
`rection to apply to the amount of information transmitted is the amount of infor-
`mation that is lost in the channel. Shannon [2] uses a correction factor called
`equivocatz’on to account for the‘ uncertainty in the received signal. Equivocation
`is defined as the conditional entropy of the message X, given Y, as shown below:
`
`H(X|D = — E PtX, Y)10g2P(XiY)
`X’Y
`= — 2 P00 2 P(XIY)10g2P(XiY)
`Y
`X
`
`(7.10)
`
`Where X is the transmitted source message, Y is the received signal, P(X, Y) is
`the joint probability of X and Y, and P(X|Y) is the conditional probability of X
`given Y. EquivOcation can be thought of as the uncertainty that message X was
`
`Sec. 7.4
`
`Shannon—Hartley Capacity Theorem
`
`391
`
`p.7
`Page 15 of 52
`
`
`
`give
`.ne,
`are
`
`the
`)WH
`
`aid,
`puld
`
`the
`
`. as-
`
`)od.
`
`glish
`>per
`the
`ies:
`
`
`
`Page 15 of 52
`
`

`

`;
`
`sent, having received Y. For an error—free channel, H(XIY) = 0, because having
`received Y, there is complete certainty about the message X. However, for a
`channel with a nonzero probability of symbol error, H(X|Y) > 0, because the
`channel introduces uncertainty. Consider a binary sequence, X, where the a priori
`source probabilities are P(X = l) = P(X = 0) = %, and where, on the average,
`the Channel produces one error in a received sequence of 100 bits (PB = 0.01).
`Using Equation (7.10), the equivocation H(XIY) is expressed as
`
`HCXIY) = ‘10— PB)10g2 (1 — PB) + PB logzPBl
`
`= —(0.99 log; 0.99 + 0.01 log 0.01)
`
`= 0.081 bit/received symbol
`
`Thus, the channel introduces 0.081 bit of uncertainty to each received symbol.
`Shannon showed that the average effective information content, Heff, at the
`receiver,
`is obtained by subtracting the equivocation from the entropy of the
`source. Therefore,
`
`Hcff = 1100 — H(XIY)
`
`(7-11)
`
`For a system transmitting equally likely binary symbols, the entropy, H(X), is l
`bit/symbol. When the symbols are received‘with PB = 0.01 the equivocation is
`0.081 bit/received symbol as was calculated above. Then using Equation (7.11),
`the effective entropy of the received signal, Hcff, is
`
`Heff = l — 0.081 = 0.919 bit/received symbol
`
`Thus, if R = 1000 binary symbols transmitted per second, fer example, the ef-
`fective information bit rate, Raff, can be expressed as
`
`Reff : RHeff‘
`
`(7.12)
`
`= 1000 symbols/s >< 0.919 bit/symbol = 919 bits/s
`
`Notice that in the extreme case where PB = 0.5,
`
`H(X|Y) = —(0.510g2 0.5 + 0.510g2 0.5)
`
`‘_ = l bit/symbol
`
`and, applying Equations (7.12) and (7.11) to the R = 1000 symbols/s example,
`yields
`. a
`
`Raff = 1000 symbols/s (1 —~ 1) = 0 bit/s ,
`
`as should be expected.
`
`Example 7.2 Apparent Contradiction in the Shannon Limit
`
`Plots of PB versus Eb/No typically display a smooth increase of PB as Eb/No is de-
`. creased. For example, the bit error probability for the curves'in Figure 7.1 shows
`PB tending to 0.5 in the limit as Eb/No approaches zero. Thus there is apparently
`always a nonvanishing information rate, regardless of how small Eb/No becomes.
`This appears to contradict the Shannon limit of Eb/No = — 1.59 dB, below which
`
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`no error-free information rate can be supported per unit bandwidth, or below which
`even an infinite bandwidth cannot support a finite information rate (see Figure 7.4).
`
`(a) Suggest a way of resolving the apparent contradiction.
`(b) Show how Shannon’s equivocation correction can resolve it for a binary PSK
`system where the source has an entropy of 1 bit/symbol. Consider that the op-
`erating point on Figure 7.1b corresponds to Eb/No = 0.1 (— 10 dB).
`
`Solution
`
`(a) The value of Eb, traditionally used in link calculations for practical systems, is
`invariably the received signal energy per transmitted symbol. However, the
`meaning of Eb in Equation (7.6) is the signal energy per bit of received infor-
`mation. The information loss caused by the noisy channel must be taken into
`account to resolve the apparent contradiction.
`
`(b) Following Equation (3.84) for BPSK,
`
`PB = Q(\/2EbNo) = Q(O.447)
`
`where Q is defined in Equation (2.42) and tabulated in Table B.1. From the
`tabulation, PE is found to be 0.33. Next, we solve for the equivocation and
`
`effective entropy:
`
`. H(X1Y) = —[(l “ 1315:)10g2(1~~ PB) + P31082PB]
`= —(0.67 log; 0.67 + 0.33 log; 0.33)
`
`0.915 bit/symbol
`
`Heff = H(X) — 111(le
`= 1 — 0.915
`
`= 0.085 bit/symbol
`
`Hence
`
`_W
`E2
`No eff # V
`Heff bits/symbol
`
`joules per hit
`_:
`0.1
`0.085 _ 1'176 watts/Hz
`
`= 0.7 dB
`
`Thus the effective value of Eb/No is equal to 0.7 dB per received information
`bit, which is well above Shannon’s limit of — 1.59 dB.
`
` (7.5 ANDWIDTH-EFFICIENCY PLANE
`
`Using Equation (7.6), we can plot normalized channel bandwidth W/ C in Hz/bits/s
`versus Eb/No, as shown in Figure_7.4. Here, with the abscissa taken as Eb/No,
`we see the true power—bandwidth trade—off at work. It can be shown [4] that well—
`designed systems tend to operate near the “knee” of this power—bandwidth trade-
`off curve for the ideal (R = C) channel. Actual systems are frequently within 10
`dB or less of the performance of the ideal. The existence of the knee means that
`
`Sec. 7.5
`
`Bandwidth-Efficiency Plane
`
`’
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`‘ 393
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`

`systems seeking to reduce the channel bandwidth they occupy or to reduce the
`signal power they require must make an increasingly unfavorable exchange in the
`other parameter. For example, from Figure 7.4, an ideal system operating at an
`Eb/No of 1.8 dB and using a normalized bandwidth of 0.5 Hz/bits/s would have
`to increase Eb/NO to 20 dB to reduce the bandwidth occupancy to 0.1 Hz/bits/s.
`Trade-offs in the other direction are similarly inequitable.
`Using Equation (7.6), we can alsoplot C/ W versus Eb/No. This relationship
`is shown plotted on the R/ W versus Eb/No plane in Figure 7.6. We shall denote
`
`R/W (bits/s/Hz)
`
`
`
`
` Capacity boundar
`for which R
`C
`
`E Region for
`
`Region for
`whic’h‘R < C
`
`\
`
`Bandwidth“
`limited \
`‘region
`
`
`
`
`
`i
`l:
`
`
`
`
`Direction of
`
`improvingPB
`
`,-
`
`18
`
`24
`
`30
`
`36
`
`Eb/NoidB)
`
`Legend
`
`
`
`1/4
`
`M =16
`I
`
`o Coherent MFSK, PB =10r5
`
`Power—
`limited
`
`
`
`region
`
`I Noncoherent MFSK, P8 = 10—5
`
`A Coherent OAMLPB = 1045
`
`Figure 7.6 Bandwidth—efficiency plane.
`
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`this plane as the bandwidth-efi‘icz’ency plane. The ordinate, R/ W, is a measure of
`how much data can be communicated in a specified bandwidth within a given
`time; it therefore reflects how efficiently the bandwidth resource is utilized. The
`abscissa is Eb/No in units of decibels. For the case where R = C in Figure 7.6,
`the curve represents a boundary that separates a region characterizing practical
`communication systems from a region where such communication systems are
`not theoretically possible. Like Figure 7.2, the bandwidth—efficiency plane in Fig-
`ure 7.6 sets the limiting performance that can be achieved by practical systems.
`Since the abscissa in Figure 7.6 is Eb/NO rather than SNR, Figure 7.6 is more
`useful for comparing digital communication modulation and coding trade-offs than
`is Figure 7.2.
`
`7.5.1 Bandwidth Efficiency of MPSK and MFSK
`Modulation
`
`On the bandwidth—efficiency plane of Figure 7.6 are plotted the operating‘points
`for coherent MPSK modulation at a bit error probability of 10—5. We assume
`Nyquist (ideal rectangular) filtering at baseband, so that the minimum double—
`sideband (DSB) bandwidth at an intermediate frequency (IF) is WIF = 1/T, where
`T is the symbol duration. Thus the bandwidth efficiency is R/ W = log; M, where
`M is the symbol set size. For realistic channels and waveforms, the performance
`must be reduced to account for the bandwidth increase required to implement
`realizable filters. Notice that for MPSK modulation, R/ W increases with increas-
`ing M. Notice also that the location of the MPSK points indicates that BPSK (M
`= 2) and quaternary PSK or QPSK (M = 4) require the same Eb/No. That is,
`for the same value of Eb/No, QPSKhas a bandwidth efficiency of 2 bits/s/Hz,
`compared to l bit/s/Hz for BPSK. This“’Unique feature stems from the fact that
`QPSK is effectively-a composite of two BPSK signals transmitted on orthogonal
`components of the carrier.
`Also plotted on the bandwidth-efficiency plane of Figure 7.6 are the operating
`points for noncoherent MFSK modulation at a bit error probability of 10—5. We
`assume that the IF transmission bandwidth is WIF = M/ T, and thus the bandwidth
`efficiency is R/ W : k/M. Notice that for 'MFSK modulation, R/ W decreases With
`increasing M. Notice also that the position of the MFSK points indicates that
`BFSK (M = 2) and quaternary FSK (M = 4) have the same bandwidth efficiency,
`even though the former requires greater Eb/No for the same error probability. The
`bandwidth efficiency varies with the modulation index (tone spacing in hertz di-
`vided by bit rate). Under the assumption that an equal increment of bandwidth
`is required for each MFSK tone the system uses, it can be seen that for M = 2,
`the bandwidth efficiency is l bit/s/2 Hz or %, and for M : 4, similarly, the R/ W
`is 2 bits/s/4 Hz or %.
`
`Operating points for coherent quadrature amplitude modulation (QAM) are
`also plotted in Figure 7.6.. Of the modulations shown, QAM is clearly the most
`bandwidth efficient; it is treated in greater detail in Section 7.9.3.
`V
`
`Sec. 7.5
`
`Bandwidth—Efficiency Plane
`
`395
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`l
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`7.5.2 Analogues between Bandwidth-Efficiency and Error
`Probability Planes
`
`The bandwidth—efficiency plane in Figure 7.6 is analogous to the error probability
`plane in Figure 7.1. The Shannon limit of the Figure 7.1 plane is analogous to the
`capacity boundary of the Figure 7.6 plane. The curves in Figure 7.1 were referred
`to as equibandwidth curves. In Figure 7.6, we can analogously describe equi-
`error—probability curves for various modulation and coding schemes. The curves,
`labeled P31, P32, and P33 , are hypothetical constructions for some arbitrary mod.
`ulation and coding scheme; the P31 curve represents the largest error probability
`of the three curves, and the P33 curve represents the smallest. The general di-
`rection in which the curves move for improved PB is indicated on the figure.
`Just as potential trade-offs among PB, Eb/No, and W were considered for
`the error probability plane, the same trade-offs can be considered on the band-
`width efficiency plane. The potential tradewoffs are seen in Figure 7.6 as changes
`in operating point in the direction shown by the arrows. Movement of the operating
`point along line 1 can be Viewed as trading PB for Eb/No, withR/ Wfixed. Similarly,
`movement along line 2 is seen as trading PB for W (or R/ W), with Eb/No fixed
`Finally, movement along line 3 illustrates trading W (or R/ W) for Eb/No, with PB
`fixed. In Figure 7.6, as in Figure 7.1, movement along line 1 can be effected by
`increasing or decreasing the available Eb/No. However, movement along line 2
`or line 3 requires changes in the system modulation or coding scheme.
`The two primary communications resources are the transmitted power and
`the channel bandwidth. In many communication systems, one of these resources
`may be more precious than the other, and hence most systems can be classified
`as either power limited or bandwidth limited. In power—limited systems, coding
`schemes can be used to save power at the expense of bandwidth, whereas in
`bandwidth-limited systems, spectrally efficient modulation techniques can be used
`to save bandwidth at the expense of power.
`
`{@POWER-LIMITED SYSTEMS
`
`For the case of power--limited systems, systems in which power is scarce but
`system bandwidthIS available (e. g., a space communication link), the following
`trade-offs might be made: (1) improved PB can be achieved by expending band—
`width (for a given Eb/No); or.(2) required Eb/No can be reduced by expending
`bandwidth (fora given PB). The error probability plane of Figure 7.1a can be very
`useful for examining these potential trade—offs. It is on such a plane that we can
`verify whether or not a candidate modulation or code offers improvement in re-
`quired Eb/No for a particular channel and for a specified PB (or whether the mod-
`ulation or code offers improvement in PB for a given Eb/No).
`
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`7.7 BAN DWIDTH-LIMHTED SYSTEMS
`
`Any digital scheme that transmits logz M bits in T seconds using a bandwidth of
`W hertz operates at a bandwidth efficiency of R/ W = (logz M)/ WT bits/s/Hz.
`From this expression it can be seen that the smaller the WT product, the more
`bandwidth efficient will be the system. Signals with small WT products are more
`often used with bandwidth-limited systems—systems in which channel bandwidth
`is constrained but power is available. For this case the usual objective is to design
`the link so as to maximize the transmitted information rate over the bandlimited
`channel, at the expense of Eb/No (while maintaining a specified value of PB). For
`bandlimited operation, bandwidth efficiency is a useful criterion of system per—
`formance, and the bandwidth—efficiency plane of Figure 7.6’is useful for examining
`potential trade—offs.
`Two regions, the bandwidth-limited region and the power-limited region, are
`shown on the bandwidth efficiency plane of Figure 7.6. Notice that the desirable
`trade-offs associated with each of these regions are not equitable. For the band-
`width~limited region, large R/ W is desired; however, as Eb/NO is increased, the
`capacity boundary curve flattens out and ever-increasing amounts of additional
`,‘ Eb/No are required to achieve improvement in R/ W. A similar relationship is at
`' work in the power—limited region. Here a savings in Eb/No is desired, but the
`/ capacity boundary curve is steep; to achieve a small reduction in required Eb/NO
`’t
`requires a large reduction in R/W.
`
`a
`
`
`
`7.8 MODULATION AND CODENG TRADE-OFFS
`
`Figure 7.7 is useful in pointing out analogies between the two performance planes,
`the error probability plane of Figure 7.1 and the bandwidth efficiency plane of
`Figure 7.6. Figure 7.7a and b represent the same planes as Figures 7.1 and 7.6,
`respectively. They have been redrawn as symmetrical, by choosing appropriate
`scales. In each case the arrows and their labels describe the general effect of
`moving an operating point in the direction of the arrow by means of ap