`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 1
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 1
`
`

`

`
`
`Printed and bound by Book-mart Press, Inc.
`
`DIGITAL COMMUNICATIONS
`
`Copyright © 1989, 1983 by McGraw-Hill, Inc. All rights reserved.
`Printed in the United States of America. Except as permitted under the
`United States Copyright Act of 1976, no part of this publication may be
`reproduced or distributed in any form or by any means, or stored in a data
`base or retrieval system, without prior written permission of the
`publisher.
`
`456789BKMBKM99876543
`
`ISBN D-I37-IJSD‘lEi?-‘i
`
`Library of Congress Cataloging-in-Publication Data
`
`Proakis. John (3.
`Digital communicstions/John G. Proakis.~2nd ed.
`p. cm.--(Mc(}rsw-Hill series in electrical engineering.
`Communicstions snd signs] processing)
`Includes bibliographies snd index.
`[SBN 0-07-050937-9
`l. Digits] communicstions.
`TK5103.7.P76
`198‘)
`Ml .3W0413ndcl9
`
`88-31455
`
`I. Title.
`
`II. Series.
`
`
`
`
`
`Petitioner Sirius X
`
`M Radio Inc. - Ex. 1021, p. 2
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 2
`
`

`

`IHARACTERIZATION OF
`LIGNALS AND
`
`gnals can be categorized in a number of different ways such as random versus
`terministic, discrete time versus continuous time, discrete amplitude versus
`lowpass versus bandpass, finite energy versus infinite
`ergy, finite average power versus infinite average power, etc. In this chapter we
`at the characterization of signals and systems that are usually encountered in
`a transmission of digital information over a communication channel. In partic-
`tr, we introduce the representation of various forms of digitally modulated
`;na1s and describe their spectral characteristics.
`We begin with the characterization of bandpass signals and systems,
`:luding the mathematical representation of bandpass stationary stochastic
`acesses. Then, we present a vector space representation of signals. We conclude
`th the representation of digitally modulated signals and their spectral charac-
`
`l REPRESENTATION OF BANDPASS
`GNALS AND SYSTEMS
`
`CHARACTERIZATION OF SIGNALS AND SYSTEMS
`
`149
`
`the
`bandpass signals and channels (systems). The modulation performed at
`transmitting end of the communication system to generate the bandpass signal
`and the demodulation performed at
`the receiving end to recover the digital
`information involve frequency translations. With no loss in generality and for
`mathematical convenience,
`it
`is desirable to reduce all bandpass Signals and
`channels to equivalent lowpass signals and channels. As a consequence,
`the
`results of the performance of the various modulation and demodulation tech-
`niques presented in the subsequent chapters are independent of carrier frequen-
`cies and channel frequency bands. The representation of bandpass signals and
`systems in terms of equivalent lowpass waveforms and the characterization of
`bandpass stationary stochastic processes are the main topics of this section.
`
`3.1.1 Representation of Bandpass Signals
`
`A real-valued signal s(t) with a frequency content concentrated in a narrow band
`of frequencies in the vicinity of a frequency f, can be expressed in the form
`
`s(t) = a(t)cos[21rfct + 0(z)]
`
`(3.1.1)
`
`where a(t) denotes the amplitude (envelope) of s(t), and 0(t) denotes the phase
`of s(t). The frequency fc is usually called the carrier of s(t) and may be any
`convenient frequency within or near the frequency band occupied by the signal.
`When the band of frequencies occupied by s(t) is small relative to fc, the signal is
`called a narrowband bandpass signal or, simply, a bandpass signal.
`By expanding the cosine function in (3.1.1) a second representation for s(t)
`is obtained, namely,
`
`s(t) = a(t)cos0(t)costrfct — a(t)sin0(t)sin27rfct
`
`= x(t)cos27rfct -— y(t)sin21rfct
`
`(3.1.2)
`
`where the signals x(t) and y(t), termed the quadrature components of s(t), are
`defined as
`
`x(t) = a(t)cos0(t)
`
`y(t) = a(t)sin0(t)
`
`(3.1.3)
`
`The frequency content of the quadrature components x(t) and y(t) is concen-
`trated at low frequencies (around f = 0, as shown below) and, hence,
`these
`components are appropriately called lowpass signals. Finally, a third representa-
`tion for s(t) is obtained from (3.1.1) by defining the complex envelope u(t) as
`
`u(t) = a(t)ejo(’)
`
`= x(t) +jy(t)
`
`(3.1.4)
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 3
`
`

`

`Use of the identity
`
`in (3.1.6) yields the result
`
`Re(€) = %(£ + £*)
`
`(3 1 7)
`
`S(f) = %f°° [u(t)ejz"fv’ + u*(t)e‘jz”f€']e‘jz”f’dt
`
`(3-1-8)
`= %[U(f-fc) + U*(-f—fc)]
`where U( f ) is the Fourier transform of u( t). Since the frequency content of the
`bandpass signal s(t) is concentrated in the vicinity of the carrier fc, the result in
`(3.1.8) indicates that the frequency content of u(t) is concentrated in the vicinity
`of f = 0. Consequently, the complex-valued waveform u(t) is basically a low-pass
`signal waveform and, hence, is called the equivalent lowpass signal.
`The energy in the signal s(t) is defined as
`
`=
`
`°° {Re[u(t)el‘2"fc']}2dz
`
`(3.1.9)
`
`When the identity in (3.1.7) is used in (3.1.9), we obtain the following result:
`]_
`co
`«3”: §/_w|u(t)|2dt
`
`+%f_°° Iu<t>12cos14m + 200)] d:
`
`(3.1.10)
`
`Consider the second integral in (3.1.10). Since the signal s(t) is narrowband, the
`real envelope u(t) E |u(t)| or, equivalently, a2( t) varies slowly relative to
`the rapid variations exhibited by the cosine function. A graphical illustration
`of the integrand in the second integral of (3.1.10) is shown in Fig. 3.1.1. The value
`of the integral is just the net area under the cosine function modulated by a2( t).
`Since the modulating waveform a2(t) varies slowly relative to the cosine func-
`tion, the net area contributed by the second integral is very small relative to the
`value of the first integral in (3.1.10) and, hence, it can be neglected. Thus, for all
`practical purposes, the energy in the bandpass signal s(t), expressed in terms of
`the equivalent lowpass signal u(t), is
`
`av= if” |u(t)|2dt
`
`(3.1.11)
`
`where |u(t)! is just the envelope a(t) of s(t).
`
`
`
`lllllllll Imm
`IIIHHHIIWW
`
`
`
`FIGURE 3.1.]
`
`The signal a2(t)cos[4wfi.t + 20(1)].
`
`3.1.2 Representation of Linear Bandpass Systems
`
`A linear filter or system may be described either by its impulse response h(tj
`by its frequency response H( f ), which is the Fourier transform of h(t). Si
`h(t) is real,
`
`Let us define C( f — fi.) as
`
`H*(-f) =H(f)
`
`Then
`
`C(f-f.)={gl(f)
`
`$3
`
`C*(-f-f..) = {2”(4)
`
`fig
`
`Using (3.1.12), we have
`
`H(f)=C(f-f.)+C*(~f-f.-)
`
`The inverse transform of H( f) in (3.1.15) yields h(t) in the form
`
`h(t) = c(t)e12"fr’ + c*(t)e‘j2”/‘"
`
`= 2Re [c(z)e12"fl']
`
`where c(t) is the inverse Fourier transform of C( f ). In general,
`response c(t) of the equivalent lowpass system is complex-valued.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 4
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 4
`
`

`

`Fourier transform of s(t) is
`S(f) = f°° s(z)e-fl"f'dt
`
`= °° {Re [u(t)e1‘2"fi']}e'fl"f'dz
`
`Re(£) = %(£ + £*)
`
`(3.1.6)
`
`(3-1-7)
`
`S(f) = %/_00 [u(t)ej2-nfct + ”*(t)e—j21rf,t]e—j21rft dt
`
`(3-1-8)
`= Hum—f.) + U*(—f—f.)}
`f ) is the Fourier transform of u(t). Since the frequency content of the
`signal s(t) is concentrated in the vicinity of the carrier fc, the result in
`.icates that the frequency content of u(t) is concentrated in the vicinity
`Zonsequently, the complex-valued waveform u(t) is basically a low-pass
`Ieform and, hence, is called the equivalent lowpass signal.
`energy in the signal s(t) is defined as
`
`°° {Re[u(t)e12"fc']}2dt
`
`(3.1.9)
`
`identity in (3.1.7) is used in (3.1.9), we obtain the following result:
`
`c5”: 2-f_°°|u(t)|2dt
`
`+ %f_°° |u(t)|2cos [4an + 200)] dt
`
`(3.1.10)
`
`be second integral in (3.1.10). Since the signal s(t) is narrowband, the
`ope u(t) E |u(t)| or, equivalently, a2(t) varies slowly relative to
`variations exhibited by the cosine function. A graphical illustration
`grand in the second integral of (3.1.10) is shown in Fig. 3.1.1. The value
`gral is just the net area under the cosine function modulated by a2(t).
`modulating waveform a2(t) varies slowly relative to the cosine func-
`et area contributed by the second integral is very small relative to the
`1e first integral in (3.1.10) and, hence, it can be neglected. Thus, for all
`nurposes, the energy in the bandpass signal s(t), expressed in terms of
`lent lowpass signal u(t), is
`1
`
`co
`
`CHARACTERIZATION OF SIGNALS AND SYSTEMS
`
`151
`
`a2“)
`
`II||||||| Hum
`HHHHIIIIW”
`
`FIGURE 3.1.1
`
`The signal a2(t)cos[4wfi.t + 20(1)].
`
`3.1.2 Representation of Linear Bandpass Systems
`
`A linear filter or system may be described either by its impulse response h(t) or
`by its frequency response H( f ), which is the Fourier transform of h(t). Since
`h(t) is real,
`
`H*(-f) = H(f)
`
`(3.1.12)
`
`Let us define C( f — f,.) as
`
`
`
`Then
`
`C(f-f.) = {:(f)
`
`fig
`
`C*(-f-f..) = {21*(_f)
`
`fig
`
`(3.1.13)
`
`(3.1.14)
`
`Using (3.1.12), we have
`
`H(f) = C(f~f..) + C*(-f-f..)
`
`(3-1-15)
`
`The inverse transform of H( f ) in (3.1.15) yields h(t) in the form
`
`h(t) = c(t)e12‘”fi-l+ C*(t)e_j2"/‘"
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 5
`
`

`

`)rthogonal signals. Consider the two signals
`
`26’
`s1(t) = VT costrft
`20?
`s2(t) = )l— sin2-rrft
`
`0 s t s T
`
`0 s t s T
`
`ither fc = l/Tor fc > 1/T, so that
`
`1
`T
`Re(p12) = 3/0 s1(t)sz(t)dt = 0
`
`(4.1.16)
`
`(4.1.17)
`
`e(p12) = 0, the two signals are orthogonal.
`[C equlvalent lowpass waveforms corresponding to sl(t) and 52(t) are
`
`26’
`u1(t)=)/— OstsT
`
`2(5’
`u2(t)=—j)/—T
`
`OSIST
`
`(4.1.18)
`
`.these s1gna1 waveforms can be represented as two-dimensional signal
`
`s1 = ((3,0)
`
`52 = (0, 4.?)
`
`(4.1.19)
`
`%respond to the signal space diagram shown in Fig. 4.1.2. Note that
`: observe that the vector representation for the equivalent lowpass signals
`um] and u2=[u21] where
`
`=1/2_(§’+j0
`
`21:0—1'1/27)
`
`(4.1.20)
`
`
`
`MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`225
`
`Hence,
`
`
`“f ' “2
`|u1l|ll2|
`
`=
`
`”12
`
`-
`_ = e
`J
`
`——j /2
`.
`
`4.1.21
`
`(
`
`)
`
`Multiphase signals. Consider the M—ary PSK signals
`1(t)—\/2gcos[27rft+ $(m—1)] m=1,2,...,M,0stsT
`2—(g)cos£r—m—1c052vzrt— 2éflsinZ—Wm—lsianzrft
`T
`M
`C
`T
`M
`0
`(4.1.22)
`
`The equivalent lowpass signal waveforms are
`
`um(t)= WNW“WM m=1,2,...,M,OstsT (4.1.23)
`'I‘hcse signal waveforms are represented by the two-dimensional vectors
`= [t/g’cos 27W») — 1),)/<§’sin 2WWW — 1)]
`m = 1,2,..., M (4.1.24)
`
`or in complex-valued form as
`
`= V[Zéilej211(m—1)/M
`
`and are described by the signal space diagram shown1n Fig. 4.1. 3 Clearly, these
`signals have equal energy Their complex--valued correlation coefficients are
`pkm=_1—Tu;:(t)um(t)dt
`k:192"~-5M9m:1927""M
`24" 0
`= eij—Io/M
`
`(4.1.25)
`
`
`
`
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 6
`
`

`

`
`
`The minimum distance dmin corresponds to the case in which |m — k| = 1, i.e.,
`adjacent signal phases. In this case
`
`aimin =
`
`277
`
`2&(1 - cos 7)
`
`(4.1.28)
`
`The special case of four phase signals (M = 4) results in the signal vectors
`
`514430)
`
`s2'=(0,i/3’)
`
`sa=(-i/Z’,0)
`
`s,=(0,—i/(?)
`
`(4.1.29)
`
`and the signal space diagram shown in Fig. 4.1.3a. These signal vectors may also
`be viewed as two pairs of orthogonal signals, i.e., the pair of vectors (sl,s3) is
`orthogonal to (s2, 54). On the other hand, the pair (51,53) is antipodal and so is
`the pair (52, 54). These four signals are called biorthogonal. The general class of
`biorthogonal signals is considered below.
`
`Multiamplitude signals. Consider the M-ary PAM signals
`
`sm(t) = Amii 2%? cos 27rfct
`
`= AmRe [u(z)e12"fv']
`
`m =1,2,...,M,0 g z s T (4.1.30)
`
`where the signal amplitude Am takes the discrete values (levels)
`
`Am=2m—1—M m=1,2,...,M
`
`The signal pulse u(t), as defined in (4.1.30) is rectangular, i.e.,
`
`u(t) = «If
`
`0 s t s T
`
`(4.1.31)
`
`(4.1.32)
`
`but other pulse shapes may be used to obtain a narrower signal spectrum.
`Clearly, these signals are one-dimensional (N = 1) and, hence, are repre-
`sented by the scalar components
`
`(4.1.34)
`
`dkm = {2&0 " Re(Pkm)l}
`={26[1- cos %4£(m — k)
`
`
`
`>1/2
`
`(4.1.27)
`
`sm1=Am\/Z’ m=1,2,...,M
`
`The distance between any pair of signals is
`
`dmk = (M1 " Sk1)2
`
`= l/EiAm _ Akl
`
`(4.1.33)
`
`
`2%?
`—-—o———————o————o——o—o———o————o——>
`s1
`s2
`s3
`0
`,
`s4
`M - 6
`
`FIGURE 4.1.4
`
`Signal-space diagrams for M-ary PAM signals.
`
`Hence, the distance between a pair of adjacent signal points, i.e., th
`distance, is
`
`dmin a 24/?
`
`Examples of signal space diagrams for M-ary PAM signals are shi
`4.1.4.
`
`QAM signals. A quadrature amplitude-modulated (QAM) signal or a
`amplitude-shift-keying (QASK) signal is represented as
`
`sm(t) - A,"c
`
`2%? cos27rfct —- A»u 3;: sin2wfi.t
`
`where Am, and Am, are the information-bearing signal amplitudes of
`ture carriers and u(t) -= “26/ T, 0 s t s T. QAM signals are tWO-l
`signals and, hence, they are represented by the vectors
`
`Sm - (@Amc’ l/gAms)
`
`The distance between any pair of signal vectors is
`2
`
`dmk a Ism _ skl
`
`= a (Ame—Akc)2+(Am.r—Aks)2
`
`take the discrete values {2;
`When the signal amplitudes
`m -= 1,2,. . ., M }, the signal space diagram is rectangular as shown i
`In this case the distance between adjacent points, i.e., the minimum
`dmln — 2V2.
`
`M-ary orthogonal signals. In Sec. 3.3.1 we indicated that multidime
`nals can be generated either by subdividing a time interval of duratic
`N distinct (nonoverlapping) time slots of duration At, or by su
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 7
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 7
`
`

`

`DIGITAL COMMUNICATIONS
`
`MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`227
`
`i the real-valued cross-correlation coefficients are
`
`Re(pkm) = cos 27'”(m—k)
`
`(4.1.26)
`
`nce, the Euclidean distance between pairs of signals is
`
`d1m= {24(1 — R<3(1okm)l}1/2
`={26’[1 — cos i; (m — k)]}1/2
`
`(4.1.27)
`
`e minimum distance dmin corresponds to the case in which |m — k| = 1, i.e.,
`acent signal phases. In this case
`
`dmin=
`
`2w
`
`2a(1 ~ cos M)
`
`(4.1.28)
`
`The special case of four phase signals (M = 4) results in the signal vectors
`
`s; = (We?)
`
`s3=(~1/47’,0)
`
`s4 =(0,—1/Z“’)
`
`(4.1.29)
`
`1 the signal space diagram shown in Fig. 4.1.30. These signal vectors may also
`viewed as two pairs of orthogonal signals, i.e., the pair of vectors (51,53) is
`hogonal to (52, 54). On the other hand, the pair (s1, s3) is antipodal and so is
`1 pair (s2, s4). These four signals are called biorthogonal. The general class of
`Irthogonal signals is considered below.
`
`lltiamplitude signals. Consider the M—ary PAM signals
`
`2—74.? cos27rf.t
`
`= AmRe[u(t)e!’2"fr']
`
`m = 1,2,..., M,0 s t g T (4.1.30)
`
`ere the signal amplitude Am takes the discrete values (levels)
`
`Am=2m—1-—M m=1,2,...,M
`
`(4.1.31)
`
`e signal pulse u(t), as defined in (4.1.30) is rectangular, i.e.,
`
`u(tt)= V? OstST
`
`(4.1.32)
`
`'. other pulse shapes may be used to obtain a narrower signal spectrum.
`Clearly, these signals are one-dimensional (N = 1) and, hence, are repre-
`1ted by the scalar components
`
`sm1=Am1/3’ m= 1,2,...,M
`
`(4.1.33)
`
`
`
`21/2
`
`s!
`0
`M =' 2
`
`f1“)
`
`
`21/?
`2%?
`21/3
`-—o———————o——o——-—o————-———H
`51
`52
`0
`s;
`5‘
`M - 4
`
`f1“)
`
`
`2%?
`—-o———o————————o——-——-4—o————————o——~——O—-D f1
`51
`s2
`51
`0
`54
`55
`9s
`M - 6
`
`FIGURE 4.1.4
`
`Signal-space diagrams for M-ary PAM signals.
`
`Hence, the distance between a pair of adjacent signal points, i.e., the minimum
`distance, is
`
`dmm - 21/?
`
`(4.1.35)
`
`Examples of signal space diagrams for M-ary PAM signals are shown in Fig.
`4.1.4.
`'
`
`QAM signals. A quadrature amplitude-modulated (QAM) signal or a quadrature
`
`amplitude-shift-keying (QASK)2signal is represented2as
`sm(t) -Am c—-os2wf,t-—~Am(iz—T-é”sin21rf.t
`- Rah/1:. +14...»mew]
`
`(4.1.36)
`
`where Am.and A,"are the information-bearing signal amplitudes of the quadra-
`ture carriers and u(t) - \/26’/T, 0 s t s T. QAM signals are two-dimensional
`signals and, hence, they are represented by the vectors
`
`- («amt/24m)
`
`(4.1.37)
`
`The distance between any pair of signal vectors is
`2
`
`dmk =
`
`'5». " ski
`
`a (Am.—A,,,)2+ (Amp/1h)2
`
`k, m- 1,2,...,M (4.1.38)
`
`(2m — 1 — M,
`take the discrete values
`When the signal amplitudes
`m = 1, 2,. . ., M }, the signal space diagram is rectangular as shown in Fig. 4.1.5.
`In this case the distance between adjacent points, i.e., the minimum distance. is
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 8
`
`

`

`
`f1(l)
`
`M=4
`
`.
`FIGURE 4.1.9
`Signal-space diagrams for M—ary simplex Signals.
`
`. = 0 or 1 for all m and j. Each component of a code word is mapped
`where c
`into an elementary binary PSK waveform as follows:
`
`cmj = 1 =9 smf(t) =
`
`
`24;
`TC
`
`cos 21rfct
`
`OStST6
`
`cmj=0=>smj(t)=—‘
`
`
`26’
`T:
`
`cos 2wfct
`
`0 _<_ t 5 Tc
`
`(4.1.50)
`
`where Tc = T/N and 6°C = J/N. Thus, the M code words {Cm} are mapped into
`a set of M waveforms {sm(t)}.
`
`
`
`f1(’)
`
`
`
`2
`
`a
`
`$2
`
`M t)
`
`'7
`
`N = 3
`
`FIGURE 4.1.10
`
`Signal-space diagrams for signals gen-
`erated from binary codes.
`
`The waveforms can be represented in vector form as
`
`sm=(sm1,sm2,...,smN)
`
`m= 1,2,...,M
`
`
`3,"! - :t «(r/N for all m and j. N is called the block length of the
`Ind it is also the dimension of the M waveforms.
`,We note that there are 2” possible waveforms that can be constructed from
`
`3” possible binary code words. We may select a subset of M < 2” signal
`v s for transmission of the information. We also observe that the 2”
`signal points correspond to the vertices of an N—dimensional hypercube
`fl“ center at the origin. Figure 4.1.10 illustrates the signal points in N = 2
`"t dimensions.
`
`, Inch of the M waveforms has energy 6’. The cross correlation between any
`.0! waveforms depends on how we select the M waveforms from the 2”
`'
`~ 0 waveforms. This topic is treated in Chap. 5. Clearly, any adjacent signal
`
`
`
`"
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 9
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 9
`
`

`

`MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`245
`
`complex-valued signals. It is interesting to note that the probability of error P2
`given in (4.2.38) is expressed as
`
`P2 = —12— erfc(
`
`
`2d2
`
`(ZN: )
`
`~
`
`(4.2.39)
`
`where d12 is the distance of the two signals. Hence, we observe that an increase
`in the distance between the two signals reduces the probabllity of error.
`
`l
`|
`
`t
`
`i
`II
`
`|I
`
`|
`i
`-
`
`4T
`
`5T
`
`i
`
`I i
`
`l
`3T
`
`A
`
`’A
`
`.
`
`i
`I
`
`I
`2T
`
`‘
`
`i
`T
`
`(0)
`
`MIN
`
`Received signal
`
`Carrier
`
`.
`
`
`Recovery
`
`
`Detector
`
`Synchronizer
`
`DIGITAL COMMUNICATIONS
`
`Mum) = Re(e1¢r- u;)
`
`m = 1,2
`
`(4.2.31)
`
`Let us limit our attention to one- and two-dimensional signals, e.g., antipodal
`and orthogonal signals. Then, the signal vectors are represented as s1 = (s11, Sn)
`and 52 = (s21, 322) or by their equivalent lowpass complex-valued signal vectors
`“1 = lfl—(Sii + 1512) and “2 = (5021 +1322)-
`If we assume that the signal u1 was transmitted, the received signal is
`
`where z is the complex-valued gaussian noise component. By substituting for r in
`
`r = ate—”’u1 + 2
`
`(4.2.32)
`
`#041) = 2015+ Nlr
`”(142) =2agpr+N2r
`
`4.2.33
`
`(
`
`)
`
`where Nm, = Re(u,';z), m = 1,2. The statistical characteristics of the noise
`components are identical to those in (4.2.27). Hence, the probability of error is
`identical to the result given by (4.2.28).
`If the computation of the error probability is based on the real-valued
`signals {Sm, m = 1, 2}, the two decision variables are
`
`“(sm) = r - sm
`
`m = 1,2
`
`(4.2.34)
`
`Again, let the signal 51 be transmitted so that r = 01s1 + n where n is the noise
`
`”(s1)= 016+ s1 - n
`(1(52) =aévpr+52'n
`
`4.2.35
`
`(
`
`)
`
`P2 = P[M(sz) > II(SI)]
`
`= 1I>[(s1 — s2) - n < —aé”(1 — p,)]
`
`(4.2.36)
`
`The gaussian noise variable y = (s1 - s2) - n has zero mean and variance
`
`of a E(y2) = dszO/z
`
`(4.2.37)
`
`P. = Ply < we — p,)} = P[y > 7-1—2
`= %erfc(
`0;le )
`
`
`2d2
`0
`
`01d2
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 10
`
`

`

`(4.2.87)
`PM 5 M 2— 1 erfc(,/y,,k8/N)
`on the probability of error for M-ary signal
`block codes. This topic will be discussed in
`
`Thus, we have an upper bound
`waveforms obtained from binary
`more detail in Chap. 5.
`
`4.2.6 Probability of Error for Multiphase Signals
`The signal waveforms considered in Secs. 4.2.2 through 4.2.5 shared the charac-
`teristic that, for a fixed information rate R, the channel bandwidth required to
`transmit the signals increases as the number M of waveforms is increased. In
`contrast, the multiphase signal waveforms considered in this section and the
`multiamplitude and QAM signals considered in the next two sections, have the
`characteristic that the channel bandwidth requirements for a fixed rate actually
`'
`the penalty in using such
`decrease with
`bandwidth-efficient waveforms is an increase in the SNR required to achieve a
`specified level of performance. In short, these bandwidth-efficient signal wave-
`forms are appropriate for channels having a large SNR.
`The general representation for a set of M-ary phase signal waveforms is
`sm(t) = Re{u(t)exp [j(2qrf,t + 2M1(m —— 1) + AH}
`m = 1,2,..., M,0 s t s T (4.2.88)
`'
`o rotate the signal constellation.
`The pulse u(t) determines the spectral characteristics of the multiphase signal. If
`u(t) is a rectangular pulse of the form
`u(t) = l/ZT?
`0 s t s T
`I
`(4.2.89)
`the signal waveforms may be expressed as
`sm(t) =
`2% Amccos2rrf,t —-
`
`215 Am sin2'n’fct
`
`(4.2.90)
`
`where
`
`Acm=cosl~2fiz(m-l) +1]
`(4.2.91)
`m= 1,2,...,M
`Am = sin [gig-(m — 1) + A]
`(4.2.92)
`m = 1,2,..., M
`shift keying (PSK). Thus, the signal
`th amplitudes Am and
`This signaling method is called phase-
`given by (4.2.90) is viewed as two quadrature carriers wi
`
`Phase
`
`selector
`
`Nuance
`
`
`phase shifter
`
`
`
`
`
`Balanced
`modulator
`
`
`
`
` Arm
`
`CURE 4.2.10
`
`diagram of modulator for multiphase PSK.
`
`
`gnalrng mterval. The
`.
`.
`.
`.
`'i m, which depend on the transmitted phase in each si
`5.2 glator for generating this PSK Signal is shown in block diagram form in Fig.
`
`
`The mapping or assignment of k information bits to the M - 2" possible
`
`_ ases may be done in a number of ways. The preferred assignment is one in
`
`.
`. chi adjacent phases differ by one binary digit as illustrated in Fig. 4.2.11. This
`
`i: ppmg rs called Gray encoding. It is important in the demodulation of the
`
`ii 3.1 because the most likely errors caused by noise involve the erroneous
`
`_ action of an. adjacent phase to the transmitted signal phase. In such a case
`.. y ammngle bit error occurs in the k-bit sequence.
`
`.
`_
`e general form of the optimum demodulator for detectin
`
`- als in an AWGN channel, as derived previously, is one that computes the
`
`'sion variables
`
`7, wd selects the signal corresponding to the largest decision variable. We observl
`
`Petitioner Sirius XM Radio Inc. - EX 1021 p 11
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 11
`
`

`

`DIGITAL COMMUNICATIONS
`
`MODULATION AN
`
`D DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`259
`
`Since each signal waveform conveys k bits of information and each
`:‘orm has energy (if, it follows that a” = ké’b, where 6”,,
`is the energy per bit.
`termOre, 6'; = J/N = ke’b/N. Therefore, the union bound may be expressed
`ms of the SNR per bit as
`
`
`(4.2.87)
`PM 5 M2" 1 erfc((/ybk8/N)
`we have an upper bound on the probability of error for M-ary signal
`‘orms obtained from binary block codes. This topic will be discussed in
`
`Probability of Error for Multiphase Signals
`
`ignal waveforms considered in Secs. 4.2.2 through 4.2.5 shared the charac-
`c that, for a fixed information rate R, the channel bandwidth required to
`lit the signals increases as the number M of waveforms is increased. In
`St, the multiphase signal waveforms considered in this section and the
`.mplitude and QAM signals considered in the next two sections, have the
`:teristic that the channel bandwidth requirements for a fixed rate actually
`lse with an increase in M. As shown below, the penalty in using such
`'idth—efficient waveforms is an increase in the SNR required to achieve a
`ed level of performance. In short, these bandwidth-efficient signal wave-
`are appropriate for channels having a large SNR.
`[‘he general representation for a set of M-ary phase signal waveforms is
`= Re{u(t)exp [j(27rfct + EMZ(m — 1) + A)”
`
`2‘-
`
`m=1,2,...,M,OSt_<. T (4.2.88)
`A is an arbitrary initial phase. Its effect is to rotate the signal constellation.
`llse u(t) determines the spectral characteristics of the multiphase signal. If
`a rectangular pulse of the form
`
`u(t)= 2%
`
`03:51"
`
`nal waveforms may be expressed as
`
`(4.2.89)
`
`Phase
`
`selector
`
`
`Information
`(Amp Am )
`sequence
` phase shifter
`
` —sin 2nfct
`
`
`Balanced
`modulator
`
`FIGURE 4.2.10
`
`Block diagram of modulator for multiphase PSK.
`
`Am, which depend on the transmitted phase in each signaling interval. The
`modulator for generating this PSK signal is shown in block diagram form in Fig.
`4.2.10.
`
`The mapping or assignment of k information bits to the M - 2" possible
`phases may be done in a number of ways. The preferred assignment is one in
`which adjacent phases differ by one binary digit as illustrated in Fig. 4.2.11. ThJS
`mapping is called Gray encoding. It is important in the demodulation of the
`signal because the most
`likely errors caused by noise involve the erroneous
`selection of an adjacent phase to the transmitted signal phase. In such a case,
`only a single bit error occurs in the k-bit sequence.
`The general form of the optimum demodulator for detecting one of M
`signals in an AWGN channel, as derived previously, is one that computes the
`decision variables
`
`1
`
`sm(t) = v 27éaAmccos27rfct —- 7- Am sin 2wfct
`
`26’
`
`(4.2.90)
`
`Acm=cos[-M-(m— 1) +A]
`
`m= 1,2,...,M
`
`(4.2.91)
`
`_?
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 12
`
`

`

`11
`
`00
`
`M=4J=0
`
`(a)
`
`01
`
`11
`
`00
`
`10
`
`110
`
`111
`
`010
`
`110
`
`M=8,)\=0
`
`(C)
`
`011
`
`001
`
`000
`
`100
`
`000
`
`100
`
`M=4,)\=1r/4
`
`(b)
`
`111
`
`101
`
`M=8,7\=1r/8
`
`01)
`
`FIGURE 4.2.11
`PSK signal constellations for M -= 4 and M = 8.
`
`that the exponential factor under the integral in (4.2.93) is independent of the
`variable of integration and, hence,
`it can be factored out. As a result,
`the
`optimum demodulator can be implemented as a single matched filter or cross
`correlator which computes the vector
`V = e7¢f0Tr(t)u*(t) dt
`
`(4.2.94)
`
`projects it onto the M unit vectors
`Vm = exp{j[2T}(m — 1) + xi}
`and selects the signal corresponding to th
`projection. Thus
`
`Um= Re(VVm*)
`Figure 4.2.12 shows a block diagr
`noise-corrupted signal components Am
`
`(4.2.95)
`m = 1,2,..., M
`e largest value obtained by this
`
`(4.2.96)
`
`m= 1,2,...,M
`am of a demodulator for recovering the
`and Am, from which the vector V 18
`
`I Carrier Irecovery
`
`
`
`
`
` Synchronizer
`
`
`
`90°
`phase shifter
`
`
`
`
`' 4.2.12
`
`diagram of demodulator for PSK signals.
`
`i»
`
`
`
`
`
`.
`
`m* = am — jaw is sin
`-. The projection of V onto the unit vectors
`.
`
`~ plished by the formation of
`the product Um = Xam + Yam,
`"
`
`..
`. .
`. . M. Equivalently the vector Vcan be followed by a phase detector wl
`Mites the phase of V, denoted by 0, and selects from the set {sm(t)}
`
`having a phase closest to 0.
`
`Having described the form of the modulator and demodulator for M
`J ~ we now evaluate the performance in terms of the probability of error i1
`
`‘ N channel. In order to compute the average probability of error, we assz
`11,0) is transmitted. Then the received waveform is
`
`
`lution of (4.2.97) into (4.2.94) yields the vector V as
`
`
`
`
`1
`
`r(t) = ae‘j¢u(t) + z(t)
`
`V = 2016+ N
`
`
`
`the noise component N is a complex-valued gaussian random vari
`mo mean and variance %E(|N|2) = 26’N0. Let V = X +jY, where
`
`X = 20163 + Re (N)
`
`Y = Im (N)
`
`, X and Y components are jointly gaussian random variables, with the j
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 13
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 13
`
`

`

`MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`261
`
` Synchronizer
`I . Carrier .
`
`
`V = x +jY
`
`
`Um = Rewm) Output
`
`K eccived
`«tunal
`
`recovery
`
`DIGITAL COMMUNICATIONS
`
`011
`
`010
`
`00
`
`110
`
`001
`
`000
`
`111
`
`100
`
`M = 8, )x = O
`
`(C)
`
`011
`
`001
`
`00
`
`010
`
`110
`
`000
`
`100
`
`10
`
`111
`
`101
`
`M = 8, K = 1r/8
`
`(d)
`
`PSK signal constellations for M = 4 and M = 8.
`
`
`
`
`
`
`
`
`
`90°
`V»?
`vectors
`phase shifter
`Reference
`
`
`
`FIGURE 4.2.12
`Muck diagram of demodulator for PSK signals.
`
`
`
`
`lormed. The projection of V onto the unit vectors K: = am — jam is simply
`fi
`wenmplished by the formation of
`the product Um = Xacm + Yam, m =
`
`i; 1.2. .
`.
`.
`, M. Equivalently the vector V can be followed by a phase detector which
`
`7, computes the phase of V, denoted by 0, and selects from the set {s,,,(t)} that
`
`5 final having a phase closest to 0.
`
`Having described the form of the modulator and demodulator for M-ary
`
`3 PSK. we now evaluate the performance in terms of the probability of error in an
`that the exponential factor under the integral in (4.2.93) is independent of the
`
`AWGN channel. In order to compute the average probability of error, we assume
`variable of integration and, hence,
`it can be factored out. As a result,
`the ’
`
`that s,(t) is transmitted. Then the received waveform is
`optimum demodulator can be implemented as a single matched filter or cross
`
`correlator which computes the vector
`
`
`butitution of (4.2.97) into (4.2.94) yields the vector V as
`
`
`:
`“
`
`
`
`V= ei¢f0Tr(t)u*(t)dt
`
`(4.2.94)
`
`projects it onto the M unit vectors
`. 2
`
`Vm=exp{}[7w(m—1)+}\]} m=1,2,...,M (4.2.95)
`
`r(t) = ae‘j¢u(t) + z(t)
`
`V = 2aé”+ N
`
`(4.2.97)
`
`(4.2.98)
`
`ore the noise component N is a complex-valued gaussian random variable
`th uro'mean and variance %E(|N|2) = 2am, Let V = X + jY, where
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 14
`
`

`

`This change in variables yields the joint probability density function
`
` r
`2
`2
`2
`2
`27702 e
`—(r +401 of —4aJrcos9)/20
`
`p ( r
`
`, 0
`
`)
`
`___.
`
`(
`
`42.102
`
`)
`
`Integration of p(r, 0) over the range of r yields 17(0). That is,
`00
`12(9) =f pm) dr
`0
`21—we’7(1 + (May 0050e7°°sz Ffl/ficosoe'xz/Z dx)
`
`0
`
`1
`
`'77 —oo
`
`(4.2.103)
`
`where y = 012(39/NO is the SNR per symbol. Figure 4.2.13 illustrates p(0) for
`several values of 7. It is observed that 12(0) becomes narrower and more peaked
`about 0 = 0 as y is increased.
`A decision error is made if the noise causes the phase to fall outside of the
`range —vr/M _<_ 0 s w/M. Thus
`PM = 1 — [WM 12(0) d0
`—'n/M
`
`(4.2.104)
`
`In general, the integral of 12(0) does not reduce to a simple form and must be
`evaluated numerically, except when M = 2 and M = 4.
`the
`For binary signaling, the PSK waveforms are antipodal and, hence,
`probability of error is given by (4.2.40). When M = 4, we have in effect two
`binary PSK signals in phase quadrature as indicated above. With coherent
`demodulation, there is no cross talk or interference between the signals on the
`two quadrature carriers and, hence,
`the bit error probability is identical
`to
`(4.2.40). That is,
`
`P2 = %erfc (fig)
`
`(4.2.105)
`
`where 7,, is the SNR per bit. On the other hand, the symbol error probability for
`M = 4 is determined by noting that
`
`(4.2.106)
`
`probability density function of 0 is obtained by a change in variables from X
`and Y to
`
`2
`
`=
`
`2
`X + Y
`R
`0 = tan‘1 Y/X
`
`(4.2.101)
`
`Pc = (1 - P2)2
`
`= [1 — %erfc(\/y;)]2
`
`[.25
`
`l .07
`
`0.89
`
`0.18
`
`p0)
`
`0.7l
`
`0.54
`
`0.36
`
`-3.l4 -2.51—l.88 —l.26 —0.63
`
`0.00
`0
`
`0.63
`
`1,26
`
`1.88
`
`2.51
`
`FIGURE 4.2.13
`Probability density function 11(0) for y = 1, 2, 4, 10.
`
`where PC is the probability of a correct decision for the 2-bit symbol. The resr
`1n (4.2.106) follows from the statistical independence of the noise on the quadr
`ture carriers. Therefore, the symbol error probability for M = 4 is
`
`n=1—g
`
`= erfc (EHI — fierfc (MM
`
`For M > 4, the symbol error probability PM is obtained by numerical
`_
`integrating (4.2.104). Figure 4.2.14 illustrates this error probability as a functi¢
`of the SNR per bit for M = 2, 4, 8, 16, and 32. The graphs clearly illustrate ti
`penalty in SNR per bit as M is increased beyond M = 4. For example,
`PM = 10's, the difference between M = 4 and M = 8 is approximately 4 d
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 15
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 15
`
`

`

`DIGITAL COMMUNICATIONS
`
`MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`263
`
`ability density function
`p(x, y) =
`
`
`1 2e”“"‘2“")2+y21/2°2
`2710
`
`(4.2.100)
`
`The phase of V, computed by the phase detector, is 0 = tan‘1 Y/X. The
`ability density function of 0 is obtained by a change in variables from X
`
`2
`
`2
`_
`R ‘ X + Y
`0 = tan‘1 Y/X
`
`(4.2.101)
`
`change in variables yields the joint probability density function
`r
`2
`2
`2
`2
`
`—(r +401 6‘ —4ad’rcos6‘)/20
`2.7.2 e
`
`0 =
`>
`
`4.1102
`
`<
`
`)
`
`gration of p(r, 0) over the range of r yields p(0). That is,
`M) =/ p0, 9) dr
`= 21—73—41 + 41w cosfle7°°520\/—2}_;/fcosae”x2/2dx)
`
`(4.2.103)
`
`re 7 = mafia/N0 is the SNR per symbol. Figure 4.2.13 illustrates p(0) for
`ral values of 7. It is observed that p(0) becomes narrower and more peaked
`.11 0 = 0 as y is increased.
`A decision error is made if the noise causes the phase to fall outside of the
`;e —7r/M s 0 S ar/M. Thus
`PM = 1 - [WM 12(0) d0
`—17/M
`
`(4.2.104)
`
`general, the integral of p(0) does not reduce to a simple form and must be
`uated numerically, except when M = 2 and M = 4.
`For binary signaling, the PSK waveforms are antipodal and, hence, the
`Dability of error is given by (4.2.40). W