Pett Ve feee a
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 1
`
`“Seegeraee
`
`a
`
`=f
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 1
`
`

`

` This book was set in Times Roman by Science Typographers, Inc.
`
`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
`reproducedor distributed in any form or by any means, or stored in a data
`base or retrieval system, without prior written permission of the
`publisher.
`
`456789 BKM BKM 99876543
`
`ISBN 0-07-05093?-4
`
`Library of Congress Cataloging-in-Publication Data
`
`Proakis, John G.
`Digital communications/John G. Proakis.—2nd ed.
`p. om.--(McGraw-Hill series in electrical engineering.
`Communications andsignal processing)
`Includes bibliographies and index.
`ISBN 0-07-050937-9
`1. Digital communications.
`TK5103,7.P76
`1989
`621.38'0413--dcl9
`
`I.Title.
`
`IT. Series.
`
`88-31455
`
`
`
`
`Petitioner Sirius X
`
`M Radio Inc. - Ex. 1021, p. 2
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 2
`
`

`

`-HARACTERIZATION OF
`IGNALS AND
`
`ndwidth is much smaller than the carrier frequency are termed narrowband
`
`znals can be categorized in a numberof 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 powerversusinfinite average power, etc. In this chapter we
`at the characterization of signals and systems that are usually encountered in
`> transmission ofdigital information over a communication channel. In partic-
`ir, we introduce the representation of various forms of digitally modulated
`mals and describe their spectral characteristics.
`We begin with the characterization of bandpass signals and systems,
`luding the mathematical representation of bandpass stationary stochastic
`ocesses. Then, we present a vector space representation of signals. We conclude
`th the representation of digitally modulated signals and their spectral charac-
`
`| 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 independentof 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 [2af,t + 6(t)]
`(3.1.1)
`where a(t) denotes the amplitude (envelope) of s(¢), and @(t) denotes the phase
`of s(t). The frequency f,
`is usually called the carrier of s(t) and may be any
`convenient frequency within or near the frequency band occupied bythesignal.
`Whenthe band of frequencies occupied by s(t) is small relative to f,, the signal is
`called a narrowband bandpass signal or, simply, a bandpass signal.
`By expandingthe cosine function in (3.1.1) a second representation for s(t)
`is obtained, namely,
`s(t) = a(t) cos 0(t) cos2af,t — a(t) sin O(t) sin 2f,t
`(3.1.2)
`= x(t) cos2af,t ~— y(t) sin2af,t
`where the signals x(t) and y(t), termed the quadrature components of s(t), are
`defined as
`
`x(t) = a(t) cos 0(t)
`y(t) = a(t) sin O(t)
`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)e®O
`= x(t) + jy(t)
`
`(3.1.4)
`
`(3.1.3)
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 3
`
`

`

`Use of the identity
`
`Re (£) = 3(é + &*)
`
`(3.1.7)
`
`€=]
`
`s(t) at
`
`in (3.1.6) yields the result
`S(f) = fo [u(t) ef2met + u*(t)ePret] es2nft dt
`(3.1.8)
`3[U(f- 7.) + U-f-f)]
`where U(f) is the Fourier transform of u(t). Since the frequency content of the
`bandpasssignal s(t) is concentrated in the vicinity of the carrier f.. 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 energyin the signal s(t) is defined as
`[oa]
`fst)
`(3.1.9)
`= fo {Re[u(t)emt]}? at
`Whenthe identity in (3.1.7) is used in (3.1.9), we obtain the following result:
`1
`se
`&= xlus)Pat
`(3.1.10)
`+57 lule)Poos [4nfs + 26(1)] at
`Consider the second integral in (3.1.10). Since the signal s(t) is narrowband, the
`real envelope a(t) = |u(t)| or, equivalently, a2(1) 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 integralis just the net area underthe cosine function modulated by a?(t).
`Since the modulating waveform a?(r) varies slowly relative to the cosine func-
`tion, the net area contributed by the secondintegralis very small relative to the
`valueof the first integral in (3.1.10) and, hence, it can be neglected. Thus, for all
`practical purposes, the energy in the bandpasssignal S(t), expressed in terms of
`the equivalent lowpasssignal u(t), is
`
`~ 6
`
`&= Ok
`where |u(t)| is just the envelope a(t) of s(t).
`
`(3.1.11)
`
`
`
`ug
`iee
`
`FIGURE3.1.1
`The signal a2(t)cos[4af.t + 20(1)].
`
`3.1.2 Representation of Linear Bandpass Systems
`A linearfilter or system may be described either by its impulse response A(t
`by its frequency response H(f), which is the Fourier transform of A(t). Si
`h(?) is real,
`
`Let us define C(f — f,) as
`
`H*(—f) = H(f)
`
` a(t)
`
`Then
`
`cif) = (0)
`
`hoo
`
`C*(-f-f.) = (-N)
`
`hob
`
`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 A(t) in the form
`
`h(t) = c(t)el?h + c*(t)e2m
`= 2Re[c(t)e?7%']
`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) = [isere dt
`=f (Re[u(t)es2*h']} eetat
`
`Re(£) = 3( + &*)
`
`(3.1.6)
`
`(3.1.7)
`
`— 0
`
`S(f) _ sf” [u(t)e/2met + u*(t)e/2rh] e—s2aft dt
`(3.1.8)
`= +[U(f-f.) + U(-f-f)]
`f) is the Fourier transform of u(t). Since the frequency contentof the
`signal s(t) is concentrated in the vicinity of the carrier f,, the result in
`icates that the frequency content of u(t) is concentrated in the vicinity
`-onsequently, the complex-valued waveform u(t)is basically a low-pass
`eform and,hence,is called the equivalent lowpass signal.
`energy in the signal s(t) is defined as
`é= i: s(t) dt
`(3.1.9)
`=f (Re[u(t)e*] }? at
`identity in (3.1.7) is used in (3.1.9), we obtain the following result:
`é= xf lu(s)iPae
`(3.1.10)
`+ sf. |u(t)|?cos [4arf,t + 20(t)] at
`he second integral in (3.1.10). Since the signal s(t) is narrowband, the
`ope a(t) = |u(z)| or, equivalently, a7(t) varies slowly relative to
`variations exhibited by the cosine function. A graphical illustration
`grand in the secondintegral 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 a(t).
`modulating waveform a?(t) varies slowly relative to the cosine func-
`et area contributed by the second integral is very small relative to the
`e first integral in (3.1.10) and, hence, it can be neglected. Thus, for all
`urposes, the energy in the bandpass signal s(r), expressed in terms of
`lent lowpass signal u(t), is
`
`
`
`CHARACTERIZATION OF SIGNALS AND SYSTEMS
`
`151
`
`a*(t)
`
`
`
`og
`Hi
`
`FIGURE3.1.1
`The signal a?(t)cos{4af.t + 20(1)].
`
`3.1.2 Representation of Linear Bandpass Systems
`
`A linearfilter or system may be described either by its impulse response A(1) or
`by its frequency response H(/), which is the Fourier transform of A(t). Since
`h(t) is real,
`
`Let us define C(f — f,) as
`
`H*(-f)=H(f)
`
`Then
`
`C(f-f.) = (gi
`
`hoo
`
`C*(-f-f.) = (7) a
`
`(3.1.12)
`
`(3.1.13)
`
`(3.1.14)
`
`Using (3.1.12), we have
`
`H(f) =C(f-f.) + CX -f-f.)
`
`(3.1.15)
`
`The inverse transform of H(/) in (3.1.15) yields A(t) in the form
`
`h(t) = (test + c*(t)e Pe
`= 2Re[c(1)e?7/]
`
`(3.1.16)
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 5
`
`

`

`wrthogonal signals. Consider the two signals
`Hence,
`
`5,(t) = /2¢S cos 2aft O<t<T
`uf + U2 +=pain/2
`
`P2= Tatu
`7
`(4.1.21)
`_
`= —j=e-%
`4.1.21
`Multiphase signals. Consider the M-ary PSK signals
`sw(t) = 38 cos [amt + 27(m —1)| m=1,2,....M,0<t<T
`26 cos a(m — 1) cos2nf,t — (22 sin aT(m — 1)sin27f,t
`
`5,(t) = Teé sin 2af.t O<t<T
`
`(4.1.16)
`
`ither f. = 1/T or f, > 1/T,so that
`Re(p,>) = al, s,(t)s,(t) dt =0
`e ( Pir) = 0, the two signals are orthogonal.
`¢ equivalent lowpass waveforms corresponding to s,(t) and s,(t) are
`
`1
`
`sT
`
`(4.1.17)
`
`2€
`u(t) => O<t<T
`
`28
`u,(t) = -j r O<1<T
`
`(4.1.18)
`
`these signal waveforms can be represented as two-dimensional signal
`
`(4.1.19)
`—_s, = (0, -vé)
`s,=(V@,0)
`perespond to the signal space diagram shown in Fig. 4.1.2. Note that
`2 observe that the vector representation for the equivalent lowpasssignals
`u,,] and u, = [u,,] where
`
`= (2 + j0
`
`n = 0-jy2é
`
`(4.1.20)
`
`Signal-space diagram for binary orthogonal signals.
`
`
`
`225
`
` MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`(4.1.22)
`
`‘The equivalent lowpass signal waveforms are
`
` m=1,2,...,M,0<t<T (4.1.23)
`u,(t) = [28neemD/M
`‘These signal waveformsare represented by the two-dimensional vectors
`= Ve cos ot(m — 1), Vé sin at(m - 1)| m=1,2,...,M (4.1.24)
`
`or in complex-valued form as
`
`= y2 Gel2nim—V)/M
`
`and are described by the signalspace diagram shownin Fig. 4.1.3. Clearly, these
`nignals have equal energy. Their complex-valued correlation coefficients are
`eeSOLO k =1,2,...,M,m=1,2,...,M
`= ei2a(m—k)/M
`(4.1.25)
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 6
`
`

`

`
`
`
`dum = {26(1 — Re(Pym)] }””
`= (26[1 — cos 5(m — k) \""
`
`
`(4.1.27)
`
`The minimum distance d,,;, corresponds to the case in which |m — k| = 1, 1e.,
`adjacent signal phases. In this case
`
`d
`
`26(1 — cos a
`
`27
`
`(4.1.28)
`
`The special case of four phase signals (M = 4) results in the signal vectors
`s,=(V8,0)
`s,=(0,Ve)
`s;=(-Vvé,0)
` s,=(0,-vé) (4.1.29)
`and the signal space diagram shownin Fig. 4.1.3a. These signal vectors may also
`be viewed as two pairs of orthogonal signals, i.e., the pair of vectors (S,,$3) is
`orthogonal to (s,,8,). On the other hand, the pair (s,,s,) is antipodal and so is
`the pair (s,,s,). These four signals are called biorthogonal. The general class of
`biorthogonal signals is considered below.
`
`Multiamplitude signals. Consider the M-ary PAM signals
`
`
`2vé
`—_—_—
`S,
`S,
`8;
`0. 8
`M=6
`
`FIGURE4.1.4
`Signal-space diagrams for M-ary PAM signals.
`
`Hence, the distance between a pair of adjacent signal points, i.e., th
`distance,is
`
`Amin = 2vé
`Examples of signal space diagrams for M-ary PAMsignals are sh
`4.1.4,
`
`QAMsignals. A quadrature amplitude-modulated (QAM)signal or a
`amplitude-shift-keying (QASK)signal is represented as
`Sm(t) ™ Ame 26 cos2mf,t — Ams - sin 2mrf,,t
`
`(4.1.31)
`
`(4.1.34) OGNtBal Wet SeDE ENEN PRN ED GeRAVEOrWe GoDE ONAN FEN 8 OOie PRES OE ee eTBeerney Nwsee we Se
`
`Sp(t) = An = cos ft
`where A,,, and A,,, are the information-bearing signal amplitudes of
` m=1,2,...,M,0<t<T (4.1.30)
`=A,,Re[u(t)e?']
`ture carriers and u(t) = ¥26/T, 0 <t < T. QAMsignals are two-
`signals and, hence, they are represented by the vectors
`wherethe signal amplitude A,, takes the discrete values (levels)
`Sin = (VFAmes VEAns)
`A, =2m-1-M m=1,2,...,M
`The distance between any pair of signal vectorsis
`Thesignal pulse u(t), as defined in (4.1.30) is rectangular,i.e.,
`u(t) = 26 O<t<T
`(4.1.32)
`dink =V[Sm — Sul”
`
`
`
`but other pulse shapes may be used to obtain a narrowersignal spectrum.
`Clearly, these signals are one-dimensional (N = 1) and, hence, are repre-
`sented by the scalar components
`Sm =AmVE m=1,2,...,M
`The distance between any pair of signals is
`
`(4.1.33)
`
`Amk = V(Smt — Sia)”
`
`= é|A,, — A,|
`
`= \é (Ame — Age)” + (Ams — Ags)?
`
`take the discrete values
`When the signal amplitudes
`m= 1,2,..., M}, the signal space diagram is rectangular as showni
`In this case the distance between adjacent points, i.e, the minimum
`dmin ™ 2vé.
`
`M-ary orthogonal signals. In Sec. 3.3.1 we indicated that multidim
`nals can be generated either by subdividing a timeinterval of duratic
`N distinct (nonoverlapping) time slots of duration At, or by sv
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 7
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 7
`
`

`

`DIGITAL COMMUNICATIONS
`
`MODULATION AND DEMODULATIONFOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`227
`
`1 the real-valued cross-correlation coefficients are
`
`2wvé
`2vé
`2vé
`2s
`
`aOO
`,
`3
`3 A(t)
`5,
`5
`6
`3;
`3, A(t)
`2
`M=2
`M=m4
`
`(4.1.26)
`
`Re (Pim) = COS aa (m-k)
`nce, the Euclidean distance between pairs ofsignals is
`dym = {26(1 — Re (Pen )] }'
`
`ave
`—_—eeiOrr
`= (rel — cos 47(m - w)}”
`Ss
`S,
`S;
`0
`S4
`Ss
`S6 Ai)
`M=6
`
`(4.1.34)
`
`(4.1.27)
`
`e minimum distance d,, corresponds to the case in which |m — k| = 1, ie.,
`acent signal phases. In this case
`
`27
`
`(4.1.28)
`dai, = 26(1 — cos a)
`The special case of four phase signals (M = 4) results in the signal vectors
`s,=(0,V)
`8; = (v8.0)
`sy = (0, -V8)
`(4.1.29)
`1 the signal space diagram shownin Fig. 4.1.3a. These signal vectors may also
`viewed as two pairs of orthogonalsignals, i.e., the pair of vectors (s,,$3) is
`hogonal to (s,,s,). On the other hand,the pair (s,,s,) is antipodal and sois
`pair (S,,8,). These four signals are called biorthogonal. The general class of
`rthogonal signals is considered below.
`
`ltiamplitude signals. Consider the M-ary PAMsignals
`Sm(t) = Am is cos 2af,.t
` m=1,2,...,M,0<1t<T (4.1.30)
`=A,,Re[u(t)e?%]
`ere the signal amplitude A,, takes the discrete values (levels)
`A, =2m-1-M m=1,2,...,M
`e signal pulse u(t), as defined in (4.1.30) is rectangular, i.e.,
`
`(4.1.31)
`
`2é
`(4.1.32)
`u(t) = V r O<t<T
`, other pulse shapes may be used to obtain a narrowersignal spectrum.
`Clearly, these signals are one-dimensional (N = 1) and, hence,are repre-
`ted by the scalar components
`Sm =AmVE m=1,2,...,M
`
`(4.1.33)
`
`FIGURE4.1.4
`Signal-space diagrams for M-ary PAM signals.
`
`Hence, the distance between a pair of adjacent signal points, i.e., the minimum
`distance,is
`
`(4.1.35)
`dain = 2VE
`Examples of signal space diagrams for M-ary PAM signals are shown in Fig.
`4.1.4,
`.
`
`QAMsignals. A quadrature amplitude-modulated (QAM)signal or a quadrature
`amplitude-shift-keying (QASK)signal is represented as
`Sm(t) ™ Ane ag cos 2mf,t — Ams\/ ae sin 27f,t
`(4.1.36)
`= Re[(Ame + JAms) u(t) e/2%*]
`where A,,, and A,,, are the information-bearing signal amplitudes of the quadra-
`ture carriers and u(t) = /26/T, 0 < t < T. QAMsignals are two-dimensional
`signals and, hence, they are represented by the vectors
`Sm ™ (VEA mer VEAms)
`The distance between any pair of signal vectorsis
`
`(4.1.37)
`
`VISm — s,|°
`Aink =
`k,m=1,2,...,M (4.1.38)
`= 8[(Ame — Ate)’ + (Ams — Aps)']
`When the signal amplitudes
`take the discrete values {2m-1-—M,
`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
`
`

`

`
`
`A
`
`f(y
`
`AG)
`Ato Ss;
`
`Ne
`
`M=4
`
`FIGURE 4.1.9
`Signal-space diagrams for M-ary simplex signals.
`
`where c,,, = 0 or 1 for all m and j. Each componentof a code word is mapped
`into an elementary binary PSK waveform as follows:
`
`Cmj = 1 = Sqj(t) = Vr cos2aft O<t<T,
`
`26
`
`Cmj = 9 S_j(t) = —
`
`
`¢2é
`T.
`
`cos 2af,t
`
`O<r<T,
`
`(4.1.50)
`
`where T, = T/N and &, = &/N. Thus, the M code words {C,,} are mappedinto
`a set of M waveforms{s,,(t)}.
`
`Ao
`
`Att)
`
`Ss AYea
`
`0
`
`~
`
`5,
`
`A(t)
`
`s)
`
`N=3
`
`FIGURE 4.1.10
`Signal-space diagramsforsignals gen-
`erated from binary codes.
`
`
`The waveformscan be represented in vector form as
`
`Sin = (Sats Smas-+9s Smy)
`m= 1,2,...,M
`es, = t V@/N for all m and j. N is called the block length of the
`
`Beand it is also the dimension of the M waveforms.
`|. We notethat there are 2” possible waveformsthat can be constructed from
`BN possible binary code words. We may select a subset of M < 2% signal
`
`\
`ms for transmission of the information. We also observe that the 2”
`ible signal points correspondto thevertices of an N-dimensional hypercube
`Bite center at the origin. Figure 4.1.10 illustrates the signal points in N = 2
`
`if dimensions.
`» Bach of the M waveformshas energy &. Thecrosscorrelation between any
`p.of waveforms depends on how weselect the M waveforms from the 2%
`ble 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
`
`

`

`DIGITAL COMMUNICATIONS
`
`(4.2.31)
` m=1,2
`u(u,,) = Re(e*reut%)
`Let us limit our attention to one- and two-dimensional signals, e.g., antipodal
`and orthogonalsignals. Then, the signal vectors are represented as s, = (54), 512)
`and s, = (55), 52)) or by their equivalent lowpass complex-valued signal vectors
`uy = V2(sy; + fog) and uy = ¥2(sq + jsp).
`If we assume that the signal u, was transmitted, the received signalis
`r=ae/*y, +z
`(4.2.32)
`
`where z is the complex-valued gaussian noise component. By substituting for r in
`
`H(u,) = 2a&+ M,
`)
`(
`p.(u,) = 2a&p, + Ny,
`where N,,, = Re(u*z), m= 1,2. The statistical characteristics of the noise
`componentsare 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 {s,,, m = 1,2}, the two decision variables are
`(4.2.34)
`B(s,,) ="res, m=1,2
`Again, let the signal s, be transmitted so that r = as, + n where n is the noise
`
`4.2.33
`
`p(s,) =a&+s,en
`H(s2) = ap, + s,° 0
`
`4.2.35
`
`(
`
`P, = P[u(s2) > w(s,)]
`(4.2.36)
`= P[(s, — 8s.) *n< —aé&(1 — p,)]
`The gaussian noise variable y = (s, — s,) * nm has zero mean and variance
`, =
`E(y*?) =d3,N)/2
`(4.2.37)
`
`y= Ply < ~ag(i—o)] = Ply > “2
`= yete|
`wEN. |
`
`
`242
`0
`
`ad?
`
`245
`complex-valued signals. It is interesting to note that the probability of error P,
`given in (4.2.38) is expressed as
`
`where d,, is the distance of the two signals. Hence, we observe that an increase
`in the distance between the two signals reduces the probability of error.
`
`‘A
`
`“A
`
`1
`|
`|
`|
`|
`|
`
`T
`
`,
`
`|
`!
`|
`
`I
`2T
`
`(a)
`
`lu(r)|
`
`|
`
`:
`
`|
`
`|
`3T
`
`|
`!
`|
`|
`|
`\
`
`I
`|
`4T
`
`t
`
`|
`|
`
`5T
`
`Received signal
`
`Carrier
`Recovery
`
`
`
`
`Output
`
`_
`2q2
`
`P= erfcly “aNe |
`
`(4.2.39)
`
`to the result for the error probability derived above for MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`
`
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 10
`
`

`

`sequence
`
` 90°
`
`Agn
`
`Balanced
`modulator
`
`IGURE 4.2.10
`diagram of modulator for multiphase PSK.
`
`
`
`
`Mam which depend on the transmitted phase in each signaling interval. The
`otator for generating this PSK signalis shown in block diagram form in Fig.
`
`
`F
`The mapping or assignment of k information bits to the M = 2* possible
`
`phases may be done in a number of ways. The preferred assignmentis one in
`
`pwh ch adjacent phasesdiffer by one binary digit as illustrated in Fig, 4.2.11. This
`2 pping is called Gray encoding. It is important in the demodulation of the
`
`
`alg nal because the most likely errors caused by noise involve the erroneous
`pelection 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 ™
`
`sion variables
`y,
`pr
`is one that computes the
`:
`’
`sly,
`se
`mals in an AWGNchannel, as derived previously,
`i
`ln = Re(e[7r(e)ur(t)exp {-i|Gx(m -1)+ a} a)
`:
`m=1,2,...,M (4.2.93
`{ nd selects the signal correspondingto the largest decision variable. We observ
`
`Pys 45
`erfc (/vk8/N)
`Thus, we have an upper bound on the probability of error for M-ary signal
`waveforms obtained from binary block codes. This topic will be discussed in
`Phase
`more detail in Chap.5.
`selector
`formation|(Acm» Asm)
`phase shifter
`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
`decrease with an increase in M. As shown below, the penalty in using such
`bandwidth-efficient waveformsis 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
`Smit) = Re{u(t)exp[sare + a(m -1)+ a)|\
`m=1,2,...,M,0<t<T (4.2.88)
`where A is an arbitrary initial phase.Itseffectis to rotate the signal constellation.
`The pulse u(t) determines the spectral characteristics of the multiphase signal. If
`u(t) isa rectangular pulse of the form
`u(t)=\ a O<t<T
`(4.2.89)
`2é
`the signal waveforms may be expressed as
`S(t) =
`26 AmeCOS2Tfct — 28An sin2nf,t
`
`(4.2.90)
`
`where
`
`(4.2.91)
`Aon = 008|5p(m ~ 1) +a] m=1,2,....M
`(4.2.92)
`Agm = Sit [arm -1)+ a] m=1,2,....M
`This signaling method is called phase-shift keying (PSK). Thus, the signal
`given by (4.2.90) is viewed as two quadrature carriers with amplitudes A,,, and
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 11
`ea
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 11
`
`

`

`MODULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`259
`
`Balanced
`
`Phase
`elector
`
`5
`
`sequence
`
`hifter
`Phase shifte
`
`Output
`to
`.
`transmitter
`
`Information|(Acm» Agm)
`
`DIGITAL COMMUNICATIONS
`
`Since each signal waveform conveys k bits of information and each
`‘orm hasenergy @,it follows that ¢ = ké,, where &,
`is the energy perbit.
`ermore, &, = &/N = ké,/N.Therefore, the union bound may be expressed
`ms of the SNRperbit as
`(4.2.87)
`P, < 45 1 orf (7,870)
`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
`gnal waveforms considered in Secs. 4.2.2 through 4.2.5 shared the charac-
`¢ 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
`ise with an increase in M. As shown below, the penalty in using such
`idth-efficient waveformsis 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.
`The general representation for a set of M-ary phase signal waveformsis
`-1(uan|ete» Sm)
`
`m=1,2,...,M,0<t<T (4.2.88)
`A is an arbitrary initial phase.Its effect is to rotate the signal constellation.
`se u(t) determines the spectral characteristics of the multiphase signal. If
`a rectangular pulse of the form
`
`u(t) = (+ O<r<T
`nal waveforms may be expressed as
`
`(4.2.89)
`
`Sm(t) = y 284,0082af,t - V 26Ans sin2af,t
`
`(4.2.90)
`
`j
`
`Aon = 00s |$7(m - 1) +A]
`
`n= 1,2,...,M
`
`(4.2.91)
`
`]
`
`:
`
`
`
`modulator
`
`
`Balanced
`modulator
`
`FIGURE 4.2.10
`Hlock diagram of modulator for multiphase PSK.
`
`A,m» 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 phasesdiffer by one binary digit as illustrated in Fig. 4.2.11. This
`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
`aclection 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 AWGNchannel, as derived previously, is one that computes the
`decision variables
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 12
`
`

`

`00
`
`110
`
`1
`
`000
`
`100
`
`M=4,2=0
`(a)
`
`M=8,X=0
`(c)
`
`011
`
`001
`
`01
`
`u
`
`00
`
`10
`
`010
`
`110
`
`000
`
`100
`
`M=4,r=17/4
`(b)
`
`101
`Wi
`M=8,r=7/8
`@)
`
`FIGURE4.2.11
`PSK signal constellations for M =4and 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= eMf'r(t)ur(s) dt
`(4.2.94)
`projects it onto the M unit vectors
`m=1,2,...,.M (4.2.95)
`= exo {iFF(m—- 0) +A}}
`and selects the signal corresponding to the largest value obtained by this
`projection. Thus
`U,, = Re(VV,*)
`m= 1,2,...,M
`(4.2.96)
`Figure 4.2.12 shows a block diagram of a demodulator for recovering the
`noise-corrupted signal components Agm and Ag,» from which the vector V is
`
`recovery
`
`phase shifter
`
`4.2.12
`diagram of demodulator for PSK signals.
`
`r(t) = aeu(t) + z(t)
`
`tution of (4.2.97) into (4.2.94) yields the vector V as
`
`V=208+N
`
`the noise component N is a complex-valued gaussian random vari
`
`gero mean and variance LE(\NI’) = 2E€N). Let V = X + jY, where
`
`
` Synchronizer
`[ Carrier 2
`
`
` 90°
`
`
`
`
`
`
`
`
`
`Bi.
`V," = a.m — J4sm 18 sin
`fmed. The projection of V onto the unit vectors
`
`Bemplished by the formation of
`the product U,, = Xa. + Yasy,
`
`m..., M. Equivalently the vector V can be followed by a phase detector w
`
`Mputes the phase of V, denoted by 9, and selects from the set {s,,,(¢)}
`
`having a phase closest to @.
`
`a Having described the form of the modulator and demodulator for M
`
`KK, we now evaluate the performance in termsof the probability of error
`WC N channel. In order to compute the average probability of error, we ass
`
`#,(¢) is transmitted. Then the received waveform is
`
`
`
`
`
`
`X =2a& + Re(N)
`
`Y =Im(N)
`
`BX and Y componentsarejointly gaussian random variables, with the |
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 13
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 13
`
`

`

`DIGITAL COMMUNICATIONS
`
`011
`
`010
`
`00
`
`110
`
`001
`
`000
`
`i11
`
`100
`
`M=8,=0
`
`(c)
`
`00
`
`011
`
`001
`
`o10
`
`110
`
`000
`
`100
`
`10
`
`lil
`10]
`M=8,r=2/8
`
`(d)
`
`PSK signal constellations for M = 4 and M = 8.
`
`
`
`
`
`fT
`
`0
`
`bstitution of (4.2.97) into (4.2.94) yields the vector V as
`V=20€+N
`
`(4.2.97)
`
`(4.2.98)
`
`biWDULATION AND DEMODULATION FOR THE ADDITIVE GAUSSIAN NOISE CHANNEL
`
`261
`
`
`
`
`
`Synchronizer
`
`Keceived
`signal
`
`recovery
`
`iw a
`
`
`
`; . Carrier .
`
`
`
`
`
`
`HN
`
`Saw
`
`g333° °=€ a5
`
`90°
`phase shifter
`
`Via
`Reference
`vectors
`
`WIGURE 4.2.12
`Wlock diagram of demodulator for PSK signals.
`
`
`
`
`formed. The projection of V onto the unit vectors Vl = Gem — J4sm is simply
`|
`- wcomplished by the formation of
`the product U,, = Xa,,, + Ya,,,. m=
`
`E 1,2,..., M. Equivalently the vector V can be followed by a phase detector which
`
`F gomputes the phase of V, denoted by @, andselects from the set {s,,(7)} that
`
`; signal having a phase closest to 0.
`
`Having described the form of the modulator and demodulator for M-ary
`
`that the exponential factor under the integral in (4.2.93) is independent of the j
`] PSK, we now evaluate the performancein terms of the probability of error in an
`
`variable of integration and, hence,
`it can be factored out. As a result,
`the 1
`AWGNchannel. In order to compute the average probability of error, we assume
`
`q that s,(¢) is transmitted. Then the received waveform is
`optimum demodulator can be implemented as a single matchedfilter or cross
`
`correlator which computes the vector
`
`r(t) = aeu(t) + z(t)
`3
`V= eltf r(t)u*(t) dt
`
`
`projects it onto the M unit vectors
`
`. 2
`’
`¥,, = exp {j|47(m 1) +a}! m=1,2,...,M (4.2.95)
`|
`
`ere the noise component N is a complex-valued gaussian random variable
`
`and selects the signal corresponding to the largest value obtained by this
`{
`th zeromean and variance 4E(|N|*) = 2@Np. Let V = X + jY, where
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 14
`
`

`

`The phase of V, computed by the phase detector, is # = tan! Y/X. The
`probability density function of @ is obtained by a change in variables from X
`and Y to
`
`(4.2.101)
`
`(4.2.106) ee eeeee rN 8 kde UEae ee edeeeee
`p(@)
`
`FIGURE 4.2.13
`Probability density function p(@) for y = 1, 2, 4, 10.
`
`_
`2
`R=vx'+Y
`6=tan | Y/X
`This changein variables yields the joint probability density function
`
` r
`2
`2
`2
`2
`p (r
`)
`202 e
`, @)=
`—(7? + 40° &? — 4a€r cos 9)/26
`Integration of p(r, @) over the range of r yields p(@). Thatis,
`ao
`It
`p(9)=f p(x, 8) ar
`= wel + (Fry cos 80720"#aefis| (4.2.103)
`
`(
`
`0
`
`2
`
`T
`
`“—o
`
`42.102
`
`)
`
`where y = a6/N, is the SNR per symbol. Figure 4.2.13 illustrates p(@) for
`several values of y. It is observed that p(@) becomes narrower and more peaked
`about @ = 0 as y is increased.
`A decision error is madeif the noise causes the phaseto fall outside of the
`range —7/M < @ < 7/M.Thus
`(4.2.104)
`Py =i f%™ p(6)do
`—2/M
`In general, the integral of p(@) 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). Thatis,
`
`“3.14
`
`-2.51 -1.88 -1.26
`
`-063
`
`0.00
`
`0.63
`
`1.26
`
`1.88
`
`2.51
`
`where P, is the probability of a correct decision for the 2-bit symbol. The rest
`in (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
`
`P,=1-P,
`(4.2.105)
`P, = erfe(y¥s)
`= erfc (Vs) — lerfc (%)]
`where y, is the SNR per bit. On the other hand, the symbolerror probability for
`For M > 4, the symbol error probability P,, is obtained by numerica!
`M = 4is determined by noting that
`integrating (4.2.104). Figure 4.2.14 illustrates this error probability as a functi
`P, = (i ~ P,)”
`of the SNR per bit for M = 2, 4, 8, 16, and 32. The graphsclearly illustrate t
`penalty in SNR per bit as M is increased beyond M = 4. For example,
`= [1 - derfe(/%)]
`Py = 107°, the difference between M = 4 and M =8is approximately 4 d
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 15
`
`Petitioner Sirius XM Radio Inc. - Ex. 1021, p. 15
`
`

`

`R= yxX?+Y?
`6 =tan-'Y/X
`changein variablesyields the joint probability density function
`r
`2
`2
`g2
`2
`
`p(r,8)
`=36
`@)=
`—(r? +407 &*—4aércos @)/20
`sration of p(r, 6) over the range of r yields p(@). Thatis,
`p(8)=f p(r,8) ar
`e) + fary £05 8etofVycos6,32/2 ax|
`(4.2.103)
`
`T
`
`“—-©
`
`is the SNR per symbol. Figure 4.2.13 illustrates p(@) for
`ral values of y. It is observed that p(@) becomes narrower and more peaked
`it 6 = 0 as y is increased.
`A decision error is madeif the noise causes the phase to fall outside of the
`ee —7/M <6 < 27/M.Thus
`(4.2.104)
`Py=1- re" p(0) 40
`—a/M
`eneral, the integral of p(@) 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
`sability of error is given by (4.2.40). When M = 4, we have in effect two
`ry PSK signals in phase quadrature as indicated above. With coherent
`\odulation, there is no cross talk or interference between the signals on the
`quadrature carriers and, hence,
`the bit error probability is identical
`to
`
`P, = herfe(/7,|
`
`(4.2.105)
`
`-3.14 -2.51 -1.88 -1.26 -0.63
`
`aa
`0.63
`
`a
`0.00
`8
`
`1.26
`
`1.88
`
`2.51
`
`3.14
`
`FIGURE 4.2.13
`
`Probability density function p(@) for y = 1, 2, 4, 10.
`
`where P, is the