Digital
`
`Communications
`
`by 8&7 tel/ite-
`
`J.J. SPILKER, JR. PhD.
`
`Chairman, Stanfbrd Telecomn-imrirations, Inc.
`
`PRENTICE-HALL, lNC.. Eng/ewood Cliffs, New Jersey
`
` 
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 1
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 1
`
`

`

`Library of Congress Cataloging in Publication Data
`Spilker, J
`J
`Digital communications by satellite.
`(Prentice-Hall information and system sciences series)
`Bibliography: p. 525
`Includes index.
`1. Artificial satellites in telecommunication.
`2. Datatransmission systems.
`I. Title.
`“(5104.864
`621 .38‘0423
`75—43878
`IS B N 0—13—214155-8
`
`© 1977 by PRENTICE-HALL, INC.
`Englewood Cliffs, New Jersey
`
`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.
`
`109876
`
`Printed in the United States of America
`
`|NC.,I.ondon
`PRENTICE-HALL INTERNATIONAL,
`PRENTICE-HALL OF AUSTRALIA PTY. LIMITED,Sydney
`PRENTICE—HALL OF CANADA, LTD., Toronto
`PRENTICEuHALL OF INDIA PRIVATE LIMITED, New Delhi
`PRENTICE-HALL OF JAPAN,
`|NC., Tokyo
`PRENTICE-I—IALL OF SOUTHEAST ASIA PTE. LTD.,Smgapore
`
` 
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 2
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 2
`
`

`

`456 MDDULAT/ON AND CODING FN DISTORTED CHANNELS
`
`VI7ERBI DECODING OF CGNI/DLUTION‘IL CODES 457
`
`It also should be noted that sateliite communications involves a sub-
`stantial time delay (20.25 sec) and often rather high data rates (2 100 Mbps).
`This combination can make Forward Error Correction (FEC) much more
`desirable than automatic request for retransmission (ARQ), because of the
`]arge costs of ARQ to store data at the transmitter until a verification signal
`is received or a request for repeat is received for a data block. In ARQ,
`blocks of data are transmitted with redundancy introduced for error detec-
`tion. If a data block is received in error, the receiver sends the transmitter
`a request for retransmission The use of PEG and ARQ together can be
`advantageous
`In this chapter we review the structure of convolutionai codes, describe
`the structure of the Viterbi decoding algorithm, and discuss the error rate
`performance of the decoding algorithm for PSK and QPSK signals both
`with and without carrier reconstruction phase noise. Many of the results
`described in this chapter were first derived by Viterbi and his coworkers in
`the cited references.
`
`15-2
`
`CONVOLUTIONAL CODE STRUCTURE
`
`A convolutional encoder with constraint length K is a K—stage shift register
`with :1 linear algebraic function generators, one for each output port. A rate
`If). code producestwo output bits for every input data bit. If one of these
`output bits is the original data bit, the code is called systematic. Figure [571
`shows the structure ofa simple nonsystematic rate If?! encoder of constraint
`length K —. 3.
`‘
`Assuming that the encoder starts in the allvzero state, the first four hits
`0100
`
`code generation
`G. = 111
`62 = 101
`
`data d;_,
`- 0110
`
`xli‘rxlr
`[0,Dl.l1.1l.[0.1i (0,1)
`
`generator denotes the tap pneutinns.
`
`
`
`
`0111
`
`Fig. 15-1 Nonsystematir: convolurional encoder with
`constraint length K = 3, and rate ‘lln = ‘lIZ. The code
`
`0110 produce an output of 00, 11, UL and 01, respectively, as shown. Clearly,
`the output of each new data bit depends on the previous bit pattern stored
`in stages 1, 2 of the shift register. These bit patterns can be labeled by the
`states defined as
`
`(15.1)
`d=ll
`crio
`b=01
`a=00
`The output bits and transitions between states can be labeled by the trellis
`diagram of Fig 15-2. The diagram starts in the all-zero state, node a. and
`makes transitions corresponding to the next data bit. These transitions are
`denoted by a solid line for a “0“ and a dotted line for a “1.“ Thus, node a
`proceeds to node a or b With output bits 00 or I].
`
`\
`\
`start >K—fl—R CO
`\
`00
`a
`DC}
`a
`00
`
`5= on
`
`
`
`\ii
`
`Fig. 15-2 Trelliscode representation for the convolu-
`tional encoder of Fig. 15-1
`
`Table 15-1 shows the optimum codes for constraint lengths K : 373.
`The code generators define the taps for the [Hall shift register; 11¢
`is the
`: number of bit errors in paths at distance d,, and d} is the upper bound on
`'. minimum free distance [Odenwaldcn [970}. Notice that these codes are all
`nonsystematic. A systematic code would have G, or G; equal to 100 V
`. .0;
`, that is, one of the code generators would have only a single tap.
`Note that some of the code structures in Table 15-1 are transparent to
`code inverston—that is, if the signs of the input bits are reversed, the coded
`output bit sequence is simply inverted. For example, if one of the parity bits
`-._ xi, is related to the data bits df by
`(15.2)
`XIIL-df®d5.1@df..z
`" where there are an odd number of terms in the sum, reversing the sign ofthe
`- d4 bits simply reverses all ofthese parity bits. Thus, if the numbersof “ones“ or
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 3
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 3
`
`

`

`458 MODULAT/ON AND EODJNG IN DISTOFTED CHANNELS
`
`Table 15-1 OPIIMUM RATE [[2 Coors (MAXIMUM MINIMUM
`DISTANCE) [Gru-rousen er AL, 1971]
`Code Trans-
`Constrm'nt
`Code
`Dfsfu'ncs
`Errors
`Distance
`parent m 180”
`
`Length X
`Generator:
`J,-
`ue
`Bound d} Phase Reversal
`
`no
`5
`1
`5
`g: : icii
`3
`no
`.,
`2
`.
`2:35:
`4
`no
`8
`4
`7
`3: Z 13.131
`5
`yes
`.
`.
`a
`can:
`.
`m
`m
`a.
`10
`2:33:21
`.
`
`
`
`'02; Z 155$?!8 w 2
`
`
`weights of both GI and G2 are odd. then the code is transparent to a sign
`inversion. That is, the decoded output bit stream has the sign ambiguity as
`the inputr This transparency is valuable if biphasemodulared PSK is used
`with its ensuing sign ambiguity for it permits decoding prior to ambiguity
`removal. Differential decoding at the decoder output removes the sign ambi-
`guity and simply increases the output error rate by a factor of less than 2.
`because decoder output errors typically occur in short bursts, Difierential
`dewding at the decoder input would double the decoder input error rate and
`thus would cause a much larger increase in the hit-error rate than a factor of
`2 because of the high slope in the output-versus-input-errorerate curve.
`
`15-3
`
`THE MAXIMUM-LIKELIHOOD DECODER
`FOR A BINARY SYMMETHlC CHANNEL
`
`Maximum-likelihood decoding could be accomplished over 1: coded two-hit
`Symbols for rate 112 codes by comparing the received 2m output sequences
`with 3114(2’") possible code paths leading to each of the 4 nodes in Fig. 15-2
`and selecting the code sequences with the largest cross-correlation} This
`calculation is extremely difiicult for large m and would result in an overly
`complex decoder structure.
`A major simplification was made by Viterbi in the likelihood calculation
`by noting that each of the 4 nodes has only two predecessors, and only the path
`
`with the highest cross-correlation weight need be retained for each node. For
`'The factor of 4 includes all possible initial Starling stares.
`
`
`
`-‘
`
`.
`
`example, the paths to node a might have weights 1t) and 6 as shown in Fig.
`15-31 At each node, the weight of the survivor path determines the new weight.
`For example, at r = ithe on path might add a weight 2 to the weight 8 of the
`previous node a. The added weight can be computed by correlating the
`received code bits r“, r2, with the parity bits for that transition p”, :12, to
`
`produce w, = [11,?” el- [22mlr where p, r are :1 for "hard” binary decisions
`in r“.
`Figure 15-4 shows a typical path structure and weights for decoding. No
`errors have been introduced in the channel. Decoding has begun with no
`information as to the initial state (node) of the Coder, Hence, all nodes at
`I = 0 have been set at zero weight Notice that at 1 : 5 all node survivor
`paths began at node a at I z D, the correct starting position, A decision on
`that t = 0 data bit, the an path corresponding to a 0 data bit, can then be
`made. Thus in this example a correct data bit decision could be made with
`a 5 symbol delay. In general, with an error-producing channel, data bit deci-
`sions can be made after computing 5K successive nodes (a decoding delay of
`SK), where K is the coder constraint length.
`If an error is made in selecting the data bit at any time instant, several
`data bits in succession may be decoded incorrectly before the correct path is
`reached. Figure 15-5 shows the surviving paths with an incorrect start position
`i (start at node n7) for the same data sequence as in Fig. 15-4. Note that in this
`; example, 6 data bits, 1 = 6 (12 code bits), have been received before the correct
`path (correct path after t = 2) has produced the highest weight.
`Some of the error correction and detection characteristics of the code
`I‘ can be established by redrawing the trellis in a state diagram (Fig. 15-6),
`
`WTERBJ DECDDHVG OF CONVOLUTIONAL CODES 455
`
`Example survivor path to nodea.
`survivor oath to node a
`/
`8+2=10
`
`time—*-
`
`E+U=6
`
`
`
`c
`
`_ / 6
`.
`I
`inltlal welght
`f — 1
`
`delete this path to nodes
`I
`i+ 1
`
`i
`
`t
`
`Fig 153 Example of the iterative likelihood calcula-
`tion showing the survivor path to node a at 1 =41 The
`alternate pattern is eliminated because of IIS lower
`weight.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 4
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 4
`
`

`

`c (T
`n5———~v
`d 0
`O
`start
`i:D
`r=1
`
`2
`v
`2
`132
`
`4
`6
`r=3
`
`i=4
`tlmB—D»
`
`i=5
`
`Fig. 154 Tvpical trellis pattern for decoded bit stream
`with code words received with no errors and unknown
`stalling node. The true path is in solid line, and the
`other paths at each node are shown in dotted lines.
`The number adjacent
`to the nails is
`its correlaliun
`metric. Ail nodes start at zero weight. Correlation is
`accomplished by replacing the codes by 11 rather
`than 0, ‘l Tiesin the metric are broken by a random
`decision.
`
`weight ualue atthis node
`
`snag; "”9 ‘3th
`slate /at q
`
`
`
`incorrect
`Starting
`state
`
`:1
`fl
`t I D
`
`.
`I“.
`6
`6
`4
`4
`0
`Cl
`0
`E
`7
`6
`5
`4
`2i
`2
`1
`Fig. 15-5 Trellis paths after an initial starting arrori
`The true path is shown by the dashed line.
`
`i
`
`r=6
`
`r=7
`
`time»
`
`where each path in the state diagram is labeled with the code hits correspond—
`ing to that pathifor cxampie, the an: path is 00. Note that thc minimum,
`weight path from a to a, other than the direct at: path. is path abca shown by
`
`480 MDEULAT/GN AND CODING 1N DISYDRTED CHANNELS
`
`I/lTEREI DECODING OF CONVOLUTIONAL CODES ‘61
`
` input data
`1
`1
`0
`1
`1
`0
`
`b
`a
`77776 ______c 777777b _____
`node
`a
`En'dEJ
`"Tiny"?iiiii n
`n
`o
`0
`T
`r
`r
`
`output
`D
`1
`1
`1
`o
`‘i
`
`i
`
`|I||
`i
`nodes 8 0a
`b D
`
`
`
`the dotted line, If mum is the correct path in three steps, the minimum alter-
`
`
`
`Fig. 15-6 Stare-diagram representation for coder of
`Fig. 15-4
`
`1 E, and has weight 5, correspond-
`nate-wcight path abca corresponds to ll, ll),
`ing to five errors or Hamming distance d -— 5. Thus, this cod: corrects any
`two errors over that path length and detects any three errors. Codes of mini-
`mum distance d ,— Ze + I can be used to correct e errors.
`The erroricorrcclion propertics of any Code can be determined by
`generating the flow diagram from a lo a for all paths, each labeied D",
`where k is the weight in that path. The flow diagram for the example code is
`given in Fig. 1577, This diagram can he reduced to the generating function
`for all paths which eventually merge with the ail-zcros path by the following
`calculations of the paths leading to each of the four nodes:
`
`d : Dd -i' D!)
`
`of
`
`r: : Dd + D!) 7 b
`
`01‘
`
`
`
`1 — D
`a' : Dii —1 b
`or
`41' = his
`and thus I) ; Dzrz + c — Dd 7 Dc
`
`(15-3)
`
`(15-4)
`
`(155)
`(15-6)
`
`Solving for a' in terms of a, we obtain from (15-3) to (1576)
`
`msg#11931), D5 +206 i
`413*
`i
`iZ"D’”5i(15.7)
`Thus, there is one path of weight 5, two of weight 6, and, in general, 2" paths
`of weight k 7i— 5.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 5
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 5
`
`

`

`“2 MODULATION AND CODING 1N DISTOHTED CHANNELS
`
`erFrBl Meow/VG 0F coNl/OLUIIONAL CODES 463
`
`
`
`
`
`[bi
`Fig. 15-? State diagram labeled according to (a) dis-
`tance from all zeros path for the coder nt Fig. 151,
`(b) distance, length, and number of input “ones"
`
`The generating function can be formed for an augmented state diagram
`in which each path in the state diagram contains a multiplier coefficient 1.
`with an exponent t’,‘ corresponding to the length of the path and a coefficient
`N" where a is I for an input data ] (a solid-line path in Fig. 15-2) and e : 0
`for a zero data-bit path_ An example augmented state diagram is given in
`Fig. 15-711 for the coder of Fig. 15-], The generating function of this ring
`merited state diagram is then
`
`Ix
`r
`D‘L’N
`a'
`7:T(D.L,N)=mé2flvDN L
`;_ D=Lw + own + L)Nz + ~-
`V‘, D(S+WI)L(SW'MJ(1 + ijNU4M) + . ..
`
`1:
`
`(15.8)
`
`Thus there is one distance 5 path from a to a’ (the D5 term) of length 3 cor-
`responding to the exponent of L. There are two distance 6 paths, one of
`length 4, and one of length 5, both of which differ from the correct path by
`two data bits (the exponent of N).
`
`
`
`15-4
`
`ERRORaRATE PERFORMANCE OF THE
`MAXIMUM-LIKELIHOOD DECODEH
`
`The error—rate performance of the decoder is calculated in this section for
`both a hard-decision input (0, l) and a softwdecision input (analog sample or
`multibit quantized sample) to the decoder. We first obtain the bound on the
`probability of a first decision—error event (PE that results in one or more
`output bit errors. By computing the number of data-bit errors for different
`error events. we then obtain a bound on the bit-error probability.
`
`First-Error-Event Probability
`A first error event occurs if the correct path to the correct node is
`excluded at the jth step in the decodingithat is, the survivor is an erroneous
`path. Since the codes are group codes, there is no loss in generality by assum-
`ing that the correct path is the allezero path era - - - ea. An erroneous decision
`then occurs at stepj in Figs [58, where the path on - - - fibre is selected as
`the survivor because of its hypothesized higher correlation metric.
`
`A
`o
`'
`5
`
`“a” James: path eliminated at step]
`”incorrect survivor of length 2 = 3
`xN
`I ob
`o \ u
`a
`\\ f
`ac
`o
`I
`c’
`o
`on'
`I
`n
`u
`u
`error path length prior to stepi
`0
`3
`2
`1
`4
`Fig.15—3 Example of a decoding decision error for the
`coder ot Fig. 15-1
`
`The probabiiity of a first error event at some arbitrary stepj with length
`l is mi). For hard-decision inputs, the probability of a first error event of
`length 3 for the one possible error path is p{3) and differs from the correct
`path in five code bits occurring with probability (Pk = 05. In general,
`the
`total set of erroneous paths and the distance, k, from the correct path have
`already been described by the expansion of T(D, L. N). If there are hard deci-
`sions, and we use the T(D, L, N) of(1578), the single possible length-3 error
`event requires three received bit errors out of the five to make the incorrect
`path have a higher metric than the correct path. Thus, the probability of a
`first error event of length {=3 and distance k : 5 is expressed by
`r- 3
`(15-9)
`m3) = on = ’2 ( f )p'cl 7 in“
`where p is the hard-decision received bit-error probability at the decoder input.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 6
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 6
`
`

`

`WTEHBI‘ DECODING OF CONVDUJTIONAL CODES 4l5
`
`_
`k
`R
`l:(k‘z-1:/2 ( i )9!“ ‘PV ’
`k
`t
`1
`k
`.
`,
`T H2)pk.2(1 — ”rel 4.. Mg” ( E)p(1 ._P)k-i
`
`e
`
`“ =
`
`k odd
`k cyan
`(1516)
`where the factor of 112 in the first term for i: even accounts for the random
`.
`.
`.
`.
`k
`.=i
`1‘ ) -_ 2mm probability
`decrsron in case of a no vote. Since p < 1/2 andz(i
`can be bounded by
`
`0,. < ”in“ _p)m
`Hence, the first-error-event probability from (1511) and (15-17) is
`(PE < zakzkpkrzu _ p)“
`
`(15.17)
`
`(15-18)
`
`For the example coder of Fig. 15-] where 4k : 2"" for leg 5, the proba-
`bility of a first-error event (P5 then becomes
`
`(1549)
`(PE < :35 arrow-‘20 7 p)“: : —][2_W%i:)_l;)
`The biteerror probability is bounded by (15-15) using (15717) to obtain
`
`6’» < Zc'tZfiJWI ’17)“: : d—Tw‘M
`Johnna—me,»
`fl'N
`where we have also used (1544). For the example code, We -_- (k 7 4)2""5,
`and the bitterror probability for hard-decision inputs is bounded by
`
`(15-20)
`
`(PD
`
`t W [ZM‘
`75
`..
`k—42J‘ 2kt—21_ raffle; 15—21
`)
`u—wpu—ni
`)
`” l
`m
`(
`(£5
`For the additive Gaussian white noise channel with soft analog decision,
`the correlation metric is
`
`2 my”
`
`x1, = l
`
`(15.22)
`
`M E
`
`where j runs over the 12 code bits in each output code symbol, and where 1'
`runs over the number nodes in each path. For a rate 112 code (:1 : 2L the
`cross-correlation metric is evaluated over M symbols (M a 5K, where K is
`the code constraint length, is usually sufficient), and is expressed by
`
`(rays + my“)
`
`(15-23)
`
` 
`
`M X
`
`45‘ MODULATION AND CODING IN DJSTORTED CHANNELS
`
`Note that, in general, at step j there is no simple way of computing
`whether the paths described in the expansion of T(D,L,N) have survived
`previous survivor decisions at earlier nodes. However, the I'irstverror-eVent
`probability can be overbounded by including all the possible paths to a
`decision error at step j whether they have been previously eliminated or not.
`Thus, since the generating function
`
`T(D) = 21:ka
`
`(15-10)
`
`gives the total number of paths a,* of distance k from the correct path, the
`probability of a first error event at some arbitrary stepj is bounded by
`(P; < mar,
`(15-1!)
`
`By a similar calculation using (15-8), the biteerror probability can be bounded
`from the expansion
`
`(15.12)
`up, L. Nun. a To). N) = :2 how“
`since the path ]cngth (3,, (exponent of L) is not needed for this‘bound for the
`distance I: error path. For the example encoder of Fig, 15-1, this expanston is
`TU), N) : EskD’UV“
`—' DSN + mm2 -i
`
`+ drown}m +
`
`(is—13)
`
`where the number of errors ek = + 1 in each error path. By differentiating
`with respect to N and setting N = l, we can weight each error path by the
`number of output bit errors. We then have
`dT(D, N)
`ck é ake,‘
`N“ = Eakepfl‘~ -:- chD"
`dN
`Thus it can be shown that the bituerror probability is bounded by
`
`(15—14)
`
`(Pi. < 2cm
`
`(is—15)
`
`in which the value of (PR depends on the type of input to the decoder (that is,
`hard or soft decision) and on the channel noise level in input bit—error rate
`(Eb/ND), respectively. The code properties determine the coefficients ck. Hence,
`we have a simple division between code property effects and channel efiects.
`
`Evaluation of (Pk
`
`For a hard-decision input, the probability of a distance k survivor deci‘
`sion error is
`
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 7
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 7
`
`

`

`465 MODULATION AND CODWG Ml DfSTCh’TED CHANNHS
`
`If a decision error is made, an incorrect path with components x0. produces
`a higher correlation metric than the correct path with components x” e 1.
`Thus, an erroneous decision is made if
`
`(I 5-24)
`Eztx’uyu , y”) > 0
`if the incorrect path x’ difi'crs from the correct path in k code bits and the
`probabiiity of this occurrence is (Pk,
`
`(pk : pr(; 12 xii)?” 7 y“ > 0) = PrL: (I; "' ”Yr > 0]
`I:
`J:
`(mm
`44§m<g:mg¢<g
`where Pr{z > 0} is the probability that z > 0. We have used x, 2 1 and
`xi : #1 i x, by definition and we have converted the double subscripts
`i, j to a single subscript r, which ranges over the same set of coefficients,
`r:n‘i+j.andk=n(M—l— 1).
`Bound on Bit-Error Probibilizy
`
`The white noise channel has noise with one-sided density N[. and signal-
`encrgy per hit of E. Thus, the variables )1” have mean *m/E where 51 :
`Erin is the energy per transmitted code bit and the variance of y” is Nufl.
`The sum 2 : Ey, therefore is also Gaussian with mean kg and variance
`anfl, and the probability of a kebit—path error from (15-25) is
`D
`— .7
`9k:Pr(z<m;J‘ Wfi
`
`ik—E‘
`==
`k
`exp (_ 19/2)“ A erfc’
`)
`(
`N.
`Jv-m :7 21:
`—
`The bit-error probability bound is then obtained from (15—15) and (15-26):
`
`15-26
`
`(15-27)
`6),, < 5:5er : fickcnc'VtLA‘f—i
`where d is the minimum distance of the code (the smallest k where fr 7‘: 0].
`
`A simpler caiculation involves the calculation of [dT(D, N)]/dN using
`(15-14) rather than all the coefficients c,“ For this purpose, we want to express
`G’,t as a power of k as in are”. First, note that the function erfe' 4/2: + y is
`bounded by
`
`erfc’ Jx + y 3 exp (%y) erf‘c’ ./ x
`
`(15-28)
`
`
`
`WTEH-‘i‘f DECODING 0F CDNVDLUNONAL CODES 437
`
`'
`3“, “5mg
`
`Next,setk:d-l-6,!:0,l,.i..andx+ =2k€N Th
`(15-26) and (15-28), we obtain
`)’
`a],
`“I
`
`.
`2k£
`(P , erfc
`‘
`4/ Na
`k
`.
`—€
`,
`7,
`201’ -i' OE,
`
`0
`n
`erfc ,/
`N
`g exp< N05) erfc
`2155'
`ght] weakened to 0b
`Thus, the bound on (9,, in (15-27) can be sli
`(15—15) and (15—29)
`y
`
`
`D
`u
`(P, < 2cm g affix/21$" ; ck exp [#1
`
`i
`242's
`d6
`“
`<
`t
`_r
`— m 'i
`(15.30)
`eerfc 1/ ND “Mmigi cke *
`A
`The expression (15-30) can be rewritten using (15-14) to replace the summa-
`tion
`
`(15-29)
`
`'
`mm from
`
`(15.31)
`
`(15-32)
`
`(15—33”)
`
`(15-331))
`
`(”'34)
`
` 
`
`[5.1, where f. = 38/” = 55/2,
`
` ”=1 U=HP (1“,...)
`
`)
`
`a an; N
`2d:
`,
`a! <erfc1f—l
`(i)
`t
`,
`Nu 9"” ND
`dN
`”
`For th
`'
`and d : 5’e example rate lf2 code of Fig,
`mi), N)
`,
`D5
` v
`dN
`
`’(1 _ 21.0)1
`Hence, the bound on bit»error probability from (15731) is
`
`(p a erfe’ ”in E M
`°
`ND “P (2N6) [1 _ 2 exp 55/2110]:
`and (l5-334r) reduces to
`
`<9 _rfc'5_W
`' < [l - 2min fibrin”:
`as compared to the result of no coding
`#
`,
`213
`1
`ME
`I? _
`_ ~
`"EC v N: " 7““ F”
`A
`
`where
`
`51'1“: x = 72rTJ‘ e'V' dy.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 8
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 8
`
`

`

`“a MODULATION AND CODING IN DISTDHTED CHANNELS
`
`Thus by comparing(15-33a),(15734) for high 5,,an this error-bound shovlrs
`that the coded signal is more than 10 log (2.5) : 3.98 dB better than t e
`uncoded signal for the same output error rate.
`_
`Figures 15-9 through l5-1] shew experimental- curves of measured bit—
`error rates indicating the approximately 4- to 6»dB Improvement tn perform-
`0 simulation 1 [32-bit-pa1h. 0T5 = 0.5l
`x
`IDJSQSImples
`
`u 921,3m5amplas } simulation 2
`Ill HAW
`
`10°
`
` i 10"
`
`
`
`
`
`
`
`
`
`
`
`
`
`i_.lIt.l.l.|_t._luul.l__t_t.uml
`9aaveragehitermrrate 10’“
`
`
`
`
` 10’?
`
` 10’3
`
`
`
`
`
`
`.__l
`(5
`
`1075
`
` 10‘“
`
`
`lui maul-1mm
`0’7
`1—3—3’4’2524 6810121416
`En/No, dB
`Fig. 15-5 Simutated nerlorrnance of a rate 112 convo-
`lurional nude with generators employing soft deuteron.
`G, : 111, G2 : 101. constraint length 3, 1fi-bit paths
`and 32m: paths. Sort decisiun 8-levell receiver‘quan-
`tizarlon is employed. The bit error rate us for maxrmnm-
`likelihood decoding. The quanlizer threshold spacing
`(0T5) is set equal to 0.50 a' where O“ is the rms noise.
`[Busiaman13, 1972]
`
`
`
`WTEHBI DECODING 0F CUNVOLUTIONAL CODES “S
`
`0
`X
`El
`
`Simulation 1 1327bit-path, 0T5 I 0.5)
`92,160 samples
`.
`.
`
`921,600 sample‘s
`SImuIaIlOl‘l 2
`
`Humi|vvw-m7iirrrrrrrn-m.l
`10‘I
`XX
`X):
`10H
`
` 10—2
`
`
`
` 10“
`
`10—5
`
`I—
`
`
`
`averagebiterrorrate
`
`i
`
`—8—6 —4 —2
`
`U
`
`M4444
`2
`4
`6
`8
`Eb W“, dB
`Fig. 15-10 Simulated performance of a rate 1l2 con-
`vnlutienal codewith generators G =11101, G =10011.
`The performance is
`for Viterbi decoding with soft
`decision, constraint
`length 5, 25-bit paths, Bilevei
`receiver quantization. The quantizer threshold spacing
`is equal to 0.50 a.
`
`
`
`
`
`
`
`
`
`
` 10'7
`‘1’wm—r..mn—.—mw~y—y—mm
` Ill
`
`1D
`
`12
`
`14
`
`15
`
`ance at (P, -: [0’5 for codes of constraint lengths 3, 5, and 7 [Heller and
`Jacobs, 1971, 8357848; Bustamante and Leong, 1972]. The codes are rate lf2
`codes with the parity tap positions defined by G,, G2 as indicated in the cap-
`tions. Eight-level soft decision is used rather than a. pure analog input; this
`soft decision experimentally produces a performance within 0.25 dB of the ana—
`log input sample decoder. Decoding is performed over M a 5K path lengths.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 9
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 9
`
`

`

`470 MODUMYION AND CODING IN orsraarw CHANNELS
`o
`simulation 1 l32-bit~pam, uTS = 050)
`x
`92,160 samples
`simulation 2
`'3
`921,600 samples
`
`1|)"
`
`uncoded quadriphase
`ltheoretieall
` 1&7
`
`10—3
`
`10"
`
`
`
`0 aireragebiterrorrate
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`,
`.
`we?
`78 its—”7:1 72 ID ‘#2
`
`.ur in;
`6
`B
`
`4
`
`10
`
`12
`
`14
`
`15
`
`Es/dea
`Fig. 15-11 Simulated performance at a rate U2 con-
`volutional code with generators 61 = 1111001, G; =
`1011011. The performance is for Viterbl decoding. $051
`dacisinn, rate ”Leonstralnt length 7, 35-bit paths.
`Eelevel receiver quantization. The quantizer threshold
`spacing is set equal lo 0.50 a.
`
`Code lnterleaving
`
`This section has assumed that the decoder input samples 01' hit daemons
`have independent noise or errors from bit to hit. This assumption. may not be
`valid if there are significant filter distortion efi‘ects, as descnbed 1:: Chap. [3,
`
` 10“
`
`Iriuut
`
`tlHm.
`
`
`
`
`tux 1041'
`
`
`
`
`
`
`
`
`
`
`wriREt assoc/N5 o; CONVDLUNCNAL [cogs 471
`
`or ifthe inphase and quadrature channels of a QPSK signal have crosstalk.
`which causes correlation between errors.
`The efiects of these adjacent or nearly adjacent errors can he reduced by
`proper interleaving of the coder output hits. For example, the coder output
`bits X“ can be transmitted with no delay, and the code bits X2; can be trans,
`mitted with a 5K hit delay simply by adding a 5K hit shift register in the
`output of the lower mod 2 adder in Fig 15-] , This simple delay operation [are
`vents adjacent received bits from affecting the same decoded data bit. Higher
`order interleaving is also possible to prevent -
`-
`- 0.0741“ .
`- error patterns from
`affecting the same decoded data bit. (See Forney and flower [197]], J. L.
`Ramsey [1970]). Higher order interleaving may require the addition of special
`sync words under to preform the deinterleaving process.
`
`15-5
`
`EFFECTS OF PHASE NOESE EN COHERENT
`DEMODULATION 0N DECODER
`PERFORMANCE
`
`The effect of phase noise in carrieretraeking phase-locked loops on error rate
`has been discussed for uncoded PSK channels in Chap. 12. A coded channel
`has a steeper curve of (Pb versus EDIND and therefore is more sensitiv: to phase
`noise. Thus. a short momentary increase in the slowly varying phase-tracking
`error it produces a momentary decrease in efiective Ebe0 and a very large
`increase in error rate.* Much of the time, the phase noise is near its mean
`value of zero to provide low error rates. However, the small fraction of time
`when the effective lib/No decreases dominates because of the steepness ofthe
`error rate curves.
`_
`The effect of slowly varying phase error d on a BPSK channel decreases
`the effective EDI/Nu to (Eb/1N0) cos2 (b. Thus, instead of an error-probability
`funclionaj l‘flla’tlonship
`
`n
`5),, add?)
`
`“545)
`
`the phase noise degrades performance to theerror probability for a giVen qS:
`a
`(15-36)
`«Prev =f(% cos“ a)
`For a first-order loop the probability density of phase error Q} for an SNR
`at In the Closed-loop bandwrdth is (see Chap. 12)
`Ellen!
`n
`(15-37)
`m5) : —— a: >>i
`21:! (,1)
`"‘Slowly“ refers to a small change in phase error d over many (> 102) bit periods.
`
`
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 10
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 10
`
`