Digital
`Communications
`by Satellite
`
`J.J. SPILKER, JR. Ph.D.
`
`Chairman, Stanford Telecommunications, Inc.
`
`PRENTICE-HALL, INC., Englewood 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. 625
`Includes index.
`1. Artificial satellites in telecommunication.
`2. Data transmission systems.
`|. Title.
`TK5104.564
`621.38°0423
`75-43878
`ISBN 0-13-214155-8
`
`© 1977 by PRENTICE-HALL, INC.
`Englewood Cliffs, New Jersey
`
`All rights reserved. No part of this book
`may be reproducedin any form or by any means
`without permission in writing from the publisher.
`
`10
`
`9
`
`8
`
`7
`
`6
`
`Printed in the United States of America
`
`INC., London
`PRENTICE-HALL INTERNATIONAL,
`PRENTICE-HALL OF AUSTRALIA PTY. LIMITED, Sydney
`PRENTICE-HALL OF CANADA, LTD., Toronto
`PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
`PRENTICE-HALL OF JAPAN,
`INC., Tokyo
`PRENTICE-HALL OF SOUTHEAST ASIA PTE. LTD., Singapore
`
` 
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 2
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 2
`
`

`

`456 MODULATION AND CODING IN DISTORTED CHANNELS
`
`It also should be noted thatsatellite communications involves a sub-
`stantial time delay (=-0.25 sec) and often rather high data rates (= 100 Mbps).
`This combination can make Forward Error Correction (FEC) much more
`desirable than automatic request for retransmission (ARQ),because of the
`large costs of ARQto 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 blockis received in error, the receiver sends the transmitter
`a request for retransmission. The use of FEC and ARQ together can be
`advantageous.
`In this chapter we review the structure of convolutional codes, describe
`the structure of the Viterbi decoding algorithm, and discuss the error rate
`performance ofthe decoding algorithm for PSK and QPSK signals both
`with and ‘without carrier reconstruction phase noise. Manyof the results
`described in this chapter were first derived by Viterbi and his co-workers in
`the cited references.
`
`15-2
`
`CONVOLUTIONAL CODE STRUCTURE
`
`A convolutional encoder with constraint length K is a K-stage shift register
`with v linear algebraic function generators, one for each output port. A rate
`1/2 code produces two output bits for every input data bit, If one of these
`output bits is the original data bit, the codeis called systematic. Figure 15-1
`showsthe structure of a simple nonsystematic rate 1/2 encoder of constraint
`length K = 3.
`;
`Assuming that the encoderstarts in the all-zero state, the first four bits
`0100
`
`code generation
`G, =111
`
`data d;
`peers,
`0110
`
`Xaje Xi
`(0, 0), (1, 1), (0, 1) (0, 1)
`
`generator denotes the tap positions. VITERBI DECODING OF CONVOLUTIONAL CODES 487
`
`Table 15-1 shows the optimum codes for constraint lengths K = 3-8.
`The code generators define the taps for the K-bit shift register; ”, is the
`: number of bit errors in pathsat distance d,, and d} is the upper bound on
`F minimum free distance [Odenwalder, 1970]. Notice that these codes are all
`nonsystematic, A systematic code would have G, or G, equal to 100... 0;
`E.
`that is, one of the code generators would haveonly a single tap.
`Note that some of the code structures in Table 15-1 are transparent to
`code inyersion—thatis, 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
`f. x,, is related to the data bits d, by
`(15-2)
`xy = dD dy Bdea
`’ where there are an odd numberoftermsin the sum,reversing the sign of the
` d, bits simply reversesall of these parity bits. Thus,if the numbersof “ones” or
`
`G, = 101
`
`
`parity bits
`
`oi
`
`Fig. 15-1 Nonsystematic convolutional encoder with
`constraint length K = 3, and rate 1/n = 1/2. The code
`
`0110 produce an output of 00, 11, 01, 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=H
`c=10
`b=01
`a=00
`The output bits and transitions between states can be labeled by thetrellis
`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 fine for a “1.” Thus, node a
`proceeds to node a or } with output bits 00 or 11.
`_ 00 *- ‘
`
`start
`
` time —>
`
`Fig. 15-2 Trellis-code representation for the convolu-
`tional encoderof Fig. 15-1
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 3
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 3
`
`

`

`VITERBI DECODING OF CONVOLUTIONAL CODES 459
`
`Example survivor path to node a.
`survivor path to node a
`a
`g+2=10
`
`,
`<a ea
`10 >. ~
`“eb
`
`a
`be
`
`time —~
`
`6+0=6 delete this path to node a
`
`468 MODULATION AND CODING IN DISTORTED CHANNELS
`
`Table 15-1 Optimum Rate 1/2 Copes (Maximum MINIMUM
`DISTANCE) [GILHOUSENET AL., 1971]
`Code Trans-
`Constraint
`Code
`Distance
`Errors
`Distance
`parent to 180°
`
`Length K
`Generators
`dy
`Ne
`Bound d~ Phase Reversal
`3
`a . iat
`5
`1
`5
`no
`6
`Gem gm
`5
`Ge too
`7 $8
`6
`rH gg em
`ee
`6 Giron =
`
`weights of both G, and G, are odd, then the code is transparent to a sign
`inversion. That is, the decoded output bit stream has the sign ambiguity as
`the input. This transparency is valuable if biphase-modulated 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 outputerror rate by a factorofless than 2,
`because decoder output errors typically occur in short bursts. Differential
`decoding at the decoder input would double the decoderinputerror rate and
`thus would cause a muchlargerincrease in the bit-error rate than a factor of
`2 because of the high slope in the output-versus-input-error-rate curve.
`
`15-3
`
`THE MAXIMUM-LIKELIHOOD DECODER
`FOR A BINARY SYMMETRIC CHANNEL
`
`Maximum-likelihood decoding could be accomplished over # coded two-bit
`symbols for rate 1/2 codes by comparing the received 2m output sequences
`with all 4(2") possible code paths leading to each of the 4 nodes in Fig, 15-2
`andselecting the code sequences with the largest cross-correlation.* This
`calculation is extremely difficult for large m and would result in an overly
`complex decoderstructure.
`A major simplification was made by Viterbi in the likelihood calculation
`by noting that each of the 4 nodeshas only two predecessors, and only the path
`
`with the highest cross-correlation weight need be retained for each node, For
`*The factor of 4 includesall possible initial starting states.
`
`
`
`
`
`initial weightgigisd i i+ .
`
`
`
`
`
`Fig. 15-3 Example ofthe iterativelikelihood calcula-
`tion showing the survivor path to node a at f= /, The
`alternate pattern is eliminated because of its lower
`weight.
`
`example, the paths to node a might have weights 10 and 6 as shown in Fig.
`15-3. At each node,the weightof the survivor path determines the new weight.
`For example, at ¢ = i the aa 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 codebits r,,, r2, With the parity bits for that transition p1,P2; to
`
`produce w,= pri: -F Pasa, Where p, r are +1 for “hard” binary decisions
`in ry.
`Figure 15-4 showsa 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
`t = 0 have been set at zero weight. Notice that at ¢ = 5 all node survivor
`paths beganat node a at ¢ = 0, the correct starting position. A decision on
`that ¢ = 0 data bit, the aa path corresponding to a 0 data bit, can then be
`made, Thusin this example a correct data bit decision could be made with
`a 5 symboldelay.In general, with an error-producing channel, data bit deci-
`B sions can be made after computing 5K successive nodes (a decoding delay of
`5K), where K is the coder constraint length.
`If an error is madein 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 showsthe surviving paths with an incorrectstart position
`& (start at node d) for the same data sequenceasin Fig. 15-4. Note that in this
`| example,6 data bits, t = 6 (12 codebits), have been received before the correct
`path (correct path after f= 2) has produced the highest weight.
`Some of the error correction and detection characteristics of the code
`: can be established by redrawingthetrellis in a state diagram (Fig. 15-6),
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 4
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 4
`
`

`

`460 MODULATION AND CODING IN DISTORTED CHANNELS
`
`
`
`
`a2
`
`input data
`0
`1
`1
`>
`node
`a
`a2
`
`!I
`nodes a 0.
`b0-
`8
`4
`2
`co
`«ce <— i
`da
`2
`6
`4
`0
`start
`f=0
`f=
`f=2
`i=3
`i=4
`time
`
`
`
`i=5
`
`f=6
`
`i=7
`
`Fig. 15-4 Typicaltrellis pattern for decodedbit stream
`with code words received with no errors and unknown
`starting 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 node fs
`its correlation
`metric. All nodes start at zero weight. Correlation is
`accomplished by replacing the codes by +1
`rather
`than 0,1. Tiesin the metric are broken by a random
`decision.
`
`the dottedline. If aaaa is the correct path in three steps, the minimum alter-
`
`correct
`starting true path
`state
`a
`
`weight value at this node
`
`6
`
`°
`
`e
`
`
`
`be
`
`ce
`
`:
`incorrect
`starting
`state
`
`
`.
`d
`0
`0
`a
`0
`4
`4
`6
`6
`8
`t= 0
`1
`2
`3
`4
`5
`6
`7
`Fig. 15-5 Trellis paths after an initial starting error.
`The true path is shown by the dashedline.
`
`time—»
`
`whereeachpath in the state diagram is labeled with the code bits correspond-
`ing to that path—for example, the aa path is 00, Note that the minimum-
`weight path from ¢ to a, other than the direct aa path, is path abca shown by
`
`VITERBI DECODING OF CONVOLUTIONAL CODES 461
`
`
`
`Fig. 15-6 State-diagram representation for coder of
`Fig. 15-4
`
`nate-weight path abea correspondsto 11, 10, 11, and has weight 5, correspond-
`ing to five errors or Hamming distance d = 5. Thus, this code corrects any
`two errors over that path length and detects any three errors. Codes of mini-
`mum distance d = 2e + I can be used to correct ¢ errors.
`The error-correction properties of any code can be determined by
`generating the flow diagram from a to a for all paths, each labeled D*,
`where & is the weight in that path. The flow diagram for the example codeis
`given in Fig. 15-7. This diagram can be reduced to the generating function
`for all paths which eventually merge with the all-zeros path by the following
`calculations of the paths leading to each ofthe four nodes:
`
`d= Dd +- Db
`
`or
`
`c= Dd+ Db—b
`
`or
`
`
`
`1—D
`a= D> —1\b
`of
`a=De
`and thus 6 = D’a + ¢ — Dd — De
`
`(15-3)
`
`(15-4)
`
`(15-5)
`(15-6)
`
`Solving for a’ in terms of a, we obtain from (15-3) to (15-6)
`
`Tp) = =P = Ds + 2S 4D + + + DHS
`(15-7)
`
`Thus, there is one path of weight 5, two of weight 6, and, in general, 2* paths
`of weight k + 5.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 5
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 5
`
`

`

`462 MODULATION AND CODING IN DISTORTED CHANNELS
`
`
`
`(b)
`Fig. 15-7 State diagram labeled according to (a) dis-
`tance from all zeros path for the coder of Fig. 15-1,
`(b) distance, length, and numberof input “ones”
`
`The generating function can be formed for an augmentedstate diagram
`in which each path in the state diagram contains a multiplier coefficient L
`with an exponent¢, correspondingto the length of the path and a coefficient
`NN‘ where ¢ is I for an inputdata | (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. 15-7b for the coder of Fig. 15-1. The generating function of this aug-
`mented state diagram is then
`
`Thus there is one distance 5 path from a to a’ (the D* term) of length 3 cor-
`responding to the exponent of L. There are two distance 6 paths, one of
`length 4, and oneof length 5, both of which differ from the correct path by
`two data bits (the exponent of N).
`
`(15-8)
`
`The probability ofa first error event at some arbitrary step j with length
`is p(é). 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 @, = @;. In general,
`the
`total set of erroneous paths and the distance, k, from the correct path have
`Pe
`New
`sk
`D°L?N
`a’
`already been described by the expansion of T(D, L, N). If there are hard deci-
`a TOL.) = 7ppn= Een L
`sions, and we use the T(D, L, N) of(15-8), the single possible length-3 error
`= D'LAN + DSA + LN? 4+
`event requires three received bit errors out of the five to make the incorrect
`+ DStm[seme] + Lypnotm foes
`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
`
`
`
`VITERBI DECODING OF CONVOLUTIONAL CODES 463
`
`15-4
`
`ERROR-RATE PERFORMANCE OF THE
`MAXIMUM-LIKELIHOOD DECODER
`
`The error-rate performance of the decoderis calculated in this section for
`both a hard-decision input (0, 1) and a soft-decision input (analog sample or
`multibit quantized sample) to the decoder. Wefirst obtain the bound on the
`probability of a first decision-error event @, that results in one or more
`outputbit 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 nodeis
`excluded at the jth step in the decoding—thatis, the survivor is an erroneous
`path, Since the codes are group codes, thereis no loss in generality by assum-
`ing that the correct pathis the all-zero path aa --- aa. An erroneousdecision
`then occurs at step jin Fig. 15-8, where the path aa --- abca is selected as
`the survivor because of its hypothesized higher correlation metric,
`
`«
`
`.
`5
`
`~~
`
`StEP/ correct path eliminated at step /
`/*— incorrect survivor of length & = 3
`‘A ef eb
`._
`.
`SOF
`ec
`.
`f
`.
`.
`.
`e
`od
`.
`.
`error path length prior to step j
`2
`1
`0
`3
`4
`Fig. 15-8 Example of a decoding decision error for the
`coderof Fig. 15-2
`
`(159)
`23) = 6, =} (7 pa — py
`where p is the hard-decision received bit-error probability at the decoderinput.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 6
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 6
`
`€
`

`

`464 MODULATION AND CODING IN DISTORTED CHANNELS
`
`Note that, in general, at step j there is no simple way of computing
`whether the paths described in the expansion of 7(D,L, N) have survived
`previous survivor decisions at earlier nodes. However, the first-error-event
`probability can be overbounded by including all the possible paths to a
`decision errorat step j whether they have been previously eliminated or not.
`Thus, since the generating function
`
`(15-10)
`T(D) = Ea,D*
`gives the total number of paths a, of distance k from the correct path, the
`probability of a first error event at some arbitrary step j is bounded by
`Op < Ea,8,
`(15-11)
`By a similar calculation using (15-8),the bit-error probability can be bounded
`from the expansion
`Gen)
`TD, L, N)lues & TD, N) = 3a,D*N"
`since the path length ¢, (exponentof L) is not needed for thisbound for the
`distance k error path. For the example encoder of Fig. 15-1, this expansionis
`T(D, N) = Xa,D*N*
`= D5N 4 2D8N2 4
`
`+++ } QEDEtSNEHE bee
`
`(85-13)
`
`where the numberoferrors e, = k + 1 in each error path. Bydifferentiating
`with respect to N and setting N = 1, we can weight each error path by the
`number of outputbit errors. We then have
`aT(D, N)
`CyB Apex
`GN hey = Ea,e,D* = Le,D*
`Thus it can be shownthat the bit-error probability is bounded by
`
`(15-14)
`
`0, < Le,P;
`
`(15-15)
`
`in which the value of @, depends on the type ofinputto the decoder (thatis,
`hard or soft decision) and on the channelnoise level in input bit-error rate
`(E,/N), respectively. The code properties determinethe coefficients ¢,. Hence,
`we have a simple division between codepropertyeffects and channel effects,
`
`Evaluation of 9,
`
`For a hard-decision input, the probability of a distance k survivor deci-
`sion error is
`
`
`
`VITERBI DECODING OF CONVOLUTIONAL CODES 465
`
`e
`
`Bm)
`
`k
`k
`7
`k odd
`ie ( i ea =p
`Tit
`EY
`4G
`fk tk
`z ii)PM — py? + we2det ( } \oa — py k even
`(15-16)
`where the factor of 1/2 in the first term for k even accounts for the random
`decision in case ofa tie vote. Since p < 1/2 and ¥ (7) = 2*, this probability
`can be bounded by
`“
`O, < 2p] — py?
`Hence, thefirst-error-event probability from (15-11) and (15-17) is
`Op, < Ea,2*p"2(1 — py?
`
`(15-17)
`
`(15-18)
`
`For the example coder of Fig. 15-1 where a, = 2*-5 for k > 5, the proba-
`bility ofa first-error event , then becomes
`
`3
`— pyev2 . [2p — py
`ae sqe_hing)
`Ft
`py? (ha (15-19)
`soph—
`Oe < Sy ak
`The bit-error probability is bounded by (15-15) using (15-17) to obtain
`
`8, < Eo,2*pe(1 — py? — DN)
`|w=1,p=2v70=9)
`dN
`where we have also used (15-14). For the example code, ¢, = (k — 4)2*-5,
`andthe bit-error probability for hard-decision inputs is bounded by
`
`(15-20)
`
`k=3
`G,< s (k — 4)2k-s2iepki2q] — py? = 2/pT — pi]? =
`[1 — 4./pC. — p)]
`Forthe additive Gaussian white noise channel! with soft analog decision,
`the correlation metric is
`Moon
`
`(15-21)
`
`(15-22)
`xy=l
`yy & XyVey
`where j runs over the n code bits in each output code symbol, and where i
`runs over the number nodes in each path. For a rate 1/2 code (n = 2), the
`cross-correlation metric is evaluated over M symbols (M = 5K, where K is
`the code constraint length, is usually sufficient), and is expressed by
`M.
`XL Gaba + X2ha)
`
`(15-23)
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 7
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 7
`
`

`

`466 MODULATION AND CODING IN DISTORTED CHANNELS
`
`Ifa decision erroris made, an incorrect path with components x; produces
`a higher correlation metric than the correct path with components x;; = !.
`Thus, an erroneous decision is made if
`(15-24)
`ELK — Pus) > 0
`If the incorrect path x’ differs from the correct path in k code bits and the
`probability of this occurrence is ©,,
`6, = P(E x Xu — Pu > 0) = P| (x ~ Dy > o|
`ke
`ie
`(15-25)
`= Pr(32»,< 0) = Pr(Sy, <0)
`where Pr{z > 0} is the probability that z > 0. We have used x, = 1 and
`x, = —1 # 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=nitj,andk =a(M + 1).
`Bound on Bit-Error Probibility
`
`erfe’ ./x + y< exp (=) erfe’ ./ x
`
`(15-28)
`
`
`
`de,\
`(de,
`dT(D, N
` IN=1, Deoxp (~es/Ny)
`(2)
`@; < erfe’ ,/Es
`(D,N)
`Wo PAN)
`‘
`aN
`The white noise channelhasnoise with one-sided density Ny and signal-
`For th
`i
`energy per bit of £,. Thus, the variables y,, have mean —./€, Where €, =
`mae - example rate 1/2 code of Fig. 15-1, where €, = E,/n = E,/2,
`E,|n is the energy per transmitted code bit and the variance of y,, is N4/2.
`The sum z = Sy,therefore is also Gaussian with mean k,/e, and variance
`aT(D, N)
`__
`DD?
` Ny
`kN,/2, and the probability of a k-bit-path error from (15-25) is
`aN
`0
`— ‘er
`~W— aDpF
`(15-32)
`
`®, = Pz <0) = J exp(2—KyeWKNseals dz
`Hence, the bound onbit-error probability from (15-31) is
`=f
`exp (= 2712) dy & erfe’
`2KE,
`(15.96
`
`JeGaemorte (ie 1826
`
`
`./98%exp(SEs) _[exp(—2/2N0)_:®, <erfe’ NP (Gxt) {l — 2exp EP2N,P (15-332)
`
`Thebit-error probability bound is then obtained from (15-15) and (15-26):
`and (15-332) reduces to
`&
`0
`B, <Ee,0, — Se, erfe’ foe
`(15-27)
`0,<eileSEIN,
`»< [T= texp(BINOF
`(15-338)
`where dis the minimum distance of the code (the smallest k where c, + 0).
`as comparedtothe result of no coding
`A simpler calculation involves the calculation of [dT(D, N)\/dN using
`(15-14) rather than all the coefficients c,. For this purpose, we want to express
`2B, 1
`E,
`@, as a power of & as in a@*. First, note that the function erfe’ ./x + y is
`°
`bounded by
`Ne = = ele
`it
`erfe x = ea I e-” dy.
`
`WTERB! DECODING OF CONVOLUTIONAL CODES 467
`
`and x + p= 2ke,/N,.
`Th
`i
`= a oy tein
`
`Next, set k=d+, €=0,1,...,
`(15-26) and (15-28), we obtain
`»
`/2ke
`6, = erfc’
`,/—=
`VW,
`
`a ada te
`_
`i
`fi
`:
`€,
`»
`/2de,
`erfe’
`,/ m <exp( Wr ) exfe Var
`ightly weakened to ob
`Thus, the bound on @, in (15-27) can besli
`(15-15) and (15-29)
`*
`i
`a
`O, < Ee,P, <erfe’ ae See exp [498]
`»
`/2de,
`de,\
`=
`Sei
`0
`(15-30)
`<< erfe’ 4/ Ft exp (S$) Shee sealM
`The expression (15-30) can be rewritten using (15-14) to replace the summa-
`tion
`
`(15-29)
`
`i
`fain From
`
`(15-31)
`
`@, = erfc’
`
`where
`
`(15-34)
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 8
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 8
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`
`
`
`i
`
`10-5
`
`r
`
`10-6
`
`14
`
`16
`
`10-7
`—-8-6-4-2 0
`
`10
`
`12
`
`Lisistisistiiiil
`2
`4
`6
`8
`E,/Ny, dB
`Fig. 15-10 Simulated performance of a rate 1/2 con-
`volutional code with generators G = 11101, G = 10011.
`The performance js
`for Viterbi decoding with soft
`decision, constraint
`length 5, 25-bit paths, 8-level
`receiver quantization. The quantizer threshold spacing
`is equal to 0,50,
`
`ance at , = 10-5 for codes of constraint lengths 3, 5, and 7 [Heller and
`Jacobs, 1971, 835-848 ; Bustamante and Leong, 1972]. The codesare rate 1/2
`codes with the parity tap positions defined by G,, G, 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 = 5K path lengths.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 9
`
`XX
`K
`
`.
`
`10-1
`
`|
`a i.
`
`2=r
`
`
`E
`E
`
`25
`2
`es
`Fol
`a
`
`simulation 1 (32-bit-path, QTS = 0.5)
`Oo
`Thus by comparing(15-332) ,(15-34) for high E,/No this errorbound ——
`92,160samples}
`%
`that the coded signal is more than 10 log (2.5) = 3.98 dB better than the
`D 921,800 samplesfSimulation 2
`
`uncoded signal for the same outputerrorrate.
`.
`10°
`TTTTTyy
`Figures 15-9 through 15-11 show experimental curves of measured bit-
`error rates indicating the approximately 4 to 6-dB improvementin perform-
`© simulation 1 (32-bit-path, OTS = 0.5)
`X
`10,752 samples
`area
`uncodedquadriphase
` 10°
`1D 921,600 samples } simulation
`
`TTT VIALE THATTETT
`
`4d
`L
`{theoretical)
`
`
`
`
`aT| 10-1
`
`buditilwu
`
`eyaveragebit-errorrate 10-4
`
`
`
`10-2
`
`
`
`
`
`
`
`
`
`
`
` 10-6
`
`468 MODULATION AND CODING IN DISTORTED CHANNELS
`
`VITERBI DECODING OF CONVOLUTIONAL CODES 463
`
`uncoded quadriphase
`(theoretical)
`
`
`
`
`
`
`
`
`
`
`
`14
`
`16
`
`
`
`
`
`
`motets bi
`0-7
`ete? OD 6
`8 7 12
`E,/No, 0B
`Fig. 15-9 Simulated performanceof a rate 1/2 convo-
`lutional code with generators employing soft decision,
`G, = 111, G, = 101, constraint length 3, 18-bit paths
`and 32-bit paths. Soft decision 8-level receiver quan-
`tization is employed. Thebit errorrate is for maximum-
`likelihood decoding. The quantizer threshold spacing
`(QTS)is set equal to 0.50 o where @ is the rms noise.
`[Bustamante, 1972]
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 9
`
`

`

`VITERBI DECODING OF CONVOLUTIONAL CODES 471
`©—simulation 1 (32-bit-path, UTS = 0.5)
`x
`92,160 samples
`simulation 2
`orif the inphase and quadrature channels of a QPSKsignal have cross-talk,
`which causes correlation between errors.
`O
`921,600 samples
`Theeffects of these adjacentor nearly adjacent errors can be reduced by
`properinterleaving of the coder output bits, For example, the coder output
`bits X,, can be transmitted with no delay, and the code bits ¥,, can be trans-
`mitted with a 5K bit delay simply by adding a 5K bit shift register in the
`output of the lower mod2 adderin Fig. 15-1. This simple delay operation pre-
`vents adjacentreceived bits from affecting the same decoded databit. Higher
`orderinterleavingis also possible to prevent -
`-
`- 0,0,0,- «+ error patterns from
`affecting the same decoded data bit. (See Forney and Bower [1971], J. L.
`Ramsey [1970]). Higher orderinterleaving may require the addition of special
`syne words underto preformthe deinterleaving process.
`
`470 MODULATION AND CODING IN DISTORTED CHANNELS
`
`10°
`
`(theoretical)
` 10-2
`
` 10-3
`15-5=EFFECTS OF PHASE NOISE IN COHERENT
`DEMODULATION ON DECODER
`PERFORMANCE
`
` 10-1 uncoded quadriphase
`
`pal
`
`po
`
`
`
`
`
`
`
`
`valid if there are significant filter distortion effects, as described in Chap. 13,
`
`
`
`
`
`
`
`
`oyaveragebiterrorrate
`
` 10-4
`
`
`
`Leeda
` 10-6
`
`14
`
`16
`
`12
`
`
`Lorsttosrilitartitrstiiiitispiiiveitinas
`10-7
`—-8-6-4-2 0
`2
`4
`6
`8
`10
`E,4/No, dB
`Fig. 15-11 Simulated performance of a rate 1/2 con-
`volutional code with generators G; = 1111001, G, =
`1011011. The performanceis for Viterbi decoding, soft
`decision, rate 1/2, constraint length 7, 35-bit paths,
`8-level receiver quantization. The quantizer threshold
`spacing is set equal to 0.50 a.
`
`Code interleaving
`
`This section has assumed that the decoder input samples or bit decisions
`have independentnoise or errors from bit to bit. This assumptionmay not be
`
`Theeffect of phase noise in carrier-tracking phase-locked loops on errorrate
`has been discussed for uncoded PSK channels in Chap. 12. A coded channel
`hasa steeper curve of @, versus E,/N, and therefore is moresensitive to phase
`noise. Thus, a short momentaryincreasein the slowly varying phase-tracking
`error ¢ produces a momentary decrease in effective E,{Ny 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 smallfraction of time
`whentheeffective E/N, decreases dominates because of the steepness of the
`error rate curves.
`.
`Theeffect of slowly varying phase error ¢ on a BPSK channeldecreases
`the effective E,/N,y to (E,/N,) cos? ¢. Thus, instead of an error-probability
`functional relationship
`
`(15-35)
`0, =£(#)
`the phase noise degrades performanceto the error probability for a given ¢:
`E,
`sd)
`=F(008? 6)
`(15-36)
`Ob)
`= f(=* cos?
`15-36
`For a first-order loop the probability density of phase error ¢ for an SNR
`a in the closed-loop bandwidth is (see Chap. 12)
`exces ¢
`(15-37)
`pb) = Talker tl
`*“Slowly”refers to a small change in phase error ¢ over many (> 102) bit periods.
`
` 
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 10
`
`Petitioner Sirius XM Radio Inc. - Ex. 1007, p. 10
`
`