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22. CHAPTER 22 - AUTOMATIC-REPEAT-REQUEST STRATEGIES

As pointed out in Chapter 1, there are two categories of techniques for controlling transmission errors

in data transmission systems: the forward-error-control (FEC) scheme and the automatic-repeat-

request (A!) scheme" #n an FEC system an error-correcting code is used" $hen the recei%er detects

the presence of errors in a recei%ed %ector it attempts to determine the error locations and then corrects

the errors"

#f the e&act locations of errors are determined, the recei%ed %ector will 'e correctly decoded: if the

recei%er fails to determine the e&act locations of errors, the recei%ed %ector will 'e decoded

incorrectly, and erroneous data will 'e deli%ered to the user (or data sin)" #n an A! system a code

with good error-detecting capa'ility is used" At the recei%er, the syndrome of the recei%ed %ector is

computed" #f the syndrome is ero, the recei%ed %ector is assumed to 'e error-free and is accepted 'y

the recei%er" At the same time, the recei%er notifies the transmitter, %ia a return channel, that the

transmitted code-word has 'een successfully recei%ed"

#f the syndrome is not ero, errors are detected in the recei%ed %ector" *hen, the transmitter is

instructed, through the return channel, to retransmit the same code-word" etransmission continues

until the code-word is successfully recei%ed" $ith this system, erroneous data are deli%ered to the data

sin only if the recei%er fails to detect the presence of errors" #f a proper linear code is used, the

pro'a'ility of an undetected error can 'e made %ery small"

$e ha%e de%oted many chapters to %arious types of error-correcting codes and decoding methods for

FEC" #n this chapter we present %arious types of A! schemes and the com'inations of A! and

FEC, called hy'rid-A! schemes" +aor references on A! are listed at the end of this chapter"

22.1. 22.1 Basic ARQ Schemes

*here are three 'asic types of A! schemes: the stop-and-wait A!, the go-'ac- A!, and the

selecti%e-repeat A!" #n a stop-and-wait A! data transmission system, the transmitter sends a code-

word to the recei%er and waits for an acnowledgment from the recei%er, as shown in Figure .."1" A

positi%e acnowledgment (AC/) from the recei%er signals that the code-word has 'een successfully

recei%ed (i"e", no errors were detected), and the transmitter sends the ne&t code-word" A negati%e

acnowledgement (A/) from the recei%er indicates that the recei%ed %ector has 'een detected in

error: the transmitter resends the code-word" etransmissions continue until an AC/ is recei%ed 'y the

transmitter" *he stop-and-wait A! scheme is simple and is used in many data communication

systems0 howe%er, this scheme is inherently inefficient 'ecause of the idle time spent waiting for an

acnowledgment for each transmitted code-word" nless the code length n is e&tremely long, the

fraction of idle time can 'e large0 howe%er, using a %ery long

F#2E .."1:3top-and-wait A!"

'loc length does not pro%ide a solution, since the pro'a'ility that a 'loc contains errors increases

rapidly with the 'loc length" 4ence, idle time is reduced at the e&pense of increased frequency ofretransmissions for each code-word" +oreo%er, a long 'loc length may 'e impractical in many

applications 'ecause of restrictions imposed 'y the data"

#na go-'ac- A! system, code-words are transmitted continuously" *he transmitter does not wait

for an acnowledgment after sending a code-word0 as soon as it has completed sending ore, it 'egins

sending the ne&t code-word, as shown in Figure .."." *he acnowledgment for a code-word arri%es

after a round-trip delay, which is defined as the time inter%al 'etween the transmission of a code-word

and the receipt of an acnowledgment for that code-word"

5uring this inter%al, -1 other code-words ha%e also 'een transmitted" $hen a A/ is recei%ed, the

transmitter 'acs up to the code-word that was negati%ely acnowledged and resents that code-word

and 6 1 succeeding code-words that were transmitted during the round-trip delay (i"e", thetransmitter pulls 'ac and resends code-words)" 7f course, a 'uffer must 'e pro%ided at the

transmitter for these code-words"

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At the recei%er, the 6 1 recei%ed %ectors following an erroneously recei%ed %ector (a recei%ed %ector

detected in error) are discarded regardless of whether they are error-free or not" *herefore, the recei%er

needs to store only one recei%ed %ector at a time" 8ecause of the continuous transmission and

retransmission of code-words, the go-'ac- A! scheme is more effecti%e than the stop-and-wait

A#, and it can 'e implemented at moderate cost" *he go-'ac- A999! scheme 'ecomes

F#2E ..".:27-'ac- A! with ;"

F#2E .."(E) denote the pro'a'ility of an error

e%ent" Clearly, for an A! system to 'e relia'le, >(E) should 'e %ery small" *herefore, the relia'ility

of an A! system is measured 'y its error pro'a'ility >(E)" 3uppose that an (n, ) linear code C is

used for error detection in an A! system" $e define the following pro'a'ilities:

>c pro'a'ility that a recei%ed %ector contains no error0

>d pro'a'ility that a recei%ed %ector contains a detecta'le error pattern0

>e pro'a'ility that a recei%ed %ector contains an undetecta'le error pattern"*hese pro'a'ilities add to 1 (i"e", >c" ? >d ? >e 1)" *he pro'a'ility p depends on the channel error

statistics, the pro'a'ilities >(@and >e depend on 'oth the channel error statistics and the choice of the

(n, ) error-detecting code C" A recei%ed %ector will 'e accepted 'y the recei%er only if it either

contains no error or contains an undetecta'le error pattern" *herefore, the pro'a'ility >(E) that the

recei%er commits an error is gi%en 'y

*he pro'a'ility >e can 'e made %ery small relati%e to >c 'y choosing the code C properly (e"g", a long

4amming code or a long dou'le-error-correcting primiti%e

8C4 code)" Consequently, the error pro'a'ility > (E) can 'e made %ery small" For a 83C with

transition pro'a'ility p, we ha%e

#n 3ection

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*he factor 5r@n may 'e interpreted as the num'er of code-words that could 'e transmitted outing the

idle time of the transmitter" $e see that the throughput can ne%er achie%e the ma&imum %alue @1;

e%en if the channel is noiseless (> 1)" For data transmission systems in which the data transmission

rate is low and the round-trip delay is short,Dcan 'e made relati%ely small compared with the code

length n" #n this case, the stop-and-wait A! pro%ides satisfactory throughput performance: howe%er,

its throughput performance 'ecomes unaccepta'le for systems in which the data transmission rate is

high and the round-trip delay is large, such as satellite communication systems"

From the analysis of throughput performance ust gi%en, we see that the selecti%e-repeat A! is the

most efficient scheme, whereas the stop-and-wait A! scheme is the least efficient one *he

throughput of the selecti%e-repeat A! does not depend on the round-trip delay of the system,

whereas the throughputs of the other two A! schemes do" #n communication systems for which the

round-trip delay is large and data rate is high, the parameter for the go-'ac- A!

F#2E .."J:*hroughput efficiencies: the ideal selecti%e-repeat A! with infinite recei%er 'uffer,

the selecti%e-repeat A! with a recei%er 'uffer of sie 1.I (the solid triangles represent

simulation results), and the go-'ac- A!"

and the parameter 5r@n for the stop-and-wait A! may 'ecome %ery large" #n this case the throughput

for the go-'ac-NA! drops rapidly as the channel error rate increases, whereas the throughput of the

stop-and-wait A! 'ecomes unaccepta'le"

*he high throughput of the selecti%e-repeat A! is achie%ed at the e&pense of e&tensi%e 'uffering at

the recei%er and more comple& logic at 'oth transmitter and recei%er" *heoretically, infinite 'uffering

is needed to achie%e the efficiency (@n)>" #f a finite 'uffer is used at the recei%er, the 'uffer may

o%erflow, which reduces the throughput of the system0 howe%er, if sufficient 'uffer (say, a 'uffer that

is capa'le of storing code-words) is used at the recei%er, e%en with a reduction in throughput, the

selecti%e-repeat A! still significantly outperforms the other two A! schemes in systems for which

data transmission rate is high and round-trip delay is large DJ, H, 1B-1." *his situation is discussed in

the ne&t section" Figures .."J to .." show the throughput efficiencies of the selecti%e-repeat A! and

the go-'ac- A i! for %arious code lengths and round-trip delays" *he channel is a 83C" 3e%eral

%ariations of the go-'ac- A! scheme ha%e 'een proposed D

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22.2.2. 22.2.2 RECIVER'S OPERATION AND ERROR RECOVERY PROCEDURE

ormally, the transmitter sends code-words continuously to the recei%er" *he recei%er checs the

syndrome of each incoming recei%ed %ector and sends an AC/ to the transmitter for each successfully

recei%ed %ector" $hen the channel is quiet the transmission proceeds smoothly, error-free code-words

are deli%ered to the user in consecuti%e order, and the recei%er 'uffer is empty" *he recei%er is said to

'e in the normal state when the recei%er 'uffer is empty and no space is reser%ed for any A/9ed

'loc"

#f the recei%ed %ector is detected in error, or a recei%ed %ector with an out-of- ordersequence num'er

is detected while the recei%er is in the normal state, the recei%er sends a A/ to the transmitter and

enters the 'loced state" #n the 'loced state the recei%er proceeds to chec the syndromes of the

incoming recei%ed %ectors, it stores those %ectors that ha%e ero syndrome at the proper locations in

the recei%er 'uffer, and reser%es proper locations for the %ectors whose syndromes are not ero until

the recei%er 'uffer is full" NB%ectors are deli%ered to the user" $hen the retransmitted code-words

arri%e after a round-trip delay, their syndromes are checed again" *he %ectors that are successfully

recei%ed (ero syndrome) are stored at the reser%ed locations in the recei%er 'uffer" 7nce the earliest

A/9ed %ector is successfully recei%ed, the recei%er outputs that %ector and all the su'sequent

consecuti%e ero-syndrome %ectors (held in the recei%er 'uffer) until the ne&t A/9ed %ector is

encountered (see Figure ..";)" #f all the %ectors in the recei%er 'uffer are released and all the reser%ed

locations are freed after the earliest A/9ed %ector has 'een successfully recei%ed, the recei%er returnsto the normal state"

F#2E .."I:*ransmission and retransmission procedure"

#f not all the %ectors in the recei%er 'uffer can 'e released to the user, or if there are still reser%ed

locations in the recei%er 'uffer for the A/9ed %ectors, the recei%er continues its error reco%ery

process until all the A/9ed %ectors are successfully recei%ed"

e&t, we descri'e the detailed operation of the recei%er in 'oth the normal and the 'loced state"

22.2.3. 22.2.3 NORMAL-STATE OPERATION

$hen a %ector with sequence num'er q is recei%ed, the recei%er checs its syndrome and computes its

forward inde& (F$L), denoted 'y f, with respect to the last accepted and deli%ered %ector" =et go 'e

the sequence num'er c)f the last accepted and deli%ered %ector" *he forward inde& f of the current

recei%ed %ector is defined

as the remainder resulting from di%iding q - go 'y

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*he flowchart shown in Figure .."K details the recei%er operations in the normal state"

22.2.4. 22.2.4 BLOCKED-STATE OPERATION

*he recei%er enters the 'loced state when either a recei%ed %ector is detected in error or a recei%ed

%ector with an out-of-order sequence num'er is detected" $hen the recei%er enters the 'loced state,

each su'sequent ero-syndrome recei%ed %ector is held in the recei%er 'uffer at a proper location, and

a space in the recei%er 'uffer is reser%ed for each erroneously recei%ed or lost %ector" $hen the 'uffer

is full, all su'sequently recei%ed new %ectors are reected and the 'uffer o%erflows"

=et %i 'e the earliest %ector that has not 'een successi%ely recei%ed" $hen a retransmitted copy of %i is

successfully recei%ed, the recei%er releases wi and the su'sequent ero-syndrome %ectors held in the

recei%er 'uffer (see Figure ..";)" 3uppose that the recei%er can release = ?1 consecuti%e %ectors

including %i (i"e", %i, w ?1," N" %i?=, where B = )"

8ecause the retransmission is selecti%e, and since %i?i to %i?= were successfully recei%ed 'efore %0 the

%ectors following the retransmitted copy of %i are new %ectors $i?, ? ?1, O, $ ??=-1" $hen

these = new %ectors arri%e their syndromes are computed" *he %ectors that ha%e ero syndrome are

then temporarily stored in the recei%er 'uffer until they are ready to 'e released, and proper locations

in the recei%er 'uffer are reser%ed for those %ectors that ha%e nonero syndromes"

F#2E .."K:ormal-state operation of the recei%er"

#f the retransmitted copy of %i is not successfully recei%ed, the ero-syndrome %ectors held in the

recei%er 'uffer cannot 'e released to the user" *herefore, there are no places in the recei%er 'uffer to

store the new %ectors, % ?, %i-E?1, ON (since the 'uffer sie is )" #n this e%ent, the 'uffer

o%erflows and the new %ectors, %iP, i??, ON will 'e reected no matter whether they are

successfully recei%ed or not"

=et go 'e the sequence num'er of the earliest %ector that has not 'een successfully recei%ed" #n the

'loced state, when a %ector with sequence num'er q is recei%ed its forward inde& with respect to the

earliest unsuccessfully recei%ed %ector, denoted 'y #f, is computed" *his forward inde& l is defined as

the remainder resulting from di%iding q - qo 'y

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*hese are the pro'a'ilities of success of the second, third, and fourth retransmissions of a code-word,

respecti%ely" $e define

*hen, the throughput efficiency of the selecti%e A! with a recei%er 'uffer of sie is lower 'ounded

as follows:

(Assuming that the num'er of 'its used for sequence num'ers is small compared with the code length

n, its effect on the throughput is ignored")

For %arious n and , the lower 'ound on the throughput gi%en 'y (.."11) is compared with the

throughput of the ideal selecti%e-repeat A! with infinite recei%er 'uffer and the throughput of the go-'ac- A!, as shown in Figures .."J to .."" $e see that the selecti%e-repeat A! with a recei%er

'uffer of sie is less efficient than the ideal selecti%e-repeat A! 'ecause of the reduction in 'uffer

sie at the recei%er0 howe%er, it significantly outperforms the go-'ac- A!, particularly for

communication systems in which the round-trip delay is large and data rate is high" #f the round-trip

delay is taen to 'e ;BB ms (suita'le for satellite channels), the data transmission rate corresponding to

11 H.J and 1.I (Figure .."J) is 1BB 'ps, the data rate corresponding to 11 .B.J and H1.

(Figure .."H) is 1"HJ +'ps, and the data rate corresponding to n 1B.J (Figure ..") is also 1"HJ

+'ps"

*he selecti%e-repeat A! descri'ed in this section can 'e e&tended for any recei%er 'uffer of sie

greater than , say .,

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%ector to determine whether an 3 or 28 type of retransmission is required" #f more than one %ector

that has 'een transmitted twice unsuccessfully is stored in the transmitter retransmission 'uffer, these

%ectors must 'e queued for successi%e independent go-'ac--type retransmissions" *he earliest

dou'le-A/9ed %ector" say ri, is retransmitted in the 28 mode, followed 'y 6 1 %ectors that were

pre%iously transmitted after wi and su'sequently discarded 'y the recei%er" *his action is repeated

until an AC/ for %1 is recei%ed" *hen, the same procedure is repeated for any su'sequent dou'le-

A/9ed %ectors in the retransmission 'uffer" *he throughput efficiency of the 3 28 A! has 'een

analyed D1." For % 1, its throughput efficiency is

where > is the pro'a'ility that a %ector will 'e successfully recei%ed" For % M 1,

For % B, the 3 ? 28 scheme 'ecomes the con%entional go-'ac- A!, and

Figure .."1< demonstrates the throughput performance of the 3 ? 2E A! for % 1 and ."

*hroughput efficiencies of other A! schemes are included

F#2E .."1

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@n) regardless of the channel error rate0 howe%er, FEC systems ha%e two draw'acs: First, when

a recei%ed %ector is detected in error it must 'e decoded and the decoded message must 'e deli%ered to

the user regardless of whether it is correct or incorrect"

8ecause the pro'a'ility of a decoding error is much greater than the pro'a'ility of an undetected error,

it is hard to achie%e high system relia'ility with FEC" 3econd, to o'tain high system relia'ility, a long

powerful code must 'e used and a large collection of error patterns must 'e corrected" *his maes

decoding hard to implement and e&pensi%e" For these reasons A! is often preferred o%er FEC for

error control in data communication systems, such as pacet-switching data networs and computer

communication networs0 howe%er, in communication (or data storage) systems where return channelsare not a%aila'le or retransmission is not possi'le for some reason, FEC is the only choice"

*he draw'acs in 'oth A! and FEC can 'e o%ercome if two error control schemes are properly

com'ined" 3uch a com'ination of the two 'asic control schemes is referred to as a hy'rid A! D., .I"

A hy'rid system consists of an FEC su'system contained in an A! system" *he function of the FEC

su'system is to reduce the frequency of retransmission 'y correcting the error patterns that occur most

frequently" *his increases the system throughout"

$hen a less frequent error pattern occurs and is detected, the recei%er requests a retransmission rather

than passing the unrelia'ly decoded message to the user" *his increases the system relia'ility" As a

result, a proper com'ination of FEC and A! pro%ides higher relia'ility than an FEC system alone

and a higher throughput than the system with A! only" Furthermore, since the decoder is designed to

correct a small collection of error patterns, it can 'e simple" *he FEC scheme can 'e incorporated with

any of the three 'asic A! schemes"

A straightforward hy'rid A! scheme is to use a code, say, an (n, ) linear code that is designed to

simultaneously correct and detect errors" $hen a recei%ed %ector is detected in error the recei%er first

attempts to locate and correct the errors" #f the num'er of errors (or the length of an error 'urst) is

within the designed error-correcting capa'ility of the code, the errors are corrected, and the decoded

message is passed to the user or sa%ed in a 'uffer until it is ready to 'e deli%ered"

#f an uncorrecta'le error pattern is detected, the recei%er reects the recei%ed %ector and requests a

retransmission" $hen the retransmitted %ector is recei%ed the recei%er again attempts to correct the

errors (if any)" #f the decoding is not successful, the recei%er again reects the recei%ed %ector and ass

for another retransmission" *his error-correction and retransmission process continues until the %ector

is successfully recei%ed or decoded" For e&ample, one may use the (1B., 'e the pro'a'ility that a recei%ed %ector will 'e decoded successfully" *his pro'a'ility dependson the designed error-correcting capa'ility of the code 'eing used" eplacing > with >r in (.."J),

(.."H), (.."), and (.."11), we o'tain the throughput efficiencies for the type-# hy'rid selecti%e-repeat

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A! with infinite recei%er 'uffer, the type-# hy'rid go-'ac- A!, the type-# hy'rid stop-and-wait

A!, and the type-# hy'rid selecti%e-repeat A! with recei%er 'uffer of sie , respecti%ely"

*he second type (or type ##) of hy'rid A! scheme is 'ased on the concept that the parity-chec digits

for error correction are sent to the recei%er only when they are needed D 'ased on Co" #f the syndrome is ero, nu is assumed to 'e error-

free and is accepted 'y the recei%er" #f the syndrome is not ero, errors are detected in >" *he erroneous

message an is then sa%ed in the recei%er 'uffer, and a A/ is sent to the transmitter" 7n recei%ing this

A/ the transmitter encodes the k-'it parity 'loc quip into a code-word, ni (f Dq (u), q (u)) of n

'its 'ased on Co, where fDcy 74 denotes the a - parity-chec digits for q (u)"

*his %ector (fDq (u), q(u)) is then transmitted (here the retransmission is a parity %ector) =et

(Dq$, cS (u)) denote the recei%ed %ector corresponding to aT (f Dq(u), (au))" $hen ;@T is recei%ed

the syndrome of > is computed 'ased on Co" #f the syndrome is ero, ii(e) is assumed to 'e error-free,

and the message a is reco%ered from J 'y in%ersion"

#f the syndrome is not ero, i) (a) and the erroneous message an (stored in the recei%er 'uffer) togetherare used for error correction 'ased on the half-rate code C1" #f the errors in ((o), H) form a correcta'le

error pattern, they are corrected" *he decoded message a is then accepted 'y the recei%er" #f the errors

in (J(u), H) form a detecta'le 'ut not a correcta'le error pattern, H is discarded and the erroneous parity

'loc J(au) is stored in the recei%er 'uffer0 also, a A/ is sent to the transmitter"

7n recei%ing the second A/ for the code-word a (f (u), an), the transmitter resends a (f (u), a)"

$hen > (f (11), H) is recei%ed the syndrome ofd is again computed 'ased on Co" #f the syndrome is

ero, H is assumed to 'e error-free and is accepted 'y the recei%er0 the erroneous parity 'loc J- (u) is

then discarded" #f the syndrome is not ero, an and the erroneous parity 'loc J (au) (stored in the

recei%er 'uffer) together are used for error correction 'ased on Ct"

#f the errors in (J(a), H) are corrected, the decoded message a is then accepted 'y the recei%er and anAC/ is sent to the transmitter0 howe%er, if the errors in (i") (H), are detecta'le 'ut not correcta'le, J (u)

is discarded, is stored in the recei%er 'uffer, and a A/ is sent to the transmitter" *he ne&t

retransmission is the parity %ector %T Dci 7, (an))" *herefore, the retransmissions are alternating

repetitions of the parity code-word (f Dq(u), q(a)) and the information code-word a (f (a), a)" *he

recei%er stores the recei%ed message a and the recei%ed parity 'loc J (aa) alternately" *he

retransmissions continue until an is finally reco%ered"

*he most important feature of the type ##-hy'rid A B is the parity retransmission for error correction

'ased on a half-rate in%erti'le code C1" *his parity-retransmission strategy can 'e incorporated with

any of the three 'asic types of

A!" #t is particularly effecti%e when it is used in conunction with the selecti%e-repeat A!" 8ecauseof the in%erti'le property of C1, the message H can 'e reconstructed uniquely from the parity 'loc

q(an) 'y in%ersion" 4ence, the parity 'loc (ail) contains the same amount of information as the

message" As a result, the o%erhead per transmission or retransmission is simply the num'er of parity-

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chec digits, a - , needed for error detection 'ased on the (a, ) code Co, which is required 'y any

A! scheme"

*herefore, when the channel is quiet or the channel error rate is low, the type-## hy'rid A! has the

same throughput efficiency as its corresponding A! scheme" $hen the channel error rate is high, the

error correction pro%ided 'y the half-rate code C1maintains the throughput high" #n 3ection .."H we

present a type-4 hy'rid A! that incorporates the parity-retransmission strategy with the selecti%e-

repeat A! with finite recei%er 'uffer" $e will show that for

a 83C, the throughput efficiency of the ideal selecti%e-repeat A! with infinite recei%er 'uffer can 'eachie%ed 'y a half-rate in%erti'le code C# with a %ery small designed error-correcting capa'ility, say t

< to H" $ith a larger t, the type-## hy'rid selecti%e-repeat A! with finite recei%er 'uffer is far

superior to the ideal selecti%e-repeat A! with infinite recei%er 'uffer"

*he decoding comple&ity for a type-## hy'rid A! is only slightly greater than that of a corresponding

type-# hy'rid A! with the same designed error-correcting capa'ility" *he e&tra circuits needed for a

type-#* hy'rid A! scheme are an in%ersion circuit 'ased on C1, which is simply a linear sequential

circuit and an error-detection circuit 'ased on Co"

*he disad%antage of the type-# hy'rid A! is that the o%erhead due to the e&tra parity-chec digits for

error correction must 'e included in each transmission or retransmission regardless of the channel error

rate" $hen the channel is quiet, this inclusion represents a waste0 howe%er, the type-## hy'rid A!remo%es this disad%antage, 'ecause is an adapti%e scheme" *his scheme is particularly attracti%e for

high-speed data communication systems for which round-trip delay is large and error rate is non-

stationary, such as satellite communication systems" #t is also attracti%e for wireless communications"

Qarious hy'rid A! schemes and their analyses can 'e found in D.

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where a(L) and '(L) are the quotient and the remainder, respecti%ely" *he code-word for an(L) is then

and '(L) is the parity-chec portion" *he following theorem pro%es the in%erti'le property of C1"

*4E7E+ .."1 o two code-words in a half-rate shortened cyclic code C1ha%e the same parity-

chec digits"

>roof =et gni (L) and 111.(L) 'e two distinct messages" 5i%iding L-si (L) and L- aii.(L) 'y the

generator polynomial g(L), respecti%ely, we o'tain

*hen, the code-words for a t (L) and 111.(L) arerespecti%ely" 3uppose that

Adding (.."1H) and (.."1), we o'tain

8ecause g(L) and Ll are relati%ely prime, -al (L) ? u.(L) must 'e di%isi'le 'y g(L): howe%er, this is

impossi'le, since *il t (L) ? la.(L) B and its degree is less than a - 'ut the degree of g(L) is - "

*herefore, #'i (L) r 1 .(L)"

8ecause the remainder >o(L) resulting from di%iding L-ari(L) 'y g(L) is unique, *heorem .."1

implies that there is one-to-one correspondence 'etween a message sr(L) and its parity chec '(L)"

*herefore, nowing '(L), we can uniquely determine u(L)"

e&t, we show how to reco%er the message an(L) from the parity H(L)" First, we multiply 'oth sides of(.."1J) 'y L :

earranging (.."1;), we o'tain

8ecause g(L) is a factor of L ? 1, we can rewrite (.1"1I) as

where h(L) (L ? 1)@g(L)" From (.1"1K) we see that the message u(L) is simply the remainder

resulting from di%iding '(L)L 'y the generator polynomial g(L)" *his operation can 'e performed 'y

using a di%ision circuit with feed'ac connection 'ased on g(L)" *he process of finding the message

u(L) from its parity-chec digits '(L) is called an in%ersion process"

A faster circuit for in%erting '(L) can 'e implemented as follows" 5i%iding L 'y g(L), we ha%e

where the remainder

+ultiplying 'oth sides of (..".B) 'y '(L) and using the equality of (.."1K), we o'tain

*he preceding e&pression suggests that we can o'tain the message u(L) 'y multiplying the parity '(L)

'yp(X) and di%iding the product '(L)p(X) 'y g(L)" $e can do this with the circuit shown in Figure

.."1H"

F#2E .."1H:#n%ersion circuit"

For e&ample, consider the (1B.

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For con%enience, we call e +u), e) the information code-word of u, and ("f Dq (>7, q (all)) the parity

code-word of u"8"" $e will use (rt (u), H) and (f Dc (e), (u)) to denote the recei%ed %ectors

corresponding to % and %T, respecti%ely" #n transmission or retransmission the information code-word %

and the parity code-word eT of the message as ha%e the same sequence num'er"

$hen an information code-word e is ready for transmission, it is num'ered and stored in the input

queue of the transmitter" After its transmission, % and its corresponding parity code-word %T are sa%ed

in the retransmission 'uffer until % is positi%ely acnowledged" $hen an AC/ is recei%ed after a

round-trip delay, 'oth % and %T are released" $hen a A/, (or no acnowledgement) is recei%ed after

a round-trip delay, %999 is sent to the recei%er for error correction (if necessary)"

After another round-trip delay, if an AC/ is recei%ed, 'oth % and %T are released0 otherwise, % is

retransmitted" *he transmitter resends % and %T alternately until % is positi%ely acnowledged, as

shown in Figure .."1"

22..1. 22..1 TRANSMISSION AND RETRANSMISSSION PROCEDURE

$hen the transmitter is sending a code-word, information, or parity it also computes the forward inde&

f* of the code-word in the retransmission 'uffer that is to 'ecome a time-out %ector" 8ased on this

forward inde& f*, the transmitter decides whether the ne&t information code-word in the input queue is

to 'e transmitted or retransmission is to 'e initiated" *he decision rule is as follows:

1" #f the current time-out code-word is positi%ely acnowledged and (or if there is no current time-

out code-word), the first information code-word in the input queue is to 'e transmitted"

." #f the current time-out code-word, say %1, is either negati%ely acnowledged or unacnowledged

and fr , a retransmission for ei is initiated" *he

F#2E .."1:*ype ## selecti%e-repeat hy'rid A! with recei%er 'uffer of sieN ;"

retransmission for %i is the parity code-word %99 if %i was pre%iously transmitted, and is a repetition of

%"0 if % was pre%iously transmitted" #f the current A/9ed (or unAC/9ed) time-out code-word is the

earliest code-word in the retransmission 'uffer that has not 'een positi%ely acnowledged (f* B), all

the information code-words in the retransmission 'uffer with forward indices are mo%ed 'ac to the

input queue for retransmission at a later time (these code-words are reected 'y the recei%er owing tothe recei%er 'uffer o%erflow)"

the first information code-word in the input queue is the ne&t to 'e transmitted"

*he transmission and retransmission operations of the type-4 hy'rid selecti%e-repeat A! with

recei%er 'uffer of sie are detailed 'y the flowchart in Figure .."1;"

22..2. 22..2 RECEIVER!S OPERATION IN THE NORMALL STATE

#n the normal state the recei%er recei%es information code-words, and the recei%er 'uffer is empty"

$hen an information code-word is recei%ed the recei%er computes its syndrome 'ased on Co and its

forward inde& f with respect to the last accepted and deli%ered information code-word" #f the

syndrome is ero and f 1, the recei%ed information code-word is accepted and deli%ered to the user"#f f M , the recei%ed %ector is regarded as an information code-word that was pre%iously accepted

and deli%ered0 it is then ignored and an AC/ is sent to the transmitter"

#f either the recei%ed information code-word is detected in error, or 1 f (i"e", its sequence

num'er is out of order)U the recei%er enters the 'loced state" *he recei%er9s operation in the normal

state is detailed 'y the flowchart in Figure .."1I"

22..3. 22..3 RECEIVER!S OPERATION IN THE BLOCK STATE

#f the recei%er enters the 'loced state owing to the detection of errors in a recei%ed information code-

word y f (o), with f 1, the erroneous message is then stored at the first location of the recei%er

'uffer and a A/ is sent to the transmitter" *he recei%er then checs the su'sequent recei%edinformation code-words and stores them in the recei%er 'uffer at the proper locations according to their

sequence num'ers until the 'uffer is full0 also, an appropriate acnowledgment is sent to the

transmitter for each recei%ed %ector"

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#f the recei%er enters the 'loced state with f M 1, then f 6 1 information code-words 'etween the

last deli%ered information code-word and the current recei%ed information code-word are lost" *he

recei%er then reser%es the first f 6 1 locations of the recei%er 'uffer for the lost %ectors and sa%es the

current recei%ed %ector (only the message part) at the (f)th location of the 'uffer" *he su'sequent

recei%ed information code-words are stored in the remaining locations of the recei%er 'uffer"

$hen the first retransmitted parity code-word, say ;;,@S (f Dq(ur), iyani)), is recei%ed it is used to

reco%er the earliest erroneously recei%ed message (or the earliest lost message) an l" #f the syndrome of

Q: is ero, uni is reco%ered 'y taing the in%ersion of i.- (tui )" #f the syndrome of % is not ero, i-)"

(ui) and Enj(stored in the 'uffer) together are used for error correction 'ased on the half-rate-error-correcting code Ci" #f the

F#2E .."1;:*ransmission and retransmission procedure of the type-## selecti%e-repeat hy'rid with

recei%er 'uffer of sie "

errors are correcta'le, u is reco%ered" $hen u @is reco%ered, the recei%er releases u and the

su'sequent error-free (or ero-syndrome) messages in consecuti%e order, say u , u1?1, O, ui?= with B

= " 8ecause the retransmission is selecti%e, and since uti?i to u ?= are successfully reco%ered,

the %ectors following ii"T@are new information code-words i?, ??1, %-B1?=-1N $hen these new

information

F#2E .."1I:ormal-state operation of the recei%er"

code-words arri%e their syndromes are checed and are temporarily stored in the recei%er 'uffer until

they are ready to 'e released to the user or to 'e corrected" #f (a i) fails to reco%er the recei%er discards

ijin the recei%er 'uffer0 also, a A/ is sent to the transmitter" 8ecause error-free messages cannot 'e

released, there are no places in the recei%er 'uffer to store the new recei%ed information code-words

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22..4. 22..4 THROUGHPUT EFFICIENCY

*o analye the throughput efficiency of the hy'rid A! descri'ed is %ery difficult0 howe%er, if we

assume that the channel is a 83C and the return channel is noiseless, we can o'tain a lower 'ound on

the throughput" Again, let > 'e the pro'a'ility that a code-word, information or parity, will 'e recei%ed

successfully (i"e", the syndrome 'ased on Co is ero)"

3uppose that C1 is capa'le of correcting any com'ination of t or fewer errors and simultaneously

detecting any com'ination d(d M t) or fewer errors" =et !i 'e the conditional pro'a'ility that a message

, 'e the pro'a'ility that Co will fail to detect the presence of errors" =et

which is the pro'a'ility that the num'er of errors in (J(u), will e&ceed the

designed error-detecting capa'ility d of C1" *hen, the pro'a'ility that the recei%er

will commit a decoding error is 'ounded as follows D

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order as >e or e%en smaller than >e" As a result, the pro'a'ility of a decoding error,

> (E), is the same order as the error pro'a'ility of a pure A!"

22.7.

22.8.

22.9. 22. *#+rid ARQ S#stems Usin -onol&tional -odes

3o far we ha%e considered only hy'rid A! systems that employ 'loc codes for 'oth error detection

and correction" 4y'rid A! systems using con%olutional codes

for error correction can also 'e de%ised" 7ne such system is a type-#t hy'rid A! system proposed in

DJ1"

Consider a rate-1@. (., 1, Xit) con%olutional code C1of memory order #n" =et g(1) (L) and g(.((L) 'e

the two generator polynomials of this code" $hen an information sequence 11"(L) of 'its is ready for

transmission it is first encoded into a code-word fm(L) in an (a, ) error-detecting code Co" *his code-

word L) is then encoded into two sequences,

and

'ased on the con%olutional code C1, each a ? #n 'its long" *he .(n ? @;1)-'it sequence %(L) o'tained

'y interlea%ing %(1) (L) and %(.) (L) is a code sequence in C#" *he sequence %(1) (L) is then

transmitted, and the sequence %(.) (L) is stored in the transmitter 'uffer for possi'le retransmission at

a later time =et 9ir (1((L) denote the recei%ed sequence corresponding to %(1) (L)" $hen Q1) (L) is

recei%ed it is di%ided 'y the generator polynomial g(1) (L)"

=et a(1) (L) and ##-K(1) (L) 'e the quotient and remainder of the di%ision, respecti%ely" #f ili)(1) (L)

B, 1(1) (L) is then checed 'ased on the error-detecting code Co" #f the syndrome s(1) (L) of)1(1) (L)

is ero, 1(1) (L) is assumed to 'e error-free and identical to the transmitted code-word u"n(L)" *hen,

the information sequence 11(L) is reco%ered from *7 (L) and accepted 'y the recei%er"

#f H"1)(11(L) B, or B) (L) B B, errors are detected in -;f@(1) (L)" *hen,, 11@(1) (L) is sa%ed in therecei%er 'uffer for reprocessing at a later time" At the same time, the recei%er requests a retransmission

'y sending a A/ to the transmitter" 7n recei%ing this A/, the transmitter sends the sequence %i f

.1 (L)" *his forms the first retransmission for ##(L)"

=et Q.) (L) 'e the recei%ed sequence corresponding to Y,,,(.) (L)" $hen Q.)(L) is recei%ed it is

di%ided 'y the generator polynomial .(..) (1)" =et u(.) (L) and l'(.) (L) 'e the quotient and

remainder of the di%ision, respecti%ely" #f h(.) (L) B, the syndrome H(.) (L) of 1(.) (L) is computed

'ased on Co" #f s(.) (L) B, then 1(.) (L) is assumed to 'e error-free and identical to lit(L)" #n this

case, the recei%er reco%ers 11(L) from 1(.) (L) and discards 1) (L)" #f ro(.) (L) B B, or H(.)(L) B, then

i, (.) (L) together with y(1) (&) (which is stored in the recei%er 'uffer) are decoded 'ased on the

con%olutional code C1using a trellis-'ased decoder, say a Qiter'i decoder"

=et inT(L) 'e the decoded sequence" *hen, anT(L) is checed 'ased on the error-detecting code Co" #f

the syndrome sT(L) of 111T(L) is ero, the recei%er accepts 1111((L)" #f the syndrome sT (L) is not

ero, then Q il (L) is discarded, and C) codes DJJ *his family of C>C codes is

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o'tained 'y puncturing a single rate-1@n (n, 1, in) con%olutional code C with rates ranging from *A* ?

1) to 1@n, where * is the punctured period"

All these punctured codes can 'e decoded with the decoder for the mother code C" A type-## hy'rid

A! system using a family of C>C codes has 'een proposed 'y 4agenauer DJ" *he principle of

this system is to transmit additional code 'its corresponding to the ne&t-lower-rate C>C code in the

family to enhance the error-correcting capa'ility of the code at the recei%ing end" *hese additional

code 'its are actually certain code 'its of the mother code that were punctured in the earlier

transmission or retransmissions" $hen these additional code 'its are recei%ed they, together with the

un-decoded code 'its stored in the recei%er 'uffer, form a lower-rate C>C code and are then decoded"

After decoding, error checing is performed 'ased on an error-detecting 'loc code Co" #f no error is

detected, the decoded sequence is assumed to 'e error-free0 otherwise, a retransmission is requested"

etransmission continues until the mother code is reached, completing a full retransmission cycle" #f

the decoding is still not successful at the end of a full retransmission cycle, then the ne&t

retransmission cycle for the same information sequence 'egins, starting with the highest-rate C>C

code0 or the decoded sequence is simply accepted at the end of decoding the mother code C" *his, of

course, degrades the relia'ility"

7ther hy'rid A! systems using con%olutional codes can 'e found in D.3#3 signal set" *he outer code C[ is a shortened (nl, 1(;) 3 code o%er GF(.m)

with generator polynomial g.(L)" *his outer code has a minimum distance of d. n. - . ? 1 and is

used to correct 1. or fewer sym'ol errors and simultaneously detect p. (p. M 1.) or fewer sym'ol

errors, where t. ? p. ? 1 d;" *he dimension 1 of the 8C+ inner code C1and the length 11. of the

3 outer code C. are chosen to satisfy the following

conditions:

with A M 1 and q M ."

*he third code, C1, is designed for parity retransmission" #t is a rate-1@. (.(B. - [), 1;. - .) shortened

3 code o'tained from the outer 3 code C. 'y deleting the first .@. - B. information sym'ols fromeach code-word in C." 8oth C. and Cr are generated 'y I.(L) and can 'e encoded and decoded 'y the

same circuits" C," is an in%erti'le code" =et a 'e an information sequence of n. - . sym'ols" =et 1(a)

denoted the sequence of n. - . parity sym'ols formed 'ased on a and C1"

*hen, (a, i"(a)) is a code-word in C1, and there is a one-to-one correspondence 'etween an and ,-

(an)" /nowing ,- (a), we can reco%er a 'y an in%ersion operation" #n fact, the in%erti'le property

implies that (,"(a), a) is also a code-word in Cr Dsee >ro'lem .."K"

=et %(L) %o %i"" %U,Ri L.-1 'e a code-word in C." 8ecause B. K (n. - .), we can di%ide %(L) into

q equal su'sections, % (L), %[(L)," wq (L), with each consisting of n. - . sym'ols, such that

whereFor 1 i q" let ,"D%i(L) 'e the parity polynomial o'tained 'y encoding the ith section Q0 (L) 'ased

on Cr" ,"D%i (L) is simply the remainder o'tained from di%iding L.-. %i (L) 'y I.(L)" *herefore,

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is a code-word in Cr" $e form the following polynomial:

*hen, D%(L) is also a code-word in C. Dsee >ro'lem .."1B" *herefore, %(L) and D%(L) form a

code-word pair in C., and one can 'e o'tained from the other 'y a unique in%ersion operation" $e call

D%(L) the parity code-word (or polynomial) of %(L)" $e will use this property in the hy'rid A 9 !

system to 'e descri'ed"

22.1".2. 22.#.2 THE CONCATENATED CODED MODULATION FEC SUBSYSTEM AND

RETRANSMISSION

*he FEC su'system is a concatenation of the 3 outer code C. and the 8C+ inner

code C1" *he outer code is interlea%ed 'y degree A, as shown in Figure ..".(('i), ('.), ON ('&)), that is formed 'ased on the

original set of code-words, '1, '., '&, and the half-rate in%erti'le code Cr" First,each code-word 11

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