ESD ACCESSION LIST AL 5823"

p. >* 8 ESD-TR-67-379 (U iu <? ^ # ** / J in ESTI Call N Cpy N. ERROR CONTROL TECHNIQUES USING BINARY SYMBOL BURST CODES MTP-55 ESD ACCESSION LIST AL 5823" SEPTEMBER 1967 K. Brayer Prepared fr DEPUTY FOR SURVEILLANCE AND CONTROL SYSTEMS AEROSPACE INSTRUMENTATION PROGRAM OFFICE ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscm Field, Bedfrd, Massachusetts This dcument h IS been apprved fr pi blic release and sa e; its dictt ibutin i s un- limited. Prject 705B Prepared by THE MITRE CORPORATION Bedfrd, Massachusetts Cntract AF19(628)-5165 AQdULtSK^

When US Gvernment drawings, specificatins, r ther data are used fr any purpse ther than a definitely related gvernment prcurement peratin, the gvernment thereby incurs n respnsibility nr any bligatin whatsever; and the fact that the gvernment may have frmulated, furnished, r in any way supplied the said drawings, specificatins, r ther data is nt t be regarded by implicatin r therwise, as in any manner licensing the hlder r any ther persn r crpratin, r cnveying any rights r permissin t manufacture, use, r sell any patented inventin that may in any way be related theret. D nt return this cpy. Retain r destry.

ESD-TR-67-379 MTP-55 ERROR CONTROL TECHNIQUES USING BINARY SYMBOL BURST CODES SEPTEMBER 1967 K. Brayer Prepared fr DEPUTY FOR SURVEILLANCE AND CONTROL SYSTEMS AEROSPACE INSTRUMENTATION PROGRAM OFFICE ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscm Field, Bedfrd, Massachusetts Prject 705B Prepared by This dc ument has been apprved fr pu blic release and sale; its di stributin i s un- limited. THE MITRE CORPORATION Bedfrd, Massachusetts Cntract AF19(628)-5165

FOREWORD This reprt was prepared by the Range Cmmunicatins Planning and Technlgy Subdepartment f The MITRE Crpratin, Bedfrd, Massachusetts, under Cntract AF 19(628)-5165. The wrk was directed by the Develpment Engineering Divisin under the Aerspace Instrumentatin Prgram Office, Air Frce Systems Cmmand, Electrnics Systems Divisin, Laurence G. Hanscm Field, Bedfrd, Massachusetts. Captain J. J. Centfanti served as the Air Frce Prject Mnitr fr this prgram, identifiable as ESD (ESSI) Prject 5932, Range Digital Data Transmissin Imprvement. REVIEW AND APPROVAL Publicatin f this technical reprt des nt cnstitute Air Frce apprval f the reprt's findings r cnclusins. It is published nly fr the exchange and stimulatin f ideas. C.#M* /* y^ctis R. HILLj^Clnel, USAF / Directr f Aerspace Instrumentatin Prgram Office 11

ABSTRACT Much has been written n the theretical descriptin f errr crrecting cdes but, due t a lack f actual channel errr patterns, little has been said f practical perfrmance. In this paper the perfrmance f three types f errr cntrl is evaluated fr the case f independent randm errrs and fr an actual channel exhibiting dense bursts. The selected cdes are burst cdes with high prbabilities f errr detectin and crrectin. 111

TABLE OF CONTENTS LIST OF ILLUSTRATIONS SECTION I SECTION H INTRODUCTION HYBRID ERROR CONTROL SYSTEM GENERAL DESCRIPTION The Hybrid Retransmissin Technique Blck Numbering Techniques Need fr Retransmissin Data Link CODE DESCRIPTION Delay Time f Cdes Specializatin t Retransmissin and Frward Errr Cntrl SECTION HI PERFORMANCE IN A MEMORY-LESS BINARY SYMMETRIC CHANNEL (BSC) RETRANSMISSION ERROR CONTROL FORWARD ERROR CORRECTION SECTION IV PERFORMANCE IN REAL CHANNELS FORWARD ERROR CONTROL RETRANSMISSION ERROR CONTROL HYBRID ERROR CONTROL SECTION V CONCLUSIONS APPENDIX I REFERENCES BIBLIOGRAPHY DEFINITION Page vi 1 2 2 3 4 5 (i 14 14 16 20 22 27 27 30 35 39 41 47 53 55

LIST OF ILLUSTRATIONS Figure N. Page 1 Prbability f Undetected Errr versus Cde Rate fr P M Symbl Cdes 11 2 Prbability f Undetected Errr versus Cde Rate fr P M Symbl Cdes 12 3 Efficiency versus Infrmatin Transmitted fr a Hybrid Cding System as a Functin f Bit Errr Prbability 21 4 Efficiency versus Blck Length as a Functin f BER in a BSC 23 5 Symbl Imprvement Factr versus Prbability f Symbl Errr 25 6 Symbl Errr Prbability versus Bit Errr Prbability 26 7 Percentage f Errrs Crrected versus Cde Rate 28 8 Percentage f Errrs Crrected versus Cde Rate 29 9 Average Efficiency f Retransmissin System n Trpscatter Data 32 10 Number f Retransmissins Required 33 11 Fixed Blck Efficiency 31 12 Fixed Blck Retransmissin Perfrmance 34 13 Average Efficiency f Hybrid System n Trpscatter Data 36 14 Percentage f Blcks Crrected by P M Cde in Hybrid System n Trpscatter 37 15 Average Number f Retransmissins by Hybrid System On Trpscatter Data 38 16 Full Duplex Cmmunicatin Circuit Cnfiguratin 42 17 Frequency f Cnsecutive Errr Occurrence 44 18 Distributin f Gaps Between Errrs 45 vi

LIST OF ILLUSTRATIONS (Cncluded) Figure N. Page 19 Message Errr Rate versus Message Size 46 20 Distributin f Lengths f Bursts 49 21 Distributin f Burst Errr Densities 50 22 Distributin f Lengths f Intervals 51 23 Distributin f Interval Errr Densities 52 VII

SECTION I INTRODUCTION In recent years, wrk in the areas f cding thrugh blck, retransmissin, and hybrid techniques was reprted in the literature. Due t a lack f real channel data fl ' 2l, the investigatin f hw these techniques perfrm in real channel en- virnments is still unknwn. Hwever, the MITRE Crpratin has cllected extensive errr pattern data at high speeds in real channel envirnments such [3, 4, 5, 61 as HF and trpscatter (Appendix 1).It is the purpse f this paper t describe and cmpare the perfrmance f techniques fr hybrid errr cntrl, frward errr cntrl, and retransmissin errr cntrl in a real channel en- virnment and t cntrast this perfrmance with that which may be expected in a channel exhibiting independent errrs. Errr cntrl methds are needed in the transmissin f digital data if the errr rate withut cntrl is unacceptable. An example f such a situatin is cmputer-t-cmputer cmmunicatins. Nt nly is nearly-perfect transmissin required in the transmissin f data, but errr-free transmissin is mandatry when cmputer prgrams are transmitted. The methd f errr cntrl described in this paper will enable the user t select the prbability with which errrs in the received message are acceptable and then t transmit the infrmatin with the highest pssible efficiency available fr the cmbinatin f mdulatin tech- nique, data rate, and envirnment in which the transmissin will take place.

SECTION II HYBRID ERROR CONTROL SYSTEM GENERAL DESCRIPTION Hybrid errr cntrl is the cmbinatin f a frward errr crrecting cde (e.g., blck cde) with the use f retransmissin errr cntrl. As a cmbinatin f tw cding prcedures, hybrid errr cntrl exhibits the advantages f bth and the disadvantages f neither. Specifically, in a randm errr envirnment, the frward errr crrecting cde will prvide errr crrectin within the cde blck and retransmissin will eliminate any residual errrs. In a high density burst* errr envirnment, the retransmissin will be used fr errr crrectin while the frward errr cntrl eliminates the errrs in the intervals between bursts. If the burst errr density increases t an extremely high level and the bursts get excessively lng, r if randm errrs ccur with a bit errr rate higher than a threshld (percentage f crrectable errr bits) derived frm the cde, the system enters cntinuus retransmissin and will fail. In failure, the cde crrects an insufficient number f errrs and burst length exceeds the blck length, causing cntinuus retransmissin. Hwever, since the cdes will be chsen frm a family, the cde pwer can be increased t ward ff failure. The hybrid errr cntrl system described herein is a cmbinatin f retransmissin and randm burst symbl errr crrecting cdes, perated as a cntinuus system. The system will be described initially in terms f retransmissin with a cde, and the specific prperties f the selected cde will be identified later. Finally, the hybrid system will be specialized t frward errr crrectin alne and retransmissin alne. *See Appendix I fr definitin f errr pattern parameters. 2

Transmissin efficiency (E) ~ id + Rd + R c > + IR(I R d + R c > + D T The Hybrid Retransmissin Technique The hybrid errr cntrl system perates accrding t the fllwing transmissin prcedure. Data is cllected at sme data center in large amunts. This data is encded with an errr detecting cde and then encded with an errr crrecting cde. The infrmatin is nw transmitted in blcks t a receiver site where the prcess f errr crrectin is activated. After errr crrectin, a final detectin is made fr residual errrs. If there are n errrs, the data is transferred t the data sink. If there are errrs, the blck identificatin number is recrded fr future use by the retransmissin system and the blck is rejected. During the finite time that it takes t decde ne blck f infrmatin, the next blck is being transmitted. If transmissin were interrupted fr retransmissin f rejected blcks, the finite delay wuld cause a system slwdwn. Thus, retransmissin requests transmitted back t the data surce initiate blck regeneratin at the cnclusin f the initial transmissin. The nly case in which there can be delay time between blcks is if the last blck must be retransmitted. The abve technique can be used fr transmissin f nn-real-time infrmatin such as cmputer prgrams r large amunts f data t be used as pre-flight and pst-flight infrmatin in space vehicle launches. If near-real-time transfer is required, a request fr retransmissin can be initiated immediately and the blck (rejected) in questin will be regenerated and retransmitted fllwing the blck in transmissin when the request is made. Actual real-time peratin cannt be achieved with this system. The perfrmance f such a hybrid system is described by the fllwing transmissin efficiency equatin: m,,.., Infrmatin t be transmitted (sec) Transmissin efficiency (E) = ^ ^ w. ;»- ' Ttal time l peratin (sec)

where I R R I D = infrmatin t be transmitted (secnds) = Data (bits)/data Rate (Bits/Sec) = errr detectin redundancy = rati f detectin parity t infrmatin bits = errr crrectin redundancy = rati f crrectin parity t detectin parity and infrmatin bits. = infrmatin rejected (Sec) - delay time (Sec) = errr cntrl delay (1 + messages transmitted) (Sec) fr interrupted systems paper), If the system perates as a cntinuus system (as will be the case in this D = errr cntrl delay. If there are n retransmissins, since I =0. E = I (2) I(l + I V R c )+D T missins, In the case where D is negligible cmpared t I and there are n retrans- E = = rate f the hybrid cde. (3) 1 + R, + R w d c If D is zer r if D < < I, the efficiency is independent f I and f the data rate. 1 Thus, it is pssible t cnsider the efficiency J as the effective rate l the 1 + R hybrid system. The effective rate is always less than the cde rate. The near- ness f E t the cde rate acts as a system perfrmance criterin. Blck Numbering Techniques Althugh it was previusly stated that the blck number is recrded, it is nt necessary, in the case f blck rejectin, t transmit blck numbers if a pwerful synchrnizatin technique is used. Since every blck has the same

number f bits and the receiving cmputer knws the sequence in advance (i.e., sequence is 1, 2,..., n), the cmputer can assign these numbers. Similarly in retransmissin, since all retransmissin blcks are placed tgether at the end f the message, the cmputer can take advantage f its prir knwledge. The nly exceptin is if there is a delay between blcks, in which case a delay message must be sent t prevent cnfusin f the receiving cmputer by the randm channel utput, r if the near-real-time mde is used. In the latter mde, the retransmitted blck may ccur immediately after the blck in the channel r ne blck later if the regeneratin time is nt sufficient. Need fr Retransmissin Data Link It might appear that the need fr a retransmissin link wuld reduce the efficiency f the system. This is nly partially true. If a link must be set up, a narrw bandwidth will be required since there is little infrmatin t transmit. It is much mre likely that the return link already exists due t a need fr cmmunicatin in tw directins. The retransmissin requests can be time-divisin multiplexed with the messages f ther users n the return link and, with careful allcatin f cmmunicatin availability, there wuld be n hardship n the regular users. This paper will cnsider efficiency n the basis f the data link nly assuming return link availability. Furthermre the retransmissin requests can be lw rate cded t prtect against errrs. Specializatin t Retransmissin Errr Cntrl In a system where there is retransmissin errr cntrl nly, tw mdificatins need t be made t the hybrid errr cntrl system. The bvius ne is that the errr crrectin redundancy is remved. Thus in the transmissin efficiency equatin, R =0. The ther mdificatin t bth the system and the transmissin efficiency equatin is with respect t delay. In the initial hybrid system, delay is due t

the use f errr crrectin and spacing between blcks. With the errr crrecting cde eliminated (R = 0), the part f the delay due t the errr crrectin is likewise eliminated. CODE DESCRIPTION The cdes which will be used are frm a class f p symbl errr crrecting [7] cdes. These cdes have been selected because f their high errr crrecting capability fr little redundancy and because f their high residual errr detecting capability. These capabilities are such that the cdes will be used fr bth R and R simultaneusly, thus simplifying the efficiency equatin. The cdes are referred t as (q, n, e) cdes, where q is the number f symbls in the alphabet expressible as p, n is the number f symbls in a cde blck, and e is the number f symbl errrs that can be crrected in a cde blck. The number p may be any prime pwer, (p = 2, 3, 5, 7..., m = 1, 2, 3, 4,...), and n is any psitive in- teger nt divisible by p. Fr the binary case, p = 2 and q = 2. Thus the alphabet used by the cde cnsists f the set f binary m-tuples. It is cnvenient in this binary subfamily f cdes t select n = 2-1. The crrectin f e symbl errrs then requires that 2e (2e< n) check symbls be included in the blck. The cdes are randm symbl errr crrecting cdes. Within a blck f n symbls, e symbls are crrectable if there are at mst e symbl errrs in the blck. With respect t bit errrs, the cdes are burst cdes since if there is ne bit errr in a symbl, it is crrected, but the cde wuld be mre efficient if there were m bit errrs in a symbl. As an example if m = 3, n = 7, and e = 2, then any tw randm bit errrs in the 21-bit blck are crrected by the cde. But the cde will als crrect tw slid bursts f three errrs in the blck r ne slid burst f six errrs (which exists in tw adjacent symbls). The parameter values which will be given special interest in this paper are: m (bits per symbl) = 8 n (symbls per blck) = 255 6

m- n (bits per blck) 2040 e (crrectable symbls) 0 t 32 2e (check symbls per blck) 0 t 64 n-2e (infrmatin symbls per blck) 255 t 191 n-2e (cde rate) 1. 0 t 0. 75 n In the encding and decding prcess, the m bit symbls f a message are treated as elements f a finite field. When dealing with binary m bit symbls, the finite field may be cnsidered t cnsist f all pssible m-tuples r m bit numbers. A special m bit arithmetric is used such that all arithmetic peratins between elements f the field yield results which are als m bit numbers and therefre cn- tained in the field. Als, it may be stated that all nn-zer members f the field may be represented by a special number, a, raised t pwer i, where i can vary frm zer t 2 m -2. The number a is referred t as a primitive element f the field. Additin is perfrmed mdul-tw with n carry. Multiplicatin in finite field arithmetic is analgus t cnventinal multiplicatin in that numbers raised t a pwer may be multiplied by adding their expnents. Thus, fr example, «2 6 8 times a equals a. The additin f expnents when multiplying is perfrmed m 14 *3 mdul 2-1. Thus, in the field with m 4, a x a wuld equal a * since 17 mdul 15 is equal t 2. The equatin belw defines a relatinship which is true fr all valid messages r cde blcks. where m 2-2 I B. a li =0, j = 0, 1, 2... 2e - 1 (4) i=0 l B. = the i th message symbl m = bits per message symbl a = primitive elements f finite field e = number f symbls which can be crrected. 7

Accrding t Equatin (4), the summatin f the prducts f all message symbls multiplied by a 1 J is equal t zer where i specifies the rder f a symbl in the message and j takes n all integer values frm 0 t 2e-l. T describe the encding and decding peratins, a simple example in which m = 3, n = 7, and e = 2 is cnsidered. Fr these parameters, a message r cde blck cnsists f seven 3-bit symbls. Since e = 2, there will be fur check symbls in each blck, and the remaining three symbls are infrmatin symbls. The prblem in encding is: given three infrmatin symbls, EL. t B 2, cmpute the fur check symbls, B 3 t B^.. Frm Equatin (4), fur simultaneus equatins crrespnding t the fur values f j can be written: j = 0; B + B l + B 2... + B r 0 j = 1; B + B a + B 2a 2... +B a 6-0 j = 2; B + B a2 + B 2 a 4 = 0 (4a) j = 3; B + B a3 * B a 6... +B «1S b = 0 Slutin f the fur simultaneus equatins abve gives the required check symbls, B t B. Nw cnsider the prblem f decding < '> >» *- > "» 1^> *'''. The re- ceived message symbls are called B'where each B-, cnsists f the riginal message symbl B. added (mdul tw) t an errr symbl V. which may r may nt be zer. The first step f the decding prcedure is t cmpute 2e numbers called syndrmes, S-, in accrdance with Equatin (5). m 2-2 S. ^ i - 0 B i alj ' j = 0, 1,..., 2e - 1 (;'.)

Substituting B! = B. + V., we btain Since, frm Equatin (4) ~m _ 2-2 E i = 0 2-2 2-2 M s. = 2_. B i lj + /. v i 1J ( 5a ) Bj a^ = 0, i=0 i=0 m 2-2 Sj = ^ V.cij. i = 0 («) If n errrs have ccurred, the V^ are all zer and the syndrmes will als be zer. Hence, if the syndrmes are fund t be all zers, the message is assumed t have been received errr-free. If ne r mre nn-zer syndrmes are btained hwever, errrs are knwn t have ccurred and the message must be crrected. If we assume that tw errrs had ccurred in ur sample message, they culd be determined frm the fllwing equatins S 0 = V l + V 2 S x = Vjc/ 1 + V 2 a 2 2i i 2i0 (6a) = V l a Y 2 a S 2 S 31. 3i 0 = V a 1 2 + V a O 1 Li 1 i-9 The fur unknwns in these equatins are V^, VJJ, «, and a ", where V^ and V are the tw errr symbls, and i.. and i crrespnd t their lcatins in the message. After slving fr the errr values and their lcatins, the syn- drmes are adjusted fr the errrs tentatively assumed t have ccurred. If the syndrmes are nw equal t zer, it is decided that the assumed errrs did 9

in fact ccur and the message is crrected accrdingly. If any f the adjusted syndrmes are nn-zer, hwever, the message is cnsidered t be uncrrectable due t excessive errrs. It was stated that if the syndrmes are zer the cde assumes the message t be errr-free r crrected. There is, f curse, a pssibility that the cde has failed t detect an errr. In this case, n retransmissin will be requested in a hybrid system and the errrs will be passed n. In Figures 1 and 2 the prbability f an undetected errr is presented fr varius cdes. The curves are based n the derivatin resulting in Equatin (10). Fr purpse f this derivatin, the fllwing terms are defined. m = number f bits/symbl n = number f symbls/blck k = number f infrmatin symbls/blck e = maximum number f errrs that can be crrected. This class f cdes fails t detect a blck in errr if, and nly if, the received cde vectr is n greater than distance e away frm sme cde vectr ther than the ne sent. Thus, assuming all n-symbl errr vectrs are equally likely, and that an errr has ccurred, the prbability f an errneus blck ging undetected (and thus als being errneusly "crrected") is Number f errr vectrs f distance ^e frm a nn-zer cde vectr Number f errr vectrs nt f distance ^e frm the zer vectr (7) where the crrect vectr is used as a zer reference. Number f ways a symbl can be nn-zer Number f vectrs with i nn-zer symbls Number f nn-zer cde vectrs 2 m -l = n ^n\ u/ ^ 2km _j 10

il. 1 1 1 1 1 1 1 I \\ \. ^N^ OM"2 10-' : 1\ \ \ \M-3 \ \ \ >v 1 1 \ \M'4 10-2 - 10-3 f 1\ \ I \M=5 \ O DC K UJ Q UJ u IO-«r i-5 f =1 u. >- \ \ r" \ \ IE Ir 0. I0"6-10-^ - \ M=7 1 \ 10-6 " 10-9 ; M<8 I>s.rt-10 1 9 8 7 6 5 4 3 2 1 t CODE RATE i^jr 1 ) 1 Figure I. Prbability f Undetected Errr versus Cde Rate fr P Symbl Cdes 11

Figure 2. Prbability f Undetected Errr versus Cde Rate fr P Symbl Cdes 12

Number f vectrs f distance = e frm a nn-zer cde vectr = (8) (2 km -l) Z On 1 i = 0 V1/ Number f vectrs f distance > e frm the zer vectr = 2 nm > ") n i (9) Z (") -' i = 0 w p^-din.'* I (") ni P {Undetected errr } = (10) 2nm m * The next sectin cntains a descriptin f a case with an actual errr distributin which shall be taken int accunt in lwering the bund n the prbability f an undetected errr. The curves demnstrate the perfrmance f the cde in failing t detect errrs as a functin f cde rate fr 2 ^ m ^ 8. The case m = 1 can be cnsidered as the trival cde where there is ne bit/symbl and ne symbl/blck and the failure prbability is bviusly 1 since there is n redundancy. Unlike many ther cdes, a bursty channel will nt reduce the perfrmance f these cdes. Since the cde crrects nly symbls, it des nt matter whether r nt ne r all bits in the symbl are in errr. The bit errr crrectin capa- bility is, in fact, better if all the bits in a symbl are in errr. As an example, if m = 8 and the cde rate is 3/4, 32 symbls can be crrected. If these errr symbls are cnsecutive, the cde can crrect 256 cnsecutive errrs. If the bursts are excessively lng, m must be increased. When used in cnjunctin with the efficiency curves in later sectins, Figures 1 and 2 will serve t give the maximum efficiency available fr a *This gives a wrst case bund n P (undetected blck errr given a symbl errr). An alternative apprach t this calculatin is presented by MitchelU 13 J

cmbinatin f envirnment, cde, data rate, length f infrmatin message, and prbability f acceptable undetected errrs. Delay Time f Cdes One f the penalties f errr crrectin is the delay intrduced int the system by parity check cding. T evaluate the delay, the three places in which delay can ccur (i.e., encding, transmissin, and decding) are cnsidered. In encding, as the infrmatin bits are made available by the surce, they are used in starting the calculatin f check symbls and "utputed" t the mdem. When the last infrmatin bit is "utputed", all check bits are nw immediately available. Thus, there is n delay in encding as lng as the cder can prcess bits at a rate higher than the channel data rate. In decding, calculatins cannt start until all symbls are available. Thus, there is a delay f ne blck in transmissin. Fr available hardware, calculatins in decding require ne blck time at mst. Thus, the verall delay due t cding fr cntinuus data transmissin is twice the blck length. If the blcks were sent in a spaced discntinuus system, the delay due t cding wuld be twice the blck length per transmitted blck. If such a system were used, this delay culd dminate the denminatr f the efficiency expressin, Equatin (1), thus drastically reducing the efficiency. Since an actual system culd have varius delays up t twice the blck length, we will assume a wrst case in ur perfrmance analysis and use the maximum value f delay in all cases. Specializatin t Retransmissin and Frward Errr Cntrl In the case f retransmissin errr cntrl, the same cdes will be used fr errr detectin nly. Since there is n crrectin, mre blcks will be retransmitted and the system efficiency will be prer. While this result is bvius, it is still instructive t see the difference. 14

In cntrast t the hybrid and retransmissin appraches where, fr a given prbability f undetected errr, the maximum efficiency is achieved, in frward errr crrectin the efficiency expressin reduces t I 1 E ~ I (1 + R) 1 + R (11) which is a cnstant. The penalty caused by this apparent cnstant efficiency is that sme percentage f the errrs will be neither detected nr crrected. If sme errrs are acceptable, this trade-ff f errrs versus cde rate may als be acceptable. 15

SECTION III PERFORMANCE IN A MEMORY-LESS BINARY SYMMETRIC CHANNEL (BSC) In this sectin, the efficiency relatin previusly presented [Equatin (1)] will be evaluated in the binary symmetric memry-less channel. As a brief review, a binary symmetric memry-less channel is a tw-state channel in which the prbability f a transitin frm ne state t the ther is given as pj, and the prbability f remaining in the same state is 1-pj, indepen- dent f past results. Thus, the prbability f an errr is p-^, independent f the nature f the input t the channel. As an example, if the message 101011 was transmitted, the prbability is p-^ that any bit may be inverted independent f what happened t any ther bit. Treating the hybrid system first and starting with the efficiency expressin E- I - I (i + R) + i R (i + R) + D T it is desired t find E versus I. If I, D and R are given, it is first necessary t find I (1 + R) which equals the rejected messages. The prbability f reject- ing a message is equal t the number f rejected messages divided by the ttal number f messages transmitted. A message will nly be rejected and thus re- transmitted if a symbl errr is detected in it. perfrmed after all pssible crrectins. Errr detectin peratins are Thus, the prbability f rejecting a message is numerically identical t the prbability f detecting a symbl errr (if n errr is detected, the message is either prperly crrected r errneusly crrected). If the same assumptins are made as were made in the calculatin f P undetected blck errr} fr the cde, it is fund that P detectin } = P j the blck is nt errneusly crrected and an un-crrectable errr pattern exists } ' ' The calculatin is best initiated by finding P {at least e + 1 symbl errrs in a blck}. 16

n P j at least e + 1 symbl errrs in a blck = y P i symbl (13) i = e + l errrs in a blck} n e 2 P i symbl errrs in a blck = 1 - j P j i symbl errrs (14) i=e+l i=0 in a blck } Given n symbls, the prbability that i symbls are in errr is p*(l - p) n-1 where p is the prbability f a symbl errr. These i symbls can be psitined in any f j j ways. Thus P i symbl errrs in a blck j ={ Ip (1 - p) (15) The prbability that a symbl is in errr is the prbability f at least ne bit errr in the symbl. Thus p = l-(l- Pl ) L (1G) Where p^ is the prbability that a bit is in errr and L is the length f the symbl in bits. Fr the cdes selected L = ni [151 Thus, (fllwing Mitchell ) n = 2 m - 1 P { crrectin failure } n. m m A i (2 -l)-i i -Z 2^ ^r ' F WI 1 )[(i-(i-pi),n i'[(i-pi) m ] i = 0 17

F(i) = Fractin f i symbl errr patterns in 2-1 cde symbls which are cr- rectable. Fr e < i -n where I n F (i) = 1 0 < i <e F (i) < 1 e < i ^n F(i) N(i) 0 lis the ttal number f i symbl errr patterns in n symbls. N(i) is W n the number f such patterns that are crrectable. The prtin f \ i ^e is knwn (i. e., all patterns). i = 0 N(i) fr Hwever, nly sme patterns are crrectable fr i > e and these patterns are a functin f the cde, channel, and the decding Il5l scheme J. Thus, we shall be cnservative and calculate the prbability f de- tectin by setting F(i) = 1 all 0 < i ^e and F(i) =0 fr e<i^n (the guaranteed crrectin). This is a reasnable apprach, since the errr is small ^ '. The first prtin f Equatin (12) is nw available. e m v (2 m -l)-i P { Detectin [ = 1 - ]T ( 2 ~\l - (1 - p^ ] [ (1 - p^ ] i=0 * 1 ' - P undetected errr } (18) T find the prbability f an undetected blck errr in a channel exhibiting [2l independent errrs, we fllw the methd used by Nesenbergs. Errrs will be undetected if sme cde vectr is transfrmed by the channel int a vectr which is less than i symbls frm sme ther cde vectr. The prbability f cmbinatins falling int this class is = PnW < 19 > i = e + 1 18

where P n (i) is the prbability f i symbl errrs in an n symbl sequence. Fr independent errrs this is P = n i=e+l x ' 2-1 / rn A. m I 2 i "M m (2 "l)-i 2^ ( h-a-pj) ] ta-pj) ] (2 ). 1 i = e + l Using the previusly derived expressin where an errr is always assumed t ccur, we find P { undetected blck errr = km x~^ /n\ i (2 -.) J.'P 1 "" 0^ (21) nm i = 0 Xl ' This value is less than the wrst-case bund since, frm the binmial therem, P < 1. Thus in a binary symmetric channel, the prbability f undetected blck errrs is less than as given previusly. P detectin } = P { retransmissin } = messages retransmitted (22) ttal messages L(l + R) I R P I retransmissin \ = = (23) '» 1(1 + R> + yi + R> l + l I _ I P { retransmissin}_ I P detectin ) R I - P retransmissinl I-P j detectin I (24) li)

Using this frmula, it is nw pssible t find E versus I fr varius values f D, p, and R (where R implies specific cde). This peratin has been per- frmed and is presented in Figure 3 fr m = 8 and e = 32 fr a range f values p. D is taken as 3. 5 sec which crrespnds t tw blcks at 1200 bits/sec. The perfrmance f the system is a mntnically increasing functin f I, ex- cept where the prbability f crrectin is zer, such that P retransmissin is ne and the system breaks dwn in bth crrectin and retransmissin. On the curves shwn, the efficiency reaches 0. 7 ut f a maximum value f 0. 75 fr _2 a 35 sec message where the prbability f a bit errr is 1 x 10. There is _ significant degradatin fr a bit errr prbability f 1. 6 x 10, but this simply means that a lwer effective rate is being achieved. This culd be imprved by ging t a cde rate less than 0. 75 but greater than 0. 4, r increasing m. RETRANSMISSION ERROR CONTROL The expressin fr efficiency in retransmissin errr cntrl is I E = Ki + Rd )+I R (i+r d ) + D T If it is assumed that all errrs are detected and retransmissins ccur fr every detectin, it is again fund that _, i.., message retransmitted P { retransmissin \ =.?. I ) ttal messages and i _ I P {retransmissin! _ I P {detectin [ I - P retransmissin} I - P jdetectinj Since all errrs are detected P detectin = P j ccurrence f a blck errr = 1-(1 - p.) (25) Thus I = (26) L (1 " Pi) 20

21

The same set f cdes will be used and, after finding the efficiency, the answer shuld have appended t it the prbability f an undetected blck errr as derived fr the BSC. In this case, L is the blck length. m L = m (2-1) D-p was taken as tw blcks in hybrid cding but, as previusly explained, D is nly ne blck in retransmissin. Frm the efficiency expressin and the expressins fr I, it is fund that when D is negligible with respect t I, ( "a)' < 1 + R >H 1 -'0- p i) L ] / ( 1 -'>i) L ) (27) Frm this expressin it is fund that, fr a given value f L, as p appraches zer the efficiency appraches the cde rate. Similarly fr a given p, the efficiency ges t zer with increasing L. These results are presented in Figure 4 alng with the results fr 104 ms. delay. FORWARD ERROR CORRECTION Using frward errr crrectin nly in the independent errr envirnment the fllwing relatinship is fund (assuming guaranteed crrectin nly). P crrectin ( = P j nt mre than e symbl errrs in a blck /2 m -l\ j^y i jrwi-p/vra-p/v 2 1^ i = 0 - (28) 22

100 300 400 500 600 TOO 800 1,000 2,000 3,000 4,000 BLOCK LENGTH L ( BITS I Figure 4. Efficiency versus Blck Length as a Functin f BER in a BSC 23

Using m = 3, 4, 5, 6, 7, 8, and cde rates (2e < n), a set f curves can be de- velped fr the percent f errrs crrected as a functin f channel bit errr rate. Hwever, the curves will lie n the vertical axis between 99 and 99. 9999 percent f the errrs crrected fr all values f p, < 10. Therefre, the re- sults will be presented as imprvement factr (rati f input errrs t utput errrs) as a functin f symbl errr prbability. A blck is in errr after decding if there are at least e + 1 symbl errrs in the blck and a false cr- rectin is nt made. Since this adjustment is nt made, the results shuld be weighted in the light f the undetected errr prbabilities. The results in Figure 5 give the imprvement factr as a functin f symbl errr prbability. The curves relating symbl errr prbability t bit errr prbability are given in Figure 6. T cnvert the symbl imprvement factrs which result frm Figure 5 t bit imprvement factrs, the fllwing relatins shuld be used. symbl imprvement factr = expected number f symbl errrs input/number f symbl errrs utput e ns M JP 1 d-p^" 1 i = 0 and p = p symbl = 1 - (1-p bit) 1 ". (30) 24

10 9 T" " PROBABILITY OF SYMBOL ERROR Figure 5. Symbl Imprvement Factr versus Prbability f Symbl Errr 25

J I I I I Mil I I I I I I III I PROBABILITY OF BIT ERROR Figure 6. Symbl Errr Prbability versus Bit Errr Prbability 26

SECTION IV PERFORMANCE IN REAL CHANNELS In this sectin, the perfrmance f the three cding systems in a trpscatter envirnment is examined. The verall distributins f the errrs measured in the envirnment is described in Appendix I alng with a statement f the mdulatin technique. The errr data was cllected in 90-minute test samples. A system which perfrms errr cntrl functins was simulated n an IBM 7030 cmputer and was perated fr each 90-minute data sample. The cmputer prgram used in this analysis perfrms all the functins f hybrid errr cntrl in terms f actual message structure and the decding and crrectin f errrs. Hwever, since the lcatins f errrs are knwn in advance, if the prgram determines that there is an excessive number f symbl errrs in a blck, n attempt t crrect is made and detectin is as- sumed. Thus, undetected errrs are nt allwed fr, and the results must be used in cnjunctin with the prbability f an undetected errr t get a true picture. FORWARD ERROR CONTROL The functin f frward errr crrectin was perfrmed by taking all data at a given data rate and crrecting as a whle. Thus the results presented in Figures 7 and 8 indicate the percent f errrs crrected as a functin f cde rate (n-2e) fr all (2e<n) acceptable cde rates fr the ttal mass f data n at the given transmissin rate. The results presented are fr m = 2, 3, 4, 5, 6, 7, and 8 fr bth the bit errr crrectin and blck errr crrectin. The pwer f the cde is seen t increase as a functin f m. Thus it is pssible t reduce the redundancy necessary fr a given amunt f errr crrectin by increasing m. The burst structure f the data indicated in the data descrip- tin is even mre evident in terms f the cde crrectin curves. If fr the 27

IB2I.259 1200 BIT/SEC DATA BLOCK ERRORS BIT ERRORS J L J I I I 0 0 I 0.2 0.3 0.4 0.5 0.6 0 7 0.8 0.9 10 Figure 7. Percentage f Errrs Crrected versus Cde Rate 2*

IB 21,258 2400BIT/SEC DATA BLOCK ERRORS Figure 8. Percentage f Errrs Crrected versus Cde Rate 29

2400 bit/sec data we cnsider the curve fr m = 4 at a cde rate f 0. 2, 70 percent f the blcks in errr are crrected but nly 20 percent f the bit errrs are crrected. Thus at this blck size f 60 bits, 30 percent f the blcks with errrs cntain 80 percent f the errrs. As m changes in value, the blck structure changes, and the fact that the cde fr m = 7 did prer than that fr m = 6 in a prtin at the range f cdes fr the 1200 bit/sec data shuld nt be surprising. At bth data rates, the pattern is brken fr m = 8. Here the cde blck becmes substantially lnger than the bursts and with respect t the cde the data appears as randm bursts, which is the mde f greatest efficiency. The curves indicate that the cdes d prly as frward bit errr crrecting cdes. Hwever, the cdes perfrm excellently as message errr crrecting cdes and can be used in the hybrid system since errr crrectin significantly reduces the number f retransmissins necessary in a hybrid system. The 2400 bit/secnd data has bursts which are f apprximately the same length as the 1200 bit/sec data but are mre than twice as dense (see Appendix I). Fr the small values f m, this difference in density shws up in errr crrectin perfrmance; fr large values f m the differences is nt s great. Hwever, it shuld be nted that fr a 2040 bit blck (m=8), 57 percent f the errrs are cntained in ne percent f the blcks. Based n these results and the excellent undetected errr perfrmance fr m=8, this value has been chsen fr use in errr detectin fr the retransmissin system and crrec- tin and detectin in the hybrid system. RETRANSMISSION ERROR CONTROL The perfrmance f retransmissin errr cntrl in the real channel is presented in tw ways. The first is efficiency versus infrmatin transferred, and the secnd is efficiency versus blck length. The results fr efficiency versus infrmatin transferred were btained by attempting t transfer the given amunt f infrmatin thrugh each 90- minute test sample and calculating the efficiency fr each test sample. The 30

results are the average acrss all the samples t btain the average efficiency, and are used t predict the expected efficiency f transfer f large masses f data by retransmissin errr cntrl nly. The results are presented in Figures 9 and 10. The blck lengths f 2040 bits and the redundancy levels have been chsen s that there can be easy cmparisn with the hybrid system which will use the mst pwerful crrectin (m=8). In retransmissin cntrl, the symbl cdes are used fr detectin nly, as previusly described, and als the results must be used in cmbinatin with the undetected errr prbabilities f the cde. The return link was cded with m = 10 using ne infrmatin symbl. Fr cmparisn with the efficiency versus blck length results presented fr the BSC, a similar set f results have been btained fr the real channel fr 10 minutes f infrmatin (Figures 11 and 12). 31

2400 BIT/SEC DATA CODE DATA RATE RATE 968-1200 90 B5 1/ / / / / y^" 968-2400 921-1200 921-2400 // 875-1200 875-2400 80 «// // // 827-1200 / / ** // // V / 827-2400 70 749-1200 / / 749-2400 / / // // 1 '/ 1 1 1 1 1 1 30 40 50 60 INFORMATION (MINI Figure 9. Average Efficiency f Retransmissin System n Trpscatter Data 32

T3 CD L- 'D cr 0) Qi c c 0) -O E D z

IA 21,255 " 6400 9600 BLOCK LENGTH (BITS) 12,800 I6J0OO 19,200 Figure II. Fixed Blck Efficiency 60 v> BO 2 ta r < 40 r t- & 30 K UJ 03 2 => 20 RUNS 1-29 DATA RATE 1200 BITS/SEC RUNS 30-71 DATA RATE 2400 BITS/SEC DELAY i 104 MS REDUNDANCY 25% 10 MIN MESSAGE 10 _L J_ 32 00 6400 9600 12,800 16,000 19,200 BLOCK LENGTH (BITS) Figure 12. Fixed Blck Retransmissin Perfrmance 34

HYBRID ERROR CONTROL Using a cmbinatin f retransmissin errr cntrl and errr crrectin (m=8), the efficiency in the hybrid mde as a functin f infrmatin transmitted has been btained (Figures 13, 14, and 15). Figure 13 indicates the average efficiency as a functin f I fr varius cde rates and data speeds in the trpscatter channel. The curves indicate bth the maximum effective rate which can be achieved fr a value f I and the data rate as a functin f the acceptable prbability f undetected errr. As befre, the curves must be read in cnjunctin with Figures 1 and 2. Fr cde rates ther than thse used, interplatin can be perfrmed. Figure 14 shws the percent f blcks in errr which are crrected by the errr crrecting cde. The dip starting at five minutes in the 2400 bit/sec curves is due t an abnrmal burst f three millin bits duratin in ne f the data runs. At first it was cnsidered that this burst be eliminated frm the data sample. Hwever, n the basis f perfrmance f the system, this burst was left in. The cde failed t crrect the errrs but the retransmissin part f the system cmpensated, with the result that the efficiency curves shw n degradatin fr the range 1-5 t 1=20. The average number f retransmissins is presented in Figure 15. The previusly mentined lw rate cding with m = 10 was used n the return link. The imprtant cnclusin frm this data is that in the actual channel (as in the BSC) it is pssible t perate at an effective rate arbitrarily clse t the cde rate and transfer large masses f data fr a given, acceptable prbability f errr. 35

1.0 CODE DATA RATE RATE.968-1200 9 5 // // // 968-2400.921-1200 921-2400 9 1 / "// //.875-1200 > z UJ u u_ u. UJ.85 "// ^".875-2400 UJ < UJ < > 8 If 1 l/.827-1200 - ^ m ^^ - 827-2400 75.749-1200.749-2400 ID CO - l I I I I I 70 C ) 10 20 30 40 50 60 7( 3 INFORMATION (MIN ) Figure 13. Average Efficiency f Hybrid System n Trpscatter Data 36

F (i) -4 l/l X n "O i a X X c 0) u > < 2 X _ ~ ci) +u a; i- u O U */> 0 Y * U u a c i_ ai n -»- 0 u> u t D -*- 8 c 0 a; i- u t <D L. Q_ 3 a31d3hh0d H0HH3 Nl SX0CH8 % 37

CO c E 1/1 K>-0 << * OK cy WW c\j OH <Tt < c < c I L c f c c I c \ III 8 8 g 8 S in m Kru^ c t c\j IONO NOdm CD CD r-^ fft (T\ CD CO pv. ^ \\ x \\ \ \ \\\? >\\ «N\\ \\\\ e 10 in <j 10 OJ INFORMATION (MIN ) X _Q i/> c c D u % >- E " z ^ 2 5! 8" < O lo 1 1 1 1 1 1 1 ^w * CJ c c <r RJ c > SN0issiwsNvai3a jaaawrin awaaav m 38

SECTION V CONCLUSIONS A methd has been develped t prvide cmbined errr crrectin and retransmissin in such a manner that the highest pssible rate f data transfer can be achieved fr a given set f cnditins. The perfrmance f the system in a binary symmetric memry-less channel and a real channel has been described. It is demnstrated that, fr a sufficiently lng message, it is pssible t perate at an efficiency that differs nly slightly frm the cde rate fr sme value f acceptable errr prbability. Unlike many ther systems, this system can be realized with present-state-f-the-art hardware and shuld prve useful in prviding reliable digital cmmunicatin between any tw remte gegraphical lcatins. 39

APPENDIX I In the spring f 1966, data transmissin tests were cnducted by The MITRE Crpratin n a U. S. Air Frce data circuit (see Figure 16). link is characterized as having three types f transmissin media: trpscatter, micrwave, and wireline. The trpscatter is multiple-hp, and the wireline cnsists f the intercnnectin f numerus leased telephne wirelines. The dminant sub-path is a trpscatter-hp f ver 500 miles. During the tests. Rixn Sebit 24B vestigial sideband AM mdems were used at data rates f 1200 and 2400 bit/sec in a 3 khz channel which was FM multiplexed with ther channels in transmissin. A ttal f 38 hurs f 1200 bit/sec and 61 hurs f 2400 bit/sec data was cllected fr a ttal f 1. 67 x 10 and 5. 20 x 10 bits respectively, in 90-minute samples. The errr pattern data was btained by ne-way transmissin f a 52- bit digital data test pattern which was cmpared with a lcally-generated test pattern at the receiving end. When there was agreement between the tw, a lgical zers were declared and when the bits were different, a lgical ne was declared. The zers and nes were recrded n magnetic tape. These magnetic tapes were then prcessed thrugh a tape-t-tape cnverter facility t prduce an IBM-cmpatible tape fr prcessing in an IBM 7030 cmputer in rder t determine the statistics f these bit-by-bit errr patterns. The data fr the tw transmissin rates was prcessed fr fur types f statistics: the ccurrence f cnsecutive errrs, the ccurence f intervals between errrs, the ccurrence f bursts, and wrd errr rate infrmatin. The extent f the data analyzed was as fllws: Table 1-1 Data Rate, bits/sec Mdem Ttal Bits Average Errr Rate (p ) 8-4 1200 Sebit 24B 1.67x10 1.39x10 2400 Sebit 24B 5.2x10 3.95 x 10" 41 The

2 O _l UJ <? 2-1 5 UJ <r ft UJ * >" r.l, ~ '%J ) <r O M 01 >- f _l O < ^- Li. <M _l UJ K k 00 in r l-uj m (O 111 X \- > Ul <rt uj H UJ z a u t- H Q s ( ) -i Y - 5 01 N UJ > in Z < UJ z S _i 1 CC * ^?5~ m GOO UJ2 a: ui t- tr UJ > OD UJ tr c c O U D U IM '0 c D _U C D E U x _0) Q. 3 Q 3 UJ r 5 42

The relative frequency f cnsecutive errrs is presented in Figure 17 fr the tw data rates. The Figure indicates that the relative ccurrence f multiple errrs fr the Trpscatter data is independent f the data rate fr these tw rates. Further, the relative ccurrence is certainly nt f an independent randm nature since, if it were, ver 98 percent f the errrs wuld ccur as single errrs while in fact nly 70 percent ccur as single errrs. This lack f independence is further illustrated in Figure 18 where the cummulative frequency at ccurrence f errr-free gaps is presented. All these curves cnfirm that this errr pattern data is widely different frm independent randm errrs at the same errr rate. The cummulative distributin f errr-free gaps fr independent randm curves is (in thery) described by where n P{c n e e} = ^ P(i _ p) k k = 0 c represents a crrect bit e represents an errr bit n represents number f cnsecutive bits The message errr rate (defined as the ccurrence f at least ne errr in a message) as a functin f message size is presented in Figure 19 fr these tw classes f data. It is interesting t nte that the value f p m fr n = 2000 in the actual data is independent f errr rate and signaling speed. This again indicates that the errrs are ccurring in bursts and, while fr the verall data p e at 2400 bits/ sec is 2. 85 times are large as p e at 1200 bits/sec, the additinal errrs which are ccurring at the higher signaling speed ccur as increased bursts density. Thus it is evident that bursts ccur as fades in time unrelated t the data rate (fr the rates measured). This cnclusin is cnfirmed by a visual bservatin f the errr patterns.

d CT Q u Id Hi t (A >* X <n 01 h- t- CQ m ( > < 4 1/5 4) U c 4) u L- <J O OD in ir UJ - > 3 O Ui t Z > " 5 9! c U c cr 4J, 03 O 10 (%) ADN3n03dd 3M1V13H 44

c CD CO t < D O C D CO CD (%) A3N3n03bd BAiivinwn I 45

~ yy sy \\ \\ Y\ ; X^ S X N \\\ \ ACTUAL 1200 BITS/SEC 2400 BITS/SEC j THEORETICAL 1200 BITS/SEC 2400 BITS/SEC I I 1 I 1 1 I 1 j - t CD LU t w 0 < t t LU CD N CD CO 0) s CO w <D > cs X \ \ \ N \ v \ N \ \ \ X \ \ \ \ - - CO (V 2 e D CO \ \ - 11 1 1 1 1 1 1 II 1 1 1 1 1 1 1 111 1 1 1 1 1 i 31VU yohh3 39VSS3VN CO I 46

The descriptin f the errr statistics presented thus far have made it clear that the errrs fall in a nn-randm clump distributin. T describe these errr bursts and the intervals between bursts (guard space) an additinal statistic shall be used. DEFINITION A burst in defined as a regin f the serial data stream where the fllwing prperties hld. A minimum number f errrs, M, are cntained in the regin and the minimum density f errrs in the regin is A. Bth f these cnditins must be satisfied fr the chsen values f M and A fr the regin t be defined as a burst. The density f errrs is defined as the rati f bits in errr t the ttal number f bits in the regin. The fllwing prperties hld fr the bursts. The burst always begins with a bit in errr and ends with a bit in errr. A burst may cntain crrect bits. Each burst is immediately preceded and fllwed by an interval in which the density f errrs is less than A. The burst prbability density functin is defined as the prbability f c- currence f a burst f size N, where N is any psitive integer. The burst size is measured in terms f the ttal number f bits in the burst. A separate burst prbability density functin may be determined fr each pair f values f A and M. e The minimum number f errrs in a burst has been chsen t be 2 fr all the data included here. The interval is defined as the regin f the serial data stream where the fllwing prperties hld. The minimum density f errrs in less than A, and the regin begins and ends in a crrect bit. An interval may cntain errrs. An interval is always immediately preceded and fllwed by a burst. Thus, each and every bit in the data stream must lie in either a burst regin r an interval regin. 47

The interval prbability density functin is defined as the prbability f ccurrence f an interval f length L, where L is any psitive integer. The interval prbability density is a jint functin f bth A and M e. In Figure 20, the distributin f bserved burst lengths is presented fr the data. The burst lengths range frm 2 (M ) t ver 1,000,000. While 90 percent are less than 1000 bits lng, the remaining bursts cntain mst f the errrs. The densities f these bursts are displayed in Figure 21. This Figure cnfirms the previus cnclusins that at 2400 bits/sec the additinal errrs serve t increase the density f the bursts. Further, while the minimum criteria (A) is chsen at 0. 01, ver 50 percent f the bursts have densities greater than 0. 1. A crrelatin was perfrmed n the burst lengths and their assciated densities and it was fund that while bursts less than 10 bits in length are generally 100 percent dense (cnsecutive errrs) and bursts ver 1,000,000 bits in length are near the minimum criteria, the ther bursts range in density between the minimum and 75 percent, indicating that there is n crrelatin between the length f bursts and densities. The value f A was chsen such that values f burst lengths and density are independent f A. The distributin seen n these curves indicates that the errrs fall int adjacent clusters f errrs with lng errr-free intervals between clusters. There are n intermediate gruping f errrs. In a theretical, randm errr distributin, the intermediate grupings wuld predminate. Figures 22 and 23 shw the distributin f interval lengths and assciated densities. The imprtant facts t nte here are that ver 60 percent f the intervals are errr free and that the remaining intervals have n mre than ne r tw randm errrs within them. 48

18 Ld CO CO CO ft CD C 19 c CO CD O CM CD -JO N - 0> O OD O IS 10 CO 0 (%) A0N3n038i 3AULv-inwn 49

en z UJ 5 (r O cc tr -»- c Q 12 D CO 00 cr CD c CN D (%) A0N3n03dd 3Aiivnnwn 50

O. g" V \\ \ \ \ \ X x 8 q. X \ a \ \ 8 - X \. Q. I z N \ Z \ \ _i \ \ < > \ \ (E \ \ UJ \ \ = 1- z \ \ - p \ \- LU W \ \- C> </> \ VJ w f A - - M CD 00 1 O O - = CM ^- CM 9-1 NO 1 II II - V) CD > 0) c CD c C D CN CN D CO 1 0, 1 5 < ~" 1 1 1 1 1 1 1 1 1 1 c 3 O K) 00 O On <DNW*>*W><M 10 en" < (%HON3naad 3Aiiv~inwn c 3 51

0) ( ) u 111 UJ in V) s </> h- m CO CM 5 (/> c Q z < tr tr tr < > c CO CM D Q en O 0> 0J P (%)ADN3n03dd 3AiivinwnD 52

REFERENCES 1. T. Kuhn, Retransmissin Errr Cntrl, IEEE Trans, n Cmm. Systems, June 1963. 2. M. Nesenbergs, Cmparisn f the 3-ut-f-7 ARQ with Bse-Chaudhuri- Hcquenghem Cding Systems, IEEE Trans, n Cmm. Systems, June 1964. 3. K. Brayer, Errr Statistics f a Multiple Link Data Transmissin Circuit, The MITRE Crpratin, MTP-40, Cntract AF 19(628)-5165, Bedfrd, Mass., Nv. 1966; als available as ESD-TR-66-337. 4. K. Brayer, Errr Patterns Measured n Transequatrial HF Cmmunicatin Links, The MITRE Crpratin, MTP-46, Cntract AF 19(628)-5165, Bedfrd, Mass., Jan. 1965; als available as ESD-TR-67-99. 5. K. Brayer and O. Cardinale, HF Channel Data Errr Statistics, Vl. I, The MITRE Crpratin, MTR-171, ESD-TR-66-290, Cntract AF19(628)- 5165, Bedfrd, Mass., April 1966. 6. K. Brayer and O. Cardinale, HF Channel Data Errr Statistics, Vl. II, The MITRE Crpratin, MTR-172, ESD-TR-66-107, Cntract AF 19(628)- 5165, Bedfrd, Mass., April 1966. 7. D. Grenstein and N. Zierler, A Class f Errr-Crrecting Cdes in p m Symbls, J. Siam, 9, June 1961. 8. E. Berlekamp, On Decding Binary Bse - Chaudhuri - Hcquenghem Cdes, IEEE Trans, n Inf. Thery, Octber 1965. 9. G. D. Frney, On Decding BCH Cdes, IEEE Trans, n Inf. Thery, Oct. 1965. 10. G. D. Frney, On Decding BCH Cdes, MIT Research Labratry f Electrnics, TR 440, Dec. 1965. 5:1

11. G. D. Frney, Generalized Minimum Distance Decding, IEEE Trans, n Inf. Thery, April 1966. 12. J. Massey, Step-by-Step Decding f the Bse-Chaudhuri-Hcquenghem Cdes, IEEE Trans, n Cmm. Systems, Oct. 1965. 13. Private Cmmunicatin with J. Terzian. 14. M. Mitchell, Inhibited Errr-Crrectin Decder Perfrmance, IEEE Trans, n Cmm. Systems, Dec. 1962. 15. M. Mitchell, Perfrmance f Errr Crrecting Cdes, IEEE Trans, n Cmm. Systems, Mar. 1962. 54

BIBLIOGRAPHY W. Bennett, F. E. Frehlich, Sme Results n the Effectiveness f Errr- Cntrl Prcedures in Digital Data Transmissin, IEEE Trans, n Cmm. Systems, Mar. 1961. E. Berlekamp, Blck Cding with Niseless Feedback, MIT PHD Thesis, September 1964. K. Brayer, O. Cardinale, The Applicatin f HF Channel Data t the Develpment f Blck Errr Crrectrs, The MITRE Crpratin MTP-32, Cntract AF 19(628)-5165, Bedfrd, Mass., June 1966; als available as ESD-TR-66-317. K. Brayer and O. Cardinale, Evaluatin f Errr Crrectin Blck Encding fr High Speed HF Data, IEEE Trans, n Cmm. Tech., June 1967. A. E. Fein et al, Time-Spread Cding and its Effectiveness in Transmissin ver Real Channels, IEEE Cnference n Cmmunicatin, June 1965. M. Hrestein, Sequential Transmissin f Digital Infrmatin with Feedback, MIT Research Labratry f Electrnics, TR 375, Sept. 1960. M. Hrstein, Sequential Transmissin Using Niseless Feedback, IEEE Trans. n Inf. Thery, July 1963. F. Jelinek, Cding fr and Decmpsitin f Tw-Way Channels, IEEE Trans, n Inf. Thery, Jan. 1964. J. J. Metzner and K. C. Mrgan, Cded Binary Decisin-Feedback Cmmunicatin Systems, IEEE Trans, n Cmm. Systems, June 1960. E. Parzan, Mdem Prbability Thery and its Applicatins, Jhn Wiley and Sns, 1965. W. W. Petersn, Errr Crrecting Cdes, Jhn Wiley and Sns, Inc., 1961. 55

L. S. Schwartz, Sme Recent Develpments in Digital Feedback Cmmunicatin Systems, IEEE Trans, n Cmm. Systems, June 1961. L. S. Schwartz, Feedback fr Errr Cntrl and Tw-Way Cmmunicatin, IEEE Trans, n Cmm. Systems, March 1963. 56

Security Classificatin DOCUMENT CONTROL DATA -R&D ""'' ylhs^itu-aiin t title, luidy f nbstrm 1 niul indexing anntatin must he entered when the verall reprt is classilieil) INATINO Activiiv (Crprate authr) The MITRE Crpratin Bedfrd, Massachusetts 2a. REPORT SECURITY C L A SSI F 1 C A I I O f Unclassified 2b. GROUP RfPORT TITLE Errr Cntrl Techniques Using Binary Symbl Burst Cdes 4 '^FSCRIPTIVE NOTES (Type l reprt and inclusive dates) N/A AU THORiS) (First name, middle initial, last name) Brayer, Kenneth 6 REPORT DATE September 1967 Ba. CONTRACT OR GRANT NO. AF 19(628)-5165 b. P ROJ F. C T NO 705B 7a. TOTAL NO. OF PAGES 61 7b. NO. OF REFS 28 9a. ORIGINATOR'S REPORT NUMBER(S) ESD-TR-67-379 9b. OTHER REPORT NO(S) (Any ther numbers that may be assigned this reprt) MTP-55 IP DISTRIBUTION STATEMENT This dcument has been apprved fr public release and sale; its distributin is unlimited. I] SUPPLEMENTARY NOTES 12 SPONSORING M,LIT A.,Y ACTIVITY Deputyfr Surveillance and Cntrl Systems, Aerspace Instrumentatin Prgram Office; Electrnic Systems Divisin, T G. HanRnm Field, Bedfrd, Mass IT ABSTRACT Much has been written n the theretical descriptin f errr crrecting cdes but, due t a lack f actual channel errr patterns, little has been said f practical perfrmance. In this paper the perfrmance f three types f errr cntrl is evaluated fr the case f independent randm errrs and fr an actual channel exhibiting dense bursts. The selected cdes are burst cdes with high prbabilities f errr detectin and crrectin. c DD FORM 1 NO V 65 1473 Security Classificatin

Security Classificatin KEY HOR OS E" «T ROLE w T ROLE WT SYSTEMS AND MECHANISMS Data Transmissin Systems v Multichannel Radi Systems Vice Cmmunicatin (Trpscatter, Micrwave, Wireline) Systems INFORMATION THEORY Cding MATHEMATICS Statistical Analysis (Trpscatter, Micrwave, Wireline Errr Lcatins) Statistical Distributins (Trpscatter, Micrwave, Wireline Errr Lcatins) Statistical Data (Trpscatter, Micrwave, Wireline Errr Lcatins) Security Classificatin