THE IMPROVEMENT OF DIGITAL HF COMMUNICATION THROUGH CODING MAY K. Brayer. Prepared for AEROSPACE INSTRUMENTATION PROGRAM OFFICE

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1 OD a. 28 i 03 W I l-h CC U* H l n P H f/5 en ESD-TR ESD RECORD COPY RETURN TO SCIENTIFIC t TECHiW <U INFORMATION DlV-Win (tsti). Bi'll.niNG 1211 THE IMPROVEMENT OF DIGITAL HF COMMUNICATION THROUGH CODING MAY 1968 MTP-75 ESD ACCESSION LIST ESTI Call N. AL Cpy N. /.. y cys. 5> & K. Brayer Prepared fr AEROSPACE INSTRUMENTATION PROGRAM OFFICE ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscm Field, Bedfrd. Massachusetts Th s dcument has been apprved fr public rel sase and talc; its distributin i s un- lim ited. Prject 705B Prepared bj THE MITRE CORPORATION Bedfrd, Massachusetts Cntract AF19(628)-5165 /ID670I* 7

2 When U.S. 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.

3 ESD-TR MTP-75 THE IMPROVEMENT OF DIGITAL HF COMMUNICATION THROUGH CODING MAY 1968 K. Brayer Prepared fr AEROSPACE INSTRUMENTATION PROGRAM OFFICE ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscm Field, Bedfrd, Massachusetts Thi s dcument h as been apprve d fr pu biic release and sa e; its di stribu tin i 5 un- limited. Prject 705B Prepared by THE MITRE CORPORATION Bedfrd, Massachusetts Cntract AF19(628)-5165

4 FOREWORD This reprt was prepared by the Cmmunicatins Techniques Department f The MITRE Crpratin, Bedfrd, Massachusetts, under Cntract AF 19(628) The wrk was directed by the Develpment Engineering Divisin under the Aerspace Instrumentatin Prgram Office, Air Frce Electrnics Systems Divisin, Laurence G. Hanscm Field, Bedfrd, Massachusetts. Rbert E. Frney served as the Air Frce Prject Engineer fr this prgram, identifiable as ESD (ESSID) Prject 5932, Range Digital Data Transmissin Imprvement. REVIEW AND APPROVAL This technical reprt has been reviewed and is apprved. OTIS R. HILL, Clnel, USAF Directr f Aerspace Instrumentatin Prgram Office 11

5 ABSTRACT In previus papers the technique f errr crrectin f digital data thrugh the use f interleaved cyclic cdes and a set f prbability functins fr the evaluatin f errr patterns have been presented. In this paper the previus results are extended t a wide range f BCH and symbl cdes. A set f simple equatins is presented fr the descriptin f an interleaved cyclic cde and its assciated delay, and a methd is presented which allws fr a significant increase in errr rate imprvement at a reductin in the delay time intrduced int the channel. It is demnstrated that the perfrmance f interleaved cyclic cdes is sufficient t crrect all types f measured HF errr patterns; that, using delay as a basis f cmparisn, nly the ttal bit interleaving is imprtant in achieving errr crrectin; and that it is pssible t get almst 100 percent errr crrectin fr delays under 3 secnds fr all channel cnditins measured. 111

6 ACKNOWLEDGMENT The authr wishes t thank E. Churchill and D.K. Leichtman wh, thrugh useful discussins, influenced the frm and cntent f this paper. iv

7 TABLE OF CONTENTS Page SECTION I SECTION n SECTION m SECTION IV SECTION V PART I: INTERLEAVED CYCLIC CODING BACKGROUND DATA DESCRIPTION PERFORMANCE OF BOSE-CHAUDHURI- HOCQUENHEM CODES,M PERFORMANCE OF P SYMBOL CODES CONCLUSIONS PART H: TANDEM INTERLEAVED CYCLIC CODING 29 SECTION I SECTION n SECTION in SECTION IV INTRODUCTION JUSTIFICATION FOR THE TANDEM CODING APPROACH TANDEM CODE PARAMETERS PERFORMANCE OF TANDEM CODING OBJECTIVES OF PERFORMANCE EVALUATION 41 CHANNELS TO BE USED IN EVALUATION 41 PERFORMANCE EVALUATION OF TANDEM CODING 42 APPENDDC 59 REFERENCES 67

8 LIST OF ILLUSTRATIONS Page Figure 1. BCH Cde Perfrmance, Test Run 12 a. n b. n c. n = d. n= e. n Figure 2. BCH Cde Perfrmance, Test Run 24 a. n= 15; 31; b. n = 127; Figure 3. BCH Cde Perfrmance, Test Run 309 a. n = b. n = c n = d. n = e. n Figure 4. BCH Cde Perfrmance, Test Run 339 a. n=15 18 b. n= c. n = d. n= e. n= M Figure 5. P Cde Perfrmance, Test Run 12 a. m= 1 22 b. m=10 22 vii

9 LIST OF ILLUSTRATIONS (Cntinued) Page M Figure 6. P Cde Perfrmance, Test Run 24 a. m = 1 b. m=10 M Figure 7. P Cde Perfrmance, Test Run 309 a. m = 1 b. m= M Figure 8. P Cde Perfrmance, Test Run 339 Figure 9. Figure 10. Figure 11. Figure 12. Figure 13. a. m = 1 b. m= 10 Tandem Cde Cnfiguratin Perfrmance f Half Rate Cdes n Typical TE-216 Data Decded Cnsecutive Errr Distributin a. Nn-Interleaved. Run 309 b. Nn-Interleaved. Run 339 c. Interleaved,Runs 309 and 339 Distributin f Gaps After Decding a. Nn-Interleaved, Run 309 b. Interleaved, Run 309 c. Nn-Interleaved, Run 339 d. Interleaved, Run 339 Cmparisn f Tandem and Single Cding vm

10 LIST OF ILLUSTRATIONS (Cncluded) Figure 14. Figure 15. Figure 16. Figure 17. Figure 18. Figure 19. Figure 20. Figure 21. Figure 22. Figure 23. Cumulative Perfrmance Curves, TE-216 Mdem Tandem Cding Cumulative Perfrmance Curves, TE-216 Mdem Single Cding Cumulative Perfrmance Curves, AN/FGC-60 Mdem Tandem Cding Cumulative Perfrmance Curves, AN/FGC-60 Mdem Single Cding Single-Cde Perfrmance as a Functin f Delay in AN/FGC-60, 600 bits/sec Channel Single-Cde Perfrmance as a Functin f Delay in AN/FGC-60, 1200 bits/sec Channel Tandem Cde Perfrmance as a Functin f Delay in AN/FGC-60, 1200 bits/sec Channel Perfrmance f Tandem Cding n Fur Channels Perfrmance f Single Cding n Fur Channels Empirical Relatinship fr Cde Perfrmance n AN/FGC-61A Data Page APPENDIX FIGURES: STATISTICS OF TYPICAL TEST RUNS Figure 24. Relative Frequency f Cnsecutive Errrs Figure 25. Distributin f Gaps Between Errrs Figure 26. Distributin f Lengths f Bursts Figure 27. Distributin f Burst Errr Densities Figure 28. Distributin f Lengths f Intervals Figure 29. Distributin f Interval Errr Densities Figure 30. Distributin f Guard Space IX

11 LIST OF TABLES Table I Table II Table III Table IV Table V Typical Test Runs Cde Perfrmance n Large Data Samples Descriptin f Channel Data Empirical Perfrmance Relatins Best BCH Cdes in Tandem Cde Evaluatin Page XI

12 PART I INTERLEAVED CYCLIC CODING - 1 -

13 SECTION I BACKGROUND In the paper, "Evaluatin f Errr Crrectin Blck Encding fr High Speed HF Data," cllected n the Eastern Test Range. a system f cding was applied t HF digital errr patterns General characteristics f the bserved data errr patterns intrduced int an HF digital data transmissin system were derived and shwn t be useful in the evaluatin and design f frward errr detectin and crrectin using interleaving with cyclic errr-crrectin cdes. The technique is applicable t digital data transmissin links that require near real-time syn- -5 chrnus peratin and lw errr rates (less than 10 bit errr rate). It is a fundamental requirement that, if a priri knwledge f the message infrma- tin transmitted is nt available at the receive terminal f the transmissin link, redundant data bits must be added t the infrmatin transmitted t achieve errr-detectin and crrectin capability. Thus errr-crrectin capability is btained at the price f transmissin delay time intrduced by errr-crrectin implementatin and by the additin f redundant data bits t prvide the necessary cding structure fr each data wrd transmitted. The errr-pattern data gathered frm an peratinal HF data trans- missin link was shwn t have characteristics that permit the direct classi- ficatin f the bserved errr patterns int predminantly randm, burst, and peridic errr categries. Randm-errr patterns were characterized by the relatively unifrm distributin f errrs thrughut the bserved data. On the ther hand, burst-errr patterns were characterized by the appearance f clusters f errrs in the data stream. Peridic errrs were demnstrated t be caused by a high errr rate n ne f the mdem tne channels relative 3 -

14 t the errr rate n the ther tne channels. Data runs with bit errr rates _3 greater than 10 were predminantly (94 percent) burst-errr type runs. -4 Of the data runs with bit errr rates less than 10, ver 72 percent were randm-errr type runs. The number f randm-errr data runs that had -3-4 bit errr rates between 10 and 10 was apprximately equal t that fr the burst-errr data runs f the same errr-rate class. The ccurrence f peridic errrs increased with increasing bit errr rate. Fr example, 83-2 percent f the runs in the 10 errr-rate decade cntained peridic errrs, -5 whereas nly 4. 5 percent f the runs in the 10 decade cntained peridic errrs. The measure f randmness f bserved errrs was established by cmparisn with distributins f knwn randm-errr data derived frm a cmputer prgram that was designed t generate randm-errr pattern data using a randm-number generatr. The magnitude f the area between the bserved errr data distributin and the crrespnding knwn randm-errr data run was used t evaluate the degree f randmness f bserved data errrs. runs. Six typical data runs were selected frm a ttal f 151 bserved data These data runs were then analyzed using the characteristic distributin functins f the errr patterns. Three f the data runs were selected as randm-errr data runs, and the ther three were selected as burst-errr data runs. Errr crrectin withut interleaving was evaluated fr the randm data runs, and it was fund that the best perfrmance is achieved using pwerful randm-errr-crrecting cdes such as a Bse- Chaudhuri (255, 123, 19) cde. The three burst-errr runs were used t evaluate perfrmance f errr crrectin with and withut interleaving. Interleaving used with a mdified Glay errr-crrecting cde gave results superir t thse btained using this cde withut interleaving. Thus a valid methd was established t permit the quantitative evaluatin f interleaver - 4 -

15 perfrmance in terms f effective randmizatin f burst-errr patterns. Criteria were als established fr prper design f the interleaver t minimize the effect f peridic errrs. There were, hwever, many questins left unanswered. Sme f these questins, which will be answered here, are: Hw d cdes f rates ther than ne-half perfrm? Hw is delay related t perfrmance? Are lng symbl cdes useful n HF errr patterns? 5 -

16 SECTION II DATA DESCRIPTION [12 3] As in previus papers the apprach used in cde evaluatin will be that f using typical runs. A typical run (Table I) is typical in the sense that it represents the type f errr pattern frequently fund at sme errr rate. Table I Typical Test Runs Data Rate Length Average Test Run N. (bits/sec) (min) Bit Errr Rate Surce 1 Antigua t Cape \Kennedy x10" 3 \ Philc x 10" 5 J AN/USC-12 f mdem 4. 3 x 10" 3 [ Antigua t ) Lped \ Ascensin x 10~ 2 ) Kineplex TE-216 / mdem A descriptin f the TE-216 can be fund in Reference [l] and f the AN/USC-12 in Reference [2]. Further infrmatin is available frm manufacturers' catalgs. The test prcedure can be fund in Reference [l]. A detailed descriptin f these test runs is in the Appendix. It is demnstrated in the Appendix that test run 12 is characterized by bursts f errrs but that

17 these bursts cntain randm and peridic errrs and are nt high-density bursts. Test run 24 cnsists f shrt, high-density bursts. Test runs 309 and 339 cntain bursts in which the errr rate within the burst is high and the intervals between bursts (guard spaces) are shrt. These test runs als cntain peridic errrs. At the cnclusin f Part II f this paper, the cding system which is fund t be best will be evaluated against all HF data previ- [3] usly reprted n

18 SECTION in PERFORMANCE OF BOSE-CHAUDHURI-HOCQUENHEM CODES [4 5] Bse-Chaudhuri-Hcquenhem (BCH) cdes are a class f cyclic, randm-errr crrecting cdes. They are described by the ntatin (n, k, e) where k infrmatin bits are encded tgether t btain n ttal bits, and f these n bits e bit errrs can be crrected. The term "cde rate" is defined by the relatin cde rate = k/n (1) The cdes are treated here as interleaved cdes. In interleaved cdes the cde wrds f length n are created by encding infrmatin bits I t generate parity bits P accrding t the fllwing: (k-l)m+l + s '"' 2m+l + s m + l + s 1 + s P... P P P (n-k-l)m + l + s 2m + l + s m+l + s 1 + s s = 0,1,.. (m-1) (2) The interleaving parameter is m. If m is ne, there is n interleaving. The fllwing example will demnstrate the value f interleaving. Example: If the mdified Glay cde is chsen, the defining parameters are (24, 12, 3). If m is 1, then 12 cnsecutive infrmatin bits are encded t create the 24-bit cde wrd. In this cde wrd if three errrs are intrduced they are crrected. If six cnsecutive errrs are intrduced there is n crrectin and actually sme errr generatin. If, hwever, m were 2, then 12 bits chsen as either the dd r the even bits frm the first 24 bits are encded tgether t get an interleaved blck f tw cde wrds. If six - 9 -

19 cnsecutive errrs ccur in the channel they will fall three each in the tw distinct cde wrds, and if there are n ther errrs in the new ttal blck f tw wrds, the errrs will be crrected. Frm the pint f view f cnsecutive errrs, interleaving redefines the cde as (mn, mk, me). The bjective f interleaving is t spread a burst f errrs int the guard space which immediately fllws it. Since the guard space has few errrs, a cde blck is created where the actual number f errrs in the cde blck is less than "me," and, since the bursts have been spread ut, errr crrectin can be achieved. The penalty fr the imprve- ment due t interleaving is in time delay. In an (n, k, e) cde all n bits must be received befre all crrectins can be perfrmed; thus there is a delay f n bit times at the data rate. The delay in the interleaved system is 2 mk [2] infrmatin bit times. The general equatin fr delay is D. _ 2 mk Infrmatin Data Rate' If the infrmatin data rate is 1200 bits/sec and the cde rate is 0. 5 then the channel rate will be 2400 bits/sec. Thus we find that Infrmatin Data Rate = (cde rate) (channel data rate); (4) als mnk mk = = mn (cde rate). (5) 10

20 Thus an alternate equatin t (3) which will be useful later is ~,.. 2 mn _ mn Channel Data Rate l/2(channel Data Rate)' Using Equatin (4) it is seen that delay has been nrmalized t a half-rate cde. The perfrmance f these cdes is described as percent f input errrs crrected. Hwever, fr ease f display, imprvement factr is defined as: Imprvement Factr = ~ r-= - -r (7), % f Errrs Crrected 100 In rder t evaluate the perfrmance f BCH cdes a cmputer prgram was develped t take the actual errr patterns measured and find the percent f errrs crrected fr varius cdes. The results are presented in Figures 1-4, shwing imprvement factr as a functin f cde rate. All curves are identified by a dublet f which the first number is n and the secnd is nrmalized delay in secnds. The channel data rate is fixed at 2400 bits/sec since all errr patterns were measured at this data rate. The cdes evaluated are BCH cdes where n = ^Q 8. (8) A hrizntal line with arrws n it indicates that 100 percent f the errrs were crrected (infinite imprvement). The curves are pltted as smth curves, but values were btained nly fr cdes which exist. - 11

21 t INFINITE IMPROVEMENT BCH CODES n = l5 IOOOO p- QIOOO U A (15,1.31)- (15,.118)- 100 : s > gc Q (I5..006) I5..03I) -i 1 i i i i_ A.3 Z CODE RATE l~) Figure la. BCH Cde Perfrmance, Test Run 12, n= 15 f INFINITE IMPROVEMENT BCH CODES n= 31 IOOOO OIOOO Ul 100 S Ul > <t a? 10 Figure lb. BCH Cde Perfrmance, Test Run 12, n = CODE RATE (f) 12

22 t INFINITE IMPROVEMENT BCH CODES n* (63,.498) (63,1.07) ; IOOO = s > QE I >0 10 CODE RATE ( _i i i i_ 4 3 ' i Figure lc. BCH Cde Perfrmance, Test Run 12, n = 63 f INFINITE IMPROVEMENT BCH CODES n»l27 _L_L OlOOO (127,.264) Figure Id. BCH Cde Perfrmance, Test Run 12, n=

23 t INFINITE IMPROVEMENT BCH CODES n=255 -(255, 4.35) -(255,2.02) -( ) > cr 106) CODE RATE ( ) Figure le. BCH Cde Perfrmance, Test Run 12, n= 255 t INFINITE IMPROVEMENT -! + ± (15, 556) / / (15, 256), / / Figure 2a. BCH Cde Perfrmance, Test Run 24, n = 15; 31; 63 CODE RATE (-r-)

24 BCH CODES t INFINITE IMPROVEMENT (127,2.17) I I I I I I I L_ BCH CODES n»255 1 t (255,.106) CODE RATE (-=-) Figure 2b. BCH Cde Perfrmance, Test Run 24, n= 127; 255 t INFINITE IMPROVEMENT BCH CODES n«is 0T O h- U z 2 " : ^(15,256) (15, 556)^ ^V(I5,.II8) /^/ ^^(I5, 08I) ^^/^/(I5, 006) IMPROV i i i i i i i i i J2I< ' CODE RATE (7-) Figure 3a, BCH Cde Perfrmance, Test Run 309, n= 15 f\j

25 t INFINITE IMPROVEMENT 1000 BCH CODES n = 3l \ J IT u < iz 100 t z LLI S K a 3 l0 i (31 ' 529»N^^^l3Ci3T (31,1.15)^' ^ CODE RATE (-^-) N -1 Figure 3b. BCH Cde Perfrmance, Test Run 309, n= 31 t INFINITE IMPROVEMENT BCH CODES n= (63,2.33L tr (63,I.07)~^C/ y(63,.l3l) r (63 498) t- z (63,026) s,0 r cr a. 5 i i J ,1 0 CODE RATE (-pp) Figure 3c. BCH Cde Perfrmance, Test Run 309, n = 63 16

26 INFINITE IMPROVEMENT BCH CODES n = l I CODE RATE (^-) Figure 3d. BCH Cde Perfrmance, Test Run 309, n= 127 t INFINITE IMPROVEMENT 1000 IT O h- O if > 10 K a. BCH CODES n = 255 (255,945) (255,435 t t 255, 106) (255,2 02) (255,.531) CODE RATE (- -) Figure 3e. BCH Cde Perfrmance, Test Run 309, n =

27 t INFINITE IMPROVEMENT BCH CODES n = l5 cr (- <t LL 100 r (15,1.31) LJ > O (T Q r _l_ J l l 1_ CODE RATE { n Figure 4a. BCH Cde Perfrmance, Test Run 339, n = 15 f INFINITE IMPROVEMENT BCH CODES n= 31 O i IMPROVEMENT _ 6 r (31,1 \5)y# /yaz\,064) ////^' 0I2) i i i > i i i i C CODE RATE (- -) Figure 4b. BCH Cde Perfrmance, Test Run 339, n = 31 r r <

28 t INFINITE IMPROVEMENT BCH CODES n = 63 r (63,2.33)/ / 1 Ji h- // / O 1 // / 100 / /\/ (63, 498)^/// A t- z /"// * * LLl //// (63, 131) s //^/ y^>v^(63, 026) 5 l0 JytS IT ^^rj^^r Q..^^S^^ z ^Z^^00^ 1 ^^ t t CODE RATE (- -> n r^ r~- N < Figure 4c. BCH Cde Perfrmance, Test Run 339, n = 63 t INFINITE IMPROVEMENT BCH CODES n=l27 a 27, 052) > a: a j i i i_ I 0 :0DE RATE (-fp) Figure 4d. BCH Cde Perfrmance, Test Run 339, n=

29 i t INFINITE IMPROVEMENT BCH CODES n=255 IOOO a. ( )/ U- 100 r (255, 53\)~T \- /(255..I06) z Ul 2 Ul 5 '«a. a Z ^f^*^0*"^ ^10 CODE RATE (- -) Figure 4e. BCH Cde Perfrmance, Test Run 339, n = 255 Cnsidering the set f curves fr any ne test run, it is interesting t nte that fr the same delay all cdes are apprximately equal in perfrmance. On test run 309 the perfrmance fr n = 127 at a delay f 2.17 (m = 41) is almst identical t that fr n = 255 at a delay f (m = 19). Thus it can be cncluded that perfrmance is a functin f the prduct mn and nt f m r n individually, if delay is cnsidered the cmmn grund upn which tw r mre cdes are cmpared. The values f m used are 1, 5, 19, 41, and 89, althugh in sme cases the upper values are nt presented wing t lack f significant imprvement in perfrmance. In all cases it is pssible t get 100 percent crrectin if the cde rate is sufficiently reduced (i. e., an increasing prtin f the channel is allcated t parity). It can be cncluded that, at a reasnable penalty in delay, the errr rate in HF digital cmmunicatin can be substantially reduced by the interleaved BCH cde technique

30 SECTION IV PERFORMANCE OF P M SYMBOL CODES. [3 Q] Symbl cdes ' are structured in a fashin similar t BCH cdes. In a symbl cde a grup f bits is taken tgether as a symbl, a set f infr- matin symbls is encded tgether t get a blck, and sme prtin f the symbls can be crrected. The parameter M is used t designate the number f bits/symbl. There is a ttal f N = 2-1 symbls/blck f which e symbls can be crrected if there are 2e parity symbls. The cde rate is as given belw: cde rate = infrmatin symbls = N - 2e ttal symbls N All previus relatins are the same when expressed in symbls, except delay: Del = 2 2 mmn _ 2 mmn ay Channel Data Rate ' 1/2 (Channel Data Rate) ( ' The factr f tw is intrduced int the delay because f the extremely cm- [9] plicated calculatins necessary in decding and is already accunted fr n the curves (Figures 5-8). The perfrmance f this system fr interleaving f m = 1 and m = 10 is presented in Figures 5-8. The perfrmance f these cdes is, even fr the high delays, inferir t that f the BCH cdes. The reasn lies in the structure f the cdes. If m = l, M = 8, N = 255 is cnsidered fr run 339, the ttal bits/blck is At a cde rate near 0. 5, a ttal f 64 symbl errrs r 512 cnsecutive bit -21 -

31 t INFINITE IMPROVEMENT M = 8,34- M>6, 63- M-7,148- M = 3,035 M-4,.10- M-5, W 2 LU > <X - 2 P M CODES CODE RATE M Figure 5a. P Cde Perfrmance, Test Run 12, m = 1 t INFINITE IMPROVEMENT 10,000 I / / 1000 m T ^"1 / in 1 / M / "M r? N / "/* 1 "^ A / a. \ i // >- / / / #/*> / 1/"^ < I / / /A>" u. t- 1 / ///^ S / /// s ill > en a. z 10 M Figure 5b. P Cde Perfrmance, Test Run 12, m= 10 s r- C-l CD If/ P M CODES II// m 10 i i I ) CODE RATE 22

32 t INFINITE IMPROVEMENT CODE RATE Figure 6a. r Cde Perfrmance, Test Run 24, m = 1 f INFINITE IMPROVEMENT "~rtr f-m^6,63 f-m-5,2 56 / / km-4,10 / 1 M'3, 35 / M Figure 6b. P Cde Perfrmance, Test Run 24, m = 10 in N S s O O 2 Z 10 s > 1 1/ /. P M CODES m»l0 li > < CODE RATE J - 23

33 t INFINITE IMPROVEMENT M-8,34 M-7,1 48 M-5, 258 M-6, 63 M'4,10 z CODE RATE Figure 7a. P Cde Perfrmance, Tesr Run 309, m = 1 t INFINITE IMPROVEMENT M = 8,34.0 M = 7,I4.8 M'6,6.3' 100 > a. 2 M=3, 35 P M CODES m = IO M Figure 7b. P Cde Perfrmance, Test Run 309, m = I 0 CODE RATE 24

34 1000 M M = 5,258 CODES m = l P M FACTOR z u Ul > 1 10 M*6,.63/"7 rl M = 4,.l/w7// M»3,035 /yw / ////('*** 8,3.4 /^^ M» 7, CD l -^Tl ; i. i i i i D CODE RATE M Figure 8a. P Cde Perfrmance, Test Run 339, m = 1 t INFINITE 1000 IMPROVEMENT M= 7,148 M = 6,6 3 M* 5, M = 4,I0 M = 8, 34 > tr a 5 10 P M M=3, 35 CODES m 10 M Figure 8b. P Cde Perfrmance, Test Run 339, m = Z I CODE RATE 25

35 errrs can be crrected. The bursts are nt slid, hwever, but are spread ut, and frequently mre than 64 symbls are in errr even thugh less than 512 bits/blck are in errr. Thus, althugh the delay is 3. 4 secnds the imprvement factr is less than 5. Since the bursts in run 339 are spread ut, a cde (BCH) that crrects bit errrs is superir. On run 24, which has the slid bursts that the symbl cde prefers t crrect, the bursts are shrt and the symbl cdes (with and withut interleaving) d as well as r better than the BCH cdes. Unfrtunately, very little f the bserved data is similar t [9] run 24. In anther study these cdes were shwn t wrk excellently with lng bursts f the type bserved in test run

36 SECTION V CONCLUSIONS The bjective f Part I has been t examine the perfrmance f easily implemented cyclic cdes with interleaving n typical samples f errr patterns that ccur in HF cmmunicatin. It has been fund that when cdes f equivalent errr-crrecting capability are used, the verall perfrmance is gverned nly by the prduct f cde length (n) and interleaver length (m) (i.e., the delay intrduced int the system). Further, it has been demnstrated that the perfrmance f symbl cdes (i. e., cdes with a greater dense-burst crrecting capability) is inferir t that f binary cdes fr a given amunt f delay. The reasn fr this is that the bursts which ccur in the channel are generally lw-density, diffuse bursts. The results btained in this wrk are extended t all previusly described HF data and t a mre sphisticated cding system in Part II f this paper. - 27

37 PART n TANDEM INTERLEAVED CYCLIC CODING

38 SECTION I INTRODUCTION The Natinal Range Divisin (NRD) presently perates high-speed (2400 bits/sec) HF cmmunicatin links t transmit data back t the Cape Kennedy cmplex frm dwn-range statins such as Ascensin Island. Because f the high errr rates nrmally experienced n these HF channels, errr crrectin must be implemented t imprve cmmunicatin. Previus [12 3] papers have described the perfrmance f simple cyclic cdes n HF errr patterns measured upn these circuits. It has been demnstrated that using half-rate cdes (50 percent f the bits transmitted are parity) and delays frm five t ten secnds it is pssible t get crrectin f 90 percent f the errrs that ccur. The methd f encding used has been interleaved cyclic cding. In interleaved cyclic encding, infrmatin bits I are encded tgether t get parity bits P accrding t the fllwing relatins: (k-l)m + l + s " ' " 2m + l + s m + l + s 1 + s P P P P (n-k-l)m+l + s " 2m + l + s m + l + s 1 + s where s = 0,1,... (m-1) n is the number f bits in a cde wrd k is the number f infrmatin bits m is the interleaving parameter As has been demnstrated in Part I, the effect f interleaving is t redefine the cde as an (mn, mk, me) cde where the number f crrectable

39 randm errrs e is multiplied by m, thus increasing the randmizatin f the errrs by placing errrs that ccur in the same burst int different cde wrds. In this paper the interleaving technique will be extended t the tandem (cncatenated) cding apprach presented by Frney (and blck diagrammed in Figure 9). In the tandem cding apprach, infrmatin frm the data surce is given t an uter cder which perfrms interleaved encding and passes the encded infrmatin and its assciated parity t the inner cder, which perfrms the same functins n the ttal data stream. The inner cder prvides the data t the cmmunicatin equipment that cnstitutes the transmit prtin f the channel. Errrs are intrduced by the actin f the HF channel and demdulatin failures, and are crrected by the decding prcedures. This system will be simulated and demnstrated t be superir t using a single interleaved cde. DATA DATA SOURCE SINK 1 1 t OUTER OUTER 1 I CODER DECODER INNER INNER CODER DECODER 0 < CHANNEL S C7\ L IWr ( ) i ERRORS J i Figure 9. Tandem Cde Cnfiguratin 32

40 SECTION II JUSTIFICATION FOR THE TANDEM CODING APPROACH [1,2] The results presented in previus papers and the results f Part I f this paper give a gd indicatin f the capability f cyclic cdes, with and withut interleaving, in terms f errr crrectin perfrmance. An example f this perfrmance is given in Table H fr all the channel data cl- lected with the tw mdems described in Part I. Table II Cde Perfrmance n Large Data Samples TE-216 Mdem AN/USC-12 Mdem Channel Data Rate 2400 bits/sec 2400 bits/sec Ttal Bits 4. 8 x x 10 8 Ttal Errrs 2.6 x x 10 5 Errr Rate 5.4x 10~ x 10~ 3 Output Errr Rate 9.3 x 10" x 10~ 6 (24,12,3) Cde; m = 211 Output Errr Rate 1.6 x 10~ x 10~ 6 (16,4,3) Cde; m =

41 Using the frmulas f Part I, the verall delay with the (24,12, 3) cde is 2 mk Infrmatin Data Rate 1200 = secnds, and the delay with the (16,4,3) cde is secnds. The verall imprve- ment factrs fr the tw cdes using the TE-216 mdem and the AN/USC-12 mdem are 58 and 337, and 361 and 809, respectively. It remains, hwever, t find a better apprach. The first indicatin [1,2] f such an apprach is in the previus papers where it is demnstrated that errr crrectin perfrmance increases rapidly as a functin f m until m»89 and thereafter increases slwly. This is demnstrated graphically in Figure 10 where a set f half-rate cdes are evaluated against the test data (previusly described in detail ). As the interleaving increases beynd M=89 it becmes desirable t find anther fast-rising perfrmance curve rather than t cntinue with ne that shws n additinal imprvement. IOOO t INFINITE IMPROVEMENT (24, 12, 3) #309 (32,16,3) #309 (16, 8,2) #309 (8,4,1) #309 (24, I2,3)#339 I6,8,2)#339 (32,I6,3)#339 (8,4,1) #339 (n,k,e) RUN NUMBER i ' i i i i i ' i 11 I i 111 i i I I WORDS INTERLEAVED Figure 10. Perfrmance f Half Rate Cdes n Typical TE-216 Data 34 -

42 It is demnstrated in Figures 11 and 12, where the cnsecutive-errr and gap functins fr test runs 309 and 339* after decding are presented with m = 1 and m = 89, that when m = 1 the cdes crrect nly the randm errrs and have n effect n the bursts. The utput errrs ccur in dense bursts with highly prnunced peridic errrs (as demnstrated by the vertical jumps in the gap functins). The curves fr m = 89, where the cde was spread thrugh the data by interleaving, shw that the remaining errrs are ramdm. Since the errrs after decding are randm, it is suggested that, using a tandem cding apprach, an inner cder that is interleaved with m ^ 89 be placed in tandem with an uter cder that will crrect the remaining randm errrs. Befre evaluating the tandem cding apprach, it is necessary t identify sme parameters f the tandem cder. 100 RUN N. 309» (15,7,2) s (31,16,3) 70 * (63,30,6) a (127,64,10) S 65 * ' (255,123,19) U 60 s ALL ABOVE COOES 2 Ul PLOTTED AT SAME 55 POINT 3, m» 1 U y s JO u. O B > 25 U 2 ^ O 1 : IS u : ( i %: 5 0 * CONSECUTIVE ERRORS r CD Figure 11a. Nn-Interleaved Decded Cnsecutive Errr Distributin-Run 309 * These test samples are described in detail in the Appendix

43 " - T r S 60r 50- RUN NO 339 x (15, 7, 2) (31, 16, 3) a (63, 30, 6) (127, 64, 10) + (255, 123, 19) ALL ABOVE CODES PLOTTED AT SAME POINT m= I 3 *0h K U S CONSECUTIVE ERRORS Figure lib. Nn-Interleaved Decded Cnsecutive Errr Distributin - Run 339 I0C< 1 RUN N 309 " (15,7,2) H ; (31,16,3) (63,30,6) 90 g RUN N 339 M " (15,7,2) y as a» (31,16,3) S a: (63,30,6) 3 80 (127,64,10) (255,123.19) O u. 75 m»89 O. s z 3 S 15 K U Figure lie Interleaved Decded Cnsecutive Errr Distributin Runs 309 and 339 K ^j CD 5 X 0 : i V CONSECUTIVE ERRORS 36

44 cr cc O u. > z u D cc < TEST RUN 309 m = 1 CURVE CODE 1 (15,7,2) 2 (31,16,3) 3 (63,30,6) 4 (127,64,10) 5 (255,123,19) _J I I I'll l l ' i i I ml -I I I I I III , ,000 1,000,000 GAP SIZE (BITS) Figure 12a. Distributin f Gaps After Nn-Interleaved Decding Run u r cr 3 u (_> > u 3 CL > < -J 5 ;J u 100 1,000 10, ,000 1,000,000 GAP SIZE (BITS) Figure 12b. Distributin f Gaps After Interleaved Decding Run

45 - 90 u Z LU 80 - ^C^xz^- cc 4 <*%/ 3 i a 2 D O r u '"5 TEST RUN 339 u. 60 m = 1 CURVE CODE >- u 4 1 (15,7,2) z LU SO v/\\ \ 3 2 (31,16,3) 40 LU (63,30,6) cc LL. 4 (127,64,10) 30 LU 5 (255,123,19) > 1-20 < _l s in C\J r- 0...I 1 1 i CNJ ,000 10, ,000 1,000,000 < GAP SIZE (BITS) K-* Figure 12c. Distributin f Gaps After Nn-Interleaved Decding Run 339 CM z LU CC cc <_> <J u_ O > U z LU O cr u. LU > < _l S /^f //f 3 54 /^ TEST RUN /// m = 89 /^/ CURVE CODE 50 - //// 1 (15,7,2) z//l 2 (31,16,3) 40 //// 3 (63,30,6) 30 HlJl 4 (127,64,10) 3 yj/l\ 5 (255,123,19) ,000 10, ,000 l.ooc,000 CM < GAP SIZE (BITS) Figure 12d. Distributin f Gaps After Interleaved Decding Run

46 SECTION HI TANDEM CODE PARAMETERS One imprtant parameter f a cde is the cde rate, which is the rati f infrmatin bits t ttal bits in a cde wrd. This parameter describes the prtin f the cmmunicatins channel that is being used fr infrmatin transmissin. The verall cde rate (rati f ttal infrmatin bits t ttal bits) f this system is the prduct f the cde rates f the tw cdes. Thus, in rder t maintain the same rati f infrmatin t ttal bits as used fr single cdes (see Part I), the tw cdes used here will f necessity have less individual errr crrectin capability than a single cde. This is easily verified in terms f Bse-Chaudhuri-Hcquenhem (BCH) cdes where, A * k n ~ P - 1 2e+ l cde rate - = ^ 1 - (12) n n n If the cde rate is increased fr a given blck length (n), the number f crrectable errrs (e) must be decreased. This will be balanced ut by taking advantage f the additinal randmizatin gained in a decder utput. The delay f the system is the sum f the delays intrduced by the tw cdes. Using the frmulas f Part I, 6 ay 2m n 2m. n. _ 1 1 (Channel Rate) + (Channel Rate). ( '

47 where the subscript i refers t the inner decder and the subscript refers t the uter decder. Thus (Channel Rate), is the cmmunicatin channel data rate. Since the number f infrmatin bits (k.) exiting the inner decder is the number f ttal bits (n ) entering the uter decder, Thus k n (Channel Rate) = (Channel Rate). - (Channel Rate).. (14) i i 2m n. 2m. n. T^ i ,,.. Delay 3 = + (15) (Channel Rate). (Channel Rate). 2n. [m + m.j 1 1 (Channel Rate). (16) where m = number f uter cde wrds interleaved m. = number f inner cde wrds interleaved I n. = ttal number f bits in a tandem cde blck I n t = I + P 1 + P 2 (17) where I P P = number f surce infrmatin bits = number f parity bits due t uter cde = number f parity bits due t inner cde

48 SECTION IV PERFORMANCE OF TANDEM CODING OBJECTIVES OF PERFORMANCE EVALUATION In this sectin the perfrmance f tandem cding n actual channel errr patterns will be evaluated. The evaluatin will be in three parts: (1) the cmparisn f tandem cding with single cding (nly ne cde) fr three f the fur typical test samples described in Part I f this paper; (2) a demnstratin f the capabilities f tandem cdes n a large sample f data; and (3) the perfrmance as a functin f delay and cde rate. The perfrmance f the mdified Glay cde, which is the best half-rate cde, will be given fr reference. CHANNELS TO BE USED IN EVALUATION The cmmunicatin channels t be used in the general evaluatin f [l 3] tandem cding have been previusly described but will be briefly reviewed here. Exact details n test prcedures and cmmunicatin equip- ment can be fund in the references. Six different mdulatin systems (mdems) were used t cllect the data. These are the Kineplex TE-202, Kineplex TE-216, AN/FGC-60, AN/FGC-61A, SC-302, and S-3000X. The TE-216 and AN/FGC-60 were used n a lped basis between Antigua and Ascensin, with bth transmissin and receptin at Antigua and reruting at Ascensin. This test was cnducted in September and Octber f The ther fur mdems were used between Pretria, Suth Africa, and Riverhead, Lng Island, New Yrk, in the spring f 1964 fr a six-week perid. transmissin was n a ne-way basis with receptin at Riverhead. The The tests were cnducted using the nrmal perating equipment and circuits n a nn- - 41

49 interference basis. All receptin used dual space diversity with rhmbic antennas. All test runs were apprximately 10 minutes lng spaced arund the clck. The test prcedure was t cntinuusly transmit a cyclic, repeating, 52-bit digital message frm the Transmitter Facility t the Receiver Facility, where the message was detected. This received test message was cmpared with the riginal test message suitably delayed t match the ttal transmissin time f the test link. The cmparisn was made using a mdul-tw adder which summed the riginal test message (delayed) and the received test message. The utput f the mdul-tw adder indicated a binary 'ne* state whenever the tw input signals were nt the same. The utput signal f the mdul-tw adder was recrded in real-time n magnetic tape. The errr patterns recrded in the field n magnetic tape were prcessed thrugh a tape cnverter which generates an IBM-cmpatible tape. An verall descriptin f the data is presented in Table HI. Additinal infrmatin n mdem characteristics is available frm manufacturers' catalgs. PERFORMANCE EVALUATION OF TANDEM CODING Tandem versus Single Cding n Typical Test Samples In Figure 13 cmparative curves f the ttal delay as a functin f cde rate fr bth single and tandem cdes are presented fr runs 309, 339, and 12 (see Appendix fr a descriptin f the test runs) fr the case f 100 percent crrectin. The pints fr single cdes are chsen frm Part I by selecting the cde that had the lwest delay fr a given cde rate f all the cdes that exhibited 100 percent crrectin. The tandem cdes were evaluated by selecting as an inner cde ne f the BCH cdes listed by Petersn

50 HU» «* Si * * g CD >H > rh CO CO CO CO CM (N CO CO O rt rh rh rh rh rh rh rh * SH X X X X X X X X >* t» CD O rh CO rh H -f t- CM rh CN rh < W CO CM CO -H rh i-h in in CO 1> t- t> l> t> t- t> t> r-t m rh rh rh rh rh rh l-h rh X X X X X X X X 13 HM CO Tf< in t * CM... H Tj" 00 C3 CO <N <N in j> ^ Test (hrs. "3 21 H H rh rh CD in HH ^ rh CM -r rh CD cr> c rh CO CM rh rh r-< p 1 3 '-. c 0) C w CD as a> CC ^ LO m Oi t- N t- CD Cd CN "* CtJ CO rh rh <-< CM rt "tf e CD 3 <H CM rh 13 4_J 13 -U 9 J, -" rh H-> rh H-> a CD CO l >i 32 CD CD * CD bd CD J4 C c rs 1 ( CD CO CD s CO O _ u l fh hi U 1 C CO Cd CD CD CD CD CD 2 CD rc CO Si 3 53 CD 5 1 C 1 CO CIS CD 3 CD CO -a L0 'H «2 3 *- rh CO 35 CD 4= >> T3 CO a a a rh ca a. i-( a 0) A >-> CD a >> es "8 c n CO U U CD CD (0 3 i c c 13 rh C CD a 1 rh CD CD CD i H_i bo!_, CD a be i H-> bc CO CO 3 rh rh c c CD CD OS cr CD 3 CD cr CD.3 5 r. C C t -1 G s ja CD O U CD XI >> 1 P> O <D rh O CD CO CD rv <rh rh <HH <rh CD ^ rh M (H CD > r^l^ O < rh c CO B u X CD 3 CO CD 0 rh csi UH 1 CN 1 CO U w < H < H U 43

51 9 itt ' ' 8 \ 1\ \ \ BEST SINGLE I \ U. 1 CODES» lift, \ " BEST 1 \ \ \ TANDEM \ \ \\ i \ CODES s q 1 CD _ 6 u Ul \ 4 \\* \ \ \\\ v^3 < >- \ \ \\ \ RUN «\ \\\ \ \ \\\ \ 3 2 vv x X \\ V?RUN 12 1 RUN 309^ W\ sa S CODE RATE(-)h) Figure 13. Cmparisn f Tandem and Single Cding fr blck length, (15 ^ ^255), interleaving the cde by m wrds (m ^89), and decding. The uter cde was similarly selected, and the prcess was repeated fr all t and m and all BCH cdes listed by Petersn f rate greater than 0. 4 (allwing the evaluatin f multiple cdes f rate greater than 0.16). Frm these cdes, thse that achieved 100 percent crrectin with the least delay fr a given cde rate were selected. It is seen in Figure 13 that this apprach will, fr 100 percent errr crrectin, reduce the system delay by a factr greater than 2 as cmpared t the delay fr single cding (see Part I). Mre interesting is the fact that test run 339 which has an -2 errr rate f 1. 2 x 10 and fr which it has never befre been pssible t btain an imprvement factr greater than 100 independent f delay at halffi rate ' - is cmpletely crrected, using this apprach, with a 7-secnd delay

52 The cdes that perfrmed best fr each f the three typical data runs were cnsistent with the results f Part I. The best inner cde was always the cde which gave the mst imprvement fr a given delay and value f m when cnsidered as a single cde, but all inner cdes acted as a functin f the mn prduct alne. Since the utput errrs f the inner decder were randm, the lngest cyclic cde (n = 255) when used as an uter decder, perfrmed best (as expected) n the remaining errrs. The pints in Figure 13 are thus the result f varius cmbinatins f all the BCH cdes cnsidered used as inner cdes and the 255-bit cyclic cde, and n interleaving was necessary with the 255-bit uter cde t crrect the remaining errrs. Perfrmance f Tandem Cding n the Measured HF Channels T evaluate the perfrmance f a cmmunicatins channel, a parameter cmmnly used is the percent f time that the channel average bit errr rate (BER) is less *han r equal t a given value. Fr cnvenience the channel data has been subdivided int equal intervals. Since the data used here had been cllected in 10-minute channel intervals, this interval length was selected fr use in the simulatin. In rder that the infrmatin bits in a decded interval will crrespnd n a ne-t-ne basis with thse in a channel interval, the prduct f the cde rate and the channel interval length is selected as the decded interval length. The effect f the use f tandem cding can best be understd by examin- ing the cumulative perfrmance f varius channels, using the tandem apprach and using the single interleaved cde apprach, fr a given cde rate. Half-rate [2] cding, which is generally used in practice, ' is examined first. The single cde used is the mdified Glay cde f length 24 bits, which crrects 3 errrs at rate 0. 5 interleaved by m- 125 fr a 2. 5-secnd delay in a channel where the infrmatin rate is 1200 bits/sec The tandem cde used has as an inner

53 cde a 63-bit length and crrects 3 errrs at rate The uter 255-bit cde crrects 10 errrs at rate The verall cde rate is The inner cde is interleaved by 34 cde wrds (m. = 34); the uter cde is nn- interleaved. The ttal delay f the tandem cde is 2.56 secnds. Using the [l 2] cdes selected, the perfrmance f the best single cde (Glay) can be cmpared with the perfrmance f tandem cding while keeping the ther parameters cnstant (cde rate, delay). The results f applying the previusly indicated cdes t the channel data are presented in Figures 14 thrugh 17. It is evident frm these figures that the tandem-cding apprach is superir t single encding. Fr example, in the channel with the TE-216 mdem at 2400 bits/sec (Figures 14 and 15) the -4 channel exhibits errr rates prer than 10 fr 80 percent f the intervals. With the Glay cde this ccurs fr 22 percent f the intervals and with tandem cding it never ccurs. Further, 83 percent f the intervals are errr-free after tandem decding, whereas with the Glay cde nly 52 percent are errr- free, and in the actual channel nly 2 percent are errr-free. Similar curves fr the AN/FGC-60 data are presented in Figures 16 and 17. In Figures 18 and 19 the perfrmance f the Glay cde as a functin f delay fr the AN/FGC bit/sec data is presented, and the same per- frmance curves are given in Figure 20 using the tandem cde set previusly defined. N curves can be presented fr the AN/FGC-60 data at 600 bits/sec, tandem-cded, since the increase f either m r m. yields a channel with n I J utput errrs. The same results ccur with the TE-216 data. The apparent incnsistency in stating that all f the TE-216 data is crrected with less than 3 secnds delay, while test sample 339 (a member f the sample) required 7 secnds fr 100 percent crrectin, is explained by the fact that 100 percent

54 s v-^. Ss v"~ TANDEM-CODE DECODED PERFORMANCE I c_> K CL I 0 CHANNEL "N PERFORMANCE, TE-216 \\ 2400 B/S \ I200B/S 10 MIN. CHANNEL INTERVALS 5 MIN INFORMATION INTERVALS ' I0" Z I0" ALL OTHER INTERVALS ERROR FREE \ \ AVERAGE BIT ERROR RATE (BER) Figure 14. Cumulative Perfrmance Curves, TE-216 Mdem Tandem Cding ^\ s \ N^. SINGLE-CODE DECODED PERFORMANCE 80 < cr 5 cr cr 3 cr JO?0 10 TE B/S 1200 B/S \v CHANNEL \ PERFORMANCE 10 MIN CHANNEL INTERVALS 5 MIN INFORMATION INTERVALS ' \ NV AVERAGE BIT ERROR RATE (BER) V \ ALL OTHER INTERVALS ERROR FREE Figure 15. Cumulative Perfrmance Curves, TE-216 Mdem Single Cding

55 a t t CD < VI CO cr cr TO \ \] \ \ \ \ V CHANNEL \ \ PERFORMANCE\ \ \ \ \ \ \ \ TANDEM-COOE DECODED \ PERFORMANCE \ ALL OTHER INTERVALS ERROR FREE i CT SO AN/FGC-6C ) I200B/S 600B/S I0MIN CHANNEL INTERVALS 5MIN INFORMATION INTERVALS N >V v. s. 10 I0" 1 10 ~ 3 I0" 4 AVERAGE BIT ERROR RATE (BER) Figure 16. Cumulative Perfrmance Curves, AN/FGC-60 Mdem Tandem Cding < CO CO u CO CO < VI CO DC cr r cr "< v "> \ \ SINGLE-CODE DECODED ^\ PERFORMANCE \> \ \ \ \ \ N \ \ CHANNEL \ \ V \ PERFORMANCEX \ \ \ \ \ \ \ ALL OTHER INTERVALS ^ \ ERROR FREE \ AN/FGC-60 I200B/S 600B/S >v I0MIN CHANNEL INTERVALS 5MIN INFORMATION INTERVALS V % ^ -I? AVERAGE BIT ERROR RATE (BER) Figure 17. Cumulative Perfrmance Curves, AN/FGC-60 Mdem Single Cding

56 BO Z 30 z u r?0 10 AN/FGC B/S ^> V N X. SINGLE-CODE \ S \ N N DECODED PERFORMANCE \0 \ \\ \ 3 SEC V, DELAY "* \ i \ ^ s \ " 4 5 X \ CHANNEL \ \ PERFORMANCE\ \ \ \ \ ^^ IOMIN CHANNEL INTERVALS 5MIN INFORMATION INTERVALS \ \ \ \ N AVERAGE BIT ERROR RATE (BER) ALL OTHER INTERVALS ERROR FREE Figure 18. Single-Cde Perfrmance as a Functin f Delay in AN/FGC-60, 600 bits/sec Channel < 00 CD < It E -J I y s u. fe z <_> r SINGLE-CODE \DEC0DED PERFORMANCE ^ N\^ Si \> \ \ L \ \ V CHANNEL \ \ PERFORMANCE\ \ \ \ \ \ \ \ AN/FGC-60. s 3 SEC ^ ^v 4 DELAY ALL OTHER INTERVALS ERROR FREE 20 I200B/S 600B/S 10 X. V, IOMIN CHANNEL INTERVALS 5 MIN INFORMATION INTERVALS ' , K) AVERAGE BIT ERROR RATE(BER) Figure 19. Single-Cde Perfrmance as a Functin f Delay in AN/FGC-60, 1200 bits/sec Channel

57 < c CO 100 TANDEM-CODE DECODED PERFORMANCE ^5 56 CO CD < 80 VI CO 70 < DELAY \> \ 3 ^v CHANNEL \ \ PERFORMANCEX \ \ \ \ \ \ \ \ ALL OTHER INTERVALS ERROR FREE U. h- JO AN/FGC-6C ) 600B/S 10 MINI CHANNEL INTERVALS 5MIN INFORMATION INTERVALS \ N, " K) " 10 AVERAGE BIT ERROR RATE(BER) Figure 20. Tandem Cde Perfrmance as a Functin f Delay in AN/FGC-60, 1200 bits/sec Channel crrectin means crrectin f all errrs in the parity as well as the infrma- tin, whereas in the large sample evaluatin, nrmal decding prcedures were used which allw fr residual errrs in the parity, since the parity is discarded after decding. The perfrmance f the same tandem cde set and f the Glay cde, using the ther fur mdems (S-3000X, TE-202, SC-302, and AN/FGC-61A), is presented in Figures 21 and 22. It is again nted that the tandem cde per- frmance is better than the Glay, which frm previus evaluatin is the best single cde. It shuld als be nted that with the S-3000X mdem, which is [12] similar t the AN/GSC-10, little imprvement can be achieved with either technique fr the delay f 2.5 secnds. The reasn is that errrs ccur in lng, dense bursts with this mdem which can nly be crrected with m fr the Glay cde and m + m 300 fr the tandem cde cnsidered. These J 1 values represent delays n the rder f ne minute

58 100 TANDEM-CODE DECODED PERFORMANCE CHANNEL PERFORMANCE S-3000X TE-202^ z tr a I0MIN CHANNEL INTERVALS 5 MIN INFORMATION INTERVALS 10" (VI < AVERAGE BIT ERROR RATE ( BER) Figure 21. Perfrmance f Tandem Cding n Fur Channels CO < < cc cc cc CHANNEL \V\ PERFORMANCE \\\ 1 \N\ S-3000X- ^\ \\V it- u<: SINGLE-CODE DECODED PERFORMANCE -SC-302 S-3000X k TE-202 AN/FGC-6IA VSC-302 <c 5 40 Z 30 u. O z 20 AN/FGC-6IA 10MIN CHANNEL INTERVALS 5 MIN INFORMATION INTERVALS 10 10" c 10 J 10"' K>" = AVERAGE BIT ERROR RATE (BER) ALL OTHER INTERVALS ERROR FREE 10" Figure 22. Perfrmance f Single Cding n Fur Channels - 51

59 Perfrmance f Tandem Cding As a Functin f Cde Rate and Delay The remaining step is t evaluate tandem cding as a functin f cde rate. This presents prblems, bth because there are large numbers f cdes t chse frm and because delay, being a functin f n., will be interrelated t cde rate, making it difficult t cmpare cdes n the basis f a cmmn delay. Perfrmance curves are nt presented, since they are meaningful nly in terms f the data used. Instead, when it was discvered that perfrmance was linear in all variables, an empirical relatinship between the average interval bit errr rate and the percent f interval errr rates belw the average was determined as a functin f delay and cde rate. Fr the S-3000X, SC-302, TE-202, AN/FGC-61A, AN/FGC-60 and TE-216 at 1200 bits/sec, the empirical relatinship was fund t be f the frm Y = (a + a R + ^J X + b + b R + b D (18) where Y = percent f interval errr rates ^abscissa X = Bit Errr Rate = abscissa a = fixed cmpnent f slpe factr b = fixed cmpnent f ffset a_ = f (cde rate) ap = f (delay) b = f (cde rate) b D = f (delay)

60 As stated previusly a, a^, b, and b are, in general, functins f bth cde rate and delay since cde rate and delay are in themselves interrelated; hwever, fr the test data cllected with the five indicated mdems, the effects f cde rate and delay were themselves separable int linear relatinships. The results f this empirical analysis are tabulated in Table IV. Fr the TE-216 mdem at 2400 bits/sec there was a wide scattering f pints fr which n relatinship in the frm f Equatin (18) culd be fund; hwever, it was fund that the relatinship fr the TE-202 usually gives perfrmance prer than the Table IV Empirical Perfrmance Relatins Data Classificatin Relatinship S-3000X Y = (7509D R )X + (7.49D R ) AN/FGC-61A Y = (66360D R )X + (13.25D R ) TE-202 Y = (25544D R )X + (15.30D R ) SC-302 Y - (414999D R )X + (15.41D R ) AN/FGC bits/sec AN/FGC bits/sec TE bits/sec Y = (98710D R )X + (11.71D R ) Y = (98710D R )X + (15.85D R ) Y = (19274D R )X + (15.29D R ) Restrictins 0 < Y ^ x 10" 6 ^ X ^ 1 x 10~ < R = cde rate < 1 0 < D = delay <2. b sec 53

61 TE-216 at 2400 bits/sec, and it is recmmended that the TE-202 curves be used as a wrst-case estimate. A set f these curves fr AN/FGC-61A data is presented in Figure 23. A subset f the BCH cdes listed by Petersn which were fund t give the best actual perfrmance as inner cdes is tabulated in Table V. Additinally, cmments n the selectin f uter cdes are Table V Best BCH Cdes in Tandem Cde Evaluatin (by practical evaluatin) Inner Cdes n k e Ntes: m. ^ 89 shuld be used with inner cdes Output errrs f inner decder cr er are sufficiently randm that the uter cde can be chsen n delay and cde rate cnsideratins nly Symbl cdes can be used as inner cdes nly if the uter cde is interleaved (m > 1)

62 100 CO a: AN/FGC-6IA \ R =.6 ^"D«2 \ R=.75 \p», 2 < > R«'.85 D«2 R"i O " I0" 2 I0~ s I0" 4 AVERAGE BIT ERROR RATE (BER) 10" i- Figure 23. Empirical Relatinship fr Cde Perfrmance n Ar4/FGC-61A Data [9] given as ntes n that table. It shuld be nted that symbl cdes ' which in Part I f this paper were fund inferir when used singly are satisfactry as inner cdes prvided that the uter cde is interleaved. As single cdes the errr crrectin capability f the symbl cdes culd easily be exceeded; thus the cdes wuld fail in bursts. In tandem cding the uter interleaved binary cde will prvide the additinal errr crrectin necessary. - 55

63 SECTION V CONCLUSION Interleaving when used in cnjunctin with a pwerful randm-errr crrecting cde will prvide an imprvement factr that is apprximately ne-tenth f the reciprcal f the input bit errr rate. This imprvement has been presented graphically in the frm f give-shaped cumulative distribu- tin curves. Fr the bserved HF channels with bit errr rates nt wrse _2 than 10, it is pssible thrugh the use f ne level f interleaved encding t crrect apprximately ne-half f the errr bursts and leave the remaining errrs in a near randm state. When the secnd level f encding is super- impsed (cncatenated) upn the first the result is that the bit errr rate at the utput f the tandem (cncatenated) system is rarely prer than 10 The decded blcks are almst all errr-free prvided that the verall cde rate is in the neighbrhd f ne half and the delay is in the range f tw t fur secnds. Thus it has been demnstrated that the divisin f redundant bits frm a single level f encding int tw cncatenated levels f encding with interleaving is an efficient methd f errr cntrl in the HF channels cnsidered. 57 -

64 APPENDIX The prbability distributin functins that describe the data presented here are: 1. Distributin f Cnsecutive Errrs 2. Distributin f Gaps 3. Distributin f Burst Lengths 4. Distributin f Burst Densities 5. Distributin f Interval Lengths 6. Distributin f Interval Densities 7. Distributin f Guard Space In the case f independent randm errrs the cumulative distributins f cnsecutive errrs and gaps are defined by the fllwing equatins: P je r c cf = ]T p j (l-p) < 19 > j = 0 r P jc r e ej = ^ < X -P) J P (20) j = 0 where c represents a crrect bit e represents an errr bit r represents the number f cnsecutive bits p is the prbability f bit errr

65 Relative distributins are cnstructed using the discrete values f j. The relative distributins f cnsecutive errrs are presented in Figure 24. If the values f p (assuming p equals average bit errr rate) fr the varius runs were inserted int Equatin (19), it wuld be fund that in all cases sin- gle errrs shuld ccur ver 98 percent f the time. Only fr run 12 des this happen, giving the first indicatin that run 12 cntains randm errrs. Test run 24 indicates a high ccurrence f shrt dense bursts f errrs, and the ther runs lie smewhere in between. The same cnclusins can be drawn frm the gap distributin functins (Figure 25). Test run 12 has a dis- tributin f gaps near that fr independent randm errrs while run 24 demn- strates a high ccurrence f shrt gaps between errrs, indicating dense bursts. Test runs 12, 309, and 339 als indicate the ccurrence f peridic errrs, as demnstrated by the high ccurrence f gaps f a specific size. Thus the initial indicatin is that run 12 exhibits independent randm errrs and peridic errrs, run 24 exhibits dense bursts, and runs 309 and 339 have peridic errrs and lie between 12 and 24 in burstiness. The abve discussin f prbability functins des nt present the cmplete picture. There is n infrmatin abut the length f bursts, nr is there infrmatin relative t the interval between bursts (guard space). A burst is 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 fr the chsen values f M and A must be satisfied fr the e 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 burst prbability density functin is defined as the prbability f ccurrence f a burst f size N where N is any psitive integer. The burst

66 100, g 80' O RUN * 309 X RUN # 339 O RUN # 24 a RUN # 12 &60 U O Ul 30 Q: li i ' * ' * ' ' CONSECUTIVE ERRORS (BITS) Figure 24. Relative Frequency f Cnsecutive Errrs 100 r 1,000 10, ,000 1,000,000 GAP LENGTH (BITS) Figure 25. Distributin f Gaps Between Errrs -61 -

67 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 fllwing prperties hld fr the burst: 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 minimum number f errrs (M ) in a burst has been chsen t be tw (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 is 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. 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. The guard space rati is defined as the rati f the interval length t burst length preceding it. The burst length and density distributins are presented in Figures 26 and 27. Examinatin f these figures verifies that run 24 is cmpsed f shrt dense bursts. fr runs 309 and 339. The distributins fr test run 12, hwever, are similar t thse Thus, it is cncluded that the randm and peridic errrs f run 12 are actually cntained within bursts. Frm Figures 28, 29, and 30 it is seen that the intervals between the bursts are generally lng, cntain few errrs, and in the case f run

68 100 r- 3 CE t- < _J A = 0 05 ME-- 2 i Mini i i i i mil i i i i mil I I i mil 100 1,000 10, BURST LENGTH (BITS) Figure 26. Distributin f Lengths f Bursts 100 i x RUN RUN \ #309 r # a* 70 >- U Z 60 / / 3 a S 50 c 40 > < / RUN #339 / / RUN #24 A «0.05 ME' 2 0 ~ '\,,,! l I 1.0 BURST ERROR DENSITY Figure 27. Distributin f Burst Errr Densities -63 -

69 100 & ME 2 RUN ^ *309 RUN ^#339 * 80 >- \ RUN O 70 N Z *I2 ^ 60 O Ul : s u_ Ul 40 > H 30 < _l ' - RUN / V* 24 / 0 i i MIIIL^S linilll i i 11 mil i i 1 1 Mill unl i i i mill i i i mill ,000 10, ,000 1,000, ,000 INTERVAL LENGTH ( BITS) Figure 28. Distributin f Lengths f Intervals 100 RUN ^ *24 /y a) it> > 90 z 60 3 RUN /(/ \ # ' 2 III A= \ ME = 2 T 0 \. /V RUN N V^7 *309 ^-^^^v RUN -^^ ^*^"^ X u -»-*rt J,» ^ * y O 50 cr U. 40 LLJ > 30 _ 1- <t _l? U / ! i i mill i mill i i i mill i i mini i i i mill i i 11 mil i! INTERVAL ERROR DENSITY CD " Figure 29. Distributin f Interval Errr Densities -64 -

70 RUN \~#309 RUN # i* 70 >- U *> u CC < 20 I ' <-> / RUN # 12 RUN #24 A= 005 C ME 2 1 i mini i i i mill i i i mill i :n ml i i mill i i mill 100 IpOO 10, ,000 1, GUARD SPACE ( INTERVAL-BURST RATIO) Figure 30. Distributin f Guard Space prvide excellent guard space prtectin. The guard space prtectin in runs 12, 339, and 309 is insufficient in that respectively ver 10 percent, 18 percent, and 26 percent f the bursts are fllwed by intervals that are actually shrter than the burst (guard space rati < 1)

71 REFERENCES 1. K. Brayer, O. Cardinale, "Evaluatin f Errr Crrectin Blck Encding fr High Speed HF Data, " IEEE Trans, n Cmmunicatin Technlgy, June K. Brayer, Design Cnsideratins fr Frward Errr Cntrl Equipment fr HF Radi, ESD-TR , August K. Brayer, Errr Patterns MeasurednTransequatrialHF Cmmunicatin Links, MTP-46, The MITRE Crpratin, Bedfrd, Mass., December (T be published, IEEE Trans, n Cmmunicatin Technlgy, April 1968) 4. R. C. Bse, D. K. Ray-Chaudhuri, "On a Class f Errr Crrecting Binary Grup Cdes, " Inf. and Cntrl, _3, (1960). 5. R. C. Bse, D. K. Ray-Chaudhuri, "Further Results n Errr Crrecting Binary Grup Cdes, " Inf. and Cntrl, 3, (1960). 6. K. Brayer, O. Cardinale, HF Channel Errr Statistics Descriptin (I), ESD-TR , June K. Brayer, O. Cardinale, HF Channel Errr Statistics Descriptin (II), ESD-TR , May M 8. D. Grenstein, N. Zierler, "A Class f Errr Crrecting Cdes in P Symbls, " J. Siam, 9, June K. Brayer, Errr Cntrl Techniques Using Binary Symbl Burst Cdes, MTP-55, The MITRE Crpratin, Bedfrd, Mass., May (T be published, IEEE Trans, n Cmmunicatin Technlgy, April 1968) 10. G. D. Frney, Jr., Cncatenated Cdes, TR 440, MIT Research Lab f Electrnics, December W. W. Petersn, Errr Crrecting Cdes, The MIT Press and Jhn Wiley & Sns, Inc., 1961, p M.S. Zimmerman, A. Kirsch, "The AN/GSC-10 (KATHRYN) Variable Rate Data Mdem fr HF Radi," IEEE Trans, n Cmmunicatin Technlgy, April

72 i Security Classificatin DOCUMENT CONTROL DATA R&D (Security classificatin f title, bdy f abstract and indexing anntatin must be entered when the verall reprt Is classified) ORIGINATING ACTIVITY (Crprate authr) The MITRE Crpratin Bedfrd, Massachusetts 2a. REPORT SECU1I TY CLASSIFICATION UNCLASSIFIED 2b. GROUP N/A 3 REPORT TITLE THE IMPROVEMENT OF DIGITAL HF COMMUNICATION THROUGH CODING 4. DESCRIPTIVE NOTES (Type f reprt and inclusive dates) N/A 5- AOTHORIS) ffirsf name, middle initial, last name) Kenneth Braver 6 REPORT DATE May » CONTRACT OR GRANT NO AF 19(628)-5165 b. PROJEC T NO 705B 7a. TOTAL NO. OF PAGES 68 9fl. ORIGINATOR'S REPORT NUMBER'S) ESD-TR b. NO. OF REFS OTHER REPORT NO<S> (Any ther numbers that may be assigned this reprt) MTP DISTRIBUTION STATEMENT This dcument has been apprved fr public release and sale; its distributin is unlimited. II SUPPLEMENTARY NOTES N/A 13 ABSTRAC T 12. SPONSORING MILITARY ACTIVITY Aerspace Ihstrumen tatin Prgram Office, Electrnic Systems Div* isin, Air Frce Systems Cmmand, L. G. Hanscm Field, Bedfrd, Massachusetts In previus papers the technique f errr crrectin f digital data thrugh the use f interleaved cyclic cdes and a set f prbability functins fr the evaluatin f errr patterns have been presented. In this paper the previus results are extended t a wide range f BCH and symbl cdes. A set f simple equatins is presented fr the descriptin f an interleaved cyclic cde and its assciated delay, and a methd is presented which allws fr a significant increase in errr rate imprvement at a reductin in the delay time intrduced int the channel. It is demnstrated that the perfrmance f interleaved cyclic cdes is sufficient t crrect all types f measured HF errr patterns; that, using delay as a basis f cmparisn, nly the ttal bit interleaving is imprtant in achieving errr crrectin; and that it is pssible t get almst 100 percent errr crrectin fr delays under 3 secnds fr all channel cnditins measured. DD F N 0 O R V M 1473 Security Classificatin

73 Security Classificatin KEY WO RDS SYSTEMS AND MECHANISMS Data Transmissin Systems Multi-Channel Radi System Vice Cmmunicatin (HF) Systems INFORMATION THEORY Cyclic Cding Cncatenated Cding Security Classificatin