NAVAL POSTGRADUATE SCHOOL THESIS

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1 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALFORNA THESS PERFORMANCE ANALYSS OF AN ALTERNATVE LNK-6/JTDS WAVEFORM TRANSMTTED OVER A CHANNEL WTH PULSE-NOSE NTERFERENCE y Cham Kok Kiang Marh 8 Thesis Advisor: Seond Reader: Clark Roertson Roerto Cristi Approved for puli release; distriution is unlimited

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3 REPORT DOCUMENTATON PAGE Form Approved OMB No Puli reporting urden for this olletion of information is estimated to average hour per response, inluding the time for reviewing instrution, searhing existing data soures, gathering and maintaining the data needed, and ompleting and reviewing the olletion of information. Send omments regarding this urden estimate or any other aspet of this olletion of information, inluding suggestions for reduing this urden, to Washington headquarters Servies, Diretorate for nformation Operations and Reports, 5 Jefferson Davis Highway, Suite 4, Arlington, VA -43, and to the Offie of Management and Budget, Paperwork Redution Projet (74-88) Washington DC 53.. AGENCY USE ONLY (Leave lank). REPORT DATE Marh 8 4. TTLE AND SUBTTLE Performane Analysis of an Alternative Link- 6/JTDS Waveform Transmitted Over a Channel with Pulse-Noise nterferene 6. AUTHOR(S) Cham Kok Kiang 7. PERFORMNG ORGANZATON NAME(S) AND ADDRESS(ES) Naval Postgraduate Shool Monterey, CA SPONSORNG /MONTORNG AGENCY NAME(S) AND ADDRESS(ES) N/A 3. REPORT TYPE AND DATES COVERED Master s Thesis 5. FUNDNG NUMBERS 8. PERFORMNG ORGANZATON REPORT NUMBER. SPONSORNG/MONTORNG AGENCY REPORT NUMBER. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflet the offiial poliy or position of the Department of Defense or the U.S. Government. a. DSTRBUTON / AVALABLTY STATEMENT. DSTRBUTON CODE Approved for puli release; distriution unlimited 3. ABSTRACT (maximum words) The Joint Tatial nformation Distriution System (JTDS) is a hyrid frequeny-hopped, diret sequene spread spetrum system that utilizes a (3,5) Reed-Solomon (RS) ode and ylial ode-shift keying modulation for the data pakets, where eah enoded symol onsists of five its. The primary drawak to JTDS is the limited data rate. n this thesis, an alternative waveform onsistent with the existing JTDS hannel waveform ut with a two-fold inrease in data rate is analyzed. The system to e onsidered uses (3,5) RS enoding as in the original JTDS, ut eah pair of five-it symols at the output of the Reed-Solomon enoder undergo serial-to-parallel onversion to two five-it symols, whih are then independently transmitted on the in-phase and quadrature omponents of the arrier using 3-ary iorthogonal keying with a diversity of two. The performane otained with the alternative waveform is ompared with that otained for the existing JTDS waveform for the relatively enign ase where additive white Gaussian noise is the only noise present as well as when pulse-noise interferene (PN) is present. Errors-and-erasures deoding as well as errors-only deoding is onsidered. Based on the analyses, we see that the proposed alternative JTDS/Link-6 waveform performs etter in AWGN as well as when PN is present. No signifiant advantage is otained using EED for the alternative waveform. There is a signifiant improvement in performane when perfet-side information is assumed. 4. SUBJECT TERMS JTDS/Link-6, M-ary Bi-Orthogonal Keying, Reed-Solomon oding, Pulse-Noise nterferene, Additive White Gaussian Noise, Error-and-Erasure deoding 5. NUMBER OF PAGES PRCE CODE 7. SECURTY CLASSFCATON OF REPORT Unlassified 8. SECURTY CLASSFCATON OF THS PAGE Unlassified i 9. SECURTY CLASSFCATON OF ABSTRACT Unlassified. LMTATON OF ABSTRACT NSN Standard Form 98 (Rev. -89) Presried y ANS Std UU

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5 Approved for puli release; distriution is unlimited PERFORMANCE ANALYSS OF AN ALTERNATVE LNK-6/JTDS WAVEFORM TRANSMTTED OVER A CHANNEL WTH PULSE-NOSE NTERFERENCE Kok Kiang Cham Civilian, Defene Siene & Tehnology Ageny (DSTA), Singapore B.Eng. (EE), Nanyang Tehnologial University, Singapore, Sumitted in partial fulfillment of the requirements for the degree of MASTER OF SCENCE N ELECTRCAL ENGNEERNG from the NAVAL POSTGRADUATE SCHOOL Marh 8 Author: Kok Kiang Cham Approved y: R. Clark Roertson Thesis Advisor Roerto Cristi Seond Reader Jeffrey B. Knorr Chairman, Department of Eletrial and Computer Engineering iii

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7 ABSTRACT The Joint Tatial nformation Distriution System (JTDS) is a hyrid frequeny-hopped, diret sequene spetrum system that utilizes a (3,5) Reed-Solomon (RS) ode and ylial ode-shift keying modulation for the data pakets, where eah enoded symol onsists of five its. The primary drawak to JTDS is the limited data rate. n this thesis, an alternative waveform onsistent with the existing JTDS hannel waveform ut with a two-fold inrease in data rate is analyzed. The system to e onsidered uses (3,5) RS enoding as in the original JTDS, ut eah pair of five-it symols at the output of the Reed-Solomon enoder undergo serial-to-parallel onversion to two five-it symols, whih are then independently transmitted on the in-phase and quadrature omponents of the arrier using 3-ary iorthogonal keying with a diversity of two. The performane otained with the alternative waveform is ompared with that otained for the existing JTDS waveform for the relatively enign ase where additive white Gaussian noise is the only noise present as well as when pulse-noise interferene (PN) is present. Errors-and-erasures deoding as well as errors-only deoding is onsidered. Based on the analyses, we see that the proposed alternative JTDS/Link-6 waveform performs etter in AWGN as well as when PN is present. No signifiant advantage is otained using EED for the alternative waveform. There is a signifiant improvement in performane when perfet-side information is assumed. v

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9 TABLE OF CONTENTS. NTRODUCTON... A. OVERVEW... B. THESS OBJECTVE... C. THESS OUTLNE.... BACKGROUND...5 A. M-ARY BORTHOGONAL SGNALS...5 B. PERFORMANCE OF M-BOK N AWGN...7 C. PERFORMANCE N AWGN WTH PULSED-NOSE NTERFERENCE...7 D. PERFORMANCE WTH DVERSTY...8 E. FORWARD ERROR CORRECTON CODNG...9 F. ERRORS-AND-ERASURES DECODNG... G. PERFECT-SDE NFORMATON...3 H. CHAPTER SUMMARY...3. V. PERFORMANCE OF ALTERNATVE JTDS/LNK-6 WAVEFORM N SNGLE-PULSE STRUCTURE 5 A. PERFORMANCE OF 3-BOK WTH (3,5) RS CODNG N AWGN...5 B. PERFORMANCE N AWGN AND PULSE-NOSE NTERFERENCE...6 C. PERFORMANCE WTH ERRORS-AND-ERASURES DECODNG N AWGN... D. PERFORMANCE WTH ERRORS-AND-ERASURES DECODNG N AWGN AND PULSE-NOSE NTERFERENCE...6 E. CHAPTER SUMMARY...3 PERFORMANCE OF THE ALTERNATVE JTDS/LNK-6 WAVEFORM WTH DVERSTY TWO (DOUBLE-PULSE STRUCTURE)..33 A. PERFORMANCE N AWGN WTH A DVERSTY OF TWO...33 B. PERFORMANCE N AWGN AND PN WTH A DVERSTY OF TWO...35 C. PERFORMANCE N AWGN AND PN WTH A DVERSTY OF TWO AND EED...39 D. PERFORMANCE WTH PERFECT-SDE NFORMATON N AWGN AND PN...49 E. CHAPTER SUMMARY...5 V. COMPARSON OF THE JTDS/LNK-6 WAVEFORM AND THE ALTERNATVE JTDS/LNK-6 WAVEFORM...53 A. COMPARSON OF JTDS/LNK-6 AND THE ALTERNATVE JTDS/LNK-6 WAVEFROM, SNGLE-PULSE STRUCTURE Comparison for AWGN...53 vii

10 . Comparison in AWGN and PN Performane using EED in AWGN and PN...56 B. COMPARSON OF THE JTDS/LNK-6 AND THE ALTERNATVE JTDS/LNK-6 WAVEFROM, DOUBLE-PULSE STRUCTURE Comparison in AWGN Performane Comparison in AWGN and PN Performane with EED in AWGN and PN Performane with PS in AWGN and PN...63 C. CHAPTER SUMMARY...64 V. CONCLUSONS AND FUTURE RESEARCH AREAS...65 A. CONCLUSONS...65 B. FUTURE RESEARCH AREAS...66 LST OF REFERENCES...67 NTAL DSTRBUTON LST...69 viii

11 LST OF FGURES Figure. Blok diagram of a M-ary iorthogonal reeiver...6 Figure. Performane of the oded and unoded alternative JTDS/Link-6 waveform....6 Figure 3. Performane of the oded and unoded alternative JTDS/Link-6 waveform for ρ= and E = 9 db....8 Figure 4. Performane of the oded and unoded alternative JTDS/Link-6 waveform for ρ=. and E = 9 (db)...9 Figure 5. Performane of 3-BOK with (3,5) RS oding in PN with E = 6 db for different values of ρ.... Figure 6. Performane of 3-BOK with (3,5) RS oding in PN with E = 9 db for different values of ρ.... Figure 7. Performane of 3-BOK with (3,5) RS oding and EED for different values of a in the presene of AWGN...6 Figure 8. Performane of 3-BOK with (3,5) RS oding and EED for ρ= and E = 6 db for different values of a....9 Figure 9. Performane of 3-BOK with (3,5) RS oding with EED in a PN environment for a =.6 and E = 6 db...3 Figure. Performane of 3-BOK with RS oding with and without EED with E = 6 db and a =.6 for different ρ...3 Figure. Performane of 3-BOK with (3,5) RS oding with and without EED with E = 6 db and a =.4 for different values of ρ...3 Figure. Performane of 3-BOK with (3,5) RS oding for oth the single-pulse and the doule-pulse struture in AWGN...34 Figure 3. Performane of 3-BOK with (3,5) RS oding and the doule-pulse struture for different ρ with E =.4 db...37 Figure 4. Performane of 3-BOK with (3,5) RS oding and the doule-pulse struture for different ρ with E = 3 db...38 Figure 5. Performane of 3-BOK with (3,5) RS oding for oth the doule-pulse struture ( E =.4 db) and the single-pulse struture ( E = 5.4 db)...39 Figure 6. Performane of 3-BOK with (3,5) RS oding and EED for the doulepulse struture with ρ =.5 and E =.5 db for different values of a...43 Figure 7. Performane of 3-BOK with (3,5) RS oding and EED for the doulepulse struture with ρ =.5 and E = 5 db for different values of a...44 Figure 8. Performane of 3-BOK with (3,5) RS oding and EED E =.5 db, a=.6 for the doule-pulse struture...45 ( ) ix

12 Figure 9. Performane of 3-BOK with (3,5) RS oding and EED ( E = 5 db, a=.6) for the doule-pulse struture...46 Figure. Performane of 3-BOK with (3,5) RS oding, EED, a =.6, for ρ = for the doule-pulse ( /.5 db) ( / 5.5 db) E N = and the single-pulse struture E N =...47 Figure. Performane of 3-BOK with (3,5) RS oding with and without EED with E =.5 db and a =.6 for different values of ρ...48 Figure. Performane of 3-BOK with (3,5) RS oding with and without EED with E = 5 db and a =.6 for different values of ρ Figure 3. Performane for 3-BOK with (3,5) RS oding with and without PS for different ρ ( E =.5 db)...5 Figure 4. Performane of 3-BOK with (3,5) RS oding and JTDS/Link-6 in AWGN Figure 5. Performane of 3-BOK with (3,5) RS oding ( E = 5.5 db) and JTDS waveform ( E = 7.6 db) for different values of ρ...55 Figure 6. Performane of the alternative JTDS/LNK-6 waveform ( a =.6, E =5.5 db) and the JTDS/Link-6 waveform with EED (threshold=4, E =7.3 db) for different values of ρ Figure 7. Performane of the alternative JTDS/Link-6 waveform ( a =.6) and the JTDS/Link-6 waveform (threshold=4) with EED at E = 7.3 db for ρ = and Figure 8. Performane of the alternative JTDS/Link-6 waveform and the JTDS waveform for the doule-pulse struture Figure 9. Performane of the alternative JTDS/Link-6 waveform ( E =.4 db) and the JTDS/Link-6 waveform ( E = 4.5 db) for the doule-pulse struture....6 Figure 3. Performane of the alternative JTDS/LNK-6 waveform ( a =.6, E =.5 db) and the JTDS/Link-6 waveform (threshold=4, E =4.4 db) with EED....6 Figure 3. Performane of the alternative JTDS/Link-6 waveform ( a =.6) and the JTDS/Link-6 waveform (threshold=4) with EED at E = 4.4 db Figure 3. Performane for the alternative JTDS/Link-6 (PS, E =.5 db) and the JTDS/Link-6 waveform (EED, threshold=4, E = 4.3 db) x

13 EXECUTVE SUMMARY Digital datalinks are the tehnology at the heart of modern wireless networks and are also the tehnologial asis of systems supporting Network Centri Warfare and Network Enaled Operations. The aility to provide real time tatial data updates to all memers of a network is ruial to ahieving information supremay and situational awareness in today s omplex war threater. The Joint Tatial nformation Distriution System (JTDS)/Link-6 is an advaned tatial datalink that is used y a numer of different ountries. t provides oth voie and data ommuniations for ommand and ontrol, navigation, relative positioning, and identifiation. JTDS/Link-6 is a time-division, multiple aess ommuniation system operating at L-and frequenies. Many tehniques are used in JTDS. These inlude the use of Reed Solomon enoding for error detetion and orretion, frequeny hopping and diret sequene spread spetrum tehniques that makes JTDS more resistant to jamming, and data enryption to make it a seure data network. Only a small fration of the availale radio andwidth in the L and is used at any one time as a result of frequeny hopping. n diret-sequene spread spetrum, eah digital symol is represented y a pseudo-random sequene for transmission. This redues the amount of ahievale data throughput per radio andwidth in proportion to the length of the pseudo-random spreading ode. Thus, while oth frequeny hopping and diret sequene spread spetrum provide JTDS with jam-resistane, the effetive data throughput is redued. As a digital system for oth data and voie, JTDS needs to handle large amounts of data - far more than the ommuniation systems now used for similar purposes. Throughput is one of the asi measures of performane for any datalink or digital ommuniations system and is linked to the type of modulation used. JTDS uses yli ode-shift keying (CCSK) and minimum-shift keying (MSK), whih is a type of ontinuous phase-shift modulation (CPSM), to modulate the digital data. The data are first enoded using a (3,5) Reed Solomon ode. These oded symols are interleaved xi

14 and modulated using a set of CCSK ode symols to produe 3-hip sequenes. The hips are transmitted using MSK. The primary drawak to JTDS is a large overhead whih results in a limited data rate. n this thesis, an alternative waveform that is onsistent with the existing JTDS hannel waveform ut with a two-fold inrease in data rate is analyzed. The system to e onsidered uses (3,5) Reed Solomon enoding as in the original JTDS, ut eah pair of the five-it symols at the output of the Reed Solomon enoder undergo a serial-toparallel onversion to two five-it symols. They are then independently transmitted on the in-phase and quadrature omponents of the arrier using 3-ary i-orthogonal keying (3-BOK) with a diversity of two. As a result, this system supports a data rate of twie that of the existing JTDS waveform and is onsistent with the diret sequene waveform generated y JTDS. The performane otained with alternative waveform is ompared with that otained with the existing JTDS waveform for the relatively enign ase where AWGN is the only noise present as well as when pulse-noise interferene (PN) is present. Errors and erasure deoding as well as errors-only deoding is onsidered. Based on the results of this thesis, the proposed alternative JTDS/Link-6 waveform has etter performane than the existing JTDS/Link-6 waveform in AWGN as well as when pulse-noise interferene (PN) is present. There is no signifiant advantage in using EED for the alternative waveform either in only AWGN or with oth AWGN and PN. There is a signifiant improvement in performane when PN is present and perfet-side information is assumed. xii

15 ACKNOWLEDGMENTS would like to express my utmost gratitude to Professor Clark Roertson of the Naval Postgraduate Shool, Monterey, California, for his guidane, patiene and ontriution to the suessful ompletion of this thesis work. would also like to thank Professor Roerto Cristi for serving as my seond reader and reviewing this thesis. would also like to thank my parents for their support and enouragement. Last, ut not least, must thank my sponsor, Defene Siene and Tehnology Ageny (DSTA), for providing me the opportunity to pursue my work here in the Naval Postgraduate Shool. xiii

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17 . NTRODUCTON A. OVERVEW Digital datalinks are the tehnology at the heart of modern wireless networks and are also the tehnologial asis of systems supporting Network Centri Warfare and Network Enaled Operations. The aility to provide real time tatial data updates to all memers of a network is ruial to ahieving information supremay and situational awareness in today s omplex war threater. The Joint Tatial nformation Distriution System (JTDS)/Link-6 is one of the most advaned tatial datalinks that is in use in today s armed fores. t provides oth voie and data ommuniations for ommand and ontrol, navigation, relative positioning, and identifiation. JTDS/Link-6 is a time-division, multiple aess ommuniation system operating at L-and frequenies []. JTDS/Link-6 uses (3,5) Reed-Solomon (RS) enoding and yli ode shift keying (CCSK) modulation for data pakets, where eah enoded symol onsists of five its and is spread into 3 hips per symol for transmission using minimum-shift keying (MSK) modulation. The primary drawak to JTDS/Link-6 is the limited data rate that an e ahieved. B. THESS OBJECTVE Numerous studies of ways to inrease the data rate of the JTDS/Link-6 throughput have een made. One example is Link-6 Enhaned Throughput (LET), whih works y replaing the spread spetrum and RS enoding of the original JTDS waveform with a RS/onvolutional oding sheme whih an adapt to required link apaility [] [3]. However, this inrease in data rate is at the expense of oth jamming resistane and transmission range. Thus, LET may not prove pratial for omat senarios. Other papers [4], [5], [6] related to JTDS inlude omparison of a CCSK waveform with an orthogonal waveform [4] and an analysis of different error-ontrol

18 oding tehniques for high-rate diret sequene spread spetrum [5]. n [6], the authors derive an analytial approximation for the proaility of symol error for a CCSK waveform with RS oding. n [7], this approximation is shown to e optimisti y aout db. To the est of the author s knowledge, the analysis of a JTDS/Link-6 ompatile waveform otained y replaing CCSK with M-ary i-orthogonal keying (MBOK) and taking into aount pulse-noise interferene has not een previously investigated. The ojetive of this thesis is to investigate an alternative physial layer hannel waveform that is ompatile with the existing JTDS hannel waveform ut with the potential to inrease the data rate y a fator of two as well as redue the required signal-to-noise ratio. The alternative JTDS/Link-6 waveform investigated utilizes a omplex MBOK waveform with ( n, k ) RS oding. MBOK an e thought of as a hyrid of M-ary orthogonal modulation and inary phase-shift keying (BPSK). The data first undergoes forward error oding (FEC) using RS oding, and the oded data undergoes serial-toparallel onversion to two five-it symols whih are independently modulated with MBOK on the in-phase () and quadrature (Q) omponents of the arrier. To e onsistent with JTDS/Link-6 waveform, we use 3-BOK and a (3,5) RS ode. The proposed waveform provides a two-fold inrease in the data rate with the same spetral effiieny as the JTDS/Link-6 waveform. The performane of the proposed alternative JTDS/Link-6 waveform is ompared to the results of the existing JTDS/Link-6 waveform in [7] oth when AWGN is the only noise present as well as with PN. The performane of the alternative waveform for oth single as well as dual diversity is examined. C. THESS OUTLNE The introdution to the thesis was presented in this hapter. The alternative waveform is disussed in Chapter. The performane analysis of the alternative waveform with RS oding and no diversity is presented in Chapter. The performane analysis of the alternative waveform with RS oding and a diversity of two is disussed

19 in Chapter V, and the performane of the alternative JTDS/Link-6 waveform and that of the JTDS/Link-6 waveform with no diversity as well as with a diversity of two, for oth an AWGN and a PN environment, are ompared in Chapter V. The thesis onlusions ased on the results otained are presented in Chapter V. 3

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21 . BACKGROUND n this hapter, we introdue some of the akground knowledge and onepts required for our susequent analysis of the alternative JTDS/Link-6 waveform onsidered in this thesis. A. M-ARY BORTHOGONAL SGNALS A set of M iorthogonal signals an e onstruted from M/ orthogonal signals y inluding the negatives of eah of the orthogonal signals. Thus, a iorthogonal set is really two sets of orthogonal odes suh that eah symol in one set has its antipodal symol in the other set. One advantage of iorthogonal modulation over orthogonal modulation for the same data is that iorthogonal modulation requires one-half as many hips per symol. Thus, the andwidth requirement for iorthogonal modulation is onehalf of that required for omparale orthogonal modulation. Sine antipodal signal vetors have etter distane properties than orthogonal ones, iorthogonal modulation performs slightly etter than orthogonal modulation [8]. The hannel waveform for omplex MBOK an e represented y () () os( ) ( ) ( ) sin ( ) s t =± A t π f t+θ ± A t π f t+θ (.) i j whih is transmitted for T = kt seonds, k is the numer of its in eah symol, and x () t represents a waveform of s k pulses of duration T / k = Ts, where i or j may or k may not e different depending on the data its. Clearly, omplex -BOK is equivalent k to transmitting -BOK independently on oth the and Q omponents of the arrier, so k omplex -BOK is atually a k -ary modulation tehnique. A lok diagram of a M-ary iorthogonal reeiver is shown in Figure. 5

22 st () A os( w +θ) A w +θ sin ( ) ( t) M / ( t) ( t) ( t) ( t) Ts s T Ts s T Ts s T Ts s T Ts s T ( ) ( ) ( ) ( ) ( ) dt dt dt dt dt X X X M / X X X X X M/ X X Choose Largest Choose Largest M / M / ( t) t Ts s T ( ) dt X M / X M/ Figure. Blok diagram of a M-ary iorthogonal reeiver. The onditional proaility density funtion for the random variales X where m=,,..., M /, that represents the integrator outputs when the noise an e m onsidered Gaussian noise are 6

23 and where ( x ) m A fx ( x ) exp for / m m m = m M, (.) πσ σ ( ) xm + A f X ( x ) exp for / m m m = M + m M, (.3) πσ σ σ = N / Ts. x n fx ( x, ) exp n n n n m = (.4) πσ σ B. PERFORMANCE OF M-BOK N AWGN When AWGN is present with power spetral density N /, the proaility of hannel symol error for MBOK in AWGN is [9] M u E s (.5) N E / s N ps = e Q( u+ ) du π where Es is the average energy per hannel symol, whih is equal to AT, where s A is the average reeived signal power, T s is the symol duration, and Q () is the Q-funtion. Equation (.5) will e used for deriving the proaility of symol and it error for the alternative JTDS/Link-6 system in the next hapter. C. PERFORMANCE N AWGN WTH PULSED-NOSE NTERFERENCE For military appliations, it is imperative that we also onsider the performane of the system when sujeted to PN. n this thesis, we onsider the performane of the alternative JTDS/Link-6 system in AWGN as well as PN. When a hannel is affeted y AWGN, the noise signal that arrives at the reeiver is assumed to e uniformly spread aross the spetrum and time-independent. When there 7

24 is PN in the hannel, the preeding assumptions may not e valid. The total noise power at the reeiver integrator outputs when oth AWGN and PN are present is given y σ X =σ o +σ (.6) where σ = N T and σ = N / ρ T, and ρ is the fration of time that a narrowand o / Gaussian noise interferer is swithed on. n the event ρ = the PN is arrage noise interferene sine it is on ontinuously. Consequently, the proaility of symol error when a signal experienes PN an e expressed as P s = Pr(nterferer is OFF) p (AWGN) s + Pr(nterfererisON) p (AWGN+PN) s (.7) P = ( ρ ) p (AWGN) +ρ p (AWGN+PN) (.8) s s s where p ( x ) represent the proaility of symol error for ondition x. The equations s assume that a symol is either ompletely free of interferene or is interfered with for an entire symol. D. PERFORMANCE WTH DVERSTY JTDS/Link-6 employs several tehniques to inrease immunity to an adversary s interferene. One of the tehniques used is diversity. While there are many ways in whih diversity an e implemented, in JTDS/Link-6, this is implemented as a simple repetition ode, referred to as either the single pulse (no diversity) or the standard doule pulse (STDP) struture (sequential diversity of two). For the STDP, the transmitter transmits the same symol twie at different arrier frequenies, thus providing redundany at the reeiver. n order for diversity to e effetive, eah redundant symol must e reeived independently []. There are four asi JTDS message formats used, of whih the STDP message provides the est jam-resistane apaility. STDP are transmitted twie at different arrier frequenies not only to ahieve 8

25 redundany for improved interferene resistane ut also to ompensate for propagation prolems or antenna overage limitations in maneuvering platforms []. When diversity of order L is employed and eah diversity signal is reeived independently, the proaility that i of L diversity reeptions are affeted y PN, where ρ represents the fration of time the hannel is affeted y PN, is represented as [] where there are in error. L i L i Pr( i of Lpulses jammed) = ρ ( ρ) i L i (.9) different ways in whih i of L diversity reeptions an e reeived Consequently, the proaility of symol error for a system with diversity L in the presene of PN is s L i= [ Pr( of signals jammed) ( )] P = i L p i (.) s whih is given expliitly y P s L L i ( ) i ps( i) (.) i= i = ρ ρ where p () i is the onditional proaility of symol error given i of L diversity s reeptions are affeted y the PN. E. FORWARD ERROR CORRECTON CODNG n a inary system that utilizes lok FEC oding, n oded its are transmitted in the time it otherwise takes to transmit k information its. At the reeiver, the deoder is ale to orret up to t its errors in every lok of n oded its. For JTDS/Link-6, the FEC used is (3,5) RS oding, a linear, non-inary ode. To maintain onsisteny with the JTDS/Link-6 waveform, the alternative JTDS/Link- 6 waveform also employs (3,5) RS oding for error detetion and orretion. RS 9

26 odes are non-inary Bose-Chaudhuri-Hoquenghem (BCH) odes. For non-inary odes, m its at a time are omined to form a symol, an ( n, k ) RS enoder takes k information symols ( mk information its) and generates n oded symols ( mn oded its). For (n, k) RS oding, the proaility of deoder error, or lok error, is upper ounded y the sum of the proailities that a reeived ode word differs from the orret ode word y i symols for all i or > t []. Therefore, n n j n j ( ) j (.) P p p E s s j=+ t t n j n j ( ) j (.3) P p p E s s j= where the inequality holds for either a perfet ode or a ounded distane deoder, t is the symol-error orreting apaility of the ode, and proaility. p s is the hannel symol error Assuming that the proaility of information symol error given j ode symol errors is approximately j/n, we otain the proaility of information symol error as n n j n j ( ) j (.4) P j p p s s s n j=+ t For as ( nk, ) RS ode, we an also express (.4) as m m m j j ( ) (.5) j=+ t j P j p p m s s s where m represents the numer of its per symol. We an approximate the proaility of it error y taking the average of the upper and lower ound on the proaility of it error given that a symol error has ourred to otain

27 P m + Ps (.6) m Either equation (.4) or (.5) an e used with (.6) to otain the proaility of symol error for the alternative JTDS/Link-6 waveform. F. ERRORS-AND-ERASURES DECODNG Error-and-erasures (EED) is one of the simplest forms of soft deision deoding. The implementation of EED is suh that, for symols that are reeived amiguously, an erasure is delared. Thus, the numer of possile outputs is the numer of symols plus an erasure. For example, in inary erasure deoding, the output of the demodulator is not inary ut ternary. The three possile outputs are it, and erasure ( e ). Suppose that a reeived ode word has a single erased it. Now all valid ode words are separated y a Hamming distane of at least dmin, where dmin is the minimum Hamming distane of the ode. n general, given i erasures in a reeived ode word, all valid ode words are separated y a Hamming distane of at least dmin e. Hene, the effetive free distane etween valid ode words is d = d i (.7) mineff min Therefore, the numer of errors j in the non-erased its of the ode word that an e orreted is given y t = d e min (.8) where x implies rounding x down. Thus, a omination of t e errors and e erasures an e orreted as long as t + e< d (.9) e min

28 Hene, twie as many erasures as errors an e orreted. ntuitively, this makes sense eause we have more information aout the erasures; the loations of erasures are known, ut the loations of errors are not. For error-and-erasures deoding, the proaility that there are a total of i errors and j erasures in a lok of n symols is given y n n i Pr( i, j) = p p p i j i j n i j s e where eah symol is assumed to e reeived independently, hannel symol erasure, (.) p e is the proaility of p s is the proaility of hannel symol error, and p is the proaility of orret hannel symol detetion. The proaility of hannel error an e otained from p = p p (.) s e Sine a lok error does not our as long as dmin > i+ j, then the proaility of orret lok deoding is given y n n i (.) P p p p t dmin i i j n i j C = s e i= i j= j This gives the proaility of lok error as whih is P E = P (.3) P p p p t dmin i i j n i j E = s e i= i j= j C n n i (.4) Using (.4), we an approximate the proaility of symol error y taking the average of the upper and lower ound on the proaility of symol error given that a lok error has ourred to otain k + Ps PE (.5) k

29 Similarly, we an approximate the proaility of it error y taking the average of the upper and lower ound on the proaility of it error given that a symol error has ourred, previously given y (.6). G. PERFECT-SDE NFORMATON For a system with a diversity of i, where the diversity reeptions are reeived independently, perfet-side information (PS) modulation an e onsidered. n the ase of the doule-pulse struture, when oth reeived symols in the repetitive pulses are not affeted y PN, they are omined and demodulated. f either of the diversity reeptions suffers from PN, the reeiver disards the PN-affeted symol and makes its deision ased on a single-pulse with AWGN. When oth diversity reeptions are affeted y PN, the reeiver reovers the signal in the normal fashion. PS requires at least a diversity of two and an improve the performane of the system in an pulse-noise environment where ρ <. H. CHAPTER SUMMARY n this hapter, we introdued iorthogonal signals and addressed the akground and onepts neessary to examine the performane of an alternative JTDS/Link-6 waveform whih onsists of omplex 3-BOK with (3,5) RS oding. The onept of diversity, EED as well as the onept of PS was introdued. n the next hapter, we examine the performane of an alternative JTDS waveform that utilizes (3,5) RS oding with MBOK modulation transmitted over oth a hannel with only AWGN as well as hannel with oth AWGN and PN. The single-pulse struture (no diversity) is onsidered in the next hapter. 3

30 THS PAGE NTENTONALLY LEFT BLANK 4

31 . PERFORMANCE OF ALTERNATVE JTDS/LNK-6 WAVEFORM N SNGLE-PULSE STRUCTURE n this hapter, we investigate the performane of an alternative JTDS/Link-6 waveform y analyzing the proaility of it error vs. E / N for AWGN as well as AWGN plus PN. The performane using EED is also to e analyzed. The analyses for this hapter onsider only the ase of no diversity. A. PERFORMANCE OF 3-BOK WTH (3,5) RS CODNG N AWGN The proaility of symol error for M-BOK is given in (.5). For the alternative JTDS system that uses (3,5) RS oding, the proaility of hannel symol error for is M u re s (3.) N re / s N ps = e Q( u+ ) du π Expressed in terms of it energy E we have M u rme (3.) N rme / N ps = e Q( u+ ) du π M u ps = e Q( u+ rmγ) du π (3.3) rmγ where m is the numer of its per symol, r = k/ n and γ = E. o Sustituting (3.3) into (.4), we otain the proaility of symol error for MBOK with RS deoding in the presene of AWGN as n n P j p p j j n j ( ) (3.4) s s s n j=+ t Using (3.4) and (.6), we otain an approximation for the proaility of it error. Using (3.3), (3.4) and (.6), we plot the results for the proaility of it error of the alternative JTDS/Link-6 waveform in Figure where r = 5/ 3 and m = 5. For 5

32 purposes of omparison, oth unoded and oded performane is plotted. The unoded performane is plotted using (.5) where Es = me. We see that at P 5 =, the oded waveform requires E / N = 4.7 db, while the unoded waveform requires E = 6.7 db. Hene, there is a oding gain of db at P 5 =. - oded unoded - P E (db) Figure. Performane of the oded and unoded alternative JTDS/Link-6 waveform. B. PERFORMANCE N AWGN AND PULSE-NOSE NTERFERENCE When the hannel also has PN, (.6), (.8) and (3.) an e used to otain the proaility of hannel symol error with AWGN and PN as either 6

33 or M u rm ps =ρ ( ) e Q u+ du N rm N π + N N E E + ρ ρe E M u rme + ( ρ ) e Q( u ) du + π N rme / N M u rm ps =ρ e Q( u+ ) du π rm + + ργ γ ργ γ M u + ( ρ) e Q( u+ rmγ ) du π rmγ (3.5) (3.6) where E γ =. Defining ( ) ( ) ζ= γ + ργ, we otain from (3.6) / M u ps =ρ e Q( u+ rmζ) du π rmζ M u + ( ρ) e Q( u+ rmγ ) du π rmγ (3.7) The proaility of information hannel symol error and it error for oth AWGN and PN are otained from (3.7), (3.4) and (.6). Results are shown in Figure 3 for E = 9 db and ρ=, and Figure 4 for E = 9 db and ρ =. for oth a oded and an unoded waveform. n Figure 3, at P E N = for the 5 =, the required / 6.8 db oded waveform, and for the unoded waveform, the required E =.8 db. There is a oding gain of 4 db. Similarly from Figure 4, for 5 P =, for the oded ase, we 7

34 required E = 9.4 db, and E = 6. db for the unoded ase. There is a oding gain of 6.8 db. From the aove, we see that oding gives etter performane than the unoded waveform. Also, we oserve that, while the asolute performane of the oded waveform is etter when ρ= in omparison to that of the oded waveform when ρ=., there is a larger oding gain etween the oded and unoded waveform for ρ=.. - oded unoded - -3 P (db) E Figure 3. Performane of the oded and unoded alternative JTDS/Link-6 waveform for ρ = and E = 9 db. 8

35 - oded unoded - -3 P E (db) Figure 4. Performane of the oded and unoded alternative JTDS/Link-6 waveform for ρ =. and E = 9 (db). The performane of 3-BOK with (3,5) RS oding in PN for different values of ρ with E = 6 db is shown in Figure 5. Taking P 5 = as a referene, we ompare the required E for different values of ρ. We see that when ρ=, whih orresponds to arrage noise interferene, the performane is etter ( E db) as ompared to ρ=. and ρ =. ( E db ). The differene in performane etween ρ=. and ρ =. is relatively small. From the figure, we see that etween ρ= and ρ =., the transition point for whih ρ = provides etter performane ours at P 3 = where / 7.8 db E N =. Note that the degradation due to PN is only aout db at 5 P =. 9

36 - - ρ = ρ =. ρ =. -3 P E (db) Figure 5. Performane of 3-BOK with (3,5) RS oding in PN with E = 6 db for different values of ρ. The performane of 3-BOK with (3,5) RS oding in PN for different values of ρ with E = 9 db is plotted in Figure 6. As expeted, when E inreases, we see an improvement in overall performane; smaller values of E are required for the same P. n this ase, when 5 P =, / 6.8 db E N = is required for ρ= and E = 9.4 db is required for ρ =.. This ompares with E = db for ρ = and E = db for ρ=. at P 5 = when E = 6 db (Figure 5). This orresponds to a degradation of.6 db ( E = 9 db) and db ( E = 6 db) due to PN. Thus, while the asolute performane due to PN improves as E inreases, relative degradation also inreases.

37 - - ρ = ρ =. ρ =. -3 P E (db) Figure 6. Performane of 3-BOK with (3,5) RS oding in PN with E = 9 db for different values of ρ. C. PERFORMANCE WTH ERRORS-AND-ERASURES DECODNG N AWGN At the MBOK demodulator, the reeiver has to deide whih of the M symols was reeived or deide that it annot make a deision with suffiient onfidene. f the output of eah integrator, V > X > V, i =,,..., M /, then the reeiver annot deide T i T with suffiient onfidene, and the symol is erased. Without loss of generality, we assume that the original signal representing symol is transmitted. With errors-and-erasures, if symol is transmitted, then the proaility of hannel symol erasure p e and proaility of orret symol detetion p are [9]

38 p = Pr( V > X > V V > X > V... V > X > V ) (3.8) e T T T T T M / T and p = Pr( X > V X > X X > X... X > X ), (3.9) T 3 M / respetively. The proaility of hannel symol error an e otained y sustituting (3.8) and (3.9) into (.). From (3.8), when the output of eah integrator V T > X i > V T, i =,,, M/, the reeiver annot deide with suffiient onfidene, and the symol is erased. Hene the proaility of symol erasure is given y V V V V ( ) T T T T p... f x, x,..., x dx dx dx... dx = (3.) e V... M / / 3 / T V X X X M M T VT VT where (,,..., ) f x x x represents the joint proaility density funtion of the XX... XM / M / random variales that model the detetor outputs. Sine the random variales that model the detetor outputs are independent, (3.) an e written as VT VT ( ) ( ) p = f x dx f X dx e V X X T VT VT VT VT ( )... ( ) f X dx f X dx X3 3 3 V XM / M / M / T (3.) V Sine T V ( ) T V = ( ) =... = T ( ) f X dx f X dx f X dx, V X X3 3 3 XM / M / M / T VT VT (3.) simplifies to V / T V M = T ( ) ( ) e X X V T V T (3.) p f x dx f X dx Sustituting (.), (.3), and (.4) into (3.), we otain ( ) M / V x A T VT x pe = exp dx exp dx V T πσ σ πσ σ (3.3)

39 whih an e evaluated to otain VT + A VT A VT pe = Q Q Q σ σ σ Alternatively, (3.4) an e written as A V T VT + A VT pe = Q Q Q σ σ σ M / M / (3.4) (3.5) Defining V = a A where < a < and T σ = N o / T, we get s A a A A + a A a A pe = Q Q Q σ σ σ M (3.6) Hene, E s E s E s pe = Q ( a) Q ( a) Q a N + o N o N o M (3.7) From (3.7), the proaility of hannel erasure with (n, k) RS oding with ode rate r is re s re s re s pe = Q ( a) Q ( a) Q a N + o N o N o M (3.8) n terms of E, (3.8) an e expressed as o rme rme rme pe = Q ( a) Q ( a) Q a N + o N o N o M (3.9) Next, we derive an expression for the proaility of orret symol detetion from (3.9). From (3.9), we have and p x x x =... f ( x, x,..., x ) dx dx... dx dx (3.) V... M / / 3 / T x X X X M M x x 3

40 p = f x VT X ( ) x x x f ( X ) dx f ( x ) dx... f ( x ) dx dx x X X3 3 3 XM / M / M / x x (3.) whih simplifies to M / p x = f ( x ) f ( x ) dx dx (3.) V X X T x Sustituting (.), (.3) and (.4) into (3.), we get ( ) x x A x p = dx dx πσ σ πσ σ M / exp exp V T (3.3) x whih an e partially evaluated to otain Letting u ( x A ) ( ) M / x A x (3.4) πσ σ σ p = exp Q dx V T = / σ in (3.4), we get M / u A V T A σ σ (3.5) p = e Q u+ du π Now, with V = a A and σ= N / T as previously, we otain the proaility of orret T hannel detetion for MBOK from (3.5) as o s M / u E s E (3.6) p = e Q u+ du s ( a π ) N N o o From (3.6), the proaility of orret hannel detetion for MBOK with FEC oding, expressed in terms of E, is o M u rme rme (3.7) p = e Q u+ du ( a π ) N N o o 4

41 Sustituting (3.9) and (3.7) into (.), we otain the proaility of hannel symol error for MBOK with EED. The proaility of lok error is otained y sustituting (3.9), (3.7) and the results for n p s into (.4), repeated here: n i t dmin i i j n i j PE = ps pe p i= i j= j (3.8) Consequently, we an otain the proaility of symol error and the proaility of it error for MBOK with (n, k) RS oding and EED in the presene of AWGN using (.5) and (.6), respetively. The performane for 3-BOK with a (3,5) RS ode and EED in AWGN for different values of a is shown in Figure 7. From the Figure, we see that performane degrades for large values of a ( a.8). There is not muh differene in performane for values of a less than.6. Sine a = implies no EED, we onlude that there is no improvement in performane using EED for the alternative JTDS/Link-6 waveform in the presene of AWGN. nstead, if a is too large, it degrades performane. 5

42 P -4-5 Figure a = a =.4 a =.6 a =.8 a = E (db) Performane of 3-BOK with (3,5) RS oding and EED for different values of a in the presene of AWGN. D. PERFORMANCE WTH ERRORS-AND-ERASURES DECODNG N AWGN AND PULSE-NOSE NTERFERENCE The proaility of hannel erasure with FEC and EED in the presene of PN an e determined from (.6), (.8) and (3.6) in terms of E as 6

43 rme rme rme pe = ( ρ) Q ( a) Q ( + a) Q a N o N o N o rme rme + ( ρ) Q ( a) Q ( + a) N + N o N + N o ρ ρ Q a rme N + N ρ o M M (3.9) As in (3.7), we express (3.9) in terms of ζ and γ to get ( ) ( ) ( ) ( ) ( ) (( ) ) ( ) pe = ρ Q a rmγ Q + a rmγ Q a rmγ ( ) (( ) ) ( ) + ρ Q a rmζ Q + a rmζ Q a rmζ M M (3.3) Similarly, we an also otain the proaility of orret hannel detetion from (.6), (.8) and (3.5) as either or M u rme p = ( ρ) rme e Q u+ du ( a π ) N N o o M u rme +ρ rme e Q u du ( a + ) π N + No N + N ρ o ρ M u ( ) ( ) p = ρ e Q u+ rmγ du ( a) rmγ π M u +ρ e Q( u+ rmζ ) du ( a) rmζ π (3.3) (3.3) 7

44 Sustituting (3.3) and (3.3) into (.), we otain the proaility of hannel symol with EED. Consequently, we an otain the proaility of lok error y sustituting (.), (3.3) and (3.3) into (.4). The average proaility of symol error as well as the proaility of it error is otained from the proaility of lok error y taking the average of their upper ound and lower ound as expressed in (.5) and (.6), respetively. The performane of 3-BOK with (3,5) RS oding for different values of a where ρ= ( E = 6 db) in the presene of AWGN and PN is shown in Figure 8. We oserve a =. and a =.4 provide almost the same performane at P 5 =, while a =.6 gives slightly etter performane. The worst performane ours for a =.8. Next, we investigate the performane of the alternative waveform with EED for different ρ when a =.6. 8

45 - - a =.8 a =.6 a =.4 a =. -3 P E (db) Figure 8. Performane of 3-BOK with (3,5) RS oding and EED for ρ= and E = 6 db for different values of a. The performane of 3-BOK with (3,5) RS oding with EED in a PN environment for various values of ρ with a =.6 and E = 6 db is shown in Figure 9. We oserve that at P 5 =, as ρ dereases, performane is degraded, ut the degradation is limited to aout db. 9

46 - - ρ = ρ =.5 ρ =.3 ρ =. ρ =. P E (db) Figure 9. Performane of 3-BOK with (3,5) RS oding with EED in a PN environment for a =.6 and E = 6 db. The performane for 3-BOK with (3,5) RS oding oth with and without EED for E = 6 db and different values of ρ in the presene of AWGN and PN is shown in Figure. At P 5 =, whenρ =. or., there is no improvement due to EED ut a degradation of. to.5 db. The only exeption is when ρ =, where we see that using EED performs slightly etter, of. db, than without EED. 3

47 No EED, ρ= EED, ρ = No EED, ρ=. EED, ρ =. No EED, ρ=. EED, ρ =. P Figure. 5 5 E (db) Performane of 3-BOK with RS oding with and without EED with E = 6 db and a =.6 for different ρ. Figure is similar to Figure exept a =.4. n this ase, there is no differene in performane for different values of ρ with or without EED. We onlude that 3-BOK with (3,5) RS oding with EED does not signifiantly improve the performane of the waveform and may instead degrade performane for ρ<. This results is somewhat surprising sine EED has een shown to signifiantly redue degradation due to PN for waveforms with inary modulation and inary oding [3]. 3

48 - - No EED, ρ = EED, ρ = No EED, ρ =. EED, ρ =. No EED, ρ =. EED, ρ =. P E (db) Figure. Performane of 3-BOK with (3,5) RS oding with and without EED with E = 6 db and a =.4 for different values of ρ. E. CHAPTER SUMMARY n this hapter, the performane of the alternative JTDS/Link-6 waveform with no diversity (single-pulse struture) was investigated oth with and without EED and for AWGN only as well as AWGN plus PN. We saw that the waveform performs etter in arrage noise interferene than PN. We also see that EED deoding does not sustantially improve performane for the reeiver when oth AWGN and PN are present. n Chapter V, we investigate the performane of the alternative JTDS/Link-6 waveform with a diversity of two (doule-pulse struture). 3

49 V. PERFORMANCE OF THE ALTERNATVE JTDS/LNK-6 WAVEFORM WTH DVERSTY TWO (DOUBLE-PULSE STRUCTURE) n this hapter, we investigate the performane of the alternative JTDS/Link-6 waveform with a diversity of two (doule-pulse struture). Diversity is widely used when the hannel is suseptile to PN and/or fading. Diversity improves the performane of the system y providing transmission redundany. A. PERFORMANCE N AWGN WTH A DVERSTY OF TWO The proaility of hannel symol error for MBOK is given in (.5). As disussed earlier, JTDS/Link-6 employs diversity in its pulse struture to otain improved performane. The doule-pulse struture improves the performane of the system while the single-pulse struture allows for higher throughput. n Chapter, we analyzed the performane of the alternative JTDS/Link-6 waveform ased on the single-pulse struture. n this hapter, we investigate the performane of the alternative waveform for the doule-pulse struture. E Sine the reeived energy per it is L times the average reeived energy per hip, = LE, the modified expression for s p with diversity an e otained from (3.). With the doule-pulse struture, giving a diversity of two ( L = ), the reeived energy per it is the omination of two hip s energy, giving twie the energy per symol reeived. Hene, the proaility of hannel symol error for MBOK with (n, k) RS oding having a diversity of L in terms of hip energy is M u LrmE (4.) N LrmE / N ps = e Q( u+ ) du π where E is the average energy per hip and soft deision demodulation is assumed. Sine L =, (4.) simplifies to 33

50 M u 4rmE (4.) N 4 rme / N ps = e Q( u+ ) du π We an otain the proaility of information symol error and information it error using (3.4) and (.6), repeated here for onveniene: n n j n j ( ) j (4.3) P i p p + m P = Ps m s s s n j=+ t (4.4) By omparing (3.) and (4.), we see that diversity gives a 3 db improvement in performane as ompared to no diversity in terms of average energy per hip. The improvement is shown in Figure. The differene in the required E etween the doule-pulse and the single-pulse is 3 db at P 5 = in AWGN. - Doule Pulse Single Pulse - P E (db) Figure. Performane of 3-BOK with (3,5) RS oding for oth the single-pulse and the doule-pulse struture in AWGN. 34

51 B. PERFORMANCE N AWGN AND PN WTH A DVERSTY OF TWO For a hannel with PN in addition to AWGN, the proaility of hannel symol error for a diversity of two is given y i ps = ρ ρ p i i ( ) hip( ) i= i (4.5) where p ( i ) is the onditional proaility of hannel hip error given that i hips hip experiene PN. The onditional proaility density funtions for the random variales X m, where m=,,..., M /, that represent the deision variales otained y soft omining of the integrator outputs are given y and where and ( xm A) fx ( x, ) exp for / m m m i = m M, (4.6) πσ () m i σm () i ( xm + A ) f X ( x, ) exp for / m m m i = M + m M, (4.7) πσ () m i σm () i x n fx ( x,, ) exp n n n n m i = πσm () i σm () i ( i) i ( i) m T (4.8) σ =σ + σ (4.9) σ =σ +σ. (4.) T With σ = N rt and / s σ = N / ρ rt, and sustituting (4.) into (4.9), we have s N N σ = i + () i m ρrts rts (4.) Comparing (4.6) (4.) to (.) (.4), we an adapt (.5) to otain the proaility of hannel hip error as 35

52 () M ( x A ) () i x σ ( ) () m (4.) m i σm () i phip i = e Q dx πσ whih an e expressed as M E phip () i = e Q u+ du π u s (4.3) N E s i + N in + N ρ ρ The onditional proaility of hannel hip error with oding and expressed in terms of hip energy is given y M rme phip () i = e Q u+ du π whih an e expressed y u (4.4) rme i N N i + N + N ρ ρ M phip () i = e Q( u+ ) du π u rm (4.5) i rm γ + γ i γ + γ ρ ρ where γ = E, γ = E, and (4.5) is sustituted into (4.5) to otain the proaility of hannel symol error. The proaility of information symol error for 3- BOK with (3,5) RS oding and a diversity of two is otained y sustituting (4.5) into (4.3). The performane for different values of ρ with E =.4 db is shown in Figure 3. The E is hosen to e.4 db sine this yields P 7 = at E = 5 db. We see that varying ρ does not degrade the performane of the reeiver 36

53 signifiantly as ompared to arrage jamming ( ρ = ). At P 5 =, degradation due to PN is only aout db, and very small ρ ( ρ =. ) results in etter performane as ompared to larger ρ. - - ρ = ρ =. ρ =. ρ =.5 ρ =. P Figure E (db) Performane of 3-BOK with (3,5) RS oding and the doule-pulse struture for different ρ with E =.4 db. 9 n Figure 4, E = 3 db and performane approahes at E = 5 db. n this ase, degradation due to PN inreases to aout db, ut asolute performane improves y aout.5 db. 37

54 ρ = ρ =. ρ =. ρ =.5 ρ =. P Figure E (db) Performane of 3-BOK with (3,5) RS oding and the doule-pulse struture for different ρ with E = 3 db. n Figure 5, we ompare the performane for reeivers oth with and without diversity where the proaility of it error approahes E = 5 db. For the reeiver without diversity, E = 5.4 db for an asymptoti limit of E 7 while with diversity, is 3 db less. The differene in performane etween the waveforms with and without diversity is aout 3 db for oth ρ = and. at 5 P =. 38

55 - - ρ =, without diversity ρ =, diversityof two ρ =., without diversity ρ =., diversityof two P E (db) Figure 5. Performane of 3-BOK with (3,5) RS oding for oth the doule-pulse struture ( E =.4 db) and the single-pulse struture ( E = 5.4 db ). C. PERFORMANCE N AWGN AND PN WTH A DVERSTY OF TWO AND EED The proaility of it error with EED an e otained using similar approah as that with no diversity. We first otain the proaility of orret hannel detetion and the proaility of hannel erasure to otain the proaility of hannel symol error. Reall from (3.) that the proaility of hannel erasure is p = VT VT f ( x ) ( ) dx f X e X X dx V T V T M / (4.6) From (4.6), (4.7), (4.8) and (4.6), the onditional proaility of hannel erasure given that i diversity reeptions experiene PN is 39

56 ( x A ) V T pe () i = exp dx V T πσ () m i σm () i VT () exp x dx πσ m i σm () i M / (4.7) whih an e evaluated to otain A V T A + V T V T pe () i = Q Q Q m() i m() i σ σ σm() i M (4.8) Defining VT a( A) = where < a <, and with (4.), we otain E s E s pe () i = Q ( a) Q ( a) in N + in N + + M E s Q a in N + (4.9) whih with oding an e expressed as rme rme pe () i = Q ( a) Q ( + a) i i N + N N + N ρ ρ rme Q a i N + N ρ M (4.) where ρ represents the fration of time the hannel is affeted y PN, r is the ode rate and E is the hip energy. 4