AIR FORCE INSTITUTE OF TECHNOLOGY

MULTIPLE ACCESS INTERFERENCE CHARACTERIZATION FOR DIRECT-SEQUENCE SPREAD-SPECTRUM COMMUNICATIONS USING CHIP WAVEFORM SHAPING THESIS Matthew G. Glen, Captan, USAF AFIT/GE/ENG/04-10 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wrght-Patterson Ar Force Base, Oho APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

The vews expressed n ths thess are those of the author and do not reflect the offcal polcy or poston of the Unted States Ar Force, Department of Defense, or the Unted States Government.

AFIT/GE/ENG/04-10 MULTIPLE ACCESS INTERFERENCE CHARACTERIZATION FOR DIRECT-SEQUENCE SPREAD-SPECTRUM COMMUNICATIONS USING CHIP WAVEFORM SHAPING THESIS Presented to the Faculty Department of Electrcal and Computer Engneerng Graduate School of Engneerng and Management Ar Force Insttute of Technology Ar Unversty Ar Educaton and Tranng Command In Partal Fulfllment of the Requrements for the Degree of Master of Scence n Electrcal Engneerng Matthew G. Glen, BS Captan, USAF March 2004 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

AFIT/GE/ENG/04-10 MULTIPLE ACCESS INTERFERENCE CHARACTERIZATION FOR DIRECT-SEQUENCE SPREAD-SPECTRUM COMMUNICATIONS USING CHIP WAVEFORM SHAPING Matthew G. Glen, BS Captan, USAF Approved: //sgned// 11 March 2004 Mchael A. Temple, Ph.D. Date Thess Advsor //sgned// 11 March 2004 Major Matthew E. Goda, Ph.D. Date Commttee Member //sgned// 11 March 2004 Major Todd B. Hale, Ph.D. Date Commttee Member

Acknowledgments I would frst lke to thank my wonderful wfe for all of her patence n dealng wth me and lovng care n rasng our son. Second, I must thank my advsor Dr. Mchael Temple. Wthout hs academc gudance and attenton to detal ths thess would not be what t s. Fnally, I wsh to thank my classmates n GE-04M for ther camaradere and all of the good tmes mxed n wth the AFIT mess. Matthew G. Glen v

Table of Contents Page Acknowledgments... v Lst of Fgures...v Lst of Tables... x Abstract... x I. Introducton...1-1 1.1 Motvaton for Multple Access Interference Characterzaton...1-1 1.2 Problem Statement and Scope...1-2 1.3 Methodology...1-2 1.4 Thess Organzaton...1-3 II. Background... 2-1 2.1 Drect Sequence Spread Spectrum Communcatons...2-1 2.1.1 Multple Access Interference....2-1 2.1.2 Sgnal and System Model....2-2 2.2 Analytcal Bt Error Rate (BER) Approxmaton...2-3 2.2.1 BER for Asynchronous and Synchronous Systems...2-7 2.2.2 Accuracy of BER Approxmatons....2-8 2.3 Chp Waveform Shapng...2-9 2.3.1 Random Codng wth Chp Waveform Shapng...2-10 2.3.2 Gold-Codng wth Chp Waveform Shapng....2-11 2.4 Summary...2-12 III. Methodology...3-1 3.1 Problem Defnton...3-1 3.2 Approach...3-1 3.3 System Boundares...3-2 3.4 System Servces...3-3 v

3.5 Performance Metrcs...3-4 3.6 Parameters...3-5 3.6.1 System...3-5 3.6.2 Workload...3-6 3.7 Factors...3-7 3.8 Evaluaton Technque...3-8 3.9 Workload...3-9 3.10 Expermental Desgn...3-9 3.10.1 System Desgn....3-9 3.10.2 Code Valdaton....3-10 3.10.3 Waveform Shapng and Code Length Tests....3-10 3.10.4 Smulaton Length...3-11 3.11 Analyze and Interpret Results...3-12 3.12 Summary...3-13 IV. Results and Analyss...4-1 4.1 Valdaton Testng...4-1 4.1.1 Sngle User Valdaton...4-1 4.1.2 Multple User Valdaton....4-2 4.1.3 Error Analyss...4-4 4.1.4 Conclusons...4-8 4.2 Impact of Gold Codes...4-8 4.3 Pulse Shapng Results...4-11 4.3.1 Synchronous System Performance...4-11 4.3.2 Asynchronous System Performance....4-13 4.4 Cross-Correlaton Analyss...4-15 4.5 Code Length Analyss...4-23 V. Conclusons... 5-1 5.1 Research Contrbutons...5-1 5.2 Summary of Fndngs...5-1 5.2.1 Random versus Gold Spreadng Codes...5-1 v

5.2.2 Chp Waveform Shapng...5-2 5.2.3 Spreadng Code Length...5-2 5.3 Recommendatons for Future Research...5-3 5.3.1 M-Ary Data and/or Spreadng Modulaton...5-3 5.3.2 Full Monte Carlo Smulatons...5-3 5.3.3 Near-Far Comparsons...5-4 5.3.4 Characterzaton Usng Delay Profle Varaton...5-4 Appendx A. Smulaton Code... A-1 A.1 ber_shape_async_all31_2.m... A-1 A.2 ber_shape_sync_br511.m... A-5 Bblography... BIB-1 v

Lst of Fgures Page Fgure 2.1. Average BER for Half Sne, Rased-Cosne and Blackman Shapes...2-11 Fgure 3.1. BPSK DSSS Recever...3-3 Fgure 3.2. Normalzed Chp Waveform Shapes...3-8 Fgure 4.1. Valdaton of Sngle User wth Random Code Sequence.....4-1 Fgure 4.2. Valdaton of Sngle User wth Gold Code Sequences. 4-2 Fgure 4.3. BER for Synchronous DS/SSMA Systems Usng Random Codes.. 4-3 Fgure 4.4. BER for Asynchronous DS/SSMA Systems Usng Random Codes. 4-4 Fgure 4.5. Quantle-Quantle Plot for K = 5 Users and E b /N o = 0 (db). 4-5 Fgure 4.6. Quantle-Quantle Plot for K = 5 Users and E b /N o = 5 (db). 4-5 Fgure 4.7. Quantle-Quantle Plot for K = 10 Users and E b /N o = 0 (db)...4-6 Fgure 4.8. Quantle-Quantle Plot for K = 10 Users and E b /N o = 5 (db)....4-6 Fgure 4.9. Resdual Plot for K = 5 Users....4-7 Fgure 4.10. Resdual Plot for K = 10 Users....4-8 Fgure 4.11. Synchronous Network: Gold vs. Randomly Coded System....4-10 Fgure 4.12. Asynchronous Network: Gold vs. Randomly Coded System...4-10 Fgure 4.13. Synchronous Random Coded System wth Chp Shapng...4-12 Fgure 4.14. Synchronous Best-Case Gold Coded System wth Chp Shapng...4-12 Fgure 4.15. Synchronous Worst-Case Gold Coded System wth Chp Shapng...4-13 Fgure 4.16. Asynchronous Random Coded System wth Chp Shapng...4-14 Fgure 4.17. Asynchronous Best-case Gold Coded System wth Chp Shapng...4-14 Fgure 4.18. Asynchronous Worst-Case Gold Coded System wth Chp Shapng...4-15 v

Page Fgure 4.19. Randomly Coded System wth Selected Delays...4-18 Fgure 4.20. Best-Case Gold Coded System wth Selected Delays...4-18 Fgure 4.21. Worst-Case Gold Coded System wth Selected Delays...4-19 Fgure 4.22. Comparson of Synchronous Systems wth N = 31 and 511...4-23 Fgure 4.23. Sngle User Performance versus DS/SSMA Systems wth N = 511...4-24 x

Lst of Tables Page Table 2.1. Error Probabltes for DS/SSMA Systems...2-8 Table 4.1. Cross-Correlaton Values for Partcular Delays...4-16 Table 4.2. Average Cross-Correlaton Values for All Possble Delays...4-17 Table 4.3. Cross-Correlaton Values for Near Average Cross-Correlaton...4-17 Table 4.4. Test Statstc Mean and Varance Values...4-20 Table 4.5. Users' 2-5 Cross-Correlaton wth Data Effects for Profle #2...4-21 Table 4.6. Users' 6-9 Cross-Correlaton wth Data Effects for Profle #2...4-22 Table 4.7. Users' 2-5 Cross-Correlaton wth Data Effects for Profle #1...4-22 Table 4.8. Users' 6-9 Cross-Correlaton wth Data Effects for Profle #1...4-22 x

AFIT/GE/ENG/04-10 Abstract The modern world has an ncreasng demand for wreless multple access communcatons; drect-sequence spread-spectrum multple access (DS/SSMA) systems comprse many of these communcaton systems. A better understandng of multple access nterference (MAI) effects on DS/SSMA system performance, specfcally ther mpact on overall system bt error rate (BER), enables system desgners to mnmze MAI degradaton and produce greater DS/SSMA system capacty. Ths research characterzes MAI effects on DS/SSMA system performance through smulaton n Matlab, and explores the mpact of multple access code selecton, chp waveform shapng, and multple access code length on BER for both synchronous and asynchronous multple access networks. In addton, the smulated DS/SSMA model permts rapd research nto the effects of addtonal factors on BER. Pror to expermental testng, model valdaton s conducted through sngle user trals and by comparson wth exstng research for smlar system desgns. For synchronous and asynchronous networks, Gold codng mproves BER by 7.5 and 4.0 db, respectvely, relatve to aperodc random spreadng codes. Synchronous network results show that chp waveform shapng provdes no sgnfcant BER mprovement for the Blackman or Lanczos shapes. However, asynchronous network results show a potental BER mprovement for Blackman and Lanczos shapes. Increasng code length from 31 to 511 resulted n a 7.5 db BER mprovement. Collectvely, these results drectly relate changes n BER to waveform cross-correlaton statstcs. x

MULTIPLE ACCESS INTERFERENCE CHARACTERIZATION FOR DIRECT-SEQUENCE SPREAD-SPECTRUM COMMUNICATIONS USING CHIP WAVEFORM SHAPING I. Introducton 1.1 Motvaton for Multple Access Interference Characterzaton The ever ncreasng demand for world-wde multple access wreless communcatons drves the need to maxmze the current system capablty and transmsson capacty. Many current systems employ drect-sequence spread-spectrum (DSSS) technques to enable multple access capablty. A clearer understandng of how multple users effect overall system performance, through characterzaton of the multple access nterference (MAI), enables more effcent use of current systems and better desgns for future systems. Although a large body of research exsts on how MAI mpacts drect-sequence spread-spectrum multple access (DS/SSMA) performance, most of ths work reles on analytcal approxmatons. Addtonally, there are numerous factors that can affect MAI contrbutons n a system between workload and envronment (e.g., number of smultaneous transmtters, type of multple access codng, and code length). Current approxmatons only account for a lmted number of these factors and can requre extensve recalculaton when factors are changed or added. The state of exstng research nto MAI n DS/SSMA systems leaves open a need for a representatve system model that s easly modfed to account for dfferent factors. 1-1

Ths need lends tself to characterzaton of MAI n DS/SSMA systems through smulaton. Smulaton of DS/SSMA system performance provdes two man benefts. Frst, smulaton allows for rapd testng of the effects that numerous factors can have on system performance and such factors can be changed or others added relatvely easly. Second, smulaton enables verfcaton of future approxmatons through a vehcle that more closely represents the actual, physcal communcaton system. 1.2 Problem Statement and Scope To ncrease the capacty and capablty of DS/SSMA systems, a greater understandng of MAI effects (as a functon of multple factors) on system performance s requred. Smulaton of a DS/SSMA system model provdes the ablty to characterze the effects of varous desgn factors and permts verfcaton of exstng and future analytcal approxmatons. Ths work provdes modelng and smulaton results for a representatve DS/SSMA system and ncreases the understandng of partcular factors' mpact on MAI. Specfcally, ths work smulates the effect of multple access code selecton, chp waveform shapng, and multple access code length on bt error rate (BER) n a DS/SSMA system. Furthermore, ths research provdes a smulaton base capable of relatvely easy modfcaton to evaluate the mpact of addtonal factors on BER. 1.3 Methodology Ths research smulates the transmtter, channel, and recever of a DS/SSMA system usng Matlab. The smulaton contans all elements of random bnary data 1-2

generaton, data modulaton, multple access codng, transmsson, recepton, despreadng, and communcaton symbol detecton and estmaton. All envronmental and system level factors that potentally mpact BER are contaned wthn the smulaton and easly modfed to support addtonal research. Theoretcal performance models and prevous research provde the bass for system model verfcaton. Smulaton s used to estmate the effect of multple access code selecton, chp waveform shapng, and multple access code length on BER. 1.4 Thess Organzaton Ths thess contans fve chapters. Chapter 1 provdes an ntroducton on the need for MAI characterzaton for DS/SSMA systems. Chapter 2 outlnes relevant background nformaton on exstng BER approxmatons and the factors to be explored n research. Research methodology s outlned n Chapter 3. Verfcaton and expermental testng results are provded n Chapter 4. Chapter 5 contans a summary of contrbutons and fndngs and outlnes possble future research. One appendx s provded contanng the Matlab code assocated wth system smulaton. 1-3

II. Background 2.1 Drect Sequence Spread Spectrum Communcatons Modern dgtal communcaton technques have kndled wdespread appeal and demand for mprovng exstng and future communcatons systems. Ths ncreased demand has forced regulatng organzatons to establsh lmts on what portons of the spectrum specfc applcatons can use. The result s lmted bandwdth for all communcatons applcatons. The desre to get more use out of the avalable bandwdth has drven varous technques to ncrease the number of users or applcatons smultaneously occupyng a specfc spectral regon. One such technque s drectsequence, spread-spectrum multple access (DS/SSMA). 2.1.1 Multple Access Interference. Drect-sequence spread-spectrum (DSSS) technques are commonly used to mplement multple access communcatons. Exstng systems usng DSSS technques nclude the Global Postonng System (GPS) and IS-95 dgtal cellular phone system [Peterson, Zemer, and Borth, 1995]. DSSS codng modulates each communcaton symbol wth a waveform consstng of multple chp ntervals. The chp modulatng waveform has a partcular pulse shape wth ether a postve or negatve orentaton as determned by the DSSS code. The undesred sgnals make up the multple access nterference (MAI) that ncreases wth an ncreasng number of users. For perfectly orthogonal DSSS codng the MAI term s dentcally zero and the system functons as f there were only a sngle user present,.e., the desred user. Although the orthogonal codng s optmal, the number of requred users, mathematcal code lmtatons, the computatonal complexty requred to generate large numbers of orthogonal codes, and 2-1

the desre to analytcally model systems typcally necesstates the selecton of nonorthogonal codes n ether the desgn or analyss of DS/SSMA systems [Gerantots and Ghaffar, 1991]. DS/SSMA systems rely on code cross-correlaton characterstcs to reject undesred sgnals and mnmze MAI. The ablty to reject undesred sgnals s known as multple access nterference suppresson. By reducng MAI n a gven system, more users can smultaneously access the same lnk. 2.1.2 Sgnal and System Model. The works of Yao and Pursley present a standard DS/SSMA system desgn. The system conssts of K users wth each usng BPSK modulaton for ther respectve data. The data modulated waveform b (t) for each user can be represented by [Yao, 1977] b ( t) = b, npt ( t nts ) s = n (2.1) where b,n s a sequence of elements, b,n ε [-1,1], and P T ( ) s a unt heght rectangular pulse of duraton T s, the symbol duraton. Spreadng code modulaton a k (t) s defned as [Pursley, 1977] ak ( t) = a j= ( k ) j p T c ( t jt c ) (2.2) where a (k) j s a sequence of elements for the k th user, a j ε [-1,1], each chp nterval has duraton T c, and the spreadng code perod s N T c. The most common pulse shape used over each chp nterval s rectangular, as represented by pt ( t jtc ) n (2.2). One of the most common spreadng technques nvolves multplyng each communcaton symbol by N coded replcatons of the chp waveform over one full code perod per symbol nterval. Ths partcular technque s the bass for DSSS systems consdered n c 2-2

ths research. Usng the data and spreadng code modulaton of (2.1) and (2.2), respectvely, the transmtted sgnal for the k th user s defned as s k ( k k c k t) = 2P a ( t) b ( t) cos( ω t + θ ) (2.3) For the equal power cases consdered here, 2 P represents the transmtted power of all sgnals, cos( ω + θ ) s the phase modulated carrer of the kth user, ω c s the modulaton c t k frequency, and θ k s the phase delay of the k th user [Pursley, 1977]. The total receved multple access sgnal s the sum of K transmtted sgnals and the channel nose and may be expressed as K r( t) = k = 1 2P a ( t τ ) b ( t τ )cos( ω t + φ ) n( t) (2.4) k k k k c k + where τ k and φ k are the tme delay and receved phase, respectvely, of the k th user. The functon n(t) represents thermal channel nose and s assumed to be addtve whte Gaussan nose (AWGN) havng two-sded spectral densty of N o /2. All subsequent dervatons presented n ths work assume the desred user's sgnal has zero tme delay and zero receved phase. All nterferng users have tme delays, relatve to the desred user, n the nterval [0, T s ] and relatve phases n [0, 2π]. In the specal case of a synchronous network all nterferng users have zero tme delay. 2.2 Analytcal Bt Error Rate (BER) Approxmaton The work of [Pursley, 1977] provdes an expresson for bt error rate (BER) n terms of the number of network users (K) and code length (N). Usng a conventonal matched flter desgn, the receved sgnal s despread and demodulated va correlaton whch generates the test statstc Z gven as [Pursley, 1977] 2-3

Z = T 0 r( t) a ( t) cos( ω t) dt. (2.5) c Assumng ω c >> T s -1, the double frequency component can be gnored and Z can be rewrtten as Z = + T 0 P b 2,0 T + K k= 1 k [ b c k, 1 n( t) a ( t) cos( ω t) dt R k, ( τ ) + b k k,0 Rˆ k, ( τ k )] cosφk (2.6) where R k, ( τ ) and Rˆ k, ( τ ) are defned as contnuous-tme, partal cross-correlaton functons and expressed as [Pursley, 1977] R Rˆ k, k, ( τ ) = ( τ ) = τ 0 T τ a ( t τ ) a ( t) dt k a ( t τ ) a ( t) dt k (2.7) The partal cross-correlatons R ( ) and R ( τ ) of (7) account for msmatch between k, τ ˆ k, symbol transton boundares of the desred user and the k th user resultng from the k th user s tme delay, and the k th user s an nterferer. R ( ) correlates the end of the frst k, τ symbol of the k th user that falls wthn the desred user s symbol nterval, and R ( τ ) correlates the begnnng of the next subsequent symbol of the k th user whch completes the rest of the desred user s symbol nterval. R k, ( τ ) and Rˆ k, ( τ ) can be wrtten n terms of the dscrete aperodc crosscorrelaton functon C k, (l) defned as [Pursley, 1977] ˆ k, 2-4

2-5 = + = = +. 0, 0 1 1 0 ) ( 1 0 ) ( 1 0 ) (, N l l N a a N l a a l C l N j j k l j l N j l j k j k (2.8) Substtutng C k, (l) of (2.8) nto (2.7) results n the followng alternate expressons for ) R (, k τ and ) ( Rˆ, k τ : ) ( )] ( 1) ( [ ) ( ) ( ˆ ) ( )] ( ) 1 ( [ ) ( ) (,,,,,,,, c k k c k k c k k c k k lt l C l C T l C R lt N l C N l C T N l C R + + = + + = τ τ τ τ (2.9) To establsh the receved sgnal-to-nterference-plus-nose rato (SINR), the power of the nose plus nterferng sgnals s calculated as the varance of Z defned n (2.6) over one symbol nterval and s gven by [Pursley, 1977] ( ) ( ) = = + = + + = + + = K k k N l o T l lt k k K k k o T o k k T N d R R T P T N d R R T P Z Var c c 1 1 0 1) ( 2, 2, 1 2, 2, 4. / ) ( ˆ ) ( 4 4 / ) ( ˆ ) ( 4 } { τ τ τ τ τ τ (2.10) In ths case, the statstcal expectaton s taken wth respect to varables φ k, τ k, and b k. Gven the desred user s data remans constant over the symbol nterval under consderaton, (2.10) only contans contrbutons from channel nose and MAI. Substtutng the ) R (, k τ and ) ( Rˆ, k τ expressons from (2.9) nto (2.10) results n a smplfed varance gven by 4 / 12 } { 1, 3 2 T N r N PT Z Var o K k k k + = = (2.11)

2-6 where [Pursley, 1977] { } = + + + + + + + + + = 1 0 2,,, 2, 2,,, 2,,. 1) ( 1) ( ) ( ) ( 1) ( 1) ( ) ( ) ( N l k k k k k k k k k l C l C l C l C N l C N l C N l C N l C r (2.12) The SINR s then determned by dvdng T 2 P by the rms nose of { } Z Var [Pursley, 1977], yeldng:. 2 ) (6 2 1/ 1, 1 3 = + = b o K k k k E N r N SINR (2.13) To facltate prelmnary system desgn, t s shown n [Pursley, 1977] that for random spreadng sequences (2.13) smplfes to 1/ 2 2 3 1 + = b o E N N K SINR (2.14) where the (K-1)/(3N) term results from takng the expected value of the summaton n (2.13) gven random sequences are employed. The equaton for probablty of bt error (P B ) for a BPSK system wthout nterferers s [Peterson, Zemer, and Borth, 1995] = o b B N E Q P 2. (2.15) Usng (2.14) an approxmate P B value can be calculated for a DS/SSMA system usng (2.15) as ( ) Q SINR P B =. (2.16) It s useful to note that f only one user s present (K = 1), (2.14) dentcally smplfes to the Q-functon argument of (2.15), the correct result for the sngle user case.

2.2.1 BER for Asynchronous and Synchronous Systems. Usng characterstc functons for DSSS sgnals, Gerantots and Ghaffar have verfed the BER approxmaton results of Pursley [Gerantots and Ghaffar, 1991; Pursley, 1977] and redefned (2.15) va dervaton as 1/ 2 K 1 No SINR = mψ + (2.17) N 2Eb where m ψ = 1/3 for rectangular pulses and s defned as T 3 c 0 c T c 2 2 m = T R d = T Rˆ ψ ψ ( τ ) τ ψ ( τ ) dτ. (2.18) 3 c 0 In (2.18) Rψ and Rˆ ψ are partal autocorrelaton functons of the pulse shape defned as [Gerantots and Ghaffar, 1991] Rˆ ψ ( τ ) = ψ ( t) ψ ( t τ ) dt R ψ T c τ ( τ ) = Rˆ ψ ( T c τ ). (2.19) Ths dervaton clarfes that (2.14) s specfcally applcable to the SINR approxmaton for asynchronous sgnals usng rectangular shaped chp waveforms [Gerantots and Ghaffar, 1991]. Contnung wth ths development a BER approxmaton for synchronous DS/SSMA systems usng rectangular shaped chp waveforms s found to be [Gerantots and Ghaffar, 1991] 1/ 2 K 1 No SINR = +. (2.20) 2N 2Eb It has been shown that BER can be approxmated for asynchronous and synchronous systems by substtutng (2.14) and (2.20), respectvely, nto the Q-functon argument of (2.16) [Gerantots and Ghaffar, 1991]. 2-7

2.2.2 Accuracy of BER Approxmatons. Usng the SINR approxmatons of (2.14) and (2.20), and ther own characterstc functon approxmatons for SINR, Gerantots and Ghaffar calculated the BER for both the synchronous and asynchronous cases usng both random aperodc spreadng codes and determnstc m-sequence codes [Gerantots and Ghaffar, 1991]. For all cases consdered, parameter values of K = 3 and N = 31 were used wth rectangular-shaped chp waveforms. Ther results for PSK modulaton are shown n Table 2.1. All values wth superscrpt G are approxmatons resultng from (2.14) and (2.20), and those wthout a superscrpt result from Gerantots and Ghaffer's characterstc functon approxmaton. Table 2.1. Error Probabltes for DS/SSMA Systems (K = 3, N = 31) [Gerantots and Ghaffar, 1991] Eb N o P G e Random Sequences m-sequences Synchronous Asynchronous Synchronous Asynchronous P e P G e G G P P P P P e e,1 e,1 e,1 e,1 8 1.37x10-3 1.57x10-3 8.15x10-4 9.13x10-4 1.03x10-3 8.04x10-4 8.13x10-4 8.49x10-4 10 2.45x10-4 3.71x10-4 9.21x10-5 1.44x10-1.43x10-4 5.99x10-5 9.17x10-5 1.08x10-4 12 3.77x10-5 9.73x10-5 7.07x10-6 2.55x10-5 1.54x10-5 1.28x10-6 7.02x10-6 1.08x10-5 14 5.98x10-6 3.19x10-5 4.46x10-7 7.69x10-6 153x10-8 3.96x10-9 4.40x10-7 1.05x10-6 16 1.16x10-6 1.35x10-6 3.01x10-8 44.1x10-7 1.78 x10-7 5.98x10-12 2.95x10-8 1.06x10-7 Four observatons from Table 2.1. are mportant for ths research. Frst, the analytcal BER approxmatons based on (2.14) and (2.20) were found to be optmstc for random codes n the synchronous and asynchronous cases, and for determnstc codes n the asynchronous case. However, the BER approxmatons based on (2.14) and (2.20) are very conservatve for synchronous systems wth determnstc spreadng codes. Second, for random sequences the asynchronous case outperforms the synchronous case 2-8

due to the ncreased randomzaton and averagng provded by the asynchronous case [Gerantots and Ghaffar, 1991]. Thrd, for determnstc sequences the synchronous case outperforms the asynchronous case, reflectng the fact that determnstc codes are typcally desgned to maxmze the rato of autocorrelaton to cross-correlaton for the synchronous case [Peterson, Zemer, and Borth, 1995]. Fnally, n both synchronous and asynchronous cases the determnstc codes outperformed the random codes. Ths result s more pronounced n the synchronous case [Gerantots and Ghaffar, 1991]. 2.3 Chp Waveform Shapng To reduce MAI effects n DS/SSMA systems, the mpact of usng chp waveform shapes other than rectangular s explored. Ths approach to reducng MAI s based on decreasng the cross-correlaton statstcs between spreadng codes as a result of varaton n the chp waveform shape. The pulse shapes consdered n ths research nclude rectangular, half-sne, rased-cosne, Blackman, Kaser, and Lanczos. As defned n [Kok and Do, 1997] each of these pulse shapes can be analytcally represented as follows: a) Blackman 2πt ψ ( t) = k 0.42 0.5cos u( t) T (2.21) c b) Half-sne πt ψ ( t) = 2 sn u( t) T (2.22) c 2-9

c) Kaser ψ ( t) = k I 0 βπ 1 I 0 t Tc / 2 ( ) T ( βπ ) c u( t) (2.23) d) Lanczos 2 sn(2πt ) ψ ( t) = k u( t) 2πt (2.24) e) Rased-cosne 2 2πt ψ ( t) = 1 cos u( t) 3 T (2.25) c where T c s the chp duraton, u(t) s the unt step functon, I 0 s a zero-order modfed Bessel functon, and k s a real-valued scalng constant. 2.3.1 Random Codng wth Chp Waveform Shapng Usng the system model from (2.1-2.4) and chp waveform defntons of (2.21), (2.22), and (2.25), Lehnert and Cho nvestgated the mpact of waveform shapng on DS/SSMA system performance usng aperodc random spreadng sequences [Lehnert, 2002; Cho and Lehnert, 1999]. All results are for systems wth K = 9 total users and N = 31 for the processng gan. Each user has equal energy, and the tme delays of the 8 nterferng users are normalzed to one symbol nterval relatve to the desred user and modeled as unformly dstrbuted random varables. The average BERs are computed usng (2.14), where a condtonal Gaussan approxmaton (CGA) s used to estmate the nterference term when estmatng SINR, whch s the argument of (2.16). The CGA s used here n place of standard Gaussan approxmaton nstrumental n the dervaton of (2.14) and (2.20) because the CGA more 2-10

accurately approxmates the MAI effect on communcaton performance n a DS/SSMA system [Cho and Lehnert, 1999]. Based on the CGA assumpton, the Blackman waveform shape was predcted to provde the best performance relatve to the half-sne, rased-cosne, and rectangular shapes. Results of ths estmaton process are reproduced n Fg. 2.1. for half-sne, rased-cosne, and Blackman pulse shapes at E b /N o values rangng from 0 to 30 db. The data clearly llustrates the Blackman chp waveform shape outperforms all other waveform shapes consdered. 10-1 10-2 Half-Sne Rased-Cosne Blackman Average BER 10-3 10-4 10-5 0 5 10 15 20 25 30 E b /N o (db) Fgure 2.1. Random Codng wth Chp Waveform Shapng: Average BER for halfsne, rased-cosne and Blackman shapes [Cho and Lehnert, 1999]. 2.3.2 Gold-Codng wth Chp Waveform Shapng. The work of Kok and Do analytcally explores the mpact of usng chp waveform shapes on DS/SSMA BER performance. All shapes evaluated are tme-lmted and normalzed for the analyss. As n the random codng case, the chp waveform shapes evaluated were rectangular, Blackman, half-sne, Kaser, Lanczos, and rased-cosne. 2-11

Usng Gold codes of length 127 and 511, whle ncreasng number of users from 5 to 70, the SINR s calculated for each pulse shape case based on aperodc autocorrelaton functons of (2.8) [Kok and Do, 1997]. Ths SINR value s then used as the argument of (2.16) to estmate BER for each case, where the Q-functon s evaluated wth the algorthm descrbed n [Parl, 1980]. Results show that all non-rectangular chp waveform shapes outperform the rectangular case. Specfcally, the system employng the Lanczos shape acheves the best BER, followed n order by Kaser, Blackman, rased-cosne, halfsne, and rectangular [Kok and Do, 1997]. 2.4 Summary The goal of ths research s to characterze the effects of pulse shapng on DS/SSMA through smulaton. Ths chapter provdes a descrpton of the DSSS system desgn and the MAI term caused by multple smultaneous transmtters. An approxmaton for expected BER for both synchronous and asynchronous systems s derved. Dscusson of the accuracy of ths BER approxmaton enables a more nformed comparson to the results of ths research. Fnally, the bascs of pulse shapng as a means of BER mprovement s dscussed for random and Gold codes, and examples of approxmated performance mprovement are provded as a baselne for comparson to ths research. 2-12

III. Methodology 3.1 Problem Defnton Ths research characterzes the multple access nterference (MAI) for drectsequence, spread-spectrum multple access (DS/SSMA) systems through smulaton, and nvestgates the mpact of code selecton, pulse shapng, and code length on bt error rate (BER). Multple access capabltes are measured n terms of BER for gven sgnal-tonose ratos as related to the average energy per bt of the desred user (E b ) dvded by the background nose power spectral densty (N o ). The effects that parametrc changes have on multple access BER are also consdered. Prevous analytcal work characterzes BER n terms of E b /N o, the number of users, and the ablty of pulse shapng over a chp nterval to reduce MAI levels. Ths research develops and uses a Matlab smulaton to model DS/SSMA system performance and to nvestgate the mpact code selecton and pulse shapng has on BER. Addtonally, the smulaton provdes a means for predctng the mpact of ncreasng code length n systems employng random and Gold codes n conjuncton wth the pulse shapng. Smulaton results are verfed aganst the sngle user communcatons baselne and research results outlned n Chapter 2. Once verfed, the code s used to characterze the mpact of synchronzaton, code selecton, code length, and pulse shapng on BER for a DS/SSMA system. 3.2 Approach Smulaton results are reported as BER vs. E b/ N o curves for both synchronous and asynchronous systems wth multple users, usng random bnary and Gold codng technques, and mplementng chp waveform shapng. Each system user transmts 3-1

communcaton waveforms usng bnary phase shft keyed (BPSK) baseband modulaton and coded spreadng waveforms consstng of the desred pulse shape over the chp ntervals. All sgnals are assumed to be receved wth equal power and addtve whte Gaussan nose (AWGN) power levels are adjusted to acheve the desred E b /N o value. Each chp nterval has duraton T c and spreadng codes are perodc wth length N. Thus, a collecton of coded chp waveforms comprses one symbol duraton T s and equals NT c. Smulaton valdaton conssts of two stages, ncludng 1) sngle user results are generated and compared wth (2-15) and 2) results are compared to the analytcal fndngs of Gerantots and Ghaffar [Gerantots and Ghaffar, 1991]. Followng code valdaton, smulaton results are used to explore the mpact of pulse shapng, code selecton, and code length on BER for DS/SSMA systems. 3.3 System Boundares The system of nterest s a drect-sequence spread-spectrum (DSSS) recever usng BPSK modulaton (data and spreadng) for multple access communcatons. Fgure 3.1 shows a dagram of the physcal system whch conssts of a receve antenna, a rado frequency (RF) bandpass flter, a despreadng mxer, an ntermedate frequency (IF) bandpass flter, and the data phase demodulator. Ths research smulates DSSS recever performance n a multple access envronment usng Matlab. Each user transmts communcaton waveforms consstng of 1) BPSK data modulaton and 2) BPSK spreadng modulaton havng the desred pulse shape over each chp nterval. All sgnals are receved wth equal power and AWGN power levels are adjusted to acheve the desred E b /N o value. Each chp nterval has duraton T c and spreadng codes are perodc 3-2

wth length N such that the symbol duraton T s equals NT c. The ablty of the despreadng mxer, IF flter and demodulator to correctly estmate communcaton symbols (whch are subsequently mapped to bts) from the desred user, n the presence of other system users, s the focus of ths research. RF Flter IF Flter Data Phase Demodulator Despreadng Mxer Fgure 3.1. BPSK DSSS Recever [Peterson, Zemer, and Borth, 1995]. 3.4 System Servces DSSS recevers are desgned to receve sgnals and estmate bts sent from the desred user whle rejectng nterferng sgnals of all undesred users. By dong so, DSSS recevers enable multple users to smultaneously communcate wthn the same spectral regon. Over each receved symbol nterval, the recever estmate results n one of two possble outcomes, ether 1) a symbol (bt) s receved and estmated correctly (correct condton), or 2) a symbol (bt) s receved and estmated ncorrectly (error condton). The source of makng errors per outcome 2 can be ether background nose or nterference from multple users (MAI). Although AWGN background nose s present for all smulatons and analyses conducted n ths research, the prmary focus of ths work s on errors due to MAI. 3-3

3.5 Performance Metrcs The ntroducton of errors n sgnal detecton and estmaton for sgnals receved through a physcal medum s nevtable and these errors are even more lkely to occur wth MAI present. The percentage of errors occurrng n DSSS recever processng s quantfed usng bt error rate (BER or P B ), defned as the number of demodulated bts that are ncorrectly estmated at the recever dvded by the total number of bts transmtted. The requred system BER establshes a lower bound on the percentage of errors a recever can make and stll correctly functon. Ths research uses BER as the prmary metrc to measure system performance. Fundamental to BER determnaton s the rato of avalable sgnal energy E b to total nterferng energy. In ths case, the total nterferng energy conssts of channel nose energy, as establshed by N o, and nterference due to multple users (N I ). The resultant energy-to-nose rato can be expressed as N o E b + N I. (3.1) For a system usng BPSK data modulaton, BER s drectly related to ths rato and s P B = Q 2E N T b (3.2) where Q s the complmentary error functon and N T = N o + N I [Lehnert, 2002]. The exstng analytcal approxmatons estmate E b /N T values whch are converted to BER for comparson wth the smulated results usng (3.2). 3-4

3.6 Parameters 3.6.1 System System parameters whch affect the demodulator s ablty to correctly estmate symbols from the desred user nclude: Channel propagaton characterstcs Recever antenna gan (or loss) Spectral wdth and center frequences of the RF and IF flters User of nterest (one of total system users) Type of spreadng code used These system parameters reflect the physcal recever as smulated. As ndcated earler, the channel s assumed to be an deal AWGN channel wth a two-sded constant power spectral densty of N o /2. Receve antenna gan affects the total amount of receved power (sgnal, nose, and nterference). For ths work, t s assumed that receved antenna gan affects all sgnal, nose, and nterference terms equally and s therefore gnored. The spectral wdth and center frequences of the RF and IF bandpass flters mpact the amount of sgnal power processed by the system. Flter center frequency s determned by whch user s desgnated as the desred user, and the flter s spectral wdth and shape are determned by the spreadng code employed. For all smulatons, t s assumed that the center frequency and spreadng code are perfectly selected for the desred user. The type of spreadng codes employed mpact the level of MAI due to varyng autocorrelaton and cross-correlaton characterstcs of dfferent codes. Random codes are employed for both sngle user and multple access valdaton tests. Ths s approprate for the sngle user valdaton because sngle user performance s ndependent of code selecton. Random 3-5

codes are employed for multple access valdaton to permt drect comparson wth the system setup used by Gerantots and Ghaffar [Gerantots and Ghaffar, 1991]. Both random and Gold codes are used to expand the research through smulaton and analyss of results obtaned wth chp waveform shapng and varyng code lengths. 3.6.2 Workload. Many workload parameters exst and potentally mpact recever BER. These parameters nclude: Sgnal structure (.e., transmtted symbol shape) Sgnal power Thermal channel nose Tme delay of nterferers relatve to the desred user Number of chp ntervals (T c ) per communcaton symbol nterval (T s ) Number of code perods per symbol nterval Number of system users Shape of chp waveforms The transmtted sgnals for ths research are baseband waveforms consstng of unformly dstrbuted random strngs of postve and negatve ones that represent random bnary waveforms. For a gven smulaton, all users have the same sgnal power. Thermal channel nose s modeled as AWGN. Relatve tme delays for nterferng users mpact BER by changng the cross-correlaton levels between the nterferng codes and the desred user's code. The number of chps per symbol nterval s commonly related to processng gan, G, and BER mproves (decreases) as G ncreases [Peterson, Zemer, and Borth, 1995]. For all smulatons exactly one code perod s contaned wthn each 3-6

symbol nterval. In DS/SSMA systems MAI levels ncrease as the number of users ncreases and BER ncreases. Fnally, chp waveform shape mpacts code crosscorrelaton values such that the dspreadng mxer output ncreases (or decreases) and results n a correspondng ncrease (or decrease) n BER. 3.7 Factors Of the parameters lsted above, only the followng factors are vared for ths research: thermal nose power, tme delay of nterferers relatve to the desred user, number of users, spreadng code length, and chp waveform shape. Sgnal power s accounted for n E b /N o wth desred values acheved by holdng E b constant and varyng the nose power spectral densty N o. E b /N o values of 0 to 10 db are smulated for all cases consdered unless otherwse specfed. The tme delay of nterferers relatve to the desred user s vared between two states, 1) tme delay equals zero for all users n a synchronous system, or 2) tme delays are unformly dstrbuted random varables n the range of [0, T s ] for an asynchronous system. The number of system users drectly mpacts MAI levels and 1) vares based on avalablty of analytc approxmatons for comparson, or 2) s held constant at K = 9 to permt comparson of smulated results wth those presented by Cho and Lehnert [Cho and Lehnert, 1999]. Spreadng code length, N, vares from 31 to 511 to llustrate the mpact of code length varaton on BER. Fnally, chp waveform shape vares based on the analytcal approxmatons presented by Lehnert, and Kok and Do. Chp waveform shapes consdered nclude, rectangular, Blackman, and Lanczos as defned n (2.22) through (2.25) [Lehnert, 2002; Kok and Do, 1997]. Fgure 3.2 llustrates the normalzed Blackman and Lanczos pulses. 3-7

2 2 1 1 0 0 2 4 6 8 10 0 0 2 4 6 8 10 T c /10 T c /10 a) b) Fgure 3.2. Normalzed Chp Waveform Shapes. a) Blackman and b) Lanczos [Kok and Do, 1997]. 3.8 Evaluaton Technque All evaluatons are based on comparson of BER results. The BERs used are ether calculated based on the BPSK sngle-user BER expresson of (2.15), approxmated based on analytcal expressons usng (2.14) and (2.20) n (2.16), observed from prevous research as outlned n Chapter 2, or estmated through smulaton from ths research. Smulaton of the physcal system provdes all expermental data used for evaluaton. Smulatons are conducted n two phases. Frst, the code s valdated aganst known analytcal equatons for a sngle user and results of prevous research for the case of multple access. Addtonal smulatons establsh BER varaton resultng from random codes verses Gold codes, multple chp waveform shapes, and spreadng code length. The mpact of factors s determned by comparng BER curves for systems employng dfferent factors n the setup or workload. Usng two smulaton phases n ths research 3-8

enables code verfcaton before explorng the mpact of addtonal factors on BER. Ths verfcaton supports the valdty of the new smulated cases. 3.9 Workload The workload vares approprately for the setup of each smulaton phase. Workloads n all phases nclude background thermal nose. Sgnal power levels used provde adequate resoluton to create BER plots for comparson wth approxmatons, prevous research, and new smulaton results. Selecton of tme delay and processng gan reflect prevous research nto BERs for DS/SSMA systems. Selecton of three chp waveform shapes enables qualty comparson wth prevous research whle lmtng the computatonal complexty. 3.10 Expermental Desgn 3.10.1 System Desgn. Matlab smulatons are used to produce BER vs. E b /N o results for both synchronous and asynchronous systems wth multple users, usng random and Gold codes, and mplementng chp waveform shapng. Each user transmts communcaton waveforms havng BPSK baseband data modulaton and coded spreadng modulaton usng the desred pulse shape over the chp ntervals. All sgnals are transmtted (receved) wth equal power and nose power s vared to acheve desred E b /N o values. Each chp nterval has duraton T c and codes are perodc wth length N such that symbol duraton T s equals NT c. All spreadng pulse shapes are scaled to normalze transmtted power to that of a system usng rectangular chp waveforms (rectangular waveform 3-9

shapng s the baselne for comparng other waveform shapes). Random spreadng codes are generated as aperodc, unformly dstrbuted sequences of postve and negatve ones. Gold codes are generated as outlned n [Peterson, Zemer, and Borth, 1995] usng an ntal regster state of all ones; generator polynomals [45] 8 and [75] 8 are used to generate N = 31-length codes and [1021] 8 and [1461] 8 are used to generate N = 511-length codes. 3.10.2 Code Valdaton. Code valdaton s conducted n two stages. Frst, usng only rectangular chp waveforms, smulaton results for K = 1 transmtter and N = 31 length codes (processng gan) are compared to the sngle user baselne calculated usng (2.15) to valdate code performance for the smplest case. Ths comparson s repeated usng both random and Gold codes to valdate both code mplementatons n the smulaton code. Second, usng only random codes and rectangular chp waveforms, the trends of smulatons usng K = 3 and N = 31 are compared to fndngs of Gerantots and Ghaffar relatve to the analytcal approxmatons for sgnal-to-nterference plus nose (SINR) n (2.15) and (2.19). 3.10.3 Waveform Shapng and Code Length Tests. Followng valdaton, smulated BER results are compared for chp waveforms havng rectangular, Blackman, and Lanczos shapes as defned n (2.21) through (2.25). Usng the verfed smulaton code, BER s estmated for spreadng codes comprsed of random codes and one full-perod of the 31-length Gold codes. A collecton of bestcase and worst-case Gold codes are used. Here, best-case Gold codes are selected as the combnaton of Gold codes havng the lowest cross correlaton values wth the desred sgnal. Lkewse, worst-case Gold codes are selected as the combnaton of Gold codes havng the hghest possble cross correlaton values wth the desred sgnal. 3-10

Fnally, to llustrate the mpact of code length on MAI, BER s smulated for systems usng N = 511-length Gold codes and compared to aperodc random spreadng codes havng N = 511 chps per symbol nterval. 3.10.4 Smulaton Length. Each bt of data that s coded, transmtted, despread, and estmated n the recever represents an ndependent test of the system's ablty to accurately demodulate the desred user's data. Subsequent bt transmssons and estmatons represent expermental repettons. Each smulaton runs untl 300 bts are estmated n error. The total number of bts transmtted, n, represents the number of expermental trals n the smulaton. Based on BER values estmated n ths research, usng a 95% confdence nterval and runnng smulatons untl 300 bt errors are detected, the sample mean observed through smulaton wll vary from the actual populaton mean due to varaton by an amount r that s at most approxmately ± 11% of the actual mean of the smulaton [Canadeo, 2003]. Varaton r s defned as PB ( 1 PB ) r = z (3.3) n where z = 1.96 for the 95% confdence nterval and P B s the effectve BER defned as the rato of the number of bts n error to n [Jan, 1991]. The above analyss s based on assumng errors are ndependent and normally dstrbuted. The propertes of an AWGN channel and the central lmt theorem ndcate that these errors should be ndependent and normally dstrbuted. To test these assumptons, control tests are run wth the number of users set at K = 5 and K = 10 usng E b /N o values of 0 and 5 db. Each test s repeated ffteen tmes, provdng ffteen ndependent BER values for each number of users consdered at a gven energy profle. 3-11

The expermental error, e j, s calculated as the dfference between measured BER and the tral average, and ths error s plotted aganst the normal quantles, defned as ( ) 0.14 x q q 0.14 = 4.91 1 (3.4) where q 0.5 = (3.5) n s the number of the tral beng plotted, and n s the total number of trals. The lnearty of ths plot verfes the assumpton of normally dstrbuted errors. Addtonally, the error, e j, s plotted aganst the average BER results. The lack of a trend n the errors verfes the ndependence of errors [Jan, 1991]. 3.11 Analyze and Interpret Results Smulaton results are used to comple BER vs. E b /N o curves for both phases of the experment. Relatve benefts of tested factors are not easly determned by drect comparson of BER plots alone due to varaton caused by randomness n some of the factors, e.g. the partcular nose realzaton. To make statstcally sgnfcant comparsons, error bars are ncluded n the BER curves. Error bar plots based on the varaton values calculated by (3.3), where the error bars are plotted an amount r above and below the observed BER, provde bounds for actual BER mean values to a 95% confdence level and allow more meanngful results through drect comparson [Jan, 1991]. For BER values to be dfferent n a statstcally sgnfcant manner, the error bars of two BER cases beng compared cannot overlap. Takng ths nto consderaton, 3-12

drect comparson of BER curves yelds meanngful nformaton about the relatve performance enhancements of factors under consderaton. Both phases of testng utlze drect comparson wth error bars present to determne f results are ether sgnfcantly smlar for valdaton or sgnfcantly dfferent for dentfyng performance enhancements. 3.12 Summary Ths chapter outlnes the expermental setup requred for 1) valdatng the DS/SSMA recever smulaton and 2) comparng the relatve performance enhancements of selected factors. The system's servce s the transmsson of data bts and metrcs are the BER vs. E b /N o curves ncludng error bars. All system components are smulated, wth smulatons conducted n two phases. The frst phase valdates the analytcal models of the DS/SSMA recever and the second phase llustrates the relatve mpact of chp waveform shapng, code selecton, and code length on BER. The analyss of results compares the BER curves to expected theoretcal BER, prevously reported BER results, or other expermental results of ths research. Relatve mpact on BER s the expected result of these comparsons. 3-13

IV. Results and Analyss 4.1 Valdaton Testng Smulaton code s valdated n two stages. Frst, smulated bt error rate (BER) for a sngle user n the presence of nose s compared to theoretcal BER calculatons from (2.14). Second, the trend of smulated BER for multple users s compared to results of Geranots and Ghaffar [Geranots and Ghaffar, 1991]. 10-1 Theoretcal Performance Smulated Performance 10-2 BER 10-3 10-4 0 1 2 3 4 5 6 7 8 E b /N o (db) Fgure 4.1. Valdaton of Sngle User wth Random Code Sequence 4.1.1 Sngle User Valdaton. Usng only rectangular chp waveforms, BER for a sngle user wth random spreadng codes s smulated for an addtve whte Gaussan nose (AWGN) channel. Error bars are generated as outlned n Chapter 3 and reflect the 95% confdence nterval for smulated results. Smulaton results for ths sngle user case are shown n Fg. 4.1 along wth theoretcal BER calculated per (2.14). As shown, the theoretcal BER curve 4-1

les wthn smulated error bar bounds, ndcatng the smulated results are consstent wth theoretcal performance. Fgure 4.2 shows the same relatonshp for the case where Gold code spreadng sequences are used n place of the random sequences. Once agan, the theoretcal result falls wthn smulated error bar bounds ndcatng smulated BER wth Gold coded sequences s consstent wth theoretcal performance. 10-1 Theoretcal Performance Smulated Performance 10-2 BER 10-3 10-4 0 1 2 3 4 5 6 7 8 E b /N o (db) Fgure 4.2. Valdaton of Sngle User wth Gold Code Sequences 4.1.2 Multple User Valdaton. Followng the system setup of Geranots and Ghaffar, the trend of smulated BER results are compared to the data from Table 2.1 [Geranots and Ghaffar, 1991]. The system ncludes K = 3 users, a spreadng code of length N = 31 chps, rectangular shaped chp waveforms, and aperodc random spreadng codes. Smulatons are run for both synchronous and asynchronous networks and results are compared wth analytcal approxmatons of (2.19) and (2.15), respectvely. 4-2

Fgure 4.3 shows smulated results for synchronous users usng E b /N o values of 8, 10, 12, 14, and 16 (db). Analytc approxmatons from (2.19) are also shown for the same energy profles. Smulaton results ndcate performance whch s poorer (hgher BER) than predcted by the analytcal approxmatons. The results presented here are consstent wth the fndngs of Geranots and Ghaffar [Geranots and Ghaffar, 1991]. Gven the analytc approxmaton falls well outsde the error bar bounds for all values of E b /N o consdered, these results are statstcally sgnfcant at the 95% confdence level. 10-3 Analytcal Approxmaton Smulated Performance 10-4 BER 10-5 10-6 8 9 10 11 12 13 14 15 16 E b /N o (db) Fgure 4.3. Smulated vs. Analytc BER for Synchronous DS/SSMA Systems Usng Random Code Sequences. A comparson of calculated BER from (2.15) wth smulaton results of an asynchronous system, usng E b /N o values of 8, 10, 12, 14, and 16 (db), s shown n Fg. 4.4. Once agan, the data ndcates that smulated BER under performs (hgher BER) the approxmatons and are consstent wth Geranots and Ghaffar's fndngs that the analytcal approxmaton s optmstc for the asynchronous case [Geranots and 4-3

Ghaffar, 1991]. Agan, the error bars add statstcal sgnfcance to ths result at the 95% confdence level. 10-3 Analytcal Approxmaton Smulated Performance 10-4 10-5 BER 10-6 10-7 10-8 8 9 10 11 12 13 14 15 16 Fgure 4.4. Smulated vs. Analytcal BER for Asynchronous DS/SSMA Systems Usng Random Code Sequences. 4.1.3 Error Analyss E b /N o (db) Error analyss was conducted to verfy the accuracy of the synchronous smulaton model. Usng only rectangular pulse shapes and random code sequences, smulatons for K = 5 and 10 users at E b /N o values of 0 and 5 (db) are repeated ffteen tmes. Expermental error, e j, s calculated as the dfference between the observed BER for each tral and the average of ffteen trals. Values of e j are plotted aganst the normal quantles as determned by (3.4) and (3.5) n Fg. 4.5, Fg. 4.6, Fg. 4.7, and Fg. 4.8. The nearly lnear nature of these plots verfes the errors are normally dstrbuted [Jan, 1991]. 4-4

0.015 0.01 0.005 Resdual Quantle 0-0.005-0.01 Resdual Quantle Lnear Approxmaton -0.015-2 -1.5-1 -0.5 0 0.5 1 1.5 2 Normal Quantle Fgure 4.5. Quantle-Quantle Plot for K = 5 Users and E b /N o = 0 (db). -3 x 10 4 3 2 Resdual Quantle 1 0-1 -2-3 Resdual Quantle Lnear Approxmaton -4-2 -1.5-1 -0.5 0 0.5 1 1.5 2 Normal Quantle Fgure 4.6. Quantle-Quantle Plot for K = 5 Users and E b /N o = 5 (db). 4-5