DESIGN AND ANALYSIS OF WIRELESS DIVERSITY SYSTEMS

Transcription

1 DESIGN AND ANALYSIS OF WIRELESS DIVERSITY SYSTEMS ZHANG SONGHUA NATIONAL UNIVERSITY OF SINGAPORE 4

2 DESIGN AND ANALYSIS OF WIRELESS DIVERSITY SYSTEMS ZHANG SONGHUA (B. Eng., Huazhong Unversty of Scence and Technology) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 4

3 Acknowledgement I would lke to express my grattude to Professor Kam Poo Yuen, my prncpal supervsor, and Professor Paul Ho, my co-supervsor, for ther gudance, support and advce over the entre study. I have receved much encouragement and stmulaton from them to work n the area of research. Ther knowledge and nsght has nspred many of the deas expressed n ths thess, and ther efforts and patence n revsng the drafts are much apprecated. Specal thanks to all my frends who have helped me n one way or another, for ther advce, help and tolerance, especally my colleagues n ECE-IR lab who have made my study here an enjoyable experence. The support of Natonal Unversty of Sngapore s hghly apprecated. To my dearest fancée, Hao Png, my mother, my father and my sster, for ther everlastng love and support, I dedcate ths thess.

4 Contents Acknowledgement..... Contents..... Lst of Fgures and Tables....v Abbrevatons. v Summary... x Chapter. Introducton. Background Motvaton Lterature Revew Contrbutons of the Thess....5 Thess Outlne 3 Chapter. BEP of coherent PSK n Nonselectve Raylegh Fadng Channels wth asynchronous Cochannel nterference 5. Introducton 5. System Model.6.3 Performance Analyss.4 Effects of Symbol Tmng Offsets.9.5 Numercal Results and Dscusson.34.6 Summary 4 Chapter 3. BEP of dfferentally detected DPSK n Nonselectve Raylegh Fadng Channels wth asynchronous Cochannel nterference Introducton 4 3. System Model Performance Analyss 47

5 3.4 Effects of Symbol Tmng Offsets Dependence of BEP on Interferng Sgnals Tmng Offset Dependence of BEP on Transmtted Symbols Numercal Results and Dscusson Summary 66 Chapter 4. BEP of Transmt-Receve Dversty System wth PSAM Introducton System Model Channel Model Channel Estmaton PSK System Recever Desgn Performance Analyss Bnary Orthogonal Sgnalng Implct PSAM Scheme Feasblty of Generalzed Quadratc Recever PSAM Channel Estmaton Based ML Detector Numercal Results and Dscusson Summary 9 Chapter 5. Space Tme Code wth Orthogonal FSK. 5. Introducton 5. Bnary orthogonal FSK System Model. 5.. Channel Estmaton Data Detecton 5

6 v 5..4 Error Performance Analyss M-ary Orthogonal FSK System Model Data Detecton Error Performance Analyss Predctor Upper Bound Unon Bound Dversty Recepton Numercal Results and Dscusson Summary.49 Chapter 6. Concluson and Suggeston for Future Work 5 6. Concluson Suggeston for Future Work...5 Appendx A. Maxmum Lkelhood Detecton of ST-MFSK 55 Appendx B. Dervaton for the condtonal representaton of (5.49).58 Appendx C. Dfferental Space Tme Block Codes.6 Bblography..75

7 v Lst of fgures and tables Fgure. Thess structure...3 Fgure. A comparson of the tme waveform of the three pulses. Fgure. Recever structure for CPSK... Fgure.3 Sgnal constellaton and decson regon.. 8 Fgure.4 BEP vs. average SNR for dfferent tmng offset.36 Fgure.5 BEP vs. average SNR for dfferent INR level.36 Fgure.6 BEP vs. average SNR for dfferent dversty orders...37 Fgure.7 BEP vs. normalzed tmng offset for dfferent pulses wth dfferent dversty orders..37 Fgure.8 BEP vs. normalzed tmng offsets of a system usng RC pulse...38 Fgure.9 BEP vs. normalzed tmng offsets of a system usng BTRC pulse 38 Fgure. BEP vs. number of nterferers for dfferent INR levels 39 Fgure. BEP vs. number of nterferers for dfferent SIR levels.39 Fgure. BEP vs. normalzed tmng offset for dfferent roll-off factors.4 Fgure 3. Recever structure for DPSK...47 Fgure 3. BEP vs. average SNR for dfferent tmng offset 6 Fgure 3.3 BEP vs. normalzed tmng offset for dfferent pulse shape and dfferent dversty orders.6 Fgure 3.4 BEP vs. normalzed tmng offset for two-user system wth rectangular pulse shapng, both analytcal and smulated...6 Fgure 3.5 BEP vs. normalzed tmng offset for two-user system wth RC pulse and BTRC pulse..6 Fgure 3.6 BEP vs. normalzed tmng offset for dfferent roll-off factors...63 Fgure 3.7 BEP vs. normalzed tmng offset for dfferent pulses wth dfferent number

8 v of nterferers but the same total nterferng power...63 Fgure 3.8 BEP vs. normalzed tmng offset for dfferent pulses wth dfferent transmtted data symbols...64 Fgure 3.9 BEP vs. normalzed tmng offset wth dfferent transmtted data symbols...64 Fgure 3. BEP vs. fadng autocorrelaton of the desred sgnal...65 Fgure 3. BEP vs. fadng autocorrelaton of the nterferng sgnal...65 Fgure 4. Transmtted frame structure 68 Fgure 4. Bnary orthogonal sgnals n rotated coordnates 84 Fgure 4.3 BEP vs. average SNR for dfferent fade rate Fgure 4.4 BEP vs. average SNR for dfferent fade rate Fgure 4.5 BEP vs. average SNR for dfferent fade rate wth optmzed frame length. Fgure 4.6 BEP vs. average SNR for dfferent fade rate wth our PSAM compare to that wth conventonal PSAM (detecton only).. Fgure 4.7 BEP vs. frame length for dfferent fade rate.3 Fgure 4.8 BEP vs. PDR for dfferent fade rate..3 Fgure 4.9 BEP vs. channel estmaton flter length for dfferent fade rate 4 Fgure 4. BEP vs. average SNR for dfferent msmatched fade rate..4 Fgure 4. BEP vs. average SNR for dfferent msmatched fade rate..5 Fgure 4. BEP vs. average SNR for dfferent number of transmt antennas...5 Fgure 4.3 BEP vs. average SNR for dfferent Tx-Rx antenna numbers wth the total number of antennas fxed 6 Fgure 4.4 BEP vs. average SNR for dfferent fade rate wth our PSAM compare to that wth conventonal PSAM (detecton only), bnary orthogonal sgnalng.6

9 v Fgure 4.5 BEP vs. average SNR, cause of performance loss..7 Fgure 4.6 BEP vs. average SNR, wth and wthout transmt weghtng...7 Fgure 4.7 BEP vs. Frame Length for BFSK 8 Fgure 5. BEP vs. average SNR of BFSK wth dfferent nterpolator sze at moderate fade rate..43 Fgure 5. BEP vs. average SNR of BFSK wth dfferent nterpolator sze at large fade rate..43 Fgure 5.3 BEP vs. average SNR for BFSK and BDPSK at small fade rate..44 Fgure 5.4 BEP vs. average SNR for BFSK and BDPSK at large fade rate...44 Fgure 5.5 BEP vs. average SNR for 4FSK and QDPSK at varous fade rates..45 Fgure 5.6 SEP vs. average SNR of MFSK 45 Fgure 5.7 BEP vs. average SNR of MFSK.46 Fgure 5.8 BEP vs. average SNR of 4FSK.46 Fgure 5.9 BEP vs. average SNR of 8FSK.47 Fgure 5. BEP vs. average SNR of 6FSK.47 Fgure 5. BEP vs. Interpolator sze for small fade rate...48 Fgure 5. BEP vs. Interpolator sze for large fade rate...48 Table C.: Dfferental encodng rule for ST-BPSK, n b Table C.: Dfferental encodng rule for ST-QPSK, n b 3/...7

10 v Abbrevatons and notatons AWGN: Addtve Whte Gaussan Nose BEP: Bt Error Probablty BTRC: Better Than Rased Cosne (pulse) CCI: Cochannel Interference cdf: cumulatve dstrbuton functon CF: Characterstc Functon CGRV: Complex Gaussan Random Varable (Vector) CSI: Channel State Informaton FSK: Frequency Shft-Keyng d: ndependent dentcally dstrbuted INR: Interference-to-Nose Rato IO: Indvdually Optmum ISI: Inter-Symbol-Interference JO: Jontly Optmum LRT: Lkelhood Rato Test MGF: Moment Generaton Functon ML: Maxmum Lkelhood MRC: Maxmum Rato Combnng MRT: Maxmum Rato Transmsson OC: Optmum Combnng pdf: probablty densty functon PDR: Plot (power) to Data (power) Rato

11 x PEP: Parwse Error Probablty PSAM: Plot Symbol Asssted Modulaton IPSAM: Implct PSAM PSD: Power Spectrum Densty PSK: Phase-Shft-Keyng BPSK: Bnary PSK CPSK: Coherent PSK QPSK: Quaternary PSK DPSK: Dfferental PSK BDPSK: Bnary DPSK QDPSK: Quaternary DPSK QR: Quadratc Recever GQR: Generalzed QR RC: Rased Cosne (pulse) REC: Rectangular (pulse) Rx: Receve (dversty) SEP: Symbol Error Probablty SINR: Sgnal-plus-Interference-to-Nose Rato SIR: Sgnal-to-Interference Rato SNR: Sgnal-to-Nose Rato Tx: Transmt (dversty) Through out ths thess, we wll use upper case boldface to represent matrx and lower case boldface to represent vector. All vectors are assumed to be column vectors unless otherwse specfed.

12 x Summary It has been recognzed that spatal dversty usng multple antennas s an effcent technque to combat the severe destructve effects of fadng and nterference on the performance of moble wreless communcaton system. Prevous works on cochannel nterference normally assume that the nterferng sgnals are synchronzed wth the desred sgnal. In the frst half of ths thess we examne the more general and realstc scenaro where the cochannel nterference s asynchronous wth the desred sgnal. In Chapter and Chapter 3, we nvestgate the performance of coherent phaseshft-keyng and dfferentally encoded and decoded phase-shft-keyng wth maxmum-rato-combnng n nonselectve Raylegh fadng channels wth multple asynchronous cochannel nterferers. Through the analytcal study of the effect of the tmng offsets between the nterferer s sgnal and the desred user s sgnal on the error performance, t s found that for system usng rectangular pulse shapng, the synchronzed model actually gves the worst error performance, whle the best error performance s acheved when all the nterferers sgnals are half-symbol-duraton delayed wth respect to the desred user s sgnal. The second half of ths thess examnes the performance of transmt dversty system wth practcal channel estmaton schemes. Two types of the transmt dversty are consdered n ths thess. In Chapter 4 we develop a plot-symbol-asssted-modulaton scheme for a maxmumrato-transmsson based transmt dversty system. Optmum transmt and receve strateges are derved and error performance are examned. In Chapter 5 we consder space-tme block codes wth orthogonal M-ary frequency-shft-keyng. The error performance s examned and compared wth that of dfferental space-tme codes.

13 Chapter Introducton. Background Durng the past few decades, the development of modern moble communcaton systems has experenced a bloomng era [-3]. Many new technologes are developed and mplemented to mprove the qualty of personal wreless communcatons. Steppng nto the new century, moble communcaton has already become an ndspensable element n our fast paced modern lfe. New wreless moble communcaton systems are expected to support more users and provde better qualty of servce for both voce and data applcatons. A prmary desgn objectve for any commercal or mltary moble communcaton system s to conserve the avalable spectrum by reusng allocated frequency channels. For ths purpose, cellular systems are wdely used n wreless communcaton networks whch dvde a geographcal area nto small cells and allow each cell to utlze specfc allocated frequency channels. The same frequency channel could then be reused n other cells that are far away from a gven cell so that the sgnal from the cochannel cells to the cell concerned would be weak enough to avod any destructve nterference. However, as the number of subscrbers ncreases, ether the sze of the cell needs to be reduced or the number of the assgned frequency channels n each cell needs to be ncreased n order to keep up wth the ncreased subscrber densty. Therefore, wth the number of the total avalable channels fxed, the

14 nterference from cochannel cells could ncrease to a level that may cause destructve effects on communcaton n the concerned cell. Another mportant ssue n wreless moble communcaton s to effcently detect the sgnal that has been corrupted from channel fadng. Unlke the conventonal wred communcaton system where the receved sgnal normally only suffers from addtve whte Gaussan nose (AWGN), n a wreless envronment the receved sgnal s typcally a combnaton of many reflected replcas of the orgnal transmtted sgnal wth dfferent power, delay and drecton of arrval. Consequently, on top of the AWGN, wreless communcaton system suffers from multplcatve random ampltude attenuaton and phase dstorton, a phenomenon known as channel fadng. Thus, developng new technques that could reduce the severe mparment caused by channel fadng s always of great mportance for any practcal desgn of hgh qualty wreless communcaton system. Among the numerous nnovatve wreless communcaton technques, spatal dversty recepton usng multple antennas s always a sgnfcant research area that has been shown to lead to tremendous mprovements n system performance. In a system where multple antennas are deployed suffcently far from one another spatally, the receved sgnal from these antennas can be vewed as undergong ndependent channel fadng process. Snce deep fades seldom occur smultaneously durng the same tme ntervals on these ndependent dversty branches, the effect of fadng can be reduced by properly weghtng and combnng the receved sgnal from these branches. Varous dversty combnng schemes have been proposed n the past, varyng n performance and complexty. For a system sufferng only from fadng and AWGN, maxmum rato combnng (MRC) has been known as the optmum combnng scheme

15 3 whch gves the receved sgnals from dfferent dversty branches a weght proportonal to the nstantaneous channel gan of that partcular branch, therefore the nstantaneous sgnal-to-nose rato (SNR) s maxmzed and the probablty of error s mnmzed. In another case, selecton combnng (SC) only chooses the branch that has the largest nstantaneous SNR and detects the sgnal based on observaton from ths one branch only. Although worse n performance when compared wth MRC, SC only processes one dversty branch at a tme and therefore the recever structure s smpler. Besdes combatng fadng, dversty technque can also suppress nterference. For example, for systems sufferng from fadng, AWGN as well as cochannel nterference (CCI), optmum combnng (OC) s proposed to mtgate the effects of both the fadng and the CCI. In addton to dversty recepton, dversty transmsson has also been consdered as an effectve technque to mprove the system performance. Accordng to the requred channel nformaton at the transmtter, transmt dversty can be categorzed nto two forms schemes that requre feedback and those do not requre feedback. For the frst type of transmt dversty, the transmtter requres the knowledge of nstantaneous channel gan so t can pre-weght the sgnal to compensate for the fadng n the same way as a conventonal dversty recever. For the second type of the transmt dversty, channel nformaton s only avalable at the recever, and the transmtter use lnear processng to spread the nformaton across the antennas, whch could also be vewed as a form of codng. One of the most-pursued form of the second type transmt dversty s space-tme codng. In general, spatal dversty s an effcent method to mprove the performance of wreless moble communcaton.

16 4. Motvaton As mentoned earler, current and future generaton wreless communcaton are expected to support more subscrbers and offer hgher transmsson data rate, or, n other word, hgher system capacty. Therefore, cochannel nterference has become an mportant ssue that must be consdered n the desgn of practcal communcaton systems. Dversty systems have been shown to be an effcent method to mtgate the destructve effect of fadng and nterference. However, the effcency of a practcal dversty system to suppress the nterference depends on the avalable amount of nformaton regardng the nterferers channel nformaton. Optmum combnng has been proposed and proven to be effcent n suppressng cochannel nterference, but t follows a smplfed and somewhat an unrealstc assumpton that the system has full channel knowledge for all the users and the sgnals of dfferent users are symbol synchronzed. For a more general and practcal stuaton where the dfferent users are asynchronous, optmum combnng s no longer mplementable. For other types of dversty combnng schemes such as MRC, lttle has been done on the performance analyss for the case wth asynchronous CCI. Therefore t s necessary to fully understand the effects of asynchronous CCI on performance of these systems. More recently, much research efforts have been gven to the desgn and analyss of new dversty schemes that offer lower error probablty and hgher capacty, one of whch s the use of multple antennas at the transmtter sde n addton to conventonal dversty at the recever sde. One potental of a combned transmt and receve (Tx-Rx) dversty system s that wth the same number of antennas utlzed by the system, a Tx-Rx dversty structure generally provdes more transmsson lnks than a conventonal receve dversty. As mentoned earler, there are generally two form of transmt dversty. One way s to provde the transmtter wth pror-

17 5 transmsson channel nformaton, so that the transmtter could use dfferent weghts on dfferent transmt antennas to pre-compensate for the channel fadng. The optmal scheme of ths type of Tx-Rx dversty s known as maxmum rato transmsson plus maxmum rato combnng (MRT-MRC) dversty. In most of the prevous works on the performance analyss of such Tx-Rx dversty systems, a basc assumpton s that the system has complete knowledge of the nstantaneous channel gan. Consequently, the error performance results obtaned n these works can only be vewed as lower bounds. To provde desgners wth more realstc results, t s mportant to consder more practcal channel estmaton strateges for Tx-Rx dversty systems, and examne ther performance n the presence of channel estmaton errors. Also wth mperfect channel estmaton, the optmum structure of ths type of the transmt-receve dversty may also assume a dfferent form other than MRT-MRC. Ths s an optmum desgn problem that worth nvestgatng. Another form of Tx-Rx dversty s to use space-tme (ST) codes. Space-tme trells codng s a recent proposal that combnes sgnal processng at the recever wth codng technques approprate to multple transmt antennas. It has been shown that specfc space-tme trells codes perform extremely well n slow-fadng envronment. However, the decodng complexty of space-tme trells codes ncreases exponentally wth transmsson rate. Recently, Alamout dscovered a remarkable scheme for transmsson usng two transmt antennas whch requres much less decodng complexty. Followng Alamout s work, orthogonal space-tme block codes are developed whch utlze sgnal processng and codng technology to acheve dversty gan from both the spatally separated antennas and orthogonal codes transmtted on these antennas. Comparng wth MRT-MRC dversty, space-tme codes do not need any pror-transmsson channel nformaton at the transmtter. Thus, no feed back s

18 6 requred for ths system. However, the channel estmaton at the recever end stll needs to be carefully examned. For any communcaton systems undergong fadng, n order to detect the transmtted sgnal from the multplcatve fadng corrupton, certan channel estmaton schemes must be employed. One popular channel estmaton method s to nsert plot symbols perodcally nto the data symbol to contnuously sample the channel and produce channel estmaton for data symbol detecton. Alternatvely, the system can also employ a non-coherent modulaton scheme such as dfferental encoded and decoded PSK, where the nformaton s embedded n the phase dfference of the adjacent symbols and the detecton s accomplshed by usng the channel s memory. Although a dfferental system provdes a smple and robust soluton for data detecton n fadng channels, when the channel fadng fluctuates fast, or, n other words, when the channel memory s short, ts performance degrades fast as well. On the other hand, orthogonal sgnalng another commonly consdered non-coherent sgnal - has been shown [4] as a modulaton scheme that possess a channel measurement component. In lght of ths fact, the channel estmaton can be refned by explotng the fadng autocorrelaton through a sequence of receved symbols. Ths encourage us to use orthogonal sgnalng n transmt dversty system and compare ts performance wth coherent sgnalng and also dfferental system.. 3 Lterature Revew The concept and fundamental performance analyss of dversty system are well documented n papers and books such as [], [3,] [5,] [6]. It has been shown that n general MRC recever provdes the optmum performance by maxmzng the

19 7 nstantaneous SNR. However, most of these fundamental analyses concern only ndependent dversty systems wth quas-statc fadng and wthout CCI. For fluctuatng fadng channels, the performance of coherent PSK sgnal remans the same as that of quas-statc fadng channels because perfect channel estmaton s assumed. However, for dfferentally encoded and detected PSK sgnals, the fadng fluctuaton plays an mportant role n the error performance as the dfferental detector reles solely on the channel autocorrelaton to recover from the fadng dstorton. Although the performance of DPSK suffers from channel fadng fluctuaton, t requres no channel estmaton mechansm, and thus the recever structure can be very smple, whereas for coherent PSK, certan channel estmaton scheme such as PSAM must be utlzed to provde channel reference for coherent detecton. The performance of DPSK sgnal n fluctuatng fadng channels s evaluated n [7] wth selecton combnng. In [8], [9], the BEP of MDPSK s studed for fluctuatng nonselectve Raylegh fadng channels wth MRC recepton. For the Rcan fadng case, the exact BEP of MDPSK and NCFSK s gven n [], where an MGF based method s adopted. In [], the same modulaton schemes are consdered and closed-form expressons for the SEP are obtaned wth post-detecton equal gan combnng. A smplfed tght bound for the smlar case can be found n []. In [3], [4], generalzaton of dversty combnng scheme and optmzaton of the recever structure for DPSK sgnalng are dscussed when the fadng statstcs are known at the recever, and the BEP performances are gven correspondngly. More recently, wth the work n [5] on calculatng the error probablty for two-dmensonal sgnal constellatons, new mathematcal tools nvolvng the Gaussan probablty ntegral and Marcum Q-functon are developed [6], [7]. These advancements n mathematcal analyss tools and technques make t possble to evaluate the error performance of

20 8 lnearly modulated sgnals over generalzed fadng channels under a unfed analytcal framework [8]. However, most of the results from ths approach are n complcated forms nvolvng numercal ntegrals, where the effects of ndvdual system parameters are dffcult to examne. For a cellular system wth CCI, MRC no longer provdes the optmum performance because only the fadng of the desred user s taken nto consderaton and compensated for by MRC. Therefore, more research nterests have been gven to OC whch explots the CSI of the CCI component as well. Compared to MRC, OC has been shown to be more effectve n suppressng nterference [9]-[3]. Although excellent n performance, the practcalty of OC s somehow questonable, as n realty t s very dffcult to obtan the requred CSI for both the desred user and the CCI. In most cases, MRC remans a more practcal choce even for systems wth CCI [4]. Most of the prevous works model CCI as a sgnal synchronzed wth the desred sgnal [4]-[8], whch s mathematcally smpler n dervaton and analyss, but practcally hard to realze on the other hand. Among the work that consders asynchronous CCI, the characterzaton of asynchronous CCI can be found n [9], and ts applcaton to the performance analyss can be found n [3], [3] for coherent PSK and DPSK, respectvely. The error performance of BPSK communcaton lnks wth multple asynchronous nterferers s studed n [3] and ts counterpart of DPSK system s gven n [33], n whch exact error probabltes are derved for sngle channel system,.e., ether non-dversty or dversty wth selecton combnng. More recently, results for the performance of BPSK n Nakagam fadng channels wth asynchronous CCI s reported n [34]. Agan, the approach n ths work s currently lmted to sngle channel systems, and the form of the BEP results s very complcated. For selecton combnng dversty system, BEP expressons of both CPSK and DPSK are gven n [35]. In [36],

21 9 performance of MPSK wth dual-dversty system usng equal gan combnng (EGC) and selecton combnng (SC) s studed. However, the extenson of the approach to hgher order dversty combnng s not addressed. In [37] a general methodology for performance analyss of a system wth asynchronous CCI s provded. Some new methods for evaluatng the outage probablty are proposed. However, the BEP analyss n ther work could only be carred out usng a semanalytcal method whch requres the help of adaptve algorthm smulaton. Another nterestng perspectve to the CCI related research s ts smlarty wth multuser detecton where the data detecton s performed for all the cochannel users [38]. By applyng the concept of multuser detecton n the CCI scenaro, the work n [39], [4] has shown that a great performance mprovement can be acheved over the popular OC. However, exact error performance analyss for multuser detecton remans rare. More recently, research nterest has been drawn to the exact performance analyss of optmum detecton for sgnals n the presence of cochannel nterference [4]-[43], where the exact BEP for a two-user system s studed usng jont-optmum (JO) (one-shot) detecton. The analyss for ndvdually-optmum (IO) (mnmum error probablty) remans unsolved n these works but the performance dfference has been shown to be very slm for most of the commonly consdered system condtons. For transmt dversty systems assumng channel nformaton at the transmtter sde, the concept of MRT has been summarzed and studed n [44]. The optmzaton of transmt and receve weght vectors s carred out so as to maxmze the nstantaneous SNR, assumng equal energes for all the entres of the receve weght vector but dfferent phase. An approxmate expresson for the bt error probablty (BEP) of bnary phase-shft-keyng (BPSK) s also obtaned for the hgh SNR scenaro. In [45], [46], mproved weghtng schemes are suggested whch remove the

22 performance degradaton due to the equal-energy assumpton n [44]. The jont optmal weghtng scheme at both the transmtter and the recever s derved n [47], whch relates the error performance analyss wth the dstrbuton of the egenvalues of a complex Wshart matrx. The exact error performance of ths optmal transmt-receve dversty system n Raylegh fadng has been studed n [48], assumng perfect CSI. In [49], the dstrbuton of the egenvalues of a non-central complex Wshart matrx s analyzed and the outage probablty for the optmal transmt-receve dversty system s studed. Ths enables the performance analyss of Tx-Rx dversty systems n a Rcan fadng envronment to be studed. Among these prevous works regardng Tx-Rx dversty system, one mportant assumpton s that perfect CSI must be avalable at both the transmtter and the recever. Thus the performance analyss results obtaned so far are only lower bound benchmarks whch could not be acheved n a realty. Therefore, to make Tx-Rx dversty a more realzable communcaton technque, t s mportant to desgn a practcal channel estmaton scheme wth the optmal transmtter/recever structure, and study the effect of channel estmaton error on the system performance. Another form of transmt dversty as ntroduced earler s space-tme codes. The systematc desgn procedure together wth performance analyss regardng Space- Tme block codng can be found n [5], [5]. The performance of specfc Space-Tme trells codes has been shown to perform extremely well n slow-fadng envronment [5]. However, the decodng complexty of ths type of codes ncreases exponentally wth transmsson rate. A smple scheme usng two transmt antennas s proposed n [5]. Despte a certan performance loss compared to the trells codes n [5], ths scheme offers farly good performance and smple decodng at the same tme. Later ths smple scheme s extended to multple transmt antennas n [53] usng the theory

23 of orthogonal desgns. Although excellent n performance, practcal mplement of these codes requres certan channel estmaton scheme such as PSAM [5], or noncoherent detecton such as dfferental detecton [54], [67], [68].. 4 Contrbutons of the Thess Ths thess provdes error performance analyss for dversty systems and also develop optmum system structure for transmt dversty system wth practcal channel estmaton schemes. For the conventonal Rx-dversty recever, we study ts error performance when asynchronous cochannel nterference s presented. We consder two extreme condtons regardng the knowledge of channel nformaton of the desred user s sgnal at the recever,.e., perfect channel estmaton for coherent PSK, and no channel estmaton for dfferental PSK. By condtonng on the tmng offsets of the nterferers, we derve error performance results that enable us to examne the effect of asynchronous CCI on the performance of the desred sgnal. Study reveals that the synchronous system s actually the worst case as far as error performance s concerned, whle the best case for the detecton of the desred sgnal s that all the nterferng sgnals are half symbol-duraton delayed. Therefore, for scentsts and engneers who need to desgn a communcaton system based on the worst case desgn, our results provde a quck performance assessment to spare them from havng to average the error probablty over all the nterferng sgnals tmng offsets. For a MRT-MRC type dversty system, we develop a practcal channel estmaton scheme usng plot-symbols-asssted-modulaton (PSAM). Based on ths partcular PSAM scheme, we derve the optmum transmt and receve weghtng strategy and study ts performance. The optmzaton of varous parameters related to

24 PSAM s also demonstrated. We then extend the proposed PSAM Tx-Rx dversty system to bnary orthogonal sgnalng, and dscuss the feasblty of usng the sequence observaton to refne the channel estmaton from the unmodulated component. We also compare our proposed PSAM Tx-Rx dversty wth deal MRT-MRC dversty where the cause of the performance dfference s carefully examned. These results gve practcal system desgners a good reference when consderng employng MRT- MRC dversty n realty. For transmt dversty usng space-tme codes, we develop an orthogonal FSK modulaton-based Alamout-type code. Channel estmaton s done by the unmodulated component of the orthogonal sgnals. The performance of ths ST-FSK system s then analytcally examned and compared wth that of dfferental ST codes. It shows that by explotng the channel measurements from adjacent symbols, FSK sgnals provde much better performance than ther dfferental counterparts when the channel fadng fluctuaton s fast. In summary, ths work provdes a comprehensve performance analyss for dgtal modulatons n dversty systems by consderng the effects of varous practcal ssues n wreless moble communcatons on error performance, namely, channel fadng fluctuaton, asynchronous CCI and mperfect channel estmaton. Also t dscusses the optmzaton problem for transmt dversty systems wth practcal channel estmaton schemes.

25 3 Fgure. Thess structure. 5 Thess Outlne Chapter presents the performance analyss of CPSK n nonselectve Raylegh fadng channels wth MRC recepton and multple asynchronous CCI. The effect of the CCI tmng offset s also examned. Three Nyqust pulses are consdered, namely, the rectangular pulse, conventonal RC pulse and the newly proposed BTRC pulse. Chapter 3 presents the performance analyss of DPSK n nonselectve Raylegh fadng channels wth MRC recepton and multple asynchronous CCI. Although the approach n Chapter s applcable for ths case, we adopt a dfferent mathematcal method for ths case whch demonstrates the effect of dversty branch correlaton. The effects of the CCI tmng offset are also examned. Smlar as CPSK case, we also compare the performance of three dfferent Nyqust pulses. Chapter 4 descrbes a practcal PSAM channel estmaton scheme for Tx-Rx dversty system and derves the optmum transmtter/recever structure for ths partcular system. Performance analyss s then gven based on the optmum desgn.

26 4 Both PSK and bnary orthogonal sgnalng are consdered. By applyng the ML detecton prncple, we fnd that the optmum transcever structure for PSAM based PSK system remans smlar to that derved for deal coherent PSK system. However, for bnary orthogonal sgnalng, the transcever utlzes both the estmated CSI from PSAM and that from ts own unmodulated component, whch s dfferent from what has been obtaned by other prevous work. An attempt of combng the proposed PSAM scheme wth the generalzed quadratc recever (GQR) s gven, where only suboptmal soluton s obtanable currently. Chapter 5 develops another type of transmt dversty usng space-tme codng wth orthogonal sgnalng. It s shown ths new modulaton scheme enables channel reference wthout plot symbols, thus no transmsson rate s sacrfced. And the detecton complexty s no more than that of dfferental ST codng, whle the performance of ths proposed system does not suffer severely from fast fadng as a dfferental system does. Chapter 6 gves some conclusons for the results obtaned and suggests some possble future extenson from the current research.

27 5 Chapter BEP of Coherent Phase-Shft-Keyng n Nonselectve Raylegh Fadng Channels wth Multple Asynchronous Cochannel Interference In ths chapter we nvestgate the error performance of BPSK and QPSK n nonselectve Raylegh fadng channels wth MRC dversty recepton and wth multple asynchronous cochannel nterferers. An ntroducton s gven n Secton.. The system model s descrbed n Secton. together wth the detector structure. In Secton.3 we carry out the performance analyss. In Secton.4 we study the effect of the asynchronous nterferers tmng offset on the BEP of the desred sgnal. Numercal results and dscusson are gven n Secton.5, and Secton.6 summarzes ths chapter.. Introducton In cellular moble communcatons, frequency reuse s necessary to ncrease spectral effcency so as to accommodate more subscrbers. In such a system, the detecton of one user s data s often corrupted by sgnals from users n other nearby cells usng the same frequency. Ths wll result n cochannel nterference, whch nevtably leads to degradaton n the performance of wreless communcatons. In addton to nterference, fadng s also a major source of performance mparment n a

28 6 moble wreless envronment. The channel fadng process ntroduces both random ampltude and random phase dstorton to the transmtted sgnal. Therefore, channel estmaton has to be carred out n order to mplement coherent detecton for modulaton schemes for whch accurate phase trackng s crucal, such as PSK.. System Model We consder a system n whch the MPSK sgnal receved from the desred user over L ndependent, dentcal dversty branches s corrupted by K asynchronous cochannel users sgnal and AWGN. The complex baseband transmtted sgnal of the desred user s ~ p j p D ( t) ESD e φ ( ) p D s g ( t pt ) TD (.) where / T s the symbol transmsson rate and g TD (t) denotes the mpulse response of the transmtter pulse shapng flter of the desred user. The average energy per symbol for the desred user s E SD. The phase φ (k D ) of the transmtted sgnal contans the kth transmtted symbol nformaton. A reasonable assumpton s that all nterferng sgnals have the same modulaton format as the desred user s sgnal. Thus the baseband transmtted sgnal of the lth nterferng user has the smlar form ~ p j p l ( t) ESl e φ ( ) p l s g ( t pt ) Tl (.) where E Sl s the average energy per symbol for the lth nterferng user sgnal. We further assume that all the users use the same transmt pulse shapng flter. Consequently, we have gtl ( t) gt ( t) forl D,,,, K.

29 7 We assume that both the desred user s sgnal and the nterferng users sgnal undergo slow nonselectve Raylegh fadng. At the recever, the receved sgnal from the th dversty branch s ~ r ( t) c~ ( t) ~ s ( t) c~ D D + l K,, l l l + t l ( t + τ ) ~ s ( t + τ ) n~ ( ) (.3) where c~ D, ( t) and c~ l, ( t) are the channel fadng process of the th dversty channel for the desred user and the lth nterferng user respectvely. The AWGN term n ~ ( t ) has zero mean and a double sde PSD of N. At the recever, the receved sgnals are matched fltered and sampled at the symbol tme of the desred user sgnal, assumng perfect symbol synchronzaton wth the desred user s symbol tme. As we assume n general an asynchronous system, the lth nterferng user s sgnal may come after an arbtrary delay τ l whch s unformly dstrbuted wthn [, T ). After matched flterng and samplng, the dscrete receved sgnal at the nput of the detector over the th dversty channel,,,l, for the kth symbol nterval [kt,(k+)t] can be represented by a decson statstc ~ r ( k ) as [3, Sec9.] ~ r ( k) p + K E SD l p e jφd ( k p) E Sl e c~ D, jφl ( k p) ( τ + pt ) g c~ l, T ( τ ) g ( τ + pt + τ ) g l R ( kt pt τ ) dτ + n~ ( k) T ( τ ) g l R ( kt pt τ τ ) dτ l.(.4) Here g R (t) s the receve flter matched to the transmt flter such that the overall cascaded mpulse response g( t) g ( t) g ( t) wthout fadng would be a pulse shape R T that fulfls the Nyqust crteron. Snce g T (t) s a unt-energy pulse, the peak ampltude of g T (t) s. The receved sgnal n a form lke (.4) s generally dffcult to manage because of the ntegral terms. Therefore, we make a commonly adopted assumpton that the fadng processes affectng the desred and nterferng sgnals

30 8 change slowly enough so that they can be consdered as constant durng the effectve length of the mpulse response g R (t) and (t) g T. Thus the fadng process nsde the ntegrand of (.4) could be approxmated by ts nstantaneous value at the samplng tme and could be factored out of the ntegral. The receved sgnal at the detector nput can now be wrtten as, ~ r ( k) K p P jφ ESD e D, Sl l, l p P ~ ~ D ( k p) jφl ( k p) c ( k) + E c ( k) e g( pt τ ) + l n~ ( k) (.5) where c~ D, ( k) and c~ l, ( k) are the pecewse-constant approxmatons to the th channel fadng process durng the kth symbol nterval [kt, (k+)t] for the desred user and the lth nterferng user, respectvely. It s obvous that n the presence of a non-zero symbol tmng offsetτ l, the effectve nterferng component comes from a sequence of transmtted symbols n a smlar form as ISI. Snce n general the Nyqust pulse shapng used n practcal communcaton system has a fast decayng waveform, we could assume that the effectve ISI components are composed of the nearest P + symbols. For the case of nonselectve Raylegh fadng channels wth even power densty spectrum, c~ D, ( k) and c~ l, ( k) are both complex Gaussan random varables whose quadrature components are d Gaussan RVs, wth mean zero, varances ~ E[ c ] D, ( k) cd and ~ E[ c ] l, ( k) cl, respectvely. The nose term ~ ( k ) n s the sampled output of the AWGN process after matched flterng from the th dversty branch, whch s a complex Gaussan random varable wth mean zero and varance ~ E n ( k) ] N /. [ We consder an ndependent dversty system where the receved sgnals from the same user at dfferent dversty branches have d channel fadng gans,.e., for arbtrary j, c~ l, ( k) and c~ l, j ( k) are d, forl D,,, K. Also, the channel fadng

31 9 gans for dfferent users are assumed to be ndependent, ether at the same dversty branch or dfferent ones,.e., c ( k) s ndependent of c ~ jh ( k ) for arbtrary l h. The l nose components n ~ ( k ) from dfferent dversty branches are assumed to be ndependent and dentcally dstrbuted, and they are ndependent from channel fadng gans of all the users from all dversty branches. The overall pulse shape g(t) we consder n ths work ncludes the followng three types. The frst one s the trangular pulse whch corresponds to the response of a matched flter to a rectangular pulse [3], [3]. Its correspondng tme functon and frequency spectrum of the rectangular pulse shapng are gven by τ g REC ( τ ) T τ < T otherwse and 4sn ( Tf / ) G REC ( f ). Tf Wth REC, the receved sgnal n (.5) could be smplfed to K ~ ~ ( ) ~ ( ( ) ( ) l ( ) l j k j k j k r k) φd ESD e cd, k + φl τ l ESl cl, ( k) e + φ τ e + l T T n~ ( k) (.6) where the effectve nterference comes from the adjacent two symbols only. The second pulse we consdered s the popular RC pulse that has been wdely used n modern dgtal communcaton systems. The RC pulse s tme functon and the correspondng frequency spectrum are gven by and g RC sn( πt / T ) cos( ( t) πt / T 4 παt α t / T ) / T

32 G RC T ( f ) T / + cosπt / α f α T α f T α + α f, T T + α f T where α s the roll-off factor and t represents the percentage excess bandwdth. It s worth notng that through out ths work, we do not consder a band-lmted system, thus the value of the roll-off factor affect the performance through the shape of the pulses when usng dfferent value of α, not through the percentage of the lost bandwdth t represents,.e., the shape of the receved sgnal s not dstorted by loss of sde-band frequency components. Trangular.8 RC BTRC Fgure. A comparson of the tme waveform of the three pulses The thrd pulse consdered n ths study s the BTRC pulse that has been proposed recently [55]. Its tme functon and frequency spectrum are gven by, g BTRC sn( πt / T ) 4βπt sn( παt / T ) + β cos( παt / T ) β ( t) πt / T 4π t + β and

33 G BTRC T T ln α T exp f α T ( f ) T ln + α T T exp f α T α f T α f T T, + α f T T + α f T where β T ln / α and α s the roll-off factor. Ths new pulse has been shown to have a better eye dagram and a better error performance than RC pulse n the presence of ISI n a baseband system [55]. A comparson of these three pulses s llustrated n Fg.., where for the RC and BTRC pulses we use a roll-off factorα. 5, where for RC pulse and BTRC pulse wth the same roll-off factor, we found that BTRC pulse has smaller sdelobes than RC, thus a better performance at the presence of symbol tmng error can be antcpated. In (.5), the frst term represents the desred sgnal component. The second term represents the CCI components from nterferng users and each of these components contans ISI terms due to the mperfect symbol synchronsm between the desred user and the nterferng users. The thrd term represents the AWGN nose n each dversty branch. As mentoned earler, at the recever, t s assumed that only the channel fadng gans for the desred user s estmated perfectly n each dversty branches. Therefore, a coherent detector s mplemented. The receved sgnals from each dversty branch are weghted by the complex conjugate fadng gan of the desred user to remove the phase dstorton. Wth equprobable transmtted symbols, the MRC recever generates the decson statstcs M L * m ( ) Re ( ) D, ( ) exp( π / ) m q k r k c k j m M

34 based on ML detecton prncple, and chooses φ( k) π l M as the detected symbol f ql ( k) Max{ qm ( k)}. The pre-detecton MRC recever s sketched n Fg... r ( ) t r ( t)..... (t) r L Matched Flter Matched Flter Matched Flter t kt t kt t kt ~ r ( k ~ r ( k ) ) rl ~ ( k ) ~ c * ( k ~ c * ( k ) ) ~ * ( k) c L L * [ r ( k) c ~ ( k) ] exp[ jπ m / M ] Re( ) (k) q m Fgure. Recever structure for CPSK. 3 Performance Analyss The BEP of BPSK and QPSK, condtoned on the set I of known transmtted symbols of all users and known tmng offsets of every nterferng user can be obtaned from the followng probablty L F( α, φ, τ ) P Re r ( k) c ( k) e <, ( k p) I * jα L { }, D, τ l φl l, p (.7) where α s one-to-one mapped to the nformaton phase φ D (k) n determnng the BEP and ts specfc value wll be gven later. Note that gven the set I, the receved sgnal n each dversty branch s a summaton of multple ndependent complex Gaussan random varables. Therefore, the receved sgnal ~ r ( k) I s also a complex Gaussan random varable wth condtonal mean E[ r~ ( k) I],

35 3 (.8a) condtonal varance E [ r ( k ) ] E + E e g ( pt τ ) + N, (.8b) K jφl ( k p) r SD cd Sl cl l l p and condtonal covarance wth ~ ( ) c D, k [ ( ) * ( ), ( )] j D k rc r cd ρ E r k c D k ESd e φ cd, (.8c) where ρ s the cross correlaton coeffcent between the channel fadng gan of the desred user s sgnal and the total receved sgnal. As we assume dentcal dversty branches, ths correlaton coeffcent holds dentcal for all. For two jontly dstrbuted complex Gaussan random varables ~ x and ~ y, wth mean zero, varance [ ~ E x ] and [ ~ E y ], and covarance x y E[ ~~ xy * ] ρ xy x y where ρ denotes the cross-correlaton coeffcent between x~ and y ~, we have the relatons [56] that when condtoned on y ~, x ~ s a condtonal Gaussan random varable wth condtonal mean E[ x y ] ρ( ) y and x y condtonal varance ~ ~ ] ) E[ x y ( ρ x. If we apply these propertes to x c~ ( k) e ~ k jα D, and r ( ), we obtan the followng condtonal mean D E[ r ( k) x ] E e x (.9a) j[ φ ( k ) + α ] SD and varance E [ ( ) ] SD cd E r k x r. (.9b) r cd Therefore we could rewrte the kth receved sgnal as: D r ( k) E e x + e (.) j[ φ ( k ) + α ] SD

36 4 where e represents the uncertanty about ~ r ( k ) when condtoned on x. It s a complex Gaussan random varable wth mean zero and a varance equals to the condtonal varance n (.9b). Substtutng ths alternatve representaton of ~ r ( k ) nto the decson statstc (.7), we have L ~ j r ( * α Re[ k) cd, ( k) e ] ESD cos( φd ( k) + α) x + E L (.) L where E Re[ e x ] *. As we assume crcular symmetry for all the channel fadng gans and AWGN components, and ndependence of the channel fadng gans between dfferent dversty branches, t s straghtforward to show that E s a real Gaussan random varable wth mean zero and varance ( r ESD cd ) x L. Snce a Gaussan random varable s completely descrbed by ts mean and varance, the probablty n (.7) could now be wrtten as F( α, φ, τ ) x P E L ( < ESD cos( φd ( k) + α) x x ) cos( φd ( k Q r cos( φd ( k Q ) + α) ESD ( E ) x ) + α ) ESD ( E ) r SD cd SD cd L L x cos( φd ( k) + α ) cos( φ ( k) + α ) < D (.) where y Q( x) exp dy x π s the Gaussan Q-functon. To remove the condton on L { x}, we average the above probablty over the dstrbuton of random varable v L x. Snce x c~ ( k) e jα D, s a complex Gaussan random varable wth mean zero and varance, t s easy to show that v has a ch-square dstrbuton [3] wth a pdf gven by cd

37 5 L v f v ( v) v exp[ ] L Γ( L) cd ( ) cd (.3) Averagng the condtonal probablty (.) over the p.d.f. of v n (.3), we get [3, eqn4.4-5] where F( α, φ, τ ) E SDd D L Q v exp[ L ( r ESD cd ) ( cd ) Γ( L) cd µ L L- j cos ( φ ( k) + α) v (L + j- )! + µ ( L )! j! j v ] dv (.4a) cos( φd + α ) ESD cd µ. (.4b) E cos ( φ + α ) + E SD D cd r Here we can wrte Eqn. (.4) n one expresson from the two results n (.) by usng the relaton SD cd L L- j L L- j (L j - )! (L j - )! µ + + µ + µ + µ ( L )! j! ( L )! j!. j j To calculate the average bt error probablty usng the result n (.4), frst we need to average over all the possble nterferng users symbol patterns. As we assume that the domnant cross-term ISI contrbuton from the lth nterferng sgnal s lmted to some P+ terms, we have F( α, φd ( k), τ ) (,, ). ( + ) F α φ τ (.5) P K M φ pattern Fnally, averagng the result over the dstrbuton of every nterferng user s tmng offset gves us T T ( ) τ τ τ F α, φ ( k) F( α, φ ( k), τ ) f ( τ ) f ( τ ) f ( τ ) dτ dτ dτ. D D K K (.6) For a system usng the rectangular pulse, an alternatve approach to derve the

38 6 average BEP s possble by drectly studyng the dstrbuton of the combnaton of the transmtted symbol and the random tmng offset. Substtutng (.6) nto (.8) then (.4), we have, for REC pulse shapng system, µ ( k + ) cos φ ( ) α E D SD cd ( + ) + K SD cos φd ( ) α cd Sl cl l E k E y (.7) where τ l yl [ cos φl ( k) ] + [ + cos φl ( k) ] T. To calculate the average BEP, one needs to average (.4) wth (.7) over the dstrbuton of y l only. Usng the total probablty theorem, the cdf of y l can be calculated from the dstrbuton of the tmng offset τ l together wth the assumpton of equprobable symbols as the followng F ( Y ) Pr{ y < Y} yl Pr{ φ ( k)}pr{ y < Y φ ( k)} φl ( k ) l l l l Y / Y < Y < BPSK Y Y / 4 Y + / Y / < Y < QPSK / 4 Y < Y / Y (.8) The correspondng pdf of y l could then be obtaned as

39 7 p yl Fyl ( Y ) ( Y ) Y / δ ( Y ) Y /(4 Y ) < Y < BPSK Y. (.9) / 4 δ ( Y ) Y /(8 Y ) + /( Y ) / < Y < QPSK /(8 Y ) < Y / Y where δ (Y ) s the Drac delta functon. As we assume ndependent CCI,.e., the transmtted symbols are ndependent and the delays of the nterferers are ndependent, the dstrbuton of the summaton K E Sl cl y l can be easly calculated from the CF of each y l. From (.9), the CF can be derved as jωy Φ ( ω) p ( Y ) e dy yl yl jω e π erf [ jω ] + 4 jω jω jω e π erf [ jω ] ( + j) e π erf [.5( + j) ω ] jω 8 ω BPSK QPSK (.) where erf [ x] e dt π x t denotes the error functon. The CF of the summaton term S K E y K Sl cl l n (.7) can now be calculated as Φ S ( ω) ESl clφ yl ( ω). Fnally by takng the nverse transform we can get the pdf of S whch lead to another form for the average probablty of (.7) as K ( ) ( ) jωs F φ ( k) + α F α, φ ( k), S ds Φ ( ω) e dω D D S l. (.)

40 8 C α B π / 4 D A Fgure.3 Sgnal constellaton and decson regon The sgnal constellaton and decson regon are sketched n Fg..3. The probablty F ( φ k α ) ( ) D + obtaned n (.6) or (.) actually denotes the probablty that the receved sgnal vector after weghtng and combnng, L * r ( ) ( ) k c k, falls nto the grey zone above whch s actually the left half of the complex plane that has been clockwse rotated by an angle α. Usng QPSK as an example here, as we assume Gray encodng of the transmtted symbol, when s transmtted, the recever wll make a wrong decson f the vector falls nto regon B, or f regon C, or f regon D. The average BEP for ths case s then ( φ ) P ( k) Pr( V B) + Pr( V C) + Pr( V D) b D [ Pr( V B) Pr( V C) ] [ Pr( V C) Pr( V D) ] π π Fα, φd ( k) + Fα, φd ( k) 4 4 Smlarly t could be shown that π π π 3π π Pb φd ( k) Fα, φd ( k) + Fα, φd ( k) 4 4,.

41 9 3π 3π Pb ( φd ( k) π ) Fα, φd ( k) π + Fα, φd ( k) π 4 4, 3π π 3π 3π 3π Pb φd ( k) Fα, φd ( k) + Fα, φd ( k) 4 4. Therefore, the average BEP for QPSK s gven by π 3π PQPSK Pb ( D ( k) ) Pb D ( k) Pb ( D ( k) ) Pb D ( k) 4 φ + φ + φ π + φ. π Fα + φd ( k) 4 (.) For BPSK system, followng a smlar dervaton, the average BEP s gven by PBPSK F D k F D k + F + ( α, φ ( ) ) ( α π, φ ( ) π ) ( α φ ( k) ) D. (.3) Equatons (.), (.3), together wth (.4), (.6) and (.), summarze the procedure of calculatng the average BEP for BPSK and QPSK n nonselectve Raylegh fadng channels wth multple asynchronous CCI usng MRC dversty recepton. It s worth notng that the average BEP s ndependent of the transmtted symbols of the desred sgnal. Ths s due to the symmetry of the sgnal constellaton and the crcular symmetry of the fadng and the AWGN component we assumed n our system model.. 4 Effects of Symbol Tmng Offsets The expresson of the average BEP obtaned n Secton.3 nvolves numercal ntegrals thus the calculaton could be very tme consumng for the case of large number of nterferers. Smlar computatonal complexty s also encountered n prevous research works concernng multple asynchronous CCI, e.g., [3], [34], [57].