Simulation of Communication Systems

Transcription

1 Simulation of Communication Systms By Xiaoyuan Wu Thsis submittd to th faculty of th Virginia Polytchnic Institut and Stat Univrsity in partial fulfillmnt of th rquirmnts for th dgr of Mastr of Scinc in Elctrical Enginring Approvd: Dr. William H. Trantr Dr. Brian D. Wornr Dr. Ira Jabcobs Octobr, 998 Blacksburg, Virginia Kywords: Simulations, Communication Systms

2 Simulation of Communication Systms By Xiaoyuan Wu Committ Chair: Dr. William H. Trantr Abstract Digital communications and computrs ar having a trmndous impact on th world today. In ordr to mt th incrasing dmand for digital communication srvics, nginrs must dsign systms in a timly and cost-ffctiv mannr. Th numbr of tchnologis availabl for providing a givn srvic is growing daily, covring transmission mdia, dvics, and softwar. Th rsulting dsign, analysis, and optimization of prformanc can b vry dmanding and difficult. Ovr th past dcads, a larg body of computr-aidd nginring tchniqus hav bn dvlopd to facilitat th dsign procss of complx tchnological systms. Ths tchniqus rly on modls of dvics and systms, both analytic and simulation, to guid th analysis and dsign throughout th lif cycl of a systm. Computr-aidd dsign, analysis, and simulation of communication systms constitut a nw and important part of this procss. This thsis studis diffrnt aspcts of th simulation of communication systms by covring som basic idas, approachs, and mthodologis within th simulation contxt. Prformanc masurmnt of a digital communication is th main focus of this thsis. Howvr, som popular visual indicators of signal quality, which ar oftn gnratd in a simulation to provid a qualitativ sns of th prformanc of a digital systm, ar also considrd. Anothr purpos of this thsis is to srv as a modl for dvloping simulations or tmplat of othr systms. In othr words, studnts larning to simulat a systm can us th work prsntd hr as a starting point.

3 Acknowldgmnt First, I would lik to xprss my most sincr thank to my advisor Dr. William Trantr, whos insight and invaluabl advic has hlpd m to rach this important milston of my lif. H has bn a mntor and frind in guiding m through my graduat carr. It has bn a grat honor for m to work undr th guidanc of Dr. Trantr. I also would lik to us this opportunity to thank Dr. Jacobs and Dr. Wornr to srv as my committ mmbrs. Without thir xprtis th compltion of this thsis would not b possibl. Thir hlp and guidanc ar dply apprciatd. I would also lik to thank my parnts and my sistr for loving m and supporting m throughout my lif. Thy hav taught m th valu of hard work and st grat xampls to m. Without thm I would not b hr today. Last but not last, I would lik to thank all my frinds hr at Virginia Tch. Without thir hlp and support, I would not b abl to complt this long journy. iii

4 TABLE OF COTETS Chaptr. Introduction.... History of Simulation-basd Analysis and Dsign.... Ovrviw of this thsis... 3 Chaptr. Rprsntation of Signals in Simulation Introduction Continuous and Discrt-Tim Signal Rprsntations of Bandpass Signals in Simulation Rprsntation of th M-ary PSK Signals Rprsntation of th M-ary FSK Signals Filtring... 8 Chaptr 3. Simulation Tchniqus Two Major Mthods of BER Masurmnt Mont Carlo Mthod Quality of an stimator Exampl: a BPSK Systm Simulation Block Analysis Mthod Discussion Quasianalytical Estimation Rstrictions and considrations for Quasianalytical tchniqu Cas Study Discussion of th Quasianalytical simulation Conclusions... 4 Chaptr 4. Cas Study iv

5 4. Introduction Mthods of Prformanc Evaluation Using Simulation Simulation of th M-ary PSK Signal Simulation Rsults QPSK Systm Simulation Rsults Conclusions Chaptr 5. Conclusion and Futur Work Conclusion Futur Work Appndix A Appndix B Appndix C v

6 LIST OF FIGURES Figur 3. Mont Carlo Estimation Systm Block Diagram... 0 Figur 3. Confidnc Intrval on BER whn stimatd valu is 0-6 basd on normal approximation... 7 Figur 3.3. BPSK Systm Block Diagram... 9 Figur 3.4 Signal Constllation Diagram for BPSK Signal with Transmittd Angl... Figur 3.5 Simulatd BER Prformanc Curv with ISI... Figur 3.6 Ey-diagram of th signal at th output of th modulator... 3 Figur 3.7 Signal Constllation of th signal at th output of th modulator... 3 Figur 3.8 Ey-diagram of th signal at th output of th transmitting filtr... 4 Figur 3.9 Signal Constllation of th signal at th output of th transmitting filtr... 4 Figur 3.0 Ey-diagram of th input of th rcivr... 5 Figur 3. Signal Constllation of th input of th rcivr... 6 Figur 3. Histogram of th signal points... 7 Figur 3.3 Complx Phas Shift twork... 7 Figur 3.4 Histogram of th signal points... 8 Figur 3.5 Signal Constllation of th output of th rcivr... 8 Figur 3.6 Block Diagram of th Convntional Mont Carlo mthod and th Block Analysis mthod Figur 3.7 Simulation Run-tim improvmnt for Block Analysis mthod... 3 Figur 3.8 Illustration of Quasianalytical mthod Figur 3.9 Th rspons of a scond ordr Chvychv II filtr Figur 3.0 All th possibl output stats for th rcivd signal vi

7 Figur 3. ISI stimation of th transmitting filtr Figur 3. BER curv simulatd using Quasianalytical tchniqus... 4 Figur 3.3 BER curv with widband transmitting filtr... 4 Figur 4. Symbol Error Probability curv of QPSK systm Figur 4. Symbol Error Probability curv of QPSK systm with widband transmitting filtr Figur 4.3 Ey-diagram of th output of th modulator Figur 4.4 Signal Constllation diagram of th output of th modulator Figur 4.5 Ey-diagram of th output of th transmitting filtr... 5 Figur 4.6 Signal Constllation of th output of th transmitting filtr... 5 Figur 4.7 Ey-diagram of th input of th rcivr... 5 Figur 4.8 Signal Constllation of th input of th rcivr... 5 Figur A. Basic Simulation Architctur Figur A. Prprocssor for th BPSK Systm Figur A.3 Postprocssor for th BPSK Systm Figur C. n-th Ordr Transposd Dirct Form II twork vii

8 Chaptr Introduction Th complxity of communication systms and signal procssing systms has grown considrably in rcnt yars du to th advancs in intgratd circuit tchnology and in th dmands placd on th systms thmslvs. This growth in complxity maks th traditional analytical mthods of analysis mor unsuitabl for analyzing and dsigning modrn communication systms. Thrfor nw and fficint dsign and analysis tools ar ndd. This dmand can b mt by using powrful computr-aidd simulation-basd tchniqus. Th simulation of communication systms is concrnd with imitating som aspcts of th bhavior of communication systms without building actual hardwar. Th digital computr is usd for this purpos. If ach lmnt of a physical communication systm is rprsntd by a mathmatical modl, th digital computr can srv as th laboratory for that systm by conncting all ths modls togthr to simulat th actual systm. This is rfrrd to as a softwar modl. Th paramtrs of th modl can b changd at will and th consquncs can b obsrvd without having to build hardwar until th prformanc of th systm is satisfactory.. History of Simulation-basd Analysis and Dsign Th simulation of control systms using analog computrs was usd as a digital tool in th 950s. Th analog computr was a simulator of continuous systms that was composd of modular componnts that wr intrconnctd via a patchboard to yild a block diagram configuration rprsnting th systm. Th linar lmnts, such as intgrators, summing junctions, and amplifirs wr ralizd using oprational amplifirs. onlinar moduls such as limitrs and arbitrary functions rprsntd in picwis linar form can b ralizd using dvics and othr nonlinar circuit lmnts. Any systm whos bhavior is dscribd by a linar or nonlinar diffrntial

9 quation with constant or with tim varying cofficints can b rducd to a block diagram whil in turn can b ralizd using componnts on an analog computr. By wiring th componnts according to th block diagram and xciting th modl with appropriat signals on can simulat th dynamic bhavior of a broad rang of linar and nonlinar systms using th analog computr. In 960s, analog computr tchniqus startd to b applid to digital simulation. Th framwork for digital simulation originatd with block-orintd languags such as CSMP, MIDAS, and SCADS. Ths simulation languags mulatd th bhavior of analog computrs on a componnt-bycomponnt basis. Ths block-orintd simulation languags draw thir motivation from th analog block diagram as a simpl and convnint way of dscribing continuous systms. In mid 960s, circuit analysis simulator such as ECAP and SPICE wr also dvlopd, which ld to advancs in numrical intgration tchniqus and topological simplification of signal flowgraphs []. SPICE is still usd xtnsivly today. Advancs in discrt-tim systms and digital signal procssing hav ld to nw approachs for digital simulation of systms. In th lat 960s and arly 970s, SYSTID, CSMP, CHAMP, LIK, and othrs wr dvlopd for aiding th analysis and dsign of satllit communication links []. In 980s, intractiv and mnu drivn simulators such as ICSSM, TOPSIM, and ICS wr dvlopd []. With ths packags, simulations wr prformd on a mainfram or a suprminicomputr, and graphics trminals ar usd to provid a limitd amount of intractiv prprocssing as wll as postprocssing. Sinc mid 980s, th simulation of communication systms has rcivd mor and mor attntions as a sparat subjct within th lctrical nginring community. In th past tn yars, computr hardwar and softwar tchnologis hav undrgon significant changs. Powrful workstations and prsonal computrs hav rplacd mainframs as major computing tools. Th currnt gnration of simulation softwar packags such as BOSS, SPW, COSSAP, and othrs offr intractiv, graphical, and usr-frindly framworks for simulation-basd analysis and dsign of communication systms [].

10 . Ovrviw of this thsis This thsis dals with major aspcts of th modling and simulation of communication systms by studying two major simulation mthods: th Mont Carlo mthod and th Quasianalytical mthod. It givs a dtaild comparison btwn thos two simulation mthods. At th sam tim this thsis also xplors anothr approach that diffrs from th convntional Mont Carlo mthod: Block Procss. Chaptr dals with many basic aspcts of th simulation of communication systms, such as discrt-tim rprsntation of diffrnt signals as wll as th lowpass rprsntation of a bandpass signal. Chaptr 3 introducs th two major simulation mthods, and an nd-to-nd Binary Phas Shift Kying (BPSK) communication systm is usd to dmonstrat th ffctivnss of th abov mthods. Chaptr 4 covrs mor sampl systms to furthr dmonstrat th strngth and th waknss of two mthods as wll as how simulation of communication systms can b applid into analyzing and dsigning ral-lif communication systms. Chaptr 5, th concluding chaptr, will summariz all th conclusions drawn from th prvious studis and it is also hopd that this thsis will srv as a springboard for th radr to furthr xplor th fild of simulation of communication systms. Anothr purpos of this thsis is to srv as a modl for dvloping simulations or tmplat of othr systms. In othr words, studnts larning to simulat a systm can us th work prsntd hr as a starting point. 3

11 Chaptr Rprsntation of Signals in Simulation. Introduction In a gnral sns, a systm is dfind as a combination and intrconnction of svral componnts to prform a dsird task []. A signal is dfind as th tim history of som quantity, usually a voltag or currnt [7]. This thsis is concrnd with th rspons of systms to random signals. Most ral lif signals and systms that w wish to simulat ar continuous. Howvr, continuoustim signals and systms ar rprsntd by quivalnt discrt-tim signals and systms for computr simulation. In this chaptr, w ar to dvlop mthods for obtaining such quivalnt discrt rprsntations.. Continuous and Discrt-Tim Signal A continuous signal is dfind as a ral or complx function of tim x(t), whr th indpndnt variabl, t, is continuous []. Thr ar a fw fundamntally important continuous signals, such as th unit stp function u(t), th unit impuls function (t), th sinc function sinc(t), and complx xponntial signals. Sinc ths signals ar wll dfind and vry commonly usd in th fild of communications, this thsis will not spnd any tim dfining and driving ths functions. Discrt-tim signal is only dfind at discrt tims t = nt s, whr n ar intgrs. In most cass, such discrt-tim signals rsult by sampling continuous signals at instants sparatd by th sampling intrval T s. According to th sampling thorm, any continuous signal s(t) with a bandwidth of W can b uniquly rprsntd by sampls of s(t) takn at a rat of f s W sampls/sc. Th minimum rat f n 4

12 = W sampls/sc is calld th yquist rat. As a rsult, any ral lif continuous signal can b sampld at or abov yquist rat. Th rsulting discrt-tim signal can b usd in simulation and this will not introduc any distortion into th systm..3 Rprsntations of Bandpass Signals in Simulation Th fundamntals of simulation of digital communication systm ar th sampling thory and th complx nvlop tchniqu to rprsnt signals and nois in a propr way [4]. In sction., w covrd th sampling thory brifly. In this sction, w will covr th complx nvlop tchniqu. Digital information-baring signals ar usually transmittd by som typ of carrir modulation. Whn th bandwidth of such a signal is much smallr than th carrir frquncy, this signal is dfind as narrowband bandpass signals. Sinc w ultimatly nd to sampl ths functions, w wish to do so in th most fficint way possibl. As mntiond prviously, any continuous signal can b uniquly rprsntd by th quivalnt discrt modl only if th sampling frquncy is gratr or qual to twic th highst frquncy. In othr words, any bandpass signal with bandwidth B and carrir frquncy f c has to b sampld at th minimum rat of *(f c +B/) according to th lowpass sampling thorm. Howvr, any lowpass signal with bandwidth of W only nds to b sampld at a rat of W. It is known that th carrir modulatd signals and bandpass systm can, with minor rstrictions, b analyzd and simulatd as if thy wr low-pass. Th tchniqu usd in th implmntation of this ida is calld complx nvlop mthod []. Any carrir modulatd signal x (t) can b rprsntd as which can b rprsntd as x( t) r( t) cos[f ct ( t)] (.3.) j ( t) jfct x( t) R[ r( t) ] (.3.) whr r (t) is th amplitud modulation, (t) is th phas modulation of th signal, and f c is th carrir frquncy. Th signal 5

13 v( t) j ( t) r( t) (.3.3) vidntly contains all th information-rlatd variations, and is of a low-pass natur. It is oftn calld th complx low-pass quivalnt or th complx nvlop of th signal. If th narrowband condition dfind arlir this sction is satisfid on can show that bandpass filtring of x (t) can b valuatd through an quivalnt low-pass filtring opration with v (t) as input. In ordr to find th quivalnt low-pass rprsntation of a carrir modulatd signal, on nds to bgin with th modulatd signal in th in-phas and quadratur form. Any bandpass signal x (t) can b xprssd as x( t) x ( t)cos f t x ( t)sin f ( t) (.3.4) d c q whr x d (t) and x q (t) ar th in-phas and quadratur componnts of th bandpass signal x(t) rspctivly. It is known that if th bandwidth B of both th in-phas and quadratur componnts ar such that B<< f, thn th Hilbrt transform of x (t) is c Th analytic signal of x (t) is thrfor c xˆ ( t) x ( t)sin f t x ( t)cos f t (.3.5) d c q c x ( t) x( t) jxˆ( t) [ x ( t) d jx ( t)] q j fct (.3.6) Th carrir modulatd signal x (t) is thn R[ ~ f t ( ) R[ ( )] ( ) j c x t x t x t ] (.3.7) Th quivalnt low-pass rprsntation or th complx nvlop is thn dfind as ~ x( t) x ( t) j fct x ( t) d jx ( t) q (.3.8).3. Rprsntation of th M-ary PSK Signals An M-ary PSK signal can b rprsntd as [3] s( t) Acos[f ct ( m ) i ] (.3.9) M 6

14 whr m M, 0 t T and i is th transmittr phas angl. Equation (.3.9) can also b rprsntd as j[ ( m) i ] M jfct s( t) AR (.3.0) From quation (.3.7), th complx nvlop is givn as [3], ~ j[ ( m) i ] M s ( t) A (.3.) Thus and x x t) Acos[ ( m ) ] (.3.) M d ( i t) Asin[ ( m ) ] (.3.3) M q ( i W will us this signal form in Chaptr 4 whn th prformanc of M-ary PSK is studid..3. Rprsntation of th M-ary FSK Signals An M-ary FSK signal can b rprsntd as [3] x t) Acos[ ( f I f ) t ] (.3.4) ( c m m or x( t) j I m ft j m j f c A R t (.3.5) whr m M ; f, which is th minimum frquncy sparation in ordr to guarant th T orthogonality btwn th signals; and signal st. I m m M. m is th phas of ach signal in th Using quation (.3.7), th complx nvlop can b drivd from quation (.3.5) as ~ j ( I m ft m ) x( t) A (.3.6) 7

15 Th complx nvlop of x (t) is thn xprssd as [3] and x x d ( m m t) Acos(I ft ) (.3.7) q ( m m t) Asin(I ft ) (.3.8) whr x d (t) and x q (t) ar th in-phas and quadratur componnts of th signal x (t) rspctivly..4 Filtring Filtring is usd in communication systms to slct a dsird signal and rjct intrfrnc and nois. An idal filtr has a transfr function whos magnitud is flat within its passband and zro outsid of this band of frquncy; its midband gain is unity and its phas is a linar function of frquncy. In simulation, th filtring of bandpass signals is usually simplifid to dal with th complx nvlops of signals and lowpass quivalnts of bandpass filtrs. In MATLAB, classical filtrs can b asily dfind with only a fw paramtrs such as th passband, stop-band, and th tolrancs allowd within ths bands. Ths filtrs ar dscribd by polynomial with diffrnt cofficints. Thn, th opration of filtring is ralizd using transposd dirct form II ntwork structur (dtails s Appndix C). Bcaus th dirct and quadratur componnts of a signal ar orthogonal to ach othr, thy nd to b procssd by th filtr opration sparatly too. 8

16 Chaptr 3 Simulation Tchniqus Th probability of symbol rror is usually an important masur of th prformanc of a digital communication systm. In th Mont Carlo simulation mthod th probability of symbol rror is dtrmind by dsigning a random xprimnt in which th communication systm transmits symbols from an alphabt of siz M total possibl symbols, whr M usually quals to k. Th masur of intrst is th avrag numbr of rrors in an arbitrarily long symbol squnc. That is, lt b th numbr of transmittd symbols, and n() b th numbr of obsrvd rrors. By th dfinition of rlativ frquncy, which is an intrprtation of probability, th probability n( ) p lim is th rror probability associatd with this systm. If M =, th symbol rror probability is usually rfrrd to as th bit rror probability. Gnrally, w us Bit-Error-Rat (BER) instad of bit rror probability in simulation. Sinc ach symbol has k bits for M = k, ach symbol rror probability has an quivalnt bit rror probability. Thrfor, w oftn us BER as a primary masurmnt of th basic rliability of a givn systm. 3. Two Major Mthods of BER Masurmnt Sinc prformanc valuation is ssntial in th analysis and dsign of communication systms, th BER stimation is on of th primary goals for simulation of digital communication systms. Th BER, which is dfind as th fractional numbr of rrors in a transmittd squnc, is crtainly informativ of th basic rliability of a systm. Thr ar many simulation-basd approachs for stimating th BER, and th Mont Carlo simulation approach discussd abov is only on approach. Othr approachs ar importanc 9

17 sampling, tail xtrapolation and quasianalytical simulation. Hr, w only look at two of thos mthods in dtail. Thy ar th Mont Carlo tchniqu and th Quasianalytical tchniqu. 3. Mont Carlo Mthod A Mont Carlo simulation run can b viwd as a statistical xprimnt that is th softwar countrpart of an xprimnt prformd on a ral communication systm. Thrfor, a Mont Carlo simulation for a digital communication systm can b dvlopd from th mathmatical modl of th systm. Th systm can b xprssd as a block diagram in which ach functional block is a signal procssing opration ralizd as a DSP algorithm. W thn pass information bits through th systm. That is, w count th numbr of succsss (rrors, in this contxt) and divid by th total numbr of trials. Th rsult is th rlativ frquncy stimat of th rror probability. This mthod rquirs no assumptions about th input procsss or th systm. In fact, in th implmntation, th systm is bypassd xcpt for th output of th digital sourc and th dcision dvic s vrsion of th sourc. Sinc th sourc output is known, comparing th two squncs at som rlativ dlay provids th mpirical basis for an rror rat. Th block diagram of a simpl communication systm is illustratd in Fig. 3.. Digital Sourc Squnc Transmittr Channl Rcivr Estimatd Squnc Dlay Error Dtctor Error Squnc Fig. 3. Mont Carlo Estimation Systm Block Diagram 0

18 In ordr to implmnt th simulation of th systm shown in Fig 3. it is ncssary to know th dlay of th systm. This knowldg is, in ffct, quivalnt to symbol synchronization. For a carrir-modulatd systm, propr dmodulation also rquirs carrir synchronization, namly, th lining up of th local oscillator phas with that of th incoming carrir. Carrir and symbol synchronization ar ssntial functions in any digital systm, and som form of synchronization is always implid. Howvr, for purposs of discussing th Mont Carlo mthod or any othr typ of stimation tchniqus, th spcific implmntation of synchronization within th simulation is not cntral to th issu. 3.. Quality of an stimator Sinc a Mont Carlo simulation can b viwd as a statistical xprimnt, th valu of any paramtr obtaind by such simulations is only an stimation of th tru valu. Thrfor it is ncssary for us to masur th quality of a Mont Carlo simulation. That is, th closnss of th stimat must b masurd in a probabilistic sns. In this papr, thr masurs of quality of a Mont Carlo simulator ar of intrst: th bias, th varianc, and th confidnc intrval Bias of a Mont Carlo Simulator For almost all cass of intrst, th simulator can b xprssd as a wightd tim-avrag: Qˆ w Y i i (3..) i whr Qˆ is th stimatd valu of a paramtr with a tru valu of Q. Sinc w normally xpct th stimation procss to improv as incrass, an ssntial attribut of an stimator is that it convrgs to th tru valu as. Such an stimator is trmd unbiasd. Thus for an unbiasd stimat Q ˆ Q as for almost vry sampl function. Sinc Qˆ is a random variabl it has an associatd distribution and pdf, say ( q; ). As w hav discussd, for an unbiasd stimat f Q ˆ

19 E( Qˆ) qf ( q; ) dq Q Qˆ (3..) It is important that a BER stimator is unbiasd, sinc for an unbiasd stimator th simulation rsult will, on th avrag, b corrct. As statd prviously, for a Mont Carlo stimator in which is th numbr of transmittd symbols, and is th numbr of obsrvd rrors, th probability pˆ is th stimatd rror probability associatd with th systm. As a rsult, w hav E( pˆ ) E( ) E( ). Sinc p E( ) p, it is clar that E( pˆ ) p. Hnc, th Mont Carlo stimator is unbiasd []. ot that th abov discussion dos not rquir that th rror vnts ar indpndnt. W thrfor hav that th Mont Carlo stimator pˆ is, in gnral, an unbiasd stimator. In th nxt sction, w will driv an xprssion for th varianc of th stimator. Th rsult obtaind will rquir indpndnc of th rror vnts. Bcaus of bandlimiting and othr important systm attributs, w will s that this constraint is not oftn mt in practic Varianc of a Mont Carlo Estimator Th varianc of an stimator, ˆ ( Qˆ) E( Q ) E ( Qˆ) q f ( q; ) dq E ( Qˆ ) Qˆ (3..3) is a masur of th disprsion about th xpctd valu. Th smallr th varianc, th bttr th stimator is, for a givn. Gnrally, ( Qˆ) 0 only as, and w ar facd with a tradoff btwn stimator varianc and sampl siz, th lattr bing dirctly proportional to runtim in th simulation contxt.

20 It is convnint to us th momnt gnrating function to dtrmin th varianc of a Mont Carlo simulator. Th momnt gnrating function is dfind such that k ( z) k p z (3..4) k Substituting p k! p ) k!( k)! k k { k} p ( p, yilds k k k k k ( z) p ( p ) z ( p z) ( p ) (3..5) k 0 k k 0 k Using th binomial thorm which stats that m m k mk ( a b) a b (3..6) k 0 m k allows, th momnt gnrating function (z), to b xprssd as ( z ) [ p z ( p )] (3..7) From (3..4) it follows that and z kp z k ) k k ( (3..8) z k k k p z ) ( ) k k ( (3..9) Ltting z=, w gt ( ) kp k E{ k} (3..0) 3

21 and k ( ) k p k kp k (.3.) k Thus E { k } () () (3..) W now us ths rsults to obtain E {k} and E { k } [8], k E{ k p } ( E{ k}) p p ( p ) (3..3) Confidnc Intrval Th confidnc intrval is th most important masur of th quality of an stimator bcaus it quantifis th masur of sprad with an associatd probability. Lt h ( ˆ), h ( ˆ ) b two functions of th stimator, such that, th intrval h, h brackts th tru valu Q. Th diffrnc h h is th width of th confidnc intrval. Th probability associatd with th condition h Q h is calld th confidnc lvl and is usually dnotd lvl ar dfind through [] Q Q. Thus, a confidnc intrval and confidnc P h ( Qˆ) Q h ( ˆ) (3..4) [ Q Typical valus for ar 0.05 and 0.0, corrsponding to 95% confidnc lvl and 99% confidnc lvl, rspctivly. Howvr quation (3..4) only dfins confidnc intrval with crtain confidnc lvl in thory. Sinc th tru valu Q is unknown in almost all cass, quation (3..4) has littl practical maning. In ordr to valuat th confidnc intrval, som kind of approximation is ndd. Th 4

22 most common mthods to valuat th confidnc intrval ar Poisson approximation and Gaussian approximation. Lt b th total numbr of symbols procssd in a Mont Carlo simulation and b th total numbr of rrors obsrvd. Th stimatd rror probability pˆ is thrfor. For any giv, has a binomial distribution. Hnc on can obtain a closd form for th confidnc intrval in trms of th cumulativ bta distribution [5]. Howvr, it is difficult to itrativly valuat th binomial distribution whn is larg. Thrfor, w us two altrnativ approachs: Poisson approximation and Gaussian approximation. It is wll known that if thr is a positiv constant such that lim pˆ, th binomial distribution can b approximatd by th Poisson distribution. Undr th limiting condition th cumulativ binomial distribution can b rplacd by th cumulativ Poisson distribution k 0 k k k k! k k k! (3..5) (3..6) As th abov two quations to b solvd for and, w can construct th confidnc intrval, with confidnc lvl. A tabl of uppr and lowr valus of for diffrnt confidnc lvl for 0 50 using th Poisson approximation can b found on pag 499 []. Th Poisson approximation givs us a usfully larg rang of valus. For largr such as 0, thr xists anothr approximation, th Gaussian approximation, which is simplr to us. It is know that as, th stimatd rror probability pˆ tnds to hav a normal distribution 5

23 with man [5] p and varianc p ( p ). Hnc w can construct a confidnc intrval in th form Pr ob{ d p [ pˆ d d [ pˆ d pˆ ˆ ( p ) d ( ) d d / pˆ ˆ ( p ) d ( ) ] / ]} (3..7) whr p is th tru valu of th BER, and d is chosn so that d / d ( ) t / dt (3..8) Lt v pˆ 0 and 0 v, and undr th assumptions that d and pˆ ( pˆ ) pˆ, which ar usually mor than amply satisfid, quation (3..3) can b xprssd as whr th confidnc intrval y, y is givn by Pr ob [ y p y ] (3..9) y 0 v d { 4 [ ( d ) / ]} (3..0) This intrval is plottd in Figur 3. for confidnc lvl of 90%, 95%, and 99%. 6

24 Fig. 3. Confidnc Intrval on BER whn stimatd valu is 0-6 basd on normal approximation [6] 3.. Exampl: a BPSK Systm Simulation In th prvious sctions, w introducd th Mont Carlo simulation mthod to valuat th prformanc of communication systms. In this sction, an nd-to-nd BPSK communication systm is usd as an xampl to furthr dmonstrat th Mont Carlo simulation mthod and its proprtis. Th objctiv of this xampl is to valuat th prformanc of a simpl BPSK communication systm with th prsnc of Additiv Whit Gaussian ois (AWG), taking into account th intrsymbol intrfrnc (ISI) causd by th transmitting filtr. A BPSK communication systm with AWG nvironmnt is a vry asy systm and consquntly, it is rlativ asy to valuat with an analytical solution. Howvr, th non-linar ffct of th transmittr filtr is not as asy to follow 7

25 with th analytical solution. A Mont Carlo approach can simulat both AWG nvironmnt and th ISI causd by th transmitting filtr vry wll and thus provid us th accurat prformanc valuation of such a systm. Morovr, a Mont Carlo simulation also can provid us wavform lvl simulation products, which will hlp us to gain mor insight of th systm Signal Rprsntation As mntiond in chaptr, it is mor fficint to rprsnt a bandpass signal with its quivalnt complx nvlop rprsntation in a simulation. A bandpass BPSK signal can b rprsntd mathmatically as following s t) Asin[f t ( m ) ] (3..) ( c i whr m, and i is th transmittr phas angl. Using th rsults drivd in quation (.3.) and (.3.3), th dirct and quadratur componnts can b xprssd as and s s D ( i t) Acos[( m ) ] (3..) Q ( i Thrfor, th lowpass rprsntation of a BPSK signal is t) Asin[( m ) ] (3..3) s ( t) s l D ( t) js Q ( t) Acos[( m ) ] i jasin[( m ) i ] (3..4) in this xampl, A is st qual to for simplicity Th Systm Modl Th block diagram of a BPSK communication systm that w ar studying in this xampl is shown in figur 3.3 Th data signal is assumd to b uniformly distributd so that th probability for a binary 0 or a binary is th sam. Th rand function in Matlab is usd to gnrat such data signals. In this 8

26 simulation, - is binary 0 and is binary in ordr to gt rid of th dc componnt of th signal. In Mont Carlo simulation, th whol simulation is conductd in th bit by bit fashion. That is, ach data bit is gnratd and passd through th systm to chck if an rror has bn mad bfor th scond data bit is gnratd. Aftr th data bit is gnratd, it is modulatd using th complx nvlop rprsntation that was drivd in th prvious sction. In this simulation, th BPSK signal is assumd to hav a random Data Signal BPSK Modulator Transmit Filtr Channl Dtctor (Match Filtr) Dlay Error Dtctor Error Fig. 3.3 BPSK systm block diagram phas angl uniformly distributd btwn 0 and. It is our intntion to show that th phas angl of th signal dos not affct th prformanc of th systm. Aftr th data signal is gnratd, it is sampld and thn passd through a transmit filtr. In this simulation, a sampling rat of 0 sampls pr bit is usd for convninc. A transmittr filtr is usd hr to limit th spctrum of th transmittd signal. In Matlab, thr is a built-in filtr command for filtring opration. Howvr, this filtring opration is a paralll opration. That is, all data nds to b usd as input of th filtr command to gt th right output of th filtr. In Mont Carlo simulation, th simulator is oprating in a sampl-by-sampl mod. Thrfor, w hav to ithr writ a squntial filtring function ourslvs or find a way to sav th final stat information of th filtr for ach sampl and us it as th initial stat for th nxt sampl. Fortunatly, both tasks ar not vry hard to achiv. Sinc saving ach stat of th filtr for vry sampl is not vry fficint, w will 9

27 us a simpl squntial filtr routin to accomplish such tasks (Dtails of th squntial filtr routin is givn in Appndix C). Th transmitting filtr usd in this simulation is a Buttrworth filtr with th bandwidth and th ordr dfind by th usr. Sinc a Buttrworth filtr is not an ISI-fr filtr, th intr-symbol intrfrnc is introducd into th systm by th transmit filtr. It is also our intntion to study th ffct of th intr-symbol intrfrnc on th systm s prformanc. Th output of th transmittr filtr is thn passd through an AWG channl bcaus it is th most commonly usd modl for nois in communication systms. Sinc E b / o valus ar input paramtrs of th simulator and th nrgy pr bit is assumd to b in this xampl, w can calculat th nois powr from thos two paramtrs. Th varianc of th nois can b asily calculatd using th following quation b (3..5) E E b k / 0 whr k is th sampling rat. Aftr th signal has bn passd through th channl, it nds to b rcivd and dtctd for rror. It is our intntion to study th ffcts of th intr-symbol intrfrnc causd by th transmitting filtr. A filtr that is only matchd with th nois portion sms to b a natural choic bcaus it minimizs th ffct of th nois. It can also b provn that in basband, such a filtr function is nothing mor than an intgrat-and-dump opration. Thrfor, in this simulation th signal dtction opration can b furthr simplifid. At th rciving nd, w only nd to accumulat 0 sampls, sum thm up and mak a hard dcision by comparing th sum of th sampls with th thrshold valu. Th actual implmntation of such task is vry asy. Howvr, thr ar som issus that nd to b addrssd hr. First, w nd to mak sur that th 0 sampls that ar intgratd ar th sampls from th sam bit. This is, in a sns, a symbol synchronization problm in a simulation. Sinc ach bit is sampld at th sam rat, this task can b asily accomplishd by lining up th first bit. In ordr to lin up th first bit, w must masur th systm dlay. Th asist way to masur th dlay is to tst diffrnt valu of dlay. Th valu that yilds th lowst BER is 0

28 th valu of th systm dlay. On can also obsrv th dlay from th y-diagram. Th filtrd signal will not cross ovr at th sam plac as th signal bfor bing filtrd. Th diffrnc btwn th two diagram is also th systm dlay. Scondly, sinc w intnd to show that th transmittr phas angl has no ffct on systm prformanc, th signal will hav an arbitrary phas angl. Th rcivd signal constllation diagram is shown blow Q S I S l Fig. 3.4 Signal constllation diagram for BPSK signal with transmitting angl It is obvious that th thrshold for dtcting signal s and s is l. It is also intuitiv that if w rotat s and s dgrs clockwis, th quadratur axis bcoms th thrshold and th rror probability will still rmain th sam. In this simulation, th scond dtction schm is usd. Aftr th dtction and dcision hav bn mad, th rcivd bit can b compard with th original transmittd bit to rach an rror statistic. Th simulator kps count th rsulting rror statistic until nough rrors hav bn accumulatd to stimat th BER with crtain confidnc lvl Simulation Rsults Figur 3.5 shows th BER stimation for this BPSK systm. It is shown in th graph that th simulatd rsults ar poorr than th thortical valu with th sol prsnc of th AWG. This rsult clarly dmonstrats th additional dgradation on th systm prformanc causd by th ISI

29 introducd by th transmit filtr. If th bandwidth of th transmit filtr is incrasd so that th filtr causs littl or no ISI, th simulatd BER curv thn should agr with th thortical valu. Sinc varying th bandwidth is a paramtric study of th systm, this graph will b gnratd using Quasianalytical tchniqu and will b shown in th lattr sction of this papr. BER vs. Eb/o with 00 rrors, Wn=/T 0 - Simulation Rsults Thortical Unfiltrd valu 0 - BE R Eb/o in db Fig. 3.5 Simulatd BER prformanc curv with ISI As analyzing and dsigning tool, Mont Carlo mthod can also gnrat th wavforms lvl simulation products such as y-diagram and signal constllation diagram at diffrnt points of th systm to hlp usr to gain mor insight of th bhavior of th systm. In this xampl, th wavforms ar monitord at four diffrnt placs, th signal bfor and aftr th transmitting filtr and th signal bfor and aftr th rcivr.

30 D/Q EYE DIAGRAM 0.5 Dirct Sampl Indx 0.5 Quadratur Sampl Indx Fig. 3.6 Ey-diagram of th input of th transmitting signal 0.8 Quadratur Channl Dirct Channl Fig. 3.7 Signal constllation of th input of th transmitting signal Both figur 3.6 and figur 3.7 show th signal aftr th modulation and bfor filtring. Th straight transition btwn th stats in y-diagram shows clarly that no distortion is prsnt at this point. 3

31 D/Q EYE DIAGRAM 0.5 Dirct Sampl Indx 0.5 Quadratur Sampl Indx Fig. 3.8 Ey-diagram of th output of th transmitting filtr Quadratur Channl Dirct Channl Fig. 3.9 Signal constllation of th output of th transmitting filtr 4

32 Th y-diagram and th signal constllation diagram of th output of th transmitting filtr ar shown in Fig. 3.8 and Fig Th ffct of th intr-symbol intrfrnc (ISI) of th transmitting filtr is clarly shown in ths two diagrams. Th y-diagram dos not hav th straight transition btwn stats any mor. At th sam tim, it is worth noticing that thr xists a transint stat from th rlaxd stat to th first stat at th first sampl point, which is not prsnt in th y-diagram of th unfiltrd signals. Th signal constllation diagram also shows th signal points ar scattrd btwn th two original signal points. D/Q EYE DIAGRAM Dirct 0-0 Quadratur Sampl Indx Sampl Indx Fig. 3.0 Ey-diagram of th input of th rcivr (E b / o =5dB) 5

33 Fig 3. Signal Constllation of th input of th rcivr Figur 3.0 and 3. show th signal constllation and th y-diagram of th signal aftr th nois has bn addd. In th y-diagram, th signal stats ar clos to ach othr du to th prsnc of th nois (closd y). In th signal constllation diagram, th signal points ar scattring around th original signal points. Th histogram diagram shown in Fig. 3. illustrats how th signal points ar distributd on th dirct and quadratur axis. Howvr, Fig. 3. is difficult to intrprt du to th prsnc of th arbitrary transmittr phas angl. In ordr to hav a convntional histogram that is asir to undrstand, w ar going to rotat all signal points th signal point is wll known to us and is shown in Fig St qual to dgrs. Th structur usd to rotat, and th rsulting Y d (t) and Y q (t) ar th dsird nw dirct and quadratur componnts of th signal. Th rsulting histogram is shown in Fig Aftr th rotation, th quadratur axis bcoms th thrshold for dtcting th two signal points s and s. Thrfor, all th rror statistics will only b shown on th dirct componnts of th signal. This is clarly illustratd in Fig Th dirct componnt has two sts of Gaussian distributd signal points with man of and. 6

34 50 umbr of signal points Dirct 50 umbr of signal points Quadratur Fig. 3. Histogram of th signal points in th signal spac X d (t) cos + _ Y d (t) sin sin X q (t) cos + + Y q (t) Fig. 3.3 Complx phas shift ntwork 7

35 50 umbr of signal points Dirct 50 umbr of signal points Quadratur Fig. 3.4 Histogram of th signal points in th signal spac Aftr th signal has bn rcivd, a hard dcision is mad on all th dtctd signals. Th dcision schm usd in this simulation was illustratd in Fig Th signal constllation of th rcivd signal is shown in Fig Quadratur Channl Dirct Channl Fig. 3.5 Th signal constllation of th rcivd signal 8

36 Th rcivd signal has th sam constllation diagram as th original signal, which mans that th dtction and hard dcision schms work vry wll. From this xampl, it is vry clar that all th wavforms gnratd by th simulator can not only hlp us undrstanding th bhavior of th systm undr diffrnt conditions, but also hlp us chcking th corrctnss of th simulator itslf. Thrfor, ths simulation products ar vry usful and important. This is a significant advantag of th Mont Carlo mthod Block Analysis Mthod As was shown in th prvious sction, a Mont Carlo simulation is run on a sampl-by-sampl basis. This structur is shown in Fig. 3.6(a). Each sampl to b producd through th simulation is sampld, filtrd, and dtctd squntially. Th procss is straightforward and asy to program. Squntial tim domain procssing is fficint for systm blocks that do not contain mmory. Whr mmory is prsnt in a systm block th sampl valu at th output of a block is a function of past input sampls as wll as th prsnt input sampl. Systm blocks that ar linar and not mmorylss ar typically xprssd using a rcursiv rlationship. Computr procssing of ths rcursion rlationships is tim consuming and lad to infficint simulations. Unfortunatly, practical communication systms us on or mor filtrs in thir implmntation and filtrs xhibit mmory. Anothr approach is ndd to incras simulation fficincy. Computing th output of a linar filtr, givn th input and unit impuls rspons of th filtr, is a convolution. W know that tim domain convolution is an infficint procss on th computr and that implmnting th procss in th frquncy domain is much mor fficint. Th implmntation involvs Fourir transforming th input signal, multiplying th rsulting transform by th Fourir transform of th unit impuls rspons, and invrs transforming th rsult. Using th Fast Fourir Transform (FFT) to implmnt th transform oprations rsults in a vry fficint implmntation, spcially if radix- FFTs can b usd. Sinc a block of sampls must b Fourir transformd in ordr to implmnt th filtr in th frquncy domain, sampl-by-sampl procssing is not 9

37 appropriat. Th transform, or frquncy domain, approach to simulation of systm blocks having mmory is rfrrd to as Block Procssing. Th basic structur of this approach is shown in Fig. 3.6(b). Th MATLAB routin filtr is wll suitd to implmnt th approach discussd abov. Although th Block Analysis mthod follows th basic approach of th Mont Carlo mthod, th improvmnt on th simulation fficincy achivd by this mthod givs us som capabilitis, such as paramtric study, othrwis can only b achivd by th Quasianalytical tchniqu. Howvr, at th sam tim, th Block Analysis mthod still kps th sam wavform lvl simulation capabilitis of th convntional Mont Carlo mthod. Input data bit Systm Output data bit (a) Th convntional Mont Carlo mthod Input data bit b u f f r Systm b u f f r Output data bit (b) Th Block Analysis mthod Fig. 3.6 Block diagram of th convntional Mont Carlo mthod and th Block analysis mthod 30

38 3..3. Th Prformanc Improvmnt of Block Analysis Mthod In this sction, w ar going to study th prformanc improvmnt achivd by th Block Analysis mthod compard with th convntional Mont Carlo mthod. To do this, w ar still going to simulat th sampl BPSK systm usd in th prvious sctions. W ar going to incras th block siz from to, whr is an arbitrarily big numbr. Th total simulation run-tim is going to b monitord and plottd. Obviously, th block siz of is th sam as th convntional Mont Carlo mthod. Thus, w can s how th simulation run-tim varis with th diffrnt block siz. Bfor th simulation can b conductd, som prliminary study nds to b don to nsur th corrctnss of this simulation. First of all, bcaus th systm is not mmory-lss, it is ncssary for us to sav th systm s stat at th nd of ach block and us it as th initial stat for th nxt block so that th Intr-Symbol Intrfrnc is not nglctd. Fortunatly, th filtr command in Matlab has this function. It accpts th initial condition as a paramtr, and outputs th final condition as anothr paramtr. Thrfor, this opration can b asily achivd. Scondly, du to th Fourir transform opration prformd by th filtr command, th block siz of powr of is prfrrd to rach th optimal prformanc rsults. Sinc th basic approach is th sam for th Block Analysis mthod and th Mont Carlo mthod, w can modify th original Mont Carlo simulation cod to construct th Block Analysis simulator. Figur 3.7 shows th prformanc improvmnt achivd by th Block Analysis mthod. Whn block siz quals to, it is th sam as th convntional Mont Carlo simulator. As th block siz incrasd, th simulation run-tim dcrasd vry quickly. It is clarly shown in th graph that th computr run-tim is rducd by a factor of 0 as th block siz incrass. 3

39 3.5 Simulation Run tim Block siz Fig. 3.7 Simulation Run-tim as a function of block siz Sinc th only plac that th Block Analysis mthod is diffrnt from th Mont Carlo mthod is th way it procsss th data, th rliability paramtrs such as bias, varianc and th confidnc intrval ar ssntially th sam as th Mont Carlo mthod. Hr, w will not covr thm in dtail again. Th bottom lin is that th Block Analysis mthod givs us an unbiasd stimation of th BER Discussion In th simulation contxt, Mont Carlo simulator rprsnts th tru situation for systms. It is vry simpl to implmnt and is indpndnt of th spcifics of th systm. Thrfor, it is on of th most fundamntal simulation mthods and should b includd in any simulation packag. Th capability to gnrat wavform lvl simulation rsults in various position of th systm also maks th Mont Carlo simulation ssntial as an analysis and dsign tool. Howvr, sinc th Mont Carlo mthod is just an stimation of th actual BER prformanc of th systm, it can not b usd blindly without any mntion of confidnc lvl. For without confidnc lvl, any BER simulation don by Mont Carlo mthod has no practical maning at all. 3

40 Although th Mont Carlo mthod is simpl to implmnt and gnrally is th closst xprimnt nxt to actually building th actual systm, th computr run-tim usually is vry long and vn impractical. Thrfor, for low BER systms, compromiss ar oftn ncssary to gt a rlativly good stimation in rasonabl tim. Th Block Analysis mthod uss th matrix capabilitis of Matlab to ffctivly rduc th simulation run-tim with big block siz. This prformanc improvmnt is vry significant, spcially for low BER systm simulations. At th sam tim, th Block Analysis mthod also maks it possibl to bring paramtric study into wavform lvl simulation, which is also vry hlpful to gain mor insight of th bhavior of diffrnt communication systms. 3.3 Quasianalytical Estimation Quasianalytical stimation tchniqu, oftn known as smianalytical mthod, is th scond simulation mthod studid in this papr. Basically, this tchniqu combins both simulation and analysis. Th simulation is usd to gnrat a noislss wavform at th rcivr. Givn this wavform and assuming that th nois is additiv and has a known pdf, on can thn calculat th probability of rror with a formula []. This is th analysis portion. In th simulation part, w lt th computr simulat th ffct of th systm distortion in th absnc of nois, and thn suprimpos th nois on th noislss wavform in th analytical part. Th simulation part can b furthr clarifid by Fig Fig. 3.8(a) shows a hypothtical transmittd bit stram, whil Fig. 3.8(b) shows th corrsponding noislss rcivd wavform at th input of th dcision dvic. Th valu of this wavform at th kth sampling instant is dnotd v k. Whn th nois is suprimposd, th rsulting rror probability is simply 33

41 P k Pr ob [ nois v Pr ob [ nois v k k ] ] vk vk f dn, n f dn, n v v k k 0 0 (3.3.) whr f n is th pdf of th nois, which is assumd to b zro man, and it is also implid that th thrshold for dtction is zro. Graphically, th distortion of v k n is simply th distortion of n shiftd by v k. Th probability of rror is just th shadd ara in Fig. 3.8(b) undr th tail. Th total probability of rror can thn b xprssd as P k P k (3.3.) whr is th total numbr of th possibl stats of v k. P k (a) v k (b) Fig. 3.8 Illustration of quasianalytical mthod. (a) Transmittd bit stram. (b) Rcivd wavform [6] 34

42 3.3. Rstrictions and considrations for Quasianalytical tchniqu Th biggst advantag gaind by using Quasianalytical mthod is tim saving. This is achivd bcaus w do not hav to wait for rror to occur as in th cas of Mont Carlo simulation. Instad, w simply calculat an intgral, which in principl taks th sam amount of tim to valuat rgardlss of th BER. Howvr, th Quasianalytical tchniqu cannot b applid to any arbitrary systm. Thr ar thr rstrictions associatd with th Quasianalytical tchniqu:. Th systm has to b linar from th point that th nois is introducd.. Th mmory of th systm nds to b prdtrmind and should b rlativly short. That is, if th amplitud of th rcivd noislss signal at any instanc is a function of only rlativly fw prcding symbols, thn a comparativly short squnc can produc all of th possibl stats of th rcivd signal v k. 3. Th probability dnsity function of th nois has to b a known function spcifid by th usr in ordr to map all th possibl stats into an rror probability. Thus, th ovrall bit rror rat can b calculatd. Any systm that satisfis th abov thr conditions can b simulatd using th Quasianalytical tchniqu. Thr is a vry simpl way to dtrmin th mmory of th systm. First, st th systm at rlax stat, and thn pass a stp function at tim t and monitor th output wavform. Sinc th systm has mmory, th output will hav a transint stat and sttl at tim t. Th systm thn is dtrmind to hav a mmory of (t-t )/T b bits, whr T b is th priod of ach bit. A small simulation program can asily accomplish this task. First a stram of zros can b snt through th systm to mak sur th systm is compltly rlaxd and at tim t, a stram of ons is snt. By monitoring th transint stat of th rspons of th systm, th mmory of th systm can b dtrmind. This opration can b furthr clarifid by th following xampl. Assum that a simpl scond ordr Chvychv II filtr is 35

43 usd in th systm and our task is to dtrmin th mmory of this filtr. By using th mthod mntiond abov, th transint rspons of th filtr can b obtaind and is shown in Fig Output of th filtr Sampls Fig. 3.9 Th rspons of a scond ordr Chvychv II filtr In this xampl, ach information bit is sampld tn tims and at th lvnth bit, a stram of ons is snt. Th graph shows that th systm s rspons sttls to a stady stat at about th 50 th sampl, which mans that th systm s mmory is about 50 sampls or 5 bits. Aftr th mmory of th systm is dcidd, a maximum lngth P squnc with lngth M Q will b gnratd, whr Q is th siz of th signal alphabt and M is systm s mmory in bits. Sinc th maximal P squnc gnrator only gnrats squncs with th xcption of th all zro cas, th all zro cas can b addd on asily. All squncs can thn b snt through th systm to produc all th possibl stats for th rcivd signal v k. This opration can b clarly dmonstratd in Fig

44 Fig. 3.0 All th possibl stats for th rcivd signal Using th sam Chvychv II filtr usd abov, sinc th mmory of th filtr is dtrmind to b 5 bits, th total numbr of all th possibl stats of th rcivd signal v k is 5 3. Fig. 3.0 clarly shows that thr ar 3 total stats for th rcivd signal, which ar disprsd along th axis. ow all th possibl stats of th rcivd signal hav bn dtrmind. Th last stp is to map all ths stats into diffrnt rror probability by using th pdf of th nois, and calculat th final rror probability of th systm. As w know that if a whit nois is passd through a filtr with transfr function H ( f ), th avrag powr at th output is 0 H ( f ) 0 P n df (3.3.3) 0 whr 0 is th two-sidd powr spctral dnsity of th input. If th filtr wr idal with bandwith B and midband gain H 0, th nois powr at th output would b 0 0B H 0 Pn (3.3.4) 37

45 Hr B is rfrrd to as th nois-quivalnt bandwidth of H ( f ). Obviously, in ordr to calculat th systm s rror probability, w nd first to obtain th nois powr at th rcivr. As a rsult, th nois-quivalnt bandwidth, B, nds to b calculatd. Onc again, a small simulation program can asily accomplish th task of dtrmining th noisquivalnt bandwidth B. From th quation (3.3.3) and (3.3.4), w can asily driv that B H ( f ) df (3.3.5) H 0 0 In th discrt form, quation (3.3.5) can b changd to B H () f s j H ( ) d (3.3.6) whr H () is th dc rspons of th digital filtr and f s is th sampling frquncy. By using Parsval s thorm and z-transform thory, quation (3.3.6) yilds h ( n) f s n B (3.3.7)! h( n) n whr h (n) is th unit impuls rspons of th digital filtr. Thortically, quation (3.3.7) rquirs infinit amount of computing tim to valuat sinc th summation is from " to ". Thrfor, an approximation is ndd. That is, aftr th impuls rspons has dclind a crtain numbr of db from th original valu, w can approximat th rspons from thn on is zro. Thus, th noisquivalnt bandwidth can b valuatd by snding a unit impuls function through th systm to obtain th unit impuls rspons of th systm. W can thn using quation (3.3.7) to calculat th nois-quivalnt bandwidth. Aftr th nois-quivalnt bandwidth is found, w can thn calculat th nois powr and th nois standard dviation at th rcivr. Thus th systm rror probability can b valuatd. For 38

46 xampl, with th systm w usd prviously in an AWG nvironmnt, th rror probability can b asily valuatd using th Q-function. P 3 3 i v Q( ki ) (3.3.8) 3.3. Cas Study Onc again, w ar going to us th simpl BPSK systm usd in th prvious sction to dmonstrat Quasianalytical mthod. From th block diagram in figur 3.3, it is obvious that this BPSK is not a mmorylss systm du to th prsnc of th transmitting filtr. Thrfor, in ordr to gt th right noislss output wavform at th output, it is ncssary to dtrmin th mmory of th systm. Onc w dtrmind th mmory of th systm in trm of bits, w can thn pass all possibl input squnc through th systm to gnrat th propr noislss output wavforms. Sinc an ovrstimation of th mmory of th systm will not affct th rsult of th simulation whil an undrstimat will caus an inaccurat stimation, stimating th systm mmory consrvativly will guarant th corrctnss of th simulation. Howvr, if th systm mmory is stimatd too consrvativly, th maximal lngth P squnc is going to bcom vry long, thus incrasing th simulation run-tim. Thrfor, it is also a trad off that try to stimat th systm s mmory corrctly and at th sam tim maintain th simulation run-tim to a minimum. Th following graphs show th stimation of th mmory of this BPSK systm. 39

47 .4. Output of th filtr Sampls Fig. 3. ISI stimation of th transmitting filtr Sinc a stp function is introducd at th 00 th sampl, th transint stat lasts about 30 sampls in this graph. Thrfor, th ISI of th filtr is about 3 bits. To nsur th corrctnss of th stimation, two mor bits wr addd. This stimation will b usd in th lattr simulation. Aftr th mmory of th systm is dtrmind, a maximum lngth P squnc gnrator can b usd to gnrat all possibl input squncs. Sinc maximum lngth P squnc gnration ar covrd xtnsivly in th litratur, this thsis will not considr it in dtail. Th abov procdurs in a sns provid us information about th rliability of th Quasianalytical stimator. Sinc th mmory of th systm is prdtrmind bfor any actual simulation was run, w could say that th stimator has no bias and zro varianc. Howvr if th stimation of th systm mmory is not prcisly corrct, th stimator thn will b biasd and it is not possibl to quantify this bias in a gnral way. Aftr th gnration of th noislss output wavforms, th BER calculation is gnrally straightforward sinc th nois is additiv and has a known pdf. In this cas, th nois is AWG, so th final calculation is just a simpl Q-function valuation. Th BER curv thn can b gnratd using th rsult from th calculation. 40

48 BER curv with Wn=/T m=5 Simulation Rsults Thortical Unfiltrd valu 0 - BER Eb/o in db Fig. 3. BER curv simulatd using Quasianalytical Tchniqu From th abov graph, w can s that th rsult obtaind using Quasianalytical tchniqu agrs with th rsult from th Mont Carlo simulation. To study mor about th ffct of th ISI on th systm s prformanc, th bandwidth of th transmitting filtr is incrasd and a nw BER is plottd to show how th systm s prformanc changs accordingly BER curv with widband transmitting filtr Bn=/(0.*T) Simulation Rsults Thortical unfiltrd valu 0 - B E R Eb/o in db Fig. 3.3 BER curv with widband transmitting filtr 4

49 As it is shown in figur 3.3, whn th bandwidth of th transmitting filtr is incrasd so that thr is littl or no ISI, th systm prformanc agrs with th thortical unfiltrd valu vry closly Discussion of th Quasianalytical simulation Thr ar many rasons that th Quasianalytical tchniqu should b implmntd. First and most important is its tim saving proprty. In a linar channl, it can provid corrct answrs xtrmly fast. This may b vry bnficial for simulations of low BER systms. It is also vry usful for paramtric studis that othrwis would b prohibitd du to th xtrmly long computation tim. Scondly, it can srv as a sanity chck for th Mont Carlo mthod. As was mntiond in th prvious sction, th Mont Carlo mthod itslf is not prfct. For xampl, random numbr gnrators ar not idal most of th tim and ar hard to chck for long squncs. Thrfor, a sanity chck is ncssary for Mont Carlo simulation rsults, and Quasianalytical mthod srvs us wll in this aspct. Howvr, Quasianalytical mthod is vry problm spcific and gnrally cannot b st up in advanc to solv gnral problms []. On th othr hand, Quasianalytical mthod is also unabl to gnrat wavform lvl simulation product with th prsnc of nois, which somtims might prohibit us to gain mor insight of th bhavior of th systm. Quasianalytical tchniqu has lots of advantags and whn usd with combination with Mont Carlo mthod, thy can provid vry usful simulation rsults for analyzing and dsigning purposs. Thrfor, thr is no doubt that Quasianalytical tchniqu should b implmntd whnvr is possibl in simulation. 3.4 Conclusions Thr ar many choics in BER stimation tchniqus. Th ons prsntd in this papr ar just som of th simulation-basd approachs for stimating th BER. ot all of ths tchniqus apply 4

50 irrspctiv of th modulation/coding schm and associatd rcivr structur. At th sam tim, all ths tchniqus ar not isolatd from ach othr. Most tims whn combind with on anothr, th bst simulation rsults can b rachd. Thrfor, on must considr which tchniqu(s) should b implmntd. A. Mont Carlo mthod. Mont Carlo runs can b viwd as th trust situation for nonlinar systm nxt to building th actual systm. For this rason, ach simulation packag should includ Mont Carlo capability, not only for BER stimation, but also calibrating and monitoring th fundamntal bhaviors of th systm. For low BER systms, th simulation run-tim could b unrasonably long and som sort of th altrnativ approach such as th Block Analysis mthod must b usd. Th implmntation of th Mont Carl mthod is vry simpl and is indpndnt of th spcifics of th systm. Thrfor, it is possibl for us to st up th basic structur in advanc for gnral purposs and mak minor modifications to accommodat ach individual systm. B. Th Quasianalytical tchniqu. Th biggst advantag of th Quasianalytical tchniqu is tim saving. In a linar channl situation, this tchniqu can provid us th corrct answr xtrmly fast. Th spd of th Quasianalytical tchniqu can also b usd to prform many paramtric studis which would b impossibl othrwis. It also can srv as sanity chck for Mont Carlo mthod du to th imprfction of th Mont Carlo mthod itslf. Howvr, th analytical part, which rlats th simulatd wavforms to th rror rat, dpnds on th actual simulation mthod. Thrfor, th implmntation of th Quasianalytical tchniqu is vry much problm spcific and gnrally cannot b st up in advanc. 43

51 Chaptr 4 Cas Study 4. Introduction In chaptr 3, w studid som basic simulation approachs and mthodologis. In this chaptr, w prsnt on mor cas study to furthr illustrat diffrnt idas and tchniqus usd in dvloping simulations. W usd a simpl simulation of a BPSK communication systm in th last chaptr as an xampl. In this chaptr, w ar going to xpand this xampl vn furthr and construct a simulator for a gnral M-ary PSK systm. Th objctiv of this cas study is to study and valuat th rror probability prformanc (BER) of th systm in an AWG nvironmnt, taking into account th ISI ffct at th transmittr. At th sam tim, visual indicators of th systm, such as y diagram and signal constllation diagram, ar also of strong intrst to us in ordr to study th diffrnc and similarity of th diffrnt lvl of PSK. 4. Mthods of Prformanc Evaluation Using Simulation Du to th combination of th arbitrary lvl of th signal alphabt and th ISI ffct, a pur analytical solution to such a problm is somwhat tdious and involvs a grat dal of rptitiv work. Morovr, such a solution dos not provid asy accss of th dsird paramtric study and valuation. Simulation is thrfor a mor dsirabl option for such a task. Howvr, a Mont Carlo simulation approach is vry tim consuming, spcially whn E b 0 is larg. On th othr hand, a Quasianalytical approach has its limitation too. First of all, as th signal alphabt bcoms larg (8- ary, 6-ary and tc.), th lngth of th rquird squnc incrass rapidly. Consquntly this approach loss its tim saving advantag quickly as M incrass. Scondly, sinc th Quasianalytical 44

52 tchniqu is vry problm spcific, modifications hav to b mad for ach M. It is impossibl to hav a gnral simulator that uss M as an input paramtr as in th cas of th Mont Carlo approachs. In ordr to rach our objctiv ffctivly, w nd to apply a combination of diffrnt mthodological approachs and simulation tchniqus. As mntiond abov, th visual indicators such as y-diagram and signal constllation diagram ar not only important but also ncssary for undrstanding and analyzing th systm in th simulation. Thrfor, a Mont Carlo simulator should b includd to gnrat such diagrams. For th othr fact of th prformanc, th BER stimation, thr is no simpl solution. Each cas has to b dalt with diffrntly. Although a Mont Carlo simulator is tim consuming, it can provid us a gnral simulator for diffrnt lvls of signal alphabt. On th othr hand, th Quasianalytical tchniqu is still vry fficint for low valu of M. Sinc all th modulation studid in this cas blongs to th sam family of modulation tchniqus, a low signal lvl systm such as a QPSK systm oftn bhavs similarly as a systm with highr signal lvl systm such as 8-ary or 6-ary PSK systm. Thrfor, vn though th Quasianalytical tchniqu loss its advantag whn M is larg, it still can provid us grat insight of th whol family of modulation tchniqus by simulating th low signal lvl systm such as a QPSK systm ffctivly. Hnc, diffrnt simulation tchniqus nd to b carfully chosn according to diffrnt problms. 4.3 Simulation of th M-ary PSK Signal As mntiond bfor, both simulation approachs hav thir own advantags. Thrfor, this simulator will hav two parts: th Mont Carlo simulator and th Quasianalytical simulator. Thus, usrs can choos diffrnt simulator accordingly. Th Mont Carlo simulation procdur is as follows.. Gnrat a symbol s in th rang of [, M].. Gnrat random transmittr phas i. 45

53 3. Construct th complx nvlop of th signal by using quation (.3.) and (.3.3) xd ( t) Acos[ ( m ) i ] M xq ( t) Asin[ ( m ) i ] M whr x d (t) and x q (t) ar in-phas and quadratur componnts of th ith symbol. A is chosn to b for convninc. 4. Sampl th in-phas and quadratur componnts, x d (t) and x q (t) rspctivly, to gnrat th sampld signal x S (t). 5. Filtr th in-phas and quadratur componnts of th sampld signal x S (t) to gnrat th transmittd signal x tr (t). 6. Gnrat nois according to th valu of E b 0, i.. signal-to-nois ratio. E 0.5* A T K K (4.3.) b b 0 Tb * whr K is th sampling rat. Thrfor, th standard dviation of th nois is (4.3.) E * b 0 Th variation of E b 0 is achivd by changing th nois varianc and kping th nrgy pr bit, E b, constant. 7. Construct th rcivd signal by summing up th transmittd signal and th nois signal. r ( t) x ( t) n ( t) (4.3.3) d tr d d r ( t) x ( t) n ( t) (4.3.4) q tr q q 46

54 whr r d (t) and r q (t) ar th in-phas and quadratur componnts of th rcivd signal s (t). 8. Calculat th output of th matchd filtr. output K k whr K is th sampling rat, which is 0 in this simulation. s( t) (4.3.5) 9. Dtrmin th rcivd symbol in trms of maximum liklihood stimation. whr s _ st k zk max( zi ) i,,, M 0. Updat th symbol rror statistics. Th procdur for th Quasianalytical approach is as following.. Estimat th mmory of th systm.. Gnrat th maximum lngth squnc. If th systm s mmory is Q symbols, for M-ary PSK modulation, th lngth of th rquird squnc is Q M. 3. Filtr all th squncs to gnrat all possibl transmittd squncs. 4. Calculat th output of th matchd filtr to obtain th noislss rsult of th systm. 5. Calculat th nois varianc for diffrnt valus of E b Calculat th symbol rror probability. 4.4 Simulation Rsults Diffrnt simulation tchniqus ar chosn for diffrnt valu of M. Whn M=4 (QPSK systm), Q th lngth of th rquird squnc is M , assuming systm s mmory is fiv symbols. 47

55 This valu is still rasonabl small. Thrfor, th Quasianalytical approach should b chosn to gnrat th symbol rror probability curv. Howvr, whn M 8, th lngth of th rquird squnc bcoms vry larg. As a rsult, th computr run-tim of a Quasianalytical simulator will incras dramatically too. Thrfor, th Mont Carlo mthod should b usd bcaus it can provid us a gnral solution for all valus of M QPSK Systm Simulation Rsults Th Quasianalytical tchniqu is chosn to gnrat th symbol rror probability for QPSK systms. Th rsult is plottd in Figur BER curv with Wn=/T m=5 Simulation Rsults Thortical Unfiltrd valu 0 - BER Eb/o in db Fig. 4. Symbol Error Probability curv of QPSK systm 48

56 Th solid lin is th thortical valu in an AWG nvironmnt without th ISI ffct. Th diffrnc btwn th simulatd rsult and th thortical unfiltrd valu clarly dmonstrats th ISI ffct on th systm prformanc. To furthr dmonstrat th ISI ffct, th bandwidth of th transmittr filtr is incrasd and anothr simulation is run to study how th prformanc of th systm will chang. Symbol Error Probability curv with Wn=0.T m=5 0 0 Simulation Rsults Thortical valu 0 - B E R Eb/o in db Fig. 4. Symbol Error Probability curv of QPSK systm It is clarly shown in Figur 4., whn th bandwidth of th transmittr filtr is incrasd so that th ISI ffct can b nglctd, th simulatd rsult agrs with th thortical valu. Bsids th symbol rror probability curv, othr visual analysis tool such as y diagrams and th signal constllation diagrams ar also our intrsts. Thrfor, th Mont Carlo simulator is usd hr to gnrat th dsird diagrams. 49

57 D/Q EYE DIAGRAM 0.5 Dir Dirct c 0 t Sampl Indx Quadratur Sampl Indx Fig. 4.3 Ey-diagram of th input of th transmittr filtr Quadratur Channl Dirct Channl Fig. 4.4 Signal Constllation Diagram of th input of th transmittr filtr 50

58 D/Q EYE DIAGRAM Dirct Sampl Indx Quadratur Sampl Indx Fig. 4.5 Ey-diagram of th output of th transmittr filtr Fig. 4.6 Signal Constllation of th output of th transmittr filtr 5

59 D/Q EYE DIAGRAM Dirct Dir ct Sampl Indx Quadratur Sampl Indx Fig. 4.7 Ey diagram of th input of th rcivr (E b / o =5 db) Fig. 4.8 Signal Constllation of th input of th rcivr (E b / o =5 db) 5

60 Th abov diagrams clarly dmonstrat th ISI ffct of th filtr and th ffct of th nois at th diffrnt nods of th systm. Th diagrams also show visually how th channl bhavs in th tim domain (y diagram) and in th signal spac (signal constllation diagram). Fig. 4.3 and Fig. 4.4 ar th y-diagram and th signal constllation diagram of th original transmittd signal rspctivly. It is clarly shown in Fig. 4.3 that th transition btwn th two stats ar straight transitions, which mans that no distortion has bn introducd at this point. Fig. 4.5 and Fig. 4.6 ar th y-diagram and signal constllation diagram of th signal aftr filtring. Th ISI introducd by filtring causs th signal bing disprsd in th signal spac which is shown in Fig Hnc, th stat transition is no longr straight. It is also worth noticing that at th bginning of th tim, thr is a transition stat coming out of zro stat, which is not shown in Fig This is bcaus that th filtr is compltly rlaxd bfor any data is introducd. Fig. 4.7 and Fig. 4.8 ar th y-diagram and signal constllation diagram of th signal aftr it has bn passd through an AWG channl. In thos two figurs, th ffct of th nois is clarly shown. In Fig. 4.7, it is shown that th y has closd somwhat du to th prsnc of th nois. If E b / o dcrass mor, th y will vntually clos compltly. Th similar ffct is also shown in Fig Th signal points ar scattrd all ovr th plac instad of bing disprsd in crtain trajctoris in Fig Similarly, whn E b / o dcrass, th signal points will b scattrd into vn widr ara. Combining all six figurs, w can obtain a clar pictur how th signal bhavs at th various points of th systm. 4.5 Conclusions In this cas study w had two objctivs; th dtrmination of th symbol rror probability and th study of systm bhavior in an AWG nvironmnt taking into account th ISI introducd by th transmittr filtr. For diffrnt lvl of signal alphabt, diffrnt tchniqu should b chosn to optimiz th fficincy of th simulation procss. For M 4, th lngth of th maximum lngth P squnc rquird for th Quasianalytical tchniqu is short. Thrfor, w should choos th Quasianalytical tchniqu to tak full advantag 53

61 of its spd to gnrat th symbol rror probability curv. For M 8, th lngth of th maximum lngth P squnc bcoms long which prvnts th Quasianalytical simulator from a rapid computation. On th othr hand, th Mont Carlo mthod fits ths situations prfctly. By using th Mont Carlo approach, it is possibl for us to dvlop a singl simulator that taks M as an input and gnrats all th dsird diagrams. This is not possibl for a Quasianalytical simulator bcaus a sparat maximum lngth P squnc gnrator nds to b constructd for ach valu of M. An important lsson givn by this cas study is that ach nw problm should b xamind carfully for fficint mans of solution. Any simulation approachs should not b accptd blindly, spcially if thy lad to larg computation tim. 54

62 Chaptr 5 Conclusion and Futur Work 5. Conclusion W studid som basic simulation tchniqus in this papr. Ths tchniqus ar primarily usd for prformanc valuation of diffrnt communication systms. Thy also can b usd for configuring and xcuting wavform lvl simulations. All thos capabilitis provid us wondrful tools for analysis of proposd dsigns. Th Mont Carlo tchniqu is vry simpl to undrstand and implmnt. It can b viwd as th trust situation nxt to building th actual systm. It is also indpndnt from th spcific problm; thrfor, it is possibl for us to dvlop a gnral modl and xchang diffrnt moduls for th spcific problm. Howvr, dspit of all th advantags that th Mont Carlo tchniqu has, it is vry infficint. It has to wait for rrors to happn, and thus it lads to larg computational burdns in most cass. Th Quasianalytical tchniqu s most ovrriding appal is its spd. It nabls us to simulat complx systms in a rlativly short amount of tim, and it also maks paramtric study possibl. Thrfor, th Quasianalytical tchniqu should b usd whnvr is possibl. Howvr, th Quasianalytical tchniqu can not b usd on th systms and undr all th situations. Thr ar constraints for th Quasianalytical tchniqu such as th distribution of th nois has to b known and th nois has to b additiv. At th sam tim, in ordr to tak th full advantag of th tim saving proprty of th Quasianalytical tchniqu, th systm mmory has to b rlativly short and th signal alphabt has to b modratly small. Th Quasianalytical tchniqu is also vry problm spcific so it is gnrally impossibl to b st up in advanc. 55

63 All th simulation tchniqus should not b isolatd from ach othr bcaus in most cass, a combination of th diffrnt tchniqus provids th bst solution for th problm. Thrfor, on must trat ach nw problm with a frsh y for th most fficint mans of solution. o tchniqu should b th univrsal or standard tchniqu. 5. Futur Work Simulation has bn widly usd in th communication industry as an analysis and dsign tool for th past dcad. Howvr, it is also clar that ach organization has dvlopd its own simulation packag and thr is no industry-wid standard, such as th SPICE packag usd for circuit simulation. In th past fw yars, MATLAB has bn adaptd widly to dvlop simulation for various communication systms. In tims to com, mor complt simulation packags, providing color graphics input-output, symbolic dbugging, with a varity of data display capabilitis ranging from tabls to charts to graphs availabl singly or in combinations, will b availabl. In addition to th dvlopmnt of softwar simulation packag, fforts ar also undrway towards th dvlopmnt of firmwar simulation systms using th hybrid hardwar/softwar computing approach. Th application of fast digital signal procssors to modrat data rats will bring us to th stag whr th softwar simulation modls of rcivrs and transmittrs will b usd as is in a firmwar implmntation of th actual systms. Th challng w ar facing today is to us computr-aidd dsign tools to gt on with th rvolution that is upon us all in information transmission and procssing. 56

64 Appndix A. Basic Structur of a Simulation It is vry hlpful to construct a simulation in a mannr that is flxibl to us and asy to maintain. In most simulations, th input and output is usually a vry larg portion of th whol program. Thrfor, if w sparat th input and output from th main program body, th simulation program will b much asir to maintain and undrstand. For this rason, th simulation should b dividd into thr parts: prprocssor, simulator, and postprocssor. Data Storag Data Storag Prprocss (St Intrinsic Paramtrs) Simulator (Excut Simulation) Postprocssor (Analyz Rsults) Fig. A. Basic Simulation Architctur Prprocssor is th initialization part of th program. It rads all th usr dfind intrinsic systm paramtrs and stors th data in data storag which can ithr b in th computr s mmory or in a fil. This is th stag in which usrs dfin th systm. Simulator is th main body of th simulation. It rads all th data from th data storag and xcuts th simulation. At th nd, simulator passs all th nd rsults into th data storag waiting for postprocssor to analyz thm. Postprocssor is th output stag of th program. It rads all th data, such as rror statistic, gnratd by simulator and outputs thm visually, such as BER curvs and signal constllations. Postprocssor can also produc som kind of usr intrfac that allows usrs to choos th dsird diagrams. Th following figurs ar th prprocssor and postprocssor usd for th BPSK simulation xampl. 57

65 EDU» initial Input "B" for BER calculation or "W" to gnrat all othr diagrams ==> w Th priod of ach information bit (0.00) ==> 0.00 Th numbr of sampls pr bit (0) ==> 0 Th phas angl of Transmittr (pi/8) ==> pi/4 Th ordr of th filtr () ==> Th total numbr of signal nds to b gnratd (500) ==> 500 Th dsird Eb/o valu (in db) to gnrat all th wavforms (5dB) ==> 5 Choos on or mor points of th systm whr you would lik to s y diagrams and signal constllation diagrams (i.. MFD to choos all) M ==> at modulator output (without filtring) F ==> at transmittr filtr output D ==> at dtctor (intgrat and dump) input mfd Fig. A. Prprocssor for th BPSK systm Th first input of this prprocssor is th choic btwn th BER curv and othr diagrams. Sinc w don t nd as many runs to gnrat y diagram and signal constllations, this input paramtr allows us to gnrat y diagrams and signal constllations much quickr. Th following paramtrs ar usrs dfind intrinsic paramtrs. Th last input of this prprocssor is whr usr would lik to s th y diagram and signal constllations. This allows usr to insrt prob at diffrnt nod of th systm to monitor th bhavior of th systm at diffrnt stags. 58

66 Fig. A.3 Postprocssor of BPSK simulation 59