Bit Error Rate Calculation for Satellite Communication Systems. Martin Dšttling 1, Simon Saunders 2

Transcription

1 Bi Error Rae Calculaion for Saellie Communicaion Sysems Marin Dšling, Simon Saunders : Insiu fÿr Hšchsfrequenzechni und Eleroni (IHE) Universiy of Karlsruhe, Kaisersra e, 768 Karlsruhe, Germany Phone: Ê57, Fax: Ê8Ê65 Marin.Doeling@eec.uni-arlsruhe.de : Cenre of Communicaion Sysem Research (CCSR) Universiy of Surrey, Guildford, Surrey GUÊ5XH, Unied Kingdom Phone: , Fax: S.Saunders@ee.surrey.ac.u ABSTRACT Simulaions are mandaory o infer he performance of land mobile saellie (LMS) communicaion sysems. For sysem availabiliy calculaions a versaile wo-level simulaion approach is proposed, which separaes linlevel and sysem-level simulaions []. The former aims a deriving saisics of coded bi error rae (BER) from propagaion models, whereas he laer uses he resuls of he lin-level simulaion o calculae he service availabiliy saisics for dedicaed operaional scenarios. This paper oulines a lin level simulaion ool, which convers ime series of effecive symbol energy per specral noise densiy (E s / ) ino raw BER, coded BER and frame error rae (). The lin-level simulaion includes E s / -o-ber conversion, random bi error generaion, de-inerleaving, error correcion, calculaion of coded BER/, as well as visualisaion of resuls. The sofware is compared o analyical upper and lower bounds on BER/ and resuls in lieraure. Moreover an exemplary lin-level simulaion is performed and discussed. The goal of his conribuion is o approximae he funcion of channel coding wihou using expensive and ime-consuming commercial sysem simulaion pacages. Thus he sofware can be easily inerfaced wih various LMS channel models. LIK-LEVEL SIMULATIO APPROACH The lin-level simulaion is based on ime series of effecive E s / as inpu, which are provided by a LMS propagaion model. For each ime series of inpu daa he mean values of bi energy per specral noise densiy (E b / ), BER and are calculaed. Addiionally saisics of inermediae resuls can be displayed and sored. Fig.Ê depics an overview of he implemenaion. The E s / values are assumed o be provided a channel symbol rae and o include all effecs of sysem noise, muli-user inerference and iner-symbol inerference. Thus E s / is relaed o he signal-o-noise raio by E s ()= B sys S Rs () isi mai + () + (), () R R where S() is he received signal power, sys () he sysem noise, B he noise bandwidh, isi () he equivalen noise due o iner-symbol inerference, mai () he muliuser inerference, R s he channel symbol rae and R c he chiprae for CDMA sysems. This generic definiion allows o invesigae ime division muliple access (TDMA), frequency division muliple access (FDMA) and code division muliple access (CDMA) sysems as well as combinaions of hese access schemes. While for TDMA/FDMA sysem noise will be he limiing facor of E s /, mai is dominan in CDMA sysems. For TDMA/FDMA schemes (R c Ê=ÊR s ) mai represens coand neighbouring channel inerference. Inpu sysem parameers Eb/ ime series general seings Resul: <Eb/ > <BER> <> Fig. : *.ma () s ebnober.m calculaion of raw BER codeword/bi error generaor de-inerleaver linsim.m cwgen.m deinerl.m errcorr.m error correcion (bloc code) calcber.m calculaion of coded BER/ c *.ma oupu.m Plo *.ma oupu.m Plo *.ma oupu.m Overview of he lin level simulaion Plo Eb/ saisics raw BER saisics coded BER/ saisics

2 The corresponding informaion bi energy per specral noise densiy (E b / ) is calculaed according o he modulaion index M and he code rae RÊ=Ê/n: Eb ()= R log M Es. () For BPSK and QPSK he raw bi error rae BER r is obained by [] BER r Eb erfc. (3) ()= The codeword generaor deermines he number of bis for each ime sep, generaes he corresponding number of codewords of size bpcw bis and generaes random bi errors according o he acual value of BER r (). Subsequenly, he bis are de-inerleaved using a bloc deinerleaver wih user-defined parameers bloc lengh and inerleaving deph. The implemenaion of coding and error correcion follows a sraighforward sraegy: for each codeword a maximum of m e bi errors are correced, where m e = ( d ) min (4) wih d min denoing he minimum Hamming disance of he bloc code. A main advanage of his approach is ha i simply based on d min, which is abulaed for various codes in open lieraure, e.êg. []. I requires no deailed nowledge of he code's properies (e.êg. ransfer funcion, sae diagram, or weigh disribuion). The mean BER and mean afer decoding is now approximaed by simply couning all remaining bi or frame errors. oe ha he decoder may decide for he correc codeword even if he number of bi errors n e is greaer han m e. Thus (4) leads o an upper bound of BER/. COMPARISO WITH AALYTICAL BOUDS O BER Bloc Codes The performance of he above simulaion approach is invesigaed by comparisons wih analyical upper bounds on. Several bounds for bloc coding can be found in [] and are lised here for convenience. For hard decision decoding he is calculaed by () n n m BER b m = e, (5) ( n m) BERb, () b,, m m ( ) where BER R E b b, ()= erfc. (6) Eq.Ê5 holds wih equaliy for perfec codes. A pair of lower and upper bounds based on d min is given by: b, () dmin d m= min + dmin m BERb, m, (7) ( dmin m) ( BERb, () ) b, 3 b, () A furher upper bound is b, 4 ( ) ( ). (8) () [ 4 BERb, () ( BERb, () )] For sof decision decoding may be used. b, 5 ( ) () Eb erfc R dmin dmin. (9) () Wihou in-deph nowledge of he code srucure, he BER for he bloc codes can no be obained. However, he mean BER can be approximaed assuming for each frame error half of he codeword bis o be in error. Thus BER b,i Ê Ê.5nÊáÊ b,i. These analyical bounds are compared o he simulaion approach oulined above. To allow comparisons wih lieraure, he Golay(3,) code is used, which is a perfec code. Fig. shows ha he simulaion is in excellen agreemen wih he exac expression (5). However, he simulaion considers he effec of inerleaving, which is no included in he analyical expression. In his example, he emporal sampling resuls in a minimum resolvable of 3ÊáÊ -4 for he simulaion, which explains he sligh deviaions from he exac soluion for low.

3 Frame Error Rae Fig. : Eb/ in db exac, Eq. 5 upper bound, Eq. 7 lower bound, Eq. 8 upper bound, Eq. 9 upper bound, Eq. simulaion simulaion and analyical bounds for he (3,) Golay bloc code Convoluional Codes For convoluional codes wih nown ransfer funcion an upper bound on he BER for sof decision Vierbi decoding is derived in []: BER R d E b c, d erfc ()< β, () d= d free where β d is a facor obained from he derivaive of he code ransfer funcion []. For large E b / values he asympoic performance is given by βd free Eb BERc, ()< erfc R dfree () If he code ransfer funcion is no nown he convoluional channel coding heorem may be used [], [3] which yields he ensemble average error rae of a convoluional code on a discree memoryless channel: ( ) [ ] KR ()/ R BERc, 3()< R R (), (3) ( R ( ) R)/ R Alhough our simulaion ool is based on a bloc-coding approach, i is applied o convoluional codes for comparison purposes. Therefore "equivalen" bloc codes are searched by assuming ha d min Ê Êd free. The codeword lengh n is approximaed by he Ploin upper bound and he Gilber-Varsharmow lower bound, which yield a lower bound n min and upper bound n max, respecively []. Fig. 3 depics he BER predicions for a RÊ=Ê/3, KÊ=Ê3 convoluional code. For his code he ransfer funcion is given in [], which allows o calculae also he upper bound in (). For he simulaion a RÊ=Ê/3, d min Ê=Ê8 bloc code wih n max Ê=Ê46 or n min Ê=Ê4 is used. The minimum resolvable BER is ÊáÊ -5. For shor convoluional codes he convoluional channel-coding heorem (3) is a very loose upper bound and can no be used for sysem simulaion. The asympoic bound () differs from he igh upper bound () for E b / less han 5ÊdB and should herefore only be used for higher values. However, if he code ransfer funcion is no nown, () is he only possible analyical bound. Fig. 3 shows ha he simulaion based on he "equivalen" bloc code wih n max ouperforms () for low E b /. The n min curve is oo opimisic. A similar comparison is performed for a RÊ=Ê3/4, KÊ=Ê7 convoluional code, as i is used by Iridium (Fig. 4). The firs five erms of he weigh disribuion for his code are aen ino accoun for he calculaion of () [4]. The resul agrees exacly wih lieraure [5]. The simulaion is again closer o he igh upper bound of () han he asympoic curve () for E b / less han 5ÊdB. The above comparison show ha he simulaion approach, alhough based on bloc coding algorihms, can also be used for convoluional coding, and is able o provide realisic and igh upper bounds on BER. This is especially imporan for low E b / values, which resul in coded BERÊ Ê -3 and in cases, where he exac code ransfer funcion is no nown. Addiionally, in conras o he analyical bounds, he effec of de-inerleaving is considered in he simulaion. - where K is he consrain lengh and R is he cu-off rae defined in [], [3]: R R E b log e. (4) ()= + Unforunaely, () holds no for small E b / values, where RÊ>ÊR (). Bi Error Rae Eb/ in db Eq. Eq. Eq. 3 simulaion, nmin simulaion, nmax Fig. 3: BER predicions for a RÊ=Ê/3, KÊ=Ê3 convoluional code

4 - Table : Simulaion parameers Bi Error Rae Eb/ in db Fig. 4: Eq. Eq. Eq. 3 simulaion, nmin simulaion, nmax BER predicions for a RÊ=Ê3/4, KÊ=Ê7 convoluional code (Iridium) RESULTS AD APPLICATIO As an applicaion he lin-level simulaion ool is used o calculae coded BER saisics for an exemplary simulaion run of he Iridium sysem in a rural area. Fig. 5 shows opography, land use and mobile rajecory. The landscape in he proximiy of Karlsruhe, Germany shows a ypical non-urban mixure of errain and land use elemens. The mobile rajecory is depiced as a whie arrow: i sars in he hilly area and eners he Rhine Valley, passing foresed, open and buil-up areas. Two differen operaional scenarios are considered: a high-speed applicaion (e.êg. high-speed rain) wih mean mobile velociy of 6m/s and a pedesrian waling a m/s. The simulaion used he LMS propagaion model, which is described in deail in [6], [7]. For boh scenarios hree E b / ime series are calculaed and compared, assuming: a) no power conrol (), b) power conrol (), and c) power conrol and saellie handover o miigae shadowing (Ê+ÊHO). open/agriculure meadow deciduous fores coniferous fores dense urban m m urban indusrial sree waer mobile pah heigh in m 3 mean mobile speed pedesrian scenario m/s high-speed scenario 6m/s ime resoluion ms 5ms sar Ð climax Ð end elevaion of visible saellies signalling delay power conrol saellie handover <8 ÊÐÊ39.6 ÊÐÊ Ð 3. Ð <8.9 Ð.9 Ð <8 <8 Ð.9 Ð ÊÐÊ45.8 ÊÐÊ Ð 5.5 Ð <8 x round rip ime 5ms processing ime 3-bi power conrol ±db dynamic range updae inerval >5ms 6dB hyseresis updae inerval >5ms The effec of signalling delay on power conrol and handover efficiency is included in he simulaions. TableÊ liss he main simulaion parameers; a comprehensive descripion of he power conrol and handover simulaion algorihm is given in [8]. The Iridium mobile lin parameers of [9] are used. A arge value of E b / Ê=Ê8.dB is assumed for line-of-sigh [9]. The resuling E b / ime series for he pedesrian and high-speed applicaion are shown in Fig. 6 and Fig. 7, respecively. For he pedesrian scenario, fading and ligh shadowing can be power conrolled; saellie handover allows o miigae some of he blocages (especially beween Ê=Ê3s and Ê=Ê45s). For he high-speed scenario, however, power conrol and handover are inefficien, since he channel sae has already changed before a sysem reacion (which is subjec o he signalling delay) occurs. Fig. 8 shows he corresponding hisograms and Fig. 9 he complemenary cumulaive disribuion funcions (CCDF). For he calculaion of he coded BER, () and () are used. An addiional modem implemenaion loss of db is considered [9]. The resuling CCDFs for () are depiced in Fig. and numerical values are given in TableÊ. Fig. 5: Topography, land use and mobile rajecory in he Rhine Valley near Karlsruhe, souhern Germany In he pedesrian scenario, he arge BER of - is exceeded by 45.4% wihou power conrol and by 9.% if power conrol is considered. This indicaes ha linlevel simulaions mus include he effec of power conrol o yield realisic BER saisics. Furhermore, i

5 is obvious ha he difference beween BER wih and wihou power conrol diminishes as he mobile speed increases. For he high-speed scenario his difference is only around %. Considering he differen curves obained for he differen mobile speeds, i can be concluded ha he simulaion is also sensiive o he consideraion of mobile speed and signalling delay. Saellie handover reduces he probabiliy of exceeding he arge BER from 9.% o.7% in he pedesrian applicaion. The handover rae is 5ÊáÊ - Hz. For he high-speed applicaion, saellie handover is no appropriae o miigae shadowing: in his paricular case, he mean BER increases. This can be explained as follows: due o he signalling delay and he high mobile speed, he channel sae o he differen saellies have already changed a handover compleion ime. Thus erroneous handover occur, i.êe. handover ha lead o a communicaion via a sub-opimal saellie. Table compares he difference beween he asympoic formula () and he upper bound of (), including he firs five erms of he code weigh disribuion. The asympoic formula is oo opimisic wih relaive errors in mean BER up o 53%. COCLUSIO A simulaion sofware o obain coded BER/ from ime series of E s / is oulined. I compares analyical upper and lower bounds on BER/ o values calculaed by a simulaion sofware. Major advanages of he simulaion sofware are he applicabiliy for low E s / values and ha no deailed informaion on he code srucure is necessary. Lin-level simulaions are performed for he Iridium mobile lin, which indicae ha mobile speed, power conrol and handover sraegies mus be included in LMS sysem simulaion, since he PDF of E b / and as a resul he CDF of coded BER are very sensiive o his parameers. Furhermore, he probabiliy densiy of E b / does no follow simple analyical disribuions, which highlighs he necessiy of realisic ime series simulaion. ACKOWLEDGEMETS Pars of his wor were carried ou while M. Dšling was visiing he CCSR, Universiy of Surrey. M. Dšling graefully acnowledges he funding by he Commission of he European Communiies in he framewor of COSTÊ55, which made his wor possible. REECES [] S. Saunders, "Mobile Projec Group Simulaion Approach Definiion", COST 55 Radiowave Propagaion Modelling for ew SaCom Services a Ku-Band and above, oordwij, The eherlands, CP639, Ocober, 998 [] J. G. Proais, Digial Communicaions, Third Ediion, McGraw-Hill, Inc., ew Yor, 995 [3] A. J. Vierbi, J. K. Omura, Principles of Digial Communicaions and Coding, McGraw-Hill, Inc., ew Yor, 979 [4] D. Haccoun, G. BŽgin, "High-Rae Puncured Convoluional Codes for Vierbi and Sequenial Decoding," IEEE Trans. on Communicaions, vol. 37, no., pp.ê3-5, 989 [5] J. B. Cain, G. C Clar, Jr., J. M. Geis, "Puncured convoluional codes of Rae (n-)/n and Simplified Maximum Lielihood Decoding," IEEE Trans. on Informaion Theory, vol. IT-5, no., pp.ê97-, 979 [6] IMST, DLR, IHE, Land Mobile Saellie Propagaion Model for on-urban Areas, Final Repor, European Space Agency Conrac o. AO/-3/96L/B, March, 998. [7] M. Dšling, A. Jahn, J. Kunisch, S. Buonomo, "A Versaile Channel Simulaor for Land Mobile Saellie Applicaions," Proc. IEEE Vehicular Technology Conf. VTC'98, 998, pp.ê3-7. [8] M. Dšling, T. Zwic, W. Wiesbec, "Invesigaion of Saellie Diversiy and Handover Sraegies in Land Mobile Saellie Sysems based on a Ray Tracing Propagaion Model," acceped for publicaion in Proc. Sixh In. Mobile Saellie Conf. IMSC'99, Oawa, Canada, 999, in press [9] J. Habeha, "Simulaion Tes Case 6: IRIDIUM Mobile Lin L-Band Sysem Requiremens", COST 55 Radiowave Propagaion Modelling for ew SaCom Services a Ku-Band and above, oordwij, The eherlands, CP636, Ocober, 998

6 Eb/ in db ime in s + HO Fig. 6: E b / ime series for he pedesrian scenario Eb/ in db ime in s + HO Fig. 7: E b / ime series for he high-speed scenario.. percenage of ime HO percenage of ime HO Eb/ in db Eb/ in db Fig. 8: Hisogram of E b / for pedesrian (lef) and high-speed (righ) scenario

7 probabiliy Eb/ > abscissa HO Eb/ in db probabiliy Eb/ > abscissa HO Eb/ in db Fig. 9: CCDF of E b / for pedesrian (lef) and high-speed (righ) scenario probabiliy BER > abscissa HO coded BER probabiliy BER > abscissa HO coded BER Fig. : CCDF of coded BER for pedesrian (lef) and high-speed (righ) scenario Table : Simulaion resuls pedesrian scenario high-speed scenario Eq.Ê Eq.Ê Eq.Ê Eq.Ê mean coded BER.3ÊáÊ -.33ÊáÊ -.9ÊáÊ -.48ÊáÊ - 3.9áÊ -.76ÊáÊ -.36ÊáÊ -.ÊáÊ - + HO 5.66ÊáÊ -3 4.ÊáÊ -3.5ÊáÊ -.5ÊáÊ - probabiliy coded BERÊ>Ê % 36.5% 4.% 34.% 9.% 7.% 9.6% 6.% + HO.7%.% 3.5% 8.6% probabiliy coded BERÊ>Ê % 44.% 44.7% 4.%.% 8.7% 3.6% 8.6% + HO.%.5% 35.3% 3.6%