Fernando Ramrez-Mireles, Moe Z. Win, and Robert A. Scholtz. be written as [Scholtz, 1993]: and Tc are the time shift values corresponding to the

Transcription

1 Sigal Selectio for the Idoor Wireless Impulse Radio hael Ferado Ramrez-Mireles, Moe Z. Wi, ad Robert. Scholtz ommuicatio Scieces Istitute Departmet of Electrical Egieerig-Systems Uiversity of Souther aliforia, Los geles, US STRT I this paper we ivestigate commuicatios usig quaterary pulse positio data modulatio over the idoor impulse radio multiple access chael disturbed with multipath. The performace of four quaterary sigal sets with dieret correlatio properties is assessed. I. INTRODUTION Reliable simultaeous commuicatios amog multiple users exchagig iformatio at rates of the order of Megabits/secod over the idoor wireless chael, overcomig multipath, fadig, shadowig, power limitatios ad iterferece, is a techical challege. ovel modulatio scheme potetially well suited for such a demadig applicatio is Impulse Radio Multiple ccess (IRM) techique proposed i [Scholtz, 993][Wi, 996a]. IRM is a Spread Spectrum (SS) scheme which uses time hoppig (TH) for the SS sequece modulatio, ad pulse positio modulatio (PPM) for the data modulatio. The commuicatios waveforms covey iformatio exclusively i the time shift values ad cosist of trais of time-shifted ultraarrow pulses. IRM is a o-costat evelope, ultra-widebad, \carrier-less" modulatio with badwidth i excess of GHz. The aalysis i [Scholtz,993] focused o commuicatios usig biary data PPM over the IRM chael disturbed with additive white Gaussia oise (WGN). I the preset paper we ivestigate commuicatios usig quaterary data PPM over the idoor wireless IRM chael disturbed with multipath i additio to WGN. The uderlyig questio is : Does multipath sigicatly iterferes with PPM data modulatio? More specically, we use the results i [Scholtz, 993] to desig four sets of quaterary data PPM sigals. Wewat to ivestigate the behavior of these sigals sets i the presece of multipath uder the assumptio of costat SNR. The research described i this paper was supported i part by the Joit Services Electroics Program uder cotract F The graduate studies of Mr. Ramrez are supported by the oacyt Grat. The authors ca be reached by at ramirezmmilly.usc.edu, wimilly.usc.edu, scholtzmilly.usc.edu II. IRM OMMUNITIONS WVEFORMS The TH-PPM sigal coveyig the user's iformatio ca be writte as [Scholtz, 993]: x(t) = X j=, p(t, jtf, cjtc, Td [j=ns ]) () where p(t) is a ultra-arrow (sub-aosecods) pulse; Tf ad Tc are the time shift values correspodig to the frame period ad the SS sequece modulatio, respectively; Td [j=ns 2 f <2< :::;< M ] d g is the time shift value correspodig to the the data modulatio; ck 2 f; ; 2; :::; Nh,g is the time hoppig sequece associated with each user, ad Ns > is the hoppig rate i hops per data symbol. Give the TH sequece, the values of Tf ad Tc, ad with the shape of the pulse xed by the geerator device ad the atea type, the sigal desig task cosists i dig the optimum values of the set of shifts: i; i= ;2;:::;Md satisfyig <i<tf,nhtcfor each data symbol i. III. HNNEL HRTERIZTION The IRM chael i the absece of multipath eects but with WGN will be called IRM-IDEL. Whe multipath is preset, it will be called IRM-MP. The eect of the IRM- IDEL ad IRM-MP chaels o the trasmitted waveform ca be characterized by the sigal correlatio fuctio of the received waveform. For the IRM-IDEL chael, the actual sigal correlatio ideal() ca be aalytically modeled by " # model() = 3 exp ", 2 # ad the sigal-to-oise-ratio (SNR) value ca be modeled as a costat. I order to calculate R ideal(), the receiver rst forms the T correlatio Rideal() = Srx(t)Sloc(t,)dt, where Srx(t)is the sigal received over the IRM-IDEL chael whe p(t) is trasmitted ad Sloc(t) is the sigal geerated locally at The quatity [j=ns] deotes \iteger part\. (2)

2 the receiver. The fuctio ideal() is the ormalized versio of Rideal(). 2 Figure (a) plots ideal(), the measured IRM-IDEL correlatio 3. Figure (b) plots model(), the aalytical IRM-IDEL correlatio model, usig a value of = :753 s. 4 IRM IDEL alytic model IRM MP samples IRM MP ve. Sigal correlatio fuctios Fig.. Sigal correlatio fuctios : (a) Measured IRM-IDEL correlatio. (b) alytical IRM-IDEL correlatio model. (c) Measured IRM-MP correlatio. The plot shows dieret realizatios (measuremets). (d) Measured IRM-MP correlatio. The plot shows the sample average take over the realizatios i (c). For the IRM-MP chael, the actual sigal correlatio fuctio MP(u; ) ca be cosidered a radom process, where u deotes a evet takig place i the sample space of a certai radom experimet (e.g. a measuremet experimet). Due to the presece of fadig, the SNR value (u) 4 = E(u)=No( with E(u) the total eergy of the received sigal ad No the power spectrum desity of the WGN) is modeled as a radom variable. I order to calculate the fuctio MP(u; ), the receiver rst forms the correlatio 5 RMP(u; ) = R T Srx(u; t)sloc(u; t, )dt, where Sloc(u; t) is the sigal geerated locally at the receiver ad Srx(u; t) is the sigal received over the IRM-MP chael whe the pulse p(t) is trasmitted. The fuctio MP(u; ) is the ormalized versio of RMP(u; ). 6 Figure (c) plots several realizatios of MP(u; ), the measured IRM-MP correlatio. Figure (d) plots the sample average of MP(u; ), the average beig take over the measuremets of gure (c). I both gures (c) ad (d), the multipath eects ca be observed clearly i the distortio 2 For ideal () to be the true sigal autocorrelatio fuctio, we eed Srx(t) to have o oise compoet. 3 Here, ideal () is cosidered a determiistic process, but actually is the average of several measuremets. 4 This value of is the oe that miimizes the squared error betwee the sigal aalytic model ad a template formed from experimetal measuremets. 5 It is assumed that the operatios ivolvig radom process are well deed. 6 For MP (u; ) to be the true sigal autocorrelatio fuctio, we eed Srx(u; t) tohave o oise compoet, ad S loc (u; t) tobeperfectly matched to Srx(u; t). Hece, we eed a receiver able to perfectly recostruct Srx(u; t). of the waveforms as well as i the presece of log tails. IV. SIGNL SELETION FOR THE IRM-IDEL HNNEL The optimum sigle-user receiver for the IRM-IDEL chael cosists of a TH despreadig operatio followed by a correlatio receiver [Scholtz, 993]. The symbol error probability Pe(; ) for this receiver depeds oly o the symbol SNR value ad the correlatio properties of the commuicatios sigal set [Weber, 987]. The same applies to the uio boud o the symbol error probability UPe(; ). For a give model() ad a particular sigal set deed by the time shifts j ;j =;2;:::;Md,we ca use the relatio j;k 4 = model(j, k) iupe(; ) toivestigate the performace of this particular sigal set. Figure 2 shows four dieret quaterary PPM sigal sets that were desiged based o model(). 7 The time shifts (i aosecods) correspodig to this sigal sets are fj opt g 4 j= = (:; :23; :46; :69) fj q,biortho g 4 j= = (:; :42; :92; 2:34) fj q,ortho g 4 j= = (:; :23; :73; :96) fj ortho g 4 j= = (:; :5; 3:; 4:5) The correspodig correlatio matrices are give i the equatios below. opt = q,biortho = q,ortho = ortho = +:,:4,:57,:2,:4 +:,:4,:57,:57,:4 +:,:4,:2,:57,:4 +: +:,:6 : :,:6 +: : : : : +:,:6 : :,:6 +: +:,:4 : :,:4 +: : : : : +:,:4 : :,:4 +: +: : : : : +: : : : : +: : : : : +: Figure 3(a) shows the performace of the four sets of sigals i the IRM-IDEL chael. 8 The curves repre- 7 The sigal set i gure 2(d) is called orthogoal for obvious reasos (see ortho ). The quasi-biorthogoal ad quasi-orthogoal sigal sets i gures 2(c) ad 2(d) receive their ame from their similarity (i correlatio properties) with the biorthogoal ad orthogoal sigal sets, respectively. The sigal set i gure 2(a) is optimum i the sese that it was desiged to miimize UPe(; ) at high values. Note the the optimum might be a fuctio of [Weber, 987]. 8 I geeral, the more egative correlatio values has, the smaller UPe(; ) is. Hece, i gure 3(a) UPe(; opt) UPe(; q,biortho) UPe(; q,ortho) UPe(; ortho )

3 set of "optimum" sigals set of "quasi biorthogoal" sigals set of"quasi orthogoal" sigals set of orthogoal sigals Fig. 2. The four sets of quaterary PPM data sigals uder study. (a) Optimum. (b) Quasi-biorthogoal. (c) Quasi-Orthogoal. (d) Orthogoal. set UPe(; ) versus for the cases = opt, q,biortho, q,ortho, ad ortho. Figure 3(b) show the performace of the four sets of sigals whe opt, q,biortho, q,ortho, ad ortho are calculated usig ideal() istead of model(). V. SIGNL SELETION FOR THE IRM-MP HNNEL To ivestigate the behavior of the four sigal sets i the presece of multipath i a idoor eviromet, we made use of sigal propagatio data recorded i a ultra-wide-bad measuremets experimet [Wi, 997]. I this experimet, multipath proles are measured at 4 dieret rooms ad hallways. I each room, 3 aosecod-log widows of multipath measuremets are recorded at 49 dieret locatios over a 3 feet by 3 feet grid. They are arraged spatially i a 7x7 square grid with 6 ich spacig. Durig each ofthe multipath prole measuremet the trasmitter, the receiver ad the eviromet are kept statioary. Oe hudred ad forty seve ormalized correlatio fuctios were calculated from the same umber of measured sigals received i three dieret oces. Due to the multipath eects, the sigal correlatios at each poit are dieret from each other. They are the sample fuctios of MP(u; ) as described before. typical set of sample fuctios measured i a sigle oce was show i gure (c). We ca exted the aalysis i the previous sectio by usig the sample fuctios of MP(u; ) istead of model(). For the four sigal sets uder cosideratio we ca dee the ormalized radom correlatio values j;k (u) = 4 MP(u; j, k) ad use this values i the uio boud o probability of error UPe(; (u)) to ivestigate the performace of each particular sigal set i the IRM-MP chael. Specically, for the four sigal sets uder cosideratio, we ca dee the ormalized radom correlatio values opt(u) j;k = MP(u; j opt, opt k ) j;k q,biortho (u) = MP(u; j q,biortho, q,biortho k ) q,ortho j;k = MP(u; j q,ortho, k q,ortho ) ortho j;k = MP(u; j ortho, k ortho ) We wat to ivestigate how, for each value of SNR cosidered, the value of UPe i the IRM-MP curve deviate from the value correspodig to the IRM-IDEL curve. For the optimum sigal, it is clear that 9 UPe(; opt) UPe((u);opt(u)) This degradatio i the probability of error i the IRM-MP case is caused by two factors: uctuatios i the SNR value, (u) ; ad severe distortios ad log tails i RMP(u; ). I this paper we will work with the ormalized sigal correlatio fuctio MP(u; ). Normalizatio will make the TO- TL SNR approximately costat for every radom evet u (i.e. (u) = ). Hece, the UPe degradatio caused by total SNR uctuatios due to fadig are ot cosidered here, ad UPe degradatio will be maily caused by the sigal correlatio fuctio distortios due to multipath. Figures 4(a) ad 4(b) show the performace of the sigal sets i the IRM-MP chael. The curves i gure 4(a) represet Eu fupe(; (u))g ad the curves i gure 4(b) represet the worst case evet max fup e(; (u))g versus for the cases (u) = opt(u), q,biortho(u), q,ortho(u), ad ortho(u). Eu fg is the expected value operator. Figure 4(a) was plotted usig the sample average take over 9 Note that (u) is the faded versio of, hece (u). lso ote that by the optimality of opt we have that UPe(; opt) UPe(; opt(u)) I [Wi, 997], measuremets over a IMR-MP chael showed that the rage of fadig is less tha 3 d. Note that Eu fupe((u);(u; ))g is the uio boud o the probability of error for a Ideal RKE (IRKE) receiver (ideal i the sese that has ulimited umber of correlators ad perfect estimates, therefore it is able to perfectly match-lter the sigal received through the IMR-MP chael.

4 (a) IRM IDEL model (b) IRM IDEL measured 2 2 Uio oud PWE 3 Uio oud PWE Ew/No (d) Ew/No (d) Fig. 3. Performace of the four sets of sigals i : (a) IRM-IDEL chael, whe model () is used. (b) IRM-IDEL chael, whe ideal () is used. (a) mea IRM MP measured (b) max IRM MP measured 2 2 Uio oud PWE 3 Uio oud PWE Ew/No (d) Ew/No (d) Fig. 4. Performace of the four sets of sigals i the IRM-MP chael, whe (a) MP (u; ) is used. The curves correspod to the sample average take over the realizatios of MP (u; ). (b) MP (u; ) is used. The curves correspod to the maximum value take over the realizatios of MP (u; ).

5 the 47 dieret realizatios of MP(u; ). Figure 4(b) was plotted takig the maximum over the dieret realizatios of MP(u; ). VI. DISUSSION OF RESULTS The purpose of the preset aalysis is to ivestigate which of the four sets of sigals has better performace uder variatios i the shape of MP(u; ) caused by multipath. This aalysis was doe uder the assumptios that the SNR is costat (i.e., i the absece of fadig), ad that fadig is relatively idepedet of which set of sigals is actually used. omparig the performace curves whe both model() ad ideal() are used, we see that for the curves i gure 3(a) UPe(; opt) < UPe(; q,biortho) <UPe(; q,ortho) < UPe(; ortho) but for the curves i gure 3(b) ad UPe(; q,biortho) < UPe(; q,ortho) < UPe(; ortho) UPe(; q,biortho) UPe(; opt) Hece the optimum sigal set is the oly oe that chage its performace sigicatively whe, istead of the model model(), the measured ideal() is used. alyzig the performace curves whe MP(u; ) is used, we see that for the curves i gure 4(a) Fially, comparig gure 3(a) with gures 4(a) ad 4(b) we observe that the quasi-biorthogoal sigal set suer less degradatio i the UPe curves tha the other sets whe multipath is preset. Therefore, the quasi-biorthogoal sigal set is the preferable sigal set for quaterary commuicatios over the IRM-MP chael. ckowledgmets The authors wish to thak Mark ares, Troy Fuqua, ad Larry Fullerto of Time Domai Systems, ad Paul Withigto of Pulso ommuicatios for several helpful discussios cocerig the techology, capabilities, ad sigal processig of impulse sigals. Refereces [Scholtz, 993] R.. Scholtz, \Multiple ccess with Time Hoppig Impulse Modulatio," Proceedigs of Milcom 93, Dec [Weber, 987]. L. Weber, Elemets of Detectio ad Sigal Desig, orrected reprit, Spriger Verlag, 987. [Wi, 996a] M. Z. Wi, R.. Scholtz, ad L. R. Fullerto, \Time- Hoppig SSM Techiques for Impulse Radio with a alog Modulated Data Subcarrier," Proc. Fourth Iteratioal Symposium o Spread Spectrum Techiques ad pplicatios ISSST'96, Maiz, Germay, September 996. [Wi, 997] M. Z. Wi ad R.. Scholtz, \haracterizatio of Ultra-Wide adwidth (UW) Wireless Idoor Propagatio haels," IEEE I'97, Jue 997. Eu fupe(; q,biortho(u))g < Eu fupe(; q,ortho(u))g < Eu fupe(; ortho(u))g ad Eu fupe(; q,biortho(u))g <EufUPe(; opt(u))g ad for the curves i gure 4(b) max fup e(; q,biortho(u))g fup e(; q,ortho(u))g fup e(; opt(u))g fup e(; ortho(u))g From this results, it is evidet that the optimum sigal actually performs worse tha the quasi-biorthogoal ad quasiorthogoal sigal sets i the IRM-MP chael. This could be attributed to the fact that the quasi-biorthogoal ad quasi-orthogoal sets of sigals were desiged usig the miimum value poit of model() ad the zero crossig poits that are closest to the origi =. swe ca see i gure (c), these poits are relatively the same for all the realizatios of MP(u; ), ad sigal desig usig these poits gives robust performace i the presece of multipath. 2 2 The results suggest, however, that multipath places fudametal limits o the ability to exted pulse-positio modulatio techiques for values of M d greater tha four.