Guard Time Optimization for Capacity Maximization of BPSK Impulse UWB Communications

Transcription

1 Guard Time Optimizatio for Capacity Maximizatio of BPSK Impulse UWB Commuicatios Abdallah Hamii, Jea-Yves Baudais, Adrea M. Toello, Jea-Fraçois Hélard To cite this versio: Abdallah Hamii, Jea-Yves Baudais, Adrea M. Toello, Jea-Fraçois Hélard. Guard Time Optimizatio for Capacity Maximizatio of BPSK Impulse UWB Commuicatios. Joural of Commuicatio, Wiley, 04, 9 (), pp.-97. <0.70/jcm >. <hal > HAL Id: hal Submitted o Sep 04 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Guard Time Optimizatio for Capacity Maximizatio of BPSK Impulse UWB Commuicatios Abdallah Hamii,3, Jea- Yves Baudais, Adrea M. Toello, ad Jea-Fraçois Hélard3 Natioal Ceter for Scietific Research (CNRS), IETR, UMR 664, F-3570 Rees, Frace DIEGM - Uiversity of Udie viadelle Scieze, Udie Italy 3 Uiversitéeuropéee de Bretage, INSA, IETR, UMR 664, F-3570 Rees, Frace abdallah.hamii@isa-rees.fr, jea-yves.baudais@isa-rees.fr, toello@uiud.it, jea-fracois.helard@isa-rees.fr. Abstract Usually, for the desig of UWB systems, the symbol duratio is chose larger tha the delay of the chael impulse respose, i order to avoid the ISI (iter symbol iterferece). However, this approach does ot maximize the system capacity. A adaptatio of the guard time (GT) is a flexible mea of exploitig system resources efficietly i a multi-path fadig eviromet. The optimal guard time legth i BPSK impulse UWB commuicatios is obtaied by exhaustively searchig for the guard time that maximizes capacity. This approach is complex sice it has to be implemeted for each chael realizatio. To reduce this complexity, i this paper we preset ew optimizatio methods. The first method assumes the fadig chael to be partitioed ito classes. The, a give GT legth for each chael class is used. The secod optimizatio method provides a guard time for each chael realizatio. However, the GT is obtaied by lookig at simplified metrics which are based o the chael delay spread, the received sigal eergy, or o a approximatio of the capacity formula. Simulatio results are performed for UWB commuicatios over WiMedia chaels ad they show that sigificat gais are achievable with the proposed guard time adaptatio w.r.t. to the use of a costat guard time. Idex Terms System capacity, pulse desig, guard time, BPSK, UWB I. INTRODUCTION The demad for ew services ad applicatios i commuicatio systems, as well as the umber of users, are steadily icreasig. This growth ivolves a great eed of data rate icrease offered by the commuicatio system. The system capacity is a importat parameter for the desig ad evaluatio of wireless etworks. Recetly, the UWB (ultra wide bad) techique has bee widely studied for wireless commuicatios [], []. UWB commuicatios offer very high data rates due to the use of a wide bad ad the robustess to multi-path fadig [3]. The system should be desiged so that capacity is maximized. The Gaussia moocycle was iitially proposed ad has bee widely used i impulsive UWB systems [4], [5]. I typical impulsive UWB system desig, the symbol duratio is larger tha the maximum Mauscript received September 9, 03; revised February 6, 04. Correspodig author abdallah.hamii@isa-rees.fr. doi:0.70/jcm Egieerig ad Techology Publishig chael impulse respose duratio so that the itersymbol iterferece ca be eglected [6], [7]. To this ed, a guard time is added after the pulse trasmissio i order to avoid ISI. The optimizatio of the guard time is ot geerally cosidered sice typically the guard time is set at a duratio loger tha the maximum chael duratio []. However, this approach is eergy ad capacity iefficiet. I fact, to maximize the capacity, the system does ot ecessarily eed a large guard time. That is, the system ca tolerate a amout of iterferece i order toreduce the guard time so that the system capacity ca be improved. A adaptatio of the guard time is a flexiblemea of exploitig system resources efficietly especially i a varyig multi-path fadig eviromet. The case of a guard iterval shorter tha the chael impulse respose has bee cosidered i OFDM systems [9], [0]. Followig a similar cocept, i this paper we report a aalysis of the guard time optimizatio i impulsive UWB trasmissio. The optimizatio problem pursues the maximizatio of capacity. Sice the chael respose varies with time ad positio, ideally the guard time should be adaptively chose. However, this method is computatioally itese. To reduce the complexity, two methods are proposed i this paper. The first method partitios the chael ito classes. Each class collects chael resposes that provide a certai average atteuatio ad delay spread. Essetially, each class is represetative of a certai eviromet as it is doe i the WiMedia chael model []. The, a sigle guard time value for each chael class ca be defied ad used. The secod method provides a adaptive guard time for all chael realizatios. However, the guard time is obtaied by lookig at simplified metrics that are based o the chael delay spread, the received sigal eergy ad o the use of a approximatio of the capacity formula. With these metrics the computatio of the guard time is simplified ad the method ca be applied to ay chael model. The remider of this paper is as follows. The descriptio of the commuicatio model is preseted i Sectio II. The typical chael model used i UWB is itroduced i Sectio III. I Sectio IV, the capacity calculatio ad the guard time optimizatio are itroduced. I additio, the statistical aalysis of the

3 optimal guard time is provided. I Sectio V, the first method for guard time optimizatio is preseted. Sectio VI itroduces a optimizatio method for the guard time desig with low complexity. Some alterative metrics are proposed to adjust the guard time i order to reduce the system complexity. Sectio VII describes the simulatio setup, umerical results ad provides a aalysis of the results. Fially, Sectio VIII cocludes this paper. II. COMMUNICATION MODEL We cosider a sigle user system model with BPSK (biary phase shift keyig) sigallig so that the trasmitted sigal ca be writte as [] () bk g(t ktb ) x(t) = where ak are the iter-symbol iterferece amplitude coefficiets, while w is the sequece of i.i.d. Gaussia oise samples with zero mea ad variace N0. ISI is geerated whe the guard time is shorter tha the chael duratio. III. CHANNEL MODEL (WIMEDIA CHANNEL) We cosider UWB chaels with frequecy selective fadig [4]. I particular, we use the model adopted by the IEEE 0.5.3a committee for the evaluatio of UWB physical layer proposals []. The model defies four classes each characterized by lie-of-sight (LOS) or olie-of-sight (NLOS), a certai mea excess delay, RMS delay spread ad distace betwee the trasmitter ad the receiver, as summarized i Table. I. k where bk = ± deotes the iformatio bit trasmitted i the frame k ad Tb is the bit period (frame duratio). We icorporate the differetial effects of the trasmissio, ad receive ateas ito g(t). g(t) is assumed to be the secod derivative of the Gaussia pulse T T t p π t p g(t) = π () T0 T0 where Tp is the moocycle pulse duratio, ad T0 is the width of the pulse. We further isert a guard time Tg betwee pulses. The bit duratio fulfills the relatio Tb = (Tp + Tg). The iter-symbol iterferece is avoided whe Tb Tp + τmax, where τmax is the maximum delay (duratio) itroduced by the chael. At the receiver side, we first deploy a badpass frot-ed filter to suppress the out of bad oise. The, the received sigal, i the sigle user case, ca be writte as z(t) = k bk (g? h)(t ktb ) + η(t) (3) where (g? h)(t) is the covolutio of the waveform filter g(t) by the impulse respose of the chael h(t). The additive oise η(t) is assumed to be a statioary zero mea Gaussia process. Further, i the followig, we cosider it to be white i the useful sigal bad. Let us suppose that the received sigal is passed first through a matched filter e(t) y(t) = (e? z)(t) (4) The optimum filter from the poit of view of SNR maximizatio is the matched filter [3]. The matched filter is adapted to the pulse ad to the chael respose. It is obtaied by correlatig the trasmit pulse ad the chael respose. We furthermore assume to use a oise whiteig filter (icluded i the impulse respose e(t)) so that the sequece of samples at the output ca be writte as y = y(tb ) = b k ak + w (5) k 04 Egieerig ad Techology Publishig 9 TABLE I: UWB CHANNEL CHARACTERISTICS Mea excess delay(s) Delay spread(s) Distace(m) LOS/NLOS CM <4 LOS CM <4 NLOS CM NLOS CM NLOS This model is a modified versio of Saleh-Valezuela model for idoor chaels, fittig the properties of measured UWB chaels. A log-ormal distributio is used for the multi-path gai magitude. I additio, idepedet fadig is assumed for each cluster ad each ray withi the cluster. The impulse respose of the multipath model is give by h(t) = G Z P z=0 p=0 α(z, p)δ(t T (z) τ (z, p)) (6) where G is the atteuatio due to log-ormal shadowig, T is the delay of cluster z, α(z, p) ad τ (z, p) represet the gai ad the delay of the multi-path compoet p of cluster z. The cluster ad the path arrival times are modelled accordig to a Poisso arrival process. The path amplitude follows a log-ormal distributio each with arrival rates ad decay factors chose to match differet usage scearios ad to fit lie-of-sight ad o-lie-ofsight cases. More details ca be foud i []. IV. GUARD TIME OPTIMIZATION A. Capacity Calculatio To evaluate the impact of the guard time legth o the system performace, we defie the optimum value of guard time that maximizes the system capacity. The capacity is defied as the maximum of the mutual iformatio (7) C = max I(, Y ) p() I the case of BPSK, the mutual iformatio is maximized for equi-probable symbols. Let us ow compute the mutual iformatio as a fuctio of the bit eergy Eb, the Gaussia oise variace N0 ad the iter-

4 symbol iterferece that is geerated whe the guard time is shorter tha the chael duratio. The mutual iformatio is I(, Y ) = S(Y ) S(Y ) where both etropies are defied as [5] S(Y ) = log πen0 () (9) ad S(Y ) = E[log p(y )] (0) where the probability p(y ) is defied as p(y ) = + π () aj αj,i y bs A + j= s= i= where bs = ±, is the umber of iterferig bits (due to a short guard time), aj is the iterferece amplitude of bit j, αi,j is the bit value equal to ±. Sice there are biary iterfererspwe have that i =... Furthermore, A = SN R = Eb /N0 The, the capacity measured i bit/s/ Hz achieved i the case of BPSK is E log C(Tg ) = Tb B + π y bs A + aj αj,i () j= s= i= / log (πen0 ) where B is the chael badwidth. The Mote Carlo itegratio is eeded to compute C(Tg). The detailed calculatio of capacity is provided i Appedix A. It follows that the optimal guard time legth ca be computed as Tg = arg max C(Tg ) Tg (3) The evaluatio of the argumet i (3) is computatioally itese because it requires a Mote Carlo itegratio for the evaluatio of the expectatio for each sigle value of guard time. Therefore, it is importat to derive a simplified solutio with lower complexity. B. Capacity Optimizatio To begi our aalysis, we cosider i this sectio the capacity of the BPSK UWB system assumig a Gaussia pulse with duratio Tp = 5 s. Ideal kowledge of the chael is assumed at the receiver side. The, the capacity is evaluated assumig the model i Sectio III. I particular, a radomly picked impulse respose withi the class CM is cosidered to obtai the capacity i Fig. 04 Egieerig ad Techology Publishig 90. The capacity has a fuctio of the SNR ad the guard time. As show i Fig., for a certai SNR the capacity depeds o the guard time. The optimal guard time varies for each SNR value although this variatio is more cotaied. I Fig., we cosider a SNR equal to 0 db ad a radomly picked chael impulse respose per class. The capacity icreases as the guard time icreases up to a give value of Tg. The maximum capacity is achieved with the optimal guard time Tg. Iterestigly, the optimal guard time is differet for each chael realizatio. For the CM chael respose, the optimal guard time value is s. This value icreases to 5 s for the CM chael, to s for the CM3 chael ad to s for the CM4 chael. The covex (although ot strictly) behaviour of the capacity ca be explaied by observig that the capacity is depedet o both the guard time ad the iterferece. Whe the guard time legth icreases there is a logarithmic icrease of the capacity with the icrease of the sigal-to iterfereceplus- oise ratio which is however couterbalaced by the liear decrease with the multiplicative factor Tb. Overall, the system capacity with the CM chael model is superior to the system capacity with other chael models. This chael has lower time dispersio so that a shorter guard time is required. Fig.. Capacity vs SNR vs Tg over the CM chael model. We ow tur our attetio to the compariso betwee the covetioal system that uses a guard time loger tha the maximum chael excess delay τmax ad the system with guard time optimally adapted. Assumig the same chael resposes of Fig., we report i Table. II the capacity achieved with the two desig methods. With opti-mal adaptatio of the guard time sigificat performace improvemets are attaiable. The gai factor is equal to 3.3 i the CM chael. This gai icreases to 4.3 i the CM chael, to 5. i the CM3 chael ad to 6.6 i the CM4 chael model. As explaied, the optimal guard time depeds o the chael realizatio. Fig. 3 shows the measured CDF (cumulative distributio fuctio) of the optimal capacity accordig to () whe the guard time is adaptively chose for each chael realizatio. Fig. 4 presets the measured CDF of the optimal guard time accordig to (3). For the CM chael model, the guard time is always shorter tha 6 s.

5 This value icreases to.5 s for the CM chael model, to 4.5 s for the CM3 chael model ad to 0 s for the CM4 chael model. I practice the adaptatio requires that the receiver calculates the value of guard time for a certai chael realizatio ad the it feeds back such a iformatio to the trasmitter. The procedure is applicable i slowly time variat chaels but it is evertheless complex. I order to simplify the approach, i the ext sectios we will describe other approaches. V. REDUCED NUMBER OF GUARD TIME VALUES I order to reduce complexity, we may wat to use a small umber of guard time legths ad adapt amog these values. The idea is to precompute a sigle guard time legth accordig to the chael class we are supposed to operate i. This is possible by choosig the guard time based o the evaluatio of the CDF of the optimal guard time preseted i Fig. 4. We propose that the optimal guard time duratio does ot vary sigificatly withi the same class. Hece, we propose, for a give chael class, to choose a sigle value of guard time for all chael realizatios. The specific guard time is chose to be the value of Tg for which the CDF of the optimal guard time is 99%. We deote this value as Tg i.e., the 99th percetile of the guard time optimal Tg. Alteratively, we ca cosider the media value of the CDF of the optimal guard time. We deote it with Tg. It follows that i both cases, the guard time is selected depedig o the class we are operatig i. Although, for illustrative examples we cosider the statistical WiMedia chael model, the desig approach ca be applied i other applicatio scearios for which a statistical chael model is available. To compare the performace, we defie the relative capacity loss w.r.t. the optimal value, as follows (99%) (50%) Fig.. Capacity vs guard time, SNR=0 db. c = C(Tg ) C(T g ) C(Tg ) (4) where T g is the sub-optimal guard time value. Fig. 5 shows the measured CDF of the relative capacity loss for both guard time selectio methods proposed i this sectio. The results are obtaied i the case of CM chael realizatios. The SNR is fixed to 0 db. The maximum relative capacity loss value is 0. with Tg ad 0. with Tg. The probability of maximum capacity loss achieved with Tg is 0% ad % with Tg. The optimal guard time is obtaied i 0% of cases with Tg ad % of cases with Tg. I the other chael classes, similar results are obtaied. (50%) Fig. 3. CDF of system capacity, SNR=0 db. (99%) (50%) (99%) (50%) TABLE II: CAPACITY COMPARISON (99%) CM CM CM3 CM4 (Tg ) s 5 C(Tg ) (τmax ) s C(τmax ) Fig. 5. CDF of relative capacity loss for two guard time values over the CM chael model, SNR=0 db. VI. ALTERNATIVE METRICS AND OPTIMIZATION METHOD Fig. 4. CDF of capacity optimal guard time, SNR=0 db. 04 Egieerig ad Techology Publishig 9

6 I this sectio, we describe aother methodology to desig the guard time that is based o the cosideratio of alterative metrics. The objective is to defie a quatity that is related to the guard time through a parameter value. The proposed metrics are the RMS delay spread, the received sigal eergy ad a approximatio of capacity that leads to the eergy of the iterferece. The relatio betwee the optimal parameter βλ, the optimal guard time ad the metric L for each chael realizatio is defied as β = fl (Tg, λ) (5) where λ is the value of the metric L, ad fl(.) the fuctio that liks λ, Tg ad β. Now, the method cosists i aalysig the capacity for a certai chael model. Also, the optimal parameters lik the capacity optimal guard time with that obtaied usig the cosidered metric are calculated. As a result, the procedure allows avoidig to compute capacity. Firstly, for a certai chael realizatio the metric λ is computed. The, the guard time is obtaied directly by the parameter β ad the value of the metric as T g = fl (β, λ) (6) The optimizatio method used is summarized as follows Defie the relatio betwee the metric ad the chael characteristics Capacity calculatio usig () Optimal guard time calculatio usig (3) Parameter value calculatio usig (5) Determie oe parameter value for all chael realizatios Guard time calculatio usig (6) Performace evaluatio ad selectio of appropriate metric Calculate the capacity attaiable with the cosidered metric Calculate the relative capacity loss betwee the capacity with optimal guard time ad the capacity with the sub/optimal guard time. The procedure above described, allows us to compare the differet metrics, extract a sub/optimal guard time value, ad compare the attaiable capacity with that achievable with the optimal guard time. We ow describe i detail the cosidered metrics. A. Delay Spread The first proposed metric is the RMS delay spread, defied as sr (τ µ) Ac (τ )dτ (7) 0 R σ= A (τ )dτ c 0 ad the average delay spread is 04 Egieerig ad Techology Publishig 9 R τ Ac (τ )dτ µ = R0 Ac (τ )dτ 0 () where Ac (τ ) = h(τ ) is the chael power delay profile. The delay spread may chage from chael realizatio to chael realizatio ad depeds o the propagatio coditios. Whe Tb σ, the system experieces egligible iter-symbol iterferece. Whe Tb is withi a order of magitude of σ, there will be some iterferece which may or may ot degrade the performace. I geeral, a sigificat fractio of the received eergy is captured withi β µ with β > 0 [6]. Therefore, i this case the metric λ i (5) is the RMS delay spread. The parameter likig the guard time ad the RMS delay spread is give as Tg (9) β = σ Tp where Tg is defied i (3) ad fl i (5) becomes a ratioal fuctio. B. Received Sigal Eergy I this sectio, we defie a parameter that liks the optimal guard time with the sigal eergy that ca be collected i a certai time widow due to the sigal spread itroduced by the multi-path compoets. I detail, the parameter β is the fractio of received sigal eergy that ca be collected i a frame of duratio Tg + Tp, Z Tg +Tp h(τ ) dτ (0) β = 0Z Tc h(τ ) dτ 0 where Tc is the legth of the chael. The parameter value β is takes values i the iterval (0, ]. The metric λ i (5) is the received eergy. I Fig. 6, we show a example of multi-path chael impulse respose together with the dotted curve that represets the CDF of the received sigal eergy as a fuctio of the frame duratio for chael realizatios belogig to CM. Sigificat sigal eergy is captured (with high probability) whe the frame duratio is shorter tha the chael maximum duratio equal to 30 s. Fig. 6. A example of multi-path chael impulse respose ad the CDF of the received sigal eergy.

7 C. System Capacity Approximatio The third parameter for the guard time optimizatio is derived by usig a capacity approximatio formula. This is obtaied whe the iterferece is cosidered Gaussia as follows SIN R log ( + ) Tb B γ CI = () [0.54, 0.7] for CM4 chaels. For SNR= 5 db, the value of β is i the iterval [0., 0.96]. These edpoits of the iterval decrease to [0.79, 0.9] for CM chaels, to [0.70, 0.] for CM3 chaels ad to [0.67, 0.7] for CM4 chaels. For SNR= 0 db, the value of β is i the iterval [0.9, ]. These edpoits of the iterval decrease to [0.4, 0.97] for CM chaels, to [0.76, 0.9] for CM3 chaels ad to [0.7, 0.6] for CM4 chaels. where γ is a gap factor that takes ito accout practical implemetatio costraits. The sub-optimal guard time is obtaied as 0 SIN R T B ) b Tg = arg max ( + () Tg γ A lower boud of () is obtaied through the Beroulli iequality ( + SIN R T B SIN R ) b ( + ) γ Tb B γ (3) A practical simplified method is to use the lower boud i (3) so that the sub-optimal guard time is Tg = arg max 0 Tg SIN R γtb B (4) Fig. 7. CDF of parameter β over the CM chael model. The computatio of the sub-optimal guard time i (4) has a advatage over (3). Firstly, the computatio of the logarithm is avoided. Secodly, (4) requires oly the evaluatio of the iterferece power for differet values of guard time istead of the computatio of the capacity, as i (3). It follows that the parameter that relates the capacity optimal guard time with the sub-optimal oe herei cosidered is defied as β3 = Tg Tg0 (5) where Tg is defied by (3). Fig.. CDF of parameter β over the CM chael model. VII. APPLICATION AND NUMERICAL RESULTS Havig defied the metrics ad the parameters that relate the capacity optimal guard time with that determied with the simplified metric, we ow report umerical results assumig agai the IEEE0.5.3a chael model preseted i Sectio III. To proceed we first study the CDF of β, β ad β3 for each chael class. Based o this, we defie a sigle value of the parameter for each chael class ad for a certai SNR. The, the guard time is adapted to a give chael realizatio by usig the cosidered metric λ ad the predefied parameter β. Fig. 7 shows the measured CDF of the parameter β accordig to (9) over CM chael realizatios. The simulatios are realized for three SNR values. These values correspod to a low, a medium ad a high SNR. For SNR= 0 db, the value of β is i the iterval [0.7, 0.]. These edpoits of the iterval decrease to [0.7, 0.] for CM chaels, to [0.6, 0.73] for CM3 chaels ad to 04 Egieerig ad Techology Publishig 93 Fig. shows the measured CDF of the parameter β accordig to (0). The value of β is i the iterval [0.65, 0.3] i the case of SNR= 0 db. For other chael classes, the value of β is i the iterval [0.55, 0.7] for CM chaels, i the iterval [0.5, 0.77] for CM3 chaels ad i the iterval [0.45, 0.73] for CM4 chaels. I the case of SNR= 5 db, the value of β is i the iterval [0.6, 0.6]. These edpoits of the iterval decrease to [0.5, 0.] for CM chaels, to [0.54, 0.0] for CM3 chaels ad to [0.4, 0.76] for CM4 chaels. I the case of SNR= 0 db, the value of β is i the iterval [0.7, 0.9]. These edpoits of the iterval decrease to [0.6, 0.5] for CM chaels, to [0.5, 0.5] for CM3 chaels ad to [0.5, 0.] for CM4 chaels. Fig. 9 presets the measured CDF of the parameter β3 accordig to (4). For SNR= 0 db, the value of β3 is i the iterval [0.45, 3]. The supremum of the iterval icreases to 3. for CM chaels, to 5.3 for CM3

8 chael ad to 6. for CM4 chaels. For SNR= 5 db, the value of β3 is i the iterval [0.4, 3]. The supremum of the iterval icreases to 3.9 for CM chaels, to 5.4 for CM3 chaels ad to 6.9 for CM4 chaels. For SNR= 0 db, the value of β3 is i the iterval [0.53, 3]. The supremum of the iterval icreases to 4 for CM chaels, to 5.5 for CM3 chaels ad to 7 for CM4 chaels. TABLE V: PARAMETER VALUES, SNR = 0 DB β β β3 CM CM CM CM Fig. 0. CDF of relative capacity loss for all three metrics over CM chael model. The SNR is fixed to 0 db. Fig. 9. CDF of parameter β3 over the CM chael model. Although the parameters deped o the chael class ad o the specific impulse respose realizatio, the value of the parameter β is less variable tha the parameter value β ad β3 for the same SNR. The value of parameter β3 is almost idetical for differet values of SNR. The best performace is offered by the use of the metric related to the RMS delay spread. However to aalyse more deeply the results, the relative capacity loss will be calculated i the ext sectio to validate the proposed approach. A. Parameter Selectio ad Performace Our strategy is to adapt the guard time legth by avoidig the calculatio of the capacity formula. To simplify further the problem, we propose to use a sigle parameter β for a certai class ad SNR. The parameter is determied by takig the media value of the correspodig CDF. We have foud that the resultig system capacity is very close to that attaiable with a capacity optimal guard time. We have tabulated the chose parameter values i Table. III, Table. IV ad Table. V respectively for three SNR levels equal to = 0 db, 5 db ad 0 db. TABLE III: PARAMETER VALUES, SNR = 0 DB β β β3 CM CM CM CM I order to compare the three metrics, the relative capacity loss is reported i Fig. 0. The results are obtaied for SNR= 0 db ad the CM chael model. Fig. 0 shows the measured CDF of the relative capacity loss whe usig the three metrics associated to the parameters β. The delay spread metric ( β) ad the sigal eergy metric (β) provide less capacity loss tha the capacity approximatio metric ( β3). The maximum relative capacity loss value is 0. with β, 0.6 with β ad 0.5 with β3. The probability of maximum capacity achieved with β ad β3 is 5% ad 9% for β3. The optimal guard time is obtaied i 5% of cases with β ad β. Although ot show, with the other chael models, the maximum relative capacity loss value with β is 0.3 i the CM chaels, 0.4 i CM3 chaels ad 0.5 i CM4 chaels. For β, the maximum relative capacity loss value is 0.6 i CM chaels, 0. i CM3 chaels ad 0.9 with the CM4 chael model. For β3, the maximum relative capacity loss value is 0.5 i CM chaels, 0.6 i CM3 chaels ad 0. i CM4 chaels. Although, all three simplified metrics provide small relative capacity losses, the best approach is to use the delay spread metric with the adjustmet parameter β. Although ot show similar coclusios are reached for other SNR values. Fially, it should be oted that the use of the delay spread metric is sigificatly less complex tha determiig the capacity optimal guard time described i Sectio IV. VIII. CONCLUSION TABLE IV: PARAMETER VALUES, SNR = 5 DB β β β3 CM CM CM Egieerig ad Techology Publishig I this paper, we have examied the problem of desigig the guard time i BPSK UWB commuicatios. We have show that the use of a guard time adjusted to the curret chael coditios is beeficial i terms of CM

9 maximizig the system capacity. Ideally, this should be doe by adjustig the guard time to the specific chael impulse respose so that the capacity is maximized. However, this requires a exhaustive guard time search which is a computatioally itese task. Therefore, we have cosidered the use of a costat guard time value for all chael realizatios belogig to a certai chael class. Aother approach is to adapt the guard time to the chael realizatio but such a adaptatio is doe by resortig o a simplified metric, amely the delay spread, the sigal eergy, or a lower boud of the capacity. The resultig guard time is the adjusted by a factor that depeds o the chael class ad operatig SNR. Numerical results for typical idoor UWB chaels have show that a sigificat gai ca be achieved w.r.t. the use of a coservative guard time legth equal to the maximum chael duratio. I particular, the metric based o the delay spread, appropriately adjusted, provides the smallest loss compared to the capacity optimal delay spread. APPENDI A: CAPACITY CALCULATION The mutual iformatio I(,Y) is a fuctio of the sigal power Es ad the oise power N0. The capacity for BPSK depeds o these parameters oly through their ratio, the SNR Es/N0. To show this, we replace Y by Y/N0 to get the model (6) Y = SN R + N, N N (0, N0 ) For otatioal simplicity, set A = SN R We have p(y + ) = exp ( (Y A) /) π (7) p(y ) = exp ( (Y + A) /) π () p(y ) = p(y + ) + p(y ) (9) exp ( (Y + bs A) /) π s= (30) where bs= {, } We ca ow compute I(, Y ) = h(y ) h(y, ) (3) As i [5], we ca show that h (Y ) = h(z ) = / log (πen0 ) We ca ow compute Z (3) h(y ) = log (p(y ))p(y ) by umerical itegratio, pluggig i (). A alterative approach, which is particularly useful for more complicated costellatios ad chael models, is to use Mote Carlo itegratio (i.e., simulatio-based empirical 04 Egieerig ad Techology Publishig the the capacity (34) C = E[log p(y )] / log (πen0 ) " # C = E log exp ( (Y + bi A) /) (35) π i= / log (πen0 ) Capacity with iterferece ) Oe bit iterferece. With oly oe iterferer, we obtai p(y ) = p(y (+, a )) + p(y (+, a )) p(y (, a )) + p(y (, a )) 4 4 where a is the iterferer amplitude (Y A + a ) p(y ) = 4 π (Y A a ) (Y + A + a ) + (Y + A a ) + + the ad p(y ) = averagig) for computig the expectatio h(y )= E[log p(y )]. For this method, we geerate i.i.d. samples Yi usig the model (6), ad the use the estimate h= log p(yi ) (33) 95 (36) (37) (Y A + a ) h(y ) = E log 4 π (Y A a ) + (3) (Y + A + a ) + (Y + A a ) + ) Two bits iterferece. With two iterferers, we obtai p(y ) = p(y (+, a, a )) + p(y (+, a, a )) + p(y (+, a, a )) + p(y (+, a, a )) (39) + p(y (, a, a )) + p(y (, a, a )) + p(y (, a, a )) + p(y (, a, a )) (Y A + a + a ) p(y ) = π (Y A + a a ) + (Y A a a ) +

10 (Y A a + a ) + (Y + A + a + a ) + (Y + A + a a ) + (Y + A a + a ) + (Y + A a a ) + the αj,i (40) [] [] (Y A + a + a ) h(y ) = E log π (Y A + a a ) + (Y A a a ) + (Y A a + a ) + (Y + A + a + a ) + (Y + A + a a ) + (4) (Y + A a + a ) + (Y + A a a ) + 3) bits iterferece. More i geeral, with biary iterferers, we have that h(y ) = E log + π (4) (y b A + aj αj,i ) s j= s= i= It follows that the capacity with BPSK ad biary iterferers ca be obtaied as C = E log + π (44) [3] [4] [5] [6] [7] [] [9] [0] [] [] [3] [4] [5] [6] I. Opperma, M. Hamalaie, ad J. Iiatti, Eds., UWB Theory ad Applicatios. Joh Wiley & Sos Ltd, 004. S. G. Glisic, Advaced Wireless Networks 4G Techologies, Joh Wiley & Sos Ltd, 006. H. Nikookar ad R. Prasad, Itroductio to Ultra Widead for Wireless Commuciatios, Spriger, 009. M. Ghavami, L. B. Michael, ad R. Koho, Ultra Wide Bad: Sigal ad Systems i Commuicatio Egieerig, S. Hito, Ed. Willey, 007. M. Wi ad R. Scholtz, Impulse radio: How it works, Commuicatios Letters, IEEE, vol., o., pp. 36 3, Feb 99. L. Yag ad G. Giaakis, Ultra-widebad commuicatios: A idea whose time has come, Sigal Processig Magazie, IEEE, vol., o. 6, pp. 6 54, Nov 004. K. Witrisal, Nocoheret autocorrelatio detectio of orthogoal multicarrier UWB sigals, i Iteratioal Coferece o UltraWidebad, IEEE, vol., Sept 00, pp MAC-PHY Iterface Specificatio, WiMedia Alliace MACPHY Workig Group Std., 009. Z. Zhao-yag ad L. Li-feg, A ovel OFDM trasmissio scheme with legth-adaptive cyclic prefix, Joural of Zhejiag Uiversity Sciece, vol. 004, pp , 004. A. Toello, S. D Alessadro, ad L. Lampe, Cyclic prefix desig ad allocatio i bit-loaded OFDM over power lie commuicatio chaels, IEEE Trasactios o Commuicatios, vol. 5, o., pp , Nov 00. J. Foerster, Chael modelig sub-comittee report fial, IEEE P0.5 Workig Group for Wireless Persoal Area Networks (WPANs), Tech. Rep., Feb A. M. Toello ad R. Rialdo, Frequecy domai chael estimatio ad detectio for impulse radio systems, i WPMC, Italy, Sep 004. J. Proakis, Digital Commuicatio, Wiley, 00. A. Saleh ad R. Valezuela, A statistical model for idoor multipath propagatio, IEEE Joural o Selected Areas i Commuicatios, vol. 5, o., pp. 37, Feb 97. U. Madhow, Fudametal of Digital Commuicatio, Cambridge Uiversity Pess, 00. A. Goldsmith, Wireless Commuicatio, Staford Uiversity, 005. (43) Abdallah Hamii received M.Sc. degree i sigal, telecommuicatios, image, etworks ad multimedia from the Istitute of Galilée i Paris, Frace, i 009. I March 03, he received PhD degree i electroics ad telecommuicatio, from both the Natioal Istitute of Applied Scieces of Rees (INSA), Frace ad the Uiversity of Udie, Italy. His mai research iterests is o the area of sigal processig ad digital / log (πen0 ) where is the umber of iterferers, aj is the amplitude of the j th iterferig bit, αj,i is a biary value correspodig to the possible combiatios of iterferers with dimesio {, } ad bs = {, }. As a example for = : 04 Egieerig ad Techology Publishig = + REFERENCES aj αj,i ) (y b A + s j= s= i= 96 commuicatios.

11 Jea-Yves Baudais received the M.Sc. degree, ad PhD degree i electrical egieerig from the Natioal Istitute of Applied Scieces of Rees (INSA), Frace, i 997 ad 00 respectively. I 00, he joied the Frech Natioal Cetre for Scietific Research (CNRS), where he is ow researcher i the Istitute for Electroics ad Telecommuicatios of Rees (IETR), Digital Commuicatio Systems team (SCN). His geeral iterests lie i the areas of sigal processig ad digital commuicatios. Curret research focuses o trasmitter desig ad receiver diversity techiques for multiuser ad multicarrier commuicatio icludig space-time codig. He is curretly the head of SCN team. Adrea M. Toello is a Aggregate Professor at the Uiversity of Udie, Italy (sice 003) where he leads the Wireless ad Power Lie Commuicatio Lab. He is also the fouder ad presidet of WiTiKee, a uiversity spioff compay workig i the field of telecommuicatios for the smart grid. From 997 to 00 he has bee with Bell Labs Lucet Techologies firstly as a Member of Techical Staff ad the as a Techical Maager at the Advaced Wireless Techology Laboratory, Whippay, NJ ad the Maagig Director of the Bell Labs Italy divisio. He obtaied the Laurea degree (996, summa cum laude) ad the Doctor of Research degree i electroics ad telecommuicatios (003) from the Uiversity of Padova, Italy. I 03, Dr. Toello received the Italia Full Professor Habilitatio. He was awarded several other recogitio amog which the Lucet Bell Labs Recogitio of Excellece award 04 Egieerig ad Techology Publishig 97 (999), the Distiguished Visitig Fellowship from the Royal Academy of Egieerig, UK (00) ad the Distiguished Lecturer Award by the IEEE Vehicular Techology Society (0-3 ad 03-5). He also received (as co-author) five best paper awards. He is the Vice-chair of the IEEE Commuicatios Society Techical Committee o Power Lie Commuicatios. He serves/ed as a Associate Editor for the IEEE Trasactios o Vehicular Techology (007-03), for the IEEE Trasactios o Commuicatios (sice 0) ad IEEE Access (sice 03). He served as the geeral chair of IEEE ISPLC 0 ad he is the geeral co-chair of IEEE Smart Grid Comm 04. Jea-Fraçois Hélard received his Dipl.- Ig. ad his Ph.D i electroics ad sigal processig from the Natioal Istitute of Applied Scieces (INSA) i Rees, Frace, i 9 ad 99, respectively. From 9 to 997, he was research egieer ad the head of chael codig for the digital broadcastig research group at Frace Telecom Research Ceter i Rees. I 997, he joied INSA, where he is curretly Professor ad Director of Research of the Istitute. He is also Deputy Director of the Rees Istitute for Electroics ad Telecommuicatios (IETR), created i 00 i associatio with the CNRS. His research iterests lie i sigal processig techiques for digital commuicatios, such as space-time ad chael codig, multicarrier modulatio, as well as spread-spectrum ad multi-user commuicatios. He is ivolved i several Europea ad atioal research projects i the fields of digital video terrestrial broadcastig, mobile radio commuicatios ad cellular etworks, power-lie ad ultra-wide-bad commuicatios, cooperative commuicatios ad relayig techiques. Prof. JF. Hélard is a seior member of IEEE, author ad co-author of more tha 00 techical papers i iteratioal scietific jourals ad cofereces, ad holds 4 Europea patets.