A General Method for SER Computation of M-PAM and M-PPM UWB Systems for Indoor Multiuser Communications

Transcription

1 A Genera Method for SER Computaton of M-PAM and M-PPM UWB Systems for Indoor Mutuser Communcatons G Durs Isttuto Superore Maro Boea Corso Trento 21 I Torno, Itay E-ma: durs@smt J Romme IMST GmH Car Fredrch Gauss Str 2 D Kamp-Lntfort, Germany E-ma: romme@mstde S Benedetto CERCOM-Potecnco d Torno Corso Duca deg Aruzz 24 I Torno, Itay E-ma: enedetto@potot Astract A genera method for the evauaton of the symo error proaty (SER) of oth M-PAM and M-PPM UWB systems, n presence of mutpath channe, mutuser and strong narrowand nterference s presented Ths method s shown to e ae to ncude a the prncpa mutaccess technques proposed so far for UWB, ke Tme Hoppng and Drect Sequence A comparson etween the performance of three of these technques s aso presented, for oth dea RAKE recever and MMSE equazer, n an ndoor communcatons scenaro I INTRODUCTION The successfu depoyment of Utra Wdeand (UWB) systems for hgh speed ndoor communcatons depends strongy on the deveopment of effcent mutaccess and moduaton technques and of ow compexty recevers, roust aganst narrowand and mutuser nterference As far as the frst ssue s concerned, severa proposas for UWB ar nterface are avaae n the terature, the prncpa ones eng Tme Hoppng (TH) [1], Drect Sequence (DS) [2] and Optca Orthogona Codes [3] A of them can e comned wth oth M-ary puse amptude and puse poston moduaton (M-PAM and M-PPM respectvey) In ths paper we present a genera method for the evauaton of the symo error proaty (SER) for M-PAM and M-PPM UWB systems The SER s evauated for oth dea RAKE recevers [4] and MMSE equazer [2], n the presence of dense mutpath [5] and strong narrowand nterference The paper s organzed as foows In Secton II and III we refy descre the genera system mode and ts tmedscrete verson In Secton IV the SER s anaytcay otaned for M-PAM systems, oth for MMSE and RAKE recevers, whe, n Secton V, the same approach eads to a smpe upper ound on the SER of M-PPM ones Fnay, n Secton VI, a performance comparson etween DS-2PAM, OOC-2PAM, TH-2PAM and TH-2PPM s presented, together wth some concudng remarks aout mutaccess codes desgn, gven the reference scenaro descred n Secton V Ths work has een partay sponsored y MIUR (Itaan Mnstry of Educaton and Research) under the projects CERCOM and PRIMO and y the European Unon under project numer IST whyesscom II SYSTEM MODEL In ths secton the prncpa characterstcs of the system w e refy descred The tme axs s dvded nto symo ntervas of T s seconds, each one further sudvded nto smaer ntervas, caed chps, of duraton T c seconds The sgnature sgna assgned to each user s a perodc sgna, wth perod equa to T For smpcty, we assume that T = N T s and T s = N c T c, wth N,N c N The perod T s onger than the symo tme, whenever the mutaccess code spans more than a snge symo The sgna transmtted y the user k can then e descred as foows: s (k) (t) = p (k) (t T) (1) p (k) (t) = N 1 j=0 N c 1 =0 ( ) g t T c jt s ;a (k) N +j and a (k) s the th symo, transmtted y user k Wth the notaton we denote the vaue assumed y the spreadng code assgned to the user k n the chp of the j symo nterva, n each perod T In ths anayss, we w assume {0, ±1}; n ths way a the prncpa mutuser technques proposed so far for UWB (TH, DS, OOC) are ncuded Note that the mutpe access code { } can e aso the comnaton of a spreadng sequence and a repetton code of ength N s The code assocated to the symo j transmtted y the user k can e wrtten n vector form as j = (2) [ j,0,c(k) j,1,,c(k) j,n c 1] T (3) The sgna g(t; a) represents the random process at the output of the moduator, whose expresson depends on the moduaton format Defnng wth N the numer of puses x(t), wth tme duraton T x T c, transmtted n one symo nterva, we w have, for M-PAM g (t;a) = E x ax(t) (4)

2 E x s the energy per puse and a A PAM = {2m 1 M} M m=1 The transmtted symos are assumed to e ndependent and equproae For M-PPM g (t;a) = E x x(t τ a ) (5) Ths tme a A PPM = {0,1,,M 1} and τ a s the assocated PPM deay We w further assume that T x + max a A P P M τ a T c (6) It s worth notng that the parameter N s a characterstc of the mutpe access and codng strategy, and determnes the reaton etween E x and E, the energy per t Assumng that N u users are actve, the receved sgna can e wrtten as N u r(t) = Ak s (k) (t) h (k) (t) + A n (t) + n(t), (7) k=1 n(t) s a whte Gaussan nose process wth two-sded power spectra densty N 0 /2 and n (t) s the narrowand nterference Furthermore, A k and A represent the attenuatons due to path oss, whch are a functon of the transmtter recever (TX-RX) dstance Fnay, h (k) (t) s the tme-nvarant, asynchronous mutpath channe mpuse response for user k Each asynchronous channe mpuse response h (k) (t) s assumed to have a maxmum deay of t (k) max seconds In the rest of the paper we w denote y q (k) (t) the convouton of the transmtted puse x(t) wth h (k) (t) III DISCRETE-TIME EQUIVALENT MODEL In order to anaytcay evauate the SER, we w construct a dscrete-tme matrx equvaent mode, otaned y sampng r(t) every T r seconds, wth T r T c The recever conssts of a dgta fter, actve on an oservaton wndow T w T s The foowng parameters need aso to e defned: N w = T w /T r, the numer of sampes n the oservaton wndow T w, on whch the recever operates N r = T c /T r, the numer of sampes per chp N = (max k t (k) max)/t c + 1, the mnmum numer of chp ntervas n whch q (k) (t) s contaned, k L = N /N c, the mnmum numer of symo ntervas n whch q (k) (t) s contaned, k N r = T w /T c, the mnmum numer of symo ntervas n whch the oservaton wndow s contaned L r = N r /N c 1, the mnmum numer of symo ntervas n whch the porton of the oservaton wndow that exceeds a t nterva s contaned Note that the notaton x ndcates the nearest nteger arger than or equa to x For the SER computaton, the system under anayss s dentca to the superposton of N susystems, transmttng n a round ron fashon on successve sots of duraton T The presence of mutpath, however, causes the receved sgnas to ose ther mutua orthogonaty, wth the consequence of ntersymo nterference n the overa system The error proaty s otaned averagng on the performance of each susystem Ths task s performed f the error proaty s computed averagng on the SER of N successve symos Wthout oss of generaty, we assume that user 1 s the reference user and that the symo a (1) n s transmtted, wth n [0,,N 1] The vector contanng the dscrete sampes of the receved sgna w e r n = [r(nt s ),r(t r +nt s ),,r((n w 1)T r +nt s )] T (8) Wth the same notaton, we aso ntroduce a Gaussan nose and narrowand nterference vector n n = [n(nt s ),n(t r +nt s ),,n((n w 1)T r +nt s )] T, (9) n,n = [n (nt s ),n (T r +nt s ),,n ((N w 1)T r +nt s )] T (10) Moreover, t s aso necessary to defne the vectors of the transmtted symos a k,n = [a (k) L +n,,a(k) n,,a (k) L r+n ]T, (11) and the spreadng ock dagona matrces S k,n L +n S k,n = L +n L r+n (12) = ( mod N ) (13) and 0 s a N c szed zero vector Fnay, the channe matrces Q k R Nw,(L +L r+1)n c w e ntroduced as foows: Q k = q (k) (L N ct c) q (k) ((L N c 1)T c) q (k) (L N ct c+t r) q (k) ((L N c 1)T c+t r) q (k) (L N ct c+(n r 1)T r) q (k) ((L N c 1)T c+(n r 1)T r) 0 q (k) (L N ct c) 0 q (k) (L N ct c+t r) q (k) (0) q (k) (T r) q (k) ((N r 1)T r) q (k) (T c) q (k) (0) 0 0 q (k) (T c+t r) q (k) (T r) 0 0 IV M-PAM Usng the prevous defntons, k=1 (14) N u r n = A k E x (k) Q k S k,n a k,n + A n,n + n n (15)

3 The dgta recever can e fuy characterzed y ts coeffcents vector Rewrtng S k,n as w n = [w 0,n,w 1,n,,w Nw 1,n] T (16) S k,n = [ ] s (k) L +n,,s(k) n,,s (k) L r+n (17) and denotng y y n the decson varae for the symo a (1) n, we otan the foowng resut: y n = w T n r n = = A 1 E x (1) P n a (1) n + n MI,n + n,n + n G,n, (18) P n = w T n Q 1 s (1) n, (19) N u L r+n n MI,n = A k E x (k) w T n Q k s (k) a (k), (20) k=1 = L +n (k,) (1,n) n,n = A w n T n,n, (21) n G,n = w n T n n (22) In order to derve a smpe cosed-form for the SER, we w further mode the ntersymo-mutuser nterference term, n MI, and the narrowand ones, n, as ndependent, ergodc, zero mean, Gaussan random process The vadty of ths assumpton w e dscussed ater We aso assume that n (t) has a power spectra densty S n (f) wth the foowng characterstcs: S n (f) = N 2, f c B 2 f f c + B 2 0, otherwse (23) f c and B are the centra frequency and the andwdth of the narrowand nterference, respectvey The fter w n concdes wth an dea RAKE recever, f w n = Q 1 s (1) n (24) and T w = T s + t (1) max On the contrary, settng the oservaton wndow to a vaue T w T s (n our anayss we w assume T w = 2T s ), the dgta fter s an MMSE recever f { w n = argmn E z R Nw 2 x a (1) n z T r n 2 } (25) In oth cases, usng standard technques [6], the SER can e wrtten as foows: {( ) P(e) = M 1 N 1 3og erfc 2 M N M N(M 2 1) n=0 } P n ( σ 2 MI,n + σ2,n + σ2 G,n ), (26) the energy per t E (1) s gven y: E (1) = E (1) x N M2 1 3og 2 M (27) and the terms σ 2 MI,n, σ2,n and σ2 G,n are the varance of n MI, n and n G, respectvey V M-PPM As stated n [1], an M-PPM sgna can e vewed as the sum of M near moduators, fed y a non near transformaton of the nformaton symos a (k) Let us denote y x a (t) = x(t τ a ), for a A PPM the waveforms transmtted y each moduator, and wth q a (k) (t) the convouton etween these waveforms and the channe mpuse response assocated to user k Wth a notaton smar to equaton (15), the receved vector can e wrtten as r n = N u a=0 k=1 A k E x (k) Q a ks k,n β a (a k,n ) + A n,n + n n (28) Q a k s the channe matrx, contanng the sampes of q a (k) (t), ke n (14) The non near transformaton β a ( ) operates over each eement of the vector a k,n n the foowng way: { β a (a (k) 1 f a (k) = a ) = (29) 0 otherwse At the recever, the sgna s passed through a ank of M parae dgta fters w n,, = 0,,M 1, whose outputs yed the decson vector ˆβ n = [ˆβ 0 (a (1) n ), ˆβ 1 (a (1) n ),, ˆβ (a (1) n )] T (30) The output of the th dgta fter, ˆβ (a (1) n ) can e evauated as n MI,n, = ˆβ (a (1) n ) = x a=0 P a n,β a (a (1) n )+ + n MI,n, + n,n, + n G,n, (31) Pn, a = w T n, Q a 1s (1) n, (32) N u L r+n A k E x (k) w T n, Q a ks (k) β a (a (k) ), a=0 k=1 = L +n (k,) (1,n) (33) n,n, = A w n, T n,n, (34) Fnay, the estmated symo â (1) n to the foowng decson rue: â (1) n n G,n, = w n, T n n (35) w e seected accordng = max A P P M ˆβ (a (1) n ) (36) Note that each dgta fter s an dea RAKE recever f w n, = Q 1s (1) n, (37)

4 or an MMSE equazer, f { w n, = argmn E z R Nw x β (a (1) n ) z T r n 2 } (38) In ths paper we present a ound on the SER of the system when M-PPM s empoyed, ased on the unon ound Ths approach, n fact, eads to a smpe cosed-form upper ound for the proaty of error aso when the transmtted sgna are not orthogona (overappng PPM technque) Gven two symos a and a, wth a a and a,a A PPM and defnng n n,a,a = n MI,n,a n MI,n,a + n,n,a n,n,a + then the SER s gven y P(e) = 1 2MN + n G,n,a n G,n,a, (39) N 1 a=0 a =0 n=0 a a erfc og2 M (Pn,a a Pn,a a ), (40) N 2σn,a,a 2 E (1) = NE (1) x /og 2 M s the energy per t A Transmtters postons VI REFERENCE SCENARIO We consder a propagaton envronment demted y a crcumference of 10 m radus, wth the recever n the center A the actve users are nsde ths area, wth a dstance of at east one meter from the recever The poston of a transmtters s randomy chosen, assumng a unform dstruton over the surface demted y the 1 and 10 m radus crcumferences Both LOS and NLOS cases are consdered B Channe mode In order to compare the performance of the prevousy descred mutpe access schemes, an adequate ndoor UWB channe shoud e ntroduced In ths paper, we w empoy the mode proposed y the IEEE a workng group [5], whch s ased on a modfcaton of the Saeh-Vaenzuea channe descrpton [7] Ths mode takes nto account the custerng phenomena oserved n severa UWB channe measurements [8] Accordng to [5], the channe mpuse response can e modeed as h (k) (t) = L H =0 h=0 α (k) (k),hδ(t T τ (k),h τ(k) a ), (41) α (k),h are the mutpath gan coeffcents, T (k) and represent the deay of the th custer and of the k th τ (k),h mutpath ray reatve to the th custer arrva tme The dstruton of custers and rays nterarrva tme s exponenta The average power deay profe shows a doue exponenta decay (for custer average power and for rays average power n each custer), and the fadng statstcs s ognorma Fnay, the sgn of each mutpath repca s ether postve or negatve, wth the same proaty In our anayss we ntroduce another random varae τ a (k), modeng the deay due to the asynchronsm etween users In partcuar, τ a (k) s assumed to e unformy dstruted over the nterva T In [5] four sets of parameters are gven, to characterze the statstca propertes of dfferent channes In partcuar, the foowng propagaton condtons are consdered: 1) LOS channe wth a TX-RX dstance etween 0 and 4 m 2) NLOS channe wth a TX-RX dstance etween 0 and 4 m 3) NLOS channe wth a TX-RX dstance etween 4 and 10 m 4) Extreme NLOS channe (RMS deay spread of 25 ns) In our anayss, a randomy generated channe w e assgned to each user accordng to the foowng rue: f the TX-RX dstance s ess than 4 m, then a channe mpuse response of type 1 or 2 (wth the same proaty) s consdered, otherwse one of type 3 or 4 Fnay, the path oss attenuatons {A k } and {A } are assumed proportona to d γ, d s the TX-RX dstance The parameter γ s set equa to 2 for LOS channe, 35 for NLOS ones C Narrowand nterference As narrowand nterferer we consder an IEEE 80211a system, a posse compettor for WPAN appcaton As shown n [2], ths sgna can e approxmated y a Gaussan narrowand process The centra frequency and the andwdth of the nterferer w then e set to 5 GHz and 200 MHz, respectvey Assumng that the UWB system, whch has a andwdth of approxmatey 3 GHz, operates at the mts set y the Federa Communcatons Commsson (FCC) of 41 dbm per MHz [9] and that the narrowand nterferer transmtted power s 100 mw, we otan a sgna to nterference rato of 26 db, gven that the two transmtters experence the same attenuaton [2] VII RESULTS AND CONCLUSIONS In ths secton we present some numerca resuts otaned empoyng the method descred n the prevous sectons The t error rate (BER) of three UWB mutaccess schemes, ased on TH, DS and OOC spreadng codes are compared For TH system oth 2PAM and 2PPM moduaton formats are consdered; t s worth pontng out that equaton (40) gves the exact BER f the moduaton scheme s nary In partcuar, we anayze two dfferent scenaros, n whch the t rate per user (PAM format) s around 200 Mt/s (T c = 07 ns, N c = 7) and 45 Mt/s (T c = 07 ns, N c = 31), respectvey; for PPM scheme the vaues are around 160 Mt/s (T c = 09 ns, N c = 7) and 35 Mt/s (T c = 09 ns, N c = 31) The decrease n t rate s caused y the ncrease n the chp ength, necessary for the overappng PPM moduaton format The mutaccess sequences are God codes for DS system and sequences ased on quadratc congruence [10] for TH The

5 OOC codes are desgned such that two puses are transmtted over each symo tme P(e) DS PAM RAKE OOC PAM RAKE TH PAM RAKE TH PPM RAKE DS PAM MMSE OOC PAM MMSE TH PAM MMSE TH PPM MMSE E /N 0 (db) Fg 1 Proaty of error of the anayzed systems Hgher t rate (N c = 7) P(e) DS PAM RAKE OOC PAM RAKE TH PAM RAKE TH PPM RAKE DS PAM MMSE OOC PAM MMSE TH PAM MMSE TH PPM MMSE E /N (db) 0 Fg 2 Proaty of error of the anayzed systems Lower t rate (N c = 31) In Fg 1 and 2 the BER pots are depcted for a the anaysed systems, for the hgher and ower t rate condton, respectvey The curves are otaned averagng over 1000 dfferent scenaros; n a of them, the mutpath channe assocated to the reference user s assumed to e a LOS one The RAKE recever shows hgh error foor for a the mutuser and moduaton schemes, due to ts ack of roustness aganst strong narrowand nterference and near-far effects On the contrary, the MMSE recever offers, as expected, much etter performance, at the cost of hgher computatona compexty The Gaussan approxmaton was successfuy adopted to derve these curves; n fact the domnant nose n ths stuaton s the narrowand nterference, that was modeed as a coored Gaussan random process Some nterestng consderatons can e derved y the comparson of Fg 1 and 2 Increasng the numer of chp per frame (wth a reducton of the t rate) eads to an mprovement n the crosscorreaton and autocorreaton propertes of the mutaccess codes and therefore to a reducton of the mutuser nterference Ths fact justfes the average 1 db gan n performance of the MMSE recevers, n the ower t rate case For the RAKE recever, however, the hgh foor s unquey determned y the effect of the strong narrowand nterference, so that a mtgaton of the mutuser nterference does not ead to an mprovement n performance The dfference etween the poston of the BER foors for the dfferent schemes at oth rates are strongy nfuenced y the shapng effect of the mutuser code on the power spectrum of the transmtted sgna For exampe, the etter performance of OOC scheme wth RAKE recever n Fg 1 can e justfed y the spectra anayss, notng that the code assgned to the reference user ntroduces a spectra attenuaton n the vcnty of the centra frequency of the narrowand nterference Furthermore, when the code does not ntroduce any shapng (ke n TH-PAM wthout repetton code, as t can e easy verfed) the foor poston s ndependent from the vaue of N c Ths consderaton suggests that an effectve strategy to optmze the performance of UWB systems wth RAKE recepton, whenever the narrowand nterference s the mtng factor, coud e ased on the desgn of spreadng codes wth desred spectra characterstcs, rather than on the optmzaton of ther auto and crosscorreaton propertes REFERENCES [1] C J L Martret and G B Gannaks, A dgta mpuse rado wth mutuser detecton for wreess ceuar systems, IEEE Trans Commun, vo 50, pp , Sept 2002 [2] Q L and L A Rusch, Mutuser detecton for ds-cdma uw n the home envronment, IEEE J Seect Areas Commun, vo 20, pp , Dec 2002 [3] G Durs and SBenedetto, Performance evauaton and comparson of dfferen moduaton schemes for uw mutaccess systems, n Proc Int Conf Comm ICC, vo 3, Anchorage, USA, 2003, pp [4] D Casso, M Wn, F Vaataro, and A F Mosh, Performance of ow-compexty rake recepton n a reastc uw channe, n Proc Int Conf Comm ICC, vo 2, New York, USA, 2002, pp [5] J Foerster, Channe modeng su-commttee report fna, IEEE P /490r1 SG3a, Fe 2002 [6] S Benedetto and E Bger, Prncpe of Dgta Transmsson wth Wreess Appcatons New York, USA: Kuwer Academc/ Penum Pushers, 1999 [7] A Saeh and R Vaenzea, A statstca mode for ndoor mutpath propagaton, IEEE J Seect Areas Commun, vo 5, pp , Fe 1987 [8] J Kunsch and J Pamp, Measurement resuts and modeng aspects for the uw rado channe, n IEEE Conference on Utra Wdeand Systems and Technooges 2002, Dgest of Papers, Batmore, USA, 2002, pp [9] Revson of Part 15 of the Commsson s Rues Regardng Utra- Wdeand Transmsson, Federa Communcatons Commsson, 1st Rep and Order, 2002 [10] T Erseghe, Utra wde and puse communcatons, PhD dssertaton, Unversty of Padova, 2002