PPM based UWB System Throughput Optimisation

Transcription

1 PPM based UWB Sysem Throughpu Opimisaion S. Zeisberg,. Müller, J. Siemes, A. Finger ommunicaions Laboraory a Dresden Universiy of Technology, 6 Dresden, Germany {zei, muellerc, siemes, finger}@ifn.e.u-dresden.de Absrac The sysem hroughpu performance of an synchronous ulra-wideband UWB communicaion sysem applying binary pulse posiion modulaion PPM is invesigaed. The opimum impulse shif is analyically derived for he AWGN channel. The informaion hroughpu of an UWB sysem in AWGN channel has been opimized by adjusing he ime overlap of he pulse shape used according o he acual signal-o-noise raio SNR. I. Inroducion Ulra-wideband echnology was firs paened for communicaions in 96 by Hoeppner []. Since hen many aemps have been repored for building up shor o medium range daa ransmission sysems applying UWB a he physical layer PHY [], [3]. Main advanages of UWB sysems are he poenial for very high daa raes a low power operaion and he possibiliy for real ime sub-decimeer posiioning [4], [5]. Recen developmens shif he echnology from miliary o civilian applicaions [6], [7], [8]. Regulaion bodies in he US and Europe are on he way o release rules for UWB sysems operaion [9], [], []. A synchronous UWB sysem applying ime spreading of bis ino chips wih orhogonal ime-hopping codes is considered here. Each chip is represened by one impulse. Before bi decision a sof inegraion of he single chip correlaion resuls is assumed. Blanking [] is no necessary due o he addiive whie Gaussian noise AWGN channel and orhogonal ime-hopping code assumpion. A correlaion receiver is assumed, where for each possibly sen symbol a correlaor exiss. If a ime overlap beween possibly ransmied symbols is inroduced in order o shoren ransmi symbol ime T S, hen a correlaion unequal o zero beween correlaor oupus is inroduced. The acual value of he correlaion depends on he symbol shape. I needs o be analyzed which loss in bi error raio resuls from ha overlap. The aim of his work is o ge he opimum value of symbol duraion T S for a maximal channel informaion hroughpu I as a funcion of a cerain impulse shape i and he signal-o-noise raio SNR. PPM, also called disjoin PPM DJPPM [3] is one of he basic modulaion schemes wih informaion coded ino ime of appearance of a cerain impulse in one ou of M possible disjoined ime slos. Maximizing he channel informaion hroughpu for DJPPM is achieved by shorening he impulse duraion τ w and increasing he symbol alphabe M [4]. Therefore he ransmi symbol ime for DJPPM can in general be given by. T S,DJPPM = τ w M An oher approach is useful in pracical cases, where he impulse widh is fixed and peak power consrains exis. This is overlapping PPM OPPM [5], ha splis he ransmi symbol ime ino M overlapping ime slos. The value of overlap in ime is considered o be τ o. Thus anoher useful variable for he ime shif beween impulses ha follow one afer he oher may be inroduced as well. This is s = τ w τ o. Summarizing he ransmi symbol ime for OPPM can be given by. T S,OPPM = s M + τ w Applying OPPM he signals a he inpu of he decision device in he opimal correlaor receiver become correlaed and he decisions will be more noise sensiive keeping he impulse energy consan. Hence, applying overlap, he bandwidh efficiency can be increased a expense of a higher sensiiviy agains AWGN. As one can expec, here should exis an opimal overlap depending on he signal-o-noise raio and he impulse shape used. Saring wih no overlap he ransmi symbol ime T S is quie large while he AWGN sensiiviy is low. The opposie exreme case is he almos oal overlap yielding minimal T S bu maximal sensiiviy agains AWGN. In beween an opimal overlap, giving he maximal channel informaion hroughpu, should exis. Deriving his opimal overlap analyically for he ypical UWB received impulse shape [6] and he special case of binary OPPM is he inenion of he following analysis subsecion. II. UWB Informaion Theory A. Basic Approach In [7] he error raio performance of PPM, MPPM and OPPM in he indoor wireless opical channel has been invesigaed for a recangular pulse shape. The error performance of M-ary modulaion was idenified assuming high SNR environmen allowing simplified analysis based on he minimum disance crierium. The general spiri of he inense This work was parly funded by he commission of he European ommuniy.

2 i a Typical received UWB single impulse chip and ime shifed replica s ρ s.6 [.54,.63] b orrelaion funcion ρ s versus ime shif s Fig.. Typical UWB received impulse and resuling correlaion funcion background ligh channel model used in [7] and may also be applied o he UWB evaluaion assuming AWGN as he only disorion of he received signal. Bu a high SNR assumpion is inapplicable as well as he impulse shape i is far more complex in he UWB case [6]. A variable may represen a ime normalizaion o keep i independen from a specific impulse duraion. However, as his pulse has infinie lengh, i mus be cu off a some poin. This poin was arbirarily chosen o be he poin where he impulse ampliude has fallen below.% of is maximum, which is approximaely a = π. i = 4π exp π If a signal described in 3 is parly overlapped wih is replica like in he OPPM case he correlaion beween wo of hose signals versus ime shif is shown in figure b. The analyical expression describes 4 [6]. ρ s + = 4π s 4π 4 s 3 exp π s 4 onsidering wo correlaors as shown in Fig.a he inpu ino he decision device will be he difference of boh correlaion pahs for he binary case wih equal symbol energy. Since hese values include he informaion as well as a noise par, he channel variance parameer σ c has o be ransformed from he inpu of he correlaor o is oupu for a laer analysis. This way he variance a he inpu of he decision device σ d becomes a funcion of he correlaion ρ s and he impulse energy E i. The full evaluaion is presened in appendix A. Fig. b shows ha he correlaor now convers binary daa ino analog signal values a cerain imes in fron of he noise source of σ d. So he discree-ime AWGN memoryless 3 pulse emplae c ou X noise in Y X n ; σ correlaor Y ime shifed pulse emplae in c ou X correlaor n ; σ D Y a receiver pah including wo correlaors b receiver pah before and afer he noise ransformaion Fig.. correlaion receiver branches and noise ransformaion

3 channel can be used furher on. The average muual informaion beween he discree inpu X = {x,...,x M } and coninues oupu Y = {, } is given [8, eq.7--8] by he capaciy of his channel in bis/channel use as [ bi sym ] M = max Px i i= py x i Px i log py x i py dy. 5 Wihin his equaion Px i represens he probabiliy of occurrence of he channel inpu symbol x i, which is defined by he informaion source. Furhermore py x i is he condiional probabiliy densiy funcion defined by he discree inpu coninuous oupu ransmission channel. B. Analysis For he channel model used wih equal energy inpu symbols he maximum of he righ hand side of 5 is achieved, if he inpu symbols are equally disribued. Thus for he binary case Px i = /. The meaning of x i in 5 is, ha i represens he expeced inpu a he decision device in case here is no disorion of he ransmied signal. In he binary case wih equal energy signals considered here his inpu is he difference of he wo correlaor oupus. E d ρ s, i = c ou x c ou x 6 This oupu depends on he correlaion beween possibly ransmied symbols which iself is a funcion of he ransmied pulse shape i and he normalized ime shif s. Inroducing E d, which describes he expeced oupu of he correlaor in he undisurbed case 6, equaion 5 becomes s = py E d py E d log dy + py py E d py E d log dy. 7 py The capaciy is a funcion of he SNR, of he ime shif s and of he pulse shape. Wih increasing SNR he capaciy increases as well. This holds for any ime shif and any impulse shape. The capaciy is expressed in bis per channel inpu symbol. In case of binary inpu i can become a maximum. To ge a crierium for he sysem efficiency his capaciy needs o be muliplied by he number of inpu symbols per ime uni. Then he normalized capaciy does incorporae he gain in ime by shorening he symbol ime compared o he orhogonal ransmission case, where he symbol ime is wice he impulse widh like in DJPPM. As he ime shif s increases he number of inpu symbols per fixed ime uni increases as well. The probabiliy densiy funcion pdf a he decision device is he sum of he wo Gaussian pdfs py E d andpy E d wih variance σ d around he expeced values a he decision device, E d and E d, respecively. By inroducing Φy g as a Gauss funcion wih he requesed properies and expeced value y m, Φy g y m = he following acual disribuions can be used wih 7: πσd exp y g y m σ d py E d =Φx +E d, py E d =Φx E d and py = Φx +E d+φx E d 9 Insering equaions 9 in 7 and rewriing he logarihmic erms as differences yields s expressed as 8 s = Φy +E d log Φy +E d dy + Φy +E d log Φy +Ed +Φy E d Φy E d log Φy E d dy dy Φy E d log Φy +Ed +Φy E d dy where he firs wo inegral erms can be easily solved analyically and resul in log πeσd.

4 / TS s τn s τn limi -6 db db 6 db db 8 db s op τn lim E i σc s numerical op= s a Throughpu versus ime shif s for various channel SNR E i σ c [db] b opimal ime shif s op versus channel E i σ c Fig. 3. Analyical and numerical resuls I is ineresing o noe, ha his parial resul represens he enropy ha was added o he signal by he addiive whie noise. The las wo inegral erms resul in log πσ d πσ d gy log gy dy wih gy = e y+e d σ d + e y E d σ d. ombining resuls and and applying some reordering yields finally 3, where he inegral erm needs o be solved numerically. s Ed = log e + log e σ e d π E d σ d e ξ cosh bξ log cosh bξ dξ 3 wih bξ = ξ E d and ξ = y. 4 σ d σd I is shown in Appendix A ha he noise a he decision device is a Gaussian process wih he variance σ d = E i σ c ρ. The par of he oupu ha depends on he ransmied symbol is shown o be E d = ρe i. As g depends only on he relaion of hese erms, and no on eiher one of hem separaely, s is a funcion of he signal o noise raio. In Fig. 3a he muual informaion normalized o a fixed ime uni versus he ime shif beween wo possible ransmi symbols is presened. This is calculaed for he impulse shape described by 3 for differen SNR. As shown in Fig. 3a here exiss a maximum in muual informaion for a cerain ime shif, which depends on he SNR. The higher he SNR, he smaller is he opimum ime shif. In Fig. 3b he opimum ime shif versus he SNR is shown for he impulse shape described by 3. I can be seen, ha wih decreasing SNR he value asympoically converges o a limi, which is.54. The limi is no equal o he minimum ime shif. This is due o he fac ha, as can be seen in Fig. b, he correlaion properies of he impulse improve only marginally as s τ s approaches he poin of minimal correlaion. A he same ime, he advanage gained by shorening he frame grows almos linear. The exac posiion of he opimum depends on he chosen lengh of one pulse. III. onclusions The opimal impulse shif posiion in case of binary overlapping PPM OPPM has been analyically derived for he AWGN channel for differen impulse shapes. I is shown ha he channel capaciy of an UWB sysem in AWGN channel can be opimized by adjusing he ime overlap of he pulse shape used according o he acual SNR. Especially in he case of very high daa rae ransmission like video in home environmen where higher SNR and small spreading is assumed his dependency should be considered in sysem design in order o opimally use he channel and hus maximize sysem informaion hroughpu.

5 IV. Acknowledgemen The auhors would like o hank all parners in he European research projec whyless.com for discussion of he subjec. References []. H. Hoeppner, Pulse communicaion sysem, U.S. Paen Nr. 9998, Sepember 96, filed 4. November 945. [] Rober J. Fonana, Recen applicaions of ulra wideband radar and communicaions sysems, o be published in Ulra-Wideband, Shor-pulse Elecromagneics, Kluwer Academic/Plenum Publishers. [3] John Mcorkle, A uorial on ulrawideband echnology, available a hp://grouper.ieee.org/groups/8/5/pub/tuorials.hml, March 6h, IEEE 8.5 uorial. [4] Paul Wihingon, Uwb applicaions, demonsraion and regulaory updae, workshop presenaion, EPT workshop on regulaory issues regarding inroducion of UWB in Europe, March, Mainz, Germany. [5] Rober Fleming and herie Kushner, Inegraed ulra-wideband localizers, in UWB onference, Washingon D.., Sepember 8 999, available a hp:// Wire/Inegraed Ulra-Wideband Localizers.pdf. [6] Timohy Jason Shepard, Decenralized hannel Managemen in Scalable Mulihop Spread-Specrum Packe Radio Neworks, Ph.D. hesis, Massachuses Insiue of Technology, 995. [7] Inc. Fanasma Neworks, High performance wireless home neworks: An ulra-wideband soluion, hp:// Whie Paper. [8] H. Luediger, S. Zeisberg, M.G. di Benedeo, and N. Blefari Melazzi, Ouline of an open radio access nework, in IEEE Proc. of European Wireless, Dresden, Germany, Sepember, IEEE, pp [9] Noice of proposed rulemaking,, F -63. [] ommission of he European ommuniy, s European Ulra Wideband Workshop, Brussels, December, hp:// [] Reg TP, German Regulaory Auhoriy for Telecommunicaions and Pos, EPT workshop on regulaory issues regarding inroducion of UWB in Europe, Meinz, March. [] S.S. Kolenchery, J.K. Townsend, J.A. Freebersyser, and G Bilbro, Performance of local power conrol in peer-o-peer impulse radio neworks wih bursy raffic, in IEEE Global Telecommunicaions onference. IEEE, 997, pp. 9 96, GLOBEOM 97. [3] Israel Bar-David and Gideon Kaplan, Informaion raes of phoon-limied overlapping pulse posiion modulaion channels, IEEE Transacions on Informaion Theory, vol. IT-3, no. 3, pp , May 984. [4] J. R. Pierce, Opical channels: pracical limis wih phoon couning, IEEE Transacions on ommunicaions, vol. OM-6, no., pp. 89 8, December 978. [5] G. M. Lee and G. W. Schroeder, Opical ppm wih muliple posiions per pulsewidh, IEEE Transacions on ommunicaions, vol. OM-5, pp , March 977. [6] Fernando Ramirez-Mireles and Rober A. Scholz, Sysem performance analysis of impulse radio modulaion, in IEEE Radio and Wireless onference. IEEE, Augus 998, pp. 67 7, RAWON 98. [7] Hyuncheon Park and John R. Barry, Modulaion analysis for wireless infrared communicaions, in IEEE Inernaional onference on ommunicaions, Seale, June 995, IEEE, pp. 8 86, I 95. [8] John G. Proakis, Digial ommunicaions, Elecrical Engineering. McGraw-Hill Inernaional Ediions, 3. ediion, 995. [9] Hors Soecker, Taschenbuch mahemaischer Formeln und moderner Verfahren, Verlag Harri Deusch, 3. ediion, 995. [] A. M. Mood, Inroducion o he Theory of Saisics, probabiliy and saisics. McGraw-Hill, 3. ediion, 974. Appendix I. Variance Transformaion In order o deermine which symbol was probably sen based on he received signal, i is common for PPM o correlae he received signal r wih emplaes. These emplaes should have he same shape as he pulse used by he sender, i. The correlaor is applied for he possible posiions of he pulse. The mos likely symbol should yield he greaes correlaion coefficien, for he binary case his resuls in he decision described in 5 T S rid x x T S ri s d 5 r consiss of he sen signal x and he channel noise n, which is assumed o be Gaussian. Inserion of 5 ino 6 yields r =x+n 6 xi d + ni d erm a erm b x x xi s d + ni s d erm a erm b 7 For he wo erms a conaining he signal, i is obvious ha one of he inegrals should represen he energy of he pulse E i, as eiher i or i s represen he signal iself. If he signals in heir possible posiions i and i s are

6 correlaed by ρ, he oher erm b should reurn he correlaion coefficien ρ of hese pulses in he respecive posiion imes he energy of he pulse. The value E d ha feeds he decision device will hus be decreased by E i ρ E d = ρe i 8 If he possible pulse posiions are correlaed, so will be he noise erms. This informaion will be los once he inegrals over he noise erms are solved in 7. I is, however, possible o calculae he variance of he overall correlaed noise which disurbs he signal. The overall noise process can be described as n D =ni + ni s 9 As shown in appendix B, he difference of wo Gaussian disribued variables wih zero mean and variance σ c can be expressed as one zero-mean Gaussian process wih variance σ difference =σ ρ. Assuming he variance of noise process n isσ, he variance of each of he wo erms in 9 can be shown o be σ b = Their correlaion is ρ bb = σ b ni nτi τ d dτ = σ c = σ c σ b δ τi i τ d dτ = σ c ni n + τ i s + τ d dτ δ τi i s + τ d dτ = E i iiτ d =E i σ c ii τ d = ρ Thus, he noise a he decision device is a gaussian process wih zero mean and variance σ d =E i σ c ρ 3 and herefore he bi error probabiliy for he binary correlaed ransmission becomes Ei E i ρ ρ P Error =Q =Q 4 Ei σ c ρ σ c II. Adding Gaussian disribued variables Le f x x andf y y be wo Gaussian disribued variables wih expecaion E[X] =E[Y ] = and variance E[X ]= E[Y ]=σ. Furher le hem be correlaed by E[X Y ] σ = ρ. According o [9], page 748, heir pdf is f xy x, y = πσ ρ exp x ρxy + y σ ρ 5 [], page 85, gives an equaion o calculae he pdf of Z = X Y : ombining 5 and 6 resuls in f xy x, y = f z z = exp f xy x, x zdx 6 x ρxx z+x z σ ρ Solving he inegral 7 and reordering erms yields z f xy x, y = exp πσ ρ σ ρ πσ ρ dx 7 omparison of equaion 8 o equaion 8 shows ha he subracion of he wo Gaussian disribued variables wih expecaion, variance σ and correlaion coefficien ρ describes again a Gaussian disribued variable wih expecaion and variance σ ρ. 8