On the Performance of Transmitted-Reference Impulse Radio

Transcription

1 MITSUBISHI LCTRIC RSARCH LABORATORIS On the erformance of Transmitted-Reference Implse Radio Sinan Gezici, Fredri Tfvesson, Andreas Molisch TR November 2004 Abstract We consider a time-hopping implse-radio system that ses transmitted-reference plses for implicit channel estimation and eqalization A hybrid receiver strctre first performs a filtering matched to the hopping seqence, and a sbseqent correlation of the data plses with the reference plses We analyze the performance of sch a system both in AWGN and in mltipath For the AWGN case, we give exact expressions for the bit error probability that tae into accont the non-gassian natre of the noise-noise crossterms arising in the correlators For the mltipath case, we analyze inter-frame interference, as well as mltipath interference from the reference plse to the data plse, providing closed-form eqations in the limit of a large nmber of mltipath components Globecom 2004 This wor may not be copied or reprodced in whole or in part for any commercial prpose ermission to copy in whole or in part withot payment of fee is granted for nonprofit edcational and research prposes provided that all sch whole or partial copies inclde the following: a notice that sch copying is by permission of Mitsbishi lectric Research Laboratories, Inc; an acnowledgment of the athors and individal contribtions to the wor; and all applicable portions of the copyright notice Copying, reprodction, or repblishing for any other prpose shall reqire a license with payment of fee to Mitsbishi lectric Research Laboratories, Inc All rights reserved Copyright c Mitsbishi lectric Research Laboratories, Inc, Broadway, Cambridge, Massachsetts 0219

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3 1 On the erformance of Transmitted-Reference Implse Radio Sinan Gezici, Stdent Member, I, Fredri Tfvesson, Member, I, and Andreas F Molisch, Senior Member, I Dept of lectrical ngineering Dept of lectroscience Mitsbishi lectric Research Labs rinceton University Lnd University 201 Broadway rinceton, NJ 08544, USA Box 118, S Lnd, Sweden Cambridge, MA 0219, USA mail: sgezici@princetoned mail: fredritfvesson@eslthse mail: andreasmolisch@ieeeorg Abstract We consider a time-hopping implse-radio system that ses transmitted-reference plses for implicit channel estimation and eqalization A hybrid receiver strctre first performs a filtering matched to the hopping seqence, and a sbseqent correlation of the data plses with the reference plses We analyze the performance of sch a system both in AWGN and in mltipath For the AWGN case, we give exact expressions for the bit error probability that tae into accont the non-gassian natre of the noise-noise crossterms arising in the correlators For the mltipath case, we analyze inter-frame interference, as well as mltipath interference from the reference plse to the data plse, providing closed-form eqations in the limit of a large nmber of mltipath components I INTRODUCTION In recent years, ltrawideband (UWB) commnications, the signal occpies more than 20 relative bandwidth, or more than 500 MHz absolte bandwidth, has raised enormos interest in the academic, indstrial, and military commnity Time-hopping implse radio (TH-IR) is a UWB modlation and mltiple-access scheme that is especially well sited for low-data-rate commnications, and has been intensively stdied since the pioneering wor of Win and Scholtz [1], [2] TH-IR is also a leading candidate for the I a standard for low-data-rate commnications In TH-IR, each symbol is represented by a seqence of short plses ach symbol dration is sbdivided into a nmber of frames, each frame carries one plse The position of the plse within the frame is determined by a psedorandom seqence, and different sers se different seqences Ths, even for nsynchronized sers, at most one plse per symbol can collide (arrive at the receiver simltaneosly) The timehopping ths garantees the mltiple-access performance The se of short plses within each frame provides the spectral spreading, reslting in the high bandwidth that is characteristic for UWB systems The short plses also reslt in a very high delay resoltion This is helpfl for the redction of fading when optimm receiver strctres (all-rae receivers) are sed [] However, it does lead to a redction of the received total energy when sboptimm receiver strctres, lie partial Rae receivers, 4 also at Lnd University, Sweden are sed [4], [5] For this reason, transmitted-reference (TR) schemes have become poplar [5]-[8] In a TR scheme, two transmitted plses are sed in each frame The first plse is not modlated (ie, does not carry information abot the data) and is called the reference plse The second plse, which is modlated, is separated by a time delay from the first plse, and is called the data plse The receiver ses plsepair correlators to recover the data, ths performing channel estimation and despreading in one simple step ach mltipath component reslts in a pea at the otpt of the mltiplier with the same phase (which is determined by the vale of the data symbol), and therefore they can be smmed by an integrator over a certain period The integrator otpt is detected in a conventional way to mae a decision on the transmitted data symbols A major drawbac of the TR scheme is the excess noise related to the mltiplication of noise contribtions in the received reference plses with the noise contribtions in the received data plses In a recent paper [9], the athors have sggested a new transceiver strctre that redces the noisenoise cross terms, and have given an approximate analysis of its performance In this paper, we perform a more indepth mathematical analysis of this strctre; however we note that the performance of conventional TR receivers can be obtained from or analysis as a special case Specifically, we analyze the impact of the non-gassian natre of the noisenoise cross terms, as well as the inter-frame interference (IFI) and the reference-plse-on-data-plse interference in mltipath environments None of those aspects have, to or nowledge, been treated in previos papers on UWB TR schemes The remainder of the paper is organized the following way: Section II describes the system setp and establishes the notation Section III analyzes the bit error probability (B) performance over AWGN channels Sbseqently, the performance in delay-dispersive mltipath channels is analyzed in Section IV, and the theoretical reslts are compared with simlation reslts A smmary and conclsions wraps p the paper II SYSTM MODL The transmit signal ses TH-IR as mltiple access format, and TR BSK as modlation format The transmit signal can

4 h _ 4o w, { X n 2 MF dt ( )* Decision Fig 1 Bilding blocs for the basic hybrid receiver Note that the sampling circit performs symbol rate sampling (')+*, $-/ -&0 21 "$ 54(687:9<;>= )+*, -/ & -& :?:@ (1) ths be written as 21 2 denotes the chip dration, is the frame dration, and is the delay between the reference plse and the data plse We assme 0 A that CB 21 B, is a positive integer The denote a (psedo-)random integer seqence with vales between D 1 and 1 -F, which determines the timehopping seqence, with being the nmber of chips per frame The denote a psedorandom seqence of G -HI@ J that ensres a zero-mean otpt and is also helpfl in the shaping of the transmit spectrm [10] according to the FCC rles [11] The fnction )K*, denotes the transmit waveform; in the following, wel assme that its spport extends only over one chip dration is the energy per transmitted symbol Note that is the symbol dration Let M denote the received signal The first step at the receiver, shown in Figre 1, is to pass M throgh a matched filter matched to the following template signal for the N th information bit: O Q *SRUTWV W X O Ÿ [Q 9<; " 9<; \)K]", $-/ -&0 21 ^@ (2) )K]", denotes the received UWB plse Then, the otpt of the matched filter can be expressed as M a` M bc O *SRUTWV Q bd-e bgf () III TRANSMISSION OVR AWGN CHANNLS When in (1) is transmitted over an AWGN channel, the received signal M can be expressed as: M i j')k]", $-/ -&0 21 "$ 54(687:9<;>= )K]", $-/ & -& :? lcmjn ^@ (4) n is the channel attenation, and is a zero mean white Gassian process with nit spectral density Depending n on whether we consider a baseband or bandpass filter, is a real or complex Gassian process, respectively The estimator in the AWGN case can be expressed as signg D 2 _ 8J(@ (5) M is the otpt of the matched filter as shown in () The following lemma expresses the probability distribtion of the decision variable nder some conditions Lemma 1: Assme that the TH seqence is constrained to the set T -ej, T 1 -/Bq-e, with Br 2 psp 21 being an integer Then, the decision variable can be expressed as M _ D 2 W 4 Stv (6) t xw )K]", bc )K]", by-e b, and 4 the conditional distribtion of given the information bit is given by z 9 n{4 W } m t D ` {~ 0 4 F > ƒ d - n -&0 ~ m t D F ~L 0 4 5~ 8 $ (7) with 0Lˆ Š ŒŽ t D roof: See [12] Assming eqiprobable information bits, the B can be calclated from (6) as R D f `& D f ` z 9a n - D f ḧ t D {4 p z 9a n D f ḧ t D {4 -Hp n f (8) Then, sing (7), the following B expression can be obtained after some maniplation: R X } t D m ` * š 7 Ũš š: š œi jž Ÿ: \ 0 ~L D f ḧ t D t D m {~L 0 {< (9) W To confirm and" illstrate the reslts, we performed Monte Carlo simlations of a TR scheme in an AWGN channel Figre 2 shows the reslts of those simlations and compares them to the evalations of (9), a good agreement between the theory and simlations is observed Also the performance of the optimm receiver is shown in the plot for comparison prposes IV TRANSMISSION OVR FRQUNCY-SLCTIV CHANNLS A Channel Model We consider the following channel model W«ª $-& 21 ^@ (10) ± is the nmber of mltipaths and is the fading coefficient of the th path with ² ª ³ The following assmptions are made in order to facilitate the theoretical analysis:

5 ª ª Ë Ë Ï ` 4o m Ë m b BR 10 0 Hybrid RAK Theory s/no (db) Fig 2 B as a fnction of the signal-to-noise ratio for optimm (Rae) receiver and the hybrid receiver Solid crves are the Monte Carlo simlations, and the crves with circle mars are the theoretical reslts; ie, evalation of (9) The Monte Carlo simlations and the theoretical crves completely overlap There is a gard interval between the symbols so that no inter-symbol interference 0 (ISI) exists The TH seqence in (1) is constrained to the set T -µj, T 1 - -µ, with being a positive integer determining the integration interval as will be defined later in this section B General Theory From (1) and (10), the received signal can be expressed as Using M in (11) and the template signal in (2), the M in () can be expressed as W d ¹ cm X n2º ^@ (12) despread signal _ and with M j')k]", $-/ -&0 21 -& 21 "$ 54(687:9<;>= )K]", $-/ -& & 21 :? lcmjn ^f (11) ¹ +ˆ O Ÿ [Q 9<; O Ÿ [Q 9<; "» " 9<; "»¼ 9<; G t ' (½g-/j S0 "» -&0 21 -& 21? 54 ¹t ' (½g-/j S0 "» -& & 21?¹J +ˆ n2º ¾ˆ x` O Ÿ [Q 9<;p " 9<; bd-e n bc (1) bg@ (14) l )K]", -/ -&0 21 ^f (15) The decision is given by the following: sign À ` }Á Y2 }à }Á }à - 2 }ÄÅ@ (16) is the integer that determines the integration interval, and the decision variable can be expressed (17) }Á Y2 }à }Á }à $- 2 aæ ¹ Æ ¹ is the signal part, is the noise-noise term and is the signal-noise term Æ The signal part ¹ can be expressed as }Á Æ ¹ ` Y2 }à }Á }à ¹ ¹ - 2 ¹ is as in (1) The noise-noise term in (17) is given by }Á ` Y2 }à n2º }Á }à n2º $- 2 ^@ (18) ^@ (19) the distribtion of which can be approximated as shown in the following lemma: Lemma 41: As - ÇÉÈ siê, is asymptotically normally distribted as Ì ` Í bg@ o b+ˆ a` ` roof: See [12] }Á Y Í }à }Á Y$O Í [Q }à The signal-noise term cm }Á ` Y2 }à }Á }à ' Í bg@ b o Í bg@ b[ o Í Ÿ bg@ b:? o bd-e o bd-e 2 in (17) is expressed as ' ¹ n2º - 2 În2º ¹ $- 2 :? o (20) ^f (21) ^f (22) n Using (22), (14), and the fact that is a white Gassian process, we obtain the distribtion of as Ï m ÅÐ ` ' ¹ b Ñ 2 ¹ b Ñ D :? ¹ b Ñ A@Òg+ˆ x` }Á Y2 }à }Á }à ¹ $-e bd-e Òg Now consider the total noise shown from (19) and (22) that and (2) ^f (24) It can be are ncorrelated Hence, the approximate distribtion of the total noise can be obtained from (20) and (2) as Ì m p m 8h (25) Ð

6 w Æ Í Í Í Í b f ä š f ˆ x` ` ' Í bg@ b o Í bg@ b[ o Í Ÿ bg@ b:? o o (26) 4 ¾ˆ x` ' ¹ b Ñ 2 ¹ b Ñ D :? bgf (27) Æ Since the decision variable in (17) is eqal to ¹ with Æ ¹ given by (18), the B is obtained as R D f ÔÓÕ D f "h w }Á Y2 }à }Á }à Ÿ Ÿ $- 2 mì p D f m [p}ö D f - ÔÓÕ D f 8h w }Á Y2 }à }Á }à $- 2 mì p D f m [-HppÖ C Special Case: No Inter-frame Interference (28) If the frames are spaced sfficiently apart, there occrs no IFI However, there can still be interference from the reference plses to the data plses, as those are typically closely spaced together Assme that T Ø 1 -eb³-åùyú ± J Then, ¹ in ¹ W ª 't $-& ¹t -Û¹B ¹ 21 :?:f (29) Æ ¹ ' ¹Æ Æ 54 ¹Æ Ð Æ Ì :?:@ (0) (1) can be expressed as follows: Hence, the signal part is given by Æ Æ,, Æ Ð Æ and Ì can be shown to be given by Æ 'Ü Í Í Í Y2Ý Í Y2Ý 5Þ Í Y2Ý Í Í Y2Ý :?:@ (1) 'Ü Í Í Í Ý Í Ý 5Þ Í Í Ý Í Ý :?:@ (2) Æ Ð 'Ü Í Í Y2Ý Í Ý Í Y2Ý Í Ý 5Þ Í Y2Ý Í Ý Í Y2Ý Í Ý :?:@ () Æ Ì 'Ü Í Í Í Þ Í Í :?:@ (4) Ü ßw }à }à t Éßw }à t Þ and t t 21 Note that D Fà for ± -Š or á D Æ From (1)-(4) Æ Æ and Ð Æ Ì can be expressed as Æ Æ ' Í Í Y2Ý Í Ý > Ü Í 5Þ Í Í Y2Ý Í Ý > Ü Í 5Þ Í :?:@ (5) Æ Ð Æ Ì 'Ü Í Í Í Í Y2Ý Í Ý Í Y2Ý Í Ý 5Þ Í Í Í Y2Ý Í Ý Í Y2Ý Í Ý :?:f 4 (6) Brà Note that when ± à, the signal part in (18) can be expressed as Æ ¹ 4 'Ü Í Í Í Þ Í Í :?:@ (7) which corresponds to the case that no collision occrs between the reference and the data plses The noise term can be shown to be distribted as in (25), 4 in (26) is now given by 4 Wx` ' ¹ b Ñ 2 ¹ b Ñ D :? Then, the B expression is obtained as R D f Óâ âõ D f Óâ âõ D Simlation Reslts D f 8h ¹Æ Æ ã äpå œ O9AŸ ; š [Q Æ Ð Æ Ì œ 9+æ Œ[Ž ; [p Ö\ç ç D f 8h ¹Æ Ð Æ Ì -ÎÆ -ÎÆ ã ä å œ O9AŸ ; š [Q ä š œ 9+æ Œ[Ž ; [-Hp Ö\ç ç bgf (8) (9) In this section, we perform compter simlations in order to stdy the properties of the proposed hybrid system and verify the theoretical analysis In the simlations, we have considered channel models from the I 80215a standard [1] Those channel models, which were designed for è f GHz bandwidth, are bandpass filtered for a simlation of a DID MHz wide system The polarity codes and the TH codes are randomly generated from the sets G -HI@ J and J, respectivelyé Figre plots the theoretical B crves verss signalto-noise ratio (SNR) for the for different I channel models, CM-1, CM-2, CM- and CM-4 For each channel, the integration interval (eqivalently in (16)) is roghly optimized and the systems are simlated with those optimal vales The nmber frames per symbol, y, is ê and the nmber of chips per frame, 1, is DID The distance between the reference and the date plse is D chips; that is, Bëa D From the figre, it is observed that the performance gets worse from CM-1 to CM-4 since the channel spread gets larger, 5 Since we consider a single-ser system, the TH code is generated from a small set for convenience In fact, the analysis holds for any vale of the TH code from the set ì"í ^ î[îî[ :ï<ðañdòfñó \ô

7 CM 1 CM 2 CM CM 4 Bit rror robability Bit rror robability SNR (db) Fig B verss SNR for different I channel models The parameters are ïwõvöe, ï$ð¾öl í^í and öeø í òîöl í : í :ø í :ø í for CM-1, CM- 2, CM- and CM-4, respectively Average Bs are obtained by means of averaging over í^í channel realizations for each model Q ò öå Fig 5 B verss at SNR ï<ð ö ^ø ïwõköå, í and öù are sed, and ò/öå^ averaging over í^í realizations of CM-1 is performed The optimal vale is Bit rror robability Theoretical, Q= Simlation, Q=1 Theoretical, Q=10 Simlation, Q=10 Theoretical, Q=20 Simlation, Q= SNR (db) ò ï<ð¾öù ^ø ïwõvöî Fig 4 B verss SNR for different vales, í and öù and a realization of CM-1 is employed which increases the effects of the IFI and reference-to-data plse interference 1 In Figre 4,, ú D Bûü and are sed, and the theoretical and simlation reslts are compared for different vales of for CM-1 We observe that as increases the theoretical and simlation reslts get closer This is expected since the approximate B expression is derived nder the condition of large vales; that is, Lemma 41 states that the noise-noise term converges in distribtion to a Gassian random variable as - ÇýÈ Using the same parameters as in the previos case, we plot, in Figre 5, the B performance of the system in CM-1 for different vales at þƒÿ db From the figre, we dedce that for small vales, the integration interval is small and therefore very little signal energy is collected As increases to larger vales, more signal energy is collected, hence the B decreases However, after a certain point, the collected signal energy becomes less significant than the collected noise-noise and/or interference terms Therefore, the B increases as we increase frther From the figre, the optimal vale is observed to be at ³I For the next simlations, B ƒ, þƒÿ and 1 the nmber of chips per symbol is ; that is, a ƒ For a fixed symbol time and energy, the nmber of frames per symbol is varied and the B is plotted in Figre 6 for CM-1 From the plot, it is observed that the frame size does not matter p to the point IFI becomes dominant After the frame dration is 1 chips ( ), and ths the IFI becomes significant (de to Bë³ and the channel spread), the B increases 1 For the final simlations, B a D, and þƒÿ Figre 7 plots the B verss, which is the distance in chips between the reference plse and the B data plse As observed from the figre, for small vales of, the B is high de to the severe interference B between the reference and the data plses For very large, the B is also high since the IFI becomes dominant in that case Therefore, the optimal vale shold minimize the total effects of the interference between the reference and data plses and the plses from different frames V CONCLUDING RMARKS In the paper we have analyzed a hybrid matched-filter TR correlation receiver We have fond that the impact of the non- Gassian natre of the noise on the B can be significant, and have given the exact closed-form eqations for both the reslting variance and the average B We have then analyzed the B in mltipath environments Lifting restrictions of previos treatments of the topic, we inclde the effects of IFI

8 6 Bit rror robability 5 x log 2 N f Fig ï$ð8ïwõköfø 6 and, and CM-1 is considered Bit rror robability 10 B verss ïõ at SNR öù db The parameters are ò/ö öù CM 1 CM Fig 7 and B is averaged over í^í realizations of CM-1 and CM- B verss at SNR ö db ï<ð+ö&ø í and ïwõhö are sed as well as interference between the reference plse and the data plse Inclding the effects of IFI is very important for optimizing the system design of TR systems Reqiring frame drations that are larger than the maximm excess delay of the channel severely restricts the frame rate, and ths the processing gain de to mltiple frames, in a TH-IR system xtremely long frame drations also mean that the pea-to-average ratio of the transmit signal becomes high, which is ndesirable both from a hardware point of view, and from a freqency-reglation point of view (note that the FCC report and order [11] limits the admissible pea-to-average ratio) It is also important to consider the case that the delay between data plse and reference plse is smaller than the maximm excess delay of the channel In many systems, the delay in the receiver is implemented by delay lines However, it is exceedingly difficlt to bild delay lines on the order of D - DID ns (typical vales for channel maximm excess delays) Therefore, interference between reference plse and data plse will occr in practice The reslts of or paper are ths important tools for system design and comptational performance of TR UWB systems with practically relevant operating parameters RFRNCS [1] R A Scholtz, Mltiple access with time-hopping implse modlation, roc I Military Commnications Conference (MILCOM 9), vol 2, pp , Bedford, MA, Oct 199 [2] M Win and R A Scholtz, Ultra -wide bandwidth time-hopping spread-spectrm implse radio for wireless mltiple -access commnications, I Transactions on Commnications, vol 48, pp , Apr 2000 [] M Win and R A Scholtz, On the energy captre of ltra -wide bandwidth signals in dense mltipath environments, I Commnications Letters, vol 2, pp , Sept, 1998 [4] D Cassioli, M Win, A F Molisch and F Vatelaro, erformance of low-complexity RAK reception in a realistic UWB channel, roc I International Conference on Commnications (ICC 2002), vol 2, pp , New Yor City, NY, Apr 28-May 2, 2002 [5] J D Choi and W Star, erformance of ltra-wideband commnications with sboptimal receivers in mltipath channels, I Jornal on Selected Areas in Commnications, vol 20, isse 9, pp , December 2002 [6] R Hoctor and H Tomlinson, Delay-hopped transmitted-reference RF commnications, roceedings of the I Conference of Ultra Wideband Systems and Technologies 2002 (UWBST 02), pp , Baltimore, MD, May 2002 [7] L Yang and G B Giannais, Optimal pilot waveform assisted modlation for ltra-wideband commnications, roc the Thirty- Sixth Asilomar Conference on Signals, Systems and Compters, vol1, pp 7-77, acific Grove, CA, Nov 2002 [8] M Ho, V S Somayazl, J Foerster and S Roy, A differential detector for an ltra-wideband commnications system, roc I Vehiclar Technology Conference (VTC 2002 Spring), pp , Birmingham, AL, May 2002 [9] F Tfvesson and A F Molisch, Ultra-wideband commnication sing hybrid matched filter correlation receivers, roc I Vehiclar Technology Conference (VTC 2004 Spring), Milan, Italy, May 17-19, 2004 [10] Y- Naache and A F Molisch, Spectral shape of UWB signals inflence of modlation format, mltiple access scheme and plse shape, roceedings of the I Vehiclar Technology Conference, (VTC 200-Spring), vol 4, pp , Jej, Korea, April 200 [11] Federal Commnications Commission, First Report and Order 02-48, 2002 [12] F Tfvesson, S Gezici and A F Molisch, Ultra-wideband commnications sing hybrid matched filter correlation receivers, in preparation, 2004 [1] A F Molisch, J R Foerster and M endergrass, Channel models for ltrawideband ersonal Area Networs, I ersonal Commnications Magazine, vol 10, pp 14-21, Dec 200