On the Inter-Symbol-Interference inseveral Ultra Wideband systems

Lorenzo Piazzo and Fabrizio AmeliINFOCOM dept. - University of Rome 'La Sapienza'

Abstract- The performance of several Ultra-WideBand(UWB) transmission schemes is analysed when the channel isfrequency selective so that Inter-Symbol-Interference (ISI) arisesat the receiver. The UWB schemes considered employ TimeHopping (TH) and/or Direct Sequence (DS) as spreading methodand a correlation receiver. Analytical expressions for the Signal toInterference power Ratio (SIR) are derived are used to comparethe schemes over several channels. Also the Bit Error Rate (BER)is discussed. The theoretical results are compared with simulationresults.

I. INTRODUCTION

Ultra-WideBand (UWB) is an emerging Spread-Spectrum(SS) technology. A number of different UWB transmissionformats were proposed. The original Impulse Radio (IR)system [1] employs Time-Hopping (TH) in order to spread thespectrum and accommodate multiple users. Other formats pro-posed employ Direct-Sequence (DS) spreading in the place ofTH [2] or in conjunction with TH [3]. In the paper we analysethe performance of several UWB schemes when the channelis frequency selective so that Inter-Symbol-Interference (ISI)arises. We derive analytical expressions for the Signal to ISIpower Ratio (SIR) and for the Bit Error Rate (BER) anduse these expressions to compare the schemes. We also runMonte Carlo simulations in order to asses the accuracy of theanalytical results.

II. SYSTEM DESCRIPTION.We consider four different UWB systems. The first employs

a TH code to spread the signal and will be referred as IR. Thesecond employs a DS code and will be referred as DS. Thethird employs both TH and DS and will be referred as DS-IR. The fourth does not employ any spreading mechanism andwill be referred as PAM. Given a symbol time T8, the symbolwaveform employed by the four systems for transmitting thek - th symbol can be compactly written as follows

gk(t) = E dk,nW(t -nTf - Ck,nTc)fNsn=O (1)

where: w(t) is the UWB monocycle; N8 is the number ofmonocycles per symbol; Tf = TI/N, is the frame time; Nh isthe number of chips per frame; T, = Tf/Nh is the chip time;Ck,n is the TH code and dk,n is the DS code. Specificallyfor PAM and DS we set Ck,n = 0 so that TH is disabled,while for IR and DS-IR Ck,n is a pseudo-random value takingvalues in {O..Nh - 1} that specifies the position of the n -th monocycle of the k - th symbol. Likewise for PAM and

IR we set dk,, = 1 so that DS is disabled, while for DSand DS-IR dk,n is a pseudo-random value taking values in{-1, 1} that specifies the polarity of the n - th monocycleof the k - th symbol. In the analysis the DS and TH codesamples are assumed independent identically distributed (iid)Random Variables (RV) with a uniform distribution in theirrespective domains. Information is transmitted by modulatingthe amplitude of the symbol waveforms yielding the followingtransmitted signal 1

(2)s(t) = :qk gk(t - kTs)k

where qk is the sequence of transmitted symbols, modelled asa sequence of iid RV uniformly distributed on a given, zeromean, unitary energy, constellation. The signal is transmittedover a channel with impulse response c(t) that models boththe propagation medium and the antennas. Letting * denoteconvolution, the received signal is r(t) = s(t) * c(t) and canbe written as in (2) by replacing gk(t) with hk(t) gk(t)*C(t)where hk(t) is the received symbol waveform. At the receiver,assuming perfect synchronisation and channel knowledge, aproperly time aligned copy of r(t) is correlated with hk(t)realising a correlation receiver and yielding the followingdecision variable

rk = Jr(t + kTs)h* (t)dt (3)

which is passed to a slicer to estimate qk. To conclude noteh(t) and qk can be real or complex: normally UWB schemesare thought as base-band systems but nothing prevents usinga modulator and a demodulator to realise a pass-band systemwith a complex channel and constellation. Also note that whenNh = 1 the DS and DS-IR schemes are both identical to thewell known DS-SS scheme which is therefore comprised as aspecial case in the following analysis.

III. SYSTEM ANALYSISWhen the channel is frequency selective the system is

affected by ISI and the SIR is a meaningful performanceparameter that we want to compute. Upon denoting by Rkn(t)the correlation of h0(t) and hk(t), by replacing r(t) in (3) onehas

rk = qkRn,k([k - n]T.)n

(4)

IUnless otherwise specified, the summations and the integrals extend from-0o to +oo.

0-7803-9206-X/05/$20.00 ©2005 IEEE 259

qkRkk(O + fRl(k- ni]T5) =Sk + ik.n$4k

In the last equation we identified a useful part, denoted by Sk,and an ISI part, denoted by ik. The SIR is defined by SIRE{ skI12}/E{Ilik12} where E{} denotes expectation. The SIRwas computed in [4] and the computations are too long to bereported here. A few details are given in the appendix. In [4] itis found that, upon introducing P(f) C(f)W(f)12 where0(f) and W(f) are the Fourier Transform (FT) of c(t) and

w t,one has:

E{sk 12} JIp(f)VX(fI f)dfj12

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(5)

E{lik 12} fs P(f)P(f - nfs)X(f, f - nfs)dfn

Pf)P(JJ f,f2 f f (6)

where the X(fj, f2) function depends upon the UWVB formatemployed and specifically:

XPAMI = Ns DN4,,Tf(f1)DN.4,Tf(f2)XDS-IR =DN1.,T,(fl - f2) XDS =DN8,,Tf (fl - f2)

XIR IAc,T(fl- f2) + NsAN,,Tc,(f1)A*Tf)

~AhT(f1)A*T(f2)AN.,Tf (f1 _-21with

eij2-rNTf - 12ATf) N(eij2,7Tf - 1) DN,T(f) = IAN,T(f)2

IV. SIR OVER A TWO PATH CHANNELS

In this section we study the SIR of the UWB systemsintroduced earlier. Here and in the following sections weconsider an ideal lowpass monocycle with W(f) =0 forIf!I > B/2 and W(f) =1 for IfI . B/2 where B =1/T,is the system bandwidth. We study the simple and instructivecase of a real channel with two paths having equal gain andrelative delay Tr. In fig. I we report the SIR as a function of,r/T, for varying values of the parameter N, and Nh whentheir product N, N5Nh = T5/T, is fixed to N, 20.The lines are obtained by numerically evaluating (5) and (6)while the data points are obtained by means of Monte Carlosimulations and one notes that the agreement is good. In thetop figure N5 1 and Nh 20 so that DS is disabled: theJR and the DS-IR systems coincide and the PAM and the DSsystems coincide too. The systems not employing TH have awidely variable SIR (ranging from +00 to 3 dB) while thesystems deploying TH have a more stable SIR lower boundedby 16 dB. In the middle figure N, = 10 and Nh = 10and one notes that JR suffers a performance degradation. Thedegradation exists as soon as N, > 1. The DS-IR performanceis unchanged while the DS scheme has SIR lower bounded by13 dB. In the bottom figure N, 20 and Nh 1 so thatTH is disabled: the IR and the PAM systems coincide andthe DS and the DS-IR systems both coincide with a classical

55

jIt ill tillI111

10 PAM ....-TH

-DSIR ~5- SIMPAM

SIMTH+SIMDSSIM DSIR

5 5 10 15 20 25 30 35 40fTc

it II Ii I It ~ ii II II PAM ~

II Ii t Ii Ii It l't'- DS It

l~~~~~~~~~~ ~~~~SIMcIDSIR lii

?5~~~~~~~~~~~~~~~~~~~~~~~~~IIi....... A-4 :t..j~~Ii i tj~t I 11* 1 t Ii I i

A~~~~~itl11111111

A4i~~~~~~~llili11111111111111111111111.1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-~~ ~ ~~~1111.

C -

0 5 10 15 20 25 35 3L5 40

-r/ Tc

Fig. 1. SIR of the UWB systems operating over a two paths channel with

N,= 20 and N, = (top) N, = 10 (middle) or N, = 20 (bottom).

DS-SS format. The DS-IR performance are unchanged while

the systems not deploying DS suffer a severe performance

degradation. We conclude that TH is required when Nh > 1

in order to yield a stable SIR, while DS is required when

N, > in order to avoid SIR degradation. Accordingly the

260

c /Tc

3

31

11

IR

6 8 10~~~~HE1/To ~ ~ ~ ~ 0

-0.5

-1

-1.5

-2

-2.5

-3.5

-4

-4.5

12

Fig. 2. SIR of the IR system operating over a two paths channel with N, = 6

and N, = 3.

DS-IR scheme is the more robust since it yields the same SIR

(equal to that of a DS-SS system) independent of N8 and Nh.

A second point that needs to be addressed is the accuracy

of the theoretical SIR prediction. While in fig. the accuracy

is good this is not always the case. As an example consider

figure 2 where the SIR of an IR system operating with N, =

6 and N., = 3 over the two paths channel is reported. The

figure has the same format of the previous figure. It is apparent

that in this case the SIR is not always well predicted when Tr

is lower than the symbol time. This mismatch is due to the

approximation of (5) which, for low values of N8, makes the

SIR prediction unreliable for these delays2. While we only

considered IR, DS and DS IR behave in a similar way.

However for systems operating with N, = and for the PAAM

system, as noted in the appendix, (5) is exact.

V. BER ANALYSIS

A fundamental parameter of any digital communication

system is the Bit Error Rate (BER). In a UWB system

operating in the presence of ISI the BER can be approximately

predicted from the SIR by assuming that the ISI term ik Of

equation (4) is Gaussian distributed. This assumption is typical

in the analysis of DS-SS systems over multipath channels, e.

g. [5]. Assuming a binary transmission (i.e. qk E

the BER is given by [6]:

BER =-erfc( )(7)22

It is well known that the ISI is Gaussian when N., is

high, since this parameter affects the number of monocycles

contributing to the ISI, and when the channel has many paths,

since also in this case there are many independent contribu-

tions to the ISI. However performing BER simulations for high

values of N, is inconvenient, since the BER is very low in this

2Another source of inaccuracy is the fact that the ideal monocycle needs

to be truncated in the simulator.

0 2 4 6 8 10 12 14 16 18 20c /Tc

IR

-0.

32

32 ..f.

0 2 4 6 8 10 12 14 16 18 20/Tc

DS-IR

-0.

.5..

43 -1.

0 2 4 6 8 10 12 14 16 18 20,r/Tc

Fig. 3. BER of the DS (top) IR (middle) and DSIR (bottom) systems

operating over a three paths channel with N, = 10 and N8, = 5.

case and long runs are needed. Therefore we consider a system

operating with N, 10 and N8 = 5. For these values of the

parameters, unreported simulations show that for the two path

channels the BER cannot be reliably predicted by means of (7)

since the resulting ISI is not Gaussian. Therefore we consider

a three paths channel. The three paths channel is obtained

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35-

30-

25-

20-

15-

10

0

-5

THEO]

.................

9a:wm

!O

I 11 CI-sim Cl-theo 11 C2-sim [ C2-theoDS-IR .0025 .0023 1 .0032 .0050IR .0143 .0179 .0405 .0523DS .0016 .0014 .0062 .0105

|__ | C3-sim ] C3-theo Iff C4-sim C4-theoDS-IR . .0060 .0053 .0074 .0031IR .0180 .0207 .0326 .0228DS .0120 .0064 .0017 .0007

TABLE ISER OVER MULTIPATH CHANNELS.

from the two path channel by adding a third real path withunitary gain and delay T/2. In figure 3 the BER is reported fora IR, DS and DS -IR system operating over that channelfor varying values of the channel duration r. From the figureit is apparent that the BER can be approximately predicted assoon as the duration T is higher than one symbol time (i.e.10T, for NC = 10). By contrast when the duration is shorterthere is a mismatch which is also due to the SIR mismatchdiscussed in the preceding section and not necessarily to thefact that the ISI is not Gaussian. The differences among thethree schemes are also apparent in the figure: the DS has anoscillating BER which can be very good or very bad dependingon the channel duration and which is due to the oscillating SIRof this scheme. By contrast the schemes deploying TH have amore stable SIR and therefore a more stable BER. Howeverone also notes that the BER of the IR system is much higherthan the one of the DS - IR system. Indeed the IR systemis inferior to the DS - IR one because, due to the lack of theDS code, noise contributions adds coherently at the decisiondevice giving rise to a higher ISI variance. Note that a similarproblem exists for IR in multiple users interference [3].As a final comment we note that by increasing the value of

N8 better agreement between the theoretical and the simulatedvalues is obtained and that the BER of the DS scheme tendsto that of the DS - IR. By contrast when NS is decreasedworse agreement is obtained. Actually when Ns is low (theextreme case being a single pulse system with N8 = 1) theBER is normally better than what predicted with the Gaussianapproximation indicating that systems operating with a fewmonocycles are an interesting option. The problem of thesesystems is that they have a high peak to average power ratiowhich makes them less appealing.

VI. SER OVER MULTIPATH CHANNELS

In order to study the BER over more realistic channelswe consider a Raleigh random multipath complex channel,having 20 paths. The paths have uniformly distributed delaysin the range 0..100T,. Each path has a random complexGaussian amplitude the variance of which is varied based onthe path delay according to an exponential power delay profile.The Symbol Error Rate (SER) of IR, DS and DS-IR when

operating with NC = 6, Ns = 3 and a QPSK constellation forfour realizations of the random channel is reported in table1 (the different realizations are numbered from 1 to 4). Boththeoretical and simulated values are reported. It can be seenthat the SER can often be accurately predicted by the equationspresented in the paper. In all cases it can be predicted within afactor 2.5, which is a good approximation. Also note that IRhas poorer performance for all channels. By contrast whichone is the best between DS and DS - IR depends on thechannel.

VII. CONCLUSIONSWe have presented an approximate analytical expression for

SIR of several UWB systems and for the DS-SS system. Wehave discussed the accuracy of the approximation and the BERprediction that can be obtained by assuming that the ISI hasa Gaussian distribution. We have seen that the BER can beaccurately predicted over Multipath channels as soon as thechannel impulse response extends over several symbols.

VIII. APPENDIXThe SIR computation is quite long and can be found in [4].

Here we only discuss the approximation made in order to getto (5). Note that

E{Isk12}== E{lqkI2}E{IRh0)2= E{IRhk(O)12} (8)where we used the fact that the information symbols have unitenergy and are independent of the DS and the TH codes. Nextnote that Rhk (0) iS the energy of the k-th received symbolwaveform. While that may vary from symbol to symbol due tothe fact that the transmitted symbol waveform is random, it isalso clear that any reasonable transmission system shall receivethe symbols almost with the same energy, in order to equalisethe error probability. Therefore we make the approximationthat all the symbols are received with the same energy andspecifically with energy equal to E{Rhkk(0)}. In this case weobtain

E{ skI } E{Rk,k (9)

from which (5) derives. Finally note that when the energy ofthe received symbol waveform is not random, like is the casefor the PAM system and for all the systems when NS = 1,the approximation is exact.

[1] R. A. Scholtz, "Multiple access with time hopping impulse modulation",Proc. of MILCOM93, Bedford, MA, pp. 11-14, 1993.[2] J. R. Foerster "The performance of a Direct Sequence spread ultra-wideband system in the presence of multipath, narrowband interference andmulti-user interference", Proc. of the UWBST, Baltimore, 2002.[3] L. Piazzo: "Performance analysis and optimisation for Impulse Radio andDirect Sequence Impulse Radio in multiuser interference", IEEE Trans. onComm., Vol. 52, no. 5, pp. 801-810, May 2004.[4] L. Piazzo: "Performance analysis and comparison of several UWB-SS formats in a multipath, frequency selective channel", INFOCOM Dept.,University of Rome, Int. Rep. no. 001-04-04, June 2004. Available:infocom.uniromal .itflorenz[5] K. Cheun: "Performance of Direct Sequence Spread Spectrum RAKEreceivers with Random Spreading Sequences", IEEE Trans. on Comm., vol.45, no. 9, pp. 1130-1143, 1997.[6] E. A. Lee and D. G. Messerschmitt, "Digital communications", Boston,MA.: Kluwer Academic, 1994.

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