Performance Measures of a UWB Mutipe-Access System: DS/CDMA versus TH/PPM Aravind Kaias and John A. Gubner Dept. of Eectrica Engineering University of Wisconsin-Madison Madison, WI 53706 akaias@wisc.edu, gubner@engr.wisc.edu Abstract Time-hopping combined with puse position moduation has been the origina proposa for utra-wideband systems. Spread spectrum techniques for mutipe access and interference suppression are being considered for utra-wideband radio systems. A digita impuse radio receiver using direct-sequence spreading is proposed in the paper and the signa processing scheme is investigated. Performance measures, such as error rate, muti-user capabiity, and system data rate for the proposed receiver structure, are derived mathematicay and compared with the impuse receiver depoying time-hopping. The direct-sequence format is shown to support more active users; i.e, provide muti-access communications at a ower biterror rate. Upper bounds for the number of permissibe active users and the system data rate are derived anayticay. 1 Introduction In the near future, there wi appear demand for high-speed wireess inks for short range communication. A typica scenario coud be in a buiding or an office environment. One famiy of techniques that aows for the creation of such high-rate, short-range systems is caed utra wideband. Utra wideband is a favor of spread spectrum communications characterized by transmitting narrow radio impuses on the order of nanoseconds) 1], ]. In this case, a signa is transmitted with a bandwidth much arger than the data moduation bandwidth and thus with a reduced power spectra density. This approach has the potentia to produce a signa that is more covert, has higher immunity to interference effects, and has improved time-of-arriva resoution. It shoud aso be noted that the gigahertz bandwidth impies that the mutipath is resovabe to the path deays on the order of nanoseconds. This significanty reduces fading effects even in indoor environments 3]. To enhance its mutipe access MA) capabiity, UWB impuse radio systems can be depoyed combining them with spread spectrum SS) techniques. Time-hopping TH) and direct-sequence DS) are popuar and simpe schemes considered for UWB systems. A ot of research done in this area has been devoted to the TH-SS UWB signas. The timehopping scheme combined with spread spectrum PPM) and its variations such as the M-ary PPM and M-ary PPM using Wash codes are investigated and anayzed in 4]-8]. Traditiona direct sequence spread spectrum has been studied etensivey, and various
forms of moduation such as binary phase shift keying BPSK) and quadrature phase shift keying QPSK), can be empoyed with this spreading scheme. In 9], the rake receiver performances for the DS and TH schemes are compared in the presence of narrowband interference and dense mutipath via simuations. TH is a part of the origina proposa for UWB communications 4]-5], but as shown in 10], DS is an estabished technique for combating muti-user interference MAI) on AWGN channes and wi outperform PPM. So it is vita to understand and study the DS-UWB impuse radio system. In this paper, a digita impuse response radio receiver for a direct-sequence scheme is proposed and its mutipe access capabiity is discussed and compared with that of the TH/PPM impuse radio UWB system. The signa processing in the receiver is anayzed and its performance in a MA environment with respect to the number of active users and system data rate are investigated. The system bit error probabiity BER) epression is derived in detai and the MA capabiity of the system as a function of the number of users is studied. The performance measures derived serve in anayticay interpreting the abiity of the UWB system to support MA communications. Upon comparing the impuse radio technoogy using the two schemes, it is seen that DS/CDMA wi outperform TH/PPM in terms of the SNR performance and can support a arger number of users whie providing a better of quaity of service. The organization of the paper is as foows. In Section II, the DS-UWB signa transmission scheme and receiver structure are presented. This is foowed by the SNR cacuations for the receiver. In Section III, the key performance measures such as the BER and bounds for the number of active users supported by the system and the maimum achievabe system data rate are derived. Finay, concusions are presented in Section IV. DS/CDMA UWB Mutipe-Access Scheme.1 Transmission Signa Format The basic transmitted CDMA waveform of user k is given by N c 1 c k) tr t) = p k) n w tr t nt c ), n=0 where w tr represents the transmitted monocyce and {p k) n } denotes the pseudo-random noise PN) sequence. Denote the period of the PN sequence of every user as N c. Let T f be the symbo period and T c be the chip period such that T f = N c T c. Hence, a typica DS format of the kth impuse radio transmitter output signa is given by s k) tr t) = P k d k) j c k) tr t jt f ), 1) j where {d k) j } represents the data symbos and P k is the transmitted power corresponding to the kth user. It is important to point out that even an idea channe and antenna system modify the shape of the transmitted monocyce w tr t) to w rec t) at the output of the receiver antenna. The w rec considered here is the second derivative of a Gaussian function and is given by w rec ) t = 1 4π ) ] t ep π ) ] t where denotes a time normaization factor. The autocorreation function for the monocyce is given by
) R = 1 4π ) + 4π 3 ) ) ] 4 ep π ) ]. For simpicity, we assume that the receiver has knowedge of this modified puse shape. Let s k) rec be the received version of the waveform transmitted by the kth user. Finay, if we consider N u transmitters, the aggregate transmitted waveform at the output of the receiver antenna ooks ike N u rt) = Pk β k s rect k) τ k ) + nt), ) k=1 where β k captures the channe attenuation of the kth user signa upon propagation, τ k represents the time asynchronism between the signa received from the kth transmitter, and nt) is additive white Gaussian noise with constant power spectra density σ n. Without oss of generaity, we consider user 1 to be the desired user for the purpose of anaysis.. Receiver Signa Processing The Digita Impuse Radio Mutipe-Access receiver DIRMA) is a form of correatormatched fiter and is based on the theory of hypothesis testing for fuy coherent data detection. For the purpose of anaysis, it is assumed that the receiver has perfect cock and sequence synchronization for the signa transmitted by the first transmitter, as a resut of which, we know the inputs for the frame cock modue. The tempate generated at the receiver again makes use of the assumption that the receiver has knowedge of the modified puse shape of the monocyce. Figure 1: DIRMA Receiver Structure. Without oss of generaity, et us consider the situation where we are detecting the th bit. Hence, the tempate generator for our scheme is different from the ones described in 4] and 5] and is given by c 1) rect T f τ 1 ), where c 1) rect) = N c 1 n=0 p1) n w rec t nt c ). The DIRMA receiver must decide if d 1) is 1 or 1. This corresponds to deciding between two hypotheses H 1 and H 1. H d : rt) = β 1 P1 d 1) c 1) rect T f τ 1 ) + n tot t) 3)
in which d is either 1 or 1. The other terms are grouped as shown beow. N u n tot t) = k= decide d 1) = 1 Pk β k s k) rect τ k ) } {{ } Mutipe-Access Noise τ1 ++1)T f + nt) }{{} Additive White Gaussian Noise τ 1 +T f rt) c 1) rect T f τ 1 ) dt }{{} puse correator output > 0. 4) Stricty speaking, the decision rue presented is not optimum when other users are present. Now, if we make the assumption that the number of users is arge, it is reasonabe to mode the MAI as a Gaussian random process as done in 4], 5], 9], and 10]. Thus n tot t) is a white Gaussian random process since it is the sum of two Gaussian random processes) and equation 4) is optimum. For this probem, the puse correator output or the test statistic) α can be rewritten as α = m + ñ, where m and ñ are given by m = τ1 ++1)T f τ 1 +T f ] β 1 P1 d 1) c 1) rect T f τ 1 ) c 1) rect T f τ 1 ) dt, and ñ = τ1 ++1)T f τ 1 +T f n tot t) c 1) rect T f τ 1 ) dt. 5) Equation 5) can be decomposed into the term due to the MAI and AWGN as n MAI = n noise = τ1 ++1)T f N u τ 1 +T f k= τ1 ++1)T f Pk β k s k) rect τ k ) c 1) rect T f τ 1 ) dt, and τ 1 +T f nt) c 1) rect T f τ 1 ) dt. 6) Now, the quantity m, is mathematicay derived to be m = N c A 1 d 1) ξ, where ξ = w rec t)] dt and A 1 = P 1 β 1. 7) 3 Comparison of Performance Measures 3.1 SNR Computations The DIRMA receiver output SNR is defined as SNR out N u ) = m E{ ñ }, where the numerator in this epression is given by 7). Without much effort, it can be shown that the denominator, E{ ñ } = σnoise + N c σsef Nu k= A k, where σ sef = T 1 f w rec s)w rec )d] ds and Ak = P k β k. In 6), it is assumed that
the mean and variance of n noise t) are given by 0 and σnoise respectivey. Then the DIRMA receiver output SNR is mathematicay given by SNR out N u ) = N c A 1 ξ) σ noise + N cσ sef Nu k= A k. 8) When ony the desired transmitter is active, N u = 1, and the epression for singe-user output SNR is SNR out N u = 1) = NcA 1ξ). σ noise 3. Error Probabiity and BER Comparisons The output SNR in 8) can aso be epressed as N c ξ SNR out N u ) = SNR 1 out1) + σsef ] 1 Nu k= k= Ak A 1 ) 1. 9) Now the epression for the probabiity of error turns out to be ] 1 1 N c ξ Nu ) Ak P e N u ) = Q SNR 1 out1) + σsef. 10) A 1 When ony the desired transmitter is active, N u = 1, and the epression for singeuser probabiity of error is P e N u = 1) = Q NcA1 ξ) 1 { } ]. In the case of TH-UWB, the system is an oversamped moduation system with N s monocyces transmitted per symbo, the moduation data changes ony every N s hops. In this anaysis we assume N s = N c. From 4], 5], and 8], the correator s tempate waveform regenerated at the DIRMA receiver is given by vt) = w rec t) w rec t δ), where δ is the PPM time shift. The SNR epression is identica to 8) ecept that ξ is repaced by the correation term, m p = w rect δ)vt) dt. Under the assumptions on the structure of w rec for eampe, the Gaussian monocyce), the autocorreation function is non-negative. Thus, based on the choice of tempate generators described, m p < ξ. Hence, upon comparing our resuts with those presented in 4] and 5], it is seen that SNR T H/P P M) out N u ) < SNR DS/CDMA) out N u ) P σ noise T H/P P M) e N u ) > P DS/CDMA) e N u ).11) Another means of comparison which immediatey foows is the usage of E b /N o, which is defined as E b /N o N u ) = SNR out N u )db] + 10 og 10 b + 1), 1) when b + 1 bits are transmitted per symbo. The theoretica BER is given by BER = ) 1 erfc SNR outn u). 3.3 Transmission Capacity and Ecess Singe-Link SNR Comparisons The ecess singe-ink SNR, P is given by P = 10 og 10 {SNR out 1)/SNR out N u )}. This is the additiona power required to accommodate every new ink in the mutiuser
system. This is one of the basic performance measures of a mutipe-access system that reates number N u of users and the SNR out N u ). Using this definition, we can rewrite the sum in 9) as ) N u A k k= A 1 = M 1 SNRoutN 1 u ){1 10 P/10) } where, M 1 = Ncξ. Now, M 1 = ξ T f σ sef R mod σ sef where, we define R mod = N c T f ) 1 as the system data rate. Under perfect power contro assumptions, i.e., A k = A 1 for a k, and with M 1 = ξ T f, σsef it reduces to R mod P ) = M 1 SNRoutN 1 u ){1 10 P/10) } {N u 1} 1, which is the system data rate, providing a simpe reation between the various parameters of interest. Foowing this anaysis is the equation for the number of users as a function of the ecess singe-ink SNR, given by N u P ) = M 1 SNRoutN 1 u ){1 10 P/10) } + 1. Note that both the epressions for N u and R mod are monotonicay increasing functions of P. To fit a bound to these parameters, we et P be as arge as possibe, i.e., P and the reations become R mod P ) im mod P ) = M 1 SNRoutN 1 u ) {N u 1} 1 R ma, P 13) N u P ) im u P ) = M 1 SNRoutN 1 u ) + 1 N ma. P 14) Hence, for a specified eve of performance as embodied in SNR out N u ), there are upper bounds on the system data rates for a given number of users) and the number of users for a given system data rate) that cannot be eceeded by the DIRMA receivers. The performance measures, R ma and N ma are higher for the DS format and serve as a basis for comparison between the two formats for MA-UWB communication systems. 4 Concusions Utra-wideband technoogy has been recenty proposed as a viabe soution for highspeed indoor short range wireess communication systems because of its robustness to severe mutipath and muti-user conditions, ow cost, and ow power impementation. With this in view, an appropriate reception scheme for direct-sequence UWB impuse radio receiver was presented in the paper. The system mutipe-access performance was anayzed in terms of the error probabiity, the data transmission rate, and number of active users supported under specified conditions. Epressions for the output SNR of the utra-wideband system were derived and the BER performance of the receiver, as a function of the number of users, was seen to be superior to the time-hopping format. In genera, this resut was true for a signa monocyce) waveforms which have nonnegative autocorreation functions. To concude, with this choice of monocyce puses, the DS-UWB supports arger number of users communicating at ower error rates and higher data transmission rates. References 1] T. Mitche, Broad is the way utra-wideband technoogy], Proc. IEEE, vo. 47, no. pp. 35 9, Jan. 001. ] S. Roy, J. R. Foerster, D. G. Leeper and V. S. Somayazuu, Utrawideband radio design: The promise of high-speed, short-range wireess connectivity, Proc. IEEE, vo. 9, no., pp. 95 311, Feb. 004.
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