Ultra Wideband Impulse Radio Systems with Multiple Pulse Types 1

Transcription

1 1 Utra Wideband Impuse Radio Systems with Mutipe Puse Types 1 Sinan Gezici, Student Member, IEEE, Zafer Sahinogu 3, Member, IEEE, Hisashi Kobayashi, Life Feow, IEEE, and H. Vincent Poor, Feow, IEEE arxiv:cs/5168v1 cs.it] 3 Oct 5 Abstract In an utra wideband UWB impuse radio IR system, a number of puses, each transmitted in an interva caed a frame, is empoyed to represent one information symbo. Conventionay, a singe type of UWB puse is used in a frames of a users. In this paper, IR systems with mutipe types of UWB puses are considered, where different types of puses can be used in different frames by different users. Both stored-reference SR and transmitted-reference TR systems are considered. First, the spectra properties of a muti-puse IR system with poarity randomization is investigated. It is shown that the average power spectra density is the average of the spectra contents of different puse shapes. Then, approximate cosed-form expressions for the bit error probabiity of a muti-puse SR-IR system are derived for RAKE receivers in asynchronous mutiuser environments. The effects of both inter-frame interference IFI and mutipeaccess interference MAI are anayzed. The theoretica and simuation resuts indicate that SR-IR systems that are more robust against IFI and MAI than a conventiona SR-IR system can be designed with mutipe types of utra-wideband puses. Finay, extensions to muti-puse TR-IR systems are briefy described. Index Terms Utra-wideband UWB, muti-puse impuse radio IR, stored-reference SR, transmitted-reference TR, performance anaysis. I. ITRODUCTIO Utra-wideband UWB technoogy hods great promise for a variety of appications such as short-range high-speed data transmission and precise ocation estimation. Commony, impuse radio IR systems, which transmit very short puses with a ow duty cyce, are empoyed to impement UWB systems 1]-4]. In an IR system, a number f of puses are transmitted per symbo, and information is usuay carried by the poarity of the puses in a coherent system, or by the difference in the poarity of the puses in a differentiaymoduated system. In the former case, it is assumed that received puse structure is known at the receiver and channe estimation can be performed; hence, RAKE receivers can be used to coect energy from different mutipath components. Since the incoming signa structure is correated by a ocay stored reference tempate signa in this case, such a system is caed a stored-reference SR system 5]. In the atter case, out of the f puses transmitted per information symbo, haf of them are used as reference puses, whereas the remaining haf are used as data puses. The reative poarity of the reference and the data puses carries the information. Since the reference 1 Manuscript received March 3, 5; revised October 15, 5. This research is supported in part by the ationa Science Foundation under grant AI , and in part by the ew Jersey Center for Wireess Teecommunications. Part of this materia was presented at the 39th Annua Conference on Information Science and Systems CISS 5. Department of Eectrica Engineering, Princeton University, Princeton, J 8544, USA, Te: , Fax: , emai: {sgezici,hisashi,poor}@princeton.edu 3 Mitsubishi Eectric Research Labs, 1 Broadway, Cambridge, MA 139, USA, e-mai: zafer@mer.com puses to be used in the demoduation are transmitted to the receiver, such a system is caed a transmitted-reference TR system 4]. In a TR system, there is no need for channe estimation since the reference and the data puses are effected by the same channe, assuming that the channe is constant for a sufficienty ong time interva, which is usuay the case for UWB systems. On the other hand, a ower throughput is expected since haf of the energy is used for non-information carrying puses. Aso since the transmitted reference is used as a noisy tempate at the receiver, more effective noise terms are generated. Considering a conventiona SR-IR system, a singe type of UWB puse is transmitted in a frames of a users 1]. In asynchronous mutiuser environments, the autocorreation function of the puse becomes an important factor in determining the effects of inter-frame interference IFI and mutipeaccess interference MAI 6]. In order to reduce those effects, UWB puses with fast decaying autocorreation functions are desirabe. However, such an autocorreation function aso resuts in a considerabe decrease in the desired signa part of the receiver output in the presence of timing itter 7]. Moreover, when there is an exact overap between a puse and an interfering puse, the interference is usuay very significant. Hence, there is not much fexibiity in choosing the puse shape in order to combat against interference effects. However, in SR-IR systems with mutipe types of UWB puses, the effects of interference can be mitigated by using different types of UWB puses with good cross-correation properties. Mutipuse SR-IR systems have recenty been proposed in 8]. However, there has been no theoretica anaysis of such systems, in terms of their spectra properties and bit error probabiity BEP performance, and no quantitative investigation of the gains that can be obtained by mutipe types of UWB puses. In this paper, we consider this probem in an asynchronous mutiuser environment and anayze the BEP performance of a generic RAKE receiver over frequency-seective channes. The resuts are vaid for arbitrary numbers of UWB puse types, and hence cover the singe-puse system as a specia case. Moreover, we aso briefy describe possibe extensions of muti-puse approach to TR-IR systems. However, no detaied anaysis is given due to space imitations. In addition to the performance anaysis of the muti-puse IR systems, the average power spectra density PSD of a generic muti-puse IR signa is derived and a simpe reationship between the Fourier transforms of the UWB puses and the average PSD of the transmitted signa is obtained. The remainder of the paper is organized as foows. Section II describes a generic transmitted signa mode, which reduces to SR- and TR-IR systems as specia cases. Then, Section III anayzes the spectra properties of this generic IR signa structure. After describing the received signa in Section IV, the performance of muti-puse SR-IR systems empoying RAKE receivers is anayzed in Section V and simuation

2 resuts are given in Section VI. The concuding remarks are made and possibe extensions are discusses in the ast section. II. TRASMITTED SIGAL MODEL The transmitted signa from the kth user in a muti-puse UWB-IR system can be expressed as s k t = 1 f i= p 1 n= s k i,n t, 1 where f is the number of puses transmitted per information symbo, p is the number of different puse types, and s k i,n t represents the UWB puses of type n transmitted for the ith information symbo of user k. ote that the signa mode in 1 can aso represent cases in which the number of puse types is ess than p, by using the same puses for different puse indices. Aso different users can have different ordering of the puses in one period, which can be usefu for reducing the effects of MAI. The number of puses per symbo, f, is assumed to be an even mutipe of p for simpicity of notation and s k i,n t is expressed as foows: s k i,n t = i+1 f p 1 =i f / p { b k 1, dk p+n pk n t p +nt f c k T k p+n c +b, dk +1 p+n pk n t p +nt f T k n ck +1 p+n T c }. In, p n k t is the UWB puse of type n for user k, T f is the frame interva, T c is the chip interva, and T n k is the distance between the two puses in a pair of type n for the kth user, considering the puses from a given type being grouped into pairs as shown in Figure 1. The time-hopping TH code for user k is denoted by c k, which is an integer taking vaues in the set {,1,..., c 1}, with c being the number of chips per frame, which prevents catastrophic coisions between different users. The poarity, or the spreading, code, { 1,+1}, changes the poarity of the puses, which smoothes the PSD of the transmitted signa 9] and provides robustness against MAI 1]. The information is represented by b k 1, and bk,, which carry the same information for an SR system, and carry the information in the difference between their vaues for a TR system. The genera signa mode in can represent SR and TR systems as specia cases: d k A. Stored Reference Impuse Radio For the SR system, b k 1, = bk, = bk p/ f, Tk n = p T f n, k, and each frame has independent TH and poarity codes. B. Transmitted Reference Impuse Radio For the TR system, b k 1, = 1 and bk, = bk p/ f. In other words, the first puse in is the reference puse and the second one is the data puse. As shown in Figure 1, this resuts in a structure in which the first p puses are the reference puses, the next p puses are the data puses, and which foows this aternating structure. ote that T n k can be chosen to be arger than p T f for the TR system if the TH sequence is constrained to a set Fig. 1 TRASMITTED SIGAL FROM A MULTI-PULSE TR-IR SYSTEM, WHERE f = 1, c = 4, T n k = T c FORn =,1, WITH = 1, AD THE TH SEQUECE IS{3,,,3,,,1,1,,1,1,}. FOR SIMPLICITY, O POLARITY CODES ARE SHOW THAT IS,d k = 1, AD b k 1, = 1 AD b k, = 1. {,1,..., h 1} with h < c. In this case, different vaues from the set p T f, p +1T f h T c ] can be chosen for different users and/or different puse types in order to provide extra robustness against the effects of interference. Aso each reference-data puse pair has the same TH and poarity codes in order to faciitate simpe deay and mutipication operation at the receiver. In other words, d k = p+n dk +1 p+n, and ck for = i f / p,...,i + 1 f / p 1, i, n =,1,..., p 1. = p+n ck +1, p+n III. PSD OF MULTI-PULSE UWB-IR SYSTEMS In order to evauate the spectra properties of the transmitted signa, the average PSD of the signa must be cacuated. Therefore, we first cacuate the autocorreation function of st as foows 4 : φ ss t+τ,t = E{st+τst} = 1 p 1 E{s i,n t+τs i,n t}, 3 f i= n= where we empoy the fact that the poarity codes are i.i.d. for different bit and puse indices. From, E{s i,n t+τs i,n t} can be cacuated, after some manipuation, as E{s i,n t+τs i,n t} = i+1 f p 1 { E p n t+τ p +nt f =i f / p c p+nt c pn t p +nt f c p+nt c }. 4 From 3 and 4, it is observed that st is not widesense stationary WSS since the autocorreation function is not independent of t. However, note that st is a zero mean cycostationary process since φ ss t + τ,t is periodic with a period of p T f 11]. Therefore, we can obtain the timeaverage autocorreation function as φ ss τ = 1 p T f pt f 1 = p T f f p 1 n= φ ss t+τ,tdt p n t+τp n tdt, 5 4 We drop the user index k in this section, for notationa convenience.

3 3 the Fourier transform of which gives the average PSD as foows: Φ ss f = 1 p 1 W n f, 6 p T s n= where W n f is the Fourier transform of p n t. ote from 6 that the average PSD of the signa is the average vaue of the squares of the Fourier transforms of the puses. The dependence on the puse spectra ony is a consequence of the puse-based poarity randomization, as considered for impuse radio systems in 9] and 1]. Moreover, we note that the muti-puse system can have more fexibiity in shaping the PSD by proper choice of the UWB puses. IV. CHAEL MODEL AD RECEIVED SIGAL We consider the foowing channe mode for user k: L 1 h k t = α k δt τ k, 7 = where α k and τ k are, respectivey, the fading coefficient and the deay of the th path for user k. Using the channe mode in 7 and the transmitted signa in 1, the received signa can be expressed as rt = K k=1 E k f i= p 1 n= i+1 f p 1 = i f p p +nt f c k T p+n c τ k k +b p +nt f T n k ck +1 T p+n c τ k with u k = { b k 1, dk p+n uk n t, dk +1 p+n uk n } +σnt, 8 L 1 n t = α k w n k t τk +τ k, 9 where w k n t is the received UWB puse of type n for user k, E k determines the received energy from user k, and nt is a zero mean white Gaussian process with unit spectra density. V. AALYSIS OF RAKE RECEIVERS FOR MULTI-PULSE SR-IR Since b k 1, = b k, = b k p/ f, and each frame has independent TH and poarity codes for the SR-IR system, the received signa in 8 can be expressed, after some manipuation, as rt = K k=1 E k f b k / f dk u k = t T f c k T c τ k +σnt, 1 with u k t given by 9. For the indices of the puse types, such as in 9, themoduo p operation is impicity assumed. In other words, for any n {,1,..., p 1}, w n t = w n+kp t for a integers k. We consider a generic RAKE receiver that can represent different combining schemes, such as equa gain or maxima ratio combining. It can be expressed as the correation of the t received signa in 8 with the foowing tempate signa, where we consider the th bit of user 1 without oss of generaity: with f 1 s 1 tempt = d 1 v 1 t T f c 1 T c, 11 = L 1 v 1 t = β w 1 t τ 1, 1 = where β denotes the RAKE combining coefficient for the th path. We assume that τ 1 =, and τ k, f T f, for k =,...,K, again without oss of generaity. ote that for a partia or seective RAKE receiver 13], the combining coefficients for those paths that are not used are set to zero. We assume that the deay spreads of the channes are not arger than one frame interva; that is τ k L 1 T f, k. In other words, the frame interva is chosen to be sufficienty arge so that the puses in one frame can interfere ony with those in the adacent frames. Using 1 and 11, the decision variabe for detecting the th bit of user 1 can be obtained as: with Y = b 1 E 1 f φ k u i v f 1 = x = φ 1 u v 1 u k +I +M +, 13 i t xv tdt, 14 where the first term in 13 is the desired signa part of the output, I is the IFI, M is the MAI, and is the output noise. For simpicity of notation, bit indices are not shown. A. Inter-frame Interference The IFI occurs when a puse of the desired user, user 1, in a given frame spis over to an adacent frame due to mutipath and consequenty interferes with the puse in that frame. The IFI for the th symbo can be expressed, using 1, 11 and 14, as the sum of the IFI to each frame of the tempate signa: f 1 E 1 I = Î, 15 f where Î = d 1 d 1 +m b1 m {±1} +m f φ u 1 = +m v1 mt f +c 1 +m c1 T c. 16 ote that due to the assumption on the deay spreads of the channes, the IFI occurs ony between adacent frames. The asymptotic distribution of the IFI in 15 is given by the foowing proposition: Proposition 5.1: Consider a random TH SR-IR system with puse-based poarity randomization, which empoys p > 1 different UWB puses. Then, the IFI at the output of the RAKE receiver, expressed by 13, is asymptoticay distributed as E 1 I, p c p 1 n= σ IFI,1 n+σ IFI, n], 17

4 4 as f p, where c σifi,1n = φ u 1 c σifi, n = φ 1 u n+1 v1 n n+1 v1 n T c +φ u 1 n 1 v1 n T c, T c φ 1 u T n v 1 c. 18 n+1 Proof: See Appendix A. Due to the FCCs reguation on peak to average ratio PAR, f cannot be chosen very sma in practice. Since we transmit a certain amount of energy in a constant symbo interva, as f gets smaer, the signa becomes peakier 7]. Therefore, the approximation for arge f / p can be quite accurate for rea systems depending on the number of puse types and the other system parameters. From Proposition 5.1, the foowing resut for a doubepuse system can be obtained. Coroary 5.1: Consider a random TH SR-IR system with puse-based poarity randomization, where the UWB puses w t and w 1 t, which are both even functions, are transmitted aternatey. For this system, the IFI in 13 is approximatey distributed as foows for arge f : I, E 1 c c φ u 1 v1 1 T c +φ u 1 v1 1 ] T c. 19 The distribution of the IFI for the case where a singe UWB puse w t is used in a the frames is given by 14] c ] I φ 1 u T c +φ 1 u T c., E 1 c v1 v1 ote that in an IFI-imited scenario, the autocorreation function of the UWB puse is the determining factor for a singe-puse system. However, for the system using mutipe types of UWB puses, the IFI is determined by the cross-correations of different puses. ote that it is possibe to design the puses so that they are orthogona and their cross-correations decay quicky, e.g. modified Hermite puses MHPs 15]. However, the autocorreation function aways causes arge vaues when there is an exact overap of the mutipath components. Aso a rapidy decaying autocorreation function, which is good for combatting the IFI, may not be very desirabe since sma timing itter in the system coud resut in a significant oss in the desired signa part of the decision variabe. Therefore, the muti-puse IR system is expected to have better IFI reection capabiity than the singepuse system. For exampe, for a system with f =, c = 3 and L =, the power of the IFI is reduced by about 3% by using the 4th and 5th order MHPs instead of using the 4th order MHP ony. B. Mutipe-Access Interference Consider the MAI term M in 13, which is the sum of the interference terms from K 1 users; that is, M = K k= Mk, where M k can be expressed as M k = Ek f 1 f = ˆM k k, with ˆM denoting the MAI from user k to the th frame of the first user. From 1, 11 and 14, ˆM k can be expressed as ˆM k = d 1 m= d k m bk m/ f φ m T u m k v 1 f +c k m c 1 T c +τ k, 1 where τ k denotes the amount of asynchronism between user k and the user of interest, user 1, since we assume τ 1 =. For a given vaue of τ k, the distribution of the MAI from user k can be obtained approximatey from the foowing proposition: Proposition 5.: Consider a random TH SR-IR system with puse-based poarity randomization, which empoys p different types of UWB puses. Then, the MAI from user k, M k, given τ k, is asymptoticay distributed as foows M k τ k as f p, where E k, p c σmai,kn,τ k = m A { witha = φ u m k v n 1 p 1 σmai,kn,τ k n= c 1 = c 1 n + 1 c τk c m n c +T c +τ k T f,...,,, 3 n+ 1 c τk T f }. Proof: The proof is simiar to that of Proposition 5.1, and is omitted due to space imitations. ote that the Gaussian approximation in Proposition 5. is different from the standard Gaussian approximation SGA used in anayzing a system with many users 17]-19]. Proposition 5. states that when the number of puses per information symbo is arge compared to the number of different puse types, the MAI from an interfering user is approximatey distributed as a Gaussian random variabe. This idea is simiar to the improved Gaussian approximation approach in 16], where the arge processing gain of a CDMA system eads to normay distributed MAI conditioned on some systems parameters. Denote the amount of asynchronism between user k and user 1 as τ k = τ k /T c T c +ǫ k, where ǫ k,t c. When a singe type of UWB puse is empoyed in the system, it can be shown from Proposition 5. that the distribution of M k is given by the foowing resut: Coroary 5.: Consider a random TH SR-IR system with puse-based poarity randomization, where the UWB puse w t is empoyed in a frames of a users. Then, the conditiona distribution of the MAI from user k is given by M k τ k, E k c c 1 φ u k v1 = c T c +ǫ k. 4 ote from Coroary 5. that the distribution of M k depends on ǫ k, instead of τ k, for a singe-puse system. This is because the probabiity that a given puse of the desired user coides with the puses of user k is the same for a deays τ k with identica ǫ k vaues, due to the random TH codes, and the same amount of average interference occurs when the same puses are used in a frames.

5 5 Denote τ = τ ]. Then, given τ, the distribution of the tota MAI M in 13 can be approximated by 1 K p 1 M τ, p c E k σmai,kn,τ k, k= n= 5 τ K for arge f / p, where σmai,k n,τk is as in 3. ote that it is not necessary to have a arge number of users, or equa energy interferers perfect power contro, for the expression in 5 to be accurate. The ony requirement is to have a arge ratio between the number of puses per symbo and the number of puse types. When the deays of the interferers are unknown and/or an average performance measure is to be obtained, then each interferer is assumed to have a uniformy distributed deay with respect to the desired user; that is, τ k, f T f, k. In this case, the performance measure, such as the BEP expression, needs to be averaged over the distribution of τ. C. Output oise The output noise in 13 is distributed as, σ s 1 temp t dt. Using the expression in 11 for s 1 temp t, we can approximate the distribution of for an SR-IR system with a singe UWB puse w t as, f σ φ 1 v, for arge vaues of f, where φ k v x = v k t xv k tdt is the autocorreation function of v k t. Simiary, for an SR-IR system empoying p types of puses, we obtain the approximate distribution of as, σ f p 1 p n= φ, for arge v 1 f / p. n D. Bit Error Probabiity Using the resuts in the previous sections, we can obtain an approximate BEP expression for the muti-puse SR-IR system as foows: E1 p 1 P eτ Q p n= φ 1 u n v n 1 p 1 ], n= σifi n+σ MAI n,τ+σ φ 1 v n 6 for arge f / p, where σifi n = E 1 c σ IFI,1 n+σ IFI, n], 7 σmai n,τ = 1 K E k σmai,k c, 8 k= τ = τ τ K ], = c f is the tota processing gain of the system, σifi,1 n and σ IFI,1 n are as in 18 and σmai,k n,τk is as in 3. If we consider a synchronous scenario, where τ k =, for k = 1,,...,K, then the unconditiona BEP is given by Pe sync = P e, with P e τ being given by 6. For an asynchronous system, we assume that τ,...,τk are i.i.d. distributed as U,T s, wheret s = f T f is the symbo interva. Hence, the unconditiona BEP can be obtained by Ts Ts Pe async = 1 Ts K 1 P e τdτ...dτ K. 9 Due to the periodicity of the puse structure, we can show that it is enough to average over an interva of ength p T f instead of f T f. Hence, P async can be expressed as P async e = e 1 p T f K 1 pt f pt f P e τdτ...dτ K. 3 In order to cacuatepe async, numerica techniques or Monte- Caro simuations can be used. For exampe, by generating m vectors according to the uniform distribution in, p T f K 1, we can approximate Pe async by Monte-Caro simuations as Pe async = 1 m m i=1 P eτ i, where τ i denotes the ith random vector of interferer deays. ote that the BEP expression in 3 becomes more accurate as f / p gets arger, without the need for arge number of users or equa energy interferers, which are needed for accurate BEP using the SGA. The SGA directy cacuates the average vaue of the variance of the tota MAI instead of averaging over a conditiona BEP expression in 6. In other words, Pe async is approximated by the expression in 6 with the ony 1 pt change of using f pt f σ MAI,k n,τk dτ k instead of σmai,k n,τk. Of course, this expression is easier to evauate than the expression in 3, especiay when there is a arge number of users. Therefore, in such a case, the SGA might be preferred if the users power eves are not very different. But for systems with sma numbers of interferers, such as an IEEE a persona area network PA, the expression in 3 is not very difficut to evauate and can resut in more accurate BEP evauations. ow consider the case in which a singe type of UWB puse w t is empoyed for a users. The BEP expression for this scenario can be obtained from 13,, 4, and Section V-C as E1 φ 1 P e ǫ Q u v1, 31 σ IFI +σmai ǫ+σ φ 1 v for arge f / p, where σ IFI = E 1 c σ MAI ǫ = 1 K k= c c 1 φ 1 u v1 = c E k φ u k T c +φ u 1 v1 v1 T c ], T c +ǫ k, 3 and ǫ = ǫ ǫ K ] characterizes the asynchronism between the interfering users and the desired user in moduo T c arithmetic. Simiar to the muti-puse case, the unconditiona BEP is given by Pe sync = P e, with P e ǫ being as in 31, for the synchronous case, and by P async e 1 T K 1 c Tc Tc P e τdτ...dτ K, 33 for the asynchronous case. From the cosed-form BEP expressions for muti-puse and singe-puse systems, we can observe that the IFI and MAI terms depend on the autocorreation function of a singe puse for singe-puse systems, whereas they depend aso on

6 6 Ampitude Bit Error Probabiity Singe Puse, Theoretica Doube Puse, Theoretica Tripe Puse, Theoretica Singe Puse, Simuation Doube Puse, Simuation Tripe Puse, Simuation 1.5 3rd order MHP 4th order MHP 5th order MHP Time ns SR db Fig. THE 3RD, 4TH AD 5TH ORDER MODIFIED HERMITE PULSES MHPS. Fig. 3 THE BEP PERFORMACE OF THE SIGLE-PULSE, DOUBLE-PULSE, AD TRIPLE-PULSE SR-IR SYSTEMS. the cross-correations of different puse types for muti-puse systems, which suggests that more fexibiity in combatting the effects of the IFI and MAI is present in muti-puse systems. In other words, by design of UWB puses with good crosscorreation properties, it is possibe to mitigate the IFI and MAI to a arger extent, as wi be investigated in the next section. VI. SIMULATIO RESULTS In this section, we compare the BEP performance of singepuse, doube-puse and a tripe-puse SR-IR systems. In the tripe-puse system, each user transmits the 3rd, 4th and 5th order MHPs 8] aternatey, whereas the doube-puse system empoys the 3rd and 4th order MHPs. For the singe-puse system, the 3rd order MHP is transmitted in a the frames see Figure. The system parameters are K = 5 users, f = 18 frames per symbo, c = 3 chips per frame, and T c = 1 ns. We consider an MAI-imited scenario, where the received energy of each interferer is 18.75dB more than that of the user of interest. A the channes have L = taps, which are generated independenty according to a channe mode with exponentiay decaying E{ α } = Ω e λ og-norma fading α Lµ,σ channe ampitudes, random signs for channe taps, and exponentia distribution for the path arrivas with a mean ˆµ. The channe parameters are λ =.5, σ = 1, and ˆµ = 1.5 ns, and µ can be cacuated from µ =.5 n 1 e λ 1 e λ σ ], for =,1,...,L 1. λl Figure 3 shows the BEP performance of a-rake receivers 13] for the singe, doube and tripe-puse systems. Both the theoretica and the simuation resuts are shown, which are in quite good agreement. From the figure, the effects of mutipe puse types on reducing the interference, hence the BEP, are observed. As the number of puse types increases, more gain is obtained. Further gains can be obtained by using a arger number of UWB puse types and/or MHPs that are severa orders apart 8]. Aso, the theoretica resuts are more accurate for smaer number of puse types, p, since the asymptotic resuts in Section V assume arge f / p vaues. VII. COCLUSIOS AD EXTESIOS In this paper, we have considered muti-puse IR systems. First, we have introduced a generic mode for an IR signa, which can represent an SR or a TR signa as specia cases. Using this mode, we have investigated the average PSD of the transmitted signa, which is important considering the power imitations imposed by the FCC. Then, we have provided a detaied BEP anaysis for a muti-puse SR-IR system, considering the effects of both the IFI and the MAI, and performed simuation studies to verify the theory. The muti-puse approach can be extended to TR-IR systems as we in order to mitigate the effects of interference. In this case, different puse types can be transmitted next to each other as shown in Figure 1. If the same deay between the reference and data puses is used for a puse types, then a conventiona TR receiver can be empoyed 4]. If different deays between reference and data puses are empoyed for different puse types, the receiver needs to perform p parae deay-andmutipy operations and combine the outputs of different branches. Theoretica and simuation studies are necessary to quantify the possibe improvements by the use of mutipe UWB puses. A. Proof of Proposition 5.1 APPEDIX Let f = r p. Then, 15 can be expressed as I = E 1 Ĩ, where Ĩ = 1 p 1 Î p+n with Î p r 1 r = n= being given by 16. It can be shown, from 16, that E{Ĩ} =, due to the i.i.d. random poarity codes. Aso from the p assumption in the proposition, it is straightforward to show that E{ĨĨ+} = for, since Ĩ and Ĩ+ incude terms with poarity codes of different indices, which are independent and zero mean by assumption. Hence, {Ĩ} r 1 =

7 7 forms a zero mean 1-dependent sequence 5. We empoy the foowing centra imit argument for dependent sequences to approximate the distribution of the IFI: Theorem 1: ] Consider a stationary d-dependent sequence of random variabes X 1,X,... with E{X 1 } = and E{ X 1 3 } <. If S n = X X n, then n S n,σ, as n, where σ = E{X1 } + d k=1 E{X 1X 1+k }. In order to appy the resuts of the theorem we first cacuate the variance of Ĩ: E{Ĩ } = 1 p 1 p = 1 p 1 p n= p 1 n 1= n = E{Î p+n}+ E{Î p+n 1 Î p+n } 34 n 1 n =1 E{Î p+n 1 Î p+n }, where the second equaity is obtained from 16 by using the fact that the poarity codes form an i.i.d. sequence. Then, after some manipuation, E{Î } can be expressed as E{Î } = 1 c c φ u 1 1 v1 and E{ÎÎ+1} can be expressed as E{ÎÎ+1} = 1 c c φ u 1 +1 v1 T c +φ u 1 +1 v1 ] T c, 35 T c φ 1 u T v 1 c In obtaining 35 and 36, we have used the expression in 16 and the facts that the poarity codes are randomy distributed in { 1,+1} and the TH codes in {,1,..., c 1}. ow considering the correation between the adacent terms of {Ĩ} r 1 =, the foowing expression can be obtained: E{ĨĨ+1} = 1 p E{Î+1 p 1Î+1 p }. 37 Theorem 1 can be invoked for {Ĩ} r 1 =, which resuts in I, E 1 E{Ĩ }+E{ĨĨ+1}]. Then, from 34-37, the distribution ofi can be approximated as in 17, as r. REFERECES 1] M. Z. Win and R. A. Schotz, Impuse radio: How it works, IEEE Communications Letters, vo., no., pp , Feb ] M. Z. Win and R. A. Schotz, Utra-wide bandwidth time-hopping spread-spectrum impuse radio for wireess mutipe-access communications, IEEE Transactions on Communications, vo. 48, pp , Apri. 3] R. A. Schotz, Mutipe access with time-hopping impuse moduation, Proc. IEEE Miitary Communications Conference MILCOM 1993, vo., pp , Boston, MA, Oct ] R. Hoctor and H. Tominson, Deay-hopped transmitted-reference RF communications, Proceedings of the IEEE Conference of Utra Wideband Systems and Technoogies UWBST, pp , Batimore, MD, May. 5] M.-H. Chung and R. A. Schotz, Comparison of transmitted- and stored-reference systems for utra-wideband communications, Proc. IEEE Miitary Communications Conference MILCOM 4, Monterey, CA, Oct. 31-ov. 3, 4. 5 A sequence {X n} n Z is caed a D-dependent sequence, if a finite dimensiona marginas X n1,...,x ni and X m1,...,x m are independent whenever m 1 n i > D. 6] S. Gezici, H. Kobayashi, H. V. Poor and A. F. Moisch, Performance evauation of impuse radio UWB systems with puse-based poarity randomization in asynchronous mutiuser environments, Proc. IEEE Wireess Communications and etworking Conference WCC 4, vo., pp , Atanta, GA, March 1-5, 4. 7] S. Gezici, H. Kobayashi, H. V. Poor and A. F. Moisch, The trade-off between processing gains of an impuse radio system in the presence of timing itter, Proc. IEEE Internationa Conference on Communications ICC 4, vo. 6, pp , Paris, France, June -4, 4. 8] H. Harada, K. Ikemoto and R. Kohno, Moduation and hopping using modified Hermite puses for UWB communications, Proc. IEEE Conference of Utra Wideband Systems and Technoogies UWBST 4, pp , Kyoto, Japan, May 4. 9] Y.-P. akache and A. F. Moisch, Spectra shape of UWB signas - Infuence of moduation format, mutipe access scheme and puse shape, Proceedings of the IEEE Vehicuar Technoogy Conference, VTC 3-Spring, vo. 4, pp , Jeu, Korea, Apri 3. 1] E. Fisher and H. V. Poor, On the tradeoff between two types of processing gain, IEEE Transactions on Communications, to appear. 11] J. G. Proakis, Digita Communications, Mc Graw Hi, ew York, 4th edition,. 1]. Lehmann and A. M. Haimovich, ew approach to contro the power spectra density of a time hopping UWB signa, Proc. the 37th Conference on Information Sciences and Systems CISS3, Batimore, MD, March 3. 13] D. Cassioi, M. Z. Win, F. Vataaro and A. F. Moisch, Performance of ow-compexity RAKE reception in a reaistic UWB channe, Proceedings of the IEEE Internationa Conference on Communications, ICC, vo., pp , ew York, Y, Apri 8-May,. 14] S. Gezici, A. F. Moisch, H. V. Poor, and H. Kobayashi, The tradeoff between processing gains of an impuse radio UWB system in the presence of timing itter, submitted to IEEE Transactions on Communications, Juy 4. 15] L. B. Michae, M. Ghavami, and R. Kohno, Mutipe puse generator for utra-wideband communication using Hermite poynomia based orthogona puses, Proc. IEEE Conference on Utra Wideband Systems and Technoogies UWBST, pp , Batimore, MD, May 1-3,. 16] R. K. Morrow, Jr. and J. S. Lehnert, Bit-to-bit error dependence in sotted DS/SSMA packet systems with random signature sequences, IEEE Transactions on Communications, vo. 37, pp , Oct ] M. B. Pursey, Performance evauation for phase-coded spreadspectrum mutipe-access communication - Part I: System anaysis, IEEE Transactions on Communications, vo. COM-5, pp , Aug ] D. E. Borth and M. B. Pursey, Anaysis of direct-sequence spread spectrum mutipe-access communication over Rician fading channes, IEEE Transactions on Communications, vo. COM-7, pp , Oct ] C. S. Gardner and J. A. Orr, Fading effects on the performance of a spread spectrum mutipe-access communication system, IEEE Transactions on Communications, vo. COM-7, pp , Jan ] W. Hoeffding and H. Robbins, The centra imit theorem for dependent random variabes, J. Duke Math., 15, pp , Sinan Gezici received the B.S. degree from Bikent University, Turkey in 1, and the M.A. degree from Princeton University in 3. He is currenty working toward the Ph.D. degree at the Department of Eectrica Engineering at Princeton University. His research interests are in the communications and signa processing fieds. Currenty, he has a particuar interest in synchronization, positioning, performance anaysis and mutiuser aspects of UWB communications. Zafer Sahinogu received his B.S. in EE from Gazi University, Turkey, M.S. in biomedica engineering and PhD with awards in EE from ew Jersey Institute of Technoogy JIT in 1998 and 1 respectivey. He worked at AT&T Shannon Research Labs in 1999, and has been with MERL since 1. His current research interests incude WK and MAC ayer issues in wireess sensor networks and impuse radio utrawideband precision ranging and positioning. He has co-authored a book-chapter on UWB geoocation, and has been author and co-author of more than 3 conference and ourna artices; and has provided significant contributions to emerging MPEG-1 standards on mobiity modeing and characterization for mutimedia service adaptation, to ZigBee on data broadcasting, routing and appication profie deveopment, to IEEE a standards on precision ranging. He is currenty the chair of ZigBee Industria Pant Monitoring Appication Profie Task Group. He hods one European and six US patents, and has pending. Hisashi Kobayashi is the Sherman Fairchid University Professor of Eectrica Engineering and Computer Science at Princeton University since 1986, when he oined the Princeton facuty as the Dean of the Schoo of Engineering and Appied Science His current research fieds incude: network security protocos, utra wideband UWB communications, wireess geoocation, hidden semi-markov mode HSMM and its computation agorithms. He is authoring with Brian L. Mark a graduate textbook Modeing and Anaysis: Foundations of System Performance Evauation to be pubished by Prentice Ha, 6.

8 He is a member of the Engineering Academy of Japan, a Life Feow of IEEE, and a Feow of IEICE of Japan. He received the 5 Eduard Rhein Technoogy Award for his pioneering work on high density digita recording technique, widey known as PRML Partia Response channe coding and Maximum Likeihood sequence detection. He was the recipient of the Humbodt Prize from Germany 1979, the IFIP s Siver Core Award 1981, and two IBM Outstanding Contribution Awards. He received his Ph.D. degree 1967 from Princeton University and BE and ME degrees 1961, 63 from the University of Tokyo, a in Eectrica Engineering. H. Vincent Poor S 7, M 77, SM 8, F 77 received the Ph.D. degree in EECS from Princeton University in From 1977 unti 199, he was on the facuty of the University of Iinois at Urbana-Champaign. Since 199 he has been on the facuty at Princeton, where he is the George Van ess Lothrop Professor in Engineering. Dr. Poor s research interests are in the areas of statistica signa processing and its appications in wireess networks and reated fieds. Among his pubications in these areas is the recent book Wireess etworks: Mutiuser Detection in Cross-Layer Design Springer: ew York, Y, 5. Dr. Poor is a member of the ationa Academy of Engineering and is a Feow of the American Academy of Arts and Sciences. He is aso a Feow of the Institute of Mathematica Statistics, the Optica Society of America, and other organizations. In 199, he served as President of the IEEE Information Theory Society, and he is currenty serving as the Editor-in-Chief of these Transactions. Recent recognition of his work incudes the Joint Paper Award of the IEEE Communications and Information Theory Societies 1, the SF Director s Award for Distinguished Teaching Schoars, a Guggenheim Feowship -3, and the IEEE Education Meda 5. 8