Designing Time-Hopping Ultra-Wide Bandwidth Receivers for Multi-User Interference Environments

Transcription

1 DESIGNING TH-UWB RECEIVERS FOR MUI 1 Designing Time-Hopping Ultra-Wide Bandwidth Reeivers for Multi-User Interferene Environments Norman C. Beaulieu and David J. Young Abstrat The multiple-user interferene (MUI) in timehopped impulse-radio ultra-wide bandwidth (UWB) systems is impulse-like and poorly approimated by a Gaussian distribution. Therefore, onventional mathed filter reeiver designs, whih are optimal for Gaussian noise, are not fully effiient for UWB appliations. Several alternative distributions for approimating the MUI proess and the MUI-plus-noise proess in UWB systems are motivated and ompared. These distributions have in ommon that they are more impulsive than the Gaussian approimation, with a greater area in the tails of the probability density funtion (pdf) ompared to a Gaussian pdf. The improved MUI and MUIplus-noise models are utilized to derive new reeiver designs for UWB appliations, whih are shown to be superior to the onventional mathed filter reeiver. Multipath propagation is abundant in UWB hannels and is eploited by a Rake reeiver. A Rake reeiver uses multiple fingers to omb the multipath rays with a onventional mathed filter implemented in eah finger. Rake strutures utilizing the new reeiver designs that are suitable for reeption of UWB signals in multipath fading hannels are provided. An optimal performane benhmark, based on an aurate theoretial model for the interferene whih fully eplains the features of the MUI pdf, is also presented. Analysis and simulation results are shown for the novel reeivers whih demonstrate that the new designs have superior performane ompared to the onventional linear reeiver when MUI is signifiant. Several adaptive reeivers are shown to always math or eeed the performane of the onventional linear reeiver in all MUI-plus-noise environments. Parameter estimation for the new reeivers also is disussed. Inde Terms Demodulation, digital reeivers, error rate, multiple-aess interferene (MAI), multiuser interferene (MUI), Rake reeiver, reeiver design, ultra-wide bandwidth (UWB). I. INTRODUCTION Ultra-wide bandwidth (UWB) wireless ommuniation systems have seen growing researh interest and industrial ativity. While UWB signaling has been used for radar and loation purposes for over 2 years, the appeal of UWB signaling for ommuniations has been more reent. Several key features make UWB attrative for a number of timely appliations. Etremely low transmitted power allows for UWB signals to underlay other users of the same radio spetrum; the United States Federal Communiations Commission (FCC) has established spetral masks for operation of UWB systems, allowing unliensed UWB systems to underlay liensed users in the same frequeny spetrum [1], and similar onventions have ourred around the world [2]. The FCC spetral masks, speified separately for indoor and outdoor appliations, are designed suh that UWB transmissions do not ause interferene with eisting narrowband users; rather, the ultra-wide bandwidth and ultra-low power signals are below the reeiver noise floor in liensed spetral regions. Thus, UWB signaling makes a wide bandwidth available for unliensed uses, bandwidth that might otherwise go unused at a partiular time and point in spae. As wireless devies beome even more prevalent, the need for simultaneous, olloated frequeny reuse, suh as offered by UWB systems, beomes paramount. Proposed appliations for UWB systems inlude wireless personal-area networks, short-range high-rate ommuniation between onsumer eletronis and omputer devies in the home, home automation, sensor networks, et. The short UWB pulse also embodies position loation and ranging apability within the modulation itself, allowing small, low-ost devies to be equipped with positioning features and further inreasing the variety of imaginable appliations. As well, the propagation harateristis of UWB signals allow for a high degree of spatial frequeny reuse, important as an inreasing number of devies are equipped with wireless features and as wireless onnetivity beomes a key onsumer epetation. Proposed UWB systems an be divided into two broad lasses, those based on a multi-band approah suh as orthogonal frequeny-division multipleing (OFDM), and those based on impulse radio (IR). IR systems [3]-[5] use an ultrashort signaling pulse transmitted at baseband, with no epliit modulation/demodulation omponents required in the transmitter or reeiver. IR-UWB systems an be further divided into systems whih use diret-sequene (DS) odes (DS-UWB) and systems whih use time-hopping (TH) odes (TH-UWB). The fous of this artile is ommuniation using time-hopped impulse radio in the presene of interfering TH-UWB users. It is ritial to distinguish between the types of UWB systems when onsidering multiple-user interferene (MUI). 1 Timehopped systems, whih will be desribed in more detail below, have substantially different MUI harateristis than DS-UWB or UWB-OFDM systems and demand different MUI models for use in system analysis and design; the almost ubiquitous Gaussian interferene model whih has been used etensively for a variety of ommuniation systems is generally not an aurate model for the MUI in TH-UWB ommuniations, as will be demonstrated. An overview of several superior reeivers designed with referene to aurate MUI models will be given. Moreover, Gaussian noise is a signifiant impairment in addition to MUI, and both the signal-to-noise ratio (SNR) and the signal-to-interferene ratio (SIR) will be relevant to system performane. An effetive UWB reeiver must work well in the ontinuum between the low-snr high-sir regime, 1 The term multiple-aess interferene (MAI) is used interhangably with the term multiple-user interferene (MUI) in the UWB literature. We will use the latter term.

2 DESIGNING TH-UWB RECEIVERS FOR MUI 2 whih will have Gaussian noise as a dominant impairment, and the high-snr low-sir regime, whih will be dominated by non-gaussian interferene. The reeivers disussed here inlude adaptive implementations that provide eellent or optimal performane in this ontinuum. UWB systems also eperiene interferene from narrowband systems. Modeling and mitigation of narrowband interferene is not onsidered in this paper but is onsidered in many other works (see, for eample, [6]-[9]). The remainder of this paper is strutured as follows. Setion II gives a brief overview of TH-UWB systems and other relevant aspets of UWB ommuniations, suh as UWB hannel models. Setion III onsiders MUI in depth, realling the salient features of MUI in TH-UWB systems, and motivating and evaluating several MUI models. Setion IV disusses several proposed reeivers that ahieve superior performane in hannel onditions where MUI is a dominant impairment. The reeivers are presented first for an additive white Gaussian noise (AWGN) hannel for larity, with detetion in multipath hannels overed in Setion V. Setion VI onsiders pratial estimation of the operating parameters required in eah reeiver design. A onlusion and summary an be found in Setion VII. II. TIME-HOPPED UWB SYSTEMS A. The UWB Signal Format Time-hopped UWB systems [3]-[5] use a very short basi signaling pulse whih will be denoted p(t). For purposes of analysis, the pulse may be onsidered to be normalized to unit energy, i.e. p2 (t)dt = 1 and to have pulse width T p, where T p is typially less than 1 ns. Fundamental to timehopped UWB systems is a frame struture with frames of length T f divided into hip slots of width T (Fig. 1). A given soure data bit is repeatedly transmitted over a number of frames, N s, in effet forming a length-n s repetition ode [3]- [5], [1]. The repetition ode allows reliable deisions to be made while the energy per transmitted pulse (hip) an be made very small, a property essential for underlaying other radio systems. In eah frame, the transmitted pulse is shifted to a different hip slot by a hopping ode { (k) i }, where i is the frame inde, k is the user inde, and (k) i {, 1..., N h 1}. Eah user s hopping ode is unique, and the use of the hopping odes avoids the situation where the signals from multiple users overlap entirely and every hip assoiated with a given data symbol eperienes a ollision with a signal hip from another user. Rather, when a hip ollision ours, adjaent frames assoiated with the same transmitted symbols and the same users will probably not also eperiene a ollision. Eah user signal in a TH-UWB system has low duty yle; i.e, the frame duration T f is muh larger than T and T p, and the symbol duration is T b = N s T f. For most of this paper, we will onsider a time-hopped binary phase-shift keying (TH-BPSK) UWB system, in whih the sign of the basi pulse p(t) is modulated aording to the data bit. Also ommon are time-hopped pulse-position modulation (TH-PPM) UWB systems, in whih the basi pulse p(t) is time-shifted for, say, a soure bit of 1, with no time T f 2T f 3T f 4T f User 2 T f 2T f 3T f 4T f T User 1 Fig. 1. The frame struture of two asynhronous TH-BPSK signals. The repeated pulses in eah user signal are employed to transmit the same data symbol. The pulses in red have eperiened a ollision; the time-hopping ode ensures that other frames assoiated with the same data symbols rarely also eperiene a ollision. shift applied for a soure bit of. The results of this paper are readily etended to TH-PPM systems. The signal pitured in Fig. 1 uses TH-BPSK modulation. The most ommonly reported pulse p(t) for studies of TH- UWB systems is the seond-order Gaussian monoyle, p(t) = 4 [ ep 2π t 2 ][ ( ) ] 2 t 1 4π (1) 6Tm T m T m where p(t) has been normalized to unit energy, and T m is a parameter ontrolling the pulse width. Families of more pratial time-limited UWB pulses are proposed in [11]. With energy per bit E b, the transmitted TH-BPSK signal for the kth user an be written as s (k) (t) = Eb N s i= d (k) i/n s p ( t it f (k) i T ) where represents the nearest integer less than or equal to and d (k) j is the jth symbol for the kth user. With N u denoting the number of users in the same overage area, the reeived signal in the absene of multipath propagation is N u r(t) = A k s (k) (t τ k ) + n(t) k=1 where A k is the real-valued hannel gain assoiated with the kth user, and τ k is the delay of the kth user relative to the desired user 1. The user delays τ k, k > 1 will be modeled as independent and uniformly distributed on [, T b ); that is, the users transmit asynhronously. Without loss of generality, user 1 is the desired user, with τ 1 =, and users k = 2,..., N u are undesired interfering users. The noise proess n(t) is modeled as additive white Gaussian noise (AWGN) with two-sided power spetral density N /2. The reeiver thus must detet the soure symbols for user 1 in the bakground of MUI from users 2,..., N u plus AWGN. The SIR, based on the output of the onventional orrelation reeiver, is defined as [3] SIR = A2 1 E bn s var{i} (2) (3)

3 DESIGNING TH-UWB RECEIVERS FOR MUI 3 where var{i} is the variane of the total interferene whih an be written as var{i} = E Nu [ 2 b k=2 A2 k p( s)p()d] ds. T f (4) Unless otherwise noted, the simulations desribed in this paper use T f = 2 ns, T =.9 ns, N h = 8, and a seondorder Gaussian monoyle pulse with T m =.2877 ns. B. The UWB Channel Wireless ommuniation systems ommonly eperiene multipath propagation, where multiple paths between transmitter and reeiver eist and the reeived signal is the superposition of signals from all paths. The signal orresponding to eah path has a unique time delay and amplitude, whih an be represented by a multipath profile of signed-amplitude versus delay. Several useful hannel models have been proposed for UWB systems [12]-[15]. The most ommon referene hannel models used for UWB analysis and simulation are those adopted by the IEEE a ommittee for the evaluation of UWB physial layer proposals [12], summarized in [13]. A key feature of these UWB hannel models is that paths are lustered. The lusters arrive aording to a Poisson proess, and within eah luster the individual rays arrive aording to an independent Poisson proess of different rate. The gain of a given path is governed by the produt of three independent random variables: a lognormal random variable representing luster fading, an independent lognormal random variable representing the fading of eah ray, and a Rademaher-distributed random variable (i.e., ±1 with equal probability) representing the inversion of the signal due to refletions. There is also overall shadowing represented by an independent lognormal random variable. The use of lognormal random variables for path gains is in ontrast to outdoor land mobile hannels, whih ommonly use a Rayleigh or Riean distribution. Due to the fine time resolution of the UWB signal, relatively few paths ombine at eah resolvable hannel delay, and the resulting sum is not well-approimated as a Gaussian random variable by the Central Limit Theorem [16]. Parameters for four models, denoted CM1 through CM4 and overing both line-of-sight and non-line-of-sight propagation, were speified by the IEEE a subommittee and an be found in [13]. III. ANALYZING MULTIPLE-USER INTERFERENCE The fous of this artile is TH-UWB ommuniation in MUI, in both single-path and multipath hannels. MUI is epeted to be a signifiant impairment in UWB systems beause of the wide appliability and epeted widespread use of UWB devies, with proposed appliations involving several devies o-loated in a small area, for eample an indoor room or offie. The short-range nature of UWB signals suggests a few dominant interferers at lose range. This has two onsequenes. One, the relative signal power of these few dominant interferers may be high. Two, the small number of interferers (oupled with the low duty yle of the UWB waveform, to be disussed further in the sequel) means that approimation of the interferene as a Gaussian proess is often inaurate. 2 Thus, modeling of MUI in TH-UWB systems is a unique and important problem. A. The Distribution of MUI in TH-UWB There are a number of ontributions in the literature to the modeling of the interferene at a TH-UWB reeiver generated by other TH-UWB transmitters. In ode-division multiple-aess (CDMA) systems, MUI an often reasonably be modeled by a Gaussian proess, and some authors have etended this assumption to TH-UWB systems. However, there are important differenes between CDMA and TH-UWB systems that make the Gaussian assumption inappropriate. Most signifiantly, a transmitted CDMA signal has a duty yle that is essentially unity; a pulse is transmitted in every hip slot. When multiple CDMA users are simultaneously transmitting, the reeiver sees a superposition of the signals from many independent users in eah hip slot. The interferene proess tends to a Gaussian proess by the Central Limit Theorem and onvergene is relatively fast with respet to the number of users. The Gaussian approimation is onvenient in terms of reeiver design and analysis sine transmission in additive Gaussian noise is a well-studied problem and, when the interfering users are onsidered purely as additive noise, the interferene-plus-noise proess remains Gaussian with straightforward definitions in terms of the mean and variane (power) of the omponent noise and interferene proesses. A TH-UWB signal, by ontrast, has a low duty yle; i.e., the frame duration is muh longer than the pulse duration, and only a single pulse is transmitted per frame. A given hip slot sees interferene from relatively few users ompared to the number of interfering user links that may be simultaneously ommuniating. Also, both the propagation harateristis of UWB signals, and the motivating appliations for UWB systems, suggest a small number of interfering users at lose range. This is in ontrast to some CDMA systems whih have a wide overage area and many ontributing interferers of lower relative power. For eample, several UWB devies may be situated within a small area in a residential living room and ontribute most of the interferene power, while the greater number of UWB devies situated in other parts of the residene or the neighborhood have muh greater path loss and thus muh smaller interfering impat on the desired link. These properties, intuitively, lead to an interferene pdf that is not Gaussian. Moreover, an interfering TH-UWB signal is impulse-like by definition ( impulse radio ), with the impulses having random arrivals due to the pseudo-random hopping ode and random user delays. The interferene proess therefore intuitively resembles impulsive noise, not Gaussian noise. The MUI in systems using both oversimplified (for the purposes of eposition) pulse models as well as pratial UWB pulses has been investigated in [17] and the pdf of the 2 In addition to a small number of interferers, the relative high powers of the interferers also hinders onvergene to a Gaussian distribution.

4 DESIGNING TH-UWB RECEIVERS FOR MUI 4 MUI is disussed in detail in [17] and [18], [19]. Several key observations are made that eplain the slow onvergene of the UWB MUI to a Gaussian distribution via the Central Limit Theorem. First, the pdf has an impulse at the origin (zero amplitude) with magnitude equal to (1 2D) Nu 1 where D is the duty yle of the UWB waveform, roughly equal to the pulse duration divided by the frame duration, and N u is the number of users in the system. It is noted in [18], [19] that the duty yle is neessarily low sine it is desireable to design the system so the frame duration T f is greater than the delay spread of the hannel, and the propagation harateristis of the hannel ditate that the delay spread of the hannel is muh greater than the pulse duration. The duty yle is also small due to the time-hopping design, where only one pulse is transmitted per frame; a low duty yle gives the pulse room to hop in the frame. It is futher observed in [17] and [18], [19] that the MUI pdf has other singularities due to zeros in the derivative of the pulse autoorrelation funtion. This an be inferred from the work of [2] [23], where the MUI has been written as a funtion of the pulse autoorrelation with a random variable in the argument of the pulse autoorrelation. Using standard tehniques [24], the pdf of the MUI an be written as a funtion of the pdf of this random variable, a funtion whih inludes the derivative of the pulse autoorrelation in its denominator. Thus, zeros in the derivative beome singularities in the MUI pdf. Suh singularities are not easily aommodated in losedform epressions for the pdf, nor would the resulting pdf be useful for optimal reeiver design. The pdf of a sum of independent random variables is given by the onvolution of the omponent pdfs; the pdf of the interferene sum over users and frames inherits some singularities from the omponent pdfs. Therefore, the onvergene to a Gaussian pdf is slow with respet to the number of terms in the sum. For UWB systems, where the number of signifiant interferers may be few, a Gaussian approimation an fail. Multipath is partiularly material in UWB systems, where short-range indoor environments have a rih set of refletion and refration surfaes to form paths. Moreover, the multipath delay profile is typially of muh greater duration than the transmitted pulse, and assumptions made on the basis of ultra-short UWB pulses must be evaluated in the light of a muh longer hannel response. A multipath hannel might be epeted to improve the onvergene to a Gaussian pdf somewhat, in that it effetively lowers the duty yle of the reeived signal for a partiular interferer so a given hip slot sees a greater number of independent interfering signals. However, an assumption of onvergene of the interferene pdf to a Gaussian distribution at a partiular finger of a Rake reeiver (to be disussed in Setion V) still may not be aurate, as will be seen in the sequel. B. MUI Modeling Sine determining an eat epression for the pdf of the MUI in a TH-UWB system is not straightforward and beause the eat distribution would not be ompatly epressed, it is neessary to onsider modeling MUI with probability distributions that are both suffiiently aurate, and tratable for reeiver design and performane analysis. The Gaussian distribution, as disussed above, is tratable but not an aurate model. A number of other distributions have been onsidered. These distributions have in ommon that they model proesses whih are more impulsive than the Gaussian proess, with pdfs that have heavier tails than the Gaussian proess (i.e., a higher probability of larger-magnitude events). Three key riteria in developing a suitable model are the auray of the model, the etensibility of the MUI model to a model for MUI plus additive Gaussian noise, and the utility of the model for synthesis of pratial reeiver designs. A number of authors have onsidered the distribution of MUI in UWB systems and performane of UWB systems in the presene of MUI, e.g. [18] [23], [25] [49]. Early results used a Gaussian approimation to the interferene in determining bit-error rates (BERs) for TH-UWB systems [3], [4], [5] [52]. However, Gaussian approimations were shown to signifiantly underestimate the BER [1], [2], [26], [27] of UWB systems for medium to large SNRs, i.e., the SNR region where MUI is the signifiant impairment, motivating non-gaussian analysis and designs. A number of non-gaussian distributions have been onsidered for IR-UWB MUI or MUI plus noise, suh as the Laplae distribution [28] [33], the Gaussian-Laplae miture distribution [18], [19], the generalized Gaussian distribution [34], [37], [38], the Gaussian miture distribution [32], [33], [44], [49], the Middleton Class- A noise distribution [25], [32], [33], and the symmetri alphastable distribution [31], [41], [42], [53], [54]. Table I lists properties for eah of these distributions. For a binary ommuniation system with a noise-plusinterferene pdf that is symmetri about the origin, the BER of a onstant threshold detetor with equal energy symbols is diretly proportional to the area of a tail region of the pdf [18], [19], [55]. The results of [2] [22] suggest that the Gaussian distribution has insuffiient area in the tails to aurately model UWB MUI (Fig. 2). A measure of the heaviness of the tails of the MUI pdf is the eess kurtosis [18], [19], [34], [36] I = E{ I 4} [E{I 2 3. (5) }] 2 A positive kurtosis indiates a distribution with heavier tails than the Gaussian distribution, suggesting that the true UWB MUI, and more appropriate models, have positive eess kurtosis. This is supported by results in [18], [19], [34], where the eess kurtosis of the total interferene I, and the interferene in eah frame, I i, is determined by simulation. The results from [19] are dupliated in Table II, where I denotes the kurtosis of the total MUI for one symbol, and Ii denotes the kurtosis of the partial MUI omponents in one frame. The distribution of the orrelator output for a partiular frame is seen to deviate signifiantly from a Gaussian distribution ( = ) even for a relatively large number of interferers. The Laplae distribution is a ommon model for impulsive noise [56], [57], and reeivers designed to operate in a MUI environment modeled by a Laplae distribution have been presented in [28] [3]. To support this, the empirial pdf of the interferene and the Laplae pdf have been plotted in [3]

5 DESIGNING TH-UWB RECEIVERS FOR MUI 5 TABLE I PROBABILITY DISTRIBUTIONS CONSIDERED FOR MODELING MUI. Name pdf Parameters Comments Gaussian f R (r) = 1 ep (r η)2 2πσ 2σ 2 Laplae f R (r) = 1 2 ep r η Gaussian- Laplae miture Generalized Gaussian f R (r) = ep(σ2 /2 2 ) ` ˆep r ` 2 Q r σ + σ +ep ` r Q r ` σ + σ 1 f R (r) = 2Γ(1+1/p)A(p,σ) ep( r η A(p,σ) p) η =mean σ 2 = variane η = mean 2 2 = variane σ from Gaussian from Laplae p = order (shape) η =mean σ 2 = variane A(p, σ) def. in IV-H poor model for MUI good approimation to MUI alone intuitive: Laplae-modeled interferene plus AWGN Laplae and Gaussian are speial ases; an adapt between the two Cauhy f R (r) = ζ π[(r µ) 2 +ζ 2 ] µ =loation ζ = sale α-stable with α = 1; pdf and opt. detetor known. Models MUI only. Infinite seond moment. Symmetri α-stable Gaussian miture Middleton Class-A unknown for general α CF Φ(ω) = ep( ζ ω α + jωµ) α = har. eponent (shape) µ =loation, ζ = dispersion f R (r) = P «L L λ q l l=1 ep r2 2πσ l 2 2σ l 2 {λ 1,..., λ L } {σ σ2 L } f R (r) = P N i= Ai q e A i! 2πσ i 2 «ep r2 2σ i 2 Cauhy and Gaussian are speial ases, but general pdf unknown and opt. detetor impratial. Infinite seond moment. eellent fit to simulation but many parameters to estimate; aommodates AWGN A, N, good fit to simulation; epressible {σ1 2,..., σ2 N } in terms of system parameters; aommodates AWGN 1 TABLE II FROM [19], EXCESS KURTOSIS OF A SIMULATED MUI PROCESS N s = 2 N s = 4 N u N h N s T f I Ii Theoretial BER of TH-BPSK System Simulation Gaussian Approimation Fig. 2. The average BER of a TH-BPSK system versus SNR for a repetition ode with N s = 2 and N s = 4 assuming seven asynhronous interferers (from [22]). and shown to have lose resemblane, partiularly in the tail region where the Gaussian pdf deviates signifiantly from the empirial pdf of the MUI (Fig. 3). MUI simulations also are provided for three and 15 interferers in [19], reprodued here in Fig. 4. Fig. 4(a) shows the Laplae distribution to be a better math to the simulated interferene for three interferers and Fig. 4(b) shows the Laplae distribution to be a good math to the simulated interferene for 15 interferers. The Laplae distribution has an eess kurtosis of 3, and based on Table II should be superior to a Gaussian approimation to model both the total interferene and the frame interferene. The Laplae distribution also has been found in [58], [59] to more losely approimate the sum of inter-hip, inter-path, and inter-symbol interferene in UWB Rake reeption than a Gaussian approimation. A model based on the addition of a Laplae proess, representing the MUI, to an AWGN proess, representing the ambient noise, is eamined in [18], [19], and optimal and suboptimal reeivers are designed based on this Gaussian- Laplaian model. The generalized Gaussian pdf is onsidered as a model for the total disturbane (MUI plus noise) in [34] [38]. The generalized Gaussian pdf has a parameter, denoted p in [37], [38], whih hanges the shape of the distribution and allows for adaptation to various hannel onditions. The results show

6 DESIGNING TH-UWB RECEIVERS FOR MUI Simulation Laplae pdf Gaussian pdf pdf fi() of MUI pdf fi() of MUI Simulated pdf Laplae pdf Gaussian pdf Fig. 3. Comparison of the pdf of the total MUI for fifteen interfering UWB signals obtained from simulation, the Laplae approimation, and the Gaussian approimation (from [3]). the generalized Gaussian pdf to be a fleible lass of pdfs to model the disturbane. When Gaussian noise is dominant, the generalized Gaussian pdf yields eatly the Gaussian pdf with p = 2. As MUI beomes more signifiant, lower values of p make the pdf better fit the observed disturbane. The Laplae distribution, useful for modeling MUI alone and disussed above, is a speial ase for p = 1. Values of p less than unity give more impulsive distributions than the Laplae distribution. Thus, the generalized Gaussian distribution offers muh fleibility in MUI-plus-noise modeling. Plots of the empirial pdf of the noise-plus-interferene for a TH-UWB system are provided in [37] for SIR = 1 db and different SNR values (Fig. 5). It is observed in [37] that for low SNR the pdf of Y i is approimately Gaussian and p 2 is appropriate. For moderate SNR the pdf an be approimated as the Laplae distribution with p = 1, while for high SNR a value of p < 1 is appropriate. The generalized Gaussian pdf also lends itself to obtaining tratable, pratial reeiver designs analytially. The symmetri alpha-stable lass of probability distributions [6] [62] has reeived reent interest for modeling UWB MUI. The Gaussian distribution is the stable distribution with α = 2 and the Cauhy distribution is stable with α = 1. Stable distributions with α < 2 are suitable for modeling data with large flutuations and have been used to model impulsive noise. The parameter α (, 2] determines the shape of the pdf, with lower α yielding more impulsive distributions with heavier tails. Eept for α = 2, stable distributions have algebrai tails and infinite variane. The use of a symmetri alpha-stable distribution to model the interferene in TH- UWB has been eamined in [31] by onsidering a smoothed pdf of simulated MUI. Other distributions ompared were the Gaussian distribution, generalized Gaussian distribution, Laplae distribution, and Cauhy distribution. The smoothing, whih removes the singularities in the empirial pdf, is justified in [31] and results in a meaningful graphi omparison, dupliated here in Fig. 6. It is seen that the alpha-stable pdf fi() of MUI (a) Simulation Laplae pdf Gaussian pdf Fig. 4. A omparison of the pdf of the MUI with Gaussian and Laplaian approimations for (a) three interferers, (b) fifteen interferers (from [19]). distribution provides an eellent math to the tail behavior of the MUI when α is estimated using the method of [31] (see Setion VI). Sine the tail behavior is ritial in determining the BER, this suggests a symmetri alpha-stable distribution is an eellent andidate for modeling UWB MUI. Referenes [32], [33] ompare the suitability of the Gaussian approimation, the Gaussian miture (GM) distribution, the Middleton Class-A (MCA) noise distribution, and the Laplae distribution for modeling MUI in TH-UWB systems. A GM distribution, whih has a pdf given by a weighted miture of Gaussian pdfs with different varianes, has been used in [44] to model MUI in an infrared UWB appliation, using the epetation-maimization (EM) algorithm [63] to determine the parameters of the model. It has been proposed in [25] to model MUI by a MCA noise distribution [64], whih also has been widely used for modeling impulsive noise. The pdfs of the total MUI under eah assumption, (b)

7 DESIGNING TH-UWB RECEIVERS FOR MUI 7 pdf fy i () of MUI plus noise pdf fy i () of MUI plus noise SNR = db Simulation Gaussian pdf Laplae pdf Gen. Gaussian with p = SNR = 16 db (a) Simulation Gaussian pdf Laplae pdf Gen. Gaussian with p = (b) pdf fi() of MUI fx() (a) Simulation Gaussian pdf Generalized Gaussian pdf Laplae pdf Cauhy pdf Alpha-stable pdf Simulation Gaussian pdf Generalized Gaussian pdf Laplae pdf Cauhy pdf Alpha-stable pdf pdf fy i () of MUI plus noise SNR = 32 db Simulation Gaussian pdf Laplae pdf Gen. Gaussian with p = Fig. 5. The simulated pdf of the amplitude of the total disturbane sample (noise plus interferene) in eah frame, plotted with the Gaussian pdf, the Laplaian pdf, and the generalized Gaussian pdf for different values of p. (a) SNR = db, (b) SNR = 16 db, () SNR = 32 db (from [37]). and the pdfs of the MUI plus noise, were ompared in [32], [33] by simulation. In addition, the predited BER under eah approimation was ompared to the eat BER analysis reported in [21]. It was noted that the Gaussian approimation did not represent the impulsive omponent and heavier tails of the simulated pdf, and that the signifiantly lighter tail region would predit the signifiantly underestimated BER results revealed in [21]. Based on the pdf omparison, the GM and () Fig. 6. Approimations to the simulated MUI pdf, (a) before smoothing, (b) after smoothing (from [31]). MCA distributions are reported in [32], [33] to be the better approimations of those onsidered, with the GM model most losely mathing the tail region (Fig. 7). The presented BER results (Fig. 8) onfirm that the GM model provides good BER estimates. However, it was found that the number of iterations used in the EM algorithm was ritial for providing aurate BER estimates, and in partiular the small number of iterations reported in [44] to estimate the pdf was insuffiient to aurately estimate the BER. It is also noted in [32], [33] that while the MCA model and GM model both an be written as a sum of Gaussian pdfs, the parameters for the MCA model an be determined from the UWB system parameters with muh less omputational ompleity than determination of the GM model parameters by the iterative EM algorithm, based on noisy hannel samples for eah SIR value and SNR value. The Laplae-based model was, in both the pdf plots and in the BER results, seen to be farther from the eat ase than the (b)

8 DESIGNING TH-UWB RECEIVERS FOR MUI 8 pdf fy i () of MUI plus noise Atual pdf Gaussian Miture Model pdf Middleton Class-A pdf Laplae pdf Gaussian pdf Fig. 7. A omparison of the pdf tails of the GMM, MCA, Laplae, and Gaussian approimations with simulation (from [32]) Aurate BER BER from Gaussian Miture Model BER from Middleton Class-A Model BER from Laplae Model BER from Gaussian Approimation Fig. 8. The average BER of a TH-PPM UWB system versus SNR with N s = 4, estimated using the different methods (from [32]). GM or MCA models, underestimating the BER. However, it an be seen in Fig. 8 that the differene between the predited BERs for the MCA and Laplae-based models and the true BER is muh smaller than the differene between the BER predited by the Gaussian approimation and the true BER. All of the impulsive-based models apture the error-rate floor behavior observable in the aurate BER, while the Gaussian model ompletely fails to apture the moderate to high SNR performane of the system. It is lear, irrespetive of the eat form of distribution, that distributions suitable for impulsive noise are a muh better fit to the MUI statistis than a Gaussian pdf. Summary omments on eah model an be found in Table I. IV. DESIGN OF TH-UWB RECEIVERS FOR SUPERIOR PERFORMANCE IN MUI In Setion III we have provided onsiderable evidene that the MUI in UWB systems is not well modeled by a Gaussian distribution. In view of these onlusions, there is potential benefit to be gained in bit error rates (or outage rates, or other system performane indiators) by designing reeivers that are appropriate for signals embedded in an aurately-modeled MUI bakground, and that are able to adapt to hannels where MUI is a dominant impairment. Moreover, a demonstrated performane enhanement obtained with suh novel reeivers will further justify the underlying models. In this setion, we start with a general eample of detetion in non-gaussian noise, and then desribe several reently proposed reeivers that offer superior performane in MUI in omparison to the onventional UWB reeiver. Realisti UWB system analysis must inlude the effets of a multipath hannel, and pratial UWB system designs must be able to ope effetively with multipath. Nonetheless, analysis, modeling, and design of UWB systems operating in singlepath hannels is useful. First, novel designs an be motivated and desribed most learly in a single-path hannel. Seond, often the key design priniples developed for the single-path ase an be readily etended to novel designs for the multipath hannel. In this setion we first desribe and evaluate all proposed reeivers in the single-path (AWGN) hannel, then the orresponding reeivers adapted for a multipath hannel will be presented in Setion V. A. Optimal Detetion in Non-Gaussian Noise As a starting point, suppose that in a generi ommuniations system we have N samples {r i } N 1 i= of a signal reeived in additive noise, with r i = Ad + n i. The binary symbol d takes a value in { 1, 1}, A is a nonnegative salar amplitude, and {n i } are additive noise samples. Let the samples r i be independent with a ommon pdf f R (r). The maimum-likelihood (ML)-optimum reeiver minimizing the overall probability of error bases its deisions on the loglikelihood ratio [56], [57] Λ = N 1 i= log f R(r i d = +1) f R (r i d = 1). (6) That is, the deision statisti is the sum of the log-likelihood ratios for eah sample onsidered individually. Considering the ase of equiprobable soure symbols for simpliity, the deision on the transmitted bit is made aording to Λ > = d = 1 Λ < = d = 1. The ase Λ = an be deided by a fair oin toss. This deision rule is valid for any pdf f R (r), i.e. any noise distribution. In the partiular ase where the noise has a Gaussian distribution with zero mean and variane σ 2, the reeived signal has pdf f R (r i ) = 1 2πσ ep ( (r i da) 2 ) 2σ 2 (7) (8)

9 DESIGNING TH-UWB RECEIVERS FOR MUI 9 and the log-likelihood ratio after simplifiation beomes Λ = 2A N 1 σ 2 r i. (9) That is, in the Gaussian ase the optimal detetor takes the sum of the samples of the reeived signal and ompares this sum with the threshold zero. (The multipliative onstant is irrelevant to the deision.) The summand r i is a partial deision statisti for the ith sample. We emphasize that while (6) and (7) are valid for any noise pdf, (9) is optimal only for the Gaussian pdf. Thus, when the bakground noise plus interferene is not Gaussian-distributed the reeiver based on (9) is not an optimal reeiver. A non-gaussian eample from [57], whih will be important in the sequel, is the ase in whih the noise samples {n i } have a Laplae distribution and therefore the reeived signal has pdf f R (r i ) = 1 ( 2 ep r ) i da (1) i= where > is a sale parameter. In this ase the log-likelihood ratio (6) is Λ = 2 N 1 ( r i 2 + A 2 r i 2 A ) 2. (11a) i= Again, the form is a sum over partial deision statistis ( r i + A r i A )/2 for eah sample. However, the partial deision statisti now is formed as a nonlinear operation on r i, whih an be written as r i 2 + A 2 r i 2 A A r i A 2 = r i A < r i < A (11b) A r i A. Thus, (11) applies a soft-limiting operation to the partial deision statistis before summation. Note that the assumed independene between the samples r i has been used in writing (9) and (11). This independene ondition will not be met preisely in a UWB system [23]. A reeiver whih takes into aount dependene between frames will, in theory, provide better performane. However, the independene assumption vastly simplifies both the analysis and reeiver design; the goal of simple and pratial reeiver designs motivates ignoring the small dependene between frames at the design phase. B. The Conventional UWB Reeiver Consider detetion of the th bit of user 1, where the users transmit signals aording to (2). The onventional singleuser orrelation reeiver uses a orrelation template waveform mathed to the desired user s signaling waveform to form r = N s 1 i= (i+1)tf it f ( r(t)p t it f (1) i T )dt. (12) (Tehniques for synhronization of UWB signals an be found, for eample, in [65]-[68].) Sine r(t) is the sum of the desired signal, MUI, and noise, all of whih are independent, we an define the partial deision statisti and write it as the sum of signal, MUI, and noise, r i = (i+1)tf it f ( ) r(t)p t it f (1) i T dt = S f + I i + N i. (13) Note that the signal part of r i is the same for all i, and an be written S f = A d (1), where A > is a salar amplitude and d (1) is the zeroth bit of the first user, i.e. the symbol to be reovered by the detetor. The amplitude A an also be thought of as the unsigned signal omponent in a single frame, A = S f = A 1 Eb N s (14) where A 1 is the hannel gain of the desired user. The onventional mathed-filter (CMF) linear reeiver forms the deision statisti with the deision rule Λ CMF = N s 1 i= Λ CMF > = d (1) = 1 Λ CMF = d (1) = 1. r i (15) (16) The onventional reeiver is the BER-optimal oherent reeiver for a signal in a bakground of Gaussian noise plus Gaussian interferene and has been widely applied to UWB systems. The terms onventional reeiver, linear reeiver, and CMF have been used to desribe this reeiver. If the reeiver filter or orrelator is viewed as being mathed to the entire symbol, then only the onventional reeiver is suh a mathed filter and the CMF terminology is unambiguous. One might not epet that one an improve upon the CMF UWB reeiver performane, espeially in a stati hannel. In partiular, any binary signaling sheme an be onverted to an equivalent binary antipodal signaling sheme [69] and, therefore, the detetion of the signal ultimately redues to a threshold omparison. The reason that a better UWB reeiver design is possible is beause of the frame struture of the UWB signal, i.e., the repetition ode. The repetition ode struture represents an inherent diversity system within the CMF UWB reeiver. It is important to note that adding together the outputs from the orrelators of the different frames is a ML struture in AWGN, but is not an ML struture in the presene of MUI whih is not Gaussian. Simply adding the outputs of the orrelators from all the frames is not an optimal proessing of the frame orrelator output signals. C. Overview of Novel Reeivers Unfortunately, two problems arise in optimal reeiver design. First, the eat pdf of the MUI in a UWB system annot be ompatly written (see Setion III) and does not lead to tratable reeiver designs. Seond, the inevitable presene of Gaussian noise in the system means that the total noise-plusinterferene has a ompliated pdf that is the onvolution of the eat MUI pdf with a Gaussian pdf, even less suggestive of

10 DESIGNING TH-UWB RECEIVERS FOR MUI 1 useful reeiver designs. Moreover, the shape of the resulting pdf depends on the UWB system parameters and the timevarying noise and interferene environment and may not be available a priori. The hallenge, therefore, is to obtain pratial reeivers that offer superior performane in MUI with no loss of optimality in AWGN, in the absene of the eat MUI-plus-noise pdf. This hallenge has been approahed in several different ways outlined in the sequel. Setion IV-D desribes the soft-limiting UWB reeiver, designed to detet a UWB signal in MUI having an approimating Laplae distribution. It will be shown that the softlimiting reeiver provides superior performane in MUI, but is not optimal in Gaussian noise. The remaining reeivers to be disussed inlude the CMF reeiver as a speial ase of the reeiver parameters, and thus, theoretially with parameter adaptation optimal for the interferene and noise onditions, an offer superior performane in MUI and no loss of performane in AWGN. That is, the reeivers of Setions IV-E through IV-J an always provide performane that meets or eeeds both the CMF reeiver and the soft-limiting UWB reeiver. Pratial or model-based parameter estimation an impat the optimality of speifi reeiver designs, however, in some hannel onditions. The adaptive soft-limiting reeiver of Setion IV-E etends the soft-limiting reeiver, mathing the soft-limiting threshold to the SNR and SIR of the hannel. The Gaussian-Laplaian miture reeiver of Setion IV-F and the simplified Gaussian- Laplaian miture reeiver of Setion IV-G are designed to detet a signal in a bakground of interferene plus AWGN, where the interferene has an approimating Laplae distribution. The p-order metri reeiver, p-order adaptive-thresholdlimiting reeiver and myriad filter reeiver of Setions IV-H and IV-I eah onsider approimating distributions to the MUI-plus-noise that ontain the Gaussian distribution as a speial ase, allowing adaptation between MUI and Gaussian environments with no loss of optimality in the Gaussian ase. Lastly, the zonal reeiver to be desribed in Setion IV-J is based on observations made for simulated MUI pdfs, and again an be adapted to provide no loss of optimality in a pure AWGN environment. Zero-threshold detetion is unaffeted by multipliation of the detetion statisti by a positive onstant. In order to unify the treatment of the various reeivers, some of the nonlinearity funtions presented in this paper have been saled from those found in the original referenes. A unified blok diagram of all the proposed reeivers is given in Fig. 9. None of the reeivers onsidered here are optimal for TH- UWB, eept in the limiting ase where MUI is absent and the only impairment is AWGN. The design of an optimal reeiver would require a tratable epression for the pdf of the interferene plus noise in a TH-UWB system. However, with reasonable additional ompleity over the CMF reeiver, eah reeiver desribed here shows superior performane in MUI and, with the eeption of the non-adaptive reeiver of Setion IV-D, no loss of performane in AWGN when optimized to the TH-UWB noise-plus-interferene onditions. Table III summarizes the advantages and disadvantages of eah r(t) r i r i p(t ) T f Pulse Generator Conventional g CMF ( ) r(t)p(t )dt Adaptive Soft-Limiting g SL ( ) T opt T opt r i Baseband Proessing & Timing r i = r i r i Gaussian-Laplaian Miture r i g GLM ( ) A A r i Simplified Gaussian-Laplaian Miture r i Fig. 9. g SGLM ( ) A A r i r i r i r i g( ) r i Aumulator Deision Rule P-Order Metri g p-omr ( ) Adaptive Threshold P-Order Metri g p-omatlr ( ) Myriad Detetor g myriad ( ) Zonal g zonal ( ) t h t l Unified blok diagram of the novel reeiver strutures. ri t l t h r i r i r i r i

11 DESIGNING TH-UWB RECEIVERS FOR MUI 11 TABLE III A SUMMARY OF SOME ADVANTAGES AND DISADVANTAGES OF EACH RECEIVER. Reeiver Advantages Disadvantages CMF Soft-Limiting Adaptive-Threshold Soft-Limiting GLM SGLM p-omr optimal in absene of interferene; less omple simple nonlinearity; good performane for low SIR/high SNR; no hannel information required beyond A 1 simple nonlinearity; performane always better than or equal to CMF nonlinearity funtion epressed in terms of funtions of SIR and SNR; intuitive miture; able to adapt over SNR/SIR simple soft-limiting nonlinearity, but adaptive to hannel onditions and better performane than soft-limiting reeiver; nonlinearity epressed in terms of SIR and SNR very good performane; nonlinearity funtion an be epressed in terms of interferene moments and SNR inferior performane in low SIR; does not adapt to noise and interferene onditions inferior to CMF for low SNR/high SIR; does not adapt to noise and interferene onditions adapted thresholds are pre-simulated and stored in lookup table ompliated nonlinearity funtion performane inferior to GLM for some noise and interferene onditions nonlinearity funtion more omple than soft-limiter p-omatlr Myriad Filter Zonal performane always better than or equal to both CMF and p-omr with same ompleity of nonlinearity eellent performane; nonlinearity epressed in terms of hannel estimates simple nonlinearity; very good performane adaptation based on pre-simulated values more omple nonlinearity; parameters require estimation of empirial harateristi funtion adapted thresholds use pre-simulated values stored in lookup table reeiver. In order to streamline the presentation of the various reeivers, disussion of algorithms for estimation of reeiver parameters from hannel data is deferred to Setion VI. D. The Soft-Limiting UWB Reeiver The first lass of reeiver designs to be onsidered are the soft-limiting designs of [28], [29], [3]. Motivated by the observation that the MUI in UWB is impulsive in nature, MUI is modeled in these reeivers by the Laplae distribution, whih is a traditional model for impulsive noise as disussed in Setion III. The BER-optimal reeiver for a onstant signal in Laplae noise was given by eample in (11), and is a standard result [56], [57], whih was proposed as a basis for novel UWB reeiver designs in [29]. The deision statisti for the onventional UWB reeiver has been given in (15), a sum of the orrelator outputs r i for eah frame. The soft-limiting detetor proposed in [29], [3] forms a deision based on a sum of transformed orrelator outputs g SL (r i ), where Λ SL = N s 1 i= g SL (r i ), A, A g SL () =, A < < A A, A. (17a) (17b) The transmitted information bit d (1) is deided aording to the rule Λ SL > = d (1) = 1 (17) Λ SL = d (1) = 1. The threshold A is the square root of the reeived signal energy in eah frame for the desired user, as defined in (14). Use of A as the threshold orresponds to the limiting threshold in (11). It is a sensible design hoie sine the signal omponent of r i has an amplitude of A ; when r i > A this is due to noise or interferene. The benefits of the soft-limiting struture in suppressing MUI are intuitive. The UWB pulse has a short duration relative to the frame duration. By design, time-hopping is used to ensure that ollisions between the desired user and a given interfering user in suessive frames are rare. Thus, many of the desired user s frames will see negligible interferene from a given interfering user. When a ollision ours, the orrelator output for the frame with a pulse ollision may have a relatively large amplitude, having a large effet in the deision statisti sum Λ CMF and yet a very small SIR making this effet deleterious. The soft-limiter moderates the effet of olliding pulses in the overall deision statisti. Indeed, in simulations where the desired signal is orrupted by MUI only, the soft-limiting reeiver shows better performane in terms of BER than the onventional reeiver, with the performane benefit inreasing at low SIR levels (Fig. 1(a)). The benefits of the soft-limiting reeiver are espeially signifiant for a small number of interfering users (Fig. 1(b)). As the number of interfering users inreases, the

12 DESIGNING TH-UWB RECEIVERS FOR MUI CMF reeiver, N s = 4 Soft-limiting reeiver, N s = 4 CMF reeiver, N s = 8 Soft-limiting reeiver, N s = SIR (db) (a) N s = 8 N u = 16 CMF reeiver Soft-limiting reeiver (a) 1 N s = 4 N u = CMF reeiver, N s = 4 Soft-limiting reeiver, N s = SIR (db) Fig. 1. The average BER versus SIR of soft-limiting and onventional TH- BPSK UWB reeivers assuming (a) fifteen asynhronous interferers, (b) three asynhronous interferers (from [3]). (b) CMF reeiver Soft-limiting reeiver Fig. 11. The average BER versus SNR of soft-limiting and onventional TH-BPSK UWB reeivers assuming (a) fifteen asynhronous interferers with N s = 8, (b) three asynhronous interferers with N s = 4 (from [3]). (b) interferene pdf beomes loser to a Gaussian pdf, and the onventional reeiver beomes loser to an optimal reeiver. This is important sine, as disussed in Setion III, the shortrange nature of UWB signals suggests a few dominant interferers loated lose to the reeiver. However, the improvement of the soft-limiting reeiver is seen throughout the SIR range and for both interferene environments onsidered. More relevant is the performane of the reeiver when both MUI and AWGN are onsidered, sine the soft-limiting reeiver will be suboptimal in an AWGN-only environment. It an be seen in Fig. 11 that there is a rossover threshold in SNR, below whih the BER for the onventional reeiver is less than that of the soft-limiting reeiver, and above whih the soft-limiting reeiver is superior. This motivates an adaptive version of the soft-limiting reeiver that always obtains superior performane, disussed in the following setion. E. The Adaptive-Threshold Soft-Limiting Reeiver The onventional reeiver is optimum when the only hannel orruption is AWGN. The soft-limiting reeiver is not optimal in this ase and it is epeted, and observed in Fig. 11, that the performane of the soft-limiting reeiver is worse than the onventional reeiver for high-sir low-snr regimes, i.e., where Gaussian noise is the dominant impairment. However, it is noted in [3] that the soft-limiting reeiver is idential to the onventional reeiver when the limiting threshold is set to infinity. Therefore, an adaptive implementation is proposed in [28], [3] in whih the threshold T opt is optimized to minimize BER. The reeiver omputes Λ ASL = N s 1 i= g ASL (r i ) (18a)