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4656 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 Robust Ultra-Wideband Signal Acquisition ¸ Ersen Ekrem, Student Member, IEEE, Mutlu Koca, Member, IEEE, and Hakan Delic, Senior Member, IEEE Abstract—Ultra-wideband (UWB) communication is envisaged to be more challenging than that of the traditional spread to be deployed in indoor environments, where the noise distribu- spectrum systems are the low duty cycle with very short tion is decidedly non-Gaussian. A critical challenge for impulse duration pulses and the low-power constraint. The former is radio UWB is the synchronization of nanosecond-long pulses. In this paper, we propose two acquisition schemes which are robust necessary to spread the UWB signals and to allow for more to uncertainties in the noise distribution and which operate efﬁcient multiple-accessing. The latter is due to the rules directly on samples taken at symbol-rate. The only channel and regulations imposed by the U.S. Federal Communications state information that the algorithms need can be obtained by Commission [5], which aim at preventing UWB signals from estimation of an aggregate channel gain, avoiding channel-related interfering with legacy systems. complications. Following Huber’s M -estimates for the Gaussian mixture noise model, the proposed robust acquisition systems The low duty cycle with very short pulses makes detection- outperform their traditional counterparts designed according based UWB synchronization algorithms difﬁcult to implement. to the Gaussian noise assumption. Necessary modiﬁcations for To allow multiple access and to remove the spectral lines operation under multiple access interference are also introduced. resulting from the pulse repetition pattern present in the Theoretical and simulation-based performance evaluations that reﬂect the asymptotic variance, normalized mean-square error transmitted UWB signal [4], long spreading codes are used and the bit error rates demonstrate the gains offered by the causing a large search space. Unlike narrowband systems, robust procedures. synchronization via linear search techniques with phase-locked Index Terms—Ultra-wideband, impulse radio, timing acquisi- loops or early-late gate synchronizers is not viable. In [6] tion, non-Gaussian noise, robust detection. and [7], alternative techniques such as bit reversal search and look-and-jump by K bins are employed, but their mean acquisition times are high. I. I NTRODUCTION The low-power constraint makes sufﬁcient energy capture I MPULSE radio ultra-wideband (IR-UWB) communication can be characterized by the huge transmission bandwidth that is obtained through the use of pulses on the order of in the samples harder for an estimation-based approach. To that end, template selection for optimal detection plays a key role at the receiver [8]. The optimal template is the chan- a few nanoseconds. The UWB technology brings out new nel output waveform without noise, which can be generated potentials for low-power, bandwidth-demanding wireless ap- with the knowledge of the channel impulse response at the plications such as the design of low-cost and low-complexity receiver. In addition to the unavailability of the channel state baseband receivers with high user capacity. Despite these information (CSI) during synchronization, it seems almost advantages, UWB also introduces unique signal processing impossible to acquire the impulse response of a UWB channel challenges, especially for timing synchronization and channel because the number of paths for 85% energy capture varies estimation [1]. Although there exist blind approaches aiming typically between approximately 20 to 120 [9]. Moreover, the to bypass explicit channel estimation and synchronization channel statistics may not be reliable enough to obtain useful tasks for simpler receiver structures, inherent performance loss information which disallows making prior assumptions. in such schemes is unavoidable [2]. In particular, because even The impact of low duty cycle and low power constraints minor misalignments may result in the lack of sufﬁcient energy on synchronization performance in UWB systems has been capture rendering symbol detection nearly impossible [3], low- investigated in [6], [8], [10]-[13], with the assumption that the complexity timing synchronization mechanisms are needed. ambient noise is additive white Gaussian (AWGN). However, The general solution to the synchronization of UWB signals as reported in [14], the indoor environments, where the UWB is to divide the task into two parts: timing acquisition and devices are envisioned to be deployed, are subject to interfer- ﬁne synchronization. The ﬁrst stage aims at reducing the ence produced by photocopiers, printers, elevators, etc. Such timing uncertainty to the order of a pulse duration while the interference exhibits bursts of high-amplitude emissions whose second makes a ﬁne tuning within pulse duration accuracy rate of occurrence depends on the device usage frequency. [4]. Two major issues causing the UWB synchronization Despite the processing gain offered by UWB systems, it is Manuscript received July 4, 2007; revised November 17, 2007 and March shown in [15] that even a tone interference with 16 MHz 7, 2008; accepted April 25, 2008. The associate editor coordinating the review bandwidth can have detrimental effects on the performance of this paper and approving it for publication was N. Arumugam. This work of transmitted-reference (TR) systems. A proper probabilistic was supported by TUB˙¨ ITAK under contracts 105E034 and 105E077. Parts of this paper were presented in IEEE WCNC, Hong Kong, March 2007 and in model that accounts for man-made interference, as well as IEEE VTC, Dublin, Ireland, April 2007. ambient and thermal noise simultaneously has to be of non- g ¸ E. Ekrem was with Boˇ azic i University, Bebek 34342 Istanbul, Turkey. Gaussian form, characterized by heavy tails that represent He is now with University of Maryland, College Park, MD 20742, USA (e- mail: ersen@umd.edu). the impulsive outliers, and it will hereafter be referred to as M. Koca and H. Delic are with Wireless Communications Laboratory, ¸ impulsive noise. Boˇ azic i University, Bebek 34342 Istanbul, Turkey (e-mail: {mutlu.koca, g ¸ delic}@boun.edu.tr). Conventional synchronization methods suffer from a per- Digital Object Identiﬁer 10.1109/T-WC.2008.070730. formance drop in case of non-Gaussian disturbance just like 1536-1276/08$25.00 c 2008 IEEE Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4657 the detectors designed solely to counteract Gaussian noise an asymptotic performance analysis. Section VI extends the as demonstrated in [16]. In this paper, we focus on the robust acquisition procedure to multiuser environments. Sec- synchronization of the signal within frame-level accuracy, tion VII presents the performance evaluation via simulations, referred to as timing acquisition, for which estimation-based and Section VIII makes some concluding remarks. approaches seem to be the strongest candidates. Taking into account the impulsive noise phenomenon, a robust UWB II. S IGNAL AND N OISE M ODEL acquisition system is proposed to combat occasional devia- In an IR-UWB communication system, every symbol is tions in the noise probability density function (PDF) from transmitted over a duration of Ts in which Nf frames, each the nominal Gaussian assumption. The main approach is to with a duration of Tf , are sent, i.e., Ts = Nf Tf . In each frame, combine minimax design for non-Gaussian noise with the a pulse p(t) with a duration of Tp is transmitted. Typically, the TR methodology. In TR systems, each symbol consists of a pulse duration is much shorter than the frame duration, i.e., modulated part (data) and an unmodulated part that serves as Tp Tf , which leads to the low-duty cycle characteristic of a template for the demodulation of the former. Because half UWB. For multiple access and channel separation, the pulse is of the signal is reserved for the unmodulated non-data part hopped by an amount of cu (n)Tc where cu (n) is the TH code and also because noisy signals are used as the template, the corresponding to uth user’s transmitted pulse in the nth frame associated bit error rate (BER) performance is poor [17]. On of the symbol and Tc is the duration of the bins to which the other hand, such a paradigm suits the timing task well as the pulses are allowed to hop. The allowable range for TH stated in [10] because of the fact that it greatly simpliﬁes the codes is [0, Nc ) where Nc = Tf /Tc . We adopt binary pulse channel estimation task. Here, as in [10], instead of allocating amplitude modulation (PAM) for the acquisition procedure, half of the transmitted symbols to the templates (unmodulated and the transmitted signal for uth user is given by signal), we use short training sequences which force the ∞ received samples to be related with the delay parameters. Then, these delay parameters are determined through the M - su (t) = bu (n)pu,s (t − nTs ) (1) n=0 estimation procedure, which is a minimax design when the uncertainty about the system stems from the PDFs. In M - where bu (n) = ±1 denotes the PAM symbols and pu,s (t) is estimation, the worst case performance of the system at the the transmitted symbol waveform of the uth user given by neighborhood of the PDF is found, and at that point the overall Nf −1 performance of the system is optimized [18]. √ pu,s (t) = E u,s p t − nTf − cu (n)Tc (2) In this paper, two data-aided robust acquisition algorithms n=0 using the robust estimation techniques mentioned above are where Eu,s is the energy of the transmitted symbol of the uth proposed. The ﬁrst method uses the received symbol waveform user. as the template while the second employs an averaged version The quasi-static channel with Lu resolvable paths observed of the received symbols so that the level of noise enhancement by user u is modeled as which mainly occurs due to cross noise terms is decreased. The robust signal acquisition procedures require an analog Lu implementation of delay line and operate with samples taken hu (t) = hu,l δ(t − τu,l ) at symbol rate with/without time-hopping (TH) codes or l=1 direct-sequence (DS) spreading. Note that all the CSI needed where hu,l and τu,l denote the path gain and the time- by the proposed receivers is the estimation of a tap gain delay of lth multipath component of the uth user’s channel resulting from the correlation of the received signals. Respec- response, respectively. The frame duration is set to satisfy tive performances of both methods are investigated through |τu,Lu −τu,1 |+Tp < Tf and cu (Nf −1) = 0 to avoid intersym- asymptotic analyses and computer simulations. It is shown bol interference (ISI). Note that interframe interference (IFI) that the averaged-template approach performs better than the is allowed by these restrictions. Although this assumption on one employing successive sampling at the expense of added the frame length is suitable for low rate UWB systems, it can complexity due to averaging in analog domain. As depicted be relaxed for high rate systems as long as guard frames are by the simulation results, both robust algorithms have superior inserted between the symbols to prevent ISI [19]. performance compared to the Gaussian-optimal maximum In this paper, the estimation of the ﬁrst path’s delay, τu,1 , is likelihood solution, which results in a linear structure. Finally, considered while it is restricted to lie in one symbol duration, the averaged-template approach is also extended to the mul- Ts . The time delay of the ﬁrst path can be expressed as tiuser environments where the multiuser interference (MUI) τu,1 = nu,f Tf + θu where nu,f = τu,1 /Tf and θu = is ﬁrst approximated with a Gaussian-distribution and then τu,1 − nu,f Tf represent the time delay at frame-level and modeled as impulsive noise. It is illustrated via simulations pulse-level, respectively. The delay of the other paths can be that the performances of both robust multiuser detectors are written as τu,l = τu,l,1 + τu,1 for l = 1, . . . , Lu where τu,l,1 similar, or in other words, that taking the impulsiveness of is the relative time delay of the lth path with respect to the MUI into account does not provide any additional robustness. ﬁrst one. The goal is to estimate nu,f from the received signal This paper is organized as follows: Section II outlines which can be written as the signal and noise model. Section III presents some pre- Nu ∞ liminaries about robust estimation. Section IV describes the r(t) = bu (n)pu,r (t − nTs − τu,1 ) + w(t) (3) proposed robust acquisition procedures. Section V carries out u=1 n=0 Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. 4658 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 Lu where pu,r (t) = l=1 hu,l pu,s (t − τu,l,1 ) is the received ML estimate with respect to this least favorable PDF such symbol waveform and Nu denotes number of users. In (3), that it can be obtained by minimizing a cost function of the w(t) is the impulsive noise modeled as a two-term Gaus- form, ρ(x) = − log fLF (x), which turns out to be sian mixture, which is more appropriate for UWB systems ⎧ x2 whose main application will be in indoor environments as ⎨ 2σ2 for |x| ≤ kσ 2 , demonstrated in [14]. The two-term Gaussian mixture is ρ(x) = (8) ⎩ k2 σ 2 2 an approximation to the more general Middleton’s Class A 2 − k|x| for |x| > kσ noise model, which consists of an inﬁnite sum of Gaussian where k is found by solving (7). Moreover, it can be PDFs with identical means and increasing variances [20], and shown that the M -estimates are asymptotically consistent it adequately describes the Class A model when impulsive and normal-distributed. The asymptotic variance of the robust interference constitutes about 0.1-1% of the total disturbance mean estimator using the PDF in (6) is given by [18] (noise plus interference) [21]. The two-term Gaussian mixture E ψ 2 (x) ψ 2 (x)fLF (x)dx noise can be expressed as ν2 = = (9) 2 2 2 2 (E [ψ (x)]) ψ (x)fLF (x)dx f (x) = (1 − )g(x; 0, σ ) + g(x; 0, κσ ) (4) where ν 2 denotes the asymptotic variance and ψ (x) is the where g(x; μ, σ 2 ) is the Gaussian PDF with mean μ and ﬁrst derivative of ψ(x), which itself is the derivative of the variance σ 2 at point x. In the mixture, the ﬁrst term accounts cost function ρ(x). for the nominal noise with higher prior probability, while the second term represents the impulsive interference components IV. ROBUST T IMING ACQUISITION FOR UWB with its heavier tails. Here, κ is the impulsive part’s relative variance with respect to the nominal noise variance, and is Templates play a critical role because for a typical UWB the relative frequency of the outliers. system, the multipath effect combined with low-power trans- Notational Convention: The subscript u, which denotes the mission make it difﬁcult to capture an estimate of the transmit- number of user is dropped for notational simplicity (e.g. b(m) ted symbol with sufﬁciently high energy. The possible lack of denotes the information bit of any user in a single user link) prior information about the channel impulse response calls for until needed in Section V, where a multiuser environment is alternative approaches for template selection. In [8] and [10], considered. the received symbols and an averaged form of the received symbols are proposed as templates for transmission without III. P RELIMINARIES and with TH, respectively. Both approaches originate from the TR methodology [17], where for each data symbol, an In this section, we brieﬂy outline the robust estimation unmodulated reference symbol is sent for demodulation. techniques, namely M -estimates, which are used throughout the paper. Robust estimation procedures are generally designed in a minimax sense such as the M -estimates by Huber [18], A. Successive Sampling which was originally developed for location parameter estima- Here, instead of allocating half of the symbol duration to a tion. In Huber’s solution, the maximal asymptotic variance of reference signal, successive symbols are used as the template the estimator is minimized by ﬁnding the least favorable PDF with the aid of a training sequence as in [8], [10]. When within a certain family of the distributions, which turns out to ISI is absent, each symbol-length waveform at the receiver possess the minimum Fisher information. This minimax pro- consists of two successively transmitted symbols [10]. By cedure is completed by determining the maximum-likelihood using each received symbol-length waveform as a template (ML) estimator of the least favorable PDF. for the next one, the mean of the samples is directly governed For the so-called -contaminated Gaussian mixture model by the frame-level delay, and thus, the estimate of the mean in (4), the PDF family can be described by the set of the training sequence can be employed as a tool for timing acquisition. F = {(1 − )g(x; 0, σ 2 ) + v(x); v(x) is a symmetric PDF}, The block diagram of the successive sampling-based ac- (5) quisition unit is given in Fig. 1. As a ﬁrst step, the received for which the least favorable PDF is the one that obeys symbol-length waveform is correlated with the previous one, the Gaussian distribution at the center and then decays with yielding exponential tails such that ⎧ Ts ⎪ √ ⎪ 1− exp − 2σ2 x2 for |x| ≤ kσ 2 , rSUC (m) = r (t + mTs ) r t + (m + 1)Ts dt, ⎨ 2πσ 0 fLF (x) = ⎪ 1− ⎪ √ which results in ⎩ 2 2 exp k 2 − k|x| σ for |x| > kσ 2 , 2πσ (6) ˜ rSUC (m) = b(m)Cg b(m − 1)˜ f + b(m + 1)(Nf − nf ) n where k, σ and are related through + ωSUC (m) (10) φ(kσ) T 2 L − Q(kσ) = (7) where Cg = t=0 f l=1 hu,l p(t − τu,l ) u dt is the energy of kσ 2(1 − ) √ ˜ the received frame and nf = nf + χ with χ being the random and√ φ(u) = (1/ 2π) exp(−u2 /2), Q(u) = uncertainty induced by the TH codes and/or the pulse-level ∞ (1/ 2π) u exp(−t2 /2)dt [18]. The M -estimate is then the ˜ uncertainty [8]. Here, χ ∈ [0, 1) so that the integer ﬂoor of nf Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4659 r(t) (k+1)Ts Robust Channel ˆ Cg Robust Frame ˆ nf dt kTs Gain Estimator Offset Estimator Delay Ts Fig. 1. Block diagram of the successive sampling based robust acquisition system. is equal to nf , the frame-offset time delay. In (10), ωSUC (m) (6) and (8), respectively, with k now satisfying is the noise sample which consists of three parts, one noise- 1 − (1 − )3 φ kσSUC noise term and two signal-noise terms. The overall noise PDF = − Q kσSUC . is given by 2(1 − )3 kσSUC The estimates for the means of the two parts of the training fSUC (x) = (1 − )3 g x; 0, σ 2 (2Er + 4Ts W σ 2 ) sequence can be found as + 2 (1 − )2 g x; 0, σ 2 ((1 + κ)Er + 4Ts W σ 2 ) ˆ Cg,SUC = + (1 − )2 g x; 0, σ 2 (2Er + 4κ2 Ts W σ 2 ) Nt /2−1 2 2 2 2 arg min ρSUC rSUC (m) − b(m)b(m + 1)Nf Cg,SUC , + 2 (1 − )g x; 0, σ ((1 + κ)Er + 4κ Ts W σ ) Cg,SUC 2 m=0 + (1 − )g x; 0, σ 2 (2κEr + 4Ts W σ 2 ) (12) 3 2 2 2 ˆ μSUC = + g x; 0, σ (2κEr + 4κ Ts W σ ) (11) Nt where Er = Nf Cg is the energy of received symbol and W is arg min ˆ ρSUC rSUC (m) − b(m)b(m + 1)Cg,SUC μSUC µSUC the bandwidth of the front-end-ﬁlter at the receiver, which is m=Nt /2 typically given by W ≈ 1/Tp . The derivation of (11) is given (13) in Appendix A. ˆ where μSUC and μSUC are deﬁned by μSUC (Nf − ˜ The primary goal is to estimate nf whose integer ﬂoor is 2˜ f ) and μSUC n ˆ (Nf − 2˜ f ). The closed form solutions n ˆ the estimate of the frame-offset, nf,SUC . Note that to these equations do not exist, but they can be computed 1) if b(m − 1) = b(m + 1), (10) reduces to rSUC (m) = iteratively via the modiﬁed residuals method [18]. Assuming b(m)b(m + 1)Nf Cg + ωSUC (m), from which Cg can be ˆ ˆ ˆ ˆ that Cg,SUC and μSUC are the estimates of Cg and μSUC at estimated; and the th step, respectively, recursive solutions of (12) and (13) 2) if b(m − 1) = −b(m + 1), (10) reduces to rSUC (m) = are given by b(m)b(m + 1)Cg (Nf − 2˜ f ) + ωSUC (m), from which n Nt /2−1 ˜ nf can be estimated. ˆ +1 ˆ Cg,SUC = Cg,SUC + α b(m)b(m + 1)Nf Therefore, a training sequence of length Nt whose ﬁrst m=0 Nt /2 elements are identically +1 and the remaining half is ˆ × ψSUC rSUC (m) − b(m)b(m + 1)Nf Cg,SUC (14) {+1, +1, −1, −1, . . . , +1, +1, −1, −1}, makes it possible to obtain the necessary equations for the estimation of Cg and Nt ˆ +1 μSUC = μSUC + α ˆ ˆ b(m)b(m + 1)Cg,SUC ˜ nf . The second half of this preamble structure is also used in [10], where timing-estimation is basically search-oriented m=Nt /2 relying on peak-picking, and it is designed for Gaussian noise. ˆ ˆ × ψSUC rSUC (m) − b(m)b(m + 1)Cg,ss μSUC Huber’s robust solution corresponds to a situation where (15) the nominal PDF is Gaussian and in the neighborhood of the nominal PDF, there exist symmetric contaminations, as in (5). where α is the step-size, taken about 0.1, and ψSUC (x) = If we consider the noise PDF in (11), there is a dominating ρSUC (x) is the ﬁrst derivative of ρSUC (x). All estimates are Gaussian term with a priori probability of (1 − )3 and ﬁve set to zero initially, and convergence is typically established more Gaussians with lower prior probabilities. Because the in less than 10 steps. Convergence is ensured so long as contaminations are symmetric and the dominating PDF is |ψSUC (x)| < 1/α [22]. Once the ﬁnal channel gain estimate Gaussian, Huber’s solution to -contaminated Gaussian models ˆ Cg,SUC is obtained through the recursions in (14), it is can be adopted here, as well. To that end, one needs to employed in (15) for the estimation of μSUC whose ﬁnal value deﬁne the nominal situation and the corresponding PDF. We ˆ at the end of the recursions is denoted by μSUC . Then, nf ˜ deﬁne the nominal case as the one in which both of the is determined, whose integer ﬂoor gives the estimate of the two successively received symbols face just thermal noise, frame-offset as and the contamination consists of all terms where either one ˆ Nf − μSUC or both of the received successive symbols contain outliers. ˆ ˆ ˜ nf,SUC = nf,SUC = 2 Therefore, the least favorable PDF and the cost function ρSUC for successive sampling can be determined by replacing σ 2 ˆ ˜ ˜ ˆ where nf,SUC denotes the value of nf calculated using μSUC 2 with σSUC σ 2 (2Er + 4Ts W σ 2 ) and with 1 − (1 − )3 in ˆ and nf,SUC represent the estimate of the frame-offset obtained Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. 4660 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 r(t) (k+1)Ts Robust Channel ˆ Cg Robust Frame ˆ nf dt kTs Gain Estimator Offset Estimator Template Estimator ¯ r(t) (Averager) Fig. 2. Block diagram of the averaged template based robust acquisition system. via successive sampling. In the next subsection, an alternative which can be derived in a similar way as (11), whose technique decreasing the degree of noise enhancement that derivation is in Appendix A. occurs due to using noisy templates is presented. The samples in (18) differ from those in (10) and a ˜ new training sequence should be designed for estimating nf . B. Averaged Template Hence, we observe that In TR systems, or in autocorrelation-based receivers, one signiﬁcant issue is the double-noise (noise-noise) terms which 1) if b(m) = b(m − 1), (18) reduces to rAVG (m) = result in noise emphasis as in our case. It is possible to lower b(m)Cg Nf + ωAVG (m) from which Cg can be esti- the noise variance of the samples by employing a template mated; that is obtained by averaging over the training sequence. The 2) if b(m) = −b(m − 1), (18) reduces to rAVG (m) = cost of performance improvement is a complexity increase ˜ Cg b(m)(Nf − 2˜ f ) + ωAVG (m) from which nf can be n that results from averaging in the analog domain. The block estimated. diagram of the corresponding acquisition system is given in Fig. 2. Suppose that during the ﬁrst half of the training Therefore, a training sequence whose ﬁrst half consists of sequence, the received symbols are averaged as only +1’s and the remaining half having a structure of Nt /2−1 {+1, −1, +1, −1, . . . , −1, +1} enables us to estimate Cg and 2 ˜ nf separately. In addition, the template can be obtained by ¯ r (t) = ¯ r(t+iTs ) = pr (t+Ts −τ1 )+pr (t−τ1 )+w(t) Nt averaging the received symbols of the ﬁrst half of the training i=0 (16) sequence. It should be noted that this preamble structure is ¯ where w(t) is the noise averaged over Nt /2 symbol-length same as the one proposed in [8] for timing-estimation under waveforms whose PDF is given by Gaussian noise. 2σ 2 2κσ 2 For the implementation of the robust estimation procedure, fw (x) = (1− )g x; 0, ¯ +(1− )g x; 0, . (17) the nominal situation is assumed to be acting when neither the Nt Nt averaged template nor the sampled symbol contains outliers. Through averaging, the noise variance of the template is Thus, the term with the a priori probability of (1 − )3 is the reduced by a factor of Nt /2 according to (17), which is crucial nominal PDF, and the rest represent the contamination with for suppressing the double-noise terms. Using the template in impulsive components. In this case, the least favorable PDF (16), the samples are obtained as and the cost function ρAVG can be found by replacing σ 2 with 2 σAVG σ 2 (Er + 4Ts W σ 2 /Nt ) and with 1 − (1 − )3 in (6) Ts rAVG (m) = r(t + mTs )¯(t)dt, m = 0, 1, . . . , Nt − 1, r 0 and (8), respectively, with k being the solution to which results in [8] rAVG (m) = Cg b(m − 1)˜ f + b(m)(Nf − nf ) + ωAVG (m) n ˜ 1 − (1 − )3 φ (kσAVG ) = − Q (kσAVG ) . (18) 2(1 − )3 kσAVG where ωAVG (m) is the sequence of noise samples having a PDF of σ2 The estimated values for the means of the two parts of the fAVG (x) = (1 − )3 g x; 0, σ 2 Er + 4Ts W Nt training sequence are given by σ2 + (1 − )2 g x; 0, σ 2 Er + 4κ2 Ts W Nt κ σ2 Nt /2−1 + (1 − )2 g x; 0, σ 2 2Er + 4Ts W ˆ Cg,AVG = arg min ρAVG r(m) − Nf Cg , Nt Nt Cg σ2 m=0 2 2 κ 2 + (1 − )g x; 0, σ 2Er + 4κ Ts W Nt −1 Nt Nt ˆ μAVG = arg min ˆ ρAVG r(m) − b(m)Cg,AVGμAVG 2 σ2 µAVG m=Nt /2 + (1 − )g x; 0, σ κEr + 4Ts W Nt 2 σ2 + g x; 0, σ 2 κEr + 4κ2 Ts W (19) Nt ˆ where μAVG = (Nf − 2˜ f ) and μAVG n (Nf − 2˜ f ). The n Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4661 2 1 10 10 1 10 0 10 0 10 −1 10 ) log (ν2 ) SUC AVG −1 log (ν2 10 −2 10 −2 10 Linear−SUC (ε = 0.1) −3 Linear−AVG (ε = 0.1) −3 10 10 Linear−SUC (ε = 0.01) Linear−AVG (ε = 0.01) Robust−SUC (ε = 0.1) Robust−AVG (ε = 0.1) Robust−SUC (ε = 0.01) Robust−AVG (ε = 0.01) −4 −4 10 10 2 3 2 3 10 10 10 10 log (κ) log (κ) Fig. 3. Asymptotic variance of successive sampling-based acquisition when Fig. 4. Asymptotic variance of averaged template-based acquisition when is ﬁxed and ((1 − ) + κ)σ2 = (0.1)2 . is ﬁxed and ((1 − ) + κ)σ2 = (0.1)2 . modiﬁed residuals method in [18] yields the iterations where N (0, σ 2 ) represents a Gaussian random variable with Nt /2−1 zero mean and variance σ 2 . Moreover, because M -estimates ˆ +1 ˆ Cg,AVG = Cg,AVG + α r(m)Nf can be viewed as a generalization of the ML-estimates, the m=0 asymptotic variance of the linear estimator designed to be ˆ optimal under AWGN but operating in impulsive noise can × ψAVG r(m) − b(m)Nf Cg,AVG , (20) be calculated via (9) by setting ψ(x) = x. In this case, Nt −1 ˆ the corresponding linear receivers use the quadratic penalty ˆ +1 ˆ μAVG = μAVG + α b(m)Cg,AVG function ρ(x) = x2 . Because the noise PDFs in (11) and (19) m=Nt /2 are complicated, the asymptotic variances of the estimators, ˆ × ψAVG r(m) − b(m)Cg,AVG μAVG ˆ (21) 2 2 νSUC and νAVG , are calculated numerically and plotted in ˆ +1 Figures 3 and 4. It is seen in these two ﬁgures that when and ˆ +1 where ψAVG (x) is the derivative of ρAVG and Cg,AVG , μAVG the total noise variance ((1− )+ κ)σ 2 are ﬁxed, the variances denote the estimates of Cg,AVG and μAVG at the th step, of the robust estimators are monotonically decreasing in κ respectively. As with successive sampling, α is the step size whereas those of the linear estimators, i.e., ML-optimal ones around 0.1, and all initial estimates are set to zero. The under AWGN, are almost invariant to changes in κ. This was recursive use of (20) produces the ﬁnal channel gain estimate, ˆ also observed and stated in [22] for robust multiuser detectors Cg,AVG , which is then employed in (21) for determining in non-Gaussian channels. μAVG , whose ﬁnal value at the end of the recursions will be 2 2 Assuming that νSUC and νAVG are available, the probability ˆ ˜ ˆ denoted by μAVG . Finally, nf,AVG is calculated using μAVG , mass functions (PMFs) of the robust estimators are derived and its integer ﬂoor gives the estimate of the frame-offset as next to calculate normalized mean square error (NMSE). First ˆ ˆ Nf − μAVG the conditional probability of frame offset’s estimate obtained nf,AVG = nf,AVG = ˆ ˜ 2 via successive sampling is considered, which is given by ˆ ˆ Nf − μSUC ˜ ˜ where nf,AVG denotes the value of nf calculated using ˜ Pr{ˆ f,SUC = x | nf } = Pr x ≤ n < x + 1 nf ˜ 2 ˆ ˆ μAVG and nf,AVG represent the estimate of the frame-offset ⎛ ⎞ ⎛ ⎞ obtained via averaged template. In the next section, asymptotic Nf − 2x − 2 − μSUC ⎠ Nf − 2x − μSUC ⎠ = Q⎝ − Q⎝ . performance analysis of these estimators is carried out. νSUC Cg Nt /2 νSUC Cg Nt /2 (22) V. A SYMPTOTIC P ERFORMANCE A NALYSIS Using the fact that μSUC = Nf − 2˜ f , (22) can be expressed n Since the M -estimates are asymptotically consistent and as normal-distributed, denoting the asymptotic variances for the 2 2 successive sampling and averaged template as νSUC and νAVG , Pr{ˆ f,SUC = x | nf } = n ˜ respectively, the estimation errors obey a normal PDF such ⎛ ⎞ ⎛ ⎞ that 2˜ f − 2x − 2 n 2˜ f − 2x n Q⎝ ⎠−Q⎝ ⎠, Nt νSUC Cg Nt /2 νSUC Cg Nt /2 2 ˆ Cg (μSUC − μSUC ) ∼ N (0, νSUC ), 2 (23) Nt ˆ 2 Cg (μAVG − μAVG ) ∼ N (0, νAVG ) ˜ ˜ where nf can be decomposed as nf = nf + ϑ with ϑ being 2 the random time-uncertainty induced by the TH codes and/or Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. 4662 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 pulse-level uncertainty, θ, which is assumed to be uniformly sequence is adopted for the multiuser environment because distributed in [0, 1). To alleviate the dependency of (23) on ϑ, of its superior performance as attested by the asymptotic expectation of (23) should be taken over ϑ resulting in variance comparison in Section IV. In the presence of MUI, 1 the averaged template becomes Pr{ˆ f,SUC = x | nf } = n Pr(ˆ f = x | nf , ϑ)dϑ n Nt /2−1 Nu Nt /2−1 ⎛ 0 ⎞ 2 2 ¯ r (t) = r(t + iTs ) = ¯ ru (t) + w(t) i 2nf − 2x Nt Nt = 2(nf − x)Q ⎝ ⎠ i=0 u=1 i=0 νSUC Cg Nt /2 ¯ where w(t) is the averaged noise whose PDF is given in (17) ⎛ ⎞ and ru (t) is given by i 2nf − 2x − 2 − (nf − x − 1)Q ⎝ ⎠ ru (t) = bu (i − 1)pu,r (t + Ts − τ1 ) + bu (i)pu,r (t − τ1 ). (26) i u u νSUC Cg Nt /2 ⎛ ⎞ The received samples are formed via 2nf − 2x + 2 − (nf − x + 1)Q ⎝ ⎠ Ts r(m) = r(t + mTs )¯(t)dt r νSUC Cg Nt /2 0 2 which is explicitly given by νSUC nf − x − 1 + √ exp −Nt Cg 2Cg πNt νSUC r(m) = Cd,g bd (m − 1)˜ d,f + bd (m)(Nf − nd,f ) n ˜ 2 nf − x + 1 + ωMUI (m) + ωMA (m) + exp −Nt Cg νSUC where the subscript d denotes the desired user, ωMUI (m) and nf − x 2 ωMA (m) denote the MUI and noise term, respectively. The − 2 exp −Nt Cg (24) PDF of ωMA (m) is given by νSUC σ2 which is the PMF of the frame-offset estimate conditioned on fMA (z) = (1 − )3 g z; 0, σ 2 Ed,r + 4Ts W ˆ the actual value of nf . Therefore, the NMSE of nf,SUC can Nt be calculated as σ2 + (1 − )2 g 0, σ 2 Ed,r + 4κ2 Ts W 2 Nt ˆ nf,SUC − nf NMSE(ˆ f,SUC ) n E ET,r σ2 Nf + (1 − )2 g z; 0, σ 2 2κ + 4Ts W Nt Nt Nf −1 1 σ2 = 2 Pr{nf = x}E (ˆ f,SUC − nf ) | nf = x n 2 + 2 g z; 0, σ 2 κEd,r + 4κ2 Ts W Nf Nt x=0 (25) σ2 + (1 − )g z; 0, σ 2 κEd,r + 4Ts W Nt where E (ˆ f,SUC − nf )2 | nf = x is the conditional mean n ET,r σ2 square error. Assuming nf to be uniform over [0, Nf − 1], + 2 (1 − )g z; 0, σ 2 2κ + 4κ2 Ts W Nt Nt (25) can be expressed as (27) NMSE(ˆ f,SUC ) = n N where ET,r = u=1 Eu,r is the sum of the received energies u Ts 1 Nf −1 ∞ 2 of all users and Eu,r = t=0 p2 (t)dt. The derivation of (27) u,r 3 Pr{ˆ f,SUC = y | nf = x} (y − x) n is given in Appendix B. Now, we turn our attention to the Nf x=0 y=−∞ MUI term whose PDF is where Pr{ˆ f,SUC = y | nf = x} is given in (24). Because n (Nu −1)Nt /2 (Nu − 1)Nt the NMSE of the frame-offset estimate obtained via averaged fMUI (z) = 2−(Nu −1)Nt /2 C ,l 2 template can be calculated with the same procedure described, l=0 its derivation is omitted. 2((Nu − 1)Nt − 2l)(Nf Cg )2 × g z; 0, , (28) 3Nt2 VI. M ULTIUSER E NVIRONMENT as shown in Appendix C, where C(n, l) denotes the number So far, a point-to-point link has been considered with the of l-groupings of n. As noted in [23], [24], the MUI in UWB assumption that user separation is accomplished via channel- systems is of non-Gaussian nature. Moreover, in [24], it is ization [10]. In this section, the performances of the proposed shown that the MUI term for TH pulse position modulation algorithms are explored when asynchronous users which are (PPM) UWB systems follows Middleton’s Class A noise assumed to transmit independent and identically distributed model. Equation (28) indicates that the MUI in the signal (i.i.d.) data symbols interfere with the desired user. The goal acquisition paradigm is also non-Gaussian. Note that in [23], is to achieve timing acquisition with only one user in the [24], the origin of MUI is the collision of independent user’s presence of both impulsive noise and multiuser interference. pulses whereas in acquisition, the correlation of same user’s The averaged template method with the associated training symbol-length waveforms contribute to the MUI term, as well. Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4663 Therefore, as long as the collision probability of two different B. Impulsive MUI user’s pulses is low, i.e., the frame length is long enough, the We propose a robust synchronizer which takes the impulsive latter dominates the MUI. In the next two subsections, we character of MUI into design consideration. To that end, ﬁrst design two robust synchronizers where one uses a Gaussian reexpress the noise term in (27) as approximation for the MUI term while the other takes the impulsiveness into consideration. σ2 fMA (z) = (1 − )3 g z; 0, σ 2 Ed,r + 4Ts W + ζ1 (z) Nt A. Gaussian Approximation and the MUI term in (28) as In this subsection, we design a robust synchronizer by (Nu −1)Nt (Nu −1)Nt approximating the MUI term as a Gaussian random variable C 2 , 4 fMUI (z) = with zero mean and variance 2(Nu −1)Nt /2 2 4(Nu − 1) 2 (Nu − 1)Nt (Nf Cg )2 σMUI = (Nf Cg ) . × g z; 0, + ζ2 (z) 3Nt 3Nt2 Although the accuracy of the Gaussian assumption for small where the distributions ζ1 (z) and ζ2 (z) denote the number of users is poor, here we incorporate this approach interference-contaminated parts of noise and MUI terms, re- to construct a comparison platform for the next robust syn- spectively. For the MUI term, the most probable Gaussian chronizer which is built using (28) instead of a Gaussian PDF in (28) is selected as the nominal part, and all others are assumption. Note that MUI is asymptotically Gaussian in the declared as interference-contaminated. Thus, the PDF of the product Nf Nc which represents the processing gain, provided overall disturbance, noise plus MUI, can be expressed as that Nf and Nc grow while the ratio Nf /Nc remains constant 2 [25]. The PDF of overall disturbances, noise plus MUI, can fDIST (z) = (1 − )g z; 0, σDIST + ζ3 (z) (30) 2 be found as fDIST,G (z) = fMA (z) ∗ g z; 0, σMUI resulting where (1 − ), the prior probability of the nominal part of in (30), is given by fDIST,G (z) = (Nu −1)Nt (Nu −1)Nt C , 3 2 σ2 2 1− = (1 − ) 3 2 4 (1 − ) g z; 0, σ Ed,r + 4Ts W + σMUI , Nt 2(Nu −1)Nt /2 σ2 2 and σDIST is given by + (1 − )2 g 0, σ 2 Ed,r + 4κ2 Ts W 2 + σMUI Nt σ2 (Nu − 1)Nt (Nf Cg )2 ET,r σ2 2 σDIST = σ 2 Ed,r + 4Ts W + . + (1 − )2 g z; 0, σ 2 2κ + 4Ts W 2 + σMUI Nt 3Nt2 Nt Nt σ2 In this case, the least favorable PDF can be found by setting + 2 g z; 0, σ 2 κEd,r + 4κ2 Ts W 2 + σMUI 2 σ 2 to σDIST and to . Moreover, k can be found through Nt σ2 φ (kσDIST ) + (1 − )g z; 0, σ 2 κEd,r + 4Ts W 2 + σMUI − Q (kσDIST ) = . Nt kσDIST 2(1 − ) ET,r σ2 + 2 (1 − )g z; 0, σ 2 2κ + 4κ2 Ts W 2 + σMUI . Again, the ﬁrst and second halves of the training sequence Nt Nt (29) ˜ are used to estimate Cd,g and nd,f according to (14)-(15) and (20)-(21) for successive sampling and averaged template-based Note that MUI introduces an additional term to the variance acquisition, respectively. of the noise PDF given in (19), thereby resulting in (29), which is independent of signal-to-noise ratio (SNR) deﬁned as the VII. S IMULATION R ESULTS ratio of symbol energy to the variance of the background noise. The MUI induces a noise ﬂoor which cannot be lowered by In this section, we test the performance of the robust increasing the transmitted power. estimators. The pulse is selected as the second derivative of the The nominal part of (29) is the ﬁrst term with a pri- Gaussian function with duration Tp = 1 ns. The CM1, CM2, ori probability (1 − )3 and all other terms are treated as CM4 channel models are implemented [9]. The frame duration interference-contaminated. In this case, the least favorable is selected to be 40 ns. Each symbol consists of Nf = 30 PDF and the cost function can be found by setting σ 2 frames resulting in Ts = 1200 ns. The chip duration is chosen 2 to σDIST,G 2 σ 2 Ed,r + 4Ts W σ 2 /Nt + σMUI , and to as Tc = 1 ns, and the TH codes are randomly generated 3 1 − (1 − ) in (6) and (8), respectively, and with k found from a uniformly distributed set of {0, 1, . . . , Nf − 1}. The through tracking level uncertainty in time delay, i.e. θ, is randomly generated from a uniform distribution on [0, Tf ). The linear φ (kσDIST,G ) 1 − (1 − )3 acquisition system that is considered as the competitor here is − Q (kσDIST,G ) = . kσDIST,G 2(1 − )3 the Gaussian-optimal ML solution as developed in [8]. Com- ˜ The estimation of Cd,g and nd,f can be carried out as indi- parison with the acquisition method in [10] is also presented. cated in (14)-(15) and (20)-(21) using the ﬁrst and second In all simulations, SNR Es / (1 − )σ 2 + κσ 2 , where 2 2 halves of the corresponding training sequence for successive (1 − )σ + κσ is the total noise variance at the input of the sampling and averaged template-based robust timing acquisi- receiver and it is ﬁxed. The NMSE is calculated through sam- tion, respectively. ple averaging. Although nf ∈ {0, . . . , Nf − 1}, the estimated Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. 4664 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 3 0 10 10 Robust−SUC (no TH) Linear−SUC (no TH) 2 10 Robust−SUC (TH) Linear−SUC (TH) 1 −1 10 Dirty Templates 10 Robust−SUC (Theoretical) 0 Linear−SUC (Theoretical) 10 NMSE NMSE −1 −2 10 10 −2 10 Linear−SUC −3 −3 10 10 Robust−SUC Linear−AVG −4 10 Robust−AVG −5 −4 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 SNR (dB) SNR (dB) Fig. 5. Simulation results for the NMSE of the successive sampling-based Fig. 7. NMSE performances of all receivers in the presence Gaussian noise, robust acquisition and the corresponding linear acquisition for = 0.1, κ = when = 0. 100. 3 10 2 10 Robust−SUC (N =30) Robust−AVG (no TH) t 2 Linear−SUC (N =30) Linear−AVG (no TH) 10 t 1 10 Robust−AVG (TH) Robust−SUC (Nt=60) Linear AVG (TH) 1 10 Linear−SUC (N =60) Dirty Templates t 0 10 Robust−AVG (Theoretical) Robust−SUC (N =120) t Linear−AVG (Theoretical) 0 10 Linear−SUC (N =120) t NMSE −1 10 NMSE −1 10 −2 10 −2 10 −3 10 −3 10 −4 10 −4 10 −5 0 5 10 15 20 25 30 10 0 5 10 15 20 25 30 SNR (dB) SNR (dB) Fig. 8. NMSE of successive sampling-based robust acquisition and corre- Fig. 6. Simulation results for the NMSE of the averaged template-based sponding linear acquisition under = 0.1, κ = 100 for various values of robust acquisition and the corresponding linear acquisition for = 0.1, κ = Nt . 100. and the ‘dirty templates’approach in [10] signiﬁcantly. For ˆ frame-offsets nf often go out of that range in simulations the SNR range of 0-20 dB, the performance of Gaussian- with linear acquisition. In such cases, instead of discarding optimal acquisition is unacceptable, justifying the design of the out-of-range frame-offset estimates, we incorporated them robust procedures. Speciﬁcally, to attain NMSE = 1 × 10−3 to reﬂect the system’s performance more realistically, with when = 0.1 and κ = 100, linear acquisition requires NMSE values exceeding unity by a large margin at low 12 dB higher SNR compared to the robust scheme with SNR. Throughout the simulations we assume that the exact averaged template. The same margin grows to 17 dB for values of the noise model parameters are known; these can be = 0.01 and κ = 1000, when the interference intensiﬁes. estimated efﬁciently via the algorithm in [26]. A simpliﬁcation (The corresponding graph is available in [27], [28].) The is possible by setting k = 1.5/σ where the estimation of σ is gap is higher for dirty template although at low SNR, the rather straightforward and no serious performance loss occurs performance is better than the Gaussian-optimal ML solution. [22]. This better performance originates from the fact that ‘dirty First, we evaluate the performance of the robust receivers templates’ in [10] is a search-based technique ensuring that under a CM4 channel for which noise parameters are selected the estimated frame-offset lies in {0, . . . , Nf − 1} contrary as = 0.1, κ = 100 and the number of training symbols to the estimation-based ones for which the estimated frame- is set to 30. The corresponding results are given in Figs. offset goes out of range due to impulses. We are concerned 5 and 6, which demonstrate that robust acquisition methods with the estimation of only frame-level time-offset and the outperform the conventional Gaussian-optimal linear system random pulse-level uncertainty results in error ﬂoors in all Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4665 2 2 10 10 Robust−AVG (CM1) Robust−AVG (Nt=30) Linear−AVG (CM1) 1 Linear−AVG (N =30) 1 Robust−AVG (CM2) 10 t 10 Robust−AVG (N =60) Linear−AVG (CM2) t Linear−AVG (Nt=60) Robust−AVG (CM4) 0 0 Linear−AVG (CM4) 10 10 Robust−AVG (Nt=120) Linear−AVG (N =120) t NMSE NMSE −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 SNR (dB) SNR (dB) Fig. 9. NMSE of averaged template-based robust acquisition and corre- Fig. 11. NMSE of averaged template-based robust acquisition and corre- sponding linear acquisition under = 0.1, κ = 100 for various values of sponding linear acquisition under = 0.1, κ = 100 for various channel Nt . models. 0 3 10 10 Robust−SUC (CM1) 2 Linear−SUC (CM1) 10 Robust−SUC (CM2) −1 10 Linear−SUC (CM2) 1 Robust−SUC (CM4) 10 Linear−SUC (CM4) −2 10 0 10 NMSE BER −1 10 −3 10 −2 Perfect Timing 10 Robust−SUC (no TH) −4 10 Robust−SUC (with TH) −3 Linear−SUC (no TH) 10 Linear−SUC (with TH) No Timing −4 −5 10 10 0 5 10 15 20 25 0 5 10 15 20 25 30 SNR (dB) SNR (dB) Fig. 12. BER of successive sampling-based robust acquisition and the Fig. 10. NMSE of successive sampling-based robust acquisition and corresponding linear acquisition under = 0.1, κ = 100. corresponding linear acquisition under = 0.1, κ = 100 for various channel models. CM4) implying the robustness of the estimators with respect cases. The error ﬂoor is higher when TH codes are employed to the channel statistics. This stems from the use of received because they induce IFI that results in an additional timing symbols as templates in TR systems which results in the same ˆ uncertainty in nf as stated in [8]. From the ﬁgures, it is clear amount of energy capture in the samples regardless of the that average-template based robust acquisition outperforms delay spread and multipath. successive sampling alone due to better noise suppression at To judge the impact of timing mismatch in an impulsive en- the expense of a complexity increase due to averaging in vironment, BER graphs are plotted assuming that the receiver analog domain. The theoretical NMSEs are also plotted for has perfect knowledge of the channel and that impulsive noise both robust and linear methods without TH in Figures 5 and occurs only during transmission of the training sequence. This 6, showing good agreement with the simulation results. In Fig. simulation set-up is selected so as to observe the impact of 7, receiver performances are presented for the case of AWGN only the timing acquisition on the BER performance [10]. only. Robust methods do not incur any noticeable performance Otherwise, both channel estimation errors (or the effect of loss in the nominal case. Then, the performances of the robust the unknown channel in blind detection) and the effect of estimators are evaluated for different lengths of the training the impulsive noise on symbol detection combine with the sequence and it can be shown in Fig. 8 and Fig. 9 that as timing acquisition error. In the BER simulations, data are the length of the training sequence increases, the performance sent through bursts which consist of 30 training symbols and improves. Fig. 10 and Fig. 11 show the performance of the 1000 information symbols. At each SNR, these bursts are estimators for different channel models (CM1, CM2, and transmitted until at least 100 symbol errors are recorded. The Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. 4666 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 0 10 PDF is similar to Gaussian. Therefore, it can be argued that as long as the ambient noise and disturbances contain outliers, −1 10 they dominate MUI in UWB systems regardless of the number of interfering users. On the other hand, robust techniques are largely ineffective (except at low SNRs where they offer a −2 10 less than 2 dBs of gain over linear acquisition) when the only source of impulsiveness is MUI and noise is white Gaussian. BER −3 10 VIII. D ISCUSSION AND C ONCLUSION In this paper, we have presented UWB timing acquisition Perfect Timing −4 Robust−AVG (no TH) schemes that combine TR methodology with robust estima- 10 Robust−AVG (with TH) Linear−AVG (no TH) tion techniques for operation in the presence of interference Linear−AVG (with TH) modeled as impulsive noise. For a single user link, successive −5 No Timing sampling- and averaged template-based robust acquisition are 10 0 5 SNR (dB) 10 15 proposed. The former suffers from noise enhancement because of noisy signal correlations, while the latter provides improve- Fig. 13. BER of averaged template-based robust acquisition and the ment through averaging. Both robust methods offer substantial corresponding linear acquisition under = 0.1, κ = 100. performance gains over their linear counterparts, i.e., ML 2 estimators designed under the Gaussian noise assumption. 10 Robust−MUI−Gauss (N =5) u Two robust acquisition techniques are proposed for mul- Robust−MUI−Impulsive (Nu=5) tiuser environments, one of them being a simple extension 10 1 Robust−MUI−Gauss (N =20) u of averaged template-based acquisition by assuming MUI as Robust−MUI−Impulsive (N =20) u Gaussian-distributed. The other is a new design that takes 0 Linear−MUI−Gauss (N =5) u the inherent impulsiveness of MUI in UWB systems into 10 Linear−MUI−Gauss (N =20) u consideration. Although the performance of the multiuser receiver is satisfactory for moderate number of users, the NMSE −1 10 speciﬁc correlation structure of MUI is not exploited here for further suppression. Synchronization methods employing MUI 10 −2 suppression in the ISI channels may be explored as future work. −3 10 A PPENDIX A −4 In this Appendix, the noise PDFs of the successive sam- 10 0 5 10 15 20 25 30 pling, (11), is derived. With following the same line of SNR (dB) arguments, (19) can be also derived, hence it is omitted. The noise term in successive sampling, ωSUC (m), consists of three Fig. 14. NMSE of the robust acquisition methods designed for the multiuser environment and of the linear counterpart when = 0.1, κ = 100. parts: ωSUC (m) ωSUC mTs performance gains obtained by the robust methods become = ωSUC,1 mTs + ωSUC,2 mTs + ωSUC,3 mTs clearer with Figures 12 and 13. The Gaussian-optimal ML where ωSUC,1 mTs , ωSUC,2 mTs and ωSUC,3 mTs are estimators cannot track the UWB signal in impulsive noise, given by whereas the robust methods can within an acceptable range of Ts SNR. Moreover, when the averaged template is employed, its ωSUC,1 mTs = rs (t + mTs )w t + (m + 1)Ts dt, performance moves within 1 dB of perfect timing when no TH 0 is employed. In these ﬁgures, the effect of IFI induced by the Ts TH codes can again be observed considering the performance ωSUC,2 mTs = rs t + (m + 1)Ts w(t + mTs )dt, 0 gaps between no-TH and TH cases. Ts Lastly, we examine the performance of the MUI receivers ωSUC,3 mTs = w(t + mTs )w t + (m + 1)Ts dt, where 5 and 20 users transmit at 5 dB below the desired 0 user. Fig. 14 depicts the rise in the NMSE error ﬂoor due respectively, where rs (t + mTs ) = b(m − 1)pr (t + Ts − to MUI. Interestingly, robust estimators designed for the τ1 ) + b(m)pr (t − τ1 ) and τ1 is the time delay of the ﬁrst Gaussian approximation-based and the impulsive model-based path. The autocorrelation function and the PDF of ωSUC (m) MUI have about the same performance. This is due to the fact are derived in [17] for Gaussian w(t) and it is shown that that in TR systems, the main performance limiting factor is the PDF of wSUC (m) can be approximated as Gaussian [17], the double-noise term irrespective of whether the system is in [29]. This approximation is valid as long as both the time- a single-user link or not. The MUI becomes comparable to bandwidth product of the system and the number of pulses per the double-noise term for large number of users, but then its symbol are large enough. Moreover, under these conditions, Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4667 noise components can be assumed to be mutually uncorrelated. represented by Gaussian mixture model with zero mean and Here, following the same theory as in [17], [29], the PDF of variances ωSUC (m) is derived for the case where w(t) is modeled as σ2 σ2 2 w(t) = wB (t) + wI (t) with wB (t), wI (t) denoting thermal E ωMA,1 (m) = ET,r (1 − )2 +2 κ , Nt Nt Gaussian and shot noise, respectively, which are zero-mean, mutually independent random processes. One can conclude 2 E ωMA,2 (m) = Ed,r (1 − )σ 2 + κσ 2 , that wSUC,1 (mTs ) and wSUC,2 (mTs ) are also mixtures whose 2 σ4 σ2 E ωMA,3 (m) = 4Ts W (1 − ) + κ2 total variance is Cg Nf (1 − )σ 2 + κσ 2 and PDF is Nt Nt fSUC,1 (x) = (1 − )g x; 0, Cg Nf σ 2 + g x; 0, Cg Nf κσ 2 . N where ET,r = u=1 Eu,r is the sum of the received energies u of all users. By deﬁning the PDFs of ωMA,1 (m), ωMA,2 (m) The shot noise wI (t) has a PDF of the form fI (x) = (1 − and ωMA,3 (m) as fMA,1 (x), fMA,2 (x) and fMA,3 (x), the )δ(x) + g(x; μ, σ 2 ) [30], [31], which can be approximated overall PDF of ωMA (m) can be determined by fMA (z) = as a Gaussian PDF. Therefore, following the analysis in [17], fMA,1 (z) ∗ fMA,2 (z) ∗ fMA,3 (z), which results in (27). [29], the autocorrelation function of ωSUC,3 (mTs ) can be found as A PPENDIX C E ωSUC,3 (mTs )ωSUC,3 (mTs − τ ) = The derivation of the MUI PDF for TH-PPM UWB systems ∞ is given in [23], [24] for an AWGN channel. Because these 2 2 2 Ts − | τ | RB (y) + RI (y) dy results are obtained for coherent receivers, they do not contain −∞ the term originating from correlation of the same user’s re- where RB (t) and RI (t) are the autocorrelation functions of ceived symbols, which is a consequence of TR-like reception. wB (t) and wI (t), respectively. Similar to ωSUC,1 and ωSUC,2 , The MUI term is given by ωSUC,3 also obeys a Gaussian mixture density because of that its autocorrelation function is governed by two process, one ωMUI (m) = ωMUI,1 (m) + ωMUI,2 (m) 2 for impulsive part leading to RB (y) and one for nominal part where 2 leading to RI (y). Therefore, again following [17], [29], its Nt /2−1 Nu variance can be found as 4Ts W σ 2 (1 − )σ 2 + κ2 σ 2 where 2 Ts ωMUI,1 (m) = ru (t)ru (t)dt, m i W is the bandwidth of the front-end ﬁlter employed in the Nt t=0 u=1,u=d i=0 receiver and can be approximated as W ≈ 1/Tp . Moreover, Nu Nu Nt /2−1 its PDF is of the form 2 Ts ωMUI,2 (m) = ru (t)rl (t)dt m i 4 2 4 Nt fSUC,3 (x)= (1 − )g x; 0, 4Ts W σ + g x; 0, 4Ts W κ σ . u=1 l=1,l=u i=0 t=0 Finally, note that in the Gaussian-mixture model, a noise sam- where is given in (26). Note that ωMUI,2 (m) consists ru (t) m ple is generated by the acting PDF in the mixture depending of cross-terms between different users and its value is gov- on whether ambient interference is present or not. Therefore, erned primarily by the collision probability of different users’ 2 as long as these three noise samples are mutually uncorrelated, pulses, which is given by 1/Nc [23], [24]. As long as Nc is they are independent as well, implying that overall noise PDF sufﬁciently large, ωMUI,2 (m) becomes negligible with respect of ωSUC is given by fSUC = fSUC,1 ∗ fSUC,2 ∗ fSUC,3 , where to ωMUI,1 (m). The reasoning behind this negligibility is that ∗ denotes linear convolution, which results in (11). the collision probability of the same user’s pulses is obviously one. Therefore, we approximately get A PPENDIX B 2 Nu Nt /2−1 ωMUI (m) ≈ ωMUI,1 (m) = βu,i In this appendix, we furnish the derivation of (27). The Nt i=0 u=1,u=d noise ωMA (m) consists of three parts as where the PDF of βu,i can be found as ωMA (m) = ωMA,1 (m) + ωMA,2 (m) + ωMA,3 (m) fβu,i (x) = h1 (x) + h2 (x) where where h1 (x) and h2 (x) are given by Nu Ts ωMA,1 (m) = ¯ ru (t)w(t)dt, m h1 (x) = 0.25 δ x − Nf Cg,u + δ x + Nf Cg,u u=1 0 0.25 2 Nu Nt /2−1 Ts h2 (x) = u x + Nf Cg,u − u x − Nf Cg,u ωMA,2 (m) = ru (t)w(t)dt, i Nf Cg,u Nt 0 u=1 i=0 and u(x) is the unit-step function. We assume that all the Ts ωMA,3 (m) = ¯ w(t)w(t)dt interfering users have same received energy, hence Cg,u = 0 Cg,l = Cg , ∀l, u ∈ {1, . . . , Nu }. The PDF of ωMUI (m) can be expressed as (Nu − 1)Nt /2-fold convolution; that is where ru (t) and ru (t) are given in (26). Following the i m analysis carried out in the previous section, it can be shown 2 2 fMUI x = fβ (x) ∗ · · · ∗ fβ (x) (31) that these noise samples are independent and their PDFs are Nt Nt Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. 4668 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 where we dropped the subscripts of βu,i . Hence, n-fold [4] S. Aedudodla, S. Vijayakumaran, and T. F. Wong, “Timing acquisition convolution of fβ (x), denoted by fβ (x), is n in ultra-wideband communication systems,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1570-1583, Sept. 2005. n [5] “FCC notice of proposed rule making, revision of part 15 of the (n−l) commission’s rules regarding ultra-wideband transmission systems,” fβ (x) = n C(n, l) hl (x) 1 ∗ h2 (x) . (32) Federal Communications Commission, Washington, DC, ET-Docket 98- l=0 153. Approximating Dirac-delta functions in h1 (x) with approxi- [6] E. A. Homier and R. A. Scholtz, “Rapid acquisition of ultra wideband signals in dense multipath,” in Proc. Conf. UWB Syst. Technol., Balti- mately zero variance Gaussian functions, we get more, MD, May 2002, pp. 105-110. l [7] E. A. Homier and R. A. Scholtz, “Hybrid ﬁxed dwell-time search 2 techniques for rapid acquisition of ultra wideband signals in dense hl (x) 1 = (0.25) l C(l, i)g x; (2i − l)Nf Cg , lσ0 (33) multipath,” in Proc. Int. Workshop UWB Syst., Oulu, Finland, Jun. 2003. i=0 [8] Z. Tian and G. B. Giannakis, “A GLRT approach to data-aided timing acquisition in UWB radios—part I: algorithms,” IEEE Trans. Wireless 2 for the lth order convolution of h1 (x), where σ0 ≈ 0. Using Commun., vol. 4, no. 6, pp. 2956-2967, Nov. 2005. the analysis in [23], we get [9] J. R. Foerster, “Channel modeling sub-committee report (ﬁnal),” Tech. Rep. P802.15-02 / 368r5-SG3a, IEEE P802.15 Working Group for n−l Wireless Personal Area Networks (WPANs), Dec. 2002. (0.25)n−l 1 [10] L. Yang and G. B. Giannakis, “Timing UWB signals with dirty tem- hn−l (x) = 2 (−1)j C(n − l, j) (n − l − 1)! Nf Cg j=0 plate,” IEEE Trans. Commun., vol. 53, no. 11, pp. 1952-1963, Nov. 2005. n−l−1 [11] S. Franz, C. Carbonelli, and U. Mitra, “Semi-blind ML synchronization x x × + n − l − 2j u + n − l − 2j for UWB signals,” in Proc. 42th Allerton Conf., Monticello, IL, Sep. Nf Cg Nf Cg 2004. (34) [12] L. Yang, Z. Tian, and G. B. Giannakis, “Non-data aided timing acqui- sition of ultra wideband transmissions using cyclostationary,” in Proc. for the (n − l)th order convolution of h2 (x). Combining (33) ICASSP, Hong Kong, Apr. 2003, pp. 121-124. and (34), (32) becomes [13] I. Maravic, J. Kusuma, and M. Vetterli, “Low-sampling rate UWB channel characterization and synchronization,” J. Commun. Netw., vol. 5, fβ (x) = n pp. 319-327, 2003. [14] K. L. Blackard, T. S. Rappaport, and C. W. Bostian, “Measurements (0.25)n n C(n, l) l n−l and models of radio frequency impulsive noise for indoor wireless (−1)j C(n − l, j)C(l, i) communications,” IEEE J. Select. Areas Commun., vol. 11, pp. 991- Nf Cg (n − l − 1)! i=0 j=0 1001, Sep. 1993. l=0 [15] Y. D. Alemseged and K. Witrisal, “Modeling and mitigation of narrow- n−l−1 x x band interference for transmitted-reference systems,” IEEE J. Select. × + n − 2i − 2j u + n − 2i − 2j Topics Signal Process., vol. 1, no. 3, pp. 456-469, Oct. 2007. Nf Cg Nf Cg [16] u ¸ N. G¨ ney, H. Delic, and M. Koca, “Robust detection of ultra-wideband 2 signals in non-gaussian noise,” IEEE Trans. Microw. Theory Tech., where we used the fact that ≈ 0, i.e., they are approxi- σ0 vol. 54, no. 4, pp. 1724-1730, Apr. 2006. mately Dirac delta functions, to evaluate the convolutions in [17] R. T. Hoctor and H. W. Tomlinson, “An overview of delay-hopped (32). Moreover, approximating the term [23] transmitted-reference RF communications,” G.E. Research and Devel- opment Center, Tech. Inform. Series, pp. 1-29, 2002. n−l n−l−1 [18] P. J. Huber, Robust Statistics. New York: Wiley, 1981. x [19] Z. Wang, and G. B. Giannakis, “Wireless multicarrier communications: (−1)j C(n − l, j) + n − 2i − 2j Nf Cg where Fourier meets Shannon,” IEEE Signal Processing Mag., vol. 17, j=0 no. 3, pp. 29-48, May 2000. x [20] D. Middleton, “Statistical-physical models of electromagnetic interfer- ×u + n − 2i − 2j ence,” IEEE Trans. Electromagn. Compat., vol. EC-19, no. 8, pp. 106- Nf Cg 127, Aug. 1977. with [21] K. S. Vastola, “Threshold detection in narrowband non-Gaussian noise,” IEEE Trans. Commun., vol. COM-32, no. 2, pp. 134-139, Feb. 1984. 2n−l (n − l − 1)! 3x2 [22] X. Wang, and H. V. Poor, “Robust multiuser detection in non-Gaussian exp − , channels,” IEEE Trans. Signal Processing, vol. 47, no. 2, pp. 289-305, 2π n−l 2(Nf Cg )2 (n − l) Feb. 1999. 3 [23] A. R. Foruzan, M. Nasiri-Kenari, and J. A. Salehi, “Performance analy- we get sis of time-hopping spread-spectrum multiple access systems: Uncoded and coded schemes,” IEEE Trans. Commun., vol. 1, no. 4. pp. 671-681, n (n − l)(Nf Cg )2 Oct. 2002. fβ (x) = (0.5)n n C(n, l)g z; 0, . [24] Y. Dhibi and T. Kaiser, “On the impulsiveness of multiuser interferences 3 in TH-PPM-UWB systems,” IEEE Trans. Signal Processing, vol. 54, l=0 (35) no. 7, pp. 2853-2857, July 2006. [25] J. Fiorina and W. Hachem, “On the asymptotic distibution of the Finally, from (31) and (35), we can conclude (28). correlation receiver output for time-hopped UWB signals,” IEEE Trans. Signal Processing, vol. 54, no. 7, pp. 2529-2545, July 2006. R EFERENCES [26] S. M. Zabin and H. V. Poor, “Efﬁcient estimation of class A noise parameters via the EM algorithm,” IEEE Trans. Inform. Theory, vol. 37, [1] L. Yang and G. B. Giannakis, “Ultra wideband communications: an idea no. 1, pp. 60-72, Jan. 1991. whose time has come,” IEEE Signal Processing Mag., vol. 21, no. 6, [27] ¸ E. Ekrem, M. Koca, and H. Delic, “Ultra-wideband signal acquisition in pp. 26-54, Nov. 2004. non-Gaussian noise via successive sampling,” in Proc. 65th IEEE Veh. [2] L. Yang, G. B. Giannakis, and A. Swami, “Non-coherent ultra-wideband Tech. Conf., Dublin, Ireland, Apr. 2007. (de)modulation”, IEEE Trans. Commun., vol. 55, no. 4. pp. 810-819, [28] ¸ E. Ekrem, M. Koca, and H. Delic, “Robust acquisition of ultra-wideband Apr. 2007. signals with averaged template,” in Proc. WCNC 2007, Hong Kong, Mar. [3] Z. Tian and G. B. Giannakis, “BER sensitivity to mistiming in ultra 2007. wideband communications—part 1: nonrandom channels,” IEEE Trans. [29] T. Q. S. Quek, M. Z. Win, and D. Dardari, “Uniﬁed analysis of Signal Processing, vol. 53, no. 4, pp. 1550-1560, Apr. 2005. UWB transmitted-reference schemes in the presence of narrowband Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply. EKREM et al.: ROBUST ULTRA-WIDEBAND SIGNAL ACQUISITION 4669 interference,” IEEE Trans. Wireless Commun., vol. 6, no. 6, pp. 2126- ¸ Hakan Delic (SM’00) received the B.S. degree 2139, June 2007. (with honors) in electrical and electronics engineer- [30] H. J. Larson and B. O. Shubert, Probabilistic Models in Eng. Sci.. ing from Boˇ azic i University, ˙ g ¸ Istanbul, Turkey, in Malabar, FL: R.E. Krieger Pub. Co., 1989. 1988, and the M.S. and the Ph.D. degrees in elec- [31] J. H. Miller and J. B. Thomas, “The detection of signals in impulsive trical engineering from the University of Virginia, noise modeled as a mixture process,” IEEE Trans. Commun., vol. COM- Charlottesville, in 1990 and 1992, respectively. He 24, no. 5, pp. 559-563, May 1976. was a Research Associate with the University of Virginia Health Sciences Center from 1992 to 1994. Ersen Ekrem received the B. S. and M. S. de- In September 1994, he joined the University of grees in electrical and electronics engineering from Louisiana at Lafayette, where he was on the Fac- Boˇ azic i University, Istanbul, Turkey, in 2006 and g ¸ ulty of the Department of Electrical and Computer 2007, respectively. Currently, he is working toward Engineering until February 1996. He was a Visiting Associate Professor the Ph. D. degree in the Department of Electrical in the Department of Electrical and Computer Engineering, University of and Computer Engineering, University of Maryland, Minnesota, Minneapolis, during the 2001-2002 academic year. Dr. Delic is ¸ College Park. His research interests include wireless g ¸ currently a professor of electrical and electronics engineering at Boˇ azic i communications and multiuser information theory. University. His research interests lie in the areas of communications and signal processing with current focus on wireless multiple access, ultra-wideband communications, OFDM, robust systems, and sensor networks. Mutlu Koca (M’01) received the B.S. degree from g ¸ Boˇ azic i University, Istanbul, Turkey, in 1996 and the M.S. and the Ph.D. degrees from the University of California, Davis (UC-Davis), in 2000 and 2001, respectively, all in electrical engineering. He was a Postgraduate Researcher and a Lecturer at UC-Davis between September and December 2001. He was a Member of the Technical Staff at Flarion Technolo- gies, Bedminster, NJ between December 2001 and December 2002, and a Researcher at the Institut de Recherche en Informatique et Systemes Aleatoires (IRISA), Rennes, France between February 2003 and March 2004. In July 2003, he joined the Department of Electrical and Electronics Engineering, g ¸ of Boˇ azic i University where he is currently an Assistant Professor. His research interests are in communication theory, error correcting codes, signal processing for communications, ultra-wideband and iterative (turbo) methods. Authorized licensed use limited to: University of Maryland College Park. Downloaded on August 19, 2009 at 12:59 from IEEE Xplore. Restrictions apply.

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