a glrt approach to data-aided timing acquisition in uwb ... · 2956 ieee transactions on wireless...

2956 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005

A GLRT Approach to Data-Aided TimingAcquisition in UWB Radios—Part I: Algorithms

Zhi Tian, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—Realizing the great potential of impulse radio com-munications depends critically on the success of timing acquisition.To this end, optimum data-aided (DA) timing offset estimatorsare derived in this paper based on the maximum likelihood (ML)criterion. Specifically, generalized likelihood ratio tests (GLRTs)are employed to detect an ultrawideband (UWB) waveform prop-agating through dense multipath and to estimate the associatedtiming and channel parameters in closed form. Capitalizing on thepulse repetition pattern, the GLRT boils down to an amplitude es-timation problem, based on which closed-form timing acquisitionestimates can be obtained without invoking any line search. Theproposed algorithms only employ digital samples collected at a lowsymbol rate, thus reducing considerably the implementation com-plexity and acquisition time. Analytical acquisition performancebounds and corroborating simulations are also provided.

Index Terms—Data-aided acquisition, generalized likelihoodratio test, timing recovery, ultrawideband communications.

I. INTRODUCTION

R ESEARCH on ultrawideband (UWB) communicationsis burgeoning for its envisioned applications on high-

speed short-range wireless access, with capability to overlayexisting channelized RF services [1], [2]. UWB impulse radiostransmit a stream of ultrashort pulses (on the order of sub-nanoseconds) at very low power density. To maintain adequatesignal energy for reliable detection, each information-bearingsymbol is transmitted over a large number of frames with onepulse per frame. The frame duration is much larger than thepulse duration, resulting in a low duty cycle UWB transmission.

Implementation of UWB radios has been perplexed by thedifficulty in synchronization due to ultrashort low-duty-cyclepulses operating at very low power density. Timing recoveryis required not only at the frame level to find when the firstframe in each symbol starts but also at the pulse level to findwhere a pulse is located within a frame. Conventional sliding-correlation-based synchronizers require a long search time and

Manuscript received January 19, 2004; revised April 25, 2004; acceptedAugust 31, 2004. The editor coordinating the review of this paper and approvingit for publication is G. M. Vitetta. The work of Z. Tian was supported by theNational Science Foundation (NSF) under Grant CCR-0238174. The work ofG. B. Giannakis was supported by the ARL/CTA under Grant DAAD19-01-2-011 and by NSF-ITR under Grant EIA-0324864. This paper was presented inpart at the 2003 IEEE Conference on Ultra Wideband Systems and Technolo-gies (UWBST’2003), Reston, VA, November 2003.

Z. Tian is with the Department of Electronics and Communications En-gineering, Michigan Technological University, Houghton, MI 49931 USA(e-mail: [email protected]).

G. B. Giannakis is with the Department of Electronics and CommunicationsEngineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TWC.2005.858356

an unreasonably high sampling rate at several gigahertz [3].The complexity issue is further exacerbated by the perfor-mance degradation incurred by dense multipath propagation,especially when time hopping (TH) is used for smoothingthe transmit spectrum and for enabling multiple access [1].Without properly accounting for the unique features of UWBtransmissions, synchronization methods that are well suited fornarrowband systems are no longer effective. Recent works havefocused on obtaining low-complexity algorithms for rapid tim-ing acquisition by making use of coarse bin search [4]–[6] andexploiting coded beacon sequences in conjunction with a cor-relator bank [8] or subspace-based spectral estimation [9], [10].Capitalizing on the cyclostationarity naturally present in UWBsignaling, non-DA timing acquisition is also possible [11], [12].Based on the maximum likelihood (ML) criterion, parameterestimation (including tap gains and delays) of UWB multipathchannels in the presence of multiple-access interference (MAI)was pursued in [13]. As this ML timing estimator has to operateat a high (subpulse) rate, it is computationally prohibitive fortiming acquisition. Relying on a special pilot symbol pattern,a rapid synchronizer was developed recently using frame-ratecross-correlation samples of neighboring noisy received wave-forms [17].

In this paper, we develop a data-aided (DA) ML estimationapproach, where only symbol-rate samples are needed fortiming acquisition. Starting with the no-TH case, we derivea generalized likelihood ratio test (GLRT) for joint detectionand frame-level timing acquisition, where channel-dependentunknowns are regarded as nuisance parameters. Interestingly,frame-level timing offset acquisition amounts to an amplitudeestimation problem accepting a closed-form solution. Basedon symbol-rate samples, the GLRT yields channel-dependentamplitude estimates, the timing acquisition solution, the symboldetection rule, as well as the associated estimation performancebounds. The GLRT reveals that the training symbol sets re-quired for estimating channel-dependent parameters and foracquiring timing information should be nonoverlapping. Judi-cious design of training sequences is desired also to optimizethe overall system performance—a goal that we will pursue ina companion paper [15].

Next, we extend our results to the TH case by properlyselecting a noisy template to generate correlator output sam-ples. Such a template inherently accounts for the unknown THpattern under mistiming, thus being able to retain the desiredsymbol-rate sample structure that links timing offset parameterswith amplitude scaling factors. Use of a noisy template alsoenjoys effective multipath energy capture and asymptoticallymaximizes the received signal-to-noise ratio (SNR), which in

1536-1276/$20.00 © 2005 IEEE

TIAN AND GIANNAKIS: GLRT APPROACH TO DATA-AIDED TIMING ACQUISITION IN UWB RADIOS—PART I 2957

turn improves timing accuracy, even for UWB radios that donot employ TH.

The rest of the paper is organized as follows. Section II de-scribes a general-form symbol-rate discrete-time UWB signalmodel with timing offset. In Section III, the timing acquisi-tion problem is formulated as a GLRT, treating the aggregatediscrete-time channel-dependent gains as deterministic nui-sance parameters. The GLRT solution to timing acquisition isextended to the TH case in Section IV, where a noisy template isadopted to capture adequate multipath energy and to enable tim-ing estimation that is robust to TH. Simulations are presentedin Section V for different algorithms under various operatingenvironments to demonstrate their mean-square timing estima-tion errors and their system-level impact on bit error rate (BER)performance. The paper concludes with summarizing remarksin Section VI.

II. SIGNAL MODEL

In UWB impulse radios, every symbol is transmitted us-ing Nf pulses over Nf frames1 with one pulse per frame.Every frame contains Nc chips. The symbol waveform ofduration Ts := TfNf is ps(t) =

∑Nf−1j=0 p(t − jTf − cjTc)2,

where p(t) is an ultrashort (so-called monocycle) pulse of widthTp at the nanosecond scale, Tf is the frame duration that may bea hundred to a thousand times Tp, Tc := Tf/Nc is the chip du-ration, and the chip sequence {cj} represents the user’s pseudo-random TH code with cj ∈ [0, Nc − 1], ∀j ∈ [0, Nf − 1].The pulse amplitude is scaled to satisfy

∫p2(t)dt = 1/Nf ,

such that the symbol waveform has unit energy∫

p2s(t)dt = 1.

We focus on pulse amplitude modulation (PAM) [2], wherethe information-bearing symbols s[n] ∈ {±1} are modeled asbinary independent and identically distributed with energy Es

spread over Nf frames. The transmitted UWB waveform isthen given by [see also Figs. 1(a) and 2(a) for an examplewith Nf = 3]

u(t) =√

Es

∞∑n=−∞

s[n]ps(t − nTs). (1)

The signal u(t) propagates through an L-path fading channel,with {αl} and {τl} representing the attenuation and the delayof the lth path, respectively, and τ0 < τ1 < · · · < τL−1. Timinginformation of interest refers to the first arriving time τ0,which is measured with reference to the receiver’s local clock.From this perspective, we isolate τ0 and model the channelimpulse response as a summation

∑L−1l=0 αlδ(t − τl,0), where

τl,0 := τl − τ0 is the relative time delay of each path. Thechannel delay spread τL−1,0 stays within a frame since we se-lect Tf ≥ τL−1,0 + Tp + (cNf−1 − c0)Tc to avoid intersymbol

1Spreading a single symbol over Nf frames in the UWB case reverses thecommonly used terminology where a frame consists of multiple symbols.

2Throughout the paper, we will use the subscript s to refer to symbol-related variables, such as symbol duration, symbol waveforms, symbol-leveltiming offset, and symbol-long correlation templates. Similarly, we will use thesubscripts f and r to denote frame- and receiver-related quantities.

Fig. 1. Transmit/receive waveforms and symbol templates in the TH-freecase: ns = 0, nf = 1, and Nf = 3. Dashed vertical lines represent symbolboundaries in the transmitted waveform and the correlation template, whilesolid vertical lines represent symbol boundaries in the received waveform.

interference [Fig. 1(b)]. The composite channel expressed asthe convolution of the multipath with the transmitted pulse p(t)is given by g(t) =

∑L−1l=0 αlp(t − τl,0), and the time-dispersed

received symbol waveform of duration Ts becomes

gs(t) =Nf−1∑j=0

g(t − jTf − cjTc) =L−1∑l=0

αlps(t − τl,0) (2)

as illustrated in Fig. 1(b) for the no-TH case and in Fig. 2(b)for the TH case with TH-induced interframe interference (IFI).The receive waveform is thus given by

r(t) =L−1∑l=0

αlu(t − τ0 − τl,0) + w(t)

=√Es

∞∑n=−∞

s[n]gs(t − nTs − τ0) + w(t) (3)

where the wide sense stationary process w(t) accounts for boththermal noise and MAI, which similar to [7] is approximated aswhite Gaussian. Letting �·� denote integer floor, we dissect τ0

based on different time scales and express it as τ0 := nsTs +nfTf + ε, where ns := �τ0/Ts� denotes the symbol-leveloffset, nf := �(τ0 − nsTs)/Tf� ∈ [0, Nf − 1] the frame-leveloffset, and ε ∈ [0, Tf ) the pulse-level offset, respectively. Thesedifferent levels of timing offsets are illustrated in Fig. 1 via anexample where τ0 = (ns = 0)Ts +(nf = 1)Tf +(ε = 2Tf/3),with Nf = 3. Using these definitions and (1), the receivedsignal can be expressed by

r(t)=√

Es

∞∑n=−∞

s[n]gs(t−nTs−nsTs−nfTf −ε)+w(t).

(4)


Fig. 2. Transmit/receive waveforms and the ideal/noisy template in the TH case (noise-free): ns = 0, nf = 1, Nf = 3, Tc = Tf /3, and the TH code

{cj}Nf−1

j=0 for each symbol is [0,2,0]. Dashed pulses in (b) are the received g(t) after channel distortion, and solid pulses are the composite gs(t) after mixingTH-shifted g(t)s.

A symbol-by-symbol sliding correlator is applied to generatesamples y[n] at the symbol rate by

y[n] :=

Ts∫0

r(t + nTs)prs(t)dt (5)

where prs(t), t ∈ [0, Ts] is the correlation template. The choiceof the template prs(t) affects detection performance and itsrobustness to mistiming. A simple template corresponds toprs(t) = ps(t), whereas the ideal template under perfect timingand channel knowledge is given by prs(t) = gs(t), which isequivalent to maximum ratio combining. In the presence ofdense multipath, the choice of prs(t) affects the received en-ergy captured in discrete-time samples {y[n]}, which in turninfluences the timing offset estimation accuracy of our digitalsynchronizers.

Given N samples {y[n]}N−1n=0 , our goal in this paper is to

acquire timing by optimally estimating ns and nf based on Mtraining symbols {s[n]}M−1

n=0 —a DA digital timing acquisitionproblem.

III. GLRT-BASED SYNCHRONIZATION APPROACH

For clarity, we first consider a point-to-point link where noTH is employed, i.e., {cj = 0}, and derive optimum acquisitionrules using the GLRT (see, e.g., [18, Ch. 6]). The results will beextended to include TH in the next section.

A. Discrete-Time Signal Model Under Mistiming

Under perfect timing (τ0 = 0), each sample y[n] of thecorrelator output serves as the detection statistic for a singlesymbol s[n]. With mistiming (τ0 �= 0), however, y[n] dependson two consecutive symbols as depicted in Fig. 1(c) and (d) [forclarity we set ns = 0 in Fig. 1(b)]. The time indices of the twocontributing symbols are determined by the symbol-level offsetnsTs, whereas portions of the corresponding two transmittedsymbol waveforms depend on the frame-level offset nfTf . We

quantify these effects in Appendix A and establish the resultingsampled model in the following proposition.Proposition 1: For a general receive correlator template

prs(t) =∑Nf−1

j=0 prf (t − jTf ) of duration Ts with prf (t) hav-ing nonzero support Trf ≤ Tf , the symbol-rate sampled outputy[n] in the no-TH case is given by

y[n] = Aε (s[n − ns](Nf − nf − λε)

+ s[n − ns − 1](nf + λε)) + w[n] (6)

where Aε :=√Es

∫ Tf

0 gs(t+Tf −ε)prf (t)dt =√Es

∫ Tf

ε g(t−ε)prf (t)dt +

√Es

∫ ε

0 g(t + Tf − ε)prf (t)dt is independentof ns and nf , and λε := (1/Aε)

√Es

∫ ε

0 g(t + Tf − ε)prf (t)dt is bounded within [0,1]. The sample w[n] denotesadditive white Gaussian noise (AWGN) with zero mean.Furthermore

1) if s[n − ns] = s[n − ns − 1], then (6) reduces to y[n] =AεNfs[n − ns] + w[n];

2) if s[n − ns] = −s[n − ns − 1], then y[n] = Aε(Nf −2νf,ε)s[n − ns] + w[n], where νf,ε := nf + λε ≈τ0/Tf ;

3) if prf (t) = p(t) and Tf ≥ τL−1,0 + 2Tp, then νf,ε =nf , ∀ε.

Proposition 1 clearly establishes the effects of different levelsof timing offset on the symbol-rate correlator output y[n]. Whentwo consecutive symbols have the same sign as in 1), thecorresponding y[n] contains no information about nf . However,the channel-dependent amplitude Aε can be estimated fromy[n] samples obeying 1) in the ML sense and the resultingAε can be used in 2) to recover τ0 and nf as ˆτ0 ≈ νf,εTf

and nf = �νf,ε� since λε ∈ [0, 1]. Estimation of nf dependson the choice of the frame-level template prf (t). When theconventional prf (t) = p(t) is used [cf. Fig. 1(c)] and the framelength Tf is suitably selected to include an extra guard timeTp in addition to the spread of the aggregate channel g(t)[cf. Fig. 1(b)], it follows from 3) that estimating νf,ε from 2)in the ML sense is identical to computing the ML estimateof nf . However, as will become clear later on, estimation


performance can be improved with a general prf (t) of widthTrf > Tp [e.g., the trapezoidal template shown in Fig. 1(d)]. Inthis case, we will pursue the ML estimate of νf,ε based on 2)and subsequently estimate nf as �νf,ε�.

B. Formulation as a Hypothesis Testing Problem

Building on Proposition 1, we now pursue estimation of boththe symbol-level timing offsets ns and νf,ε that contain theframe-level timing offset nf . Suppose that a total of M train-ing symbols {s[n]}M−1

n=0 are received during [τ0, τ0 + MTs].Without knowing when the UWB stream starts, a synchronizercollects N observations y := [y[0], . . . , y[N − 1]]T over theinterval [0, NTs]. We formulate the following binary hypothesistesting problem for detecting y[n] along with estimating ns andνf,ε, i.e.,

H1 :y[n] = Aε(s[n−ns](Nf −νf,ε)+s[n−ns−1]νf,ε)

+w[n], n=ns, . . . , ns+M−1≤N−1 (7)

H0 :y[n] = w[n], n=0, 1, . . . , N−1. (8)

Here, s[n] is deterministic and known, Aε, νf,ε, and ns areunknown nuisance parameters, and w[n] has variance σ2

w. Thisis a classical detection problem with unknown parameters inAWGN, for which the GLRT is well motivated. Utilizing theprobability density function (pdf) p(·) of y conditioned on thetwo hypotheses, the GLRT rejects H0 if

J0(y;Aε, νf,ε, ns) :=p(y; Aε, νf,ε, ns,H1)

p(y;H0)≥ τFA (9)

where Aε, νf,ε, and ns are the corresponding ML estimates(MLEs) under H1, and TFA is a threshold set by the desiredprobability of false alarms (FA). When seeking MLEs of theunknown parameters, we not only assume that νf,ε and ns aredeterministic but also treat the channel-dependent amplitudeAε as deterministic but unknown, which leads to a condi-tional ML (CML) approach [20]. In contrast, an unconstrainedML regards the unknown Aε as a random variable, whichleads to Aε estimators that depend on the underlying channelstatistics.

Before proceeding with the GLRT, it is appropriate to clarifythe role of the training sequence (TS) pattern in this hypothesis-testing problem. The correlator output samples under H1, de-noted as ys := {y[n] : y[n] ∈ H1} = [y[ns], . . . , y[ns + M −1]]T , clearly depend on ns. For binary PAM, ys can besplit into two groups: one group, denoted by y+(ns) :={y[n] : n ∈ G+(n;ns)}, is indexed by all the ns in the setG+(n;ns) := {n : s[n − ns] = s[n − ns − 1]}, while the sec-ond group, y−(ns) := {y[n] : n ∈ G−(n;ns)}, is associatedwith the rest of ns in the set G−(n;ns) := {n : s[n − ns] =−s[n − ns − 1]}. For a given ns, this partitioning into twononoverlapping sets G+(n;ns) and G−(n;ns) is solely deter-mined by whether the two successive symbols involved in thecorrelator output under mistiming [cf. (6)] have identical oropposite signs. The two data sets have sizes M+ := |G+(n;ns)|

and M− := |G−(n;ns)|, respectively, which obviously dependon the TS pattern, but not on the symbol offset ns. Dependingon whether s[n] = s[n − 1] or s[n] = −s[n − 1], H1 in (7) canbe rewritten as

H1 : y[n] ={

AεNfs[n] + w[n], n ∈ G+(n;ns)Aε(Nf − 2νf,ε)s[n] + w[n], n ∈ G−(n;ns).

(10)

Since y+(ns) does not contain νf,ε, it will not play a rolein finding the MLE νf,ε. On the other hand, for any y[n] ∈y−(ns), the channel-dependent parameter Aε and the frame-level timing parameter νf,ε are always coupled in a nonidenti-fiable manner. Indeed, the true pair (Aε, νf,ε) and another pair(cAε, ((c − 1)Nf + 2νf,ε)/2c), with any c ≥ 1 − 2νf,ε/Nf ≥0, give rise to the same y−(ns). These observations motivate usto estimate Aε and νf,ε from the two disjoint data sets y+(ns)and y−(ns), separately. Compared with working directly onthe entire sample set ys = y−(ns)

⋃y+(ns), this separate

approach will retain ML optimality of νf,ε since we do notdisregard any useful data. One may question the optimalityof Aε since the unused portion y−(ns) also contains channel-related information. We maintain that y−(ns) does not offer ad-ditional information on Aε since it cannot resolve the ambiguitybetween Aε and (Nf − 2νf,ε) without knowing νf,ε, which hasto be estimated from y−(ns) itself. This optimality claim willbe rigorously proved in the following subsection.

C. CML Timing Acquisition Using Symbol-Rate Samples

With Aε, ns, νf,ε being deterministic but unknown and w[n]being AWGN, straightforward manipulation leads to the log-likelihood ratio (LLR)

J(y;Aε, νf,ε, ns)

=2σ2w log

p(y;Aε, νf,ε, ns,H1)p(y;H0)

=2Aε

ns+M−1∑n=ns

y[n](s[n−ns](Nf −νf,ε)+s[n−ns−1]νf,ε)

−A2ε

ns+M−1∑n=ns

(s[n−ns](Nf −νf,ε)+s[n−ns−1]νf,ε)2 .

(11)

The nuisance parameters in this LLR are Aε, νf,ε, and ns,whose MLEs will be derived in an alternating fashion. Let usfirst fix ns and estimate Aε from y+(ns) and νf,ε from y−(ns),as suggested by the separation property we discussed earlier.The MLE of ns will then be found after substituting Aε andνf,ε in (11) with their ns-dependent MLEs.

To perform separate ML estimation, we decompose the LLRin (11) into J(y;Aε, νf,ε, ns) = J+(y+(ns);Aε, νf,ε, ns)+J−(y−(ns);Aε, νf,ε, ns), where J+(·) and J−(·) consist ofthe subsets of summands corresponding to G−(n;ns) and


G+(n;ns), respectively. When estimating Aε, only J+(·) isrelevant, which we write as a function of Aε in the form

J+ (Aε;ns,y+(ns))

= 2AεNf

∑n∈G+(n;ns)

y[n]s[n − ns] − E+MA2

εN2f (12)

where E+M :=

∑n∈G+(n;ns) s2[n − ns] is the symbol energy

corresponding to y+(ns). In a binary transmission, E+M = M+

equals the cardinality of G+(n;ns), which is independent ofns. The LLR in (12) corresponds to the problem of detecting adeterministic signal known except for its amplitude in AWGN[18]. It is straightforward to find from (12) the MLE of Aε

conditioned on ns, the associated Cramer–Rao bound (CRB),and the resulting maximum J+(·) at the optimum Aε(ns), allin closed-form

Aε(ns)

=1

NfE+M

∑n∈G+(n;ns)

y[n]s[n − ns] (13)

CRB(Aε;ns)

= − 2σ2w

∂2J+(Aε;ns,y+(ns))∂A2

ε

=σ2

w

N2f E

+M

(14)

J+(ns;y+(ns))

:= J+

(ns; Aε(ns),y+(ns)

)

=1E+

M

∑

n∈G+(n;ns)

y[n]s[n − ns]

2

. (15)

The Aε(ns) estimate in (13) takes the usual cross correlationform between the input sequence and the shifted output se-quence. The estimation quality, measured by the CRB(Aε;ns)in (14), is inversely proportional to the number of trainingsymbols in the set G+(n;ns) as expected, while the resultingmaximum LLR in (15) is proportional to the cross-correlationenergy.

Moving on to find the MLE of νf,ε, we focus on y−(ns) andwrite the corresponding J−(·) as

J− (νf,ε;Aε, ns,y+(ns)) = 2Aε(Nf − 2νf,ε)

×∑

n∈G−(n;ns)

y[n]s[n − ns] − E−MA2

ε(Nf − 2νf,ε)2

(16)

where E−M :=

∑n∈G−(n;ns) s2[n − ns]. Since Aε has been

found via (13), the LLR in (16) corresponds to the prob-lem of detecting a deterministic signal known except for themistiming-dependent scale factor (Nf − 2νf,ε) in AWGN. By

maximizing (16), we reach the MLE of νf,ε conditioned on Aε

and ns, the associated CRB, and the resulting maximum J−(·)at the optimum νf,ε(ns), also in closed-form

νf,ε(Aε, ns)

=Nf

2− 1

2AεE−M

∑n∈G−(n;ns)

y[n]s[n − ns] (17)

CRB(νf,ε;Aε, ns)

= − 2σ2w

∂2J−(νf,ε;Aε,ns,y−(ns))

∂ν2f,ε

=σ2

w

4A2εE−

M

(18)

J− (ns;y−(ns))

:= J− (ns; νf,ε(Aε, ns),y−(ns))

=1E−

M

∑

n∈G−(n;ns)

y[n]s[n − ns]

2

. (19)

Similar to (13), the νf,ε estimate in (17) is determined bythe cross correlation over the set G−(n;ns), while again theCRB in (18) is inversely proportional to the number of trainingsymbols in this set since E−

M = M−. In addition, νf,ε andits associated conditional CRB also depend on the aggregatechannel-dependent parameter Aε. On the other hand, note from(19) that J−(·) at the optimum νf,ε(Aε, ns) does not dependon Aε. Therefore, no Aε estimate can be obtained from y−(ns)due to the ambiguity between Aε and νf,ε. In fact, the overallLLR for optimizing Aε is the same as J+(·) induced by y+(ns)because

J(Aε; νf,ε, ns,y)

= J+ (Aε; νf,ε, ns,y+(ns)) + J− (Aε; νf,ε, ns,y−(ns))

= J+ (Aε, ns,y+(ns)) + C (20)

where C := J−(ns;y−(ns)) is a constant independent of Aε.Thus, we have proved that the MLE of Aε based on the entiredata set ys coincides with that based on y+(ns) only. In fact,a more tedious derivation that directly maximizes (11) withrespect to Aε yields

A(ns) = arg maxAε

J(Aε;ns,y)

=

∑ns+M−1n=ns

y[n] (s[n − ns] + s[n − ns − 1])

NfEM + Nf

∑ns+M−1n=ns

s[n − ns]s[n − ns − 1]

(21)

where EM := E+M + E−

M . Seemingly different from the Aε esti-mate obtained from y+(ns) in (13), the Aε obtained from ys in(21) can be reduced to exactly that in (13), utilizing the binarynature of {s[n]}. In a nutshell, we have established that theseparate estimation approach retains ML optimality of Aε.


Having obtained the ns-dependent MLEs of Aε and νf,ε in(13) and (17), the next task is to derive the MLE of ns. Notethat either J+(ns;y+(ns)) or J−(ns;y−(ns)) could serve asan objective function based on which ns can be estimated usinga line search. However, since both y+(ns) and y−(ns) containuseful information about ns, the MLE utilizing all the samplesy should be designed from the LLR in (11), which is nothingbut the sum of J+(·) and J−(·). Thus, we reach from (15) and(19) the MLE of ns as

ns = arg maxns∈[0,N−M ]

J(ns,y)

= arg maxns∈[0,N−M ]

[J+ (ns;y+(ns)) + J− (ns;y−(ns))]

= arg maxns∈[0,N−M ]

1E+

M

∑

n∈G+(n;ns)

y[n]s[n−ns]

2

+1E−

M

∑

n∈G−(n;ns)

y[n]s[n−ns]

2.

(22)

So far, we have obtained the MLEs of all nuisance para-meters, where Aε depends on ns and νf,ε depends on bothAε and ns. In implementing our timing algorithm, one shouldfollow these dependencies and proceed in a reverse order toestimate ns first, Aε afterwards, and finally νf,ε, according to(22), (13), and (17). The estimate of nf itself is then given bythe integer floor of νf,ε. The UWB waveform is detected whenthe optimum LLR exceeds a threshold TFA predescribed by adesired probability of false alarms PFA

detection : J(ns;y) ≥ 2σ2w ln TFA (23)

where TFA = σwQ−1(PFA) and Q(x) := (1/√

2π)∫ ∞x exp(−u2/2)du denotes the complementary error function.

The implementation advantages of our symbol-rate GLRTtiming acquisition approach are evident. In addition to requir-ing very low sampling rate of one sample per symbol, thisapproach does not involve discrete-time line search, as mostcorrelation-based synchronizers do at least on a frame-by-framebasis [17]. The frame-level timing offset information manifestsin the amplitudes of two TS subsets and is obtained usingsimple amplitude estimators based on digital cross-correlationoperations.

IV. ROBUST TIMING RECOVERY WITH NOISY TEMPLATE

When TH is employed, the discrete-time model of y[n] in(6) no longer holds true for all templates prs(t) made of Nf

equally spaced replicas of a frame-template prf (t). Because ofthe different TH codes used in different frames, each receivertemplate prf (t) correlates with a different portion of the TH-dependent aggregate channel gs(t). As a result, there does not

exist a single channel-dependent amplitude Aε common to allframes. In fact, when the TH codes repeat from symbol tosymbol, the channel-dependent Aε takes on Nf possible values,reflecting the periodicity of the TH codes over frames. As such,the portions of y[n] contributed from two consecutive symbolscannot be simply linked to the acquisition offset parameter νf,ε,rendering Proposition 1 invalid. On the other hand, Proposition1 still applies when some properly selected template is usedto account for the TH effect. We will present such a templateprs(t) = prs(t) next and discuss the corresponding asymptoticGLRT timing acquisition solution under TH.

A. Sample Model for an Ideal Template

With perfect timing (τ0 = 0), the ML optimum correlator isa filter matched to the aggregate symbol-long channel gs(t),which must be known to the receiver. Under mistiming, theoptimum symbol-by-symbol correlation template is given by

prs(t) ={

gs(t − τ0), t ∈ [τ0, Ts)gs(t + Ts − τ0), t ∈ [0, τ0)

(24)

where τ0 := nfTf + ε is the timing offset within a symbolperiod and prs(t) is nothing but a circularly shifted (by τ0)version of gs(t) to confine it within [0, Ts). As depicted inFig. 2, prs(t) is in essence the aggregate Ts-long channel that isviewed with reference to the receiver’s clock; see the Ts-apartmistimed symbol boundaries in Fig. 2 (dashed vertical lines) att = kTs, ∀k. On the other hand, gs(t) can be properly locatedby the receiver only when τ0 is known; see the (unknown)actual received symbol boundaries in Fig. 2(b) (solid verticallines) at t = τ0 + kTs, ∀k ≥ ns.

Because prs(t) inherently contains the correct TH code ineach frame without knowing τ0, it hops in accordance with thereceived waveform to correlate with the same portion of r(t)in each frame, subject to the impact of IFI. As a result, theaggregate channel-dependent amplitude Aε is approximatelythe same for all frames, making (6) approximately valid. Wequantify the sample model of y[n] resulting from the use ofprs(t) = prs(t) in Appendix B and establish the followingproposition under TH.Proposition 2: For the correlator template prs(t) in (24),

the symbol-rate sample output y[n] in the presence of TH isgiven by

y[n] = Aε

(s[n − ns](Nf − nf − λε)

+ s[n − ns − 1](nf + λε))

+ w[n] (25)

where Aε = (√Es/Nf )

∫ Ts

0 g2s(t)dt ≈

√Es

∫ Tf

0 g2(t)dt andthe bounded λε entails the effects of both the pulse-levelmistiming ε and the IFI.

Similar to the no-TH case in (6), each sample y[n] is aweighted linear combination of two consecutive symbols, withthe combining weights determined by νf,ε := nf + λε, whoseinteger portion corresponds to the frame-level timing offsetparameter nf . Accordingly, the channel-dependent unknownsAε and νf,ε (thus nf ) can be obtained via amplitude estimation


in the ML sense provided that prs(t) is available. However,the ideal prs(t) contains the channel-related information thatis unknown during the synchronization phase. Interestingly, wewill see that the receiver can acquire a noisy template ˆprs(t)from the received waveform r(t) that approaches prs(t) givena sufficient number of training symbols M . Based on this noisytemplate and using y[n] samples as in (25), we will developa GLRT-based solution to timing acquisition in the presenceof TH, starting from the construction of the desired noisytemplate ˆprs(t).

B. CML Timing Acquisition in the Presence of TH

In order to find ˆprs(t) from r(t), we focus on segments ofr(t) that correspond to the subset of consecutive symbol pairs(s[n], s[n − 1]) with n ∈ G+(n;ns); see, e.g., the segment ofr(t) corresponding to y[n] in Fig. 2(b). For the other subsethaving n ∈ G−(n;ns), the corresponding segments of r(t) ex-perience a change of symbol signs every Ts, thus being unableto retain the symbol-independent prs(t); see also the segmentof r(t) corresponding to y[n + 1] in Fig. 2(b). In fact, it isconvenient to put all training symbols in G+(n;ns) (i.e., all1s) as a preamble to estimate ns, prs(t), and other channel-dependent nuisance parameters, followed by a second blockG−(n;ns) to be used as a post-amble to recover nf . IdentifyingG+(n;ns) relies on the symbol-level timing offset parameterns, which we must obtain before proceeding to find ˆprs(t).

To elaborate on the estimation of ns in the presence of TH,we revisit Proposition 1 to note that ns does not affect thesymbol amplitudes in (6), regardless of TH. Indeed, for anygeneral template prs(t), (6) retains the same form as far asns is concerned, except that νf,ε does not simply depend onnf when TH is present. The GLRT approach in Section III,fortunately, relies on νf,ε rather than nf to obtain the MLE ofns. This observation indicates that ns can be obtained using thederived GLRT rule even under TH for any correlation templatesatisfying Proposition 1. Specifically, ns can be estimated viaa line search to maximize the objective function in either (15),(19), or (22), all of which involve only cross-correlation opera-tions independent of Aε or nf . Based on our TS placement, wewill use the first subset of (less than) M+ training symbols toestimate ns as

ns = arg maxn∈[0,N−M+]

J+(ns,y)

= arg maxn∈[0,N−M+]

1E+

M

ns+M+−1∑

n=ns

y[n]s[n−ns]

2

. (26)

Here, the sample vector y is generated by correlating the TH-dependent r(t) with a general template prs(t), e.g., prs(t) =ps(t) for simplicity.

Having acquired ns, we can observe within G+(n;ns) asegment of MsTs-long received waveform r1(t), t ∈ [(ns +1)Ts, (ns + Ms)Ts], which contains s[n] = 1, ∀n. For nota-tional convenience, we introduce a window function WTs

(t),which equals 1 for t ∈ [0, Ts) and 0 otherwise. With the re-

TABLE IROBUST GLRT TIMING ACQUISITION USING NOISY TEMPLATE

maining unknown timing offset τ0 = nfTf + ε, we recall r(t)in (4), and write r1(t) as

r1(t) =√Es

ns+Ms−1∑k=ns+1

gs(t − kTs − τ0) + w(t)

∣∣∣∣∣(ns+Ms)Ts

t=(ns+1)Ts

=√

Es

ns+Ms−1∑k=ns+1

prs(t − kTs)

+ns+Ms−1∑k=ns+1

w(t)WTs(t − kTs). (27)

Apparently, the noise-free part of r1(t) consists of prs(t)replicas with spacing Ts. Hence, prs(t) can be estimatedfrom r1(t) by averaging over Ms − 1 consecutive Ts-longreceived waveform segments to yield a noisy template ˆprs(t),t ∈ [0, Ts) as

ˆprs(t) =1

Ms − 1

ns+Ms−1∑k=ns+1

r1(t + kTs)WTs(t). (28)

Being an unbiased sample mean of waveforms {r1(t +kTs)WTs

(t)}, ˆprs(t) asymptotically equals the correspondingensemble mean prs(t) as Ms → ∞. As a result, any optimumestimator built upon the ideal correlation template prs(t) willretain asymptotic optimality when the practical noisy templateˆprs(t) in (28) is used instead.

Using ˆprs(t), the CML parameter estimators in Section IIIcan be directly applied to (25) in Proposition 2 to obtain theMLEs of Aε and nf in the presence of TH. We summarize theoverall procedure of GLRT-based timing synchronization underTH in Table I. This GLRT-based algorithm enjoys the samebenefits as that in Section III in terms of symbol-rate samplingand low-complexity amplitude estimation without resorting toa line search over nf ∈ [0, Nf − 1]. In addition, it is robust toTH by invoking a TH-dependent noisy template. To appreciatethe TH-dependent feature of this noisy ˆprs(t) template, wenote that the TH code cannot be explicitly used during theacquisition phase because of the lack of code synchronization.A couple of remarks are in order.Remark 1: The noisy template ˆprs(t) is useful in a UWB

correlator receiver even in the absence of TH or dense multipatheffects. As shown in Appendix B, the ideal template prs(t)maximizes the channel-dependent scalar Aε, an indicator ofmultipath energy capture, which in turn minimizes the timing


acquisition CRB in (18), thus improving estimation accuracy.This establishes that ˆprs(t) is asymptotically optimum in theML sense for symbol-by-symbol receiver processing.Remark 2: Employing ˆprs(t), Proposition 1 applies to all

transmission formats utilizing Nf equally spaced repetitivepulses to form a symbol, as long as there is no IFI in the receivedwaveform. This class includes not only PAM- and PPM-basedUWB but also direct-sequence (DS) UWB, operating in bothdense multipath and low-scattering environments3. Similarly,Proposition 2 has merits in approximately modeling the corre-lator output of the same class of UWB transmissions, with theapproximation entailing a small degree of IFI. IFI could arisewhen TH is used or when Tf is set to be less than the channeldelay spread to increase the date rate.Remark 3: There is recent work on channel and timing

estimation at low sampling rates based on the notion of finiteinnovation rates [9], [10]. Such work aims at recovering thedelays and gains of (a few) dominant channel taps using asubspace-based harmonic retrieval approach operating in thefrequency domain. As such, the sampling rates of innovationare lower than yet still comparable to the Nyquist rate. In thesimulation curves in [10], considerable degradation of timingestimation performance can be witnessed when the samplingrate is 1/(8Tp). Our approach here aims at acquiring the timearrival of the first channel path without explicit channel esti-mation. The sampling rate of one sample per symbol (1/Ts �1/Tp) is much lower than the Nyquist rate.

V. SIMULATIONS

Computer simulations are performed to test the proposedCML timing acquisition algorithms. In all test cases, the prop-agation channels are generated randomly according to [19],where rays arrive in several clusters within an observation win-dow. The cluster arrival times are modeled as Poisson variableswith cluster arrival rate Λ. Returns within each cluster alsoarrive according to a Poisson process with arrival rate λ. Theamplitude of each arriving ray is a real-valued random variablewith a double-sided Rayleigh distribution having exponentiallydecaying mean square value with parameters Γ and γ. Parame-ters of this channel model are chosen as Γ = 30 ns, γ = 5 ns,1/Λ = 2 ns, and 1/λ = 0.5 ns. The diminishing tail of themultipath channel power profile is cut off to make the maximumdelay spread of the multipath channel to be 99 ns. We selectp(t) as the second derivative of the Gaussian function, whichhas a pulse width Tp = 1 ns and normalized energy. The frameduration is chosen to be Tf = 100 ns. Each symbol durationcontains Nf = 25 frames. In all cases, Nε and ε are uniformlydistributed over [0, Nf − 1] and [0, Tf ), respectively. The chipduration is Tc = 1.0 ns and the TH code is chosen randomlyover the range [0,90].

We present the acquisition performance of our GLRT-basedtiming offset estimators under different operating scenarios.

3In low-scattering environments, since the inherent multipath diversity pro-vided by the channel is relatively small, the advantage of using noisy templatesfor collecting diversity gains at low cost has to be weighed against the noiseenhancement effect incurred by lack of (computationally expensive yet noisy)channel estimates.

Fig. 3. Performance of GLRT timing acquisition with no TH and smalltracking errors. (a) MSE. (b) BER.

Two correlation templates are compared: one is the simplesuboptimal template prs(t) = ps(t), and the other is the asymp-totically optimal template prs(t) = ˆprs(t). Performance met-rics of interest include the normalized acquisition estimationmean square error (MSE) E{|(nf − nf )/Nf |2}, and the BERwhen an optimum detector with perfect channel knowledge isavailable to recover information-bearing symbols, subject toresidual timing errors.

Fig. 3 depicts the MSE and BER values versus SNR Es/σ2w

when TH is not employed and the pulse-level offset ε is small.Two different sizes of training symbol sequence, M = 6 andM = 30, are considered. Fig. 4 depicts the MSE and BERperformance under large ε but no TH, while Fig. 5 depictsthe case with TH and ε = 0. With the template ps(t), theGLRT estimator in Section III-C can tolerate ε �= 0 to somedegree, but is not robust to TH. Fig. 6 considers the acquisitionperformance with large ε errors in the presence of TH. Thetemplate ps(t) does not work well in this case, but the GLRTestimator in Section IV-B with the noisy template ˆprs(t) is ableto acquire timing at a reasonably low SNR. The noisy ˆprs(t)


Fig. 4. MSE performance of robust GLRT timing acquisition with largetracking errors but no TH.

Fig. 5. MSE performance of robust GLRT timing acquisition with TH but notracking errors.

template considerably outperforms the ps(t) template in all theoperating conditions tested because of its capability to benefitfrom the large diversity gain in dense multipath.

Since the unknown error ε has not been treated during theacquisition phase, there is a noise floor in the MSE curves inFigs. 4–6, which is caused by the bounded unknown λε ∈ [0, 1]in (6) and (25). Nevertheless, tracking errors only have a smallimpact on BER in dense multipath channels [14]. As Figs. 3(b)and 6(b) illustrate, there are small BER gaps between ourreceivers using GLRT-based timing acquisition and an idealreceiver knowing the perfect timing information. Operating ata practical sampling rate of 1/Ts, our GLRT timing acquisitionalgorithms not only enjoy very low computational complexitybut also offer good acquisition accuracy when proper correla-tion templates are used.

VI. CONCLUDING SUMMARY

Taking on a conditional maximum likelihood (CML) ap-proach, this paper formulates the data-aided (DA) timing ac-

Fig. 6. Performance of robust GLRT timing acquisition with TH and largetracking errors. (a) MSE. (b) BER.

quisition task for ultrawideband (UWB) communications as ageneralized likelihood ratio test (GLRT) problem, where theeffective discrete-time channel gains are viewed as nuisanceparameters and estimated as by-products. Relying on symbol-rate samples, our synchronization solution boils down to anamplitude estimation problem, which permits a closed-formsolution based on cross correlation among symbol-rate samples.This is in contrast to other digital synchronizers that requireeither chip-rate sampling or at least a line search. For com-petitive acquisition performance in dense multipath, the GLRTalgorithms should be coupled with properly selected correlationtemplates that can effectively collect multipath energy in thepresence of timing offsets. To this end, a noisy template isdesigned that not only offers asymptotically optimal estima-tion performance but also makes the GLRT robust to timehopping (TH). Performance bounds of DA acquisition havebeen established through Cramer–Rao bound (CRB), whichwe will explore in a companion paper to provide judicioustraining sequence design for improved acquisition performanceand enhanced overall system capacity.


APPENDIX ADISCRETE-TIME MODEL OF SYMBOL-RATE

CORRELATOR OUTPUT SAMPLES

Consider a correlator receiver with a general correlationtemplate prs(t) :=

∑Nf−1j=0 prf (t − jTf ) made of Nf identical

prf (t) with spacing Tf . Without knowing the mistiming, thereceiver correlates the received waveform r(t) [cf. (4)]with prs(t) for every Ts to generate samples {y[n]},starting from t = 0. Thus, y[n] =

∫ Ts

0 r(t + nTs)prs(t)dt =√Es

∑∞k=−∞s[k − ns]

∫ Ts

0 gs(t − kTs−nfTf −ε) prs(t)dt +w[n], where w[n] :=

∫ Ts

0 w(t + nTs)prs(t)dt is the noisesample. Since gs(t) has a finite nonzero support on t ∈ [0, Ts),the possible nonzero summands in y[n] are associated onlywith either k = n or k = n − 1, yielding

y[n]= w[n]

+√Ess[n−ns−1]

nf Tf+ε∫0

gs(t+Ts−nfTf −ε)prs(t)dt

+√Ess[n−ns]

Ts∫nf Tf+ε

gs(t−nfTf −ε)prs(t)dt. (29)

A key step in simplifying (29) is to express a finite-durationsignal, namely, gs(t) in this case, in terms of its overlap withan infinite-duration version. To do so, let us introduce an ex-tended aggregate channel gs(t) :=

∑∞j=−∞ g(t − jTf ), which

is periodic with period Tf . Noting that gs(t) overlaps with gs(t)within [0, Ts], we can replace gs(t) by gs(t) in (29). Becausegs(t) is periodic for t ∈ (−∞,∞), any arguments inside gs(·)that are integer multiples of Tf can be simply omitted; thus,the integrands in (29) are independent of nf but still dependon ε. As a result, the weights of the two contributing sym-bols become λ0 :=

√Es

∫ Ts

nf Tf+ε gs(t − ε)prs(t)dt and λ1 :=√Es

∫ nf Tf+ε

0 gs(t − ε)prs(t)dt. We can thus simplify (29) toy[n] = s[n − ns]λ0 + s[n − ns − 1]λ1 + w[n]. Next, we willshow how λ0 and λ1 are related with the frame-level timingoffset parameter nf .

Let us start with λ1. The following equalities hold afterchange of variables:

λ1 =√

Es

nf Tf+ε∫0

gs(t−ε)Nf−1∑j=0

prf (t−jTf )dt

=Nf−1∑j=0

√Es

nf Tf+ε−jTf∫−jTf

gs(t+jTf −ε)prf (t)dt. (30)

Because prf (t) is zero outside t ∈ [0, Tf ], each summand in(30) may contribute to λ1 only when its integration range

overlaps with [0, Tf ], reducing the possible values for the indexj to j ∈ [0, nf ]. Specifically

λ1 =√

Es

nf−1∑j=0

Tf∫0

gs(t + jTf − ε)prf (t)dt

+√

Es

ε∫0

gs(t + nfTf − ε)prf (t)dt. (31)

Because gs(t) :=∑Nf−1

i=0 g(t − iTf ) is made of Nf replicas ofg(t), the first nf integrals in λ1 are identical and we denote

them as Aε :=√Es

∫ Tf

0 g(t + Tf − ε)prf (t)dt. The last termin λ1 is equivalent to

√Es

∫ ε

0 g(t + Tf − ε)prf (t)dt becauseonly the j = (nf − 1)th pulse g(t − jTf ) in gs(t) falls withinthe integration range of [0, ε]. It bears the same integrand asthat in Aε, but the integration range is narrower. As such, the ε-dependent scalar λε := (1/Aε)

√Es

∫ ε

0 g(t + Tf − ε)prf (t)dtis bounded within [0,1]. If there is no tracking error (ε = 0),then λε = 0. With these definitions, (31) can be expressed asλ1 = Aε(nf + λε).

Similar derivations apply to λ0, yielding λ0 = Aε(Nfnf −nf − λε). Together, we reach

y[n] = Aε (s[n − ns](Nf − nf − λε)

+ s[n − ns − 1](nf + λε)) + w[n]. (32)

In (32), the unknown channel impulse response g(t) is con-tained in the scalars Aε and λε in a discrete-time form.

APPENDIX BSAMPLE MODEL USING AN IDEAL TEMPLATE

UNDER MISTIMING

To find the output sample model of a correlator using theideal template prs(t) in (24), we replace prs(t) by prs(t) in (29)to reach the noise-free component y(n) of each sample, i.e.,

y(n) =√Ess[n − ns]

Ts∫nf Tf+ε

g2s(t − nfTf − ε)dt

+√

Ess[n − ns − 1]

nf Tf+ε∫0

g2s(t − nfTf − ε + Ts)dt

=√Ess[n − ns]

Ts−(nf Tf+ε)∫0

g2s(t)dt

+√Ess[n − ns − 1]

Ts∫Ts−(nf Tf+ε)

g2s(t)dt. (33)


ξτ0 =√

Es

Nf Tf∫(Nf−nf )Tf−ε

Nf−1∑i=0

g(t − iTf − ciTc)Nf−1∑j=0

g(t − jTf − cjTc)dt

=√Es

Nf−1∑i=Nf−nf−2

(Nf−i)g(t+Tf−(ci−1−ci)Tc)Tf−ciTc∫(Nf−nf−i)Tf−ciTc−ε

i+1∑j=i−1

g(t)g (t − (j − i)Tf − (cj − ci)Tc) dt

=√Es

Nf−1∑i=Nf−nf

Tf∫0

g2(t)dt

︸︷︷︸Egf nf

+√

Es

Nf−nf−1∑i=Nf−nf−2

Tf∫(Nf−nf−i)Tf−ciTc−ε

g2(t)dt −Tf∫

Tf−cNf −1Tc

g2(t)dt

︸︷︷︸Egf λε

+√

Es

Nf−2∑i=Nf−nf

Tf∫0

g(t) [g (t − Tf − (ci+1 − ci)Tc) + g (t + Tf − (ci−1 − ci)Tc)] dt

+√Es

Nf−nf−1∑i=Nf−nf−2

Tf∫(Nf−nf−i)Tf−ciTc−ε

g(t) [g (t − Tf − (ci+1 − ci)Tc) + g (t + Tf − (ci−1 − ci)Tc)] dt

+√Es

Tf−cNf −1Tc∫0

g(t)g(t + Tf −

(cNf−2 − cNf−1

)Tc

)dt

=√EsEgf (nf + λε + ζnf,ε) (34)

We first derive the effective channel-dependent amplitudeAε attained by such a correlation template, [i.e., (34) shownat the top of the page]. Analogous to the signal structure in(32), we observe that Aε is equivalent to the sample am-plitude when there is no modulation or change of symbolsigns, while the portion of sample amplitude contributed froms[n − ns − 1] reflects the frame-level acquisition error. Basedon this observation, we deduce that the effective gain Aε

in (33) is given by Aε = (1/Nf ) y(n)|s[n−ns]=s[n−ns−1]=1=(1/Nf )

√Es

∫ Ts

0 g2s(t) dt, where Egs :=

∫ Ts

0 g2s(t)dt repre-

sents the maximum energy collected within a symbol whenfree of noise, simply because the received (unknown) waveformtemplate gs(t) itself is used as the correlation template toperform optimum matched filtering. The result on Aε can beintuitively understood from Fig. 2. Since the template matchesexactly the received waveform and the integration range is thesame as the waveform period Ts, such a template collects theenergy of an entire symbol.

Next, we focus on the effective channel-dependent weightinggain of s[n − ns − 1] while the gain of s[n − ns] is simplygiven by Aε minus that of s[n − ns − 1]. The quantity ofinterest is ξτ0 :=

√Es

∫ Ts

Ts−(nf Tf+ε) g2s(t)dt. Defining the pulse

energy after multipath spreading as Egf :=∫ Tf

0 g2(t)dt, themanipulations in (34) stand.

Using (34) and scaling it by 1/√

Egf to keep the noisevariance at σ2

w, the symbol-rate signal model in (33) becomes

y[n] =√

EsEgfs[n − ns](Nf − nf − λε − ζnf ,ε)

+√EsEgfs[n − ns − 1](nf + λε + ζnf ,ε) + w[n]. (35)

The following observations can be made.

• The no-TH case ({ci = 0}): It holds that Egs = NfEg

and ζnf ,ε = 0 because the latter involves correlating g(t)with g(t ± Tf ) whereas the nonzero support of g(t) is Tf .Furthermore, λε is reduced to λε = (1/Egf )

∫ ε

0 g2(t)dt ≤1, corroborating with (6).

• The TH case: Due to the IFI induced by pulse overlapping,ζnf ,ε is no longer zero; thus, the λε term in (6) shouldbe replaced by λε := λε + ζnf ,ε to reach (25). As longas IFI is small, the value of ζnf ,ε in (35) is relativelysmall, especially when Tp � Tf and Nf is large. Besides,the channel taps in the overlapping region do not matchand could be faded independently; therefore, the impact oftheir correlation in the fading case is trivial.

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Zhi Tian (M’98–S’98–M’00) received the B.E. de-gree in electrical engineering (automation) from theUniversity of Science and Technology of China,Hefei, in 1994, the M.S. and Ph.D. degrees fromGeorge Mason University (GMU), Fairfax, VA, in1998 and 2000, respectively.

From 1995 to 2000, she was a Graduate ResearchAssistant at the Center of Excellence in Command,Control, Communications and Intelligence (C3I),GMU. Since August 2000, she has been an AssistantProfessor at the Department of Electrical and Com-

puter Engineering, Michigan Technological University, Houghton. Her currentresearch focuses on signal processing for wireless communications, particularlyon ultrawideband systems.

Dr. Tian serves as an Associate Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS. She was the recipient of the 2003 NationalScience Foundation (NSF) CAREER award.

Georgios B. Giannakis (S’84–M’86–SM’91–F’97)received the Diploma in electrical engineering fromthe National Technical University of Athens, Greece,in 1981, and the M.Sc. degree in electrical engineer-ing, the M.Sc. degree in mathematics, and the Ph.D.degree in electrical engineering from the Universityof Southern California (USC), Los Angeles, in 1983,1986, and 1986, respectively.

After lecturing for one year at USC, he joinedthe University of Virginia, Charlottesville, in 1987,where he became a Professor of electrical engineer-

ing in 1997. Since 1999, he has been a Professor at the Department of Electricaland Computer Engineering, University of Minnesota, Minneapolis, wherehe now holds an ADC Chair in Wireless Telecommunications. His generalinterests span the areas of communications and signal processing, estimationand detection theory, time-series analysis, and system identification—subjectson which he has published more than 220 journal papers, 380 conferencepapers, and two edited books. Current research focuses on transmitter and re-ceiver diversity techniques for single- and multiuser fading communicationchannels, complex-field and space-time coding, multi-carrier ultrawide bandwireless communication systems, cross-layer designs, and sensor networks.

Dr. Giannakis is the (co)recipient of six paper awards from the IEEE SignalProcessing (SP) and Communications Societies (1992, 1998, 2000, 2001, 2003,and 2004). He also received the IEEE-Signal Processing Society’s TechnicalAchievement Award in 2000 and European Association for Signal, Speechand Image Processing (EURASIP) Technical Achievement Award in 2005.He served as Editor-in-Chief for the IEEE SIGNAL PROCESSING LETTERS,as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING

and the IEEE SIGNAL PROCESSING LETTERS, as Secretary of the SignalProcessing Conference Board, as member of the Signal Processing PublicationsBoard, as member and Vice-Chair of the Statistical Signal and Array ProcessingTechnical Committee, as Chair of the Signal Processing for CommunicationsTechnical Committee, and as a member of the IEEE Fellows Election Commit-tee. He has also served as a member of the IEEE-Signal Processing Society’sBoard of Governors, the Editorial Board for the PROCEEDINGS OF THE IEEE,and the Steering Committee of the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS.

a glrt approach to data-aided timing acquisition in uwb ... · 2956 ieee transactions on wireless...

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