EURASIP Journal on Applied Signal Processing 2005:3, 426–438c© 2005 Hindawi Publishing Corporation

Cramer-Rao Lower Bounds for the Synchronizationof UWB Signals

Jian ZhangNational ICT Australia (NICTA) and the Department of Telecommunications Engineering, Research School of Information Sciencesand Engineering, The Australian National University, Canberra ACT 0200, AustraliaEmail: andrew.zhang@nicta.com.au

Rodney A. KennedyNational ICT Australia (NICTA) and the Department of Telecommunications Engineering, Research School of Information Sciencesand Engineering, The Australian National University, Canberra ACT 0200, AustraliaEmail: rodney.kennedy@anu.edu.au

Thushara D. AbhayapalaNational ICT Australia (NICTA) and the Department of Telecommunications Engineering, Research School of Information Sciencesand Engineering, The Australian National University, Canberra ACT 0200, AustraliaEmail: thushara.abhayapala@anu.edu.au

Received 30 September 2003; Revised 10 February 2004

We present Cramer-Rao lower bounds (CRLBs) for the synchronization of UWB signals which should be tight lower bounds forthe theoretical performance limits of UWB synchronizers. The CRLBs are investigated for both single-pulse systems and time-hopping systems in AWGN and multipath channels. Insights are given into the relationship between CRLBs for different Gaussianmonocycles. An approximation method of the CRLBs is discussed when nuisance parameters exist. CRLBs in multipath channelsare studied and formulated for three scenarios depending on the way multipath interference is treated. We find that a larger numberof multipaths implies higher CRLBs and inferior performance of the synchronizers, and multipath interference on CRLBs cannotbe eliminated completely except in very special cases. As every estimate of time delay could not be perfect, the least influence ofthe synchronization error on the performance of receivers is quantified.

Keywords and phrases: ultra-wideband, synchronization, Cramer-Rao lower bounds.

1. INTRODUCTION

Ultra-wideband (UWB) is a promising technique in the ap-plication of short-range high-speed wireless communicationand precise location tracking. Typically, ultranarrow pulses,such as Gaussian monocycles [1], are modulated to trans-mit information. These pulses could be narrower than 1nanosecond. This brings very stringent synchronization re-quirements.

A UWB signal is basically a baseband signal withoutphase and carrier information, hence time delay estimationis the main task of a synchronizer. This synchronizer couldbe one in a simple single-pulse UWB system; however, dueto the power limitation imposed by FCC [2], UWB pulsesare generally combined with spread spectrum techniques, es-pecially time hopping (TH), to achieve multiuser access, toensure sufficient received energy, and to mitigate interfer-ence in existing wireless systems. Similar to traditional spreadspectrum systems, the synchronization of a TH-UWB sys-

tem is accomplished in two steps: code acquisition followedby code tracking. The former, involving the optimization ofsearch strategies, tries to determine the phase of the incom-ing pseudonoise (PN) sequence within a fraction of chipwidth. The latter refers to the process of achieving and main-taining fine alignment of the chip boundaries of the incom-ing and locally generated PN sequences.

As UWB pulses are very narrow, very stringent synchro-nization requirements are incurred, and timing errors usu-ally imply marked degradation of receiver performance [3].Abundant research on the design and performance of syn-chronization systems have been reported in the literature, forexample, [4, 5, 6]. These techniques can be transplanted intoUWB systems with some modifications to meet the strin-gent timing requirement, as discussed in [7, 8, 9, 10]. Dif-ferent to them, in this paper, we try to find some generalperformance limitations for UWB synchronizers, and pro-vide guidelines for the system design within acceptable per-formance region.

CRLBs for the Synchronization of UWB Signals 427

It is known that in the presence of noise, perfect synchro-nization cannot be achieved. For UWB systems with strin-gent timing requirements, it is of special interest to character-ize this synchronization error and its influence on the per-formance of detectors. This task becomes even more urgentwhen we realize that the radiated power of UWB signals isso low that the channel estimates could contain large errorsand the performance of synchronizers could be largely dete-riorated. Under these conditions, is it still possible for UWBsynchronizers to reach a satisfying accuracy of timing lock-ing? Some common performance parameters to evaluate syn-chronizers are tracking time, S-curve behavior, and probabil-ity of success. However, in order to provide benchmarks foractual UWB synchronizers, we are more interested in under-standing their theoretical performance limits. In the theoryof parameter estimation, Cramer-Rao lower bound (CRLB)is most widely used in evaluating the performance of esti-mates.

The CRLB [11] is a fundamental lower bound on thevariance of any unbiased estimator. The analysis of CRLB fortraditional systems is well founded [5, 12, 13, 14, 15, 16, 17,18, 19], but for UWB, there is no systematic work reportedyet to our knowledge. The evaluation of the CRLB is gen-erally mathematically quite difficult when the observed sig-nal contains, besides the parameter to be estimated, somenuisance parameters that are unknown [14, 19]. These nui-sance parameters could be the transmitted data and, some-times, multipath gains and delays which arise in fading chan-nels. When the nuisance parameters are present, the modi-fied CRLB (MCRB) [13, 14, 15], and the asymptotic CRLB(ACRLB) [14], are good approximations to the true CRLB athigher signal-to-noise ratio (SNR), and the lower-SNR limitof the CRLB is approximated in [18] by applying a TaylorSeries expansion.

This paper is concerned with evaluating the CRLB forUWB signals. Both single-pulse systems and TH systems areconsidered. For TH, the CRLB should be a lower bound forthe performance of code tracking. The structure of this pa-per is as follows. In Section 2, the problem is formulated.In Section 3, considering AWGN channels, the CRLBs forsingle-pulse systems are investigated in both cases of knownand unknown transmitted data. Some insights into the re-lationship between CRLBs for different Gaussian monocy-cles are given explicitly. We also highlight an oversight in thelower-SNR approximation method [18] and provide a possi-ble solution to remedy this problem by tightly locating therange of SNR γs. These results can be readily extended toa TH-UWB system in AWGN channels with minor modi-fications. In Section 4, we extend this work to more prac-tical multipath channels while considering an unmodulatedTH system. Depending on the way multipath interference istreated in a practical synchronizer, three scenarios are ana-lyzed when multipath interference contributes as an increaseof noise variance or multiple synchronization parameters. InSection 5, the influence of synchronization error on the per-formance of receiver is quantified, which may be the leastinfluence a UWB correlator receiver can expect. Finally, nu-merical results are given in Section 6 to verify some analyt-

ical results and illustrate the effect of pulse truncation onCRLBs.

2. PROBLEM FORMULATION

Binary pulse position modulation (BPPM) and binary phaseshift keying modulation (BPSK, or antipodal modulation)are considered here. Let s(t) be the transmitted UWB signal.In a single-pulse system, s(t) = ∑

i biω(t − iTs) for BPSK,and s(t) = ∑

i ω(t − iTs − biTd) for BPPM, where ω(t) isa UWB pulse, bi ∈ {−1, +1} is the ith transmitted data, Ts

is the symbol period, and Td is the time offset of BPPM. In

an unmodulated TH system, s(t) =∑i si(t) =∑

i

∑Nf

j=1 ω(t −iTs− jT f −cjTc) where si(t) is the ith transmitted symbol, Tf

is the frame width, Nf is the number of frames in a symbol,Tc is the chip width, and cj are the TH codes.

The UWB pulses considered are series of Gaussianmonocycles ω(t;n, tp), which are scaled and/or differen-tiated versions of the basic Gaussian waveform ω0(t) =exp(−2πt2), that is, ω(t;n, tp) = ω(n)

0 (t/tp), where the super-script (n) stands for n-order differentiation with respect to t,and tp parameterizes the width of the pulse.

To ensure equal energy of monocycles, a coefficientε(n, tp) is introduced, and let ω(t) = ε(n, tp)ω(t;n, tp). De-note the energy of ω(t) as Ep and symbol SNR as γs, thenε(n, tp), depending on n and tp, satisfies

ε2(n, tp) = Ep∫ +∞

−∞ ω2(t;n, tp

)dt. (1)

When passing through a pure AWGN channel n(t), thereceived signal r(t) becomes

r(t) = s(t − τ) + n(t), (2)

where every sample of n(t) is Gaussian distributed with zeromean and variance σ2

0 , and τ is the timing delay to be esti-mated.

When passing through a frequency-selective fading chan-nel, h(t) =∑L

�=1 a�δ(t − τ�), the received signal is given by

r(t) =L∑

�=1

a�s(t − τ�

)+ n(t), (3)

where a� and τ� are real multipath gains and delays, respec-tively. Note that the time delay τ between the transmitter andthe receiver is merged into τ� .

Due to the low-duty cycle of UWB signals, we assume thereceived signal is free of intersymbol interference (ISI) unlessindicated otherwise. For the effect of ISI and the design oftraining sequence accordingly, the readers can refer to [20,21].

For the AWGN model in (2), for the purpose of formingestimates based on K independent observations, the receivedsignal can be represented as a vector model

r = s(b, τ) + n, (4)

428 EURASIP Journal on Applied Signal Processing

where r = [r1, . . . , rK ], s = [s1, . . . , sK ], and n = [n1, . . . ,nK ]are the sample vectors of the received signal r(t), the trans-mitted signal s(t − τ), and the noise n(t), respectively, andb = {bi} are the transmitted data sequence.

Suppose an unbiased estimate τ of the time delay τ canbe generated from (4). Then the estimation error variance islower bounded by the CRLB Er[(τ − τ)2] ≥ CRLB(τ), where

CRLB(τ) =(

Er|τ[− d2

dτ2ln(p(

r∣∣τ))])−1

. (5)

In (5), the conditional pdf p(r|τ) is the likelihood functionof τ, and the expectation Er|τ[·] is with respect to p(r|τ).

The likelihood function p(r|τ) can be obtained by aver-aging the joint likelihood function p(r|b, τ) over the a prioridistribution of the data b : p(r|τ) = Eb[p(r|b, τ)]. When bis known, p(r|τ) = p(r|b, τ).

Since the additive noise n(t) is white and zero mean, thejoint conditional pdf p(r|b, τ) can be expressed as

p(r|b, τ) =K∏k=1

1√2πσ0

exp(− 1

2σ20

(rk − sk

)2)

=(

1√2πσ0

)Kexp

(− 1

2σ20

K∑k=1

(rk − sk

)2).

(6)

Applying the signal orthogonal expressions [6, page 335]or letting the number of samples K go to infinity [11, page274] (or from the standpoint of generating sufficient statis-tics), we have

K∑k=1

(rk − sk

)2 =∫To

[r(t)− s(t − τ)

]2dt, (7)

where To is the observation period.Now, a continuous-time equivalent of p(r|b, τ) can be

developed. Considering the subsequent operations of loga-rithm and differentiation, only terms related to b and τ willbe retained. Then the evaluation of p(r|b, τ) is equivalent toevaluating the likelihood function

Λ(b, τ)=exp

(1

2σ20

(2∫To

r(t)s(t − τ)dt −∫To

s2(t − τ)dt))

.

(8)The process from (4) to (8) can be applied to the multi-

path model (3) with minor modifications.

3. CRLB FOR SINGLE-PULSE SYSTEMSIN AWGN CHANNELS

3.1. CRLB with known transmitted dataThe CRLB with known b, further derived from (8) or directlyfrom [15], has the form

CRLB(τ; b) = σ20∫

Tos2(t − τ)dt

, (9)

where s(t−τ) denotes first partial differentiation with respectto τ.

Assuming that the pulse is strictly restricted within asymbol period, and To = NTs, where N is the number ofsymbols contained in the observation period (one pulse persymbol in this case), then for both BPSK and BPPM, thedenominator in (9) equals N

∫Tsω2(t − τ)dt. For a specific

monocycle, the lower variance bound becomes

CRLB(τ; b) = 1Nγs

∫Tsω2(t − τ;n, tp

)dt∫

Tsω2(t − τ;n, tp

)dt

, (10)

where the symbol SNR is γs = Ep/σ20 .

If the symbol period Ts is large enough so that most ofthe energy of the pulse concentrates within Ts, we can express(10) in the frequency domain:

CRLB(τ; b) = 1Nγs

∫ +∞−∞∣∣W(

f ;n, tp)∣∣2

df∫ +∞−∞ f 2

∣∣W(f ;n, tp

)∣∣2df

, (11)

where W( f ;n, tp) is the Fourier transform of ω(t;n, tp).According to the properties of the Fourier transform of

derivatives of functions, we find explicit relationships exist-ing between the CRLBs of monocycles with different n butthe same tp, that is,

CRLB(τ; b)nCRLB(τ; b)n+1

=∫ +∞−∞∣∣W(

f ;n, tp)∣∣2

df · ∫ +∞−∞ f 4

∣∣W(f ;n, tp

)∣∣2df( ∫ +∞

−∞ f 2∣∣W(

f ;n, tp)∣∣2

df)2

> 1,(12)

where the inequality follows from an application of Schwarz’sinequality. This inequality implies that monocycles withhigher-order differentiation have the potential for better per-formance in the sense of lower synchronization error vari-ance.

For monocycles with different tp but with the same n, theratio between their CRLBs can be found as

CRLB(τ; b)tp1

CRLB(τ; b)tp2

=(tp1

tp2

)2

, (13)

which implies that monocycles with smaller tp (narrower ef-fective pulse width) have the potential for better synchro-nization performance.

3.2. CRLB with unknown randomly transmitted data

For PPM, the uncertainty of time jitter introduced by modu-lation will cause large synchronization error when the trans-mitted data is random and unknown. When further methodsare adopted to solve this problem, the CRLB analysis in thesecases will usually be based on a system model similar to theone with known data. Hence we only consider BPSK-UWBsignals in this subsection.

CRLBs for the Synchronization of UWB Signals 429

For BPSK, the likelihood function in (8) becomes

Λ(b, τ) = exp

( N∑i=1

1σ2

0

(bi y(τ)− b2

i γs))

, (14)

where y(τ) = ∫Tsr(t)ω(t − τ)dt.

Dropping the constant term γs∑N

i=1(b2i ) = Nγs, we ob-

tain the log-likelihood function of p(r|τ) as

L(r; τ) = ln p(r|τ)

= ln Eb[Λ(b, τ)

]=

N∑i=1

ln Ebi

[exp

(1σ2

0bi y(τ)

)]

= N ln cosh

(1σ2

0y(τ)

).

(15)

By differentiating L(r; τ) twice with respect to τ, we get

∂2L(r; τ)∂τ2

= N

σ20

tanh

(y(τ)

σ20

)y(τ)

+N

σ40

(1− tanh2

(y(τ)

σ20

))y2(τ),

(16)

where y(τ) and y(τ) denote first and second derivatives ofy(τ) with respect to τ.

Due to the nonlinear function tanh(·) in (16), an analyt-ical solution for Er|τ[∂2L(r; τ)/∂τ2] is infeasible.

Since the pulse energy is restricted to be very low bythe FCC [2] (the maximum power of a transmitted pulsewith bandwidth 7 GHz is only 0.5 mW), one can refer to thelower-SNR limit of CRLB in [18], applying a Taylor exten-sion of the likelihood function p(r|b, τ), to obtain a simi-lar result for UWB. One thing we wish to emphasize here isthat, in [18], the statistical property of the likelihood func-tion L(u, τ) (original notation in [18]) is somewhat ignored.Due to L(u, τ) containing Gaussian variables with variancecomparable to the reciprocal of symbol SNR, more care isneeded when dropping the higher-order terms in Taylor ex-tension according to the lower symbol SNR assumption. Asimilar problem arises in an alternative method we introducebelow, where this ambiguity is revealed further, and resolvedby tightly locating the value of the symbol SNR.

The alternative method we suggest is also based on ap-proximation. The basic idea is to find best-fitting functionsfor ln(cosh(·)) in a piecewise fashion. To make analysistractable, these functions are polynomials with order smallerthan 3. But they should not be constructed by only consider-ing the goodness of fit due to the succeeding expectation op-eration. This is because y(τ) is a random variable and whenwe deal with the expectation operation, we have to makesure that all the possible samples of y(τ) are involved. Al-though integrating these polynomials in segments is feasi-ble, it cannot produce a closed form result and is still a nu-merical method. Instead, we try to construct each polyno-mial in which the variable space supports the sampling space.

It seems impossible as the pdf of y(τ) distributes in the en-tire one-dimension real space. We overcome this obstacle byassuring that most of the samples (say, 99%) are located inthe interval of interest.

With this criterion in mind, we find that a three-segmentapproximation is a good choice by studying the shape ofthe waveform ln(cosh(x)). A detailed discussion is shown inAppendix A. Examples of such three lower-order (≤ 2) poly-nomials are

ln(

cosh(x)) ≈

0.3x2 + 0.14x − 0.018, |x| < 1.5,

0.000034x2 + x − 0.69, 1.5 ≤ |x| ≤ 2.5,

x − 0.69, |x| > 2.5.

(17)

The root mean squared approximation errors are 0.0081,0.0091, 0.0031 for the three pieces, respectively. The rangesof corresponding SNR γs are [−∞,−6.25] dB, [10.3, 10.8] dB,and [10.8, +∞] dB, respectively, which can be determined ac-cording to the way addressed in Appendix A.

Due to nonexistence of a polynomial with goodnessof fit and a fully covered sampling space simultaneously,an appropriate interval corresponding to SNR range of[−6.25, 10.3] dB cannot be found.

Representing a general 2-order polynomial function asln(cosh(x)) ≈ f (x) = ax2 + b|x| + c|x=y(τ)/σ2

0, |x| ∈ [x1, x2],

where 0 ≤ x1 < x2, we derive the CRLB based on it below.The reciprocal of the CRLB can be calculated as

−Er|τ

[∂2L(r; τ)

∂τ2

]= −N Er|τ

[2axx + 2ax2 + bx

], (18)

where we utilize

d2|x|dτ2

= d2

dτ2

(√x2) = d

dτx = x. (19)

As shown in Appendix B, these expectations are given by

Er|τ[x2] = 1

σ20

∫Ts

ω2(t − τ)dt,

Er|τ[xx] = −γs + 1

σ20

∫Ts

ω2(t − τ)dt,

Er|τ[x2] = − 1

σ20

∫Ts

ω2(t − τ)dt.

(20)

Then for a specific monocycle ω(t;n, tp), the CRLB is

CRLB(τ) = 1N(2aγs + b

)γs

∫Tsω2(t − τ;n, tp

)dt∫

Tsω2(t − τ;n, tp

)dt. (21)

By substituting a and b with the coefficients of polynomialsin (17), the CRLBs for different γs are readily obtainable. Itis clear that the relationship between CRLBs for monocycleswith different n or tp is identical to that when the transmitteddata is known.

430 EURASIP Journal on Applied Signal Processing

By comparing (10) and (21), we find that CRLB(τ;b)/ CRLB(τ) = 2aγs + b. Referring to (17), it is obviousthat the CRLB with unknown data is always larger than thatwith known data for the lower SNR case, and converges toCRLB(τ; b) in the higher-SNR case, which coincides with theattributes of ACRB given in [14].

4. CRLB FOR TH-UWB SYSTEMSIN SELECTIVE FADING CHANNELS

When the channel is AWGN, the analysis and results inSection 3 can be applied to TH-UWB systems with minormodifications. The change can be merged into the symbolSNR γs, that is, γs equals the ratio between the energy of Nf

pulses and the noise variance σ20 for TH-UWB systems. In

this section, we will focus on selective fading channels.Synchronization in selective fading channels is a chal-

lenging task. The performance largely depends on theschemes and algorithms. Based on the way multipath signalsare treated, these systems can be divided into three categories.Accordingly, we consider the CRLB for each of them. SinceCRLB with unknown data is straightforward but computa-tionally complex as derived in Section 3, we only consider thecase of known data b here.

4.1. Passive methods: regarding multipathsignals as interference

This refers to methods that apply general synchronizers, suchas early/late gates, while treating multipath signals as inter-ference [22, 23], or partly utilizing multipath energy [24], orusing a whitening filter before a synchronizer [25]. The effectof multipath interference on synchronizers has been studiedin [22, 23, 26, 27, 28, 29]. From the viewpoint of CRLB, allthese methods can be generalized to a model in which only aspecific multipath is of interest. Mathematically, we can rep-resent this model as

r(t) = ams(t − τm

)+

L∑�=1, � �=m

a�s(t − τ�

)︸ ︷︷ ︸

interference

+n(t), (22)

where am and τm are the parameters to be estimated.Generally, the received signal r(t) first passes through a

PN code correlator si(t − τm), where τm is the preestimateddelay, so that the energy of all pulses in a symbol are collectedto make an estimation. Then the model in (22) can be furtherwritten as

r f(τm) = am

N∑i=1

φ(τm + iTs − τm

)

+N∑i=1

L∑�=1, � �=m

a�φ(τm + iTs − τ�

)︸ ︷︷ ︸

na

+n f(τm),

(23)

where

r f(τm) = N∑

i=1

∫iTs

r(t)si(t − τm

)dt,

φ(v) =∫iTs

s(t − v)si(t)dv,

n f(τm) = N

∫iTs

n(t)si(t − τm

)dt.

(24)

Successful detection requires sampling r f (t) at τm = iTs + τmaccurately.

Each sample of n f (τm) is Gaussian distributed with zeromean and variance σ2

n f= NNf Epσ

20 . The component na,

containing interchip interference and ISI, is hard to modeland evaluate without prior knowledge of TH codes and mul-tipath delay. To make the analysis mathematical tractable,here we assume that na is Gaussian distributed1 with meanmna and variance σ2

na . In Appendix C, more information isgiven on the parameters of this distribution.

Recall that when considering the CRLB for TH-UWBsynchronizers in the phase of code tracking, it is reason-able to assume that φ(t − τm)|t=τm is restricted in an inter-val [−Tφ,Tφ], where Tφ is smaller than half of the frame pe-riod (Tφ < Tf /2). Then the sum of φ(t − τm) for N symbols,∑N

i=1 φ(t + iTs− τm), equals Nφ(t− τm). Now, the estimationproblem can be reformed as

r f (t) = amNφ(t − τm

)+ na + n f (t), (25)

which is a problem of multiple parameters estimation in aGaussian interference.

Although am and τm are correlated via the mean powerprofile of fading, they are usually treated as unknown and de-terministic parameters, and nonrandom parameter estima-tion techniques are applied, as the statistical relationship be-tween them are hardly predictable. This means they are nota function of each other any more. Strictly speaking, τm isthe only synchronization parameter, and CRLB(τm) can beobtained when regarding am as a nuisance parameter. How-ever, it is known that joint estimation of τm and am usuallygives lower CRLB for τm than that in a separate-estimationcase [11, 14, 19]. Hence we will focus on joint estimation andgenerate CRLB(am) as a byproduct.

Representing (25) as a vector form rf = amNΦ + na + nf

and applying the similar process from (6) to (8), the jointlog-likelihood function L(rf ; am, τm) = ln p(rf ; am, τm) canbe obtained as

L(

rf ; am, τm)

= − N

2(σ2na +σ2

n f

) ∫2Tφ

(Na2

mφ2(t − τm

)− 2amr f (t)φ(t−τm

)+ 2mnaamφ

(t − τm

))dt.

(26)

1In [11, page 309], a general equation is provided for the CRLB of anyunbiased estimate in colored noise. But a closed-form solution is not readilyavailable.

CRLBs for the Synchronization of UWB Signals 431

Lower bounds on the variances of estimates for the com-ponents of am and τm are given in terms of the diagonal el-ements of the inverse of the Fisher information matrix J−1

[11]. In this example,

J =

−E

[∂2L

(rf ; am, τm

)∂a2

m

]−E

[∂2L

(rf ; am, τm

)∂am∂τm

]

−E

[∂2L

(rf ; am, τm

)∂am∂τm

]−E

[∂2L

(rf ; am, τm

)∂τ2

m

]

,

(27)

where the expectation E[·] is with respect to p(rf ; am, τm).Noting φ(t − τm) and am are mutually independent, the

elements of J can be calculated as

J11 = C∫

2Tφ

φ2(t − τm)dt,

J12 = J21 = Cam

∫2Tφ

φ(t − τm

)φ(t − τm

)dt,

J22 = Ca2m

∫2Tφ

φ2(t − τm)dt,

(28)

where C is a constant defined as C � N2/(σ2na + σ2

n f).

The crossterms J12 and J21 will vanish if we extend theobservation period Tφ till φ(Tφ) ≈ 0. Then the CRLBs for τmand am are

CRLB(τm) = 1

J22=(Ca2

m

∫2Tφ

φ2(t − τm)dt)−1

,

CRLB(am) = 1

J11=(C∫

2Tφ

φ2(t − τm)dt)−1

.

(29)

It is clear that the estimation of the time delay τm dependson the amplitude of the multipath given that C is the samefor all multipath signals, while the estimation of am could beindependent of τm supposing we extend the observation pe-riod appropriately. For the performance of synchronizer, themultipath interference contributes as an increased estimatevariance.2

Depending on the Gaussian approximation for the mul-tipath interference na, C may be related to a specific mono-cycle, hence the relationship between CRLB for differentmonocycles cannot be claimed directly.

Finally, we wish to say a little more on the relationshipbetween our model in this section and practical systems. Inthe literature on synchronizers for spread spectrum systemssuch as CDMA, we can always find the terms fading band-width, tracking loop bandwidth, and predetection bandwidthand discussions on how the relationship between them af-fects the performance of synchronizers in a multipath chan-nel (e.g., [26, 27, 28]). Briefly, the relationship between thesebandwidths determines the degree of multipath interference

2The multipath interference also very much likely causes a biased esti-mation according to [27].

entering the final decision part of the synchronizer. Consid-ering our model, the effect can be regarded as a reduction ofnoise variance σ2

na .

4.2. Positive joint detection of multipath signals

We refer to the method of jointly detecting fading ampli-tude and delay of all the multipath signals as a positiveone. For CDMA, this method has been well studied in lit-erature such as [16, 17, 25]. And the derivation of CRLBfor CDMA systems can been found in [16, 17, 30]. Here,following the process in Section 3, we study the CRLB us-ing joint detection for a UWB system where the parametersa = [a1, . . . , a� , . . . , aL]1×L and τ = [τ1, . . . , τ� , . . . , τL]1×L tobe estimated are treated as unknown but deterministic.

Beginning with (3), similar to the derivation from (4) to(8), we can obtain the log-likelihood function L(r; τ, a) as

L(r; τ, a) = 1σ2

0

∫To

r(t)∑�

a�s(t − τ�

)dt

− 12σ2

0

∫To

[∑�

a�s(t − τ�

)]2

dt.

(30)

After some manipulation, the Fisher information matrixJ has the structure

J =(Jττ Jτa

Jaτ Jaa

), (31)

where Jττ , Jτa, Jaτ , and Jaa are all L× L matrices with [�,m]thelements:

Jττ[�,m] = 1σ2

0

∫To

a�ams(t − τ�

)s(t − τm

)dt,

Jaa[�,m] = 1σ2

0

∫To

s(t − τ�

)s(t − τm

)dt,

Jτa[�,m] = Jaτ[m, �] = − 1σ2

0

∫To

a� s(t − τ�

)s(t − τm

)dt,

(32)

respectively.The CRLB for τm is just the mth diagonal element of the

inverse of J. Use m = 1 as an example and rewrite the matrixJ as

J =(J11 BC D

); (33)

we have

CRLB(τ1) = J−1

11 + J−111 B

(D− CJ−1

11 B)−1

CJ−111 (34)

= J−111 + J−2

11 BJ−111 C (35)

≥ J−111 , (36)

where J11 is called the Schur complement of J11 [31, page 175].Since J is nonnegative definite, the Schur complement matrixJ11 is also nonnegative definite, and so is J−1

11 . At the same

432 EURASIP Journal on Applied Signal Processing

time, B is the transpose of C since J is a symmetric matrixin this case. Thus we get BJ11C ≥ 0 and the inequality in(36) follows immediately. Whenever J11 > 0, we can get theinequality in (36) more readily according to

CRLB(τ1) = (J11 − BD−1C

)−1> J−1

11 . (37)

As J−111 can be regarded as the CRLB in an AWGN channel

with a known scalar of amplitude, this inequality implies thatthe CRLB in joint detection is always larger than that in thesingle-parameter estimation in an AWGN channel. Then aninteresting question arises, whether more multipath meanshigher CRLB and inferior performance of synchronizers ac-cordingly.

We consider a channel with L− 1 multipath signals. TheFisher information matrix J′ can be written as

J′ =(J11 BC D′

), (38)

with

D′ =(

D1 00† 0

), (39)

where 0 is an (L−2)×1 zero vector and † stands for transposeoperation. Then the CRLB with L− 1 multipath is

CRLB(τ1)L−1 =

(J11 − BD′−1C

)−1. (40)

Comparing BD−1C and BD′−1C gives

BD−1C− BD′−1C = B

(D−1 −

(D−1

1 00† 0

))C (41)

≥ 0, (42)

where the inequality in (42) yields from the fact that D−1 −D′−1 is a nonnegative definite matrix as can be proven ac-cording to the property of partitioned nonnegative definitematrices (see, e.g., [31, page 178] and let D−1 = A in equa-tion (6.10)).

Since J11 > 0, we have

CRLB(τ1)L > CRLB

(τ1)L−1, (43)

which shows that more multipath does lead to higher CRLBand inferior performance of synchronizers. Since the numberof multipaths is closely relevant to the bandwidth of mono-cycles, we conclude that narrower monocycles will very likelycause larger CRLBs. We did not say “absolutely” because allother variables besides D during this derivation are assumedunchanged, but it could be unrealistic when different mono-cycles are applied.

Another key factor with influence on CRLB is the choiceof TH codes. When the autocorrelation of TH codes is ideal,both the CRLBs in this subsection and Section 4.2 will be thesame and similar to the one in an AWGN channel.

4.3. Active methods: cancellation of interference?

From Sections 4.1 and 4.2, we have seen that the performanceof synchronizers is deteriorated by the multipath interfer-ence. It is natural to ask whether the multipath interferencecan be mitigated or fully eliminated before entering the deci-sion part of a synchronizer.

As shown for CDMA systems in [27], it is possible to re-move part of multipath interference in UWB systems. How-ever, unless the correlation of TH codes is ideal, the total re-moval of multipath interference is impossible due to the ex-istence of n(t). This is because, from Section 4.2, we see thatany estimate of parameters, including amplitude and delay,even though unbiased, may still have a nonzero variance inthe present of noise. The CRLB can generally be achievedby maximum likelihood estimation asymptotically (when thenumber of observation samples goes to infinity), and the es-timation error becomes Gaussian distributed with zero meanand variance equivalent to the CRLB [5, 11]. Therefore, thefinal signal with a pair of synchronization parameters of in-terest contains the sum of 2(L− 1) Gaussian variables, whichhas a variance larger than the variance of n(t). Since CRLB isproportional to the variance of (interference and) noise, theCRLBs for this pair of parameters will be larger than thosein a single-path channel. So no matter how perfect the struc-ture and algorithm to remove multipath signal are, the effectof multipath interference can only be mitigated but cannotbe canceled thoroughly. This result also partly explains whymore multipath generally leads to higher CRLBs.

However, there are some special cases when multipath in-terference becomes negligible. For example, when the max-imal multipath delay is smaller than the frame period in asingle-pulse system, multipath signals do not interfere witheach other due to the low duty cycle of UWB signal struc-ture.

5. INFLUENCE OF SYNCHRONIZATION ERROR ON BER

As every estimate of time delay could not be perfect, we usean example to show the influence of synchronization erroron the performance of receivers in UWB systems.

We consider a BPSK modulated single-pulse signal in anAWGN channel like that in Section 3. A correlator receiver[32, 33] is used to detect the signal.

The conditional bit error ratio (BER), depending on thesynchronization error eτ , is given by

Pe(eτ) = Q

ρ(eτ)√

Epσ0

, (44)

where we have assumed that the observation periodequals a symbol period such that N = 1, Q(x) �∫ +∞x exp(−t2/2)/

√2π dt, and ρ(eτ) = ∫Ts

ω(t)ω(t − eτ)dt.Recall that the best theoretically achievable eτ is Gaussian

distributed with zero mean and variance equivalent to theCRLB (denoted by σ2

c ). In the best case, σ2c = σ2

0 /(N∫Tsω2(t−

τ)dt) from (9) is the smallest. Averaging Pe(eτ) over eτ ,

CRLBs for the Synchronization of UWB Signals 433

−2−4−6−8−10

SNR γs (dB)

0.2

0.4

0.6

0.8

1

1.2

N×C

RLB

n = 2n = 3n = 4

151050

SNR γs (dB)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

N×C

RLB

n = 2n = 3n = 4

Figure 1: CRLB versus symbol SNR γs for n-order monocycles with tp = 2 nanoseconds; n = 2, 3, 4.

we get the mean BER

Pe = E[Pe(eτ)]

=∫ +∞

−∞1√

2πσcQ

√√√√ρ2(eτ)

Epσ20

exp

(−e2

τ

2σ2c

)deτ .

(45)

Statistically, this is the best achievable performance undercertain SNR. This equation can be evaluated numerically byMonte Carlo simulation which requires high computationalcomplexity. Alternatively, we invoke the Hermite-Gaussianquadrature [34], and Pe can be accurately approximated by

Pe 1√π

Nh∑n=1

HxnQ

ρ(√

2σcxn)√

Epσ0

, (46)

where Nh is the order of the Hermite polynomial HNh(·), andxn and Hxn are the zeros (abscissas) and weight factors of Nh-order Hermite polynomial, respectively. These values are tab-ulated in many mathematical handbooks (e.g., [35]). In ex-periments, we find that 16 coefficients (Nh = 16) are enoughto generate accurate approximation results.

Further define a variable η as the degrading ratio betweenPe and Pe(0) = Q(√γs), which is the BER in the case of

perfect synchronization. We show the values of η for differ-ent monocycles in Section 6 to compare the synchronizationerror robustness of monocycles.

6. NUMERICAL RESULTS

Since for multipath channels, the CRLBs depend on the THcodes and detailed fading channel models, we only show nu-merical results on the CRLBs in pure AWGN channels in thispaper.

In Figures 1, 2, and 3, the CRLBs for different monocyclesin the case of known data b are demonstrated. Since in prac-tice, a transmitted monocycle is usually the truncated por-tion of a whole pulse w(t;n, tp), this effect of truncation isconsidered by varying the actual width of pulse in (10).

From Figure 1, we can see that CRLBs are inversely pro-portional to the symbol SNR and the observation periodNTs. The relationship between CRLBs for monocycles withdifferent n order coincides with the analytical results in (12).This can be further observed in Figure 2, which also depictsthe effect of truncated pulses on CRLB. The CRLBs changelittle even when the truncated portion narrows to 1.6 tp(symmetric with respect to t = 0). However, with the widthof truncated pulse decreasing further, the CRLBs become or-derless. Figure 3 shows the effect of tp on the CRLBs and is adirect verification of (13).

434 EURASIP Journal on Applied Signal Processing

987654321

Order of differentiation n

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

N×

SNR

s×

CR

LB

0.8tp1.6tp2.4tp

Figure 2: CRLB versus n-order for monocycles with tp = 1nanoseconds; different lines correspond to different widths of trun-cation.

Figure 4 demonstrates the influence of synchronizationerror on the performance of receivers. It is plotted from (46)using Hermite-Gaussian approximation. The influence is no-table when the observation window in the stage of synchro-nization has small width (NTs) and weakens with N increas-ing (CRLBs decreasing). The figure also indicates that syn-chronization errors of different monocycles have very closeinfluence on BER, although the data in experiments showsthat the influence of monocycles with larger n is a little worsewhen SNR γs is small and changes toward the opposite withSNR increasing.

7. CONCLUSIONS

We have derived the Cramer-Rao lower bounds (CRLBs) forthe synchronization of UWB signals for both single-pulsesystems and time-hopping systems in AWGN and multi-path channels. Insights are given on the relationship betweenCRLBs for different Gaussian monocycles. The CRLBs inAWGN channels are discussed in both cases of known andunknown transmitted data. An approximation method of theCRLB is introduced when nuisance parameters, unknowntransmitted data, exist. An oversight in the lower-SNR ap-proximation method [18] is highlighted, and a possible solu-tion is provided by tightly locating the range of SNR γs. TheCRLBs in multipath channels are studied for three scenar-ios depending on the way multipath interference is treated ina practical synchronizer, where multipath interference con-tributes as an increase of noise variance or multiple syn-chronization parameters. It is found that larger number ofmultipaths implies higher CRLBs and inferior performanceof synchronizers, and multipath interference on CRLBs can-not be eliminated completely except in very limited cases.

21.510.5

Parameter tp (ns)

0.00570.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

N×

SNR

s×

CR

LB

0.8tp1.6tp2.4tp

Figure 3: CRLB for a 3-order (n = 3) monocycle with different pa-rameters tp; different lines correspond to different widths of trun-cation.

The least influence of synchronization error on the perfor-mance of receivers is quantified. The influence is notablewhen observation window (NTs) in the stage of synchro-nization is small, and weakens with N increasing (CRLBsdecreasing). Synchronization errors of different monocycleshave very close influence on BER.

APPENDICES

A. APPROXIMATION OF ln(cosh(y(τ)/σ20 ))

Here we show how to approximate ln(cosh(y(τ)/σ20 )) as low-

order polynomials in a piecewise fashion and determine thecorresponding range of symbol SNR γs.

From y(τ) = ∫Tsr(t)ω(t − τ)dt, y(τ) has Gaussian dis-

tribution N (E1, γsσ40 ), where |E1| = Ep in the case of perfect

synchronization, otherwise |E1| < Ep. As the estimate is usu-ally clustered tightly around the true value in our case, andE1 changes smoothly for UWB monocycles, we assume that|E1| ≈ Ep (this can also be obtained from the assumptionof unbiased estimation of τ). Then y(τ)/σ2

0 is also Gaussiandistributed N (γs, γs) or N (−γs, γs). For Gaussian distribu-tion, we know that when the distance between a sample andthe mean is larger than about 2.6

√variance, the probability

of appearance can be assumed to be zero. Let the interval ofinterest be [x1 ≤ y(τ)/σ2

0 ≤ x2]; to ensure that most of thesamples are in this interval, γs should satisfy the followingequations:

−2.6√γs + γs ≥

∣∣x1∣∣,

2.6√γs + γs ≤

∣∣x2∣∣,∣∣x1

∣∣ + 5.2√γs ≤

∣∣x2∣∣.

(A.1)

CRLBs for the Synchronization of UWB Signals 435

20151050

SNR γs (dB)

1.055

1.06

1.065

1.07

1.075

1.08

1.085

1.09

1.095

Rat

ioη

n = 1n = 2n = 3n = 4

(a)

20151050

SNR γs (dB)

1.055

1.06

1.065

1.07

1.075

1.08

1.085

1.09

1.095

Rat

ioη

n = 1n = 2n = 3n = 4

(b)

Figure 4: The degrading ratio η versus SNR γs for monocycles. Two observation periods (NTs) in a synchronizer are compared with(a) N = 10 and (b) N = 50. The time t is normalized with respect to tp.

Briefly, two guidelines for determining the interval [x1, x2]are (1) to ensure that this variable space be larger than thesampling space for a specific polynomial and SNR γs, andcover the range of γs as widely as possible; (2) although twointervals can overlap, each interval should be fully covered bya single polynomial.

Considering the waveform of ln(cosh(x)), from x = 0 toa small x2, it has a very different shape with other segmentsand has to be approximated separately by a polynomial. Thisimplies that there is only one interval covering the segmentcontaining the point zero. For this interval, only x2 needs tobe determined since ln cosh(·) is an even function, and thedistributions N (γs, γs) and N (−γs, γs) are symmetric withrespect to 0. For the same reason, it is enough to consider thepositive value of x1 and x2 for other segments hereafter. Notethat γs should be at least 6.76 for x1 > 0, this implies thatx2 > 13.52.

A well-known fact is that ln(cosh(x)) can be accuratelyapproximated by x2/2 when |x| 1, and by |x| − 0.69when |x| � 1. But this simple scheme is not good enoughto be realistic. For example, for a value x2 as large as 0.5,

the approximation error is already 0.005, while the corre-sponding maximum SNR γs is only 0.0324 = −15 dB whichis of little interest in practice.

To summarize the description above, we find that a three-segment approximation is a good choice. Although the con-struction of these approximations is not unique, they can berepresented as a general 2-order polynomial function f (x),which leads to a general CRLB expression as shown in (21).

B. DERIVATION OF Er|τ[·]

First we derive the autocorrelation of r(t) which will be usedin subsequent calculation:

Er|τ[r(t1)r(t2)]

= Er|τ[(s(t1 − τ

)+ n(t1))(

s(t2 − τ

)+ n(t2))]

= Er|τ[s(t1 − τ

)s(t2 − τ

)]+ σ2

0δ(t1 − t2

),

(B.1)

where in the last equality, we utilize the assumption that sig-nal and noise are mutually independent and n(t) is AWGN

436 EURASIP Journal on Applied Signal Processing

with zero mean and covariance σ20δ(t1− t2). Note that the ex-

pectation with respect to p(r|τ) is equivalent to average overthe data b and noise n(t). Recalling that the convolution be-tween r(t) and ω(t) in y(τ) is only within one symbol periodTs, in the case of ISI-free, we have

Er|τ[s(t1 − τ

)s(t2 − τ

)] = ω(t1 − τ

)ω(t2 − τ

),

Er|τ[r(t1)r(t2)] = σ2

0δ(t1 − t2

)+ ω

(t1 − τ

)ω(t2 − τ

).

(B.2)

Then expectations on y(τ) can be calculated as

Er|τ[y(τ) y(τ)

]=∫Ts

∫Ts

Er|τ[r(t1)r(t2)]ω(t1 − τ

)ω(t2 − τ

)dt1dt2

=[∫

Ts

ω2(t)dt + σ20

]·∫Ts

ω(t − τ)ω(t − τ)dt

= (γs + 1)σ2

0

∫Ts

ω(t − τ)ω(t − τ)dt,

Er|τ[y(τ)

]=∫Ts

Er|τ[r(t)

]ω(t − τ)dt

=∫Ts

ω(t − τ)ω(t − τ)dt,

Er|τ[y2(τ)

]=∫Ts

∫Ts

Er|τ[r(t1)r(t2)]ω(t1 − τ

)ω(t2 − τ

)dt1dt2

= σ20

∫Ts

ω2(t − τ)dt +[∫

Ts

ω(t − τ)ω(t − τ)dt]2

.

(B.3)

Assume that the energy of a pulse outside Ts is negligible,these results can be further simplified due to∫

Ts

ω(t − τ)ω(t − τ)dt = 0,∫Ts

ω(t − τ)ω(t − τ)dt

= −∫Ts

ω(t − τ)d(ω(t − τ)

)= −ω(t − τ)ω(t − τ)|Ts︸ ︷︷ ︸

=0

−∫Ts

ω2(t − τ)dt

= −∫Ts

ω2(t − τ)dt.

(B.4)

According to the linear relationship between x and y(τ), theexpectations in terms of x are straightforward.

C. GAUSSIAN APPROXIMATION OF MULTIPATHINTERFERENCE

The key assumption we make is, for each multipath withindex �� �=m, φ(t)t �=0 is identically independently distributed

with mean mφ and variance σ2φ . As the number of multipaths

L in a dense UWB channel is very large, we invoke the centrallimit theorem so that every sample variable of na(t) is Gaus-sian distributed with

mean mna =L∑

�=1, � �=ma�mφ,

variance σ2na =

L∑�=1, � �=m

a2�σ

2φ.

(C.1)

The distribution of φ(t)t �=0 and the values of mφ and σ2φ

can be determined according to a general model describingeach sample and probability in detail or some specificallychosen TH codes and multipath delays.

ACKNOWLEDGMENT

The authors would like to thank Professor Zhi Ding of theUniversity of California, Davis, for his invaluable suggestionswhich inspired the research presented in this paper.

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Jian Zhang received the B.S. degree fromXi’an Jiaotong University, China, in 1996,the M.S. degree from Nanjing Universityof Posts and Telecommunications, China,in 1999, and the Ph.D. degree from TheAustralian National University, Australia,in 2004, all in telecommunications engi-neering. From 1999 to 2001, he was em-ployed with Zhongxing Telecom Corpora-tion (ZTE), China, as a Research and Devel-opment Engineer. His current research interests are in the areas ofadaptive signal processing and wireless communications, with em-phasis on ultra-wideband systems.

Rodney A. Kennedy received the B.E. (withhonors) degree in electrical engineeringfrom the University of New South Wales,Australia, in 1982, the M.E. degree in digitalcontrol theory from the University of New-castle, Australia, in 1986, and the Ph.D. de-gree from the Department of Systems En-gineering, The Australian National Univer-sity, Canberra, in 1988. He has been theHead of the Department of Telecommuni-cation Engineering, Research School of Information Sciences andEngineering, The Australian National University, since 1994. Hisresearch interests are in the fields of digital and wireless communi-cations, digital signal processing, and acoustical signal processing.He is currently an interim Program Leader for the Wireless SignalProcessing Program, part of the National ICT Australia (NICTA)Research Centre. He has shared responsibility in supervising about25 research students and has coauthored approximately 180 papers.He is on a number of editorial boards for international journals andmonograph series.

438 EURASIP Journal on Applied Signal Processing

Thushara D. Abhayapala was born inColombo, Sri Lanka, on March 1, 1967.He received the B.E. degree with honors ininterdisciplinary systems engineering fromthe Faculty of Engineering and InformationTechnology, The Australian National Uni-versity (ANU), Canberra, in 1994. He re-ceived the Ph.D. degree in telecommunica-tions from the Research School of Infor-mation Sciences and Engineering, ANU, in1999. From 1995 to 1997, he was employed as a Research En-gineer at the Arthur C. Clarke Centre for Modern Technolo-gies, Moratuwa, Sri Lanka. From December 1999 to July 2002, heworked as a Research Fellow at the Department of Telecommuni-cations Engineering, and as a Lecturer at the Faculty of Engineer-ing and Information Technology, both at ANU. Since July 2002,he has been a Fellow at the Research School of Information Sci-ences and Engineering, ANU. Currently he also holds an affiliationwith the Wireless Signal Processing Program, Canberra, ResearchLab, National ICT Australia (NICTA), Canberra, as a Senior Re-searcher. His research interests are in the areas of space-time signalprocessing for wireless communication systems, spatio-temporalchannel modeling, MIMO capacity analysis, space-time receiverdesign, equalization, UWB systems, array signal processing, broad-band beamforming, and acoustic signal processing. Dr. Abhayapalais currently an Associate Editor for the EURASIP Journal on Wire-less Communications and Networking.