Cramér-Rao Lower Bounds for the Synchronization of UWB Signals

We present Cramér-Rao lower bounds (CRLBs) for the synchronization of UWB signals which should be tight lower bounds for the 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 Gaussian monocycles. An approximation method of the CRLBs is discussed when nuisance parameters exist. CRLBs in multipath channels are studied and formulated for three scenarios depending on the way multipath interference is treated. We find that a larger number of multipaths implies higher CRLBs and inferior performance of the synchronizers, and multipath interference on CRLBs cannot be eliminated completely except in very special cases. As every estimate of time delay could not be perfect, the least influence of the synchronization error on the performance of receivers is quantified.


I. INTRODUCTION
Ultra Wideband (UWB) is a promising technique in the application of short-range high-speed wireless communication and precise location tracking. Typically, ultra narrow pulses, such as Gaussian monocycles [1], are modulated to transmit information. These pulses could be narrower than 1 nanosecond. This brings very stringent synchronization requirements.
A UWB signal is basically a baseband signal without phase and carrier information, hence time delay estimation is the main task of a synchronizer. This synchronizer could be one in a simple single-pulse UWB system, however, due to the power limitation imposed by FCC [2], UWB pulses are generally combined with spread spectrum techniques, especially timehopping (TH), to achieve multiuser access, to ensure sufficient received energy and to mitigate interference to existing wireless systems. Similar to traditional spread spectrum systems, the synchronization of a time-hopping UWB system is accomplished in two steps: code acquisition followed by code tracking. The former, involving the optimization of search strategies, tries to determine the phase of the incoming pseudonoise (PN) sequence within a fraction of chip width. The latter refers to the process of achieving and maintaining fine alignment of the chip boundaries of the incoming and locally generated PN sequences.
As UWB pulses are very narrow, very stringent synchronization requirements are incurred, and timing errors usually imply marked degradation of receiver performance [3]. Abundant research on the design and performance of synchronization systems have been reported in the literature, e.g., [4]- [6]. These techniques can be transplanted into UWB systems with some modifications to meet the stringent timing requirement, as discussed in [7]- [10]. Different to them, in this paper, we try to find some general performance limitations for UWB synchronizers, and provide guidelines for the system design within acceptable performance region.
It is known that in the presence of noise, perfect synchronization cannot be achieved. For UWB systems with stringent timing requirements, it is of special interest to characterize this synchronization error and its influence on the performance of detectors. This task becomes even more urgent when we realize that the radiated power of UWB signals is so low that the channel estimates could contain large errors and the performance of synchronizers could be largely deteriorated. Under these conditions, is it still possible for UWB synchronizers to reach a satisfying accuracy of timing locking? Some common performance parameters to evaluate synchronizers are tracking time, S-curve behavior and probability of success. However, in order to provide benchmarks for actual UWB synchronizers, we are more interested in understanding their theoretical performance limits. In the theory of parameter estimation, Cramér-Rao Lower Bound (CRLB) is most widely used in evaluating the performance of estimates.
The CRLB [11] is a fundamental lower bound on the variance of any unbiased estimator. The analysis of CRLB for traditional systems is well founded [5], [12]- [19], but for UWB, there is no systematic work reported yet to our knowledge. The evaluation of the CRLB is generally mathematically quite difficult when the observed signal contains, besides the parameter to be estimated, also some nuisance parameters that are unknown [14], [19]. These nuisance parameters could be the transmitted data and sometimes, multipath gains and delays which arise in fading channels. When the nuisance parameters are present, the modified CRLB (MCRB) [13]- [15], and the asymptotic CRLB (ACRLB) [14], are good approximations to the true CRLB at higher signal-to-noise ratio (SNR), and the lower-SNR limit of the CRLB is approximated in [18] by applying a Taylor Series expansion.
This paper is concerned with evaluating the CRLB for UWB signals. Both single-pulse systems and time-hopping systems are considered. For time-hopping, the CRLB should be a lower bound for the performance of code tracking. The structure of this paper is as follows. In Section II, the problem is formulated. In Section III, considering AWGN channels, the CRLBs for single-pulse systems are investigated in both cases of known and unknown transmitted data. Some insights into the relationship between CRLBs for different Gaussian monocycles are given explicitly. We also highlight an oversight in the lower-SNR approximation method [18] and provide a possible solution to remedy this problem by tightly locating the range of SNR γ s . These results can be readily extended to a TH UWB system in AWGN channels with minor modifications. In Section IV, we extend this work to more practical multipath channels while considering an unmodulated TH system. Depending on the way multipath interference is treated in a practical synchronizer, three scenarios are analyzed when multipath interference contributes as an increase of noise variance or multiple synchronization parameters. In Section V, the influence of synchronization error on the performance of receiver is quantified, which may be the least influence a UWB correlator receiver can expect. Finally, numerical results are given in Section VI to verify some analytical results and illustrate the effect of pulse truncation on CRLBs.

II. PROBLEM FORMULATION
Binary pulse position modulation (BPPM) and binary phase shift keying modulation (BPSK, or antipodal modulation) are considered here. Let s(t) be the transmitted UWB signal. In a single-pulse system, is the ith transmitted symbol, T f is the frame width, N f is the number of frames in a symbol, T c is the chip width, and c j are the time-hopping codes.
The UWB pulses considered are series of Gaussian monocycles ω(t; n, t p ), which are scaled and/or differentiated versions of the basic Gaussian waveform ω 0 (t) = exp(−2πt 2 ), that is, ω(t; n, t p ) = ω (n) 0 (t/t p ), where the superscript (n) stands for n-order differentiation with respect to t, and t p parameterizes the width of the pulse.
To ensure equal energy of monocycles, a coefficient ε(n, t p ) is introduced, and let ω(t) = ε(n, t p )ω(t; n, t p ). Denote the energy of ω(t) as E p and symbol SNR as γ s , then ε(n, t p ), depending on n and t p , satisfies When passing through a pure AWGN channel n(t), the received signal r(t) becomes where every sample of n(t) is Gaussian distributed with zero mean and variance σ 2 0 , and τ is the timing delay to be estimated.
When passing through a frequency selective fading channel, h(t) = L =1 a δ(t − τ ), the received signal is given by where a and τ are real multipath gains and delays, respectively. Note the time delay τ between transmitter and receiver is merged into τ . Due to the low duty cycle of UWB signals, we assume the received signal is free of intersymbol interference (ISI) unless indicated otherwise. For the effect of ISI and the design of training sequence accordingly, the readers can refer to [20], [21].
For the AWGN model in (2), for the purpose of forming estimates based on K independent observations, the received signal can be represented as a vector model where r = [r 1 , · · · , r K ], s = [s 1 , · · · , s K ] and n = [n 1 , · · · , n K ] are the sample vectors of the received signal r(t), the transmitted signal s(t − τ ) and the noise n(t), respectively, and b = {b i } are the transmitted data sequence. Suppose an unbiased estimateτ of the time delay τ can be generated from (4), then the estimation error variance is lower bounded by In (5), the conditional pdf p(r|τ ) is the likelihood function of τ , and the expectation E r|τ [·] is with respect to p(r|τ ). The likelihood function p(r|τ ) can be obtained by averaging the joint likelihood function p(r|b, τ ) over the a priori distribution of the data b: Since the additive noise n(t) is white and zero mean, the joint conditional pdf p(r|b, τ ) can be expressed as Applying the signal orthogonal expressions [6, p.335] or letting the number of samples K go to infinity [11, p.274] (or from the standpoint of generating sufficient statistics), we have where T o is the observation period. Now, a continuous-time equivalent of p(r|b, τ ) can be developed. Considering the subsequent operations of logarithm and differentiation, only terms related to b and τ will be retained. Then the evaluation of p(r|b, τ ) is equivalent to evaluating the likelihood function The process from (4) to (8) can be applied to the multipath model (3) with minor modifications.

A. CRLB with Known Transmitted Data
The CRLB with known b, further derived from (8) or directly from [15], has the form whereṡ(t − τ ) denotes first partial differentiation with respect to τ . Assuming that the pulse is strictly restricted within a symbol period, and T o = N T s , where N is the number of symbols contained in the observation period (one pulse per symbol in this case), then for both BPSK and BPPM, the denominator in (9) equals N Tsω For a specific monocycle, the lower variance bound becomes where the symbol SNR is γ s = E p /σ 2 0 . If the symbol period T s is large enough so that most of the energy of the pulse concentrates within T s , we can express (10) in frequency domain where W (f ; n, t p ) is the Fourier Transform of ω(t; n, t p ).
According to the properties of the Fourier Transform of derivatives of functions, we find explicit relationships exist between the CRLBs of monocycles with different n but same t p , that is, where the inequality follows from an application of Schwarz's inequality. This inequality implies that monocycles with higher order differentiation have the potential for better performance in the sense of lower synchronization error variance. For monocycles with different t p but same n, the ratio between their CRLBs can be found as which implies that monocycles with smaller t p (narrower effective pulse width) have the potential for better synchronization performance.

B. CRLB with Unknown Randomly Transmitted Data
For PPM, the uncertainty of time jitter introduced by modulation will cause large synchronization error when the transmitted data is random and unknown. When further methods are adopted to solve this problem, the CRLB analysis in these cases will usually be based on a system model similar to the one with known data. Hence we only consider BPSK UWB signals in this subsection.
For BPSK, the likelihood function in (8) becomes where By differentiating L(r; τ ) twice with respect to τ , we get whereẏ(τ ) andÿ(τ ) denote first and second derivative of y(τ ) with respect to τ . Due to the nonlinear function tanh(·) in (17), an analytical Since the pulse energy is restricted to be very low by the FCC [2] (the maximum power of a transmitted pulse with bandwidth 7GHz is only 0.5mW), one can refer to the lower-SNR limit of CRLB in [18], applying a Taylor extension of the likelihood function p(r|b, τ ), to obtain a similar result for UWB. One thing we wish to emphasize here is, in [18], the statistical property of the likelihood function L(u, τ ) (original notation in [18]) is somewhat ignored. Due to L(u, τ ) containing Gaussian variables with variance comparable to the reciprocal of symbol SNR, more care is needed when dropping the higher order terms in Taylor extension according to the lower symbol SNR assumption. A similar problem arises in an alternative method we introduce below, where this ambiguity is revealed further, and resolved by tightly locating the value of the symbol SNR.
The alternative method we suggest is also based on approximation. The basic idea is to find best-fitting functions for ln(cosh(·)) in a piecewise fashion. To make analysis tractable, these functions are polynomials with order smaller than 3. But they should not be constructed by only considering the goodness of fit due to the succeeding expectation operation. This is because, y(τ ) is a random variable and when we deal with the expectation operation, we have to make sure that all the possible samples of y(τ ) are involved. Although integrating these polynomials in segments is feasible, it can not produce a closed form result and is still a numerical method. Instead, we try to construct each polynomial in which the variable space supports the sampling space. It seems impossible as the pdf of y(τ ) distributes in the entire onedimension real space. We overcome this obstacle by assuring that most of samples (say, 99%) are located in the interval of interest.
With this criterion in mind, we find a three-segments approximation is a good choice by studying the shape of the waveform ln(cosh(x)). A detailed discussion is shown in Appendix A. Examples of such three lower order (≤ 2) polynomials are Due to non-existence of a polynomial with goodness of fit and a fully covered sampling space simultaneously, an appropriate interval corresponding to SNR range of (−6.25, 10.3)dB can not be found.
Represent a general 2-order polynomial function as The reciprocal of the CRLB can be calculated as where we utilize As shown in Appendix B, these expectations are given by Then for a specific monocycle ω(t; n, t p ), the CRLB is By substituting a and b with the coefficients of polynomials in (18), the CRLBs for different γ s are readily obtainable. It is clear that the relationship between CRLBs for monocycles with different n or t p is identical to that when the transmitted data is known. By comparing (10) and (24), we find CRLB(τ ; b)/CRLB(τ ) = 2aγ s + b. Referring to (18), it is obvious that the CRLB with unknown data is always larger than that with known data for the lower SNR case, and converges to CRLB(τ ; b) in the higher SNR case, which coincides with the attributes of ACRB given in [14].

SELECTIVE-FADING CHANNELS
When the channel is AWGN, the analysis and results in Section III can be applied to time-hopping UWB systems with minor modifications. The change can be merged into the symbol SNR γ s , that is, γ s equals to the ratio between the energy of N f pulses and the noise variance σ 2 0 for TH UWB systems. In this section, we will focus on selective fading channels.
Synchronization in selective fading channels is a challenging task. The performance largely depends on the schemes and algorithms. Based on the way multipath signals are treated, these systems can be divided into three categories. Accordingly, we consider the CRLB for each of them. Since CRLB with unknown data is straightforward but computationally complex as derived in Section III, we only consider the case of known data b here.

A. Passive methods: Regarding Multipath signals as Interference
This refers to methods that apply general synchronizers, such as early/late gates, while treating multipath signals as interference [22], [23], or partly utilizing multipath energy [24], or using a whitening filter before a synchronizer [25]. The effect of multipath interference on synchronizers has been studied in [22], [23], [26]- [29]. From the viewpoint of CRLB, all these methods can be generalized to a model in which only a specific multipath is of interest. Mathematically, we can represent this model as where a m and τ m are the parameters to be estimated. Generally, the received signal r(t) first passes through a PN code correlator s i (t −τ m ), whereτ m is the pre-estimated delay, so that the energy of all pulses in a symbol are collected to make an estimation. Then the model in (25) can be further written as where Successful detection requires sampling r f (t) atτ m = iT s +τ m accurately.
Each sample of n f (τ m ) is Gaussian distributed with zero mean and variance σ 2 n f = N N f E p σ 2 0 . The component n a , containing interchip interference and ISI, is hard to model and evaluate without prior knowledge of TH codes and multipath delay. To make the analysis mathematical tractable, here we assume n a is Gaussian distributed 1 with mean m na and variance σ 2 na . In Appendix C, more information is given on the parameters of this distribution.
Recall that when considering the CRLB for TH UWB synchronizers in the phase of code tracking, it is reasonable to where T φ is smaller than half of the frame period ( τ m ). Now, the estimation problem can be reformed as which is a problem of multiple parameters estimation in a Gaussian interference.
Although a m and τ m are correlated via the mean power profile of fading, they are usually treated as unknown and deterministic parameters and nonrandom parameter estimation techniques are applied, as the statistical relationship between them are hardly predictable. This means they are not a function of each other any more. Strictly speaking, τ m is the only synchronization parameter, and CRLB(τ m ) can be obtained when regarding a m as a nuisance parameter. However, it is known that joint estimation of τ m and a m usually gives lower CRLB for τ m than that in a separate-estimation case [11], [14], [19]. Hence we will focus on joint estimation and generate CRLB(a m ) as a byproduct.
Representing (30) as a vector form r f = a m N Φ + n a + n f and applying the similar process from (6) to (8), the joint loglikelihood function L(r f ; a m , τ m ) = ln p(r f ; a m , τ m ) can be obtained as Lower bounds on the variances of estimates for the components of a m and τ m are given in terms of the diagonal elements of the inverse of the Fisher information matrix J −1 [11]. In this example, the elements of J equal where the expectation E[·] is with respect to p(r f ; a m , τ m ). Note φ(t − τ m ) and a m are mutually independent, the 1 In [11, p309], a general equation is provided for the CRLB of any unbiased estimate in colored noise. But a closed form solution is not readily available. elements of J can be calculated as where C is a constant defined as C N 2 /(σ 2 n a + σ 2 n f ). The cross terms J 12 and J 21 will vanish if we extend the observation period T φ till φ(T φ ) ≈ 0. Then the CRLBs for τ m and a m are It is clear that the estimation of time delay τ m depends on the amplitude of the multipath given that C is the same for all multipath signals, while the estimation of a m could be independent of τ m supposing we extend the observation period appropriately. For the performance of synchronizer, the multipath interference contributes as an increased estimate variance 2 . Depending on the Gaussian approximation for the multipath interference n a , C may be related to a specific monocycle, hence the relationship between CRLB for different monocycles can not be claimed directly.
Finally, we wish to say a little more on the relationship between our model in this section and practical systems. In the literature on synchronizers for spread spectrum systems such as CDMA, we can always find the terms of fading bandwidth, tracking loop bandwidth and predetection bandwidth and discussions on how the relationship between them affecting the performance of synchronizers in a multipath channel (e.g. [26]- [28]). Briefly, the relationship between these bandwidths determines the degree of multipath interference entering the final decision part of the synchronizer. Considering our model, the effect can be regarded as a reduction of noise variance σ 2 n a .

B. Positive Joint-Detection of Multipath Signals
We refer to the method of jointly detecting fading amplitude and delay of all the multipath signals as a positive one. For CDMA, this method has been well studied in literature such as [16], [17], [25]. And the derivation of CRLB for CDMA systems can been found in [16], [17], [30]. Here, following the process in Section III, we study the CRLB using joint detection for a UWB system where the parameters a = [a 1 , . . . , a , . . . , a L ] 1×L and τ = [τ 1 , . . . , τ , . . . , τ L ] 1×L to be 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 After some manipulation, the Fisher Information Matrix J has the structure where J τ τ , J τ a , J aτ and J aa are all L × L matrices with respectively.
The CRLB for τ m is just the m th diagonal element of the inverse of J. Use m = 1 as an example and rewrite the matrix J as we have where J 11 is called the Schur complement of J 11 [31, p.175]. Since J is nonnegative definite, the Schur complement matrix J 11 is also nonnegative definite, so is J 11 −1 . At the same time, B is the transpose of C since J is a symmetric matrix in this case. Thus we get B J 11 C ≥ 0 and the inequality in (46) follows immediately. Whenever J 11 > 0, we can get the inequality in (46) more readily according to As J −1 11 can be regarded as the CRLB in an AWGN channel with a known scalar of amplitude, this inequality implies the CRLB in joint detection is always larger than that in the single parameter estimation in an AWGN channel. Then an interesting question arises, whether more multipath means higher CRLB and inferior performance of synchronizer accordingly?
Let us consider a channel with L − 1 multipath signals. The Fisher Information Matrix J can be written as where 0 is a (L − 2) × 1 zero vector and † stands for transpose operation. Then the CRLB with L − 1 multipath is Comparing BD −1 C and BD −1 C gives where the inequality in (52) yields from that D −1 − D −1 is a nonnegative definite matrix as can be proven according to the property of partitioned nonnegative definite matrices (e.g., see [31, p178] and let D −1 = A in equation (6.10)).
Since J 11 > 0, we have which shows that more multipath does lead to higher CRLB and inferior performance of synchronizer. Since the number of multipath is closely relevant to the bandwidth of monocycles, we conclude that narrower monocycles will very likely cause larger CRLBs. We did not say "absolutely" because all other variables besides D during this derivation are assumed unchanged, but it could be unrealistic when different monocycles are applied.
Another key factor with influence on CRLB is the choice of TH codes. When the autocorrelation of TH codes is ideal, both the CRLBs in this subsection and last subsection will be the same and similar to the one in an AWGN channel.

C. Active methods: Cancellation of interference?
From the last two subsections, we have seen that the performance of synchronizers is deteriorated by the multipath interference. It is natural to ask whether the multipath interference can be mitigated or fully eliminated before entering the decision part of a synchronizer?
As shown for CDMA systems in [27], it is possible to remove part of multipath interference in UWB systems. However, unless the correlation of TH codes is ideal, the total removal of multipath interference is impossible due to the existence of n(t). This is because, from Section IV-B, we see that any estimate of parameters, including amplitude and delay, even though unbiased, may still have a nonzero variance in the present of noise. The CRLB can generally be achieved by Maximum Likelihood estimation asymptotically (when the number of observation samples goes to infinity), and the estimation error becomes Gaussian distributed with zero mean and variance equivalent to the CRLB [5], [11]. Therefore, the final signal with a pair of synchronization parameters of interest contains the sum of 2(L−1) Gaussian variables, which has a variance larger than the variance of n(t). Since CRLB is proportional to the variance of (interference and) noise, the CRLBs for this pair of parameters will be larger than those in a single path channel. So no matter how perfect the structure and algorithm to remove multipath signal are, the effect of multipath interference can only be mitigated but can not be cancelled thoroughly. This result also partly explains why more multipath generally leads to higher CRLBs.
However, there are some special cases when multipath interference becomes negligible. For example, when the maximal multipath delay is smaller than the frame period in a single pulse system, multipath signals do not interfere with each other due to the low duty cycle of UWB signal structure.

V. INFLUENCE OF SYNCHRONIZATION ERROR ON BER
As every estimate of time delay could not be perfect, we use an example to show the influence of synchronization error on the performance of receivers in UWB systems.
We consider a BPSK modulated single-pulse signal in an AWGN channel like that in Section III. A correlator receiver [32], [33] is used to detect the signal.
The conditional bit-error-ratio (BER), depending on the synchronization error e τ , is given by where we have assumed that the observation period equals a symbol period such that N = 1, Recall that the best theoretically achievable e τ is Gaussian distributed with zero mean and variance equivalent to the CRLB (denoted by σ 2 c ). In the best case, σ 2 (9) is the smallest. Averaging P e (e τ ) over e τ , we get the mean BER Statistically, this is the best achievable performance under certain SNR. This equation can be evaluated numerically by Monte Carlo simulation which requires high computational complexity. Alternatively, we invoke the Hermite-Gauss quadrature [34], and P e can be accurately approximated by where N h is the order of the Hermite polynomial H N h (·), x n and H x n are the zeros (abscissas) and weight factors of N h -order Hermite polynomial, respectively. These values are tabulated in many mathematical handbooks (e.g., [35]). In experiments, we find 16 coefficients (N h = 16) are enough to generate accurate approximation results.
Further define a variable η as the degrading ratio between P e and P e (0) = Q( √ γ s ), which is the BER in the case of perfect synchronization. We show the values of η for different monocycles in Section VI to compare the synchronization error robustness of monocycles.

VI. NUMERICAL RESULTS
Since for multipath channels, the CRLBs depend on the time-hopping codes and detailed fading channel models, we only show numerical results on the CRLBs in pure AWGN channels in this paper.
In Fig. 1 -Fig. 3, the CRLBs for different monocycles in the case of known data b are demonstrated. Since in practice, a transmitted monocycle is usually the truncated portion of a whole pulse w(t; n, t p ), this effect of truncation is considered by varying the actual width of pulse in (10).
From Fig. 1, we can see CRLBs are inversely proportional to symbol SNR and the observation period N T s . The relationship between CRLBs for monocycles with different order n coincides with the analytical results in (13). This can be further observed in Fig. 2, which also depicts the effect of truncated pulses on CRLB. The CRLBs change little even when the truncated portion narrows to 1.6t p (symmetric with respect to t = 0). However, with the width of truncated pulse decreasing further, the CRLBs become orderless. Fig. 3 shows the effect of t p on the CRLBs, and is a direct verification of (14). Fig. 4 demonstrates the influence of synchronization error on the performance of receivers. It is plotted from (56) using Hermite Gaussian approximation. The influence is notable when the observation window in the stage of synchronization has small width (N T s ), and weakens with N increasing (CRLBs decreasing). The figure also indicates that synchronization errors of different monocycles have very close influence on BER, although the data in experiments shows the influence of monocycles with larger n is a little worse when SNR γ s is small, and changes toward opposite with SNR increasing.

VII. CONCLUSIONS
We have derived the Cramér-Rao lower bounds (CRLBs) for the synchronization of UWB signals for both single pulse systems and time-hopping systems in AWGN and multipath channels. Insights are given on the relationship between CRLBs for different Gaussian monocycles. The CRLBs in AWGN channels are discussed in both cases of known and unknown transmitted data. An approximation method of the CRLB is introduced when nuisance parameters, unknown transmitted data, exist. An oversight in the lower-SNR approximation method [18] is highlighted, and a possible solution is provided by tightly locating the range of SNR γ s . The CRLBs in multipath channels are studied for three scenarios depending on the way multipath interference treated in a practical synchronizer, where multipath interference contributes as an increase of noise variance or multiple synchronization parameters. It is found that larger number of multipath implies higher CRLBs and inferior performance of synchronizers, and multipath interference on CRLBs can not be eliminated completely except in very limited cases. The least influence of synchronization error on the performance of receivers is quantified. The influence is notable when observation window (N T s ) in the stage of synchronization is small, and weakens with N increasing (CRLBs decreasing). Synchronization errors of different monocycles have very close influence on BER.
ACKNOWLEDGEMENT The authors would like to thank Prof. Zhi Ding of the University of California, Davis, for his invaluable suggestions which inspired the research presented in this paper. 8 APPENDIX A: APPROXIMATION OF ln(cosh(y(τ )/σ 2 0 )) Here we show how to approximate ln(cosh(y(τ )/σ 2 0 )) as low order polynomials in a piecewise fashion and determine the corresponding range of symbol SNR γ s .
From y(τ ) = T s r(t)ω(t − τ )dt, y(τ ) has Gaussian distribution N (E 1 , γ s σ 4 0 ), where |E 1 | = E p in the case of perfect synchronization, otherwise |E 1 | < E p . As the estimate is usually clustered tightly around the true value in our case, and E 1 changes smoothly for UWB monocycles, we assume |E 1 | ≈ E p (This can also be obtained from the assumption of unbiased estimation of τ ). Then y(τ )/σ 2 0 is also Gaussian distributed N (γ s , γ s ) or N (−γ s , γ s ). For Gaussian distribution, we know that when the distance between a sample and the mean is larger than about 2.6 √ variance, the probability of appearance can be assumed to be zero. Let the interval of interest be [x 1 ≤ y(τ )/σ 2 0 ≤ x 2 ], to ensure most of samples be in this interval, γ s should satisfy the following equations Briefly, two guidelines for determining interval [x 1 , x 2 ] are: 1. to ensure this variable space be larger than the sampling space for a specific polynomial and SNR γ s , and cover the range of γ s as widely as possible; 2. Although two intervals can overlap, each interval should be fully covered by a single polynomial.
Considering the waveform of ln(cosh(x)), from x = 0 to a samll x 2 , it has a very different shape with other segments and has to be approximated separately by a polynomial. This implies there is only one interval covering the segment containing the point zero. For this interval, only x 2 need be determined since ln cosh(·) is an even function, and the distributions N (γ s , γ s ) and N (−γ s , γ s ) are symmetric with respect to 0. For the same reason, it is enough to consider the positive value of x 1 and x 2 for other segments hereafter. Note γ s should be at least 6.76 for x 1 > 0, this implies x 2 > 13.52.
A well known fact is that ln(cosh(x)) can be accurately approximated by x 2 /2 when |x| 1, and by |x| − 0.69 when |x| 1. But this simple scheme is not good enough to be realistic. For example, for a value x 2 as large as 0.5, the approximation error is already 0.005, while the corresponding maximum SNR γ s is only 0.0324 = −15dB which is of little interest in practice.
Summarize the description above, we find a three-segments approximation is a good choice. Although the construction of these approximations is not unique, they can be represented as a general 2-order polynomial function f (x), which leads to a general CRLB expression as shown in (24).

APPENDIX B: DERIVATION OF E r|τ [·]
First we derive the autocorrelation of r(t) which will be used in subsequent calculation.
where in the last equality, we utilize the assumption that signal and noise are mutually independent and n(t) is AWGN with zero mean and covariance σ 2 0 δ(t 1 − t 2 ). Note the expectation with respect to p(r|τ ) is equivalent to average over the data b and noise n(t). Recall that the convolution between r(t) and ω(t) in y(τ ) is only within one symbol period T s , in the case of ISI-free, we have and E r|τ [r(t 1 )r(t 2 )] = σ 2 0 δ(t 1 − t 2 ) + ω(t 1 − τ )ω(t 2 − τ ).

(B-3)
Then expectations on y(τ ) can be calculated as Assume that the energy of a pulse outside T s is negligible, these results can be further simplified due to and Ts The key assumption we make is, for each multipath with index =m , φ(t) t =0 is identically independently distributed with mean m φ and variance σ 2 φ . As the number of multipath L in a dense UWB channel is very large, we invoke the Central Limit Theorem so that every sample variable of n a (t) is Gaussian distributed with mean m n a = L =1, =m a m φ variance σ 2 na = L =1, =m a 2 σ 2 φ . (C-1) The distribution of φ(t) t =0 and the values of m φ and σ 2 φ can be determined according to a general model describing each sample and probability in detail or some specifically chosen TH codes and multipath delays.      Fig. 4. The degrading ratio η versus SNR γs for monocycles. Two observation periods (N T s ) in a synchronizer are compared with N = 10 (left) and N = 50 (right). The time t is normalized with respect to t p .