Application of Cramer-Rao bounds in Synchronization of DS-UWB

Jun Chen, Zheng Zhou Wireless Network Lab, Beijing University of Posts and Telecommunications Inner Box.96, Beijing University of Posts and Telecommunications, Beijing 100876, P.R.China [email protected], [email protected] Abstract1 Synchronization process is essential especially in UWB communication system. It owes a lot to its extremely short transmitted pulse shape and the ultra dense multipath interference. A close-form expression of Modified CramerRao bounds (MCRBs) for DS-UWB both in data-aided (DA) and non-data aided (NDA) acquisition scheme is derived. IEEE defined standard channel models are adopted in the presence of multi-access interference (MAI) during derivation. Insights are also given into the comparison between this derived limit of theoretical performance and classical sliding correlation acquisition algorithm with computer simulation in fixed observation window. It is found that the derivation under prior assumptions is a feasible tight lower bound and the increase of multi-path delay in environment weakens the estimation performance of timing delay.

1. Introduction Ultra-Wideband (UWB) technology has emerged as a promising one for many applications in high speed wireless communication, location, and tracking etc. To realize these unique features of UWB radio, clock synchronization constitutes a major challenge, the difficulty of which is emphasized due to the impulselike low-power UWB transmit-waveforms. The process of synchronization algorithm in UWB communication system typically includes two successive stages same as traditional spread spectrum (SS) communication system. Step1 is acquisition, it makes a coarse alignment (less than a fraction of a chip) between the received signal and local generated template signal. The challenge in acquisition process is that more acquisition positions need to be searched or more precise acquisition is needed. Step2 is tracking. 1

This research was supported by National Natural Science Fund of China (60372097, 60432040, 60572020, 60572158), Beijing Municipal Natural Science Fund (4052021), University IT Research Center Project (INHA UWB-ITRC), Korea.

1-4244-0697-8/07/.00 ©2007 IEEE

This process maintains the two codes in fine synchronization by means of a closed loop operation. Synchronization problem about UWB communication system have been investigated by many scholars [1,2,3]. But most of these efforts focus on special acquisition algorithms. Some of them are derived directly from acquisition algorithms used in spread spectrum communication system. The modifications are made to meet the stringent timing requirement of UWB ultra short impulse feature [4]. The process of synchronization in communication system actually is the estimation process to find the propagation delay τ from received signal. If this parameter is treated as unknown but deterministic, the maximum-likelihood (ML) criterion is applied widely for estimation [5]. As the request of ML, the prior knowledge of parameter τ, p(X|τ) needs to be provided. A well-known result in parameter estimation is Cramer-Rao bounds (CRBs) on the mean square error, it gives the minimal achievable variance for any unbiased estimator. One of the most important applications of the CRBs is that it provides the asymptotic optimality property of maximum likelihood estimators. According to the issued papers [12,13] have been discussed, the synchronization theoretical performance limits which aims at providing benchmarks based on the actual TH-PPM-UWB scheme is proposed in [14]. This paper considers the standard spread-spectrum multiple-access (SSMA), direct spread (DS) and binary pulse amplitude modulated (PAM) based UWB system. The rest of paper is organized as follows. Section Ⅱ describes the system model used throughout this paper and recall the basic concept of Cramer-Rao bounds (CRBs) and its modified form. In addition, we discuss the application of CRBs and MCRBs in general AWGN and frequency-selective fading channel communication system. The CRBs and MCRBs used in data-aided (DA) and no-data-aided (NDA) acquisition scheme in DS-UWB are described in

Section Ⅲ. It aims at deriving simplified expressions for Mean-Square Error (MSE) as the function of Signal-Noise ratio (SNR). Then, Section Ⅳ presents some interesting numerical results of the analysis. Finally, some concluding remarks are made in Section Ⅴ.

th cluster, Tn is the arrival time of the n-th cluster, and τnk is the delay of the k-th multi-path contribution within the n-th cluster. The IEEE suggested an initial set of values for different 4 channel environments from CM1 to CM4. Figure-1 show the CM1, the Line Of Sight (LOS) scenario. -3

2. Problem formulation

1

Impulse response realizations

x 10

0.8

2.1 System structure

0.6

str( k ) ( u, t ) =

∞

Ns

j =−∞

n=0

∑ d (jk) (u)∑cn( k) (u) ⋅ωtr ( t − jTf − nTs )

(1)

Where: ωtr(t) represents the transmitted monocycle waveform with the time duration Tm, {dj(k)(u)} are the modulated data symbols for the k-th user, {cn(k)(u)} are the spreading chips. That means one bit transmitted in system as Ns monocycles, where Ns=Tf/Ts is the spread spectrum process gain. Ts is chip time. Given that no data modulation is done during the acquisition stage, that is, dj(k) = 1,∨j. Without loss of generality, any k can be assumed in this paper. The multipath fading channel is described conventionally as a tapped-delay line, with L taps attenuation {αl}l=0…L, and delays {τl}l=0…L, the impulse response is h(t)= ∑ l αl·δ(t-τl). For UWB scheme, the modified Saleh-Valenzuela model is proposed based on the observation that usually multi-path contributions generated by the same pulse arrive at the receiver grouped into clusters. The arrival time of clusters is modeled as a Poisson arrival process with rate Λ: p(Tl|Tl-1)= Λ·exp[-Λ(Tl－Tl-1)], l>0 With in each cluster, subsequent multi-path contributions also arrive according to a Poisson process with rate λ: p(τk.l|τ(k-1),l)= λ·exp[-λ(τk.l－τ(k-1),l)],k>0 The channel impulse response of the improved IEEE model can be expressed as follows: [7] N K (n)

h ( t ) = X ∑ ∑ α nk δ ( t − Tn − τ nk )

(2)

n =1 k =1

Where X is log-normal random variable representing the amplitude gain of the channel, N is the number of observed clusters, K(n) is the number of multi-path contributions received within the n-th cluster, αnk is the coefficient of the k-th multi-path contribution of the n-

Amplitude Gain

0.4

Direct-Sequence Spread Spectrum (DS-SS) is a well-know digital modulation method. According to its basic principles here, followed discussion mainly focuses on its extension to UWB scheme. The signal str(k)(u,t) expresses a typical transmitted signal of the k-th user [6]:

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

0.2

0.4

0.6

0.8 1 Time(s)

1.2

1.4

1.6

1.8 -7

x 10

Figure-1: Channel impulse response (CM1).

As the request of quasi-static, the channel attenuations and delays remain invariant during one transmission burst, but are allowed to change across bursts. In other words, the propagation fluctuations within an observation time T>>Ts×Ns and pathdependent distortions can be neglected. At the receiver, we only concentrate on one user while the other users are viewed as noise. The waveform arriving at the receiver is given by: r ( t ) = X ETX

N K (n)

∑∑ ∑α j

n =1 k =1

nk

a j p0 ( t − jTs − τ nk ) + + m(t ) + n ( t )

(3) Where: ETX is the transmitted energy per pulse, aj is the amplitude of the j-th transmitted pulse, Ts is the average pulse repetition period, p0(t) is the energy normalized waveform of the basic pulse, m(t) represents the MAI caused by other users and n(t) is thermal noise, other coefficients are same as the definitions above.

2.2 Cramer-Rao bounds (CRBs) and its modified form It assumes that a sequence of observations [x1, x2....xn] = X, the probability density function of X for a given τ is p(X|τ), an estimate of a parameter τˆ( X ) would be extracted from it. The score function V(X,τ) is the derivative of the log likelihood function, ∂ V ( X ,τ ) = ln p ( X | τ ) . The covariance matrix ∂τ J(τ)=Var{V(X,τ)}, where : τ=[τ1,τ2,…τm], is called the

Fisher information matrix (FIM). If the expectation E[τˆi ( X )] = τ i , it means that the estimate is unbiased. The Cramer-Rao bounds on the mean square error defined as: E{[τˆ( X ) − τ ]2 } ≥ CRBs (τ ) = J (τ ) −1 . Note that E[V(X,τ)]=0, the Fisher information matrix is simplified to ∂2 J (τ ) = E[V 2 ( X ,τ )] = − E[ 2 ln p( X | τ )] . ∂τ Actually, the calculation of CRBs in many practical cases is impossible for insuperable integration and expectation computation, the modified Cramer-Rao bound (MCRBs) is defined in [8] aims at overcoming this obstacles. If p(X|u,τ) is the conditional probability density function of X given u and τ, MCRBs are same as the form of CRBs: ∂ MCRBs (τ ) = E X ,u {[ ln p ( X | u ,τ )]2 }−1 . ∂τ Generally, MCRBs(τ) is less than CRBs(τ), MCRBs(τ)≤CRBs(τ), which means that MCRBs(τ) is losser than CRBs(τ) but tight enough to use in practical applications [8].

2.3 CRBs and MCRBs used in general AWGN and frequency-selective fading channel For ordinary AWGN channel communication system, the received signal r(t) would be: r(t)=s(tτ)+n(t), where n(t) is Gaussian distributed with zero mean and variance σ2, and τ is the timing delay to be estimated. The conditional pdf p(r|τ) based on N independent observations becomes: (r − s (t ,τ , d )) 2 . 1 N p(r | τ ) = ( ) exp(−∑ [ n n 2 ]) 2σ 2πσ N The likelihood function with infinite N became: Λ(τ ) = exp{−

1 2 (r(t ) − s(t,τ , d ))2dt} = exp{ ∫ r(t ) ⋅ s(t,τ , d )dt} N0 ∫T0 N0 T0

Where: N0=2σ2, T0 is observation interval, d is the transmitted data sequence, the second equation yields from the fact that the r2(t) included in exponential factor does not involve the signal parameter τ and the integral of s2(t,τ,d) is the energy of signal during the observation interval and would be seen as a constant. The needed estimate parameter is the value which makes the likelihood function Λ(τ) become maximum. From the Λ(τ), we also find the basic structure of various synchronization schemes, like the classical sliding-correlation acquisition structure with the correlation between sliding transmit waveform template and generated digital samples, shown in Figure-2.

t = iT f 1 Tf (.)dt Tf ∫0

Zi arg(max( Z i )) × Δt i

Figure-2: Block diagram of a single dwell time serial search acquisition system with non-coherent detection.

In local generated correlation template, if the transmitted data {di} is already exactly known between the peers of the communication, the synchronization scheme refers as data-aided (DA) mode. The performance of parameter estimations partially rely on a special designed pilot symbol pattern [9]. On the contrary, if only the prior distribution knowledge of the data {di} is known or even this information does not known at all, the synchronization process under this condition is called no-data-aided (NDA) or blind synchronization mode [10,11]. MCRBs used for estimation more than one parameter is appropriate for this case. For frequency-selective fading channel, the received signal r(t) takes the following form: r(t)= ∑ j αj·s(tτj)+n(t). where: αj∈[α1,…, αL] and τj∈[τ1,…, τL] are attenuation and delay of the jth path respectively. All these parameters need to be estimated. Assuming that the number of multipath L in a dense fading channel is very large and each of them is independently and identically distributed (i.i.d), according to the central limit theorem, each sample of multipath interference is Gaussian distributed value. Same as the derivation of AWGN channel case, the likelihood function Λ(τ) is given by: Λ (τ ) = exp{− 1 (r (t ) − ∑ j α j s (t ,τ j )) 2 dt} . N 0 ∫T0 The same derivation will be adopted in the NDA case with minor modifications

3. MCRBs for DS-UWB DA and NDA acquisition scheme In DS-UWB DA scheme, the transmitted symbols {di} for synchronization are known deterministic sequence and assume that the number of transmitted symbols referred as training sequence is M. In NDA scheme, for analytically realizable, {di} is unknown as random sequence but assumed to be i.i.d and has the distribution function: p(d)=∏i(δ(di)+δ(di -1)) /2. Where δ(.) means the Dirac delta pulse. The likelihood function of (τ, a) derived from the Equation (3) as:

M N ( ) 1 [r(t) −X ETX ∑∑∑αnk aj p0 ( t − jTs −τnk )]2 dt} ∫ T 0 N0 j =1 n=1 k =1 Kn

(4) Where: T0=[0,M×Ns×Ts) represents the duration of observation window. The feasibility of equation is based on two approximations. First, with the help of low duty cycle of UWB signals, the inter-symbol interference (ISI) at the received signal is neglected. Second, if the number of users is large enough and each of them has comparable transmission power, according to the central limit theorem, the MAI is thought of as white Gaussian noise. The marginal likelihood function for τ used in DA scheme can be derived from (4), Λ(τ)=∫Λ(τ,d)·p(d)dt. For sake of simplicity, (4) can be rewritten as follow:

derivative Gaussian pulse (see Figure-3): Scholtz’s monocycle [14] for analysis and omit the distortion of waveform imposed by the transmitter and receiver’s antenna. ⎡ ⎣⎢

Where: equivalence ( ≡ ) indicates that irrelevant constants have been dropped, γs= 2ERX/N0 is symbol Ts with SNR, f ( p ,q ) = a 2j p0′ (t − τ p ) p0′ (t − τ q )dt

2 ⎛ t ⎞ ⎤ ⎜ ⎟ ⎥ ⎝ τ m ⎠ ⎥⎦

1

M N×K ( n) 1 Λ(τ , a) = exp{− ∫ [r(t) − ERX ∑ ∑ a jαnk p0 ( t − jTs −τ nk )]2 dt} N0 T0 j =1 nk =1

(5) Where: ERX=X2ETX is the total received energy for one transmitted pulse. The Fisher information component for parameters (θp, θq) as defined above is: ⎛ ∂ 2 ln Λ (τ , a ) ∂ 2 ln Λ (τ , a ) ⎞ ] −Ε r ,a [ ]⎟ ⎜ −Ε r ,a [ ∂τ p ∂τ q ∂τ p ∂aq (6) ⎜ ⎟ J =⎜ 2 2 ⎟ ∂ ln Λ (τ , a ) ∂ ln Λ (τ , a ) ⎜ −Ε r ,a [ ] −Ε r ,a [ ]⎟ ⎜ ⎟ a τ ∂ ∂ ∂a p ∂aq p q ⎝ ⎠ Where: p, q is integer and belongs to [1,N×K(n)]. In this paper, only timing delay τ is considered to be analyzed, taking Equation (5) into (6) and after straightforward algebraic manipulations, the result is: (7) J (τ p ,τ q ) ≡ α pα q γ s MN s f ( p , q )

2 ⎡ ⎛ t ⎞ ⎤ ⎟ ⎥ exp ⎢ − 2π τ ⎢⎣ ⎝ m ⎠ ⎦⎥

ω rec ( t ) = ⎢1 − 4π ⎜

received monocycle Wrec (t)

Λ(τ , d) = exp{−

0.5

0

-0.5 -3

-2

-1

0

1

2

t in seconds

3 -10

x 10

Figure-3: A typical idealized received monocycle ωrec(t) at the output of the antenna subsystem as a function of time in seconds. Shaping parameter: τm=0.25ns, impulse duration: Tm=0.5ns.

Figure-4 plots the acquisition performance in the mean square error (MSE) versus signal-to-noise ratio (Ex/N0) for four different IEEE standard channel models, CM1-CM4. For each scheme, classical peakpicking method finding the maximum output of a sliding correlator with the transmit-waveform template (local generated) in the presence of dense multi-path shown in Figure-2 is used to compare with the derived MCRBs. In order to improve the acquisition probability, 13-bit length Bake code [1 +1, +1, -1, -1, +1, +1, -1, +1, -1, +1] is used as spreading code sequence in preamble. -3

∫

15

x 10

MCRBs CM4 CM3 CM2 CM1

p0’(t)=dp0(t)/dt. Since the duration of impulse waveform is confined over [0,Tg], Tg>|τp-τq|>0 is deduced as the assumption of Equation (7). Furthermore, as the definition of DS-UWB in Section Ⅱ, the amplitude of the j-th transmitted pulse aj is +1 or -1, then aj2=1, Ts f ( p,q ) = p0′ (t − τ p ) p0′ (t − τ q )dt . That means, the

∫

Mean Square Error

0

10

5

0

MCRBs for DA and NDA synchronization in this case has the same value: (8) MCRBs(τ)=J(τp, τq)-1

0 0

4. Numerical results At the receiver, we assume that the true transformed pulse shapes is known. This time we choice the second

2

4

6

8

10

12

14

16

Ex/No(dB)

Figure-4: MCRBs(τ) or MSE versus symbol SNR γs

18

20

5. Conclusions We have investigated the estimation of timing delay for DS-UWB system operating in a modified S-V multipath environment and in the presence of MAI. Furthermore, the derived close-form Modified CramerRao bounds (MCRBs) used as theoretical performance limits for both DA and NDA acquisition scheme compared with classical sliding correlation acquisition algorithm by computer simulation in fixed observation window. It is found that the derivation under prior assumptions is a feasible tight lower bound and the increase of multi-path delay and path attenuation in environment weaken the estimation performance of timing delay. However, MCRBs derived at the assumption that without taking into accounts the intersymbol interference (ISI) at the received signal. It is not appropriate for real communication environments and need more deliberate analysis in further research.

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[8]. N. D'Andrea, U. Mengali and R. Reggiannini, "The modified Cramer-Rao bound and its application to synchronization problems," IEEE Transactions on Communications, Volume: 42, Issue: 234 Part=2, Feb/Mar/Apr 1994, Page(s): 1391-1399. [9]. Z. Tian and G. B. Giannakis, "Training sequence design for data-aided timing acquisition in UWB radios," Communications, 2004 IEEE International Conference on, Volume: 6, 20-24 June 2004, Page(s): 3399- 3403 Vol.6. [10]. Y. Liuqing and G. B. Giannakis, "Blind UWB timing with a dirty template," Acoustics, Speech, and Signal Processing, 2004. Proceedings. (ICASSP'04). IEEE International Conference on, vol 4, Page: iv-509. [11]. TianZhi and WuLin, "Timing Acquisition with Noisy Template for Ultra-Wideband Communications in Dense Multipath", EURASIP Journal on Applied Signal Processing 2005:3, pp.439-454, 2005. [12]. ZhangJian and Kennedy Rodney and Abhayapala Thushara, "Cramer-Rao Lower Bounds for the Synchronization of UWB Signals", EURASIP Journal on Applied Signal Processing 2005:3,pp.426-438,2005. [13]. V. Lottici, A. D'Andrea and U. Mengali, "Channel estimation for ultra-wideband communications", Selected Areas in Communications, IEEE Journal on, Volume: 20, Issue: 9, Dec 2002, Page(s): 1638-1645. [14]. M. Z. Win and R. A. Scholtz, "Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications", Communications, IEEE Transactions on, Volume: 48, Issue: 4, Apr 2000, Page(s): 679-689.