LNCS 4523, Pp

中国科技论文在线 http://www.paper.edu.cn Jitter Distribution Evaluation and Suppression Method in UWB Systems

Weihua Zhang, Hanbing Shen, Zhiquan Bai, and Kyung Sup Kwak

UWB Wireless Communications Research Center (INHA UWB-ITRC), Inha University, 402-751, Incheon, Korea {zhweihua2000}@hotmail.com

Abstract. For the extreme short duration of the baseband Impulse Ra- dio (IR) Ultra-Wideband (UWB) transmission waveforms, even small amounts of timing jitter can bring out considerable system performance degradation. Thus jitter performance evaluation and corresponding suppression methods are very important for UWB high rate transceiver. We analyzed the jitter distribution function in UWB systems, also a jitter suppression method is proposed by modiﬁcation of template waveforms. Performance of our proposed waveforms and traditional template waveforms are compared by analysis and simulation, and the results verify our jitter suppression method works.1

1 Introduction

Ultra-wideband (UWB) is proposed as the primary candidate for the physical layer technology of next high speed short range wireless communication for per- sonal area networks (PAN) [1]. This radio technology is based on the radiation of waveforms formed by a sequence of very short, about nanosecond or below nanosecond, base band pulses [2]. The impulsive nature means that synchro- nization error (or jitter) inﬂuences the correlation of the received waveform and the template waveform generated in the receiver, thus deteriorates the system perform. Some past researches about this problem have been proposed: The eﬀects of timing jitter on the bit error rate (BER) of UWB have been analyzed [3][4]. To suppressing this degradation system clock with higher stability is proposed which has the capability of precisely positioning the sub-nanosecond pulses with Gaussian distributed root-mean-squared (RMS) jitter of 10ps and within a 50ns time window [5]. However, even if the system clocks are very stable and introduce very little jitter, other issues such as relative velocities between transmitter and receiver introduce additional degradation [6].

1 This research was supported by the MIC (Ministry of Information and Communi- cation), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Advancement). (IITA-2006-(C1090-0603-0019)).

Y.-H. Lee et al. (Eds.): ICESS 2007, LNCS 4523, pp. 810–817, 2007. 转载 c Springer-Verlag Berlin Heidelberg 2007 中国科技论文在线 http://www.paper.edu.cn Jitter Distribution Evaluation and Suppression Method in UWB Systems 811

We offered an approach to suppress the system BER degradation introduced by timing jitter in the paper. This approach is based on modification of the receiver template according to the jitter distribution function to gain more correlation with the jittered received signals. We also offered two evaluation methods of jitter distribution function which can be adopted in our proposed schemes.

2SystemModel

2.1 Signal Model of Transmission and Receiving Generic transmission signal s(t) of BPSK and PPM systems for the n-th bit can be given by:

− ∞ Ns 1 Eb BPSK: s(t)= bnwt(t − nTf − jTh) Ns n=−∞ j=0 − , (1) ∞ Ns 1 Eb δ PPM: s(t)= wt t − nTf − jTh − (1 − bn) Ns 2 n=−∞ j=0

where bn ∈{−1, +1} is the n-th transmitted binary data bit. Eb is the energy of one bit signal. wt(t) is the transmitted waveform with unit energy. Ns is the number of pulses per bit. Th is the nominal interval between two pulses. Tf = NsTh , which is the average time duration of one transmission symbol. δ is the modulation index if the modulation is PPM. The received signals are given by:

− ∞ Ns 1 Eb BPSK: s(t)=α bnwr(t−nTf −jTh −τj(n))+n(t) Ns n=−∞ j=0 (2) ∞ Ns−1 Eb δ PPM: s(t)=α wr t−nTf −jTh − (1−bn)−τj(n) +n(t) Ns 2 n=−∞ j=0

where α is attenuation amplitude in the transmission path and n(t) is additive white Gaussian noise (AWGN) with double sided spectrum of N0/2. wr(t)is the received signal waveform with unit energy corresponding to wt(t). τj (n)is transmission delay for the j-th pulse of the n-th bit.

2.2 Timing Jitter Jitter is simply the time diﬀerence between when a pre-deﬁned event should have occurred and when it actually did occur [7]. In a UWB correlator receiver, the template signal can be given by:

− ∞ Ns 1 v(t)= wr (t−nTf −jTh −τˆj(n))+n(t), (3) n=−∞ j=0 中国科技论文在线 http://www.paper.edu.cn 812 W. Zhang et al.

whereτ ˆj (n) is the estimate value of τj(n). The time oﬀset between these two delays j(n)=ˆτj(n) − τj(n) is called timing jitter. The exact value of j(n) can not be easily evaluated, whereas its distribution function can be evaluated. Timing jitter distribution characteristics models the timing jitter j(n)asa sequence of Gaussian random variables with zero means and standard deviation of σ . The probability density function (pdf) can be given by: 1 τ 2 f(σ, τ )=√ exp − . (4) 2πσ 2σ2

σ is also the average timing error of the jitters from the ideal template timing position, and it is the only parameter deﬁning the jitter distribution.

3 Templates with Jitter Suppression Abilities

3.1 Pulse Shape of Proposed Templates

We assume that the template is jitter free and all the jitters come only from the received waveforms in analysis and simulation of this paper. To improve the system perform, higher correlation coeﬃcient is required. Thereby a novel template is proposed to replace the original template waveform wr(t). We set

∞

wˆr(t)= f(σ, τ )wr (t − τ)dτ (5) −∞

as the novel template waveform. The correlation coeﬃcient between timing shifted received waveform and proposed template is given by:

∞ ∞ w (t − τ)ˆw (t)dt w (t − τ)ˆw (t)dt ˆ −∞ r r −∞ r r Rwj (τ)= ∞ ∞ = ∞ . (6) 2 − 2 2 −∞ wr (t τ)dt −∞ wˆr (t)dt −∞ wˆr (t)dt

The correlation between the timing shifted received waveform and jitter free template waveform is R(τ).Thus the integral correlation coeﬃcient between the jitter free template waveform and the jittered received waveform is given by:

∞ ∞ ∞ ∞

ρ= f(σ, τ )R(τ)dτ= f(σ, τ ) wr(t − τ)wr(t)dtdτ= wˆr(t)wr(t)dt. (7) −∞ −∞ −∞ −∞

The integral correlation coeﬃcient between the jitter free proposed template waveform and the jittered received waveform is given by:

∞ ∞ ˆ 2 ρˆ = f(σ, τ )R(τ)dτ = wˆr (t)dt. (8) −∞ −∞

Next we will compare the correlation coeﬃcients ρ andρ ˆ.Wehaveˆρ ≥ 0in(8). 中国科技论文在线 http://www.paper.edu.cn Jitter Distribution Evaluation and Suppression Method in UWB Systems 813

Fig. 1. General structure of proposed jitter suppression UWB correlator receiver

If ρ<0,ˆρ ≥ ρ deservedly. If ρ ≥ 0, we use the method below to compare two correlation coeﬃcients:

∞ 2 2 ρ −∞wˆr(t)wr(t)dt = ∞ . (9) 2 ρˆ −∞wˆr (t)dt Applying the Schwartz inequality to (9), we obtain:

2 ≤ 2 · 2 2 wˆr(t)wr(t)dt wˆr (t)dt wr (t)dt = wˆr (t)dt. (10)

So we haveρ ˆ ≥ ρ,consequently. The conclusion is that our proposed template waveform has higher correlation value with the jittered received signal waveform than the original template waveform in any jitter distribution schemes.

3.2 Receiver Structure and Novel Template Waveforms Functions The structure of the proposed receiver is given in Fig.1, where timing jitter distribution evaluator and an integrator used to realize our proposed template are added to the template waveform generator. If the jitter free pulse waveform is given by t 2 t wr(t)= 1 − 4π exp −2π , (11) τm τm

the proposed template waveforms can be deduced as: − 2 − 2 2 − 2 2πt wˆr(t)= f(σ, τ )wr (t τ)dτ = μ τm +4πσ 4πt exp 2 2 , (12) τm +4πσ where μ is attenuation parameters which can be used to adjust the amplitude of the template waveform. Fig.2(a) and Fig.2(b) give out the samples of our proposed 中国科技论文在线 http://www.paper.edu.cn 814 W. Zhang et al.

jitter robust template waveforms, where the original jitter free waveform is same as (11), and proposed waveforms corresponding to diﬀerent jitter distribution parameters (σ =0.1ns, σ =0.2ns and σ =0.5ns) are given, respectively.

Energy Normalized Proposed Template Waveforms for BPSK(τ =0.25ns) Energy Normalized Proposed Template Waveforms for PPM(τ =0.25ns) m m 1 1 Original w (t) Original w (t) rx rx σ Proposed Waveform with σ=0.025ns 0.8 Proposed Waveform with =0.025ns σ 0.8 Proposed Waveform with σ=0.05ns Proposed Waveform with =0.05ns σ Proposed Waveform with σ=0.1ns Proposed Waveform with =0.1ns 0.6

0.6 0.4

0.2 0.4

0 0.2 −0.2 Normalized Amplitude Normalized Amplitude 0 −0.4

−0.6 −0.2

−0.8

−0.4 −1 −5 −4 −3 −2 −1 0 1 2 3 4 5 −6 −4 −2 0 2 4 6 −10 −10 Time (s) x 10 Time (s) x 10 (a) BPSK system (b) PPM system

Fig. 2. Original template and the proposed jitter robust templates

4 Jitter Evaluation Methods

Timing jitter distribution evaluator is the key function module in proposed jitter suppression scheme. As jitter distribution is a Gaussian i.i.d function, the target of jitter distribution evaluation is to achieve the unique Gaussian distribution parameter σ. There are two diﬀerent schemes to evaluate this parameter σ: pilot impulse sequence (PIS) and waveform parameter evaluation (WPE):

4.1 Pilot Impulse Sequence (PIS) Method This method depends on such a scheme: the transmitter sends ﬁxed pilot sequence of impulse-like waveforms periodically, which is given by: − Nt 1 ST (t)= δ(t − nTd), (13) n=0

where Nt is the number of impulse-like waveforms, and Td is the ﬁxed time interval of the waveforms. Received impulses sequence is given by: − Nt 1 SR(t)=α δ(t − nTd − τ − (n)) + n(t), (14) n=0 where α is the amplitude weight of the channel and τ is the time delay which is introduced by channel, (n) is timing jitter. n(t) is Gaussian distributed noise.The receiver produces a sequence of same impulse-like waveforms with the same pe- riodic: − Nt 1 RR(t)= δ(t − nTd − τ). (15) n=0 中国科技论文在线 http://www.paper.edu.cn Jitter Distribution Evaluation and Suppression Method in UWB Systems 815

Fig. 3. PIS jitter distribution evaluator Fig. 4. WPE jitter distribution evaluator

Without considering the noise, the n-th timing jitter e(n) can be evaluated by comparing RR(t)andSR(t)inthen-th time slice: (n+1)Td+τ ∼ m(n, τd)= SR(n, t)RR(t + τd)dt = Rδ(τd − (n)) nTd+τ e(n) = arg max (m(n, τd)) . (16) τd

After achieving enough statistic of e(n), jitter distribution function can be achieved. The system block is given in Fig.3. The PIS method is suitable for systems with any distribution functions and parameters. The disadvantage rests with request for system clock with high definition and stability. The definition and stability of the system clock confirms the evaluation veracity.

4.2 Waveform Parameter Evaluation(WPE) Method We can design jitter resist template waveforms of different s. The template achieving the best BER is the optimal template. This method also requires a long WPE duration in which transmitter sends a series of waveforms with predefined information. If this predefined information is all ”1”, the transmitted waveform can be given by: − Nt 1 ST (t)= wt(t − nTd). (17) n=0

The receiver produces Ns diﬀerent template waveforms, each is designed with diﬀerent jitter resist parameter f(σj,τj) by the proposed method. The j-th template waveform is given by: ∞

wˆr(j, t)= f(σj ,τj)wr(t − τj )dτj . (18) −∞

σj increases from 0 to a certain value with a step of Δσj, a statistic mechanism will count the corresponding BER, the evaluated distribution parameter σe is:

σe =argmax(BER(σj)) . (19) τj 中国科技论文在线 http://www.paper.edu.cn 816 W. Zhang et al.

BPSK UWB BER vs. E /N b 0 BPSK UWB BER vs. Root Mean Square Jitter 0 10 0.5

0.45

0.4

0.35 −1 10

0.3 no jitter original template with σ = 0.025ns novel template with σ = 0.025ns 0.25 BER BER original template with σ = 0.05ns novel template with σ = 0.05ns 0.2 original template with σ = 0.1ns −2 10 novel template with σ = 0.1ns 0.15 original template with E /N = −5dB b 0 novel template with E /N = −5dB b 0 0.1 original template with E /N = 0dB b 0 novel template with E /N = 0dB b 0 0.05 original template with E /N = 5dB b 0 novel template with E /N = 5dB b 0 −3 10 0 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −10 E /N (dB) Root Mean Square Jitter /sigma(s) x 10 b 0

Fig. 5. BER vs. Eb/N0 of BPSK systems Fig. 6. BER vs. σ of BPSK systems

The system block figure is given in Fig. 4. WPE does not require a firm restric- tion on definition and stability of system clock. Also impulse-like waveforms are not imperative. However, it can only deal with a pre-known jitter distribution function, e.g. Gaussian distribution.

5 Computer Simulations

Based on the developed framework above, the system performance is evaluated and compared by computer simulations in this section. Our simulations are categorized into two systems, a BPSK system and an orthogonal PPM system. We utilize the waveform of (11) and setting the time duration of the waveform as 0.5ns, τm as 0.25ns and the nominal interval between two pulses Th as 2ns in both modulation schemes. To realize orthogonal PPM system time shift δ is set to equal to the pulse width 0.5ns. The pulses are assumed to be transmitted through an AWGN channel for simplification in our analysis and simulation. Multi-user and multi-path are not considered, either. The preliminary simulation results are presented in Fig.5 ∼ 8. Fig.5 and 6 are BER performance of BPSK systems. Fig.7 and 8 are of orthogonal PPM systems. At the same time, performances are compared by different Eb/N0 and different jitter distribution parameters. Fig.5 and 7 compare the BER vs. Eb/N0 performance of conventional and proposed systems in different jitter distribution scenarios. Fig.6 and 8 compare the characteristics of BER vs. RMSJ of conventional and proposed systems in scenarios of different Eb/N0. Firstly, all scenarios show that our proposed jitter robust template waveforms outperform the original waveforms. Secondly, as Eb/N0 increases (Fig.5 and 7), the BER reaches a static value; this is because that in the case of high Eb/N0, it is the jitter not noise degrades BER. Also we found that as the jitter distribution parameter increases (Fig.6 and 8), more gain can be achieved by proposed schemes. This means that proposed schemes can perform better than the conventional ones when the jitter is severe. 中国科技论文在线 http://www.paper.edu.cn Jitter Distribution Evaluation and Suppression Method in UWB Systems 817

PPM UWB BER vs. E /N b 0 PPM UWB BER vs. Root Mean Square Jitter 0 10 0.5

0.45

0.4

0.35 −1 10 no jitter 0.3 original template with σ = 0.025ns novel template with σ = 0.025ns original template with σ = 0.05ns 0.25 BER novel template with σ = 0.05ns BER original template with σ = 0.1ns 0.2 novel template with σ = 0.1ns −2 original template with E /N = −5dB 10 b 0 0.15 novel template with E /N = −5dB b 0 original template with E /N = 0dB b 0 0.1 novel template with E /N = 0dB b 0 original template with E /N = 5dB 0.05 b 0 novel template with E /N = 5dB b 0

−3 10 0 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 E /N (dB) Root Mean Square Jitter /sigma(s) −10 b 0 x 10

Fig. 7. BER vs. Eb/N0 of PPM systems Fig. 8. BER vs. σ of PPM systems

6 Conclusions

We analyzed the properties of timing jitter and oﬀer a novel jitter robust receiver template design method in UWB systems in this paper. Also we propose two jitter evaluation methods to perform proposed schemes. By analysis, we analyzed the performance of proposed waveforms and compared the proposed schemes with the conventional ones by computer simulation. Simulation results veriﬁed the proposed waveform with jitter suppression abilities outperform the conventional ones. Proposed template waveform design method can be used in other modulation schemes, for example pulse shape modulation (PSM) and is expected to achieve the same results.

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