Research ArticleComparing Digital Phase-Locked Loop and Kalman Filter forClock Tracking in Ultrawideband Location System

Qian Gao,1,2 Chong Shen ,1,2 and Kun Zhang1,2,3

1State Key Laboratory of Marine Resources Utilization in South China Sea, Hainan University, Haikou, Hainan 570228, China2College of Information Science and Technology, Hainan University, Haikou, Hainan 570228, China3College of Ocean Information Engineering, Hainan Tropical Ocean University, Sanya, Hainan 572022, China

Correspondence should be addressed to Chong Shen; sc hainu@163.com

Received 5 January 2018; Accepted 8 March 2018; Published 18 April 2018

Academic Editor: Jose R. C. Piqueira

Copyright © 2018 QianGao et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For timing and synchronization system, digital phase-locked loop (DPLL) and Kalman filter all have been widely used as the clocktracking and clock correction schemes for the similar structure and properties. This paper compares the two schemes used forultrawideband (UWB) location system. The improved Kalman filter is more immune to interference.

1. Introduction

Impulse radio ultrawideband (IR-UWB) [1] is considered tobe promising for indoor location. To estimate the tags loca-tion using time difference of arrival- (TDOA-) based local-ization, the anchors’ local clocks are required to be fullysynchronized with each other [2], but the anchors’ clocksare varied with the running time and temperature drift [3].The anchors must be synchronized periodically [4]. Thelocation system is as Figure 1 shows. There are four anchors:anchor 1 is selected as the reference anchor and the otherthree anchors are the passive anchors. The reference anchorsends the clock synchronization packets to its passive anchor,which are represented by the orange lines in Figure 1. Theclock synchronization algorithm (Algorithm 1) that we usedis one-way message dissemination [5]. The clock variancebetween the passive anchor’s local clock and its referenceanchor is tracked. The data’s arrival time from tag to anchorsis corrected for the same time base between the reference andpassive anchors. Then the TDOA algorithm effectively getsthe tag’s location. So how to track the clock variance betweenreference-passive anchors is important for UWB location.

Traditionally, we model the clock time as a continuousfunction of clock skew (frequency difference) 𝛾 and the clockoffset (phase difference) 𝜃 [6].

𝐶𝑚 (𝑡) = 𝑡,

𝐶𝑠 (𝑡) = 𝛾 ⋅ 𝑡 + 𝜃,(1)

where 𝐶𝑚(𝑡) denotes a reference clock of the sending anchorand 𝐶𝑠(𝑡) denotes the local clock of the receiving anchor. Indigital clocks, time is recorded by counting the number ofperiods of a repeating clock signal. At each rising clock edgeof the periodic signal, an integer time counter is incremented.

The main problem of network synchronization is to re-solve the observed time in (1).The algorithms considered hereuse a one-waymessage dissemination approach at the level ofdiscrete clock ticks.

Suppose that the anchors all have the features of trans-mitting and receiving the clock check packets (CCP) withthe time stamps, and the initial master anchor transmits aCCP with period 𝑇, as shown in Figure 2. In the 𝑗th round ofbroadcast message, reference anchor broadcasts a synchroni-zation message CCP at 𝑇1,𝑗 and the passive anchor recordsits time 𝑇2,𝑗 at the reception of that message. Δ 𝑗 denotes theinterval between receiving a signal and the following initiallocal clock tick caused by the clock offset. According to [5],the timing model of the 𝑗th broadcast message is given by

𝑇2,𝑗 ≈ 𝛾 ⋅ 𝑇1,𝑗 + 𝜃 + 𝜓𝑗, (2)where 𝜓𝑗 is the random variable delay in the transmission.

HindawiJournal of Electrical and Computer EngineeringVolume 2018, Article ID 5873239, 5 pageshttps://doi.org/10.1155/2018/5873239

2 Journal of Electrical and Computer Engineering

Anchor 1 Anchor 2

Anchor 3

Tag

Anchor 4

Tag

Location engine

Ethernet/WLAN

Clock andsynchronization

Location broadcast

Figure 1: IR-UWB location system diagram.

Reference anchor

Passive anchor

T

T1,j−1 T1,j

Δ k−1 Δj

T2,jT2,j−1

Ts

Figure 2: Space-time of reference-passive anchors.

The clock tracking is implemented with the main “pro-cess” function taking two inputs: (1) The slave anchor CCPreceiving time with its time base. (2)Themaster anchor CCPtransmitting time with its time base and the CCP time offlight (TOF).

According to Figure 3, the clock tracking process usesCCP receiving time and CCP transmitting time and the bestestimated time between the master unit and the slave unit.If given the master and slave anchors’ (𝑋,𝑌, 𝑍) coordinates,the CCP TOF will be obtained by dividing the distance bythe speed. At last, clock tracking process draws the real rela-tive clock offset and the best estimated relative clock offsetbetween master and slave units. Digital phase-locked loop(DPLL) and Kalman filter both have been widely used as theclock tracking and clock correction schemes for the simi-lar structure and properties. This paper compares the twoschemes used for UWB location system.

2. Digital Phase-Locked Loop

Digital phase-locked loop (DPLL) is a digital closed-loopautomatic control system that can follow the frequency andphase of the input signals [7, 8]. For UWB location system,we consider a second-order DPLL based on ZC-DPLL, asFigure 4 shows.

Assume that 𝑠(𝑡) is the input signal, 𝑛(𝑡) is zero meanadditive white Gaussian noise, and𝑇0 is the input signal clockperiod without correction. The input signal 𝑠(𝑡) with 𝑛(𝑡)is sampled at 𝑡𝑘 by digital clock to output the loop phase

CCP reception time

CCP transmission time

CCP estimated time

Real reception clock offset

Estimated clock offsetZ−1

Z−1

Z−1

−−

−

−

−

Figure 3: The relative clock variation by clock tracking.

𝑃 = 𝐴 ∗ 𝑃 ∗ trans(𝐴) + 𝑄;𝐽 = 1/(𝑅 + 𝐻 ∗ 𝑃 ∗ trans(𝐻));OM = measuredError ∗ 𝐽 ∗measuredError;if (OM > threshold) && (counter > 20),outlier = 1;measuredError = 0.0;

outlier counter = outlier counter + 1;if outlier counter > 8,

outlier counter = 0;counter = 0;end𝑥0 = 𝑥 0 + 𝑑𝑡;EstimatedTime(𝑖) = 𝑥0;else𝐾 = 𝑃 ∗ trans(𝐻) ∗ inv(𝐻 ∗ 𝑃 ∗ trans(𝐻) + 𝑅);𝑥 = 𝑥 + 𝐾 ∗measuredError;

Algorithm 1

error 𝑧𝑘. Ignore the impact of the quantizer; the sequence {𝑧𝑘}directly comes into the digital filter; by smoothing, the digitalfilter outputs a more reliable correcting sequence {𝑦𝑘} to digi-tal clock: 𝑦𝑘 = 𝐷(𝑧)𝑧𝑘. The second-order 𝑧 operator functionis

𝐷(𝑧) = 𝐺1 + 𝐺2 (1 − 𝑧−1)−1

, (3)

where 𝐺1 and 𝐺2 are the loop gain factors. Assume that theloop gains of second-order DPLL are 𝐾0𝑓 and 𝐾1𝑓, respec-tively.

𝑦𝑘 = 𝐾0𝑓𝑧𝑘 + 𝐾1𝑓𝑘

∑𝑖=0

𝑧𝑖. (4)

In DPLL, correction signal 𝑦𝑘 is used to control the nextperiod:𝑇𝑘+1 = 𝑇0−𝑦𝑘. Adjust𝑇𝑘 until the loop into the lockedstate 𝑇𝑘 is the sample interval: 𝑇𝑘 = 𝑡𝑘 − 𝑡𝑘−1, 𝑘 = 1, 2, . . ..

The sampling time 𝑡𝑘 is deduced:

��𝑘+1 = ��𝑘 + 𝑇0 + 𝐾0𝑓𝑧𝑘 + 𝐾1𝑓𝑘

∑𝑖=0

𝑧𝑖. (5)

Journal of Electrical and Computer Engineering 3

Band-

VCO

s(t)

n(t)x(t) zk

T

G1

Kof

K1f

G2yk

−passfilter

Figure 4: DPLL structure block diagram.

3. Kalman Filter

Kalman filter is the solution by the minimum mean squareerror (MMSE) of the optimal linear filtering [9, 10]. Itestimates the current signal value according to the previousestimation and a recent observation data. In the concreteimplementation process, the (𝑘 + 1)th period clock skew andclock drift of the master-slave clock is estimated according tothe 𝑘th sync cycle information.𝑇 is the clock synchronizationperiod;𝑈𝜃,𝑘 and𝑈𝛾,𝑘 are the correction of the clock skew andclock offset at the 𝑘th clock period, respectively. 𝜃𝑘 and 𝛾𝑘are the 𝑘𝑇 clock skew and clock offset, respectively. At themoment of (𝑘 + 1)𝑇, the clock relations between the adjacentclock periods are

𝜃𝑘+1 = 𝜃𝑘 − 𝑈𝜃,𝑘 + (𝛾𝑘 − 𝑈𝛾,𝑘) 𝑇 + 𝜔𝜃,𝑘

𝛾𝑘+1 = 𝛾𝑘 − 𝑈𝛾,𝑘 + 𝜔𝛾,(6)

where 𝜔𝜃,𝑘 is the clock skew variance and 𝜔𝛾,𝑘 is the clockoffset variance. Assume that 𝜔𝑘 = [𝜔𝜃,𝑘 𝜔𝛾,𝑘]𝑇; its additivecovariance matrix is 𝑄. We define the vector and matrix asfollows:

𝑥𝑘 = [𝜃𝑘; 𝛾𝑘]𝑇 ,

𝑢𝑘 = [𝑈𝜃,𝑘; 𝑈𝛾,𝑘]𝑇.

(7)

Kalman filter equations by iteration are as follows.

(1) Estimation

𝑥𝑘+1|𝑘 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘, (8)

where 𝐴 = [ 1 𝑇0 1 ], 𝐵 = [ −1 −𝑇0 −1 ], 𝑥𝑘 is the state to be estimated,and 𝑢𝑘 is the input control vector.

(2) MMSE Matrix of the Estimation

𝑃𝑘+1|𝑘 = 𝐴𝑃𝑘𝐴𝑇 + 𝑄, (9)

where 𝑃𝑘 is the MMSE matrix of the estimated 𝑥𝑘.

(3) Kalman Filter Gain Matrix

𝐾𝑘+1

= 𝑃𝑘+1|𝑘 (𝐻𝑘+1)𝑇 (𝑅𝑘+1 + 𝐻𝑘+1𝑃𝑘+1|𝑘 (𝐻𝑘+1)

𝑇)−1

,(10)

xk

nk

xk|k−1

zk = xk − xk|k−1 + nk

T

T

T

Kok

K1k

K0kzk +k

∑i=0

K1izi

Figure 5: Kalman filter structure block diagram.

where 𝑅𝑘+1 is the covariance matrix of the observation noiseand the measurement matrix𝐻𝑘+1 is a unit matrix.

(4) Correction

𝑥𝑘+1 = 𝑥𝑘+1|𝑘 + 𝐾𝑘+1 (𝑧𝑘+1 − 𝐻𝑘+1𝑥𝑘+1|𝑘) . (11)

(5) MMSE Matrix

𝑃𝑘+1 = (1 − 𝐾𝑘+1) 𝑃𝑘+1|𝑘. (12)

After Kalman filtering, the correction is 𝑥𝑘+1 = [𝜃𝑘+1;𝛾𝑘+1]𝑇 at the (𝑘+1)th clock period. 𝑢𝑘+1 = 𝑥𝑘+1 is set to make

up for the clock skew and clock offset. So the slave anchor’sclock base will make up to the same clock base when the tag’sdata arrives.

Relative to DPLL, we define 𝑋𝑘 = [𝑥𝑘; 𝑥𝑘] and definethe Kalman gain vector as 𝐾𝑘 = [𝐾0𝑘; 𝐾1𝑘]. 𝑃0 =[𝑇20 /12 0; 0 𝑇20𝑓

2Δ𝑘], where 𝑇20 /12 is the variance of the

initial phase 𝑋0; 𝐸[(𝑋0)2] = 𝑇20𝑓2Δ𝑘; 𝑓2Δ𝑘 = 𝐸[(𝑓Δ/𝑓0)2]

is the normalized variance values of 𝑓Δ with zero averagedistribution.The error is 𝑧𝑘 = 𝑥𝑘−𝑥𝑘|𝑘−1+𝑛𝑘 = 𝑦𝑘−𝐻𝑋𝑘|𝑘−1.

According to (4)–(7), as 𝑘 = 0, 1, 2, . . ., we will get

𝑥𝑘+1 = 𝑥𝑘 + 𝐾0𝑘𝑧𝑘𝑘

∑𝑖=0

𝐾1𝑖𝑧𝑖. (13)

Comparing (5) and (13), the recursive types are verysimilar. The Kalman filter structure is depicted in Figure 5.

4. Comparison and Analysis

According to the above descriptions about Kalman filter andDPLL, we compare the two schemes for UWB indoor loca-tion. In DPLL, correction sequences as the output of thesignal through digital filtering control the digital clock perioduntil the loop is locked. Kalman filter also abstracts theneeded signal through the feedback loop, which uses theformer data to estimate the current data.The two schemes usethe error 𝑧𝑘 through gain factor𝐾 to find the optimal estima-tion. By comparing (5) and (13), we just need to adjust gainfactor𝐾 so as to get the similar results.

We define the passive anchor clock variance error be-tween the real clock variance and the optimal estimated clock

4 Journal of Electrical and Computer Engineering

variance as 𝑒𝑘 = 𝑥𝑘 − 𝑥𝑘|𝑘−1. Using the reference-passiveanchors data backhaul sending time, data receiving time, andthe optimal estimated time with the data flight time, the loca-tion engine in the server will calculate the relative clock offsetvariance.

The paper uses Matlab for simulation. Assume that theanchors’ coordinates are anchor1 (1.1, 1.17, 1.93) and anchor2(11.3, 1.17, 1.21). The TOF of the reference anchor1 to thepassive anchor2 is TOF1,2 = 0.000000034123193 s. Forsecond-order DPLL, variable loop gains with lower bounds𝐾0𝑓 = 0.2, 𝐾1𝑓 = 0.05, 𝐸𝑏/𝑁0 = 10.6 dB, 𝜎2𝑛/𝑇

20 = ��2𝑛/𝑇

20 =

0.001, and 𝑓Δ/𝑓0 = 0.1. The clock synchronization periodis 150ms. The measurement noise variance is 3𝑒 − 12; theprocess noise variance is 5𝑒 − 12. Alternatively, a more com-plex multistage in-lock and out-of-lock detection algorithmmay be employed, which trades off acquisition, tracking,and false lock performance according to the system require-ments. Such tradeoff issues are beyond the scope of this paper[11, 12].

As Figure 6 shows, the green line is the clock variancedifference by Kalman filter and the black line is that by DPLL.They all tend to be stable over time, but the properties ofKalman filter are significantly better than those of DPLL.Kalman filter requires shorter capture time and smaller error.

By the theory of hypothesis, in order to give sufficientinformation for the DPLL to stay locked for continuedreal-time location system operation with good performance(including coping with a certain packet error/loss rate),we need to send the clock synchronization message morefrequently than forKalmanfilter. It reduces the air-occupancyneeded for clock synchronization messages, which allowsmore air-time for receiving blink messages. This essentiallyincreases the system tag capacity, especially in the lower datarate and longer preamble modes.

5. Improved Algorithms on Kalman Filter

As stated above, Kalman filter is better for clock synchroniza-tion indoor UWB location system. Its calculation is based onsuch an assumption: all measurements are composed of thereal signal and additive Gaussian noise. If these assumptionsare correct, Kalman filter will effectively get signal from themeasurements containing noise. But if the reference anchor’sclock check packets collide with tag’s data packets with TOAor some other mistake challenges in [13], the assumptionsare incorrect. Kalman filter will treat the collision or mis-take as credible clock variance data, and it calculates by thesedata. And Kalman filter itself is a kind of low-pass filter;its response and correcting speed are slower. Therefore, theerrors generated by the collision will for a long time seriouslydegrade the performance of the clock synchronization algo-rithm.

This paper proposes a method of monitoring and avoid-ing the wrong of collisions. Kalman filter gain is as (10) shows,defining an information matrix 𝐽 as

𝐽 = (𝑅𝑘+1 + 𝐻𝑘+1𝑃𝑘+1|𝑘 (𝐻𝑘+1)𝑇)−1

. (14)

𝐽 is used to represent the difference between estimated clockerror and actual clock error. This information will be used

Kalman filterDPLL

100 200 300 400 5000

Time (s)

−10

−8

−6

−4

−2

0

2

4

6

8

10

Erro

r (ns

)

Figure 6: The difference between estimated time and real time.

300 400 500100 2000

Time (s)

−60

−40

−20

0

20

40

60Er

ror (

ns)

Real clock offsetEstimated clock offset

Figure 7: The relative clock offset with big disturbance by Kalmanfilter.

to prompt how well the current input fits the current state offilter.

OM𝑘+1 = (𝑥𝑘+1 − 𝑥𝑘+1) ∗ 𝐽 ∗ (𝑥𝑘+1 − 𝑥𝑘+1) . (15)

If the OM (outlier metric) rises above a preset thresholdwhich is an empirical value, the current input is untrusted.The improved Kalman filter does not update current state butdiscards this data directly to avoid error packet having a bigimpact for filter output.

In Figures 7 and 8, the blue lines are the real clock offsetsand the red lines are the estimated clock offset by Kalmanfilter. We set a big data mistake at 150 s; the estimated clockoffset is unable to keep pace with the real clock offset andup and down shocks with Kalman filter in Figure 7. With the

Journal of Electrical and Computer Engineering 5

Real clock offsetEstimated clock offset

100 200 300 400 5000

Time (s)

−60

−40

−20

0

20

40

60

Erro

r (ns

)

Figure 8:The relative clock offset with big disturbance by improvedKalman filter.

resolution of the trustless input, the estimated clock offset issmooth in Figure 8. By comparing Figures 7 and 8, it is clearlyseen that the improved Kalman filter enhances the capacity ofresisting disturbance.

6. Conclusion

We have compared DPLL and Kalman filter for UWB indoorlocation network clock synchronization, and the analysisresults show that Kalman filter copes better with clock errorsand has better lock performance. And the improved Kalmanfilter is more immune to interference as the simulation resultsshow.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by Major Research and De-velopment Plan of Hainan Province (ZDYF2016002), theNational Natural Science Foundation of China (61461017),Hainan Province Natural Science Foundation of InnovationTeam Project (2017CXTD004), and Innovative ResearchProject of Postgraduates in Hainan Province (Hyb2017-04).

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[4] D. Zachariah, S. Dwivedi, P. Handel, and P. Stoica, “Scalable andPassive Wireless Network Clock Synchronization in LOS Envi-ronments,” IEEE Transactions onWireless Communications, vol.16, no. 6, pp. 3536–3546, 2017.

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