Ranging Without Time Stamps ExchangingMohammad Reza Gholami, Satyam Dwivedi, Magnus Jansson, and Peter Handel

ACCESS Linnaeus Centre, Department of Signal Processing,

KTH–Royal Institute of Technology, Stockholm, Sweden

Abstract

We investigate the range estimate between two wireless nodes without time stamps exchanging.Considering practical aspects of oscillator clocks, we propose a new model for ranging in whichthe measurement errors include the sum of two distributions, namely, uniform and Gaussian. Wethen derive an approximate maximum likelihood estimator (AMLE), which poses a difficult globaloptimization problem. To avoid the difficulty in solving the complex AMLE, we propose a simpleestimator based on the method of moments. Numerical resultsshow a promising performance forthe proposed technique.

Introduction

•Wireless sensor networks: a significant growing technologyin different areas

•Ranging based on (two-way time-of-arrival) TW-TOA by exchanging the time stamps

•Ranging based on TW-TOA is severely affected by an imperfectclock

•Affine function to model the local clock, parameterized by clock offset and clockskew, the distance estimate between two nodes is mainly affected by an imperfectclock skew

• In this study: no continuous clock reading, no communication during ranging

System Model

•TW-TOA between a master node equipped with a time-to-digital convertor (TDC)and a slave node•The clock of the slave node:Cs(t) = θ0+wt, clock offsetθ0, clock skeww = 1+ρ•The relation between the clock skew and the oscillator frequency offset:fs = fo±∆ f +ξ (t),with the frequency of the slave nodefs, nominal frequencyfo,offset∆ f , perturbationξ (t)

⇒Ts =1

fo±∆ f+ξ (t) ≈ To(1∓ρ)+ζ (t), ρ , ∆ f/ fo, ζ (t) =−ξ (t)/ fo

•TW-TOA measurements:zk =dc +

T Ds2 +nk, k = 1, . . . ,N, nk ∼ N (0,σ2)

c the speed of propagation,d the Euclidian distance between two nodes.T D

s : processing delay in the slave nodenominal value for the delayT D

s = DT0 with D as an integer•The total delay:T D

s = wDTo+ εk with εk as a delay in detecting the signal presence.The arrived signal at the slave node may be detected afterεk.•A natural way to modelεk: εk ∼ U (0,wTo), for high SNRs, with high probabilitytime-of-arrival detection happens in the period that signal arrives.•The ranging model:zk =

dc +

wDTo2 + εk

2 +nk, k = 1, . . . ,N

Ranging Algorithms

1-Maximum Likelihood Estimator (MLE)The measurements vector:z , [z1, . . . ,zN]

T . The pdf ofz indexed:

pZ(z;θ) =N

∏k=1

∫ wT0

x=0pZk(zk|x, θ)pεk(x)dx =

N

∏k=1

∫ wT0

x=0

1√2πσwT0

exp

(−(αk− x/2)2

2σ2

)dx

with αk , zk −d/c−wDT0/2 andθ , [d,w,σ ]. No closed-form expression for thepZ(z;θ).Using

∫ x0 exp(−πt2)dt ≈ 1

2 tanh(

39x2 − 111

2 arctan(

35x111

)), an approximate MLE (AMLE):

maximizeσ ; w; d

−N(wT0)+N

∑k=1

log

[tanh

(39αk

2√

2πσ− 111

2arctan(

35αk

111√

2πσ

)

− tanh

(39βk

2√

2πσ− 111

2arctan(

35βk

111√

2πσ

)]

whereβk = αk−wT0/2•AMLE poses a difficult global optimization problem (Fig. 1)

40

45

50

55

60

0.9

1

1.1

1.2

−1500

−1000

−500

0

dw

AMLEcostfunction

(a)

40

45

50

55

60

2

4

6

8

x 10−9

−1200

−1000

−800

−600

−400

−200

0

dσ

AMLEcostfunction

(b)

0.9

1

1.1

1.2

2

3

4

5

6

7

x 10−9

−40

−35

−30

−25

−20

−15

wσ

AMLEcostfunction

(c)

Fig. 1. AMLE cost function for (a) d and w for fixed σ = 1/c, (b) d and σ for fixed w = 1.0001, and (c) w and σ forfixed d = 50.

We now obtain the AMLE expression by solving an opti- where . Hence, an estimate of the

2-A Low Complexity Estimator Based on Method of MomentConsider the relations between the unknown parameters and the following statistics:

µ1 , Ezk =dc+

(D2+

14

)wT0

µ2 , E(zk −µ1)2 = σ2+

(wT0)2

48

µ4 , E(zk −µ1)4 = 3σ4+σ2(wT0)

2

8+(wT0)

4

1280

The statisticsµ1, µ2, andµ4 can be approximated by the means of ensemble averaging

µ1 ≈∑N

k=1zk

N= S1, µ2 ≈

∑Nk=1(zk−S1)

2

N= S2, µ4 ≈

∑Nk=1(zk −S1)

4

N= S4.

From the statisticsµ2 andµ3

a(wTo)4 = µ4−3µ2

2

with a , 1/1280−3/482.An estimate of the clock skew

w =1To

(∣∣∣∣S4−3S2

2

a

∣∣∣∣)1/4

Alternatively, an estimate of the clock skew consideringwK = (1+ρ)K ≈ 1+Kρ ,

w = 1+aT 4

o −S4+3S22

4aT 4o

It can be shown that the estimator is asymptotically unbiased, i.e.,Ew ≈ 1+ aT 4

o −a(wT0)4

4aT 4o

≈ w, N ≫ 1Estimates of the distance and the varianceσ2:

d = c

(S1−

(D2+

14

)wT0

)

σ2 =

∣∣∣∣S2−(wT0)

2

48

∣∣∣∣

Numerical Results & Conclusions

•Simulation parameters: the distance between a master and a slave noded = 30 [m]f0 = 100 MHz,ρ = 0.0001 corresponding to 10 KHz frequency offset,D = 10•Comparison between the proposed technique, traditional approach, andcounter-based technique•TW-TOA measurements for 10 ms• 1000 realizations of noise

0 0.5 1 1.5 2 2.5 3 3.5

x 10−9

29.9

30

30.1

30.2

30.3

30.4

30.5

30.6

30.7

30.8

0.5 1 1.5 2

x 10−9

30.75

30.7505

30.751

30.7515

Standard deviation of noise, σ

Meanofthedistanceestimate[m]

Proposed

Traditional

Counter-based

(a)

0 0.5 1 1.5 2 2.5 3 3.5

x 10−9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Counter−based

Standard deviation of noise, σ

RMSE[m]

Proposed

Traditional

(b)

Fig. 2. Comparison between different approaches, (a) the

mean of distance estimate and (b) the RMSE of the esti-

mate.

•The proposed technique shows aconsiderable gain, especially forhigh SNRs

0 100 200 300 400 500

29.5

30

30.5

31

31.5

32

32.5

33

33.5

34

N

Ed[m]

Theoretical

Actual

Fig. 3. The mean of the distance estimate versus the num-

ber of samplesN .

•After a sufficient number ofsamples, e.g., 100 correspondingto 1 micro second, the estimate isvery close to the theoretical value

Contact Information:mohrg@kth.se