1. Ranging Without Time Stamps Exchanging
Mohammad Reza Gholami, Satyam Dwivedi, Magnus Jansson, and Peter H¨andel
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 which
the measurement errors include the sum of two distributions, namely, uniform and Gaussian. We
then derive an approximate maximum likelihood estimator (AMLE), which poses a difficult global
optimization problem. To avoid the difficulty in solving the complex AMLE, we propose a simple
estimator based on the method of moments. Numerical results show a promising performance for
the proposed technique.
Introduction
• Wireless sensor networks: a significant growing technology in 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 imperfect clock
• Affine function to model the local clock, parameterized by clock offset and clock
skew, the distance estimate between two nodes is mainly affected by an imperfect
clock 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 offset θ0, clock skew w = 1+ρ
• The relation between the clock skew and the oscillator frequency offset:
fs = fo ±∆ f +ξ(t),with the frequency of the slave node fs, nominal frequency fo,
offset ∆ f, perturbation ξ(t)
⇒Ts = 1
fo±∆f+ξ(t) ≈ To(1∓ρ)+ζ(t), ρ ∆ f/fo, ζ(t) = −ξ(t)/fo
• TW-TOA measurements: zk = d
c + TD
s
2 +nk, k = 1,...,N, nk ∼ N (0,σ2
)
c the speed of propagation, d the Euclidian distance between two nodes.
TD
s : processing delay in the slave node
nominal value for the delay TD
s = DT0 with D as an integer
• The total delay: TD
s = wDTo +εk with εk as a delay in detecting the signal presence.
The arrived signal at the slave node may be detected after εk.
• A natural way to model εk: εk ∼ U (0,wTo), for high SNRs, with high probability
time-of-arrival detection happens in the period that signal arrives.
• The ranging model: zk = d
c + wDTo
2 + εk
2 +nk, k = 1,...,N
Ranging Algorithms
1-Maximum Likelihood Estimator (MLE)
The measurements vector: z [z1,...,zN]T
. The pdf of z indexed:
pZ(z;θ) =
N
∏
k=1
wT0
x=0
pZk
(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 and θ [d,w,σ]. No closed-form expression for the
pZ(z;θ).
Using x
0 exp(−πt2
)dt ≈ 1
2 tanh 39x
2 − 111
2 arctan 35x
111 , an approximate MLE (AMLE):
maximize
σ; w; d
−N(wT0)+
N
∑
k=1
log tanh
39αk
2
√
2πσ
−
111
2
arctan(
35αk
111
√
2πσ
−tanh
39βk
2
√
2πσ
−
111
2
arctan(
35βk
111
√
2πσ
where βk = 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
d
w
AMLEcostfunction
(a)
40
45
50
55
60
2
4
6
8
x 10
−9
−1200
−1000
−800
−600
−400
−200
0
d
σ
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 for
fixed 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 Moment
Consider the relations between the unknown parameters and the following statistics:
µ1 Ezk =
d
c
+
D
2
+
1
4
wT0
µ2 E(zk − µ1)2
= σ2
+
(wT0)2
48
µ4 E(zk − µ1)4
= 3σ4
+σ2(wT0)2
8
+
(wT0)4
1280
The statistics µ1, µ2, and µ4 can be approximated by the means of ensemble averaging
µ1 ≈
∑N
k=1 zk
N
= S1, µ2 ≈
∑N
k=1(zk −S1)2
N
= S2, µ4 ≈
∑N
k=1(zk −S1)4
N
= S4.
From the statistics µ2 and µ3
a(wTo)4
= µ4 −3µ2
2
with a 1/1280−3/482
.
An estimate of the clock skew
w =
1
To
S4 −3S2
2
a
1/4
Alternatively, an estimate of the clock skew considering wK
= (1+ρ)K
≈ 1+Kρ,
w = 1+
aT4
o −S4 +3S2
2
4aT4
o
It can be shown that the estimator is asymptotically unbiased, i.e.,
E ˜w ≈ 1+ aT4
o −a(wT0)4
4aT4
o
≈ w, N ≫ 1
Estimates of the distance and the variance σ2
:
d = c S1 −
D
2
+
1
4
wT0
σ2
= S2 −
(wT0)2
48
Numerical Results & Conclusions
• Simulation parameters: the distance between a master and a slave node d = 30 [m]
f0 = 100 MHz, ρ = 0.0001 corresponding to 10 KHz frequency offset, D = 10
• Comparison between the proposed technique, traditional approach, and
counter-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 a
considerable gain, especially for
high 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 samples N.
• After a sufficient number of
samples, e.g., 100 corresponding
to 1 micro second, the estimate is
very close to the theoretical value
Contact Information:
mohrg@kth.se