Slide set presented for the Wireless Communication module at Jacobs University Bremen, Fall 2015. Teacher: Dr. Stefano Severi, assistant: Andrei Stoica

1. Wireless Localization: Ranging
Stefano Severi and Giuseppe Abreu
s.severi@jacobs-university.de
School of Engineering & Science - Jacobs University Bremen
October 7, 2015

2. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Questions and Help
First Aid Kit
For the matlab part:
Andrei Stoica
Room 100b, Res I
email: r.stoica
tel. 3203
Book appointment in
advance
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 2/34

3. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Calendar & Deadlines
Assignments and Lab Reports
All lectures will be given in Seminar Room in Research I from
13:30 to 16:30. Modification of this schedule can be anyway
agreed with the students.
Wed 30 Sep Lecture I
Wed 7 Oct 13:00 Deadline Report I
Wed 7 Oct Lecture II
Wed 7 Oct Lecture III
Wed 14 Oct 13:00 Deadline Report II
Wed 21 Oct 13:00 Deadline Report III
The date for final exam has yet to be defined - please refer to
Prof. Henkel.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 3/34

4. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Lab Report Submission
How and when
The lab reports must be submitted individually.
The lab reports must be submitted only electronically via
email to both Dr. Severi (s.severi@jacobs-university.de)
and Mr. Stoica (r.stoica@jacobs-university.de) by the
previously depicted deadlines.
The 3 reports will constitute the 12,5% of the final grade.
The grade of each report will be therefore multiplied by
0,0416 to compute the final grade.
Failing to submit a report within the deadline will result in a
grade 5.
Do not wait for the last day before the deadline to ask for an
appointment and/or clarification!
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 4/34

5. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
The Network
Anchor and Target Nodes
anchor nodes
ΘA {θ1, · · ·, θA}.
target nodes
ΘT {θA+1, · · ·, θN }.
[ηxN] matrix network
Θ [ΘAΘT ].
θ1
θ2
θ3
θ4
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 5/34

6. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Distances and Measurements
The Ranging Error
True distance
dij (θx:i − θx:j)2 + (θy:i − θy:j)2 = (θi − θj), (θi − θj) = θi−θj
* The red part is valid only for η = 2, i.e. bidimensional case.
eij ranging error
˜dij dij + eij measured
distance
D euclidean distances
matrix [NxN].
θi θj
dij
The symbol ˆ denotes estimated quantities and ˜ denotes
measured quantities.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 6/34

7. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
LOS and NLOS Conditions
An Example
Source: Dardari et al., Ranging With Ultrawide Bandwidth Signals in Multipath Environments
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 7/34

8. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
RSSI Ranging
Power-based Ranging
Received Signal Strength
Pr ∝ Pt − 10 γ log10(d) + S
S large-scale fading variation typically N(0, σ2
S)
d distance
γ path-loss factor (typically between 2 and 6)
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 8/34

9. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
RSSI Ranging
GSM example
GSM Line-of-Sight link budget
Pr(dB) = Pt(dB) − 20 log10(d) − 20 log10(f) − 20 log10(4π
c )
100 200 300 400 500 600 700 800 900 1000
−90
−80
−70
−60
−50
−40
Line-of-Sight Link Budget
Transmitted Power = 1W, frequency = 900 Mhz
ReceivedPower[dB]
Distance [m]
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 9/34

10. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
RSSI Ranging
Pros and Cons
Pros
¨ No need for synchronisation.
¨ No expensive hardware needed.
Cons
¨ Severely affected by multipath even in LOS.
¨ Subject to errors in NLOS environments.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 10/34

11. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Time-based Ranging
Time-of-Flight
Distance estimated from Time-of-Flight
τf d/c,
where c = 299792458 m/s
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 11/34

12. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Time-based Ranging
Clock Errors
Source: Verdone et al., Wireless Sensor and Actuator Networks: Technologies, Analysis and Design.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 12/34

13. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
ToA Ranging
One-Way
τf = t2 − t1.
time according
to node A
time according
to node B
t1
t2
τf
Effects of synchronisation error [clock offset] could be catastrophic!
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 13/34

14. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
ToA Ranging
Effect of Synchronization Error
If we consider a clock offset ετ between the two clocks, we
have:
˜d = (τf + ετ ) · c,
that can be rewritten as:
˜d = τf · c
d
+ ετ · c
εd
=
Now let’s estimate εd is ετ is . . .
1 ms −→ ετ · c = 0, 001 · 299792458 = 299,79 km,
1 µs −→ 299,79 m,
1 ns −→ 29,98 cm.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 14/34

15. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
ToA Ranging
Two-Way
round-trip time τRT = 2 τf + τd.
time according
to node A
time according
to node B
t1
t2
τf
t3
t4
τf
τd
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 15/34

16. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
ToA Ranging
Two-Way
τf =
(τRT − τd)
2
Effect of synchronisation (offset) error mitigated
τd is assumed known a-priori
τd ∼ ms
τf ∼ ns
a 0, 00001% error (one over one million!) on the holding time τd could
lead to catastrophic error in distance estimate.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 16/34

17. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
ToA Ranging
Differential Time of Arrival
The two-way procedure is repeated, but holding time at node B is first τdB
and then 2τdB .
t1
t2
τf
t3
t4
τf
τdB
t1
t2
τf
2τdB
τf
t3
t4
τf = t4 − t1 − (t4 − t1)/2.
Clock drift can be considered negligible within the interval [t1, t4], although
clock jitter could still affects the ranging.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 17/34

18. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Error on ToA
Gaussian Distribution
Even ignoring errors introduced by clocks, time-based distance estimation
is still subject to the imperfect detection of time or arrival of transmitted
packet in presence of noisy channel and multipath propagation.
Under this perspective, we can model the time-based estimation as a
process following a normal distribution, with mean d and variance
proportional to the inverse of the SNR γ, whose PDF is:
fn( ˆd; d, γ) =
1
2π(kγ)−2
exp −
ˆd − d
2(kγ)−2
, (1)
where k is a proportionality constant to model the variance.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 18/34

19. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Difference Ranging
Basic Principle
x(t) = A0 cos (2πf0t + ϕA).
y(t) = B0 cos (2πf0t + ϕB).
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 19/34

20. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Difference Ranging
Measurement Cycle
ϕ1 = ϕB − ϕA.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 20/34

21. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Difference Ranging
First Measurement Cycle
Relationship between phase and distance:
ϕ1 = 2π
2d
λ1
− N1 = 2π
2f1d
c
− N1 ,
λ1 =
c
f1
,
N1 number of integer part of wavelength,
N1 =
2d
λ1
.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 21/34

22. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Difference Ranging
Second Measurement Cycle
f2 = f1 + ∆f,
ϕ2 = 2π
2d
λ2
− N2 = 2π
2f2d
c
− N2 ,
for N1 = N2 we have:
∆ϕ = ϕ2 − ϕ1 =
4πd∆f
c
,
that leads to
d =
c
4π
∆ϕ
∆f
.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 22/34

23. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Difference Ranging
Pros and Cons
Pros
¨ Very robust to NLOS environment.
¨ No need for robust synchronization.
Cons
¨ Maximum ranging dMAX =
c
2∆f
.
¨ Phase estimation subject to Tikhonov error.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 23/34

24. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Error on Phase Measurement
Tikhonov Distribution
For medium and low SNR we model the estimation of a true phase ϕ as
follows:
ˆϕ = ϕ + nT, (2)
where nT is the estimation error and consequently the ˆϕ’s are
Tikhonov-distributed [ABREU08] random variables with mean ϕ and their
PDF is given by:
ft( ˆϕ; ϕ, γ) =
exp(γ cos( ˆϕ − ϕ))
2πI0(γ)
. (3)
where γ expresses the SNR of the system and Ij(·) is the j-th order
Besseli function.
[ABREU08] G. T. F. de Abreu, “On the generation of Tikhonov variates,” Communication, IEEE Transactions on,
vol. 56, no. 7, pp. 1157-1168, July 2008.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 24/34

25. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Error on Phase Measurement
Besseli Function of j-th order
The j-th order Besseli function is:
Ij(γ) =
∞
m=0
(−1)m
m! Γ(m + j + 1)
γ
2
2m+j
, (4)
where Γ(·) is the Gamma function defined as:
Γ(n) = (n − 1)! (5)
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 25/34

26. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Error on Phase Difference
The estimation error on the difference of two phases ϕ1 and ϕ2, both
transmitted over sinusoidal tones with SNR γ, is still characterized by a
Tikhonov distribution:
ft(∆ϕ; ∆ϕ, γ/2), (6)
where ∆ϕ = ϕ2 − ϕ1 and the reference SNR is γ/2.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 26/34

27. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
On-Field Experience
Observe, ask and learn
Mr. Stoica will guide you through an on-field ranging campaign and
classification
Lab Tips
To best exploit this experience:
Pay carefully attention to Mr. Stoica explanation;
Make questions about any unclear item;
Repeat what you have learn on your own computer: it will be later useful
to write down the report.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 27/34

28. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Based Ranging
On-Field Measurements
1 collect the phase measurements from sensors
2 import the data in Matlab
3 estimate the measured distances
4 compute the error distribution
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 28/34

29. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase-Based Ranging
On-Field Measurements
Data from measurements come according to the following format:




ϕA1f1 . . . ϕAN f1 | ϕB1f1 . . . ϕBN f1
ϕA1f2 . . . ϕAN f2 | ϕB1f2 . . . ϕBN f2
. . .
ϕA1fM
. . . ϕAN fM
| ϕB1fM
. . . ϕBN fM



 (7)
where M is the number of frequencies used and N is the number of
samples per each frequencies.
In the left side of the matrix are store phases sent by the initiator, on the
right side the received phases.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 29/34

30. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase Difference vs Frequency
On-Field Measurements
Having in mind that
d =
c
4π
∆ϕ
∆f
.
we focus on the radio ∆ϕ/∆f.
1 plot the measured phase differences as a function of the frequencies
Matlab Tip
In order to linearly plot the phase differences as function of the
frequencies, use the following command:
unwrap(angle(exp(1j*phi)))
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 30/34

31. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Phase Difference vs Frequency
On-Field Measurements
Due to measurements errors, the relationship between phase differences
and frequencies is not linear
1 linearize that relationship with linear regression (best fitting line)
2 get both graphically and analytically the ratio ∆ϕ/∆f
3 computed the estimated distances
Matlab Tip
For linear regression an useful command is:
p = polyfit(X,Y,N)
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 31/34

32. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Error Distribution
On-Field Measurements
Compute the error phase distribution, i.e. the distribution of the difference
between the measured phase differences and the estimated phase
differences, at each frequency
1 plot the distribution of the phase estimation error for a given frequency
using histogram
2 find the parameter of the Tikhonov distribution that best fits the
distribution of phase estimation error
3 plot the Tikhonov distribution on top of the histogram
Matlab Tip
For the histogram use
[p,r] = ecdf(data); ecdfhist(p,r,nBins);
whiles for the Tikhonov distribution you must implement eq. (3).
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 32/34

33. Localization &
Positioning
Preliminaries
Ranging
RSSI
Time Based
One-Way ToA
Two-Way ToA
DToA
Phase-Difference Ranging
Error Estimation
Experience
Report 1/3
Ranging
Complete the lab experience writing (one per group) a report
with:
1 the plots described in the previous slide (only for one frequency)
2 the description of the selected parameter of the best fitting Tikhonov
distribution (γ and θ).
3 a clear explanation of the whole experience.
Please print and deliver the report within the aforementioned deadline to
s.severi@jacobs-university.de,
r.stoica@jacobs-university.de.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 33/34

34. Thank you!