1. Positioning in Wireless Networks
- Non-cooperative and Cooperative Algorithms -
Giuseppe Destino
Centre for Wireless Communications
University of Oulu
October 12, 2012
1
10. Network Planning and Expansion
• Location-based RSS map
• Use mobile nodes for monitoring
• Adaptive network expansion
10
11. Cognitive Radio in the TV-White Space
• Location-based primary user database
• Allocate free TV-white space based on location information (FCC’ 10)
• 50 m, minimum distance between primary and secondary users
11
13. System Architecture
System Centric Node Centric
• Measurements convey to the • Measurements convey to the
radio access network mobile nodes
• Centralized calculations • Local calculations
13
14. Network
Nodes
• NA anchors, known fixed location A4 A3
• NT targets, unknown location
Topology
X
• Star-like, non-cooperative scheme
• Mesh, cooperative scheme
Syncronization
A1 A2
• Global, synchronous
• Local, asynchronous
14
15. Internode Interaction
- Measurement system -
Power Profile
• Channel Impulse Response (CIR), wideband signal
• Power Delay Profile (PDP), wideband signal
Angle
• Angle-of-Arrival (AoA), multiple-antenna
Distance (Ranging)
• Received Signal Strength Index (RSSI), always available
• Time-of-Arrival (ToA), technology dependent, asynchronous network
• Time-Difference-of-Arrival (TDoA), technology dependent, synchronous
network
15
16. Source of Errors
- Example of an indoor propagation channel -
S-V Indoor Propagation Model
III-A we will see, from another point
5: 300 ns.
Room 1 Room 3
Rate, X
individual rays in about 200 power
ts similar to those in Figs.3 and 4, we
in the range of 5-10 ns. The range
rom the• Noise our ray-resolving al-
fact that
with our • Multipaths sensitivity, is
measurements Room 2 Room 4
ny weak rays, in particular, those fall-
• Blockage
. The higher the sensitivity, the more
ld find, andMobilitythe larger the value
• hence,
ime, theprobability distribution of the Fig. 8. Four spatially averaged power profiles within various rooms.
ould be increased for small values of dashed lines correspond to exponential power decay profile of the
and the clusters.
opriate choiceof X is strongly coupled
istribution of the P ’ s . We find that.a
1/ X = 5 ns‘coupled with Rayleigh- rooms, we find that, on the average, rays within a clu
16
17. The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Ra
Ranging with Bluethooth
- Measurement result with AP Class 1 and MT Class 2 -
(a) (b)
• Fig. 3
Connection-based RSSI (dB)
10 260 This i
Link Quality (8-bit quantity)
Distance vs. RSSI Distance vs. LQ
5
250
which
240
0 230
power
220 maine
-5
210 for the
-10 200
2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18
Distance (meter) Distance (meter)
• From
much
Inquiry-based RX power level (dBm)
(c) (d)
Transmit Power Level (dBm)
20
15 Distance vs. TPL
-40
-45 Distance vs. RX power level ings o
10 -50 tion.
5 -55
0 -60 rather
-5
-10
-65
-70
at our
-15 -75 which
-20 -80
2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 Class
Distance (meter) Distance (meter)
the A
[Hossain] A. Hossain and W.-S. Soh, A comprehensive study of bluetooth signal parameters for our m
Figure 3: Relationship between various Bluetooth signal pa-
localization’, in Proc. IEEE 18th International Symposium on Personal, Indoor and Mobile
Radio Communications, pp. 1-5, September 2007
rameters & distance. • Our B
17
18. RSSI Ranging with Wi-Fi
826 IEEE JOURNAL OF SELECTED TO
- Cardbus Wi-Fi, corridor environment -
and therefore
where the valu
and describes
ence of averag
In the same
distance const
[Mazuelas] S. Mazuelas, A.4. RelationLorenzo, P. Fernandez, RSSI in aE. Garcia, J. Blas, and E. Abril,
Fig. Bahillo, R. between distance and F. Lago, corridor.
Robust indoor positioning provided by real-time RSSI values in unmodified WLAN networks,
Therefore,
IEEE Journal of Selected Topics in Signal Processing, vol. 3, pp. 821 - 831, October 2009. a fe
Thus, we can impose certain constraints to the distance esti- 18
19. Frequency Diversity of RSSI
- TelosB Platform, IEEE 802.15.4 Compliant, d = 2m -
−55
−60
−65
RSS (dBm)
−70
−75
−80
8 9 0 2 4 6 8 10 12 14 16 18
Channel
Fig.
[Zhang] D. Zhang, Y. Liu, X. Guo, M. Gao, and L. Ni, On distinguishing the multiple radio paths in
RSS-based ranging, in Proc. IEEE INFOCOM 2012, pp. 2201 - 2209, March 2012. path
ent environ- Fig. 2. RSS measurement in different channels: λ1 ,
node distance=2m ceiv
19
20. Ranging based on Time Measurements
- Time-of-Arrival and Time-Difference-of-Arrival -
ToA TDoA
2 1 2 1
(TT x − TT x ) − (TRx − TRx )
d=c δ = c(TRx1 − TRx2 )
2
• Asynchronous method • Asynchronous Tx-Rx
• Two-way-communication • Synchronous Rx-Rx
• One-way-communication
20
21. Ranging Error
6 - IR UWBEURASIP Journal - Wireless Communications and Networking
Technology on
Line-of-Sight (LOS)
100 80
B = 0.5 B=1 B=2 B=4 B=6
90
Average range error (cm)
80
70
70
Path loss (dB)
60
50 60
40
30
20 50
10
0 40
0 20 40 0 20 40 0 20 40 0 20 40 0 20 40
8 EURASIP Journal on Wireless Communications and Networking
Ground-truth range (m)
Non-Line-of-Sight (NLOS)
(a) NIST North, LOS, fc = 5 GHz
400 130
B = 0.5 B=1 B=2 B=4 B=6
350
Average range error (cm)
300 110
100
Path loss Path loss (dB)
80
250
90 B = 0.5 B=1 B=2 B=4 B=6
Average range error (cm)
200
80 90
70
70
(dB)
150
60
100 70
50 60
50
40
30
0 50
200 20 40 0 20 40 0 20 40 0 20 40 0 20 40 50
10 Ground-truth range (m)
[Gentile] C. Gentile, and A. Kik, “A Comprehensive Evaluation of Indoor Ranging Using
0 40
Ultra-Wideband Technology”, NIST North, NLOS, fc =20 GHz 0
0 20 40 0 (a) 40 0 5 40
20 EURASIP Journal on Wireless Communications and
20 40 0 20 40
Networking, vol. 2007, pages 10.
Ground-truth range (m)
(b) Child Care, LOS, fc = 5 GHz
21
23. Positioning via Connectivity
- Proximity Positioning -
System model
• Single target A4 A3
• Connectivity/proximity
information
Position estimation
X
• Logic intersection (1,4) (1,2,3,4)
• Centroid:
NA
wi a i (1,2,4)
i=1
ˆ
z= NA
wi A1 A2
i=1
wi ∝ 1/di .
23
24. Finger-Printing
- Signal Space based Positioning -
System model
• Single target
• Measurement phase
• Real-time localization A A
1 4 3
Position estimation 0.8
• Fingerprint: fpq = (ˆp , φq , f (rx ))
i
z i
0.6
X
• Database: Ω {fpq }, |Ω| = P QNA
i
0.4
• Real-time measurement: 0.2
i AN
˜
s {f (rx )}i=1
• Search method, e.g. Nearest neighbor 0
A1 A2
0.2
NA 0.2 0 0.2 0.4 0.6 0.8 1 1.2
ˆ
z = arg min (˜pq − fpq )2
fi i
i
fpq ∈Ω
i=1
s.t. ˜pq = (ˆp , φq , f (rx ))
fi z i
24
25. Triangulation
- Angle-based and Positioning -
X
System model
• Single target
• AoA measurements
A1 H A2
Position estimation
−1
˜
pML (θ) = pR + GT Σ−1 Gθ
ˆ ˜
Gθ Σ−1 θ − θ R )
θ θ θ
− sin(θR1 )/dR1 cos(θR1 )/dR1
. .
Gθ . .
. .
− sin(θR1 )/dRNA cos(θRNA )/dRNA
25
26. Multilateration
- TDoA-based Positioning -
System model A4 A3
Anchor node
• NA ≥ η + 1 anchors Target node
• Single target
• Differential distance estimation
• Syncrhonous system
Z1
Position estimation
z − ai F − z − aR F = ∆diR ,
∀i ∈ IA R A1 A2
26
27. Trilateration
- ToA-based Positioning -
System model
• NA ≥ η + 1 anchors A4 A3
• Single/Multi-targets NT ≥ 1 Anchor node
Target node
• Distance estimation
Position estimation
• Single target Z1
z − aj F = dij , ∀j ∈ IA
• Multi-target
zi − aj = dij , ∀i ∈ IT , j ∈ IA
F A1 A2
.
.
.
zi − zj = dij , ∀i ∈ IT , j ∈ IT
F
27
28. Trilateration Using AoA
- Hybrid Angle-Distance Positioning -
System model
• NA ≥ η + 1 anchors
• Multi-targets NT ≥ 1 X
• Differential-AoA {βi }k
i=1
N (N +1)/2+2N
• k> 2
Position estimation
!
• Differential angle β c
O !
A1 A2
• Angle-to-distance
dXA1 = 2
2ro − (1 − cos(2β))
• Position estimation
zi − aj = dij , ∀i ∈ IT , j ∈ IA
F
.
.
.
zi − zj = dij , ∀i ∈ IT , j ∈ IT
F
28
30. Range-based Position Estimation Problem
- System model -
Network
• NA anchor nodes
• NT target nodes
• Connectivity range RMAX
Measurement
1, dij ≤ RMAX
• connectivity, cij
0, dij > RMAX
˜ ˜
• ranging, dij ∼ fij (dij |dij ), if cij = 1
˜ dij , LOS
• channel, E{dij } =
dij + bij , NLOS, bij > 0
30
31. Rigid-Bar Model
c d
Graph f
• Connectivity
• Distance values
a b
e
Is the graph unique?
31
32. Mirroring Ambiguity
- Symmetric ambiguity -
The white node is subject to a flip-ambiguity.
32
33. Swing Ambiguity
- Folding in the η-dimensions -
c d c d
f f
a b a b
e e
d
f
c
c
e
e
f
a b a b
d
Equivalent graphs with incongruent nodes
Notice the distances between (c,e) and (c,f)
33
35. Position Information
Theorem: Generalized Information Matrix Decomposition (Destino, ’12)
In a network with NA = η + 1 anchors, NT targets and connectivity C, the
position error bound to the location of the k-th node is given by the inverse of
NA k−1
Sk ζnk Υnk + ζnk Υnk − QT G−1 Qk ,
k k−1
n=1 n=NA +1
equivocation
anchor-to-target information
target-to-target information
where Gk−1 and Qk are obtained by partitioning Fd as
Gk−1 Qk
Fd = ˘d .
QT
k Fk ,
• ζnk , Ranging Information Intensity (RII)
• Υnk , Ranging Direction Matrix (RDM)
35
36. Equivocation Matrix
Theorem: Decomposition of the Equivocation Matrix (Destino, ’12)
Consider a network with NA anchors and NT targets, the equivocation matrix
of the k-th target node, denoted by Ek , with k = N can be decomposed as
k−1 k−1 k−1
e
Ek = ζik Υik + κkj Υkj ,
ik ik
i=ma i=ma j=ma
j=i
link uncertainty coupling uncertainty
where ma = NA + 1 and sa = max (i, j) + 1.
36
37. Impact of the Information Coupling
- Benefits of node cooperation -
Investigation of the Information Coupling
- cooperative network -
10
Anchor node A1
Target node
8 Error with coupling
Error w/o coupling
6
4
y-coordinate, [m]
2
Z1
A2
0
−2 Z3
Z4
−4 Z2
−6
−8 A3
−10
−10 −8 −6 −4 −2 0 2 4 6 8 10
x-coordinate, [m]
Decoupling by disconnection
(c46 = 0, c56 = 0) → (ζ46 = 0, ζ56 = 0) → κ75 = 0.
67
37
43. Maximum-Likelihood - Weighted Least Squares
• Independent measurements
• Anchor-to-target ranging
• Gaussian model
Non-Cooperative Maximum Likelihood Formulation:
NA N ˜
(dij − ai − zj 2 )2
ˆ
ˆ
z = max K exp −
2
2σij
ˆ∈RηNT
z i=1 j=NA +1
anchor-to-target
Non-Cooperative Weighted Least-squares Formulation:
NA N
2
ˆ
z = min wij ˜ ˆ
dij − ai − zj F
ˆ∈RηNT i=1 j=N +1
z
A
anchor-to-target
43
44. Illustration of the Log-Likelihood Function
- 2-D, 4-Anchors and 1-Target -
Perfect Measurements Noisy Measurements
44
45. The Least-Square Formulation Revised
NA 2 NA
˜
di − ai − z
ˆ F = ( ai − z F + ρi − ai − z F )2
ˆ
i=1 i=1
When the variables ρi s are not harmful?
45
46. Linear Algebra Intuition
- The null space -
Property: Noise in the Null-space
Let n denote a perturbation vector and assume that n lies in the
null-space of A, i.e. n ∈ N (A). Then,
A(x + n) = Ax
46
52. Maximum-Likelihood - Weighted Least Squares
• Independent measurements
• Target-to-target cooperation
• Gaussian model
Cooperative Maximum Likelihood Formulation:
NA N ˜ N N ˜
(dij − ai − zj 2 )2
ˆ (dij − zi − zj 2 )2
ˆ
ˆ
z = max K exp −
exp −
2
2σij 2
2σij
ˆ∈RηNT
z i=1 j=NA +1 j=NA +1 j=NA +1
j=i
anchor-to-target target-to-target
Cooperative Weighted Least-squares Formulation:
NA N N N
2 2
ˆ
z = min wij ˜ ˆ
dij − ai − zj F + wij ˜ ˆ
dij − zi − zj F
ˆ∈RηNT i=1 j=N +1
z i=NA +1 j=NA +1
A
j=i
anchor-to-target target-to-target
52
53. Facts of the WLS Optimization Problem
Number of local minima grows with:
• number of nodes N ,
• the lack of connections,
• the noise.
Robustness to measurement errors can be achieved by:
• adding constraints (hard mitigation method),
• using weights (soft mitigation method).
Optimization complexity grows with:
• number of nodes N ,
• the lack of connections,
• the lack of a priori information,
• number of costraints.
53
54. Global Optimization
- Smoothing continuation method -
Optimization via GDC Technique
- sum of Gaussian functions -
0
smoothing parameter, λ → 0
−1
λ
Objecitve function, g −2
−3
−4
−5
−6
−7
Estimated minimum
Smoothed objective
Original objective
−8
−5 0 5 10
Optimization variable, x
• Smooth, g(x) → g λ (x)
• Minimize, xm = min g λ (x)
x
• Continue (minimum tracking), λ < λ and x0 = xm
54
55. Efficient Implementation of the Optimization Method
- Range-Global Distance Continuation -
• Smoothed function and gradient in closed-forms
• The first smoothed function is convex
• Random initialization
• Minimization without matrix inversions (BFGS)
• Decreasing smoothing paramters
55
56. Cooperative Positioning
- R-GDC optimization performance -
R-GDC Performance in Large Scale Networks
- cooperative network -
0.65
R-GDC
0.6 PEBLOS
0.55
0.5
Location accuracy, ε [m]
0.45
¯
0.4
0.35 NT = 50
0.3
0.25 NT = 100
0.2 NT = 200
0.15
0.1
0.05
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Meshness ratio, m
Network: Area (14.14 × 14.14) [m2 ], 4 anchors (square location), NT targets (inside the anchors).
Noise: σd = 0.3 [m]. s Location accuracy: ε =
¯ ˆ
E{(Z − Z)2 } [m].
56
57. Weighing Function
- Heuristic strategies -
“Weights are chosen to reflect differing levels of concern
about the size of the squared error terms.
Higher weights to more reliable measurements, less to others
and zero the unmeasured.”
• Inverse of the noise variance (optimal in zero-mean Gaussian model)
• Inverse of the squared-ranging (simple and effective in small scenarios)
• Locally weighted Scatterplot Smoothing (LOESS)-based
(emphasize shorter connections rather than long ones)
• Channel-based (feasible if propagation model is available)
57
58. Ranging in a Realistic Environment
Ranging Error Distribution
25
TF404 eLab LOS d7,12: LOS
NLOS
NLOS2 d2,12: NLOS
7
2
12
d4,12: NLOS
A2
20 Fitting
4
5
15
A1
pdf
8
10
11
6
10
5
y A3
9
x TF407 TF406 TF405
0
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Ranging error (in meters)
• Ranging statistics are spatial-time variant
• Long distance can be more accurate than short ones, e.g d2,12 vs d4,14
• Distributions are not generally Gaussian (see S-V model)
• Channel-statistics are not practical with off-the-shelf devices
58
59. Stochastic-Geometric Weighing Function
... wij is the confidence that the true distance dij is within a confidence bound
¯
of ±γ of the mean estimate dij , weighted by a penalty Pij on the hypothesis
that the samples {dij,k } are obtained under LOS conditions.
¯ ¯
wij = Pr dij − γ ≤ dij + cij ≤ dij + γ , ∀eij ∈ E
¯ ¯
= Pr dij − γ ≤ dij ≤ dij + γ · Pr {¯ij = 0}
c
¯ij − γ ≤ dij ≤ dij + γ · Pij
= Pr d ¯
D
= Dispersion · Penalty = wij · Pij
where
Pij → 1 Hypothesis of LOS is true
Pij → 0 Hypothesis of LOS is false
59
60. Stochastic (Dispersion) Weighing Function
- The intuition behind: sort categorical data -
Pool of objects Scale
Which unit:? [γ]
Object characteristics
• shape, σ
• density, K
Metric
• weight: w = f (σ, K; γ)
60
61. Maximum Entropy Criteria
Our context:
• Categories → (K, σ )
ˆ
• Sample → zij = (Kij , σij )
ˆ
• Weight of zij → wij
• Population → Z = [Kmin , Kmax ] × [ˆmin , σmax ]
σ ˆ
Diversity of the the objects by the weights is
Kmax ˆ
σmax
H(γ) = w(S, r; γ) · ln (w(S, r; γ)) dS
r=Kmin σ
ˆ min
• Uncertainty analysis: measure wij and compute H (diversity)
• Weight optimization: compute wij that maximizes H (diversity)
γopt = arg max H(γ)
γ∈R+
61
62. Geometric (Penalty) Weights: Concept
- Handling NLOS with scarce information -
Rationale:
• Higher confidence = higher weights
• Pij → 1 ⇒ LOS; Pij → 0 ⇒ NLOS;
˜
• Kij → 1 ⇒ insufficient LOS/NLOS information in {dij }
Concept: Relate likelihood of NLOS with a geometric effect captured by
neighboring nodes, e.g., obtuseness of triangles.
q
q
q
i j
i j
i j
62
68. Thank You
Ph.D. Thesis “Positioning in Wireless Networks”
Defence: 16/11/2012, Op-Sali L10
Author: Giuseppe Destino, University of Oulu
Advisor: Prof. Giuseppe Abreu, Jacobs Universtiy, Germany
Supervisor: Prof. Jari Iinatti, University of Oulu
68