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

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

2. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Open Question
Facing the Ranging Error
What happens when all measurements are error-affected?
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Positive Ranging Error
YAxis
X Axis
Anchors
Target
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 2/33

3. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Error on Ranging
Forced to Face it!
Ranging Error
We have observed that Ranging Error, even in perfect
Line-of-Sight conditions, can not be avoided and neither can
be negligible.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 3/33

4. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Positioning
Localization over Geographic Support
Localization Algorithm
The goal is to obtain a quite accurate estimation of a target
position even in presence of ranging error.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 4/33

5. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Positioning
Networks
Different type of Localization Networks
Multihop Network Fully Connected Network
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 5/33

6. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Positioning
Different type of Localization Scenarios
C(Θ) is the convex hull of ΘA, i.e. the smallest convex set
that contains the anchor nodes.
Target inside C(Θ) Target outside C(Θ)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 6/33

7. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Preliminaries Definitions
{θa1
, · · ·, θaN
} is the ordered coordinate vectors of the aN
known nodes (a-priori known location).
{θ1, · · ·, θnT
} is the ordered coordinate vectors of the nT
target nodes (unknown location).
ΦA = [φa1
, · · · , φηaN
]T
and Φ = [φ1, · · · , φηnT
]T
are the
stacked vectors whose first and second nT elements are
given by {θi:x} and {θi:y}, respectively, where T
denotes
transpose.
dij θi − θj = θi − θj, θi − θj is the true distance
between the i-th and j-th nodes.
eij ∼ N(0, σ2
ij) is the Gaussian ranging error on the
distance estimate between the i-th and j-th nodes.
˜dij = dij + eij is the measured distance
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 7/33

8. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Least Squares
Effect of Ranging Error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Least-Square Example without Ranging Error
YAxis
X Axis
Anchors
Target
LS without ranging error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Least-Square Example with Ranging Error
YAxis
X Axis
Anchors
Target
LS with ranging error
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 8/33

9. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Least Squares
Exploiting the Cost Function
ˆΘ arg min
i∈ΘA j∈ΘT
( ˆdij − ˜dij)2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2
LS without ranging error
(3D error function)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
LS with ranging error
(3D error function)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 9/33

10. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Least Squares
Exploiting the Cost Function
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Least-Square Example without Ranging Error
YAxis
X Axis
Anchors
Target
LS without ranging error
(error function)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Least-Square Example with Ranging Error
YAxis
X Axis
Anchors
Target
LS with ranging error
(error function)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 10/33

11. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Least Squares
Derivation
Problem definition:
ˆΘ arg min
i∈ΘA j∈ΘT
( ˆdij − ˜dij)2
, (1)
or, equivalently:
ˆΘ arg min
i∈ΘA j∈ΘT
( θi − ˆθj − ˜dij)2
. (2)
We have now a non-linear system, often quite complicated to
be solved.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 11/33

12. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Linear Least Squares
Linearization
Non-linear Least Squares algorithms use mathematical solutions
to find the parameters (i.e. ΘT ) that minimize the cost function.
They typically need an initial guess to start the iterative process
(successive refinement of the estimated parameters) and in such a
class of problem a closed-form solution does not exist.
A possible alternative is given by the Linear Least Squares.
The Linearization Idea
We look for an algebraic manipulation that allows to transform the
non-linear system into a linear system, therefore much simpler to
be treated and solved (and later implemented) using a very
efficient matrix form.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 12/33

13. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Linear Least Squares
Single-Target Linearization Example
When considering one single target per time at θ, solving (2)
is equivalent to solve the following system:
θi − ˆθ = ˜di, i = 1, . . . , n (3)
and, in turn, to:
(θi:x − θx)2
+ (θi:y − θy)2
= ˜d2
i , i = 1, . . . , n. (4)
describing a system of n equations that can be linearized
pivoting one arbitrary equation as follows:
(θi:x −θx)2
−(θn:x −θx)2
+(θi:y −θy)2
−(θn:y −θy)2
= ˜d2
i − ˜d2
n, (5)
with i = 1, . . . , (n − 1).
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 13/33

14. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Linear Least Squares
Single Target Linearization Example
Expanding and properly grouping all the terms, we obtain:
2θx(θn:x−θi:x)+2θy(θn:y−θi:y) = θ2
n:x−θ2
i:x+θ2
n:y−θ2
i:y+ ˜d2
i − ˜d2
n, (6)
with i = 1, . . . , (n − 1),
that is an overdetermined linear system that we can describe in
matrix form as follows:
A ˆθ = b. (7)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 14/33

15. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Linear Least Squares
Single Target Linearization Example
The matrix of eq. (7) are related to the elements of eq. (6) as
follows:
A = 2




(θn:x − θ1:x) (θn:y − θ1:y)
(θn:x − θ2:x) (θn:y − θ2:y)
. . . . . .
(θn:x − θ(n−1):x) (θn:y − θ(n−1):y)



 , (8)
and
b =





θ2
θ2
n:x−1:x + θ2
n:y − θ2
1:y + ˜d2
1 − ˜d2
n
θ2
θ2
n:x−2:x + θ2
n:y − θ2
2:y + ˜d2
2 − ˜d2
n
. . .
θ2
n:x − θ2
(n−1):x + θ2
n:y − θ2
(n−1):y + ˜d2
(n−1) − ˜d2
n





. (9)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 15/33

16. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Linear Least Squares
Multiple Target Nodes Linearization
Eq. (7) can be generalized for multiple target nodes:
A ˆΘT = b. (10)
In general A is not invertible, therefore we need to solve the
equivalent normal equation linear system:
AT
A ˆΘT = AT
b, (11)
then, using the psesudoinverse of AT
A, we get to the final
formulation:
ˆΘT = (AT
A)−1
AT
Moore-Penrose
pseudoinverse
b. (12)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 16/33

17. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Linear Least Squares
Matlab Implementation
A Smart Algebraic Implementation
The backslash operator Ab is used by matlab to solve linear
systems and in the considered case is equivalent to the explicit
formulation pinv(A)*b.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 17/33

18. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
MultiDimensional Scaling
The Fundamental Idea
How MultiDimensional Scaling works:
The distance matrix D can be seen as a dissimilarities
matrix between nodes.
Exploiting this information it is possible to map the nodes
into an η-dimensional space.
This mapping is unique but subject to rotation.
It is possible to exploit the information on the a-priori known
nodes (anchors) to get back to the true network realization.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 18/33

19. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
MultiDimensional Scaling
Classic Formulation
Define the Gramian Matrix
KΘ ΘT
Θ. (13)
It can be shown being a rotation of the positive semidefinite
and double-centered
KD −
1
2
J · D 2
· J, (14)
where m denotes the m-th element-wise power, the matrix
J is given by
J I −
1
N
(1 · 1T
), (15)
I is identity matrix and 1 is a vector whose entries are all 1.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 19/33

20. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
MultiDimensional Scaling
Classic Formulation
Question: Why we want to use KD rather than KΘ?
Answer: Because we do not know Θ (is the goal of our
localization process) but we have some knowledge about
D. We have infact measured ˜D.
We can now decompose KD as follows:
KD UD · ΛD · UT
D, (16)
where UD and ΛD are, respectively, the eigenvector and
eigenvalue matrices of KD.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 20/33

21. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
MultiDimensional Scaling
Classic Formulation
A rotated and translated matrix containing all the nodes
location can now be obtained from equation (17):
ΘR
= UD
N×η
·ΛD
1
2
η×η
T
, (17)
where · m×n denotes the m-by-n upper-left partition and
the symbol R
denotes the rotation.
Since at least η + 1 column of Θ are known, it is possible to
get an estimated matrix ˆΘ from ΘR
via Procrustes
transformation.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 21/33

22. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Analyzing MDS
The MultiDimensional Scaling:
In the above scenario (network + ranging error) has the
same performances than LS.
Zero error in absence of ranging error.
It is now presented as a possible new theoretical approach.
It is the base for one of the most powerful localization
algorithm Super MDS.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 22/33

23. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Smacof
Cooperation without confidence box
σ(X(0)
) = i<j wij( ˆd
(0)
ij − ˜dij)2
is the original stress function
ˆd
(m)
ij is the estimated distance dij after the m-th iteration
ˆX(m)
is the estimate of X = [Φ; ΦA] after the m-th iteration
ˆX(m−1)
is the solution at the previous iteration
τ( ˆX(m)
, ˆX(m−1)
) is the majored convex function at the m-th iteration
The majorized convex function
This algorithm attempts to find the minimum of a non-convex function by
majorizing the initial stress function and then tracking the global minimum
of the so-called majorized convex function τ( ˆX(m)
, ˆX(m-1)
) ,which is then
successively constructed from the previous solution ˆX(m-1)
obtained in the
iterative procedure.
The expression of the majorized convex function is:
τ( ˆX(m)
, ˆX(m−1)
) = 1+tr ˆX(m)T
V ˆX(m)
−2tr ˆX(m)T
B( ˆX(m−1)
) ˆX(m−1)
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 23/33

24. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Smacof
Cooperation without confidence boxes
The entries of the matrix V are given by
vij =
i=1
i=j
wij i = j
wij i = j
while the i-th element of the matrix B( ˆX(m−1)
) is defined as follow:
bij =



i=1
i=j
wij
˜dij
ˆd
(m−1)
ij
i = j
wij
˜dij
ˆd
(m−1)
ij
i = j
Since τ( ˆX(m)
, ˆX(m−1)
) is a quadratic convex function, in order to
compute ˆX(m)
is sufficient to solve the following equation:
∂τ( ˆX(m)
, ˆX(m−1)
)
∂ ˆX(m)
= 0
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 24/33

25. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Smacof
Cooperation without confidence boxes
After the m-th iteration the estimated locations are given by:
ˆX(m)
= V†
B( ˆX(m−1)
) ˆX(m−1)
where V†
is the Moore-Penrose Pseudoinverse of V.
The iterative process ends when σ(X(m)
) − σ(X(m−1)
) < or a
certain iteration limit is reached.
Accuracy plus Confidence
The Smacof algorithm is known for its accuracy and efficiency (it is
applicable also to non-convex functions).
It doesn’t provide an estimate of the confidence associated with
any position estimate.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 25/33

26. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
CRLB
A Concise and General Bound on the Ranging Variance
A general description of the error model:
σ2
ij σ2
0 ·
dij
d0
α
α ≥ 0 is the path-loss factor.
d0 is the reference distance.
σ2
0 is the ranging variance at d0.
The Generality of the Bound
Lower bounds on the ranging errors obtained from distance
estimates using either narrowband or ultra-wideband radios is
σ ≥ β√
SNR
where β is a coefficient depending on the speed of light and on the
signal’s duration, center-frequency and bandwitdh.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 26/33

27. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Target Specific Approach
The Covariance Matrix
The idea: a node cooperates only with the subset of other
nodes which knows its own position (anchor nodes).
ˆθi is the estimate of the location
of the i-th target
Associated with each ˆθi there is
the η-by-η covariance matrix:
Ωθi
E (ˆθi − θi)(ˆθi − θi)T
The CRLB relates this covariance matrix to the inverse FIM
by
Ωθi
F−1
θi
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 27/33

28. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Target Specific Approach
The Fisher Information Matrix
The likelihood function of the Target specific approach is
given by:
f(θi; ¯di)
NA
j=1
1
σij
√
2π
exp −
( θi−θj − ¯dij)2
2σ2
ij
The anchor-to-target paths is
¯dij
nij
k=1
˜dk
Associated Fisher Information
Matrix: the qp−th element is
given by:
Fθi
−E ∂2 ln f(θi; ¯di)
∂2θi
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 28/33

29. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Bidimensional Scenario
Regular Planar Network
FIM under Target-specific Approach
Fθi
Fθi:xx Fθi:xy
Fθi:xy Fθi:yy
Fθi:xy =
j∈Ni
∆θij:x ∆θij:y
¯σ2
ij d2
ij
+
α2
∆θij:x ∆θij:y
2d4
ij
.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 29/33

30. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Error Confidence/Ellipse
The Uncertainty Ellipse
The Error Ellipse express the confidence region of a target position
estimate, under a given error probability, for a generic χ2
-error
distribution.
For each target the Covariance Matrix is: Ωθi =


σ2
x σxy
σxy σ2
y


−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.17
0.24
0.42
0.3
5% Uncertainty Ellipses in a Regular Planar Network
YAxis
X Axis
σ = 0.01
σ = 0.03
Anchors
Target
The scale factors sf is givenby the
inverse of the CDF of the χ2
at the
given confidence probability
The length of the axis is given by
sf · eigval(Ωθi )
The rotation angle β is given by
β = 1
2 atan 1
sfx
2σxy
σx
2−σ2
y
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 30/33

31. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Report 3/3
Positioning
Complete the lab experience writing (one per group) a report
with:
1 create the map of the network environment using the
coordinates of anchor and target given,
2 run in Matlab LS and MDS, obtain the different target
estimated positions using the distances ˆd provided,
3 estimate the average ranging error and variance of the each
target distance link,
4 Obtain the Covariance matrix Ωθ and draw the Error ellipse
using the mean of the target’s position estimates and
inverse variance of the target’s distance estimates.
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 31/33

32. Follow-Up Lect I
and II
Positioning
Preliminaries
Localization
Algorithms
Preliminaries
Least Squares
MDS
Smacof
Cramèr-Rao
lower bound
Target Specific Approach
Bidimensional Scenario
Confidence
Degree
Report 3/3
Positioning
5 Save the code, the maps and the plot the target estimated
position and the fisher ellipse of both LS and MDS in
different figures.
6 write a short (max 2 pages) description of this experiment.
Please print and deliver the report within the aforementioned deadline to
s.severi@jacobs-university.de,
r.stoica@jacobs-university.de.
Matlab Tip
To perform the report, use the command provided in the Matlabtips.m file
Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 32/33

33. Thank you!