Wireless Localization: Positioning
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The document discusses wireless localization techniques focusing on positioning algorithms and the impacts of ranging error on target estimation accuracy. It highlights methods such as least squares, multidimensional scaling, and emphasizes the challenges arising from errors in distance measurements. The presentation also covers network types, algorithms for solving nonlinear systems, and a theoretical foundation for enhanced localization strategies.
![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](https://image.slidesharecdn.com/lecture032015-160126091341/85/Wireless-Localization-Positioning-7-320.jpg)
![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](https://image.slidesharecdn.com/lecture032015-160126091341/85/Wireless-Localization-Positioning-23-320.jpg)
Wireless Localization: Positioning
- 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!