Optimal Jammer Placement in Wireless Localization NetworksSinan Gezici∗, Suat Bayram♭, Mohammad Reza Gholami♮, and Magnus Jansson♮

∗ Department of Electrical and Electronics Engineering, Bilkent University, 06800, Ankara, Turkey

♭ Department of Electrical and Electronics Engineering, Turgut Ozal University, 06010, Ankara, Turkey

♮ ACCESS Linnaeus Centre, Department of Signal Processing, KTH–Royal Institute of Technology, Stockholm, Sweden

Abstract

The optimal jammer placement problem is proposed for a wireless localization network, wherethe aim is to degrade the accuracy of locating target nodes asmuch as possible. In particular, theoptimal location of a jammer node is obtained in order to maximize the minimum of the Cramer-Rao lower bounds for a number of target nodes under location related constraints for the jammernode. Theoretical results are derived to specify scenariosin which the jammer node should belocated as close to a certain target node as possible, or the optimal location of the jammer node isdetermined by two or three of the target nodes. In addition, explicit expressions for the optimallocation of the jammer node are derived in the presence of twotarget nodes. Numerical examplesare presented to illustrate the theoretical results.

Introduction

•Position information: vital for many location aware services/applications

•The main aim in wireless localization networks: achieving high localization accuracy

•Wireless sensor localization: a well investigated topic

•The effects of jamming on wireless localization networks: arather new topic - littleattention in the literature

•Recent work [1]: the optimal power allocation strategies for the jammer nodes in or-der to maximize the average or the minimum Cramr-Rao lower bounds (CRLBs) ofthe target nodes.

•Current work: optimal jammer placement problem based on maximization the mini-mum CRLB

System Model

•NA anchor nodes at known locationsyi ∈ R2 andNT target nodes atxi ∈ R

2

•A jammer node atz∈ R2

•The jammer node is assumed to transmit zero-mean Gaussian noise•Ai , { j ∈ {1, . . . ,NA} | anchor nodej is connected to target nodei}, i ∈ {1, . . . ,NT}

Received signal:r i j(t) =Li j

∑k=1

αki js(t − τk

i j)+ γi√

PJ vi(t)+ni j(t), t ∈ [0,Tobs], j ∈ Ai

ni j(t),√

PJ vi(t): independent zero-mean white Gaussian random processes

• τki j ,

‖y j−xi‖+bki j

c bki j ≥ 0 & Ai , A

Li ∪A

NLi

•bi j , [b2i j . . .b

Li j

i j ] if j ∈ A Li & bi j , [b1

i j . . .bLi j

i j ]T if j ∈ A NL

i

• θ i , [xTi bT

iAi(1) · · · bTiAi(|Ai|)]

T

•CRLB: E{‖xi −xi‖2} ≥ tr{[F−1i ]2×2} with [F−1

i ]2×2 = Ji(xi,PJ)−1

• Ref.[2] : Ji(xi,PJ) = ∑j∈A L

i

λij

N0/2+PJ|γi|2φ ijφ

Tij λi j ,

4π2β 2|α1i j |2∫ ∞−∞ |S( f )|2d f

c2 (1−ξ j)

φ i j , [cosϕi j sinϕi j ]T β =

√∫ ∞−∞ f 2|S( f )|2d f∫ ∞−∞ |S( f )|2d f 0≤ ξ j ≤ 1: the path-overlap coefficient

ϕi j : the angle between target nodei and anchor nodej

• CRLBi = tr{Ji(xi,PJ)−1}= ri(PJ|γi|2+N0/2) r i , tr{[∑ j∈A L

iλi jφ i jφ

Ti j ]

−1}

Optimal Jammer Placement

Generic Formulation and Analysis

maximizez

mini∈{1,...,NT}

r i

(

PJ|γi|2+N0

2

)

subject to‖z−xi‖ ≥ ε , i = 1, . . . ,NT

Assumption:|γi|2 = Ki

(

d0‖z−xi‖

)νfor ‖z−xi‖> d0

maximizez

mini∈{1,...,NT}

r i

(

KiPJ

‖z−xi‖ν +N0

2

)

subject to‖z−xi‖ ≥ ε , i = 1, . . . ,NT

(1)

Ki , Ki(d0)ν

Proposition 1: If there exists a target node that satisfies the following inequality,

rℓ

(

KℓPJ

εν +N0

2

)

≤ mini∈{1,...,NT}

i 6=ℓ

r i

(

KiPJ

(‖xi −xℓ‖+ ε)ν +N0

2

)

(2)

and if set{z : ‖z−xℓ‖= ε & ‖z−xi‖ ≥ ε , i = 1, . . . , ℓ−1, ℓ+1, . . . ,NT} isnon-empty,then the solution of (1), denoted by zopt, satisfies‖zopt−xℓ‖= ε ; that is,the jammer node isplaced at a distance ofε from theℓth target node.

The optimization problem in (1) in the presence of two targetnodesℓ1 andℓ2 only:

maximizez

mini∈{ℓ1, ℓ2}

r i

(

KiPJ

‖z−xi‖ν +N0

2

)

subject to‖z−xℓ1‖ ≥ ε , ‖z−xℓ2‖ ≥ ε(3)

ℓ1, ℓ2 ∈ {1, . . . ,NT} andℓ1 6= ℓ2. zoptℓ1,ℓ2

and CRLBℓ1,ℓ2: the optimizer and the optimalvalue of (3)Proposition 2: LetCRLBk,i be the minimum ofCRLBℓ1,ℓ2 for ℓ1, ℓ2 ∈ {1, . . . ,NT} andℓ1 6= ℓ2, and let zopt

k,i denote the corresponding jammer location (i.e., the optimizer of(3) for ℓ1 = k andℓ2 = i). Then, an optimal jammer location obtained from (1) isequal to zopt

k,i if zoptk,i is an element of set

{

z : ‖z−xm‖ ≥ ε , m∈ {1, . . . ,NT}\{k, i}}

and

rm

(

KmPJ

‖zoptk,i −xm‖ν

+N0

2

)

≥ CRLBk,i (4)

for m∈ {1, . . . ,NT}\{k, i}.⇒ the optimal jammer location is mainly determined by two of the target nodes sincethe others have larger CRLBs when the jammer node is placed atthe optimal locationaccording to those two jammer nodes only.——————————–Special cases———————————–1-Two Target NodesProposition 3: For the case of two target nodes (i.e., NT = 2), the solution zopt of (1)satisfies one of the following conditions:(i) if ‖x1−x2‖< 2ε , then‖zopt−x1‖= ‖zopt−x2‖= ε(ii) otherwise,(a) if r1

(

K1PJεν + N0

2

)

≤ r2

(

K2PJ(‖x1−x2‖−ε)ν +

N02

)

, then‖zopt−x1‖= ε and

‖zopt−x2‖= ‖x1−x2‖− ε(b) if r2

(

K2PJ

εν + N02

)

≤ r1

(

K1PJ

(‖x1−x2‖−ε)ν +N02

)

, then‖zopt−x1‖= ‖x1−x2‖− ε and

‖zopt−x2‖= ε(c) otherwise,‖zopt−x1‖= d∗ and‖zopt−x2‖= ‖x1−x2‖−d∗, where d∗ is the uniquesolution of the following equation over d∈ (ε , ‖x1−x2‖− ε).

r1

(

K1PJ

dν +N0

2

)

= r2

(

K2PJ

(‖x1−x2‖−d)ν +N0

2

)

(5)

2-Infinitesimally Small εProposition 4: Suppose NT ≥ 3 andε → 0. The max-min CRLB in the presence of

target nodesℓ1, ℓ2, andℓ3: CRLBℓ1,ℓ2,ℓ3 = maxz minm∈{ℓ1,ℓ2,ℓ3} rm

(

KmPJ‖z−xm‖ν +

N02

)

. Also,let target nodes i, j, and k achieve the minimum ofCRLBℓ1,ℓ2,ℓ3 forℓ1, ℓ2, ℓ3 ∈ {1, . . . ,NT} and let zopt

i, j,k denote the jammer location corresponding toCRLBi,j,k. Then, the optimal location for the jammer node (i.e., the optimizer of (1) inthe absence of the distance constraints) is equal to zopt

i, j,k, and at least two of the CRLBsof the target nodes are equalized by the optimal solution.

Numerical Examples

•Simulation parameters:ε = 1m,PJ = 6, N0 = 2, ν = 2, andKi = 1 for i = 1, . . . ,NT.LOS scenarios,λi j = 100N0‖xi −y j‖−2/2 (free space propagation model)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Optimal location of the jammer node

x [m]

y[m]

Anchor nodeTarget node

Target 1

Target 2

Target 3

Target 4

ε

ε

ε

ε

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Optimal location of the jammer node

x [m]

y[m]

Anchor nodeTarget node

Target 1

Target 2

Target 3

Target 4ε

ε

ε

ε

Fig. 1. The network consisting of anchor nodes at [0 0], [10 0], [0 10], and [10 10]m. Left figure: target nodes at [2 6], [5 2], [8 3], and [9 6]m.Right figure target nodes at [2 5], [4 1], [8 8], and [9 2]m.

V. N E• Left Fig: Proposition 2 fork= 1&i = 3→ the solution determined by subnetworkconsisting of target 1 and 3. Solution:zopt

1,3 = [5.0605 4.4697] m, CRLB1,3 = 0.8053m2

•Right Fig: Subnetwork consisting of taget nodes 1,3, and 4 archives the minimummax min CRLB among all other possible subnetworks with threetarget nodes.Solution: CRLB1,3,4 = 0.7983m2 andzopt

1,3,4 = [5.5115 5.5717]mReferences[1] S. Gezici, M. R. Gholami, S. Bayram, and M. Jansson, “Optimal jamming of wireless localizationsystems,” in IEEE International Conference on Communications (ICC) Workshops, June 2015.[2] Y. Shen, and MZ. Win, “Fundamental limits of wideband localizationPart I: A general framework,”IEEE Transactions on Information Theory, pp. 4956-4980, 2010.Emails: gezici@ee.bilkent.edu.tr, sbayram@turgutozal.edu.tr, mohrg@kth.se, janssonm@kth.se