Blind Estimation of Transmit Power in Wireless Networks

Transcription

1 Bln Estmaton of Transmt Power n Wreless Networks Murtaza Zafer (IBM Research), Bongjun Ko (IBM Research), Chatschk Bskan (IBM Research) an Ivan Ho (Imperal College, UK)

2 Transmt-power Estmaton: Problem Synopss p 1 (x 1,y 1 ) P? m 1 (x,y) T m N p N (x N,y N ) m 2 p 2 (x 2,y 2 ) m 3 p 3 (x 3,y 3 ) Noe T s a wreless transmtter wth Tx power = P (unknown), at a poston (x, y) (unknown) Noes {m 1,, m N } are montors that measure receve power {p } Goal gven {p } an {(x,y )} (montor locatons), estmate unknown P an (x, y). Dffculty: Bln estmaton no pror knowlege (statstcal or otherwse) of the locaton or transmt power of the transmtter.

3 Motvaton Applcatons Sgnal jammng attack etecton n MANET. Noe ms-confguraton etecton. Prmary user etecton n cogntve rao networks. Event ntensty etecton n sensor network. Power-aware rao resource control wth unknown transmt power. Locaton entfcaton of wreless users.

4 Wreless sgnal attenuaton P = transmsson power P r = receve power = stance between the transmtter an recever α= attenuaton factor, (α > 1) k = normalzng constant P T Receve power Determnstc moel Receve power P r Stochastc moel P r = kp α Dstance () H = P = H e W Dstance () kp α ; lognormal r.v.

5 Estmaton uner etermnstc moel Determnstc propagaton moel: P r = kp α P T P r Sngle montor measurement P P 1 T 1 1 Montor 1 best estmate of transmt power: P* = P 1

6 Two montors P T Transmtter P = kp / α P 1 P 2 12 Montor 1 Montor 2 2 Locus of transmtter : usng Lower boun of P*: By the trangular nequalty: P * = 1 1 ( k/ P 1) α + ( k/ P2) α α

7 Multple montors Multple montor scenaro 1/ α ( = 1); 1 2 c 2 P = P 1 1/ α ( = 2); 2 3 c 3 P = P 2 P 1/ α N 1 N = = N N PN 1 Wth multple montors versty n measurements System of equatons wth unknowns (x,y,p) We shoul be able to solve these equatons to obtan exact P? Answer: Yes an No!! ( c 1)

8 Mult-montor estmaton uner etermnstc moel Theorem I: There s a unque soluton (P*, x*, y*) except when the montors are place on an arc of a crcle. 3 r,1 1 1 T (x, y) Proof: (x r, y r ) T r r, A locaton (x, y) s a soluton f an only f t satsfes 1 / 2 =c 1,, N-1 / N = c N-1 The actual locaton (x r, y r ) s one soluton; thus r,1 / r,2 =c 1,, r,n-1 / r,n = c N-1 There exsts another soluton at (x, y) f an only f, r,1 / r,2 = 1 / 2, ; equvalently,

9 Determnstc moel Multple montor scenaro Corollary 1: Two montors always has multple solutons 1 2

10 Determnstc moel Multple montor scenaro Corollary 1: Two montors always has multple solutons Corollary 2: Three montors always has multple solutons In general, for any regular polygon placement of montors the transmsson power cannot be unquely etermne! For all non-crcular placement of montors, transmsson power can be unquely etermne.

11 Stochastc attenuaton moel Sgnal propagaton moel: lognormal fang P = H kp α H = e W ; lognormal r.v. P = transmsson power P = receve power at montor = stance between the transmtter an montor H = lognormal ranom varable H unknown to the montor represents the aggregate effect of ranomness n the envronment; eg: mult-path fang

12 Stochastc attenuaton moel Let z = ln(p ) ; where p s receve power We are gven (z, x, y )for = 1,.., N Let Z = ln(p), an ) θ = ( Z, x, y) T P (x,y) The jont probablty ensty functon m 1 m 2 m N p 1 (x 1,y 1 ) p 2 (x 2,y 2 ) p N (x N,y N ) Maxmum Lkelhoo Estmate ML estmate (Z*,x*,y*) s the value that maxmzes the jont probablty ensty functon ) ( Z *, x*, y*) = arg max f ( ; θ ) z ) θ

13 ML estmate uner stochastc moel = 2 ( x x) + ( y y ) 2 stance between some locaton (x,y) an montor * = 2 ( x x*) + ( y y *) 2 stance between estmate Tx. locaton (x,y) an montor (x*,y*) s the soluton to the mnmzaton above, where the objectve functon s sample varance of {ln(p α )} P* s proportonal to the geometrc mean of {p (* ) α }

14 Asymptotc Optmalty of ML estmate

15 Performance Evaluaton Synthetc ata set N = 2 to 20 place unformly at ranom n a sk of raus R. Receve power s generate by... lognormal fang moel for each montor. Performance measure: average over estmaton for 1000 transmtters. Emprcal ata set Sensor network measurement ata at U of Mchgan. Total 44 sensor evces. Receve powers are measure between all pars of evces. α = 2.3, an σ B = Ranomly choose N=3,4,,10 montors out of 44 evces. Estmators MLE-Coop-fmn : MLE wth fmnsearch for locaton estmaton. MLE-Coop-gr: MLE wth locaton estmaton among gr ponts. MLE-eal: MLE wth known transmtter locaton. MLE-Par: Average of par-wse MLEs. Performance metrc

16 Evaluaton Synthetc ata set Emprcal ata set (MLE-Coop-gr)

17 Mult-transmtter estmaton K: # of transmtters N: # of montors,j : stance between Tx I an montor j Queston: - How many transmtters are out there (at least)? - What are ther transmsson powers? - How many montors o we nee?

18 Concluson an Open Problems Bln estmaton of transmsson power Stue estmators for etermnstc an stochastc sgnal propagaton Utlze spatal versty n measurements Obtane asymptotcally optmal ML estmate Presente numercal results quantfyng the performance Open problems Non-truste montorng scenaro Montorng uner heterogeneous channel characterstcs Reference: I. W. Ho, B. Ko, M. Zafer, C. Bskan, an K. Leung, Cooperatve Transmt-Power Estmaton n MANETs, WCNC M. Zafer, B. Ko an I. W. Ho, Cooperatve transmt-power estmaton uner wreless fang, ACM Mobhoc I. W. Ho, B. Ko, an M. Zafer, Bln Estmaton of Transmt-Power for Multple Wreless Sources, MILCOM 2008.

19 Thank you.