On Dropping Noisy Packets in Kalman Filtering over a Wireless Fading Channel

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Rosamund Gibbs
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1 2005 merica Cotrol Coferece Jue 8-10, Portla, OR, US FrC02.2 O Droppig Noisy Packets i Kalma Filterig over a Wireless Faig Chael Yasami Mostofi a Richar M. Murray Califoria Istitute of Techology Pasaea, C 91125, US bstract We cosier the wireless estimatio problem i faig eviromets which is iheret to may mobile sesor etworks. Previous works o wireless estimatio i sesor etworks have extee the same esig priciples of ata etworks by carryig out their aalysis uer the assumptio that the erroeous packets are simply roppe. I this paper, we show that proper use of the iformatio cotaie i all the packets, icluig the erroeous oes, ca improve the performace cosierably. We erive a aalytical expressio to evaluate the performace improvemet, for oe class of commuicatio oise characteristics, whe the iformatio of all the packets is use i the receiver. We also prove that keepig all the packets ca prevet the potetial istability that is itrouce if erroeous packets are roppe. This suggests that applyig the same esig priciples of ata etworks to sesor etworks ca lea to performace egraatio as sesor etwork applicatios are typically elay sesitive a therefore may require ifferet esig strategies. The work also highlights the importace of sharig iformatio betwee the physical a applicatio layers sice extractig the iformatio cotaie i all the packets woul ecessitate a cross layer esig approach. I. INTRODUCTION There has recetly bee cosierable iterest i sesor etworks 1, 2. Such etworks have a wie rage of applicatios such as evirometal moitorig, surveillace, security, smart homes a factories, target trackig a military. Commuicatio plays a key role i the overall performace of sesor etworks as both sesor measuremets a cotrol commas are trasmitte over wireless liks. Cosierig the impact of commuicatio chaels o wireless estimatio/cotrol is a emergig area of research. t this poit, it is ot yet clear what the right trasmitter/receiver esig strategies are for such applicatios. Estimatio/cotrol of a rapily-chagig yamic system is a elay-sesitive applicatio. Therefore, the commuicatio protocols a esigs suitable for other alreayexistig applicatios like ata etworks may ot be etirely applicable to sesor etworks. Data etworks are ot as sesitive to elays sice the applicatio is ot real time. The receiver, therefore, ca affor to rop erroeous packets a wait for retrasmissio. Cotrol applicatios, o the other ha, are typically elay sesitive as we are racig agaist the yamics of the system. Therefore, applyig the same esig strategies of ata etworks may ot be appropriate. uthors i 3 have looke at the impact of packet loss o wireless Kalma filterig. uthors i 4 extee the work /05/$ CC 4596 of 3 to multiple sesors. Both of these works assume that the receiver is roppig the erroeous packets. The mai questio this paper aresses is whether iscarig erroeous packets 1 is a appropriate esig strategy for elay-sesitive wireless estimatio/cotrol applicatios. I other wors, we are itereste i kowig if, a how much, we ca improve the performace by usig all the receive packets i the estimatio process. Results from 5 have show that the useful rate (useful rate oly cosiers the bits that are receive correctly of a system shoul be above a give threshol to esure stability of the cotrol process. Droppig the erroeous packets ca result i a rop of the overall useful rate, icreasig the chace of istability. uthors i 3 also cofirme this by fiig a maximum tolerable packet loss probability beyo which the Kalma filterig process woul go ustable. I this paper we cosier a mobile sesig/estimatio process usig Kalma filterig. Istea of roppig erroeous packets, we use the iformatio cotaie i all the receive packets. This chages the traitioal form of Kalma filterig as chael yamics are ow itrouce i the estimatio process. We cosier oe class of commuicatio oise characteristics. We erive a expressio for the average error variace for the case of Kalma filterig over faig chaels aalytically. We also show the stability of the estimatio process whe all the packets are use i the receiver. This allows us to compare the performace with the sceario whe the erroeous packets are roppe. Whe keepig all the packets i the receiver, the physical layer woul ee to share iformatio o the quality of the commuicatio lik with the applicatio layer for estimatio. Therefore, the paper also emphasizes the importace of cross-layer esigs i sesor etworks. The aalysis ca be a startig poit for establishig esig strategies i such etworks. II. SYSTEM MODEL Cosier a mobile sesor observig a system with the followig liear yamics: x k+1 x k + w k y k Cx k + v k, 1 Erroeous packets refer to those that are ot receive %100 accurately i the receiver. However, a erroeous packet may still carry some iformatio about the trasmitte sesor measuremet. (1

2 where x k a y k represet the state a observatio respectively. w k a v k represet zero-mea process a observatio oises with variaces of Q a R respectively. We assume scalar quatities i this paper to facilitate mathematical erivatios. We are itereste i estimatig ustable yamics a therefore we take > 1. y k is the quatize, trasforme ito a packet of bits a trasmitte over a mobile faig chael: ŷ k y k + k, (2 This is ifferet from typical Kalma filterig as is a fuctio of h k through f( a therefore is a raom process. To focus o the impact of the commuicatio chael, i this paper we assume, R 0, Q 0 a C 1. IV. SCENRIO 1: DROPPING ERRONEOUS PCKETS I this case, if the receiver gets a packet that cotais a error, it rops it a will ot use it i the estimatio process. This traslates ito the followig f fuctio: where ŷ k is the receiver versio of the observatio a k represets zero-mea commuicatio oise. Let G k represet the variace of k. G k is a fuctio of,the istataeous receive Sigal to Noise Ratio at k th trasmissio. is a stochastic process a its istributio is a fuctio of the eviromet a level of the mobility of the sesor: G k f(. (3 Fuctio f is a ecreasig fuctio that epes o the trasmitter/receiver esig priciples as well as the trasmissio eviromet. We cosier a receiver that iscars the packets at SNR 0, which meas f(0. Furthermore we take f( 0, which assumes that quatizatio oise is egligible. Depeig o the eviromet, woul have ifferet istributios. I a arrowba faig eviromet, we will have h k 2 σ 2 x σ where σ 2 2 x a σ 2 represet the trasmitte sigal power a receiver oise power respectively. h k is the value of the chael at k th trasmissio. I a eviromet with o LOS path, it is reasoable to assume that h k has Rayleigh faig istributio which results i a expoetial istributio for 6. We ivestigate the impact of such eviromets o the estimatio process. Furthermore, we take h k a therefore to be iepeet from oe trasmissio to the ext. This will be the case as log as the time iterval betwee cosecutive trasmissios is bigger tha chael coherece time 6. III. ESTIMTION The istat oe estimates the state base o the receive observatio usig a Kalma filter 7. Let ˆx k represet the estimate of x k at the receiver: ˆx k E x k ŷ 0, ŷ 1,...,ŷ k 1. (4 Let represet variace of the estimatio error of the k th trasmissio: E (x k ˆx k 2. (5 I the presece of commuicatio oise, we will have the followig equatio for upatig estimatio error variace: Pk 2C2 R + C 2 + Q. (6 + f( 4597 { 0 SNRk SNR f rop,ieal ( Thresh. (7 else SNR Thresh. is the miimum that woul result i packet loss probability of zero i theory 2. I practice, however, a small fractio of those packets that are perceive to cotai o error, may be erroeous. We take this ito accout i Sectio VI. f rop,ieal ca moel the strategy use i 3 a 4 for roppig packets. There they showe that there is a critical average packet loss probability above which the estimatio error variace woul get uboue. Here we briefly relate that to SNR. Eq. 6 will be as follows i this case: η k η k +1 2 (1 η k { 1 SNRk SNR Thresh. 0 else where η k Prob{ SNR Thresh. }, which will be as follows for a expoetially istribute : η k e SNR Thresh., (9 where 3 1. The, +1 P 0 2 (1 e SNR Thresh. k+1. (10 Therefore, if SNR > SNR Thresh. lim l( 2 k (11 if SNR < SNR Thresh. lim k l( We ca see From Eq. 11 that epeig o the average SNR of the eviromet, the estimatio process may get ustable i this sceario. This motivates the work presete i the ext sectios. For average SNR above the miimum require, the estimatio error may ot coverge to zero asymptotically (as is preicte by Eq. 11 i practice. We explore this i more etails i Sectio VI. 2 The receiver may ot ecie o roppig packets irectly base o. For istace it may use a checksum bit i the simplest sceario. Sice ay other use measure is a fuctio of, we fi it useful to express f rop,ieal as a fuctio of this fuametal quatity. Fuctio f rop,ieal ca have ifferet forms as a fuctio of, epeig o the packet rop mechaism. Eq. 7 assumes that the receiver rops the packets whe <SNR Thresh.. 3 this assumes that is a wie-sese statioary process. (8

3 V. SCENRIO 2: KEEPING LL THE PCKETS I this sectio we ivestigate the case where the receiver keeps all the packets. I geeral, fuctio f woul have ifferet forms epeig o the eviromet a trasmitter/receiver esig. We pursue the aalysis with the followig f: for 0. f keep ( (12. Performace Evaluatio I this sectio we will fi a expressio for.usig Eq.6a12wewillhave, γ k, (13 where γ k with 1 γ k. Itisshowi ppeix that averagig Eq. 13 over γ k will result i the followig (isert c 1, a a 2 i Eq. 30: +1,0 2 S(, (14 where S(ζ e ζ Expit(ζ, for a arbitrary ζ, with Expit represetig expoetial itegral: Expit(ζ e t ζ t t. +1,i refers to the case that +1 is average over γ k, γ k 1,..., γ k i a +1 +1,k. Isertig as a fuctio of 1, we will have, +1,0 2 S( (1 + γ k (15 1 We ee to take a average over 4 γ k 1. ppeix B shows that the followig equality hols for a q > 0 a a expoetially istribute raom variable γ with γ 1 : q S( (1 + qγ S( q q 1 1 S( 1 1 i 1 > 1 (16 Usig Eq. 16 for i 1, we will have, 1 +1,1 2 S( S( 1 1. ( It ca be see from Eq. 14 a 17 that we will have the followig after m +1 steps of averagig: m +1,m B i,m S(. (18 m The goal is to fi B i,m for m k. Let D k (i, m S( m. The, +1,m m B i,m D k (i, m (19 4 We ca take averages over γ i sfor0 i k oe at a time sice they are assume iepeet (see Sectio II with B 0,0 2. Substitutig m as a fuctio of m 1 a averagig over γ k m 1 (usig Eq. 16 will result i the followig for 1 m k 1: +1,m+1 m + m+1 z1 m+1 B i,m m B i,m m B i,m B z 1,m D k (i +1,m+1 D k (0,m+1 D k (0,m+1 ξ z D k (z,m +1 z0 B z,m+1d k (z,m +1, (20 where ξ i 1 1 a the last equality is writte usig Eq. 19. Therefore for 0 z m +1, { m B i,m ξ B z,m+1 i+1 z 0 (21 z 0 B z 1,m ξ z The +1,k will be as follows as a fuctio of P 0 : +1 +1,k k B i,k e P 0 Expit(, (22 P 0 where B i,k for 0 i k is calculate usig B i,k 1 for 0 i k 1 (usig Eq. 21. This meas that +1 is calculate usig the coefficiets of. B. Stability of the Estimatio Process I this part we show that the average error variace always stays boue as log as SNR 0.Wehave, (23 + Let Z r represet average of Z with respect to r for a arbitrary raom fuctio Z a a raom variable r. Z represets average of Z with respect to all its variables. First we take a average of +1 with respect to.itis easy to show that the fuctio Z(r r r+u is cocave for ay r 0 a u 0. Therefore we will have the followig by first applyig the Jese s iequality a the averagig over : +1 Pk Pk e / Expit(/. (24 The last equality of Eq. 24 is prove i ppeix. Next we examie the coitios uer which the upper bou of

4 +1 is smaller tha.wewat, 2 e / Expit(/ <. (25 Call α 1, α 2 a ζ α1 α 2, the we are itereste i fiig those coitios uer which g(ζ 2 ζe ζ Expit(ζ < 1 (26 Fuctio g has the followig characteristics: 1 g(0 0 2 g( 2 3 g(ζ is a icreasig fuctio of ζ for ζ 0 where 1 a 2 ca be cofirme easily a 3 is prove i ppeix C. If < 1 the Eq. 26 will always hol as expecte. Here, however, we are itereste i estimatio of ustable processes, i.e. > 1. Therefore for a ζ 0 > 0, we will have g(ζ 0 1meaig that, Therefore, ζ<ζ 0 g(ζ < 1 (27 if > +1 < (28 ζ 0 It ca be see from Eq. 28 that as is icreasig, oce it reaches ζ 0, it will ecrease. Therefore, uless, the average estimatio error variace stays boue. Case of correspos to the case of SNR 0. This is the oly case where the estimatio process will get ustable, as oe woul expect. Such a case results i zero Sigal to Noise Ratio i every trasmissio, which makes it a ope loop process. However, for ay SNR > 0, the process is stable. VI. COMPRING THE TWO SCENRIOS To cofirm the mathematical erivatios of the past sectios, we simulate wireless Kalma filterig of a system with liear yamics i a faig eviromet. The followig parameters are chose for this example: 4,.1 a x 0 5. First we look at sceario 2, the case i which iformatio of all the packets is use i the receiver. Soli lies of Fig. 1 show the performace evaluate through aalysis (Eq. 22 for sceario 2. To cofirm mathematical erivatios of sceario 2, we also simulate this system. We ca see that the simulatio results (stars of Fig. 1 cofirm the mathematical erivatios. Fig. 1 shows average error variace at ifferet SNR. s ca be see the error stays boue eve for SNR as low as -40B if the iformatio i all the packets is use properly i the receiver. To see how much we lose by roppig the erroeous packets, ext we compare the performace of the two aforemetioe scearios. For sceario 1, case of roppig erroeous packets, fuctio f rop,ieal of Sectio IV assume that as log as SNR is above SNR Thresh.,the 4599 commuicatio oise is egligible. s we wat to compare the two scearios, we have to use a more practical versio of fuctio f for sceario 1. This will result i the followig fuctio: f rop ( { SNR Thresh. (29 else Fig. 2 shows the performace of the two scearios as a fuctio of SNR a for ifferet SNR Thresh. s. We ca see that roppig erroeous packets results i istability. Performace loss icreases as SNR Thresh. icreases. I this simulatio, for sceario#1, we cosiere the cases where the iformatio o the quality of the lik is use by the estimator for those packets that are kept i the receiver. If such a cross-layer feeback is ot available, the the performace of sceario#1 woul be eve worse. The graph cofirms that keepig all the packets a sharig iformatio of the physical layer (commuicatio oise variace with the applicatio layer ca improve the performace cosierably. VII. CONCLUSION I this paper, we cosiere the wireless sesig/estimatio problem i faig eviromets which is key to may mobile sesor etworks. We showe that applyig the same esig strategies of ata etworks to sesor etworks may lea to performace egraatio sice sesor etwork applicatios are typically elay sesitive a therefore may require ifferet esig priciples. I particular, we looke at the impact of roppig erroeous packets, which is oe i ata etworks, o the estimatio quality i sesor-etwork applicatios. We showe that proper use of the iformatio cotaie i all the packets ca improve the performace cosierably compare to the case of roppig erroeous packets. We erive a mathematical expressio to evaluate the performace whe the receiver uses all the packets for oe class of commuicatio oise characteristics. We also prove that keepig all the packets ca prevet the potetial istability that is itrouce if erroeous packets are roppe. Our mathematical results were also cofirme by simulatios. Fially our aalysis emphasize the importace of sharig iformatio betwee the physical a applicatio layers. VIII. FUTURE WORK We are workig o exteig the work presete i this paper to ifferet f fuctios that woul characterize commuicatio oise of ifferet eviromets a protocols. Furthermore, we are workig o exteig the work to more geeral cases where R 0, Q 0a C 1a to the vector case. REFERENCES 1 C. Chog a S. Kumar, Sesor etworks: evolutio, opportuities a challeges, Proceeigs of the IEEE, vol. 91, issue 8, ug. 2003, pages:

5 Sceario 2 Sceario 2 B. Siopoli, C. Sharp, L. Scheato, S. Schaffert a S. Sastry, Distribute cotrol applicatios withi sesor etworks, Proceeigs of the IEEE, vol. 91, issue 8, ug. 2003, Pages: B.Siopoli, L. Scheato, M. Fraceschetti, K. Poolla, M. Jora, S. Sastry, Kalma filterig with itermittet observatios, Proceeigs of the 42 IEEE Coferece o Decisio a Cotrol, Dec. 9-12, 2003, Volume: 1, Pages: X. Liu a. J. Golsmith, Kalma Filterig with Partial Observatio Losses, 43r IEEE Coferece o Decisio a Cotrol, N. C. Martis a M.. Dahleh, Fuametal limitatios of feeback cotrol with commuicatio costraits, 6 William Jakes, Microwave Mobile Commuicatios. IEEE Press, T.Kailath,.H.Saye,B.Hassibi,Liear Estimatio. Pretice Hall iformatio a system scieces series IX. PPENDIX Let γ be a expoetially istribute raom variable with γ 1. The we will have the followig for arbitrary a 0, c 0 a 0: a e 0 γ a c+γ a a c e c e c c+γ γ e z z z Expit( c (30 X. PPENDIX B Let γ be a expoetially istribute raom variable with γ 1.Leta c represet positive scalars where <. The we will have, S(γ + c e γ+c Expit(γ + c 0 e ( γ+c Expit(γ + cγ e ( γ+c Expit(γ + c γ γ0 + e γ 0 γ+c γ e c Expit(c + e c Expit( c S(c c S( (31 Isertig c q a will result i Eq. 16. XI. PPENDIX C Let g(ζ 2 ζe ζ Expit(ζ for ζ 0. The we will have, g(ζ 2 e ζ (1 + ζexpit(ζ 1. (32 ζ Sice e c c e c ζ > e c (c+2 (c+1 2 c c > resultig i ζ for a arbitrary c 0, we will have, e c (c +2 c for ζ 0, (33 (c +1 2 Expit(ζ > e ζ 1+ζ for ζ 0 g(ζ ζ 2 e ζ (1 + ζexpit(ζ 1 > 0 ζ 0. (34 Hece 5 g(ζ is a icreasig fuctio of ζ for ζ 0. at k20 Fig. 1 Performace of sceario 2 (keepig all the packets 10 6 alysis Sim. SNR 40B SNR40B SNR20B SNR30B SNR0B SNR10B SNR 20B SNR 10B SNR 30B k Fig. 2 Comparig the two scearios Sceario2: keepig all packets Sceario1: SNR Thresh 40B Sceario1: SNR 10 5 Thresh 30B Sceario1: SNR Thresh 20B Sceario1: SNR Thresh 10B Sceario1: SNR 0B Thresh SNR 5 g(ζ ζ is oly zero at ζ. 4600

CDS 270-2: Lecture 6-3 Optimum Receiver Design for Estimation over Wireless Links

CDS 270-2: Lecture 6-3 Optimum Receiver Design for Estimation over Wireless Links CDS 70-: Lecture 6-3 Otimum Receiver Desig for stimatio over Wireless Lis Goals: Yasami Mostofi May 5, 006 To uderstad imact of wireless commuicatio imairmets o estimatio over wireless To lear o-traditioal