Recursive Estimation and Identification of Time-Varying Long- Term Fading Channels

Transcription

1 Recursive Esimaion and Idenificaion of ime-varying Long- erm Fading Channels Mohammed M. Olama, Kiran K. Jaladhi, Seddi M. Djouadi, and Charalambos D. Charalambous 2 Universiy of ennessee Deparmen of Elecrical and Compuer Engineering 508 Middle Dr. Knoxville, N 37996, USA molama@u.edu, jaladhi@u.edu, djouadi@ece.u.edu 2 Universiy of Cyprus Deparmen of Elecrical and Compuer Engineering 75, Kallipoleos Sree P.O.Box , Nicosia, Cyprus chadcha@ucy.ac.cy Absrac: his paper is concerned wih modeling of ime-varying wireless long-erm fading channels, parameer esimaion, and idenificaion from received signal srengh daa. Wireless channels are represened by sochasic differenial equaions, whose parameers and sae variables are esimaed using he expecaion maximizaion algorihm and Kalman filering, respecively. he laer are carried ou solely from received signal srengh daa. hese algorihms esimae he channel pah-loss and idenify he channel parameers recursively. Numerical resuls showing he viabiliy of he proposed channel esimaion and idenificaion algorihms are presened.

2 I. INRODUCION his paper is concerned wih he developmen of ime-varying (V) long-erm fading (LF) wireless channel models based on sysem idenificaion and esimaion algorihms o exrac various parameers of he LF channel using received signal measuremens. V wireless channel models capure boh he space and ime variaions of wireless sysems, which are due o he relaive mobiliy of he receiver and/or ransmier and scaerers []-[3]. In he V models he saisics of channel are ime-varying. his conrass wih he majoriy of published wor ha mainly deals wih ime-invarian (saic) random models or simple free space model, where he channel saisics do no depend on ime [4]-[6]. In ime-invarian models, channel parameers are random bu do no depend on ime, and remain consan hroughou he observaion and esimaion phase. his conrass wih V models, where he channel dynamics become V random (sochasic) processes []-[3]. he V LF channel model is discussed in [2] and represened by sochasic differenial equaions (SDEs). We propose o esimae he V power pah-loss of he LF channel and is parameers from received signal srengh daa, which are usually available or easy o obain in any wireless newor. he expecaion maximizaion (EM) algorihm [7] and Kalman filering [8] are employed in he idenificaion and esimaion of he channel parameers and pah-loss. he proposed idenificaion and esimaion algorihms are recursive and only based on received signal measuremens. Numerical resuls are provided o deermine he performance of he proposed esimaion algorihm. he paper is organized as follows. In Secion II, he V LF mahemaical channel model is inroduced. In Secion III, he EM algorihm ogeher wih he Kalman filer, o esimae he channel parameers as well as he channel power loss from signal measuremens, is developed. In Secion IV, numerical resuls are presened. Finally, Secion V provides he conclusion. 2

3 II. V LF WIRELESS CHANNEL MAHEMAICAL MODEL Wireless channels suffer from shor-erm fading (SF) due o mulipah, and LF due o shadowing depending on he geographical area. In suburban areas, which are populaed wih less obsacles lie vehicles, buildings, mounains and so forh, is communicaion signal undergoes phenomenal LF (lognormal shadowing) [5]. For such propagaion environmens, he random process power loss (PL) in db, { } 0, 0 X (, τ ), which is a funcion of boh ime and space τ τ represened by he ime-delay τ, is generaed by a mean-revering version of a general linear ime-varying SDE given by [2], [3]: 0 ( ) dx (, τ) = β(, τ) ( γ(, τ) X (, τ) d + δ(, τ) dw ( ), X (, τ) N ( PLd ( )[ db]; ) 2 = σ 0 () { } 0 where W() is he sandard Brownian moion (zero drif, uni variance) which is assumed o be independen of ( ) 0, variance κ, and ( )[ ] X τ, N ( μ; κ ) denoes a Gaussian random variable wih mean μ and PL d db is he average PL in db. he parameer (, ) γ τ models he average V PL a disance d from ransmier, which corresponds o PL( d)[ db ] a d indexed by. his model racs and converges o his value as ime progresses. he insananeous drif (, )( (, ) X(, )) represens he local mean while β (, ) β τ γ τ τ deviaion. Noe ha β (, ) mean value associaed wih (). τ represens he local sandard τ can be seleced o conrol he speed of adjusmen owards a specific In [2] and [3] his model is shown o capure he spaio-emporal variaions of he propagaion { } 0 environmen as he random parameers β(, τ), γ (, τ), δ (, τ) can be used o model he V characerisics of he LF channel. he received signal, y( ), a any ime can be expressed as: y() = s( ) H( ) + v( ) (2) 3

4 where s() is he informaion signal, v( ) is he channel disurbance a he receiver, and H( ) is he signal aenuaion coefficien defined by (, τ ) H () e X, where ln(0) / 20 = [5]. he general spaio-emporal lognormal model in () and (2) can be realized by a sochasic sae space sysem given by: (, τ) = (, τ) (, τ) + (, τ) ( ) X (, ) () = () τ + () () X A X B w y s e D v where A (, τ ) = β(, τ), B(, τ ) δ (, τ) β(, τ) γ (, τ) = and w () [ dw ] = (). he above sysem parameers and sae variable values are esimaed from received signal measuremens. he EM algorihm and Kalman filering are employed in he sysem parameers and sae esimaion, respecively. hese algorihms are inroduced nex. III. WIRELESS CHANNEL ESIMAION VIA HE EM ALGORIHM AND KALMAN FILERING his secion describes he procedure employed o esimae he channel model parameers and saes associaed wih he sae space model in, based on he EM algorihm [7] ogeher wih Kalman filering [8]. Since he esimaion and idenificaion processes are carried ou in discree insans, we consider a sampled version of he sae space model in discree-ime as: x = + Ax + Bw y = se + Dv x (4) where n x R is a sae vecor, d y R is a measuremen vecor, m w R is a sae noise, and d v R is a measuremen noise. Noe ha he sae space model is nonlinear since he oupu equaion in (4) is nonlinear. he channel parameers { A, B, D} θ = are unnown and esimaed ogeher wih he pahloss represened by he sysem saes x from a finie se of received signal measuremen daa, 4

5 N {,,..., } Y = y y y. he proposed mehodology is recursive and based on he EM algorihm 2 N combined wih he exended Kalman filer (EKF). he laer is used due o he nonlinear oupu equaion. A. Channel Sae Esimaion: he EKF he EKF approach is based on linearizing he nonlinear sysem model (4) around he previous esimae. I esimaes he channel saes measuremensy. I is described by he following equaions [8]: x for given sysem parameer { A, B, D} θ = and x ( ) ( ) xˆ = Axˆ + P C D y C Axˆ xˆ 2 = Axˆ d e C s se x = = dx x = xˆ x = xˆ (5) where = 0,,2,..., N, and P is given by: P = P + A B A 2 P = C D C + B B P A B P = AP A + B 2 (6) where B = BB, 2 D = DD. he channel parameers θ = { A, B, D} are esimaed based on 2 he EM algorihm, which is inroduced nex. B. Channel Parameer Esimaion: he EM Algorihm he EM algorihm uses a ban of Kalman filers o yield a maximum lielihood (ML) parameer esimae of he sae space model. I is an ieraive scheme for compuing he ML esimae of he sysem parameers θ, given he se of daa Y. Specifically, each ieraion of he EM algorihm consiss of wo seps: he expecaion sep and he maximizaion sep [9]. he filered expecaion sep only uses filers for he firs and second order saisics. he algorihm 5

6 yields parameer esimaes wih nondecreasing values of he lielihood funcion, and converges under mild assumpions [0]. he expecaion sep evaluaes he condiional expecaion of he log-lielihood funcion given he complee daa as: (, ˆ df θ Λ θ θ) = Eθ log Y l dfˆ θ (7) F denoes a family of probabiliy measures induced by he sysem parameers where { θ ; θ Θ} θ, and ˆ θ denoes he esimaed sysem parameers a ime-sep. he maximizaion sep finds: ( ˆ ) ˆ θ arg max Λ θ, θ (8) + θ Θ he expecaion and maximizaion seps are repeaed unil he sequence of model parameers converge o he real parameers. he EM algorihm is given by [7], [9]: ˆ A = E xx Y E xx Y = = ( xx ) A ( xx ) ( ) + ( ) ( yy ) ( yx ) C = ( ) + ( ) ˆ 2 B = E (( x Ax )( x Ax ) ) Y E Y = = = xx A A x x A ˆ 2 D = E ( ( y Cx)( y Cx) ) Y E Y = = C y x C x x C (9) where E( ) denoes he expecaion operaor, and = 0,,2,..., N. he sysem parameers { ˆ, ˆ } 2, ˆ 2 A B D are compued from he following condiional expecaions [7]: () (2) L = E xqx Y ; L = E x Qx Y = = (4) L = E xrx + x Rx Y ; L = E xsy + ysx Y = = (0) where Q, R and S are given by: ee i j + ejei ee i j ee i Q= ; R= ; S = ; i, j =,2,... n; =,2,.. d ()

7 in which e i is he uni vecor in he Euclidean space; ha is e i = in he ih posiion, and 0 elsewhere. For insance, consider he case n = d =, hen E xx Y is: = E x x Y L R he oher erms in (9) can be compued similarly. () (2) (4) he condiional expecaions {,,, } follows: ) Filer esimae of () L : = = (2) = 2 L L L L can be esimaed from measuremens Y as () L = E xqx Y = () () = r ( N P ) r ( N P ) x P r + 2x P r x N x () () () 2 () 2 = = + x B AP N P A B x (3) where r( ) denoes he marix race. In (3), () r and () N saisfy he following recursions: ( ) 2 ( ) r = A P C D C A r + P Qx P N P C D y C x () () r = Ar () r0 = 0m () 2 () () 2 () 2 () 2 N = B AP N P A B 2Q () N0 = 0m m (4) 2) Filer esimae of (2) L : L = E x Qx Y = E x Qx Y + E x Qx Y E x Qx Y herefore, θ{ 0 0 } θ θ{ } (5) (2) = = 3) Filer esimae of (2) L can be obained from L : () L. 7

8 L = E ( xrx + x R x) Y = = r ( N P ) r ( N P ) x P r + 2x P r x N x 2 2 = = + x B AP N P A B x (6) In his case, r and N saisfy he following recursions: ( ) ( ) ( 2 2 ) r = A P C D C A r P N P C D y C x + P R+ P B A P R A x r r N = B AP N P AB 2RP AB 2B AP R N0 = 0 m m = Ar 0 = 0m (7) 4) Filer esimae of where (4) L : L = E x Sy + y S x Y = x P r x P r ( ) ( ) (8) (4) (4) (4) = = (4) r saisfy he following recursions: r = ( A P C D C A ) r + 2P Sy (4) (4) r = Ar (4) r0 = 0m (4) 2 (4) (9) Using he filers for L ( i =,2,3,4) and he exended Kalman filer described in (5) and (6), () i he sysem parameers { A, B, D} θ = are esimaed hrough he EM algorihm described in (9). Numerical resuls ha show he applicabiliy of he above algorihm are discussed nex. IV. NUMERICAL RESULS In his secion, he accuracy of he EM algorihm ogeher wih he exended Kalman filer o esimae channel parameers, as well as channel PL from he received signal measuremens, is deermined. he measuremen daa are generaed by he sysem parameers: 2/ 0π γ (, τ) = γm ( τ) + 0.5e sin, δ (, τ) = 5, β(, τ) = 0.2 (20) 8

9 where γ ( ) m τ is he average PL a a specific locaion τ and is chosen o be 25 db, is he observaion inerval, and he variances of he sae and measuremen noises are 0-2 and 0-6, respecively. Figure shows he acual and esimaed received signal using he EM algorihm ogeher wih he exended Kalman filer for 500 sampled daa. From Figure, i can be noiced ha he received signal have been esimaed wih very good accuracy. Figure 2 shows he received signal esimaes roo mean square error (RMSE) for 00 runs. I can be noiced ha i aes jus few ieraions (less han 5) for he filer o converge, and he seady sae performance of he proposed channel esimaion algorihm using he EM ogeher wih Kalman filering is excellen. Since our sochasic model in is firs order, he compuaional cos of he proposed esimaion algorihm is very low and can be implemened on-line. Moreover, he filers of he expecaion sep are recursive and decoupled and hence easy o implemen in parallel on a muliprocessor sysem [9]. V. CONCLUSION his paper develops a general scheme for exracing mahemaical LF channel models from noisy received signal measuremens. he proposed esimaion algorihm is recursive and consiss of filering based on he exended Kalman filer o remove noise from daa, and idenificaion based on he EM algorihm o deermine he parameers of he model which bes describe he measuremens. he proposed esimaion and parameer idenificaion algorihms esimae he pah-loss and he channel parameers. Performance of he laer is invesigaed hrough a numerical example ha shows excellen resuls. herefore he proposed algorihms have good poenial for real-ime applicaions. Fuure wor includes combining idenificaion and esimaion wih oher performance requiremens, such as, power conrol, admission conrol, and base saion assignmen. 9

10 REFERENCES [] C.D. Charalambous, S.M. Djouadi, and S.Z. Denic, Sochasic power conrol for wireless newors via SDE s: Probabilisic QoS measures, IEEE rans. on Informaion heory, vol. 5, No. 2, pp , Dec [2] M.M. Olama, S.M. Djouadi, and C.D. Charalambous, Sochasic power conrol for imevarying long-erm fading wireless newors, EURASIP Journal on Applied Signal Processing, vol. 2006, Aricle ID 89864, 3 pages, [3] M.M. Olama, S.M. Shajaa, S.M. Djouadi and C.D. Charalambous, Sochasic power conrol for ime-varying long erm fading wireless channels, Proceedings of he American Conrol Conference, pp , June 8-0, [4] J. Proais, Digial Communicaions, 4 h Ediion, McGraw Hill, 200. [5].S. Rappapor, Wireless Communicaions: Principles and Pracice, Prenice Hall, 2 nd Ediion, [6] P.M. Shanar, Error raes in generalized shadowed fading channels, Wireless Personal Communicaions, vol. 28, no. 3, pp , [7] C.D. Charalambous and A. Logoheis, Maximum-lielihood parameer esimaion from incomplee daa via he sensiiviy equaions: he coninuous-ime case, IEEE ransacion on Auomaic Conrol, vol. 45, no. 5, pp , May [8] G. Bishop and G. Welch, An inroducion o he Kalman filers, Universiy of Norh Carolina, 200. [9] R.J. Ellio and V. Krishnamurhy, New finie-dimensional filers for parameer esimaion of discree-ime linear Guassian models, IEEE rans. On Auomaic Conrol, vol. 44, no. 5, pp , 999. [0] C.F.J. Wu, On he convergence properies of he EM algorihm, Annals of Saisics, vol., pp ,

11 Received Signal Esimaed 0.2 Real Samples Figure. Real and esimaed received signal for he channel model. 0.6 RMSE Samples Figure 2. Received signal esimaes RMSE for 00 runs using he EM algorihm ogeher wih he exended Kalman filer.