Time of Arrival Based Location Estimation for Cooperative Relay Networks

Transcription

1 2010 IEEE 21 st Intemational Symposium on Personal Indoor and Mobile Radio Communications Time of Arrival Based Location Estimation for Cooperative Relay Networks Hasari Celebi *, Mohamed Abdallah *, Syed I. Hussain *, Khalid A. Qaraqe *, Mohamed-Slim Alouini t *Electrical and Computer Engineering, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar hasari.celebi,mohamed.abdallah,syed.hussain,khalid.qaraqe@qatar. tamu.edu t Electrical Engineering Program, KAUST, Thuwal, Saudi Arabia mohamed.alouini@kaust.edu.sa Abstract-In this paper, we investigate the performance of a cooperative relay network performing location estimation through time of arrival (TOA). We derive Cramer-Rao lower bound (CRLB) for the location estimates using the relay network. The analysis is extended to obtain average CRLB considering the signal fluctuations in both relay and direct links. The effects of the channel fading of both relay and direct links and amplification factor and location of the relay node on average CRLB are investigated. Simulation results show that the channel fading of both relay and direct links and amplification factor and location of relay node attect the accuracy of TOA based location estimation. Index Terms-Location estimation, time of arrival, time delay estimate, cooperative ranging, cooperative communications. I. INTRODUCTION Cooperative communications is a technique to create a virtual antenna array using several distributed single antenna nodes in the system [1]. It helps in increasing the area of coverage without the need of increased transmission power. As the destination receives multiple copies of the source's signal, it also improves the diversity order [2]. The performance of cooperative networks has been thoroughly investigated in the past for various system models, protocols, forwarding techniques and fading environments, e.g [3], [4], [5]. Location estimation is another crucial process in cooperative relay networks likewise to the other type of wireless communications networks [6]-[10]. For instance, the range and location information can be used for network authentication, localization or cluster forming in cooperative networks. In wireless sensor networks, cooperative localization problem has been studied [11]-[13]. However, one of the main differences between cooperative localization in wireless sensor networks and localization in cooperative relay networks is that the nodes in cooperative relay networks relay signal according to different relay strategies. In our problem, relays are used to improve the signal quality while they can be used for location estimation. As a result, we need to understand the effects of using relay strategies on the performance of localization. Therefore, in this paper, we study the fundamental limits for the Time of Arrival (TOA) based location estimation in cooperative relay networks. The Cramer-Rao lower bound (CRLB) of location estimate for the considered system is derived. More specifically, the instantaneous CRLBs for the TOA based location estimate are derived. These bounds are extended to obtain the average CRLBs that take signal fluctuations in both relay and direct links into account. The effects of the channel fading of both relay and direct links and amplification factor and location of relay node on average CRLB are investigated through computer simulations. The remainder of the paper is organized as follows. In Section IT, the system model for the cooperative relay network is provided. In Section III, CRLB of time delay estimate is presented. In Section N, numerical results and discussions are provided. Finally, the conclusions are presented in Section V. II. SYSTEM MODEL We consider a system in which a source node S is present at an unknown point (x, y). The destination D which may be a base station or a wireless access point located at (Xd, Yd) wants to determine the location of the source. Normally, triangulation method requires minimum three reference signals (i.e. destinatin node and two relay nodes). However, it is possible to estimate the location of S using only destination node D and relay node R by employing Hybrid TONAOA location estimation method [14]. In this method, it is assumed that the destination node D have the AOA knowledge of S. As a result, we assume that there exists only one relay node in our case. However, the analysis can be easily extended to multiple relay node case. In our model, the source transmits its signal to the destination through a relay R present between the source and the destination at some known locations (xr' Yr) and through direct link to D. The S --+ R distance is db R --+ D distance is d2 do do di +d 2 whereas S --+ D distance is and We assume slow and flat Rayleigh fading environment with the first and the second hop channel coefficient as hi and h2' respectively, and the direct link channel gain is ho. The noise terms in each hop are ni, n2 and no respectively. The noise is assumed to be additive white Gaussian with zero mean and unit variance (i.e.no 1). The system model is shown in Fig. 1. The baseband transmit signal s(t) at the source node with absolute bandwidth of B is given by s(t) L blp(t - lts), (1) I where bl is the known real data for the lt h symbol, p( t) is the pulse signal with energy Ep and duration Tp, i. e., p(t) for t f/. [0, Tp] and Ts is the symbol duration. The baseband /10/$ IEEE 871

2 representation of the received signal Yr(t) at the relay node is given by Yr(t) JP:h1S(t -71) + n1(t), (2) where 71 dd c 1 is the delay for the S -+ R link and Ps is the available transmit power at the source. At the relay node, different relaying strategies can be employed. Due to ease in its implementation, we consider amplify-and-forward (AF) strategy, however, the analysis can be easily extended to other strategies. We assume that the channel state information (CSI) of the first hop is available at the relay. The amplification gain a at the relay is set in such a way that it reciprocates the average channel gain of the first hop and maintains a constant output power. If the power available at the relay is Pr, then a Pslh'2+No' Ihl12 1E(lhlI2) where is the average channel gain in the first hop and No is the noise variance of the first hop which is assumed to be 1 in the system model. After amplifying Yr (t) signal with the gain a, the received signal Yd(t) at the destination node takes the following form (3) where 72 d2/ c is the delay for the R -+ D link. After some straightforward manipulations, (3) is reduced to the following form Yd(t) JP:ah1h2s(t -7T) + nt(t), (4) where 7 T is the total time delay over sourcerelay-destination link, which is referred to as the relay link in this paper. The noise term nt(t) is a zero mean additive white Gaussian noise with variance of O'f CN(o,O'f)) a2h + 1 (i. e., Similarly, the signal received at D through the direct link is Yo(t) JP:hos(t -70) + no(t), (5) where 70 is the time delay of the direct link. The problem is boiled down to the estimation of (x, y) using TOA of yo(t) and Yd(t) at the destination node, which is discussed in the following section. III. CRAMER-RAO LOWER BOUND In this section, we present the location estimation of (x, y) through the TOA of the direct and the relayed signals at the destination. We derive both instantenous and average CRLB expression of the location estimate considering the signal fluctuations. Since we assume that the destination is aware of the location of the relay, 72 can be considered as a known parameter. Hence, the estimate of 7 T is actually used to determine 71 only. Similarly, the destination will also estimate 70 from the directly received signal. Let 0 [x ylt represents the vector of unknown signal parameters, where a, hi> h2, (Xd, Yd) and (xr, Yr) are assumed to be known. The observation interval 1 C is the speed of light. (x"y,) Fig. 1. System model for time of arrival based location estimation using cooperative relaying. [0, Tl is considered and it can be expressed as T NTs, where N is the number of observation symbol. The delay estimates for the direct and the relay links can be approximated as [15] TO , T , (6) where 100 '" N(O, N(O, a) are Gaussian distributed estimation errors of the direct and the relay link, respectively. ';5) and 101 '" The estimated delays in each link are related to the source location as follows 70 ( v(x -Xd)2 + (y-yd)2), 71 (V(X-Xr)2+(Y-Yr)2). Now, the Fisher information matrix (FIM) can be determined by [16] (7) Je lee [: 0 Inf(rIO)(: o Inf(rIO)) T ], (8) where f (r 10) is the joint pdf of TO and T1 given 0 and lee is the expectation operator conditioned over O. Since TO and T1 are independent of each other and the estimation errors are Gaussian distributed, we can write f(rio) ex exp [ (TO -70)2]. exp [ - 2 r (T1-7t}2] In the above, f( rio) is a function of r only, but as per (7), it is in turn a function of O. Therefore, we use the following chain rule in order to obtain the FIM : 0 Inf(rIO) ;. : r Inf(rlr). (10) Hence, FIM can now be written as Je ; [IE,. [! lnf(rlr) (: r Inf(rlr)) T ] 1 (;) T, (9) (11) 872

3 where at [ cos cos' ] a(j c sin sin ', (12) such that,i. tan-l( ) and.1. tan-l(jcjb::.) which 'of' X-Xd tp X-Xr are illustrated in Fig. 1 and let [: T In f( TIT) C! In f( TIT)) T ] diag(:5 : ). ' (13) Finally, the CRLB of the estimates of x and y can be given as c 2(C2 '>osm c2. 'f/ '>lsm 2,1.) 'f' CRLB. x c2 sin2(, ), (e5 cos2 ' + e cos2 ) CRLB. sin2(' - ) _ and (15) y where e5 and e are the CRLBs for estimates TO and Tl, respectively and they are defined in the following subsection. A. Average CRLB (14) The above CRLBs are based on instantaneous values of the parameters used. In this section, we derive average values of CRLBs to have more insight about the location estimation through a relay. In (14) and (15), e5 represents the CRLB of the estimate of TO through the direct link, which can be given as [7] 2 1 eo ---, (16) 'Yo Eo where 'Yo Psh5 is the instantaneous SNR of the direct link and Eo Jo T [8'(t-TO)]2 dt. Similarly, e is the CRLB of the estimate of Tl through the relay link. It is important to note that though is known, the T2 variance in the estimate of Tl depends upon the total noise of the relay link i. e. (Tf a2 h + 1. Hence, e can be written in terms of end-to-end SNR 'Yr of the relay link [see Appendix] 2 1 el -, (17) 'YrEr )] 1';2 such that - where Er Jo T [8'(t-T T 2 dt and 'Yr "Y 'Yl Psh is the SNR of the first hop and 'Y2 Prh is the SNR of the second hop in the relay link. It is well known that the end-to-end SNR of a relay link is half of the harmonic mean of both hops individual SNRs [3]. Hence (18) 'Yr 'Yl 'Y2 Since, the channel coefficients are Rayleigh distributed, per hop SNRs would be exponentially distributed. Therefore, pdf of the direct link SNR or individual per hop SNRs can be given as 1 / - p "Y ib) -e-"y "Yi, i0,1,2. (19) 'Yi where 'Vi is the average SNR of each link. Now, replacing (16)-(19) in (14), the average CRLB for x can be evaluated as c2 [sin2' sm (' - ) 'Yo Eo _ 0 sin sin Avg. CRLBx _- -e-"y/"yld'y + -e-"y/"y2d'y --_- -e-"y/"yod'y+ 'VI Er 0 'Y 'V2Er 0 'Y. (20) It is obvious that due to the presence of 'Y in the denominator, each integral in the above becomes infinite. However, if z is an infinitesimally small positive number i.e. z 0+, the above will approach to A CRLB - c2el(z) [sin2' Sin2 ( vg. x )l -. sm (' - ) 'VoEo Er 'Yl 'Y2 where El(z) is the exponential integral. Similarly, the average CRLB for i) can be written as A vg. sm (' - ) 'VoEo Er 'Yl 'Y2 (21) CRLB -- c2el(z) [cos2' COS2 ( )l ii (22) Based on the relations derived for the CRLB of the location estimation of the (x,y) coordinates, we note the following: In order to minimize the estimation error for both location estimates, the difference between the relay angle and destination angle relative to the source location should be equal to 90. It is also clear that a difference equal to zero, or equivalently the relay is collinear with the source and destination, results in CRLB equal to 00 and hence the estimation of the source location will fail. The CRLB of the location estimates of x and y is equal when, -45 and 45. Choosing any set of angles with 90 difference will results in different CRLB for x and y. In particular, setting 0 and ' 90 results in being the CRLB for x and y depends solely on ( and (, respectively. Note that and ( are the average CRLB for TO and TIo respectively. Therefore, in the rest of the paper, we will assume that the CRLB for x and y are equal unless otherwise stated. IV. RESULTS In this section, numerical results are presented in order to verify the analysis. More specifically, the effects of channel gain of relay and direct links and amplification factor and location of relay node on the CRLB are investigated. The following simulation parameters are used throughout the simulations. For the pulse shape, the following Gaussian second order derivative pulse shape is employed p(t) A (1-4 t) e- 21rt2/(.2, (23) where A and ( are parameters that are used to adjust the pulse energy and the pulse width, respectively. A is selected in order 873

4 to generate pulses with unit energy. For the given pulse shape, pulse width is defined as Tp 2.5( [10], where ( 1/ B and B IMHz. The number of training symbol N that is considered is 1. The results are obtained over 1000 channel realizations. It is assumed that h, hi, and h2 are Rayleigh random variables. Finally, we assume that the angles are set to <p 45 and 7/J -45, respectively. In Fig. 2 the effects of the channel variances on the CRLB are investigated. We define the parameters K where the channel variances are defined as follows; ah1 i+jt and a 2 :;}K' It is evident that the value of K 1 for which a o a 1 a 2 achieves the best CRLB for all values of SNR. In Fig. 3 the effects of channel distortion incurred by the second hop of the relay signal are investigated. In particular, we capture this effect by varying the second hop SNR using the power transmitted at the relay where setting Pr 00 is equivalent to perfect second hop channel. The parameters used are Pr [ ] and Ps 1, K 1. The figure shows that increasing the second hop SNR improves the CRLB as expected. However, such improvement is noticeably diminishing as we increase the SNR especially for high values. In particular, increasing Pr from 1 to 4 results in 2 db improvement in the RMSE while increasing Pr from 4 to 9 results in a maximum improvement of 0.4 db for low SNR values and almost negligible improvements in high SNR values. Fig. 4 shows how CRLB is affected with the variations in the relay location. We keep <p 45 and 7/J -45 while moving the relay along a straight line maintaining this constraint. It is evident from the figure that as the relay moves away from the source the CRLB degrades. In fact, due to the angle constraint considered to make both CRLBs the same, moving the relay away from the source also increases the R -+ D distance. Hence, SNRs in both hops are reduced and location estimation becomes inaccurate. This can be overcomed by increasing the transmit power of the relay. The curves drawn at Pr [1 4 9] while keeping Ps 1 and K 1, show that an increase in the relay transmit power improves the location estimation accuracy to an order of few meters to few tens of meters. V. CONCLUSIONS In this paper, we propose a TOA based location estimation for determining the two-dimensional coordinates of source node in cooperative relay networks. In this method, the signals received over direct and relay links are used to perform location estimation. We derive both instantenous and average CRLB for this scheme in order to take the signal fluctuations in both relay and direct links into account. The simulation results show that channel fading in both relay and direct links affects the TOA based location estimation accuracy. It is also observed that amplify-and-forward relay strategy can be used to improve the accuracy of the TOA based location estimation. In addition, the location of relay node on the Fig SNR (db) Average CRLB vs SNR for different K values. 10 ' _Pr1 Pr4 -B--- Pr9 --f-pr 10 L L o 5 10 SNR (db) Fig. 3. Average CRLB vs SNR for different Pr values. average CRLB affects the location estimation accuracy of source node. Finding the optimal location of relay node that minimizes location estimation error can be considered as a future work. Furthermore, the current analysis can be extended considering other type of relay strategies and multiple relay scenarios. VI. ApPENDIX The CRLB expression of time delay estimate through the relay link a is derived as follows. Let () [TT] represent the vector of unknown signal parameters, where a, hi, and h2 are assumed to be known. Note that the only unknown parameter is reduced to Tl since T2 is assumed to be known, i. e. () h]. The observation interval [0, T] is considered and it can be expressed as T NTs,

5 10 ' r:.-:-:-:-::-:-:-r:-:.. :-:..... '-. :-:.. :-:... ;:-:- :-: :-:-:-::-:T: :-:-::-:-:-:-::-:-:-:T,:::r:::il : P,1 ---"1- P, 4. --a-- P,9 10 ' L- L- L S R Distance "d, " (m) Fig. 4. Average CRLB vs dl for different Pr values. where N is the number of observation symbol. Then, the log- [10] likelihood function for Tl is given by 1 i T [II] A(Tl) Cl [Yd(t)-ahlh2vP:s(t-TT)]2dt, (24) at 0 where Cl is constant independent of Tl. The CRLB for unbiased estimate of Tl is obtained after straightforward derivation, which is given by where CRLB(f1) i I'rEr (25) (26) Er I T [Sl(t-TT)]2dt. (27) Note that (26) can be written in terms of 1'1 and 1'2, which are the SNR of the first and second ho s in the relay link, respectively. By replacing a Pslh'2+No in (26) and after some straightforward manipulatlons, (26) takes the following form 1 I'r..!.. +..! ')'1 ')'2 ')'1 ')'2 where 1'1 Pshr and 1'2 P rh. Since 1'11'2 approximated as 1'1 + 1'2 I' n2 I'r --'---'--- VII. ACKNOWLEDGEMENTS (28)» 1, (28) is (29) This work is supported by Qatar National Research Fund (QNRF) grant through National Priority Research Program (NPRP) No QNRF is an initiative of Qatar Foundation, Doha Qatar. REFERENCES [I] 1. N. Laneman, D. N. C. Tse, and G. Womell, "Cooperative diversity in wireless networks: efficient protocols and outage behaviour," IEEE Trans. on Info. Theory, vol. 50, no. 12, pp , Dec [2] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversity, part i: System description," IEEE Trans. Commun., vol. 51, no. II, pp , Nov [3] M. Hasna and M.-S. Alouini, "End-to-end performance of transmission systems with relays over rayleigh fading channels," IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. II26-1l31, Nov [4] N. Beaulieu and A. Abu-Dayya, "Analysis of equal gain diversity on nakagami fading channles," IEEE Trans. Commun., vol. 39, no. 2, pp , Feb [5] X. C. A. Ribiero and G. Giannakis, "Symbol error probabilities for general cooperative links;' IEEE Trans. Wireless Commun., vol. 4, no. 3, pp , May [6] S. Gezici, "A survey on wireless position estimation," Springer Wireless Personal Communications, Special Issue on Towards Global and Seamless Personal Navigation, Oct [7] H. Celebi, "Location awareness in cognitive radio networks," Ph.D. dissertation, University of South Florida, FL, Aug [8] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, "Localization via UWB Radios," IEEE Signal Processing Mag., vol. 22, no. 4, pp , July [9] Y. Qi, H. Kobayashi, and H. Suda, "Analysis of wireless geolocation in a non-line-of-sight environment," IEEE Transactions on Wireless Communications, vol. 5, no. 3, pp , [12] [l3] [14] [15] [16] S. Gezici, H. Celebi, H. Vincent Poor and H. Arslan, "Fundamental limits on time delay estimation in cognitive radio systems," IEEE Trans. on Wireless Communications, vol. 8, no. I, pp , Jan N. Patwari, 1. Ash, S. Kyperountas, A. Hero Iii, R. Moses, and N. Correal, "Locating the nodes: cooperative localization in wireless sensor networks," IEEE Signal processing magazine, vol. 22, no. 4, pp , E. Larsson, "Cramer-Rao bound analysis of distributed positioning in sensor networks," IEEE Signal processing letters, vol. II, no. 3, pp , R. Moses, D. Krishnamurthy, and R. Patterson, "A self-localization method for wireless sensor networks," EURASIP Journal on Applied Signal Processing, pp , M. Zhaounia, M. Landolsi, and R. Bouallegue, "Hybrid TONAOA Approximate Maximum Likelihood Mobile Localization," EURASIP Journal of Electrical and Computer Engineering, vol. 2010, S. Gezici and Z. Sahinoglu, "Ranging in a single-input multiple-output (SIMO) system," IEEE Communications Letters, vol. 12, no. 3, pp ,2008. H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. John Wiley and Sons,