RSS-Based Localization in Environments with Different Path Loss Exponent for Each Link Junichi Shirahama† Tomoaki Ohtsuki†† Department of Electrical Engineering, Tokyo University of Science 2641 Yamazaki, Noda, Chiba 278-8510 Japan Department of Information and Computer Science, Keio University 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan E-mail:††
[email protected]
I. A BSTRACT The path loss exponent is very important parameter for localization using receive signal strength (RSS). In actual environments, path loss exponent for each link (target to each receive node) differs. However, the conventional localization methods use the same path loss exponent for all links. Hence, there are some mismatches between the real path loss exponent and the one used to estimate. We proposed the localization method that considers all the combinations of path loss exponents for each link and estimates the target location by averaging the target locations derived with all the combinations. However, the amount of calculation is huge. In this paper we propose RSS-based localization in environments with different path loss exponent for each link. The proposed method is a grid-based centralized localization using RSS. First the proposed method sets the minimum distance di,min and maximum distance di,max for each node i by using the RSS of each receive node i and the minimum and maximum path loss exponents set before estimation. Next, it calculates the distance di,(k,l) between the candidate target position (k, l) and each receive node i. If di,min ≤ di,(k,l) ≤ di,max , vote the grid (k, l). These processes are performed for all the receive nodes over the search area. Finally, the grid point with most voting is estimated to be the target location. According to the simulation results, we show that the proposed method achieves the higher localization accuracy than the conventional localization method using the same path loss exponent for all the links when the distribution of the path loss exponents over the field is uniform distribution. II. I NTRODUCTION Recently, sensor networks are attracting much attention as a means to gather information [1]. The sensor networks use a lot of sensor nodes. One of the applications of the sensor networks is monitoring targets. The sensor networks classify, detect, locate, and track targets in the networks. However, the sensor node is required to be small and low power consumption. Therefore, efficient collaborative signal processing algorithms that consume less energy for computation and less bandwidth for communication are needed [2]. There are several ways to estimate target location. In [3] 978-1-4244-1645-5/08/$25.00 ©2008 IEEE
angle of arrival (AOA) is used to estimate the target location. In [4] time difference of arrival (TDOA) is used to estimate the target location. In [5]- [7] received signal strength (RSS) is used to estimate the target location. The path loss exponent is very important parameter for localization using RSS. In actual environments, the path loss exponent for each link (target to each receive node) differs. Some factors causing the differences are the reflection and non-line of sight (NLOS) transmission. However, the conventional localization methods use the same path loss exponent for all links. Hence, there are some mismatches between the real path loss exponent and the one used to estimate. We proposed the localization method that considers all the combinations of path loss exponents for each link and estimates the target location by averaging the target locations derived with all the combinations [11]. However, the amount of calculation is huge. In this paper we propose RSS-based localization in environments with different path loss exponent for each link. The proposed method is a grid-based centralized localization using RSS. First the proposed method sets the minimum distance di,min and maximum distance di,max for each node i by using the RSS of each receive node i and the minimum and maximum path loss exponents set before estimation. Next, it calculates the distance di,(k,l) between the candidate target position (k, l) and each receive node i. If di,min ≤ di,(k,l) ≤ di,max , vote the grid (k, l). These processes are performed for all the receive nodes over the search area. The grid point with maximum voting is expected to be the target position. Thus, the coordinated of the grid points with most voting are averaged and the averaged one is decided to be the target location. By computer simulation, we show that the proposed method achieves the higher localization accuracy than the conventional localization method using the same path loss exponent for all the links when the distribution of the path loss exponents over the field is uniform distribution. III. S YSTEM MODEL The RSS of the receive node i (i = 1, · · · , N ) can be often modeled by the following equation [8]. di + Xs [dBm] (1) P L(di ) = P0 − 10ri log10 d0
1509
where di is the distance between the receive node i and the target, P0 is the received power in dBm at a reference distance d0 , P L(di ) is the received power in dBm at a distance of di (di > 1), ri is the path loss exponent for the link i, and Xs is a Gaussian random variable N (0, σ 2 ) representing lognormal shadow fading effects in multipath environments [8]. d0 is set to 1 m in this paper.
grid point
IV. C ONVENTIONAL L OCALIZATION METHOD
3
3
3
4
4
5
4
4
4
5
5
4
4
5
5
4
A. Using the Same Path Loss Exponent to All Links [5] 1) Use the path loss exponent r obtained by the experiment and calculate the square error denoted by (2), where [xk , yl ], (k = 1, · · · , X, l = 1, · · · , Y ) is the grid point in the search area X × Y . 2) Search all the grid points. 3) ρ minimizing the square error becomes the target locaˆ tion ρ.
^ y) ^ calculate (x, Fig. 1.
l(ρ|xk , yl , r) =
N
Ri − Rˆi
2
(2)
i=1
Rˆi =
S (xk − xi )2 + (yl − yi )2
r
(3)
S = 10P L(d0 )/10
(4)
where N is the number of sensor nodes and ρT = [xt , yt ] is the target location. B. Using Different Path Loss Exponent to Each Link [11] Search the path loss exponents from rmin to rmax at interval rint for each link. 1) Using the path loss exponent ri , calculate the square error denoted by (5). 2) Search all the grid points and the combinations of path loss exponents. 3) ρ minimizing the square error becomes the target locaˆ tion ρ.
l(ρ|xk , yl , r1 , r2 , · · · , rN ) =
N
Ri − Rˆi
2
(5)
i=1
Rˆi =
S (xk − xi )2 + (yl − yi )2
r i (6)
For instance, when N = 5, rmin = 2, rmax = 4, rint = 1, this method has to examine 35 combinations of path loss exponents for each link to each grid point.
Example of the proposed method
for(k = 0; k < X + 1; k++){ for(l = 0; l < Y + 1; l++){ for(i = 0; i < N ; i++){ if((di,min < di,(k,l) < di,max ){ vote the grid point (k, l) ; } } } } where r1 S , r = rmax (7) di,min = Ri r1 S , r = rmin (8) di,max = Ri di,(k,l) is the distance between the receive node i and the grid point (k, l). • Step2 Estimate the target location. x1 + x2 + · · · + xK y1 + y2 + · · · + yK , (ˆ x, yˆ) = K K (9) where K is the number of grid points that have the maximum vote. We show the example in Fig. 1. If N = 5, 5 grid points have the maximum vote and the target location is estimated using (9).
V. P ROPOSED M ETHOD
VI. S ET THE O PTIMAL rmin AND rmax We first evaluate the effects of the optimal values of rmin and rmax on the localization accuracy by the computer simulation.
The path loss exponent of each link is an unknown parameter. However, according to the experiment, we can get the distribution of path loss exponent in the field roughly. • Step1 Search all the area and perform the following processes.
A. Simulation Parameters The field is 10 m × 10 m, the number of receive nodes is N = 8, the number of target is one, and transmit energy is 1 mW.
1510
σ2 = 0.5 σ2 = 1.0 σ2 = 1.5 σ2 = 2.0 σ2 = 2.5
3
1.8 1.6
2
RMSE(m)
RMSE (m)
2.5
2
1.5
1 0.4
0.8
1.4 1.2 1 0.8
optimal
0.6
σ = 0.5 2 σ = 1.0 2 σ = 1.5 2 σ = 2.0 2 σ = 2.5
1
1.2
1.4
optimal
1.6
0.6 0.4
Max. deviation from mean
0.6
0.8
1
1.2
1.4
1.6
Max. deviation from mean
Fig. 2. Attenuation range versus RMSE (path loss exponents are generated with uniform distribution (2, 4])
Fig. 3. Attenuation range versus RMSE (path loss exponents are generated with N (3, 0.33))
B. Simulation Results In Fig. 2 we show the maximum deviation from the mean versus RMSE when the path loss exponents using uniform distribution (2, 4], where the maximum deviation from the mean (abscissa axis) decides the rmin and rmax . When the maximum deviation from the mean is j, rmin = 3 − j and rmax = 3 + j. For example, if the maximum deviation from the mean is equal to 0.4, rmin and rmax become 2.6 and 3.4, respectively. According to the simulation results, the optimal rmin and rmax are 1.9 and 4.1, respectively. In Fig. 3 we show the maximum deviation from the mean versus RMSE when the path loss exponents are generated using N (3, 0.33). According to the simulation result, the optimal rmin and rmax are 2.3 and 3.7. We use these rmin and rmax for estimation. Note that in practical systems it is hard to get the accurate information about the values and the distribution of the path loss exponents. However, we know that the path loss exponent generally takes the value from 2 to 5. In addition we can get rough information by prior measurements in not all but some links. We can set suboptimum values of rmin and rmax . Note also that the performance of the conventional localization method with RSS degrades a lot if the correct information on the path loss exponent is not available. VII. P ERFORMANCE C OMPARISON BETWEEN P ROPOSED M ETHOD AND C ONVENTIONAL M ETHOD A. Simulation Parameters The field is 10 m × 10 m, the number of targets is one, transmit power is 0 dBm. We consider the following 2 types of distribution of path loss exponents. 1) Uniform distribution (2, 4] 2) Gaussian N (3, 0.33)
B. Simulation Results In Fig. 4 we show the number of receive nodes versus the RMSE when the path loss exponents are generated with uniform distribution (2, 4]. “Ideal” shows the localization result when the path loss exponent of each link is known. “Common” shows the localization result when the path loss exponent of each link assumed to be same. We use the path loss exponent of 3 (the average of the uniform distribution (2, 4]) for all the links in “Common”. We can see that the proposed method achieves better RMSE performance than “Common”. This is because the proposed method votes the grids that can be possible with the combinations of possible path loss exponents and average them. Thus, the proposed method can take the most probable one. Fig. 5 shows the variance of shadow fading versus RMSE when the number of receive nodes is 8 and the path loss exponents are generated with uniform distribution (2, 4]. We can see that the proposed method has better performance than the conventional one for any value of the variance of shadow fading. This is because the proposed method votes the grids that can be possible with the combinations of possible path loss exponents and average them. Thus, the proposed method can take the most probable one. We can also see that the improvement of the proposed method over the “Common” becomes smaller as σ 2 becomes larger. This is because when σ 2 is large, the effects of shadowing is also large and dominant on the RMSE performance, compared to the effects of the mismatch of the path loss exponents. In Fig. 6 we show the number of receive nodes versus RMSE when the path loss exponents are generated with N (3, 0.33). The path loss exponent of “Common” is set to 3. We can see that the proposed method has a little bit worse performance than “Common”. This is because when the
1511
3
3.5 ideal Common Proposed
2.5
2 1.5 1
1.5
0.5 4
5
6
7
8
9
10
11
0
12
The number of receive nodes
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.5
1
1.5
4
5
6
7
8
9
10
11
12
The number of receive nodes
Fig. 4. The number of receive nodes versus RMSE (path loss exponents are generated with uniform distribution (2, 4])
RMSE (m)
2
1
0.5 0
ideal Common Proposed
3
RMSE (m)
RMSE (m)
2.5
Fig. 6. The number of receive nodes versus RMSE (path loss exponents are generated with N (3, 0.33))
that the proposed method has worse performance than “Common”. This reason is the same as that for Fig. 6: When the distribution of the path loss exponents is normal distribution, Ideal each link has high possibility to have the path loss exponent Common close to the mean, compared to the uniform distribution case. Proposed The proposed method votes the grids with low possibility, because it votes the grids that can be possible with any combinations of path loss exponents within the predefined range. On the other hand “Common” uses the common path loss exponent that is the same as the mean of the distribution in this simulation. Thus, the conventional method with setting the common path loss exponent to the mean achieves the better RMSE performance. Note that when “Common” uses the path loss exponent not equal to the average, the performance of 2 2.5 “Common” degrades a lot.
σ2 VIII. C ONCLUSION Fig. 5. The variance of shadow fading versus RMSE (path loss exponents are generated with uniform distribution (2, 4])
distribution of the path loss exponents is normal distribution, each link has high possibility to have the path loss exponent close to the mean, compared to the uniform distribution case. The proposed method votes the grids with low possibility, because it votes the grids that can be possible with any combinations of path loss exponents within the predefined range. On the other hand “Common” uses the common path loss exponent that is the same as the mean of the distribution in this simulation. Thus, the conventional method with setting the common path loss exponent to the mean achieves the better RMSE performance. Fig. 7 shows the variance of shadow fading versus RMSE when the number of receive nodes is 8 and the path loss exponents are generated with N (3, 0.33). We can see again
In this paper we propose the method of target localization using the values of maximum and minimum path loss exponents in the area. The proposed method is a grid-based centralized localization using RSS. Through the simulation results, we showed that the proposed method achieves the higher localization accuracy than the conventional localization method using the same path loss exponent for all the links when the distribution of the path loss exponents over the field is uniform distribution. When the distribution of the path loss exponents over the field is normal distribution, the proposed method has worse performance, though the degradation is not so large.
1512
R EFERENCES [1] J. Pister, “Smart Dust-Hardware limits to wireless sensor networks,” IEEE Int. Conf. on Distributed Computing Systems, vol. 53, pp. 44–53, May 2003.
1.2
Ideal Common Proposed
RMSE (m)
1 0.8 0.6 0.4 0.2 0 0.5
1
1.5
2
2.5
2
σ
Fig. 7. The variance of shadow fading versus RMSE (path loss exponents are generated with N (3, 0.33))
[2] D. Li, K. D. Wong, Y.H. Hu, and A.M. Sayeed, “Detection, classification, and tracking of targets,” Proc. IEEE Signal Process. Mag., vol. 19, pp. 17–29, Mar. 2002. [3] L.M. Kaplan, Q. Le, and P. Molnar, “Maximum likelihood methods for bearings-only target localization,” Proc. IEEE ICASSP, vol. 5, pp. 3001 – 3004, May 2001. [4] P. Svaizer, M. Matassoni, and M. Omologo, “Acoustic source location in a three-dimensional space using crosspower spectrum phase,” Proc. IEEE ICASSP, vol. 1, pp. 231–234, Apr. 1997. [5] X. Sheng and Y.H. Hu, “Maximum likelihood multiple-source localization using acoustic energy measurements with wireless sensor networks,” IEEE Trans. Signal Processing, vol. 53 , no. 1, pp. 44–53 Jan. 2005 [6] Y.H. Hu and D. Li, “Energy based collaborative source localization using acoustic micro-sensor array,” IEEE. Multimedia Signal Processing. pp. 371–375, Dec. 2002 [7] M.G. Rabbat and R.D. Nowak, “Decentralized source localization and tracking [wireless sensor networks],” IEEE ICASSP’04, vol. 3, pp. 921– 924, May 2004 [8] X. Li, “Performance study of RSS-based location estimation techniques for wireless sensor networks,” IEEE MILCOM’05, vol. 2, pp.1064–1068, Oct. 2005 [9] P.P. Joshi and T.C. Jannett, “Performance-Guided Reconfiguration of Wireless Sensor Networks that Use Binary Data for Target Localization,” IEEE ITNG’06, pp. 562–565, Apr. 2006 [10] H. Shi, X. Li, Y. Shang, and D Ma, “Cramer-Rao bound analysis of quantized RSSI based localization in wireless sensor networks,” IEEE ICPADS ‘05 vol.2, pp.32–26, July 2005 [11] I. Yamada and T. Ohtsuki, “An indoor location estimation method based on local electric wave propagation parameter values,” IEICE Tech. Rep. SN2006 38-58, pp.1–6, Dec. 2006
1513
