1

A Scalable UWB Based Scheme for Localization in Wireless Networks Ananth Subramanian and Joo Ghee Lim Abstract— The promise of location-aware applications has led to the design and development of various localization schemes. In this paper we propose a distributed localization scheme for nodes with emphasis on power measurements (RSSI) using UWB. The localization algorithm is distributed and follow an indirect approach where nodes get to determine their location with the help of their neighbors. The UWB based scheme proposed is robust and scalable.

II. L OCALIZATION THROUGH POWER MEASUREMENTS USING UWB The free space propagation model is normally used to model the received signal strength when the transmitter and the receiver have a clear line of sight communication between them. According to this model, the following holds: Pr = C × Pt /dm

I. I NTRODUCTION The promise of location-aware applications has led to the design and development of various localization schemes [1][9]. However simple GPS based schemes do not work well in providing accurate location information when the nodes are indoor or in the presence of obstacles that block the line-ofsight from GPS satellites. Hence, various GPS-less schemes [3]-[9] have also been proposed. Current location estimation techniques can generally be divided in to two main categories - Direct and Inferred. In the direct approach to location estimation, there is usually a finite set of reference nodes or base stations, with known locations, and all other nodes (with unknown locations) communicate directly with reference nodes to find their locations. Some approaches to this end can be found in [2]-[6]. One disadvantage of the direct approach is the trade-off between the number of reference nodes and the signal coverage. In order to determine location, the coverage of the signals from the reference nodes must overlap. Using a smaller number of reference nodes will require longer transmission ranges, which will aggravate the multi-path problems and increase measurement inaccuracies and power consumption levels. However, having reference nodes with smaller coverage will result in higher density of reference nodes to be deployed, leading to scalability and cost issues. The main idea of the inferred approach is to use the known, recently estimated locations of some sensor nodes to determine or infer the location of their neighbors. A node will normally receive ranging and location information from its neighbors to find its own location. The result will then be transmitted to its own neighbors for them to perform the same operation (see [7], [8]). The merit of the inferred approach over the direct one is that few or no reference nodes are needed. However, this approach could be prone to propagation errors. In this paper we propose a distributed localization scheme for nodes with some emphasis on power measurements (RSSI) using UWB. Ananth Subramanian is with the Institute for Infocomm Research and Joo Ghee Lim is with University of New South Wales. E-mail: [email protected], [email protected]

(1)

where Pr is the received power, Pt is the transmitted power, d is the distance between the transmitter and the receiver in meters, m is the path loss coefficient, and the constant C depends on the wavelength of transmission and the antenna gains. In this section, we will show how to get good estimates of the received power Pr in order to subsequently estimate the distance d. Later we will propose a scalable localization protocol that in a distributed manner localizes all the nodes in a network. A. Channel Model for UWB reception The receiver node observes a linear combination of all transmitted data by the sender node, each distorted by ISI, under white Gaussian noise. Assuming the maximum number of channel taps to be L, the received signal at time n is y(n) =

L X

ck (n)s(n − k) + v(n)

(2)

k=1

where s(n) is the transmitted sequence, ck (n) is the kth tap of the channel at time n. Collecting the channel taps into a column vector hn : hn = col{c1 (n) c2 (n) · · · cL (n)}

(3)

we can rewrite (2) as y(n) = sn hn + v(n)

(4)

where the L row data vector sn contains the transmitted data. The goal is to track the vector hn , estimate the channel taps, and recover the power level Pr . We use the model in [10] where all channel taps are assumed independent and the variability of the wireless channel over time is reflected in the autocorrelation function of a complex Gaussian process. It is shown in [11] that the theoretical power spectral density associated with either the in-phase or quadrature portion of each channel tap has the well-known Ushaped bandlimited form : ( q f 2 1/(πf d 1 − ( fd ) ) |f | ≤ fd (5) S(f ) = 0 elsewhere

where fd is the maximum Doppler frequency. The corresponding normalized discrete-time autocorrelation of each tap is R[n] = J0 (2πfm |n|) where J0 (.) is the zeroth-order Bessel function of first kind and fm = fd T is the Doppler frequency normalized by the sampling rate 1/T [11].

and the corresponding error covariance matrices, ∆

= E(xn − x ˆn )(xn − x ˆn )T

Pn

∆

= E(xn − x ˆn|n )(xn − x ˆn|n )T .

Pn|n

Then the {ˆ xn , x ˆn|n } can be constructed recursively as follows: B. State-Space Model The time variation of the vector process {hn } can be approximated by the following AR process of order q [13], hn =

q X

A(l)hn−l + G0 wn

(6)

x ˆn+1|n x ˆn+1|n+1 en+1

= Fx ˆn|n ,

n≥0

(13)

= x ˆn+1|n + Pn+1|n+1 H = yn+1 − H x ˆn+1|n

T

−1 Rn+1 en+1

(14) (15)

where

l=1

where wn is a zero-mean i.i.d circular complex Gaussian vector process. Assuming Gaussian wide-sense stationary uncorrelated scattering fading (WSSUS), the matrices A(l), l = 1, · · · , q, and G0 turn out to be diagonal. For the selection of their entries, various criteria of optimality can be adopted, such as requiring the AR model of order q in (6) to provide a “best-fit” to the real channel auto correlation function of (5). In this paper we adopt the AR model presented in [12]. This model has the attractive property that its sampled auto correlation function perfectly matches the desired sampled auto correlation function of (5) up to lag q. This is achieved by solving a set of q Yule-Walker equations [12]. The multichannel AR model (6) can be rewritten in statespace form as : xn = F xn−1 + Gwn (7) where · F =

xn = col{hn hn−1 · · · hn−q+1 } A(1) I(q−1)N L

and

A(2) . . . h

G=

G0

A(q) 0(q−1)N L×N L

¸

i

0(q−1)N L×N L

(8)

Hence consider a state-space description of the form: xn+1 yn

= =

F xn + Gun , n ≥ 0, Hxn + vn ,

(9) (10)

where {x0 , un , vn } denote uncorrelated zero-mean random variables with covariance matrices ! Ã µ ¶ µ ¶T Π0 0 0 x0 x0 u 0 Qk δkj 0 j E uk = (11) vk vj 0 0 Rk δkj that satisfy Π0 > 0, Rk > 0, Qk ≥ 0.

(12)

Here, δkj is the Kronecker delta function that is equal to unity when k = j and zero otherwise. The Kalman filter provides the optimal linear least-mean-squares (l.l.m.s., for short) estimate of the state variable given prior observations. More specifically, introduce the following predicted and filtered estimates: x ˆn|n−1

=

∆

l.l.m.s. estimate of xn given {y0 , y1 , . . . , yn−1 }

x ˆn|n

=

∆

l.l.m.s. estimate of xk given {y0 , y1 , . . . , yn }

Pn+1 Pn+1|n+1 Re,n+1

= FPn|n F T + GQn GT = Pn+1 − Pn+1 H

T

−1 Re,n+1 HPn+1 T

= Rn+1 + HPn+1 H

(16) (17) (18)

and with initial conditions x ˆ0|0

=

−1 T −1 P0|0 H0 R0 y0

(19)

P0|0

=

T −1 −1 (Π−1 0 + H0 R0 H0 )

(20)

III. P OSITION ESTIMATION USING UWB The power level of the periodic UWB signal transmitted is computed as: N 1 X Pt = |s(n)|2 N 1 Note that the received signal has noise in it that we filter out through the Kalman filter as described in the previous section. The filtered version of the received signal is then given by the following: ˆn yˆ(n) = sn h ˆ n is given by the Kalman filter. The received power where h level is then computed as: Pr =

N 1 X |ˆ y (n)|2 N 1

With known constants of C and m, the distance d from the transmitter to the receiver can be computed (see eq. (1)). The constants C and m are calibrated through measurements. A. Localization algorithm In the beginning, only a particular node known as the anchor node knows its location and it will initiate its own movement. It will follow a mobility path consisting of a predetermined set of 4 points from where it will then broadcast each of its 4 positions along with separate periodic UWB transmissions. The anchor node location and the periodic UWB transmission are broadcast from all the four points that the anchor moves to. After moving through the 4 points, the anchor node will return to its original position ending its journey. The choice of these points is to ensure that as many nodes as possible around the anchor node will have an adequate chance of receiving the

Transmitted UWB pulse

Received signal at a distance of 2 meters

0.3

0.4

0.2

0.3 0.2

0.1

Amplitude

Amplitude

0.1 0

−0.1

0 −0.1

−0.2

−0.2

−0.3

−0.4

−0.3

0

1000

2000

3000

4000

−0.4 0.5

5000

Samples

(a)Transmitted UWB pulse.

Received signal at a distance of 5 meters 0.2

−60

0.15

−70

0.1

Amplitude

Power spectral density in dB

Power spectral density of the received signal at a distance of 2 meters

−80

−90

0.05

0

−100

−0.05

−110

−0.1

0

2

4 6 Frequency in Hz

3 4

x 10

(b)Received UWB pulse at a distance of 2 meters indoor.

−50

−120

1 1.5 2 2.5 samples at a sampling frequency of 1 Tera Hz

8

10 9

x 10

(c)Power spectral density of the received UWB pulse at 2 meters indoor.

−0.15

0

2000 4000 6000 8000 samples at a sampling frequency of 0.2 Tera Hz

10000

(d)Received UWB pulse at a distance of 5 meters indoor.

Fig. 1. Transmitted and received UWB signals.

broadcast messages. Nodes within the transmission range of anchor node will acquire the location of the anchor node along with the periodic UWB transmission. After a time interval, these nodes will use the broadcast messages they have received to obtain filtered versions of their respective received periodic UWB signals along with the channel estimates, and hence their distances from anchor locations as explained above. Note that a set of nodes would have received the location information from at-least three of the four anchor node locations, and hence know their distances from at least three anchor node locations. Such nodes can estimate their position using triangulation. As noted, the individual distances that are used for triangulation are estimated through power measurements. Now the nodes that have estimated their positions in turn broadcast their respective location information and a periodic UWB signal to their respective neighbors. Collision is avoided using CSMA/CD. When a particular node receives location information and the periodic UWB transmission from three distinct neighbors, it can then estimate its own position using the distances it computes through power measurements from

these neighbors and then performing triangulation. Note that only the anchor node had to perform a mobile operation. The other nodes in the network could be stationary. As more nodes around the anchor acquire their position, they will help to localize the nodes around them. This operation will propagate throughout the entire network until all the nodes are localized. Effect of mobility of all nodes on localization accuracy: If at any point of time, the nodes that have learnt their location information are allowed to subsequently begin their own mobile operation by going through a similar mobility path as that of the anchor though with a different initial heading θ, simulation results indicate that there is increased location accuracy of the remaining nodes (see results in the simulation section). The algorithm remains the same as that mentioned above, where, nodes that have estimated their positions in turn broadcast their respective location information from all the four locations they have moved, to their respective neighbors. Collision is avoided using CSMA/CD. Note that there is a possibility of a node with unknown location receiving information from more locations as each of the node with known location

transmits from four points on its mobility path. When a particular node with unknown location receives information from at-least three distinct locations, it can then estimate its own position using the distances it computes through power measurements from these neighbors and then performing triangulation. As more nodes around the anchor acquire their position and begin to move, they will help to localize the nodes around them. This operation will propagate throughout the entire network until all the nodes are localized. IV. S IMULATIONS To evaluate and analyze the performance of our localization scheme, we conducted several simulation experiments in GloMoSim [14]. The simulation model consists of a 2-dimensional space with nodes randomly placed within a region. A single anchor node is placed in the center of the network region. Each node has a radio range, R of about 10m with a free space propagation path-loss model. The variance in error of distance d was captured as approx. 0.5. As noted in the previous section, fig. 1 provides the actual measurements of the received UWB pulses at different distances from the transmitter. Note that these measurements are used to calibrate the constants C and m by least squares through Pt and Pr . The performance metric that we use to evaluate the performance of our localization scheme is the distance error between the true and estimated position of the nodes, normalized to the transmission range, R. The performance metric is given by p q = ((xtrue − xest )2 + (ytrue − yest )2 )/R Fig. 2 shows the frequency distribution of error in position for 100 nodes in a region of size 50 × 50m2 , averaged over 100 simulation trials. In fig. 2, we increase the number of nodes deployed from 80 to 200 keeping the network area fixed at 50 × 50m2 . Fig. 3 shows the effects of network area size on our localization scheme. The network area was increased from 2000m2 to 5000m2 , with the node density kept constant at 0.4 nodes per m2 . Fig. 4 shows the mean position estimation errors of the nodes along with standard deviation for varying node density and fig. 5 shows the same for varying network area with constant node density. Figure 6 shows the effect of having mobility for all the nodes as that of the anchor node alone on the localization accuracy. V. C ONCLUSIONS In this paper we proposed a distributed localization scheme for nodes with emphasis on power measurements (RSSI) using UWB. The localization algorithm follows an indirect approach where nodes get to determine their location with the help of their neighbors. The UWB based scheme proposed is scalable. Acknowledgement The authors wish to acknowledge Mr. Low Zhen Ning and Ms. Ohnmar Kyaw for their assistance in providing actual UWB measurements of fig. 1.

Fig. 2. Histogram of percentage of nodes vs. position estimation error for varying node density.

Fig. 3. Histogram of percentage of nodes vs. position estimation error for varying area size.

R EFERENCES [1] P. Bahl and V. Padmanabhan, “V.RADAR: An inbuilding RF-based user location and tracking system”, Proc. IEEE INFOCOM, Tel-Aviv, Israel, Mar. 2000. [2] A. Smith, H. Balakrishnan, M. Goraczko, and N. Priyantha, “Tracking moving devices with the cricket location system”, Proc. MobiSys, 2004. [3] P. Pathirana, N. Bulusu, S. Jha, and A. Savkin, “Node localization using mobile robots in delay-tolerant sensor networks”, IEEE Transactions on Mobile Computing, to appear, 2004. [4] I. Getting, “The Global Positioning System”, IEEE Spectrum, Dec. 1993. [5] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-less low cost outdoor localization for very small devices”, IEEE Personal Communications Magazine, vol. 7, no. 5, pp. 28–34, Oct. 2000. [6] J. Hightower and G. Borriello, “Location systems for ubiquitous computing”. IEEE Computer, vol. 34, no. 8, pp. 57–66, Aug. 2001. [7] C. Savarese, J. M. Rabaey and J. Beutal, “Locationing in distributed adhoc wireless sensor networks”, Proc. ICASSP, Salt Lake City, UT, May 2001. [8] L. Doherty, K. Pister and L. El Ghaoui, “Convex position estimation in wireless sensor networks”, Proc. Infocomm, Anchorage, AK, Apr. 2001. [9] A. Savvides, C. Han, and M. B. Srivastava, “Dynamic Fine-Grained Localization in Ad-Hoc Networks of Sensors”, Proc. ACM MobiCom, 2001. [10] P. A. Bello, Characterization of randomly time-variant linear channels, IEEE Trans. Communications Systems, vol. CS-11, pp. 360–393, Dec.1963. [11] W. C. Jakes, Microwave Mobile Communications, New York: Wiley, 1974. [12] K. E. Baddour, N. C. Beaulieu, Autoregressive models for fading chan-

Fig. 4. Average mean and standard deviation for position estimation errors vs. number of nodes.

Fig. 5. Average mean and standard deviation for position estimation errors vs. area size. Fig. 6. Distribution of nodes with different position errors - Comparison between anchor only mobility and all node mobility. nel simulation, Proc. IEEE Global Telecommunication Conf., vol. 2, pp. 1187–1192, 2001. [13] M. K. Tsatsanis, G. B. Giannakis, and G. Zhou, Estimation and equalization of fading channels with random coefficients, Signal Processing, vol. 53, no. 2–3, pp. 211–229, Sep. 1996. [14] X. Zheng, R. Bagrodia, M. Gerla, “GloMoSim: a Library for Parallel Simulation of Large-scale Wireless Networks”, Proc. of the 12th Workshop on Parallel and Distributed Simulations – PADS ’98, Banff, Alberta, Canada, May 1998.