We study the problem of source localization in multihop wireless sensor networks. A fully distributed algorithm based on sensor measurements of time of arrivals (TOAs) is proposed. In contrast to centralized methods where all TOA measurements are transmitted via certain routes to a central location (the sink) for processing, the proposed method distributes the processing among the relay nodes on the routes to the sink. Fusion strategies are proposed so that the raw and intermediate data are progressively processed, and only the refined results are further relayed. As a result, the proposed scheme has improved flexibility and scalability since it does not impose any special requirements on the sink node. The proposed distributed strategy also has the potential to save energy and bandwidth due to reduced radio transmissions.

1. Introduction

In centralized wireless sensor network (WSN) source localization, the transmission of all the raw measurements to a central point is required and the processing is performed in a centralized manner. These schemes have the potential to achieve optimum solutions due to the availability of the full data set at the processing point. However, the centralized systems typically have low energy efficiency and low scalability due to excessive radio transmissions and dependency on a central processing node.

Distributed methods based on clustering have been proposed [1]. In [2], the intermediate result is cycled through the network on specially designed routes, and each node makes a small adjustment to the received estimate based on deepest descent on a cost function, then passes updated estimate to one of its neighbours.

In this paper, we propose a fully distributed source localization algorithm based on TOA measurements in multihop wireless sensor networks. The proposed scheme distributes the localization process among the relay nodes along the routes to the (arbitury) node where the localization estimate is needed. Whenever determined feasible, the intermediate estimate is produced or refined by incorporating TOAs and intermediate estimates available at the local node. Data fusion strategies are proposed so that the raw and intermediate data are progressively processed and only the refined results are further relayed.

2. The Data Model and Centralized Processing

As in [3] and illustrated in Figure 1, for a signal emitted by the source at time T (unknown), the TOA measurement at node i (𝑖=1,…,𝑁,𝑁β‰₯3) is modelled by where πœ€π‘– denotes measurement noise and c the propagation speed. The noise samples are assumed to be uncorrelated with zero mean and variance 𝜎2. No assumption is made about the probability of density function of the noise. The distance between the source and a node located at (π‘₯𝑖,𝑦𝑖) is given by 𝑅𝑖=(π‘₯βˆ’π‘₯𝑖)2+(π‘¦βˆ’π‘¦π‘–)2.

We assume that a nominal source position (π‘₯𝑛,𝑦𝑛) is available. This nominal position, which is close to the true source position (π‘₯,𝑦), could have been obtained from the previous measurements. Such a situation is typical when the source is being tracked [3]. Hence, to estimate the source position we require an estimation of [(π‘₯βˆ’π‘₯𝑛)(π‘¦βˆ’π‘¦π‘›)]𝑇=[𝛿π‘₯𝛿𝑦]𝑇, where T denotes matrix transpose.

Using a first-order Taylor expansion of 𝑅𝑖 about (π‘₯𝑛,𝑦𝑛) and defining 𝑅𝑛𝑖=(π‘₯π‘›βˆ’π‘₯𝑖)2+(π‘¦π‘›βˆ’π‘¦π‘–)2, we have π‘…π‘–β‰ˆπ‘…π‘›π‘–+(π‘₯π‘›βˆ’π‘₯𝑖)𝑅𝑛𝑖𝛿π‘₯+(π‘¦π‘›βˆ’π‘¦π‘–)𝑅𝑛𝑖𝛿𝑦. Therefore, (1) can be expressed as where cos𝛼𝑖=(π‘₯π‘›βˆ’π‘₯𝑖)𝑅𝑛𝑖,sin𝛼𝑖=(π‘¦π‘›βˆ’π‘¦π‘–)𝑅𝑛𝑖,, and πœπ‘–=π‘‘π‘–βˆ’π‘…π‘›π‘–/𝑐.

By collecting all TOA measurements at one point, the centralized processing scheme formulates the model π‘₯=π‡πœ½+𝐰, where 𝜏𝐱=[1𝜏2β‹―πœπ‘]𝑇,]𝜽=[𝑇𝛿π‘₯π›Ώπ‘¦π‘‡πœ€,𝐰=[1πœ€2β‹―πœ€π‘]𝑇, and Based on this linear model, the unknowns 𝑇,𝛿π‘₯, and 𝛿𝑦 can be solved by the best linear unbiased estimator (BLUE) πœƒ=[H𝑇H]βˆ’1𝐇𝑇𝐱, and the estimation variances are given by the diagonal items of the covariance matrix π‚πœ½=𝜎2[H𝑇H]βˆ’1, that is, [𝜎2𝑇,𝜎2𝛿π‘₯,𝜎2𝛿𝑦]=[π‚πœƒ(1,1),π‚πœƒ(2,2),π‚πœƒ(3,3)].

Note that, in order to perform the estimation, the sink, where the processing is performed, has to know the locations of all participating nodes as well as their TOA measurements. This could cause severe communication/energy problems in situations when, for example, nodes are moving and therefore being required to constantly report their locations to the sink.

3. The Proposed Distributed Processing

As with the centralized processing, we assume that the sink is the point where the final estimate is needed. However, in the proposed scheme, the sink is just an ordinary node which does not have additional energy or computing capacity and does not perform any additional work as compared to other nodes. Only the nodes which have a TOA measurement or have received any measurements or intermediate estimates (IEs) from the up-stream nodes participate in the process. The principle is, for each participating node, to process the available measurements and/or IEs whenever possible and to send the result to the next-hop node (i.e., the down-stream node). The fusion result from the sink node is the final estimate. Note that the proposed scheme does not require any special arrangements on other functionalities of the network such as routing; TOA measurements and IEs are fused naturally while travelling on any established routes to the sink.

Different fusion rules have to be used depending on the information available at a node. Basically, three or more TOA measurements can be fused into one IE where a TOA measurement is defined as the location of the node and its measured πœπ‘–, that is, TOAβ‰œ(π‘₯𝑖,𝑦𝑖,πœπ‘–). The IE is defined as the estimates of the unknowns and their corresponding variances, that is, ξ‚Šξ‚Šξ‚Šπ›ΏIE=(𝛿π‘₯,𝛿𝑦,𝑇,𝜎2𝛿π‘₯,𝜎2𝛿𝑦,𝜎2𝑇). An IE can be further fused with one measurement, two measurements, or one or more IEs. Fusion is performed at a local node after receiving all TOAs and/or IEs from up-stream nodes. When an IE is obtained, all follow-on transmission on this particular route will involve one IE only. Figure 2 depicts a typical fusion scenario where the arrows represent data transmissions, and rectangular blocks indicate performed fusion functions which are described in detail below.

3.1. Fuse_TOAs()

This function fuses N (β‰₯3) TOA measurements into one IE. The formulas used here are exactly the same as in Section 2.

3.2. Fuse_IEs()

When N (>1) IEs are available, they can be combined according to the principle of BLUE to yield an updated IE. That is, for ξ‚Ώξ‚Ώξ‚Šπ›Ώ(𝛿π‘₯𝑛,𝛿𝑦𝑛,𝑇𝑛,𝜎2𝛿π‘₯𝑛,𝜎2𝛿𝑦𝑛,𝜎2𝑇𝑛), where 𝑛=1,…,𝑁, the updated IE is given by 𝛿π‘₯_new=𝜎2𝛿π‘₯_newβˆ‘π‘π‘›=1𝛿π‘₯𝑛/𝜎2𝛿π‘₯𝑛,𝛿𝑦_new=𝜎2𝛿𝑦_newβˆ‘π‘π‘›=1𝛿𝑦𝑛/𝜎2𝛿𝑦𝑛, and ξ‚Šπ‘‡π‘›_new=𝜎2𝑇_newβˆ‘π‘π‘›=1ξ‚Šπ‘‡π‘›/𝜎2𝑇𝑛, where 𝜎2𝛿π‘₯_newβˆ‘=(𝑁𝑛=11/𝜎2𝛿π‘₯𝑛)βˆ’1,𝜎2𝛿𝑦_newβˆ‘=(𝑁𝑛=11/𝜎2𝛿𝑦𝑛)βˆ’1, and 𝜎2𝑇_new=(βˆ‘π‘π‘›=11/𝜎2𝑇𝑛)βˆ’1.

As can be seen from the above equations, the BLUE is actually a weighted average of the estimates, giving the more accurate (with smaller variances) ones larger weights.

3.3. Fuse_IE_with_one_TOA()

We now fuse one intermediate estimate ξ‚Šξ‚ŠξIE0=(𝛿π‘₯,𝛿𝑦,𝑇,𝜎2𝛿π‘₯,𝜎2𝛿𝑦,𝜎2𝑇) with one TOA (π‘₯𝑖,𝑦𝑖,πœπ‘–) measured by a certain node 𝑛𝑖. In other words, we refine IE0 with the new information provided by the TOA measurement. In order to do this, we approximate T by its estimated value 𝑇 and refine the estimates of 𝛿π‘₯ and 𝛿𝑦 only. By denoting ξπ‘‡β‰ˆπ‘‡+πœ€π‘‡, where πœ€π‘‡ is the error in the estimate of T, the TOA measurement (2) can be expressed by πœπ‘–βˆ’ξπ‘‡β‰ˆ(cos𝛼𝑖/𝑐)𝛿π‘₯+(sin𝛼𝑖/𝑐)𝛿𝑦+πœ€π‘–+πœ€π‘‡ which can be further simplified as where π‘Ž=cos𝛼𝑖/𝑐,𝑏=sin𝛼𝑖/𝑐, and When the error terms (πœ€π‘– and πœ€π‘‡) are ignored, (4) can be represented by a straight line in the 𝛿π‘₯βˆ’π›Ώπ‘¦ plane. The estimate (ξ‚Šξ‚Šπ›Ώπ‘₯,𝛿𝑦) is a point on the same plane, and we propose to use a weighted average of it and its projection onto line (4) (ξ‚Šπ›Ώπ‘₯ξ…ž,ξ‚Šπ›Ώπ‘¦ξ…ž) as the new estimate. As far as the node 𝑛𝑖 can see, (ξ‚Šπ›Ώπ‘₯ξ…ž,ξ‚Šπ›Ώπ‘¦ξ…ž) lies on the circle centred at (π‘₯𝑖,𝑦𝑖) with radius π‘Ÿ=π‘π‘§π‘œ, where c is the propagation speed. It can be seen from (5) that the variance of r is given by 𝜎2π‘Ÿ=𝑐2(𝜎2+𝜎2𝑇). Since it is too complex to obtain the exact expressions for the variances of ξ‚Šπ›Ώπ‘₯ξ…ž and ξ‚Šπ›Ώπ‘¦ξ…ž, in order to determine their weights in forming the new estimates, we will use 𝜎2π‘Ÿ to approximate their variances, that is, 𝜎2𝛿π‘₯β€²β‰ˆπœŽ2π›Ώπ‘¦β€²β‰ˆπœŽ2π‘Ÿ. This approximation is in line with the new information the TOA measurement provides on the source location.

By denoting 𝐾=π‘Ž2+𝑏2, the projection coordinates can be expressed by leading to new estimates with updated variances 𝜎2𝛿π‘₯_new=(1/𝜎2𝛿π‘₯+1/𝜎2𝛿π‘₯β€²)βˆ’1 and 𝜎2𝛿𝑦_new=(1/𝜎2𝛿𝑦+1/𝜎2𝛿𝑦′)βˆ’1.

3.4. Fuse_IE_with_two_TOAs()

Here, we have two TOA measurements (π‘₯1,𝑦1,𝜏1) and (π‘₯2,𝑦2,𝜏2) along with one IE ξ‚Šξ‚ŠξIE0=(𝛿π‘₯,𝛿𝑦,𝑇,𝜎2𝛿π‘₯,𝜎2𝛿𝑦,𝜎2𝑇) available. We first use the BLUE principle to fuse the two TOAs to obtain initial estimates ξ‚Šπ›Ώπ‘₯ξ…ž and ξ‚Šπ›Ώπ‘¦ξ…ž. The result is then further fused with IE0. As in the previous section, by denoting ξπ‘‡β‰ˆπ‘‡+πœ€π‘‡, for 𝑖=1,2, we have 𝑧𝑖=π‘Žπ‘–π›Ώπ‘₯+𝑏𝑖𝛿𝑦+πœ€π‘–+πœ€π‘‡, where π‘Žπ‘–=cos𝛼𝑖/𝑐, 𝑏𝑖=sin𝛼𝑖/𝑐, and 𝑧𝑖=πœπ‘–βˆ’ξπ‘‡.

By defining 𝐱=[𝑧1𝑧2]𝑇,𝐇=[π‘Ž1𝑏1π‘Ž2𝑏2],𝜽=[𝛿π‘₯𝛿𝑦]𝑇, and 𝐰=[πœ€1+πœ€π‘‡πœ€2+πœ€π‘‡]𝑇, we have π‘₯=π‡πœ½+𝐰 which immediately leads to the following BLUE [3]: with covariance matrix π‚πœ½=[π‡π‘‡π‚βˆ’1𝐇]βˆ’1, where, with E denoting expectation, That is, [𝜎2𝛿π‘₯β€²,𝜎2𝛿𝑦′]=[π‚πœƒ(1,1),π‚πœƒ(2,2)]. This fusion function is now summarized as follows. Step 1, use (8) to obtain ξ‚Šπ›Ώπ‘₯ξ…ž and ξ‚Šπ›Ώπ‘¦ξ…ž. Step 2, fuse (ξ‚Šπ›Ώπ‘₯ξ…ž,ξ‚Šπ›Ώπ‘¦ξ…ž) and (ξ‚Šξ‚Šπ›Ώπ‘₯,𝛿𝑦) using (7).

4. Simulation

In this section, we conduct simulation to verify the proposed distributed source localization scheme. The tree routing protocol and network setup process including parent-child relationship and address assignment algorithms of the ZigBee [4] standard are followed in the simulation.

100 ZigBee nodes are randomly deployed in a square region with size 500 m by 500 m. The root node (the coordinator in ZigBee’s terms) is acting as the sink and is located at the centre of the lower edge of the region. The nominal location of the sound source (target) (with speed 𝑐=344m/s) is the centre of the region, and it has an effective range 𝑅=200m. When a node is less than R meters away from the target, it can make a TOA measurement. For each simulation scenario, the source takes a small move in both x and y directions, that is, (𝛿π‘₯,𝛿𝑦)=(0.5m,0.5m), and 1000 runs are conducted with independent and identically distributed zero-mean Gaussian noise samples generated in each run. The estimation accuracy is measured by the mean square error over all the runs: βˆ‘MSE=(1/1000)1000π‘Ÿ=1((π‘₯βˆ’Μ‚π‘₯π‘Ÿ)2+(π‘¦βˆ’Μ‚π‘¦π‘Ÿ)2), where (π‘₯π‘Ÿ,π‘¦π‘Ÿ) is the estimate of the true source location (π‘₯,𝑦) in the rth run.

Figure 3 shows the estimation errors with varying noise variance. It can be seen that the proposed distributed scheme works, and its performance attains that of the centralized best linear unbiased estimator when the noise is reduced to a certain level.

In summary, the benefit of the proposed scheme is two-fold. First, it has improved flexibility and scalability over centralized schemes since it shares the processing among multiple nodes and does not pose any special requirements on the sink node. Second, it has the potential to save energy/bandwidth consumption of the network due to reduced radio transmissions.