2. For in (14), the L is divided into two parts: B and 2A while in (15) it is taken as a whole. According to
, finer division will lead to better results. If R keeps increasing, it not difficult to find that more division of ‘B’ and ‘2A’ will become a whole ‘L’, surely the localization error increases as well.
From the aspect of one-dimensional space, there are different cases when the value of R varies, such as the three cases illustrated above, and each case corresponds to an interval of value segment of the sensing radius R. According to the above equations, we operate the theoretical analysis and the results are shown in
. During each segment, an optimal solution can be achieved and the general trend of optimal solution is monotonic, just as can be seen from
. Then we may conclude from the figure that when R increases, the minimal error at each interval will increase at the same time. To simply the analysis which may be very complicated in a 2D space, we try to start from the one-dimensional space to reveal the certain trends. The analysis in the view of one-dimensional space further verifies the correctness of Theorem 1, indicating that finer division of the field will lead to more patches, thus better localization accuracy can be achieved. Different from the one-dimensional space that larger R will deteriorate the division, when the value of R changes in a 2D space, a small increase of R can lead to a giant increase in patches, which nearly grows in the exponential manner. So the situation is much more different in a 2D space, which we can see from the simulation.
Theoretical analysis of localization error in a one-dimensional space.
In this section, we carry out the simulations to evaluate the performance of the proposed algorithm and verify the impact of different sensing radius in a 2D space. In all the cases, the field is a 10
square, divided into grids and sensors are deployed in all the vertices of the grids. Length of the grid is fixed, which we set to be 1m here, while the sensing radius R ranges from 1m to 10 m. All results were averages of 1000 independent runs.
Our algorithm proposed takes advantage of the proximity feature to ensure accuracy. It seems as if larger sensing radius will lower the resolution for a sensor can only locate a target in a larger area, however, this is not that true. For if we view this from another aspect, a larger sensing range can also make sure more sensors can detect a target at the same time, according to (1), a small increase in a single sensor node’s uncertainty may lead to a much more decrease in uncertainty when we take a network of sensors that works in a collaborative way into account. Theoretically, if the sensing radius is more than L/2, sensor nodes’ sensing ranges begins to intersect in the space. With larger radius, more sensors will detect the target, and the whole sensing field will be partitioned with finer degree, more patches will be formed, the average size of patch decreases as well, then the localization error determined by patch decreases at the same time. This is because though less information provided by a single sensor, a collaborative network of binary sensors can performance better which is balanced by a quadratic increase in the number of patches. The connectivity feature is fully utilized in this method, and for comparison, the performance of Centroid method and APIT method
based on the connectivity principle are also shown here. The Centroid method simply averages the coordinates of sensors that detect the target as the estimated location of the target, while the APIT method performs location estimation by isolating the environment into triangular regions between beaconing nodes. As can be seen from
, as the connectivity degree increases, more sensors far away from the target will be involved, localization accuracy in the Centroid method will decrease. While better connectivity will improve the performance of both the APIT method and our method, a finer division of the sensing field of our method outperforms the APIT method.
Performance comparison under different connectivity degrees.
Later, we consider a more practical scene that edge effect can’t be ignored. When the sensing radius is above half of the length of the area, namely, U/2, the central part of the sensing region begins to form a large patch, size of which increases with sensing radius, as can be seen from
. An extreme example is when the sensing radius is big enough for every sensor to cover the space, then the whole region will become a single patch, information behind these data is far from enough to locate a target. Under this circumstance, the number of patches does not grow as the sensing radius increases, so the trend of localization error will also change.
Patch forming with sensing radius: 2m, 7m and 10m.
When we come to a practical scene with noises, the situation will be different. Surely, the error of observation is highly related to the accuracy of localization for incorrect observation will lead to a wrong sensing vector
. Since each
is corresponding to a patch, a wrong sensing vector will lead to the wrong patch, thus comes to lower localization accuracy. Let e% denotes the number of sensor nodes which can detect the target that make wrong observations,
shows the experimental results with various values of e when the data are not filtered with location information of sensors. It is easy to observe that higher sensing error will lead to higher localization error.
Localization error under different observation error € with varying sensing radius.
Added with location information of the sensors, we can correct the reading of sensors that are affected by noise to some extent as the case in shown in
. Localization accuracy will be much better improved for the mapping will be much more precise, as is shown in
Localization error under different observation error € with varying sensing radius after data filtering.
We can see from the above simulation results that, theoretically, in an ideal environment, the optimal sensing radius is about U/2, where U is the side length of the space of interest. But we can conclude from the simulation results when sensing radius grows from U/3 to U/2, the location accuracy won’t improve at a significant level while the computation cost will increase a lot and more vulnerable to noise in a practical scene. So taken the factors of cost and fault-tolerance into account, we can set the optimal sensing radius to be U/3, then both good localization accuracy and low cost can be achieved.
In this paper, we consider a simple approach to locate the source of interference in a passive way with a binary sensor network. We consider the problem of locating it with different sensing radius and analyze the geometrical influence on patch forming. We are dedicated to explore the basic bounds of localization errors from a one-dimensional space in order to gain basic rules. Results show that when sensing radius is about half the length of the space, best localization accuracy can be achieved, taking the cost into account, we find a third of the length can be chosen as the best sensing radius, leading a way to more efficient utilization of binary sensor networks in the field of interference localization.
Qihui Wu received his B.S. degree in communications engineering, M.S. degree and Ph.D. degree in communications and information systems from Institute of Communications Engineering, Nanjing, China, in 1994, 1997 and 2000, respectively. From 2003 to 2005, he was a Postdoctoral Research Associate at Southeast University, Nanjing, China. From 2005 to 2007, he was an Associate Professor with the College of Communications Engineering, PLA University of Science and Technology, Nanjing, China, where he is currently a Professor and Ph.D. supervisor. From March 2011 to September 2011, he was an Advanced Visiting Scholar in Stevens Institute of Technology, Hoboken, USA. His current research interests span the areas of wireless communications and signal processing, with emphasis on system design of software defined radio, cognitive radio, and spectrum management.
Liang Yue received the B.S. degree in Changsha University of Science and Technology, Changsha, China, in 2013. He is currently pursuing the M.S. degree in communications and information system at College of Communications Engineering, PLA University of Science and Technology, Nanjing, China. His research interests are wireless sensor networks and localization.
Long Wang received the B.S. degree in electronic information science and technology from Tsinghua University, Beijing, China, in 2010, and the M.S. degree in communication and information systems from the PLA University of Science and Technology, Nanjing, China, in 2013. His current research interests include femtocell networks, wireless sensor networks, and big data analytics.
Guoru Ding received the B.S. degree (Hons.) in electrical engineering from Xidian University, Xi’an, China, in 2008. He is currently pursuing the Ph.D. degree in communications and information systems at College of Communications Engineering, PLA University of Science and Technology, Nanjing,China. His research interests include cognitive radio networks, machine learning, statistical signal processing, and big data analytics over wireless networks. Mr. Ding currently serves as a Technical Program Committee (TPC) member of IEEE GLOBECOM 2014 and IEEE VTC 2014-Fall. He is an invited reviewer for more than 10 journals such as IEEE Signal Processing Magazine, IEEE Communications Magazine, IEEE TRANSACTIONS ON COMMUNICATIONS, and IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, etc. He was a recipient of the Best Paper Award from IEEE WCSP 2009. He is a ACM student member and was a voting member of IEEE 1900.7 White Space Radio Working Group.
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