Detection of mobile targets on the plane and in space using heterogeneous sensor networks

Abstract

Detection of targets moving within a field of interest is a fundamental service Wireless Sensor Network (WSN) service. The WSN’s target detection performance is directly related to the placement of the sensors within the field of interest. In this paper, we address the problem of deterministic sensor deployment on the plane and in space, for the purpose of detecting mobile targets. We map the target detection problem to a line-set intersection problem and derive analytic expressions for the probability of detecting mobile targets. Compared to previous works, our mapping allows us to consider sensors with heterogeneous sensing capabilities, thus analyzing sensor networks that employ multiple sensing modalities. We show that the complexity of evaluating the target detection probability grows exponentially with the network size and, hence, derive appropriate lower and upper bounds. We also show that maximizing the lower bound on the probability for target detection on the plane and in space, is analogous to the problem of minimizing the average symbol error probability in two-dimensional and three-dimensional digital modulation schemes, respectively, over additive white Gaussian noise. These problems can be addressed using the circle packing problem for the plane, and the sphere packing problem for space. Using the analogy to digital modulation schemes, we derive sensor constellations from well known signal constellations with low average symbol error probability.

Appendix

In this section, we prove the monotonicity of m ₂(d _i,j ) as it is expressed in Corollary 1.

Proof

To prove the monotonicity of m ₂(d _i,j ), we must show that m ₂(d _i,j ) becomes smaller as the distance d _i,j among the sensing area ${\mathcal{A}}_i, {\mathcal{A}}_j$ increases. For cases where L _in , L _out have an analytical form, this can be easily shown by computing the first derivative of (L _in  − L _out ), with respect to d _i,j . As an example, when the two sensing areas are discs of radius r, we can analytically express L _in , L _out as follows:

$$ L_{in}= 2\left(\pi r + arc \tan \left(\frac{2r}{d_{i,j}}\right) r + 2 \sqrt{ \frac{d^2_{i,j}}{4}-r^2 } \right),\,\,L_{out}= 2(\pi r +d_{i,j}). $$

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Computation of the derivative of (L _in  − L _out ) verifies the monotonicity of m ₂(d _i,j ) (this is an elementary mathematic exercise not presented here due to space constraints). For the case where (L _in  − L _out ) does not have a closed analytic form, we can illustrate the monotonicity of m ₂(d _i,j ) by considering its equivalence to the set of lines intersecting both sets. The argument in our proof is that as d _i,j increases, a smaller “number” of lines will intersect both sets and, hence, m ₂(d _i,j ) becomes smaller.

Let a ₁, a ₂ denote the slopes of the lines of the inner string that wraps around ${\mathcal{A}}_i, {\mathcal{A}}_j,$ as shown in Fig. 21(a). Any line that is crossing both ${\mathcal{A}}_i, {\mathcal{A}}_j$ must have a slope a with a ₁ ≤a ≤a ₂. As an example, all lines that pass through the intersection point T with slope a ₁ ≤a ≤a ₂ intersect both sensing areas. The measure of the set of lines crossing both sensing areas is monotonically related to the range of (a ₂ − a ₁) that is the greater the difference between the slopes a ₁, a ₂ the larger the “number” (measure) of lines that cross both sets (more lines out of all possible trajectories satisfy the a ₁≤a ≤a ₂ condition).

Fig. 21

(a) Any line intersecting both sensing areas has a slope a ₁ ≤a ≤a ₂, (b) when d _i,j increases, the slope difference a ₂ − a ₁ decreases and, hence, a smaller number of lines intersects both sensing areas. Therefore, the measure m ₂(d _i,j ) of the set of lines intersecting two sensing areas is a monotonically decreasing function of d _i,j

As the distance d _i,j between the sets ${\mathcal{A}}_i, {\mathcal{A}}_j$ increases, the slope difference (a ₂ − a ₁) decreases and, hence the “number” of lines intersecting both sets also decreases. Therefore, m ₂(d _i,j ) which expresses the measure of the set of lines intersecting both sensing areas, also decreases with the decrease of the slope difference, or equivalently with the increase of the pairwise distance d _i,j . In Fig. 21(b), we show the reduction in the slope difference (a ₂ − a ₁) that reduces the set of lines that intersect both sets. $\hfill\square$