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Optimal sensor placement for source localization based on RSSD

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Abstract

Source localization based on the received signal strength (RSS) has received great interest due to its low cost and simple implementation. In this paper we consider the source localization problem based on the received signal strength difference (RSSD) with unknown transmitted power of the source using spatially separated sensors. It is well- known that the relative sensor-source geometry (SSG) plays a significant role in localization performance. For this issue, the fisher information matrix (FIM) which inherently is a function of relative SSG is derived. Then for different scenarios the SSG based on the maximization of determinant of FIM is investigated to obtain the optimal sensor placement. Finally, computer simulations are used to study the performance of various sensor placements. Both theoretical analysis and simulation results reveal the ability of the proposed sensor- source geometries.

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Notes

  1. We have to mention that for \(N \geqslant 6\) with equal sensor ranges, the optimal geometry is not unique and equiangular sensor separation is a special case of infinitely many optimal geometries.

  2. Although some places in the Fig. 9b have better performance respect to the corresponding places in Fig. 9a but in average the sensor configuration in Fig. 9a gives a better performance for all region.

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Authors and Affiliations

  1. Department of Electrical Engineering, Yazd University, Yazd, Iran

    Ali Heydari & MasoudReza Aghabozorgi

  2. Department of Electrical Engineering, Shiraz University, Shiraz, Iran

    Mehrzad Biguesh

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  1. Ali Heydari
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  2. MasoudReza Aghabozorgi
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  3. Mehrzad Biguesh
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Correspondence to MasoudReza Aghabozorgi.

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Appendix A

Appendix A

Using the matrix inversion lemma [35]

$$\begin{aligned} {\left( {{\mathbf {A}} + {\mathbf {B}\mathbf {C}\mathbf {D}}} \right) ^{ - 1}} = {{\mathbf {A}}^{ - 1}}{{ - }}{{\mathbf {A}}^{ - 1}}{\mathbf {B}}{\left( {{{\mathbf {C}}^{ - 1}}{{ + \mathbf {D}}}{{\mathbf {A}}^{ - 1}}{\mathbf {B}}} \right) ^{ - 1}}{\mathbf {D}}{{\mathbf {A}}^{ - 1}}, \end{aligned}$$
(34)

where \({\mathbf {A}} = {\text{diag}}\left[ {\sigma _1^2,\,...,\,\sigma _{N - 1}^2} \right] ,\,{\mathbf {B}} = {\left[ {\begin{array}{*{20}{c}} 1&{...}&1 \end{array}} \right] ^T},\,\,{\mathbf {C}} = \sigma _N^2,\,\,{\text{and}}\,{\mathbf {D}} = {{\mathbf {B}}^T}\) Hence the inverse of \({\varvec{\varSigma }}\) in Eq (8) can be written as

$$\begin{aligned} {{{{\varvec{\varSigma }}}}^{ - 1}} = {{\mathbf {A}}^{ - 1}} - \alpha \,{\mathbf {s}}\,{{\mathbf {s}}^T}, \end{aligned}$$
(35)

where \(\alpha = \left( \sum _{i=1}^N {\frac{1}{\sigma _i^2}} \right) ^{-1}\) and \({\mathbf {s}} = {\left[ {\begin{array}{*{20}{c}} \frac{1}{\sigma _1^2}&{...}&\frac{1}{\sigma _{N-1}^2}\end{array}} \right] ^T}\).

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Heydari, A., Aghabozorgi, M. & Biguesh, M. Optimal sensor placement for source localization based on RSSD. Wireless Netw 26, 5151–5162 (2020). https://doi.org/10.1007/s11276-020-02380-6

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