Nothing Special   »   [go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

Backscatter communication system efficiency with diffusing surfaces

  • Published:
Annals of Telecommunications Aims and scope Submit manuscript

Abstract

In an ambient backscatter communication system, the waves generated by a source are reflected by a tag, in a variable manner in time. Therefore, the tag can transmit a message to a reader, without generating any radio wave and without battery. As a consequence, such a communication system is a promising technology for ultra-low energy wireless communications. In the simplest implementation of such a system, the tag sends a binary message by oscillating between two states and the reader detects the bits by comparing the two distinct received powers. In this paper, for the first time, we propose to analyze the impact of the shape of diffusing flat panel surfaces that diffuse in all directions, on an ambient backscatter communication system. We establish the analytical closed form expression of the power contrast in the presence of flat panels, by considering a rectangular surface and a disk-shaped surface, and we show that diffusing surfaces improve the power contrast. Moreover, our approach allows us to express the contrast to noise ratio, and therefore to establish the BER performance. Furthermore, we show that it makes it possible to improve the energetic performance, thanks to diffusing surfaces. For any configuration characterized by a fixed source, tag and reader, we moreover determine the precise locations of diffusing surfaces, which induce a maximum efficiency of the surfaces, whatever the wavelength. Furthermore, we show that it becomes possible to easily determine an optimal frequency which maximizes the contrast power, thanks to the expression of the contrast power.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Griffin J (2009) High-frequency modulated-backscatter communication using multiple antennas. PhD Dissertation, Georgia Institute of Technology. Available at: https://smartech.gatech.edu/bitstream/handle/1853/28087/

  2. Rachedi K, Phan-Huy D-T, Selmene N, Ourir A, Gautier M, Gati A, Galindo-Serrano A, Fara R, de Rosny J (2019) Demo abstract : real-time ambient backscatter demonstration. In Proc. IEEE INFOCOM 2019, pp 987–988

  3. Fara R, Phan-Huy D-T, Di Renzo M (2020) Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers. Accepted to IEEE WCNC 2020

  4. Gati A, et al (2019) Key technologies to accelerate the ICT Green evolution - An operator’s point of view submitted to IEEE. Available at: https://arxiv.org/abs/1903.09627

  5. Van Huynh N, Hoang DT, Lu X, Niyato D, Wang P, Kim DI (2018) Ambient backscatter communications: a contemporary survey. IEEE Commun Surv Tutorials 20(4):2889–2922 Fourthquarter

  6. Rachedi K (2019) Antennes compactes reconfigurables en diagramme de rayonnement pour la modulation spatiale MIMO et introduction aux communications numériques par rétrodiffuseurs. PhD Dissertation, Sorbonne Université. Available at: https://theses.hal.science/tel-03139838

  7. Fara R (2021) Ambient backscatter communications in future mobile networks. PhD Dissertation. Université Paris-Saclay. English. Available at: https://theses.hal.science/tel-03434254

  8. Liu V, Parks A, Talla V, Gollakota S, Wetherall D, Smith JR (2013) Ambient backscatter: wireless communication out of thin air. In: Proc. of ACM SIGCOMM 2013. Hong Kong, China, pp 39–50

  9. Wang G, Gao F, Fan R, Tellambura C (2016) Ambient backscatter communication systems: detection and performance analysis. IEEE Trans Commun 64(11):4836–4846

    Article  Google Scholar 

  10. Yang G, Liang Y, Zhang R, Pei Y (2018) Modulation in the air: backscatter communication over ambient OFDM carrier. IEEE Trans Commun 66(3):1219–1233

    Article  Google Scholar 

  11. Devineni JK, Dhillon HS (2019) Non-coherent detection and bit error rate for an ambient backscatter link in time-selective fading. arXiv:1908.05657, available at: https://arxiv.org/pdf/1908.05657.pdf

  12. Renzo MD et al (2019) Smart radio environments empowered by reconfigurable AI meta-surfaces: an idea whose time has come. J Wireless Com Network 2019:129

    Article  Google Scholar 

  13. Fara R, Phan-Huy D-T, Ourir A, Di Renzo M, De Rosny J (2021) Reconfigurable intelligent surface-assisted ambient backscatter communications – experimental assessment. 2021 IEEE international conference on communications workshops (ICC Workshops), pp 1–7

  14. Fara R, Ratajczak P, Phan-Huy D-T, Ourir A, Di Renzo M, De Rosny J (2022) A prototype of reconfigurable intelligent surface with continuous control of the reflection phase. To appear in IEEE Wireless Communications Magazine

  15. Kelif J-M, Phan-Huy D-T (2021) Backscatter communication system with dumb diffusing surface. IEEE international symposium on personal, indoor and mobile radio communications PIMRC 2021

  16. Recommendation ITU-R P.2040-1 (07/2015) Effects of building materials and structures on radiowave propagation above about 100 MHz-P Series-Radiowave

  17. Kelif J-M, Phan-Huy D-T, Ratajczak P Diffusing surfaces impact on backscatter communication system. IEEE 1\(^{st}\) international conference on 6G networking, 6GNet 2022, Paris

  18. Devineni JK, Dhillon HS, (2018) Exact bit error rate analysis of ambient backscatter systems under fading channels. (2018) IEEE 88th vehicular technology conference (VTC-Fall). Chicago, IL, USA, pp 1–6

Download references

Author information

Authors and Affiliations

  1. Chatillon, France

    Jean-Marc Kelif & Dinh-Thuy Phan-Huy

  2. Sophia Antipolis, France

    Philippe Ratajczak

Authors
  1. Jean-Marc Kelif
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dinh-Thuy Phan-Huy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philippe Ratajczak
    View author publications

    You can also search for this author in PubMed Google Scholar

Consortia

Orange Innovation/Networks

Corresponding author

Correspondence to Jean-Marc Kelif.

Ethics declarations

Conflict of interest

Not applicable

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A. System geometric expressions

We denote \(\psi = (\widehat{\overrightarrow{r_{0}},\overrightarrow{z}})\), \(\phi = (\widehat{\overrightarrow{r_{0}},\overrightarrow{x}})\), \(\psi ^{\prime } = (\widehat{\overrightarrow{{r^{\prime }}_{0}},\overrightarrow{z}})\) and \(\phi ^{\prime } = (\widehat{\overrightarrow{{r^{\prime }}_{0}},\overrightarrow{x}})\). Considering a vector \(\overrightarrow{r_{j}}\) = \(\overrightarrow{SM_{j}}\) where \(M_{j}\) is located at any location of the surface \(W_{1}\), we can write the expression

$$\begin{aligned} \overrightarrow{r_{0}} + \overrightarrow{z} + \overrightarrow{x} = \overrightarrow{r_{j}} \end{aligned}$$
(A1)

where \(\overrightarrow{z}\) and \(\overrightarrow{x}\) represent the projections of \(\overrightarrow{r_{j}}\) on the axis z and x axis. Considering \(x_{j}\) and \(z_{j}\) the coordinates of \(\overrightarrow{r_{j}}\) according the axes x and z, we have

$$\begin{aligned} {r_{j}^{2}}= & {} {r_{0}^{2}} + {z_{j}^{2}} + {x_{j}^{2}} + 2 r_{0} z_{j} \cos \psi + 2 r_{0} x_{j} \cos \phi \\{} & {} + 2 z_{j} x_{j} \cos (\widehat{\overrightarrow{z},\overrightarrow{x}})\\= & {} {r_{0}^{2}} \left( 1+ 2 \frac{z_{j} \cos \psi }{r_{0}} + 2 \frac{x_{j} \cos \phi }{r_{0}} + \frac{{z_{j}^{2}} + {x_{j}^{2}} }{ {r_{0}^{2}}}\right) \end{aligned}$$

Therefore

$$\begin{aligned} r_{j}\approx & {} r_{0}+ 2 \frac{1}{2} z_{j} \cos \psi + 2 \frac{1}{2} x_{j} \cos \phi + \frac{{z_{j}^{2}}+ {x_{j}^{2}}}{2 r_{0}}\nonumber \\\approx & {} r_{0}+ z_{j} \cos \psi + x_{j} \cos \phi \end{aligned}$$
(A2)

due to \(\cos (\widehat{\overrightarrow{z},\overrightarrow{x}}) = 0\) and considering that \({z_{j}^{2}} + {x_{j}^{2}}<< {r_{0}^{2}}\). We do the analog development considering the ray \(r^{\prime }_{j}\) :

$$\begin{aligned} \overrightarrow{r^{\prime }_{j}} = \overrightarrow{- r^{\prime }_{0}} + \overrightarrow{z} + \overrightarrow{x} \end{aligned}$$
(A3)

So we have

$$\begin{aligned} {r^{\prime }}_{j}^{2}= & {} {r^{\prime }}_{0}^{2} + {z_{j}^{2}} + {x_{j}^{2}}\nonumber \\- & {} 2 {r^{\prime }}_{0} z_{j} \cos \psi ^{\prime } - 2 {r^{\prime }}_{0} x_{j} \cos \phi ^{\prime } + 2 z_{j} x_{j} \cos (\widehat{\overrightarrow{z},\overrightarrow{x}})\nonumber \\= & {} {r^{\prime }}_{0}^{2} \left( 1- 2 \frac{z_{j} \cos \psi ^{\prime }}{{r^{\prime }}_{0}} - 2 \frac{x_{j} \cos \phi ^{\prime }}{{r^{\prime }}_{0}} + \frac{{z_{j}^{2}} + {x_{j}^{2}} }{{r^{\prime }}_{0}^{2}}\right) \end{aligned}$$
(A4)

Therefore

$$\begin{aligned} {r^{\prime }}_{j}\approx & {} {r^{\prime }}_{0} - 2 \frac{1}{2} z_{j} \cos \psi ^{\prime } - 2 \frac{1}{2} x_{j} \cos \phi ^{\prime } + \frac{{z_{j}^{2}}+ {x_{j}^{2}}}{2 {r^{\prime }}_{0}}\nonumber \\\approx & {} {r^{\prime }}_{0} - z_{j} \cos \psi ^{\prime } - x_{j} \cos \phi ^{\prime } \end{aligned}$$
(A5)

since \(\cos (\widehat{\overrightarrow{z},\overrightarrow{x}}) = 0\) and considering that \({z_{j}^{2}} + {x_{j}^{2}}<< {r^{\prime }}_{0}^{2}\). So we can write in the first order

$$\begin{aligned} r_{j} + {r^{\prime }}_{j}= & {} r_{0}\!+\! {r^{\prime }}_{0} \!+\! z_{j} \cos \psi \!+\! x_{j} \cos \phi - z_{j} \cos \psi ^{\prime } - x_{j} \cos \phi ^{\prime }\nonumber \\= & {} r_{0} + {r^{\prime }}_{0} + z_{j} (\cos \psi - \cos \psi ^{\prime }) + x_{j} (\cos \phi - \cos \phi ^{\prime })\nonumber \\= & {} r_{0} + {r^{\prime }}_{0} + z_{j} \gamma + x_{j} \zeta \end{aligned}$$
(A6)

From (A2) and (A5) \(r_{0} {r^{\prime }}_{0} \approx r_{j} {r^{\prime }}_{j}, \forall\) j = 0...N. We have

$$\begin{aligned} Y_{dif}= & {} \sum _{j=1}^{N} \sqrt{\frac{K^{sar}P}{{r_{j}^{2}}}} \sqrt{\frac{1}{{r^{\prime }}_{j}^{2}}} e^{iwt}e^{-i k(r_{j}+{r^{\prime }}_{j})}\nonumber \\\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt}e^{-i k(r_{0}+{r^{\prime }}_{0})} \sum _{j} e^{-i(k z_{j} \gamma + k x_{j} \zeta )} \end{aligned}$$
(A7)

Therefore

$$\begin{aligned} Y_{dif}\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt}e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{-L}^{L} \int _{-H}^{H} \exp -i(k \gamma z) \exp -i(k \zeta x) dzdx\nonumber \\\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt}e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{-L}^{L} \exp -i(k \gamma z) dz \int _{-H}^{H} \exp -i(k \zeta x) dx\nonumber \\\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \frac{e^{-i(k\gamma L)}- e^{i(k \gamma L)}}{-ik \gamma } \frac{e^{-i(k\zeta H)}- e^{i(k \zeta H)}}{-ik \zeta }\nonumber \\\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \frac{4}{k^{2} \gamma \zeta } \sin (k\gamma L) \sin (k\zeta H) \end{aligned}$$
(A8)

Therefore \(Y_{dif}\) can be approximated by the following expression

$$\begin{aligned} Y_{dif} = \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})} F_{plan} \end{aligned}$$
(A9)

where

$$\begin{aligned} F_{plan} = \frac{4}{k^{2} \gamma \zeta } \sin (k\gamma L) \sin (k\zeta H) \end{aligned}$$
(A10)

If the diffusing surface is a disk of radius \(R_{d}\), we can write

$$\begin{aligned} Y_{dif}\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt}e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{Surf} \exp -i k(\gamma z + \zeta x) dzdx \nonumber \\\approx & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt}e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{0}^{R_{d}} \int _{0}^{2 \pi } \exp -i k( \Omega \rho \cos \theta ) \rho d \rho d \theta \end{aligned}$$
(A11)

Where \(\overrightarrow{\Omega } (\zeta , 0, \gamma )\) and \(\overrightarrow{\rho } (x, 0, z)\). So we can write \(\gamma z + \zeta x\) as the scalar product of \(\overrightarrow{\rho }\) and \(\overrightarrow{\Omega }\). And we can express \(\overrightarrow{\rho }.\overrightarrow{\Omega }\) = \(\rho \Omega \cos \theta\) where \(\rho\) and \(\Omega\) are the module of \(\overrightarrow{\rho }\) and \(\overrightarrow{\Omega }\) and where \(\theta = \widehat{(\overrightarrow{\rho },\overrightarrow{\Omega })}\). So we can express the approximation

$$\begin{aligned} Y_{dif}= & {} \sqrt{\frac{K^{sar}P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2}}} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{0}^{R_{d}} \int _{0}^{2 \pi } \exp -i k( \Omega \rho \cos \theta ) \rho d \rho d \theta \nonumber \\= & {} A_{dif} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{0}^{R_{d}} \int _{0}^{2 \pi } \exp -i k( \Omega \rho \cos \theta ) \rho d \rho d \theta \nonumber \\= & {} A_{dif} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})}\nonumber \\{} & {} \times \int _{0}^{R_{d}} \rho d \rho 2 \pi \times \frac{1}{2 \pi } \int _{0}^{2 \pi } \exp -i (k \Omega \rho \cos \theta ) d \theta \nonumber \\= & {} A_{dif} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})} \int _{0}^{R_{d}} 2 \pi \times J_{0}(k\Omega \rho ) \rho d \rho \nonumber \\= & {} A_{dif} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})} 2 \pi \int _{0}^{k\Omega R_{d}} \frac{x}{k\Omega } J_{0}(x) \frac{dx}{k\Omega } \end{aligned}$$
(A12)

Since \(x J_{0}(x) = \frac{d}{dx}(x J_{1}(x))\), where \(J_{0} (x)\) and \(J_{1} (x)\) are the Bessel functions defined as \(J_{0} (x) = \frac{1}{2 \pi } \int _{0}^{2 \pi } e^{-i (x \cos \theta )} d \theta\) and \(J_{1} (x) = \frac{1}{2 \pi } \int _{0}^{2 \pi } e^{-i (x \cos \theta -\theta )} d \theta\). So we have

$$\begin{aligned} Y_{dif} = A_{dif} e^{iwt} e^{-i k(r_{0}+{r^{\prime }}_{0})} \frac{2 \pi R_{d}}{k\Omega } J_{1}(k\Omega R_{d}) \end{aligned}$$
(A13)

Therefore, in the case of a disk, \(F_{plan}\) is replaced by \(F_{disk}\) in (A9)

$$\begin{aligned} F_{disk} = \frac{2 \pi R_{d}}{k\Omega } J_{1}(k\Omega R_{d}) \end{aligned}$$
(A14)

Appendix B. Power received by the reader directly from the source

Considering P the transmit power of the source, \(S_{R}\) the power density received by the reader coming from the source, \(A_{R}\) the apparent area of the reader, \(G^{s}\) the gain of the source, we can express

$$\begin{aligned} P_{R}= & {} S_{R} A_{R}\nonumber \\= & {} \frac{P}{4 \pi r^{2}} G^{s} A_{R} \end{aligned}$$
(B1)

Considering \(G^{r}\) the gain of the reader, we have

$$\begin{aligned} G^{r} = A_{R} \frac{4 \pi }{\lambda ^{2}} \end{aligned}$$
(B2)

where \(\lambda\) is the wavelength of the transmitted signal. Therefore

$$\begin{aligned} A_{R} = G^{r} \frac{\lambda ^{2}}{4 \pi } \end{aligned}$$
(B3)

As a consequence we can write

$$\begin{aligned} P_{R} = \frac{P}{r^{2}} G^{s} G^{r} \frac{ \lambda ^{2}}{(4 \pi )^{2}} \end{aligned}$$
(B4)

Therefore

$$\begin{aligned} P_{R} = \frac{K^{sr}}{r^{2} } P \end{aligned}$$
(B5)

where

$$\begin{aligned} K^{sr} = \frac{G^{s} G^{r}\lambda ^{2}}{(4 \pi )^{2}} \end{aligned}$$
(B6)

Appendix C. Power reflected by the tag or the surface, and received by the reader

1.1 1) Power reflected by the tag

Considering \(P_{t}\) the power reflected by the tag coming from the source, \(S_{t}\) the power density received by the tag coming from the source, \(A_{t}\) the apparent area of the tag, we can express

$$\begin{aligned} P_{t}= & {} S_{t} A_{t} \nonumber \\= & {} \frac{P}{4 \pi {r_{t}^{2}}} G^{s} A_{t} \end{aligned}$$
(C1)

Considering \(G^{t}\) the gain of the tag, we have

$$\begin{aligned} G^{t} = A_{t} \frac{4 \pi }{\lambda ^{2}} \end{aligned}$$
(C2)

where \(\lambda\) is the wavelength of the transmitted signal. Therefore

$$\begin{aligned} A_{t} = G^{t} \frac{\lambda ^{2}}{4 \pi } \end{aligned}$$
(C3)

As a consequence we can write

$$\begin{aligned} P_{t} = \frac{P}{4 \pi {r_{t}^{2}}} G^{s} G^{t} \frac{ \lambda ^{2}}{4 \pi } \end{aligned}$$
(C4)

1.2 2) Power received by the reader from the tag

Considering \({P_{R}^{T}}\) the power received by the reader coming from the tag, \({S_{R}^{T}}\) the power density received by the reader coming from the tag, \(A_{R}\) the apparent area of the reader, \(G^{r}\) the gain of the reader we can express

$$\begin{aligned} {P_{R}^{T}}= & {} {S_{R}^{T}} A_{R}\nonumber \\= & {} \frac{P_{t}}{4 \pi {r^{\prime }}_{t}^{2}} G^{t} A_{R}\nonumber \\= & {} \frac{P_{t}}{4 \pi {r^{\prime }}_{t}^{2}} G^{t} G^{r} \frac{ \lambda ^{2}}{4 \pi }\nonumber \\= & {} \frac{\frac{P}{{r_{t}^{2}}} G^{s} G^{t} \frac{ \lambda ^{2}}{(4 \pi )^{2}} }{4 \pi {r^{\prime }}_{t}^{2}} G^{t} G^{r} \frac{ \lambda ^{2}}{4 \pi }\nonumber \\= & {} \frac{P}{{r_{t}^{2}} {r^{\prime }}_{t}^{2} } G^{s} (G^{t})^{2} G^{r} \frac{ \lambda ^{4}}{(4 \pi )^{4}} \end{aligned}$$
(C5)

since we have

$$\begin{aligned} G^{r} = A_{R} \frac{4 \pi }{\lambda ^{2}}. \end{aligned}$$
(C6)

and thus

$$\begin{aligned} A_{R} = G^{r} \frac{\lambda ^{2}}{4 \pi }. \end{aligned}$$
(C7)

Therefore

$$\begin{aligned} {P_{R}^{T}} = \frac{P}{{r_{t}^{2}} {r^{\prime }}_{t}^{2} } G^{s} (G^{t})^{2} G^{r} \frac{ \lambda ^{4}}{(4 \pi )^{4}} \end{aligned}$$
(C8)

Therefore

$$\begin{aligned} {P_{R}^{T}} = \frac{K^{str}}{{r_{t}^{2}} {r^{\prime }}_{t}^{2} } P \end{aligned}$$
(C9)

where

$$\begin{aligned} K^{str} = \frac{G^{s} (G^{t})^{2} G^{r}\lambda ^{4}}{(4 \pi )^{4}} \end{aligned}$$
(C10)

1.3 3) Power received by the reader from the surface

In an analog way, we can express the power \({P_{R}^{d}}\), received by the reader coming from the surface

$$\begin{aligned} {P_{R}^{d}} = \frac{P}{{r_{0}^{2}} {r^{\prime }}_{0}^{2} } G^{s} (G^{d})^{2} G^{r} \frac{ \lambda ^{4}}{(4 \pi )^{4}} \end{aligned}$$
(C11)

Therefore

$$\begin{aligned} {P_{R}^{d}} = \frac{K^{sar}}{{r_{0}^{2}} {r^{\prime }}_{0}^{2} } P \end{aligned}$$
(C12)

where

$$\begin{aligned} K^{sar} = \frac{G^{s} (G^{d})^{2} G^{r}\lambda ^{4}}{(4 \pi )^{4}} \end{aligned}$$
(C13)

Appendix D. Location of the diffusing surface

Let consider the coordinates of the source S(0, 0, 0) the reader \(R(x_{R},y_{R},z_{R})\) and the center of the diffusing surface \(O_{1}(x,y,z)\). We have \(r_{0}(x, y, z)\) and \(r^{\prime }_{0}(x^{\prime }, y^{\prime }, z^{\prime })\) where

$$\begin{aligned} \left\{ \begin{array}{lll} x^{\prime } = x-x_{R}\\ y^{\prime } = y-y_{R} r\\ z^{\prime } = z-z_{R}. \end{array}\right. \end{aligned}$$
(D1)

From

$$\begin{aligned} \left\{ \begin{array}{ll} \cos \psi =\cos \psi ^{\prime }\\ \cos \phi =\cos \phi ^{\prime } \end{array}\right. \end{aligned}$$

we can write

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{x}{r_{0}} = \frac{x-x_{R}}{{r^{\prime }}_{0}}\\ \frac{z}{r_{0}} = \frac{z-z_{R}}{{r^{\prime }}_{0}}. \end{array}\right. \end{aligned}$$
(D2)

Let notice that in our system, the coordinate z is not equal to zero. Otherwise, it would imply \(z_{R}=0\) (which is not the case in our system (Fig. 3)). The coordinate x can be equal to zero. In this case, \(x_{R}=0\). Therefore, from (D2)

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{x^{2}}{{r_{0}^{2}}} = \frac{x^{\prime 2}}{{r^{\prime }}_{0}^{2}}\\ \frac{z^{2}}{{r_{0}^{2}}} = \frac{(z-z_{R})^{2}}{{r^{\prime }}_{0}^{2}}. \end{array}\right. \end{aligned}$$

So we have

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{x}{z} = \frac{x-x_{R}}{z-z_{R}}\\ z^{2}((y-y_{R})^{2}+(x-x_{R})^{2}) = (z-z_{R})^{2}(y^{2}+x^{2}), \end{array}\right. \end{aligned}$$

which can be expressed

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{x}{z} = \frac{x-x_{R}}{z-z_{R}}\\ z^{2}((y-y_{R})^{2}+ (x\frac{z-z_{R}}{z})^{2}) = (z-z_{R})^{2}(y^{2}+x^{2}). \end{array}\right. \end{aligned}$$

Therefore

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{x}{z} = \frac{x-x_{R}}{z-z_{R}}\\ y-y_{R} = \pm y\frac{z-z_{R}}{z}. \end{array}\right. \end{aligned}$$
(D3)

So we have two cases.

1.1 1) The first case is expressed

$$\begin{aligned} \left\{ \begin{array}{ll} z = \frac{z_{R}}{x_{R}}x\\ z = \frac{z_{R}}{y_{R}}y \end{array}\right. \end{aligned}$$

where \(x_{R}\) and \(y_{R}\) are not equal to 0. Therefore, denoting \(a =\frac{x_{R}}{z_{R}}\) and \(b =\frac{y_{R}}{z_{R}}\)

$$\begin{aligned} \left\{ \begin{array}{ll} x = a z \\ y = b z. \end{array}\right. \end{aligned}$$

In this case, the set of locations of \(O_{1}\) is a line, intersection of 2 plans. The coordinates of a point of this line are expressed as z(a, b, 1) where z is a real number.

1.2 2) The second case is expressed

$$\begin{aligned} \left\{ \begin{array}{ll} x = a z \\ y = \frac{y_{R}}{2z - z_{R}}z. \end{array}\right. \end{aligned}$$

In this case, the set of locations of \(O_{1}\) is given by the intersection of the plan \(z = \frac{1}{a}x\) and the hyperbole \(z = \frac{z_{R}}{2y - y_{R}}y\). It can be expressed \(O_{1}(az, \frac{y_{R} z}{2z - z_{R}}, z)\) where z is a real number (not equal to \(\frac{z_{R}}{2}\)). However if \(z=\frac{z_{R}}{2}\) from (D3) we have \(y_{R}=0\). If \(x_{R}\) = 0, therefore x = 0, so we have

$$\begin{aligned} \left\{ \begin{array}{ll} x = 0 \\ z^{2} (y-y_{R})^{2} = (z-z_{R})^{2} y^{2}. \end{array}\right. \end{aligned}$$

This case is equivalent to the precedent case, with the plan given by \(x= 0\).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kelif, JM., Phan-Huy, DT., Ratajczak, P. et al. Backscatter communication system efficiency with diffusing surfaces. Ann. Telecommun. 78, 561–576 (2023). https://doi.org/10.1007/s12243-023-00955-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12243-023-00955-w

Keywords

Search

Navigation