# Error Performance Estimation of Modulated Retroreflective Transdermal Optical Wireless Links with Diversity under Generalized Pointing Errors

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## Abstract

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## 1. Introduction

- A novel system and channel model for MRR TOW links is developed that incorporates the basic characteristics of this type of link.
- The impact of the more realistic stochastic pointing errors with nonzero boresight is introduced and estimated in the MRR TOW area for the first time.
- A spatial-diversity technique with optimal combining is considered by employing multiple out-of-body receiver apertures, and thus, MRR TOW data transmissions of enhanced reliability are achieved.
- An outage bit error rate analysis is performed that evaluates the effectiveness of the spatial-diversity technique versus the joint impact of transdermal pathloss and generalized pointing errors with nonzero boresight.
- An average SNR estimation is performed that reveals the feasibility of the proposed MRR TOW system architecture.
- Novel analytical mathematical expressions are derived along with proper corresponding illustrated results that can be utilized in the design of MRR TOW links.

## 2. System and Channel Model

#### 2.1. Signal Model

_{m}represents the m-th signal copy arriving at the m-th receiver’s aperture with m = 1, 2, ..., M; η

_{m}is the corresponding effective photodiode’s photo-current conversion ratio; x is the information bit signal; h

_{m}denotes the m-th total channel state; and n is the additive noise that is modeled as a zero-mean complex Gaussian process with variance σ

^{2}[5,12,25].

_{m}is the dermal thickness of the m-th transmitted or reflected transdermal path, and α(λ

_{m}) is its skin-attenuation coefficient, with λ

_{m}being the corresponding operational wavelength. For operational wavelengths between 400 nm and 1800 nm, we obtain [11,24,25]:

_{m}values are expressed in nm and the values of a

_{i}, b

_{i}and c

_{i}, with i = 1, 2, ...,8, are obtained from ([19], Table 1). It is noteworthy that the accuracy of the latter expression is higher than 99.7% [19,25].

#### 2.2. Generalized Pointing Errors with Nonzero Boresight

_{m}in length, the corresponding fraction of the collected power at the m-th receiver circular aperture with radius r

_{m}, can be approximated as [12,27]:

_{eq,m}represents the equivalent beam width in the m-th receiver aperture, which is given as ${w}_{eq,m}={\left[\frac{\sqrt{\pi}\mathrm{erf}\left({v}_{m}\right){w}_{\delta ,m}^{2}}{2{v}_{m}\mathrm{exp}(-{v}_{m}^{2})}\right]}^{1/2}$, with ${v}_{m}=\frac{\sqrt{\pi}{r}_{m}}{\sqrt{2}}{w}_{\delta ,m}$ and erf(.) denoting the error function ([31], Equation (8.250.1)). Additionally, in TOW links we get [6,11,18]:

_{m}from the implanted retroreflective modulator, with θ

_{m}being the corresponding divergence angle [4,25]. Moreover, ${A}_{0,m}={\mathrm{erf}}^{2}\left({v}_{m}\right)$ is the fraction of the collected power at r

_{m}=0, [5,17]. Furthermore, R

_{m}is the radial displacement at the m-th receiver, which is expressed as ${R}_{m}=\left|{\overrightarrow{R}}_{m}\right|=\sqrt{{R}_{x,m}^{2}+{R}_{y,m}^{2}}$, where ${\overrightarrow{R}}_{m}={\left[{R}_{x,m},{R}_{y,m}\right]}^{T}$ is the radial displacement vector with R

_{x,m}, R

_{y,m}being the displacements located along the horizontal and elevation axes at the detector plane, respectively. It should be noted that these variables are considered as nonzero mean Gaussian distributed random variables, i.e., ${R}_{x,m}~N\left({\mu}_{x,m},{\sigma}_{x,m}^{2}\right)$, ${R}_{y,m}~N\left({\mu}_{y,m},{\sigma}_{y,m}^{2}\right)$, where μ

_{x,m}, μ

_{y,m}, denote their mean values and σ

_{x,m}, σ

_{y,m}are the jitters for horizontal and elevation displacements, respectively, [12,28]. Here, it should be also mentioned that the approximation of Equation (5) is valid for ${w}_{\delta ,m}/{r}_{m}6$, which is true in typical optical wireless links, [27,28,30].

_{m}is obtained as [17,28,29]:

#### 2.3. Joint Impact of Pathloss and Generalized Pointing Errors with Nonzero Boresight

#### 2.4. SISO MRR TOW Links

#### 2.5. SIMO MRR TOW Links with Spatial Diversity and OC

## 3. Analytical Results

_{m}, was assumed to be equal to 7 mm or 8 mm, while the divergence angle, θ

_{m}, was set to 20°. Furthermore, unless otherwise stated, it was assumed that ${P}_{s}$ = 1 μw/MHz, while ${N}_{0}$ is fixed at ${\left(1.3\mathrm{pA}/\sqrt{\mathrm{Hz}}\right)}^{2}$ [37]. Additionally, for each TOW link, the operational wavelength, λ

_{m}, was selected to be equal to 1.1μm, since according to [19], this is the optimal transmission wavelength value for TOW links. It became evident that the choice of the appropriate operational wavelength was even more critical for MRR TOW links, where the light signal traverses the skin channel twice, which makes it even more susceptible to skin-induced attenuation and transdermal pathloss. Regarding pointing errors, considering the above parameter values, we obtained ${w}_{\delta ,m}/{r}_{m}>6$, which made the approximation in Equation (5) valid. Additionally, for $M=1$, we assumed zero-boresight pointing errors either with $\left({\delta}_{m},{\mu}_{x,m}/{r}_{m},{\mu}_{y,m}/{r}_{m},{\sigma}_{x,m}/{r}_{m},{\sigma}_{y,m}/{r}_{m}\right)=\left(7\mathrm{mm},0,0,4,4\right);$ i.e., ${\xi}_{1}=1.14$; or $\left({\delta}_{m},{\mu}_{x,m}/{r}_{m},{\mu}_{y,m}/{r}_{m},{\sigma}_{x,m}/{r}_{m},{\sigma}_{y,m}/{r}_{m}\right)=\left(8\mathrm{mm},0,0,4,4\right)$; i.e., ${\xi}_{1}=1.30$. For $M=2$, we also assumed NZB pointing errors for the second photodetector with $\left({\delta}_{m},{\mu}_{x,m}/{r}_{m},{\mu}_{y,m}/{r}_{m},{\sigma}_{x,m}/{r}_{m},{\sigma}_{y,m}/{r}_{m}\right)=\left(7\mathrm{mm},2,1,4.5,4.5\right)$; i.e., ${\xi}_{2}=0.96$; $\left({\delta}_{m},{\mu}_{x,m}/{r}_{m},{\mu}_{y,m}/{r}_{m},{\sigma}_{x,m}/{r}_{m},{\sigma}_{y,m}/{r}_{m}\right)=\left(7\mathrm{mm},2,1,5,5\right)$; i.e., ${\xi}_{2}=0.99$; $\left({\delta}_{m},{\mu}_{x,m}/{r}_{m},{\mu}_{y,m}/{r}_{m},{\sigma}_{x,m}/{r}_{m},{\sigma}_{y,m}/{r}_{m}\right)=\left(8\mathrm{mm},2,1,4.5,4.5\right)$; i.e., ${\xi}_{2}=0.87$; and $\left({\delta}_{m},{\mu}_{x,m}/{r}_{m},{\mu}_{y,m}/{r}_{m},{\sigma}_{x,m}/{r}_{m},{\sigma}_{y,m}/{r}_{m}\right)=\left(8\mathrm{mm},2,1,5,5\right)$; i.e., ${\xi}_{2}=1.10$. Finally, for $M=3$, we correspondingly assumed ${\xi}_{2}={\xi}_{3}=0.96$, 0.99, 0.87 or 1.10. Under these assumptions and settings, the following analytical and simulation results are presented.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gil, Y.; Rotter, N.; Arnon, S. Feasibility of retroreflective transdermal optical wireless communication. Appl. Opt.
**2012**, 51, 4232–4239. [Google Scholar] [CrossRef] [PubMed] - Guillory, K.S.; Misener, A.K.; Pungor, A. Hybrid RF/IR transcutaneous telemetry for power and high-bandwidth data. In Proceedings of the 26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, San Francisco, CA, USA, 1–5 September 2004; IEEE: Piscataway, NJ, USA, 2004; pp. 4338–4340. [Google Scholar]
- Abualhoul, M.Y.; Svenmarker, P.; Wang, Q.; Andersson, J.Y.; Johansson, A.J. Free space optical link for biomedical applications. In Proceedings of the 2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, San Diego, CA, USA, 28 August–1 September 2012; IEEE: Piscataway, NJ, USA, 2012; pp. 1667–1670. [Google Scholar]
- Varotsos, G.K.; Nistazakis, H.E.; Aidinis, K.; Jaber, F.; Rahman, K.K.M.; Tsigopoulos, A.D.; Christofilakis, V. Average BER Estimation of Retroreflective Transdermal Optical Wireless Links with Diversity, Attenuation and Spatial Jitter. In Proceedings of the 2020 International Conference on Modern Circuits and Systems Technologies (MOCAST), Bremen, Germany, 7–9 September 2020; IEEE: Piscataway, NJ, USA, 2020; pp. 1–4. [Google Scholar]
- Varotsos, G.K.; Nistazakis, H.E.; Aidinis, K.; Jaber, F.; Rahman, K.K.M. Transdermal subcarrier L-PSK or DBPSK optical wireless links with time diversity, skin attenuation and spatial jitter. J. Modern Opt.
**2020**, 67, 14. [Google Scholar] [CrossRef] - Varotsos, G.K.; Nistazakis, H.E.; Aidinis, K.; Jaber, F.; Rahman, K.K.M. Signal Intensity Estimation in Transdermal Optical Wireless Links with Stochastic Pointing Errors Effect. Technologies
**2020**, 8, 60. [Google Scholar] [CrossRef] - Parmentier, S.; Fontaine, R.; Roy, Y. Laser diode used in 16 Mb/s, 10 mW optical transcutaneous telemetry system. In Proceedings of the Biomedical Circuits and Systems Conference, BioCAS, Baltimore, MD, USA, 20–22 November 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 377–380. [Google Scholar]
- Liu, T.; Bihr, U.; Anis, S.M.; Ortmanns, M. Optical transcutaneous link for low power, high data rate telemetry. In Proceedings of the 2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), San Diego, CA, USA, 28 August–1 September 2012; IEEE: Piscataway, NJ, USA, 2012; pp. 3535–3538. [Google Scholar]
- Liu, T.; Anders, J.; Ortmanns, M. System level model for transcutaneous optical telemetric link. In Proceedings of the 2013 IEEE International Symposium on Circuits and Systems (ISCAS), Beijing, China, 19–23 May 2013; IEEE: Piscataway, NJ, USA, 2013; pp. 865–868. [Google Scholar]
- Liu, T.; Bihr, U.; Becker, J.; Anders, J.; Ortmanns, M. In vivo verification of a 100 Mbps transcutaneous optical telemetric link. In Proceedings of the Biomedical Circuits and Systems Conference (BioCAS), Lausanne, Switzerland, 22–24 October 2014; IEEE: Piscataway, NJ, USA, 2014; pp. 580–583. [Google Scholar]
- Trevlakis, S.E.; Boulogeorgos, A.A.A.; Sofotasios, P.C.; Muhaidat, S.; Karagiannidis, G.K. Optical wireless cochlear implants. Biomed. Opt. Express
**2019**, 10, 707–730. [Google Scholar] [CrossRef] [PubMed] - Varotsos, G.K.; Nistazakis, H.E.; Aidinis, K.; Jaber, F.; Rahman, K.K. Transdermal Optical Wireless Links with Multiple Receivers in the Presence of Skin-Induced Attenuation and Pointing Errors. Computation
**2019**, 7, 33. [Google Scholar] [CrossRef] [Green Version] - Ackermann, D.M.; Smith, B.; Kilgore, K.L.; Peckham, P.H. Design of a high speed transcutaneous optical telemetry link. In Proceedings of the 2006 International Conference of the IEEE Engineering in Medicine and Biology Society, New York, NY, USA, 30 August–3 September 2006; IEEE: Piscataway, NJ, USA, 2006; pp. 2932–2935. [Google Scholar]
- Abita, J.L.; Schneider, W. Transdermal Optical Communications; John Hopkins APL Tech: Laurel, MD, USA, 2004; Volume 25, pp. 261–268. [Google Scholar]
- Ackermann, D.M.; Smith, B.; Wang, X.F.; Kilgore, K.L.; Peckham, P.H. Designing the optical interface of a transcutaneous optical telemetry link. IEEE Trans. Biomed. Eng.
**2008**, 55, 1365–1373. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ritter, R.; Handwerker, J.; Liu, T.; Ortmanns, M. Telemetry for implantable medical devices: Part 1-media properties and standards. IEEE Solid-State Circuits Mag.
**2014**, 6, 47–51. [Google Scholar] [CrossRef] - Varotsos, G.K.; Nistazakis, H.E.; Tombras, G.S.; Aidinis, K.; Jaber, F.; Rahman, M. On the use of diversity in transdermal optical wireless links with nonzero boresight pointing errors for outage performance estimation. In Proceedings of the 2019 International Conference on Modern Circuits and Systems Technologies (MOCAST), Thessaloniki, Greece, 7–9 May 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 1–4. [Google Scholar]
- Varotsos, G.K.; Nistazakis, H.E.; Aidinis, K.; Roumelas, G.D.; Jaber, F.; Rahman, K.K.M. Modulated Retro-Reflector Transdermal Optical Wireless Communication Systems with Wavelength Diversity over Skin-Induced Attenuation and Pointing Errors. In Proceedings of the 2020 IEE International Symposium on Signal Processing and Information Technology (ISSPIT), Ajman, United Arab Emirates, 10–12 December 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 1–5. [Google Scholar]
- Trevlakis, S.; Boulogeorgos, A.A.; Karagiannidis, G. Signal Quality Assessment for Transdermal Optical Wireless Communications under Pointing Errors. Technologies
**2018**, 6, 109. [Google Scholar] [CrossRef] [Green Version] - Varotsos, G.K.; Nistazakis, H.E.; Petkovic, M.I.; Djordjevic, G.T.; Tombras, G.S. SIMO Optical Wireless Links with Nonzero Boresight Pointing Errors over M modeled Turbulence Channels. Elsevier Opt. Commun.
**2017**, 403, 391–400. [Google Scholar] [CrossRef] - Varotsos, G.K.; Nistazakis, H.E.; Gappmair, W.; Sandalidis, H.G.; Tombras, G.S. SIMO subcarrier PSK FSO links with phase noise and non-zero boresight pointing errors over turbulence channels. IET Commun.
**2019**, 13, 831–836. [Google Scholar] [CrossRef] - Navidpour, S.M.; Uysal, M.; Kavehrad, M. BER performanceof free-space optical transmission with spatial diversity. IEEE Trans. Wirel. Commun
**2007**, 6, 2813–2819. [Google Scholar] - Ghassemlooy, Z.; Arnon, S.; Uysal, M.; Xu, Z.; Cheng, J. Emerging optical wireless communications-advances and challenges. IEEE J. Sel. Areas Commun.
**2015**, 33, 1738–1749. [Google Scholar] [CrossRef] - Trevlakis, S.E.; Boulogeorgos, A.A.A.; Karagiannidis, G.K. On the impact of misalignment fading in transdermal optical wireless communications. In Proceedings of the 7th International Conference on Modern Circuits and Systems Technologies (MOCAST), Thessaloniki, Greece, 7–9 May 2018; IEEE: Piscataway, NJ, USA, 2018; pp. 1–4. [Google Scholar]
- Trevlakis, S.E.; Boulogeorgos, A.A.A.; Karagiannidis, G.K. Outage Performance of Transdermal Optical Wireless Links in the Presence of Pointing Errors. In Proceedings of the 2018 IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Kalamata, Greece, 25–28 June 2018; IEEE: Piscataway, NJ, USA, 2018; pp. 1–5. [Google Scholar]
- Li, X.; Zhao, X.; Zhang, P.; Yang, W.; Wang, T.; Jiang, H. Probability density function of turbulence fading in MRR free space optical link and its applications in MRR free space optical communications. IET Commun.
**2017**, 11, 2476–2481. [Google Scholar] [CrossRef] - Farid, A.A.; Hranilovic, S. Outage capacity optimization for free space optical links with pointing errors. IEEE/OSA J. Lightwave Technol.
**2007**, 25, 1702–1710. [Google Scholar] [CrossRef] [Green Version] - Yang, F.; Cheng, J.; Tsiftsis, T.A. Free-space optical communication with nonzero boresight pointing errors. IEEE Trans. Commun.
**2014**, 62, 713–725. [Google Scholar] [CrossRef] - Boluda-Ruiz, R.; Garcia-Zambrana, A.; Castillo-Vazquez, B.; Castillo-Vazquez, C. Impact of nonzeroboresight pointing error on ergodic capacity of MIMO FSO communication systems. Opt. Express
**2016**, 24, 3513–3534. [Google Scholar] [CrossRef] [PubMed] - Boluda-Ruiz, R.; García-Zambrana, A.; Castillo-Vazquez, C.; Castillo-Vazquez, B. Novel approximation of misalignment fading modeled by Beckmann distribution on free-space optical links. Opt. Express
**2016**, 24, 22635–22649. [Google Scholar] [CrossRef] [PubMed] - Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 6th ed.; Academic Press: New York, NY, USA, 2000. [Google Scholar]
- Beckmann, P.; Spizzichino, A. The Scattering of Electromagnetic Waves from Rough Surfaces; Artech House: Norwood, MA, USA, 1987. [Google Scholar]
- Helstrom, C.W. Probability and Stochastic Processes for Engineers; Macmillan Coll Division: Stuttgart, Germany, 1991. [Google Scholar]
- The Wolfarm Functions Site. 2008. Available online: https://functions.wolfram.com/ (accessed on 30 November 2020).
- Chiani, M.; Dardari, D.; Simon, M.K. New exponential bounds and approximations for the computation of error probability in fading channels. Trans. Wireless Commun.
**2003**, 2, 840–845. [Google Scholar] [CrossRef] [Green Version] - Alouini, M.-S.; Simon, M.K. An MGF-based performance analysis of generalized selection combining over Rayleigh fading channels. IEEE Trans. Commun.
**2000**, 48, 401–415. [Google Scholar] [CrossRef] - Maxim Integrated Products. 155 Mbps Low-Noise Transimpedance Amplifier. Available online: http://pdf.datasheetcatalog.com/datasheets2/44/444242_1.pdf (accessed on 30 November 2020).

**Figure 2.**Average BER evolution over a wide range of electrical average SNR for various link configurations of ${\delta}_{m}=7\mathrm{mm}$ with spatial diversity and OC along withvarying generalized PE.

**Figure 3.**Average BER evolution over a wide range of electrical average SNR for various link configurations of ${\delta}_{m}=8\mathrm{mm}$ with spatial diversity and OC along withvarying generalized PE.

**Figure 4.**Average SNR evolution over a wide range of power spectral density of the transmitted signal for various generalized pointing errors strength and skin thicknesses.

i | a_{i} | b_{i} | c_{i} |
---|---|---|---|

1 | 10 | 0.35 | 0.065 |

2 | 4.5 | 0.42 | 0.25 |

3 | 13.48 | −1.5 | 50.12 |

4 | 14.7 | 1442 | 49.35 |

5 | 7.435 | 1499 | 75.88 |

6 | 48 | 3322 | 1033 |

7 | 594.1 | −183 | 285.9 |

8 | 11.47 | −618.5 | 1054 |

Parameter | Value(s) |
---|---|

θ_{m} | 20° |

${P}_{s}$ | 1 μW/MHz–20 μW/MHz |

${N}_{0}$ | ${\left(1.3\mathrm{pA}/\sqrt{\mathrm{Hz}}\right)}^{2}$ |

μ | 30–60 dB |

η_{m} | 0.8 |

$M$ | $\left\{1,2,3\right\}$ |

λ_{m} | 1.1 μm |

r_{m} | 0.5 mm |

δ_{m} | $\left\{7\mathrm{mm},8\mathrm{mm}\right\}$ |

${\mu}_{x,m}/{r}_{m}$ | {$0,2\}$ |

${\mu}_{y,m}/{r}_{m}$ | $\left\{0,1\right\}$ |

${\sigma}_{x,m}/{r}_{m}$ | $\left\{4,4.5,5\right\}$ |

${\sigma}_{y,m}/{r}_{m}$ | $\left\{4,4.5,5\right\}$ |

${\xi}_{1}$ | $\left\{1.14,1.30\right\}$ |

${\xi}_{2}$ | {$0.87,0.96,0.99,1.10\}$ |

${\xi}_{3}$ | {$0.87,0.96,0.99,1.10\}$ |

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## Share and Cite

**MDPI and ACS Style**

Varotsos, G.K.; Nistazakis, H.E.; Aidinis, K.; Jaber, F.; Nasor, M.; Rahman, K.K.M.
Error Performance Estimation of Modulated Retroreflective Transdermal Optical Wireless Links with Diversity under Generalized Pointing Errors. *Telecom* **2021**, *2*, 167-180.
https://doi.org/10.3390/telecom2020011

**AMA Style**

Varotsos GK, Nistazakis HE, Aidinis K, Jaber F, Nasor M, Rahman KKM.
Error Performance Estimation of Modulated Retroreflective Transdermal Optical Wireless Links with Diversity under Generalized Pointing Errors. *Telecom*. 2021; 2(2):167-180.
https://doi.org/10.3390/telecom2020011

**Chicago/Turabian Style**

Varotsos, George K., Hector E. Nistazakis, Konstantinos Aidinis, Fadi Jaber, Mohd Nasor, and Kanhira Kadavath Mujeeb Rahman.
2021. "Error Performance Estimation of Modulated Retroreflective Transdermal Optical Wireless Links with Diversity under Generalized Pointing Errors" *Telecom* 2, no. 2: 167-180.
https://doi.org/10.3390/telecom2020011