## Mathematic models for a ray tracing method and its applications in wireless optical communications |

Optics Express, Vol. 18, Issue 17, pp. 18431-18437 (2010)

http://dx.doi.org/10.1364/OE.18.018431

Acrobat PDF (860 KB)

### Abstract

This paper presents a new ray tracing method, which contains a whole set of mathematic models, and its validity is verified by simulations. In addition, both theoretical analysis and simulation results show that the computational complexity of the method is much lower than that of previous ones. Therefore, the method can be used to rapidly calculate the impulse response of wireless optical channels for complicated systems.

© 2010 OSA

## 1. Introduction

1. Wikipedia. “Ray_tracing_(physics)”. http://en.wikipedia.org/wiki/Ray_tracing_(physics).

2. F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Ray tracing algorithms for fast calculation of the channel impulse response on diffuse IR-wireless indoor channels,” Opt. Eng. **39**(10), 1510–1512 (2000). [CrossRef]

3. J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Comm. **11**(3), 367–379 (1993). [CrossRef]

*K*is the number of reflections considered,

*N*is the number of rays, and

4. O. González, S. Rodriguez, R. Perez-Jimenez, B. R. Mendoza, and A. Ayala, “Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm,” IEEE Trans. Commun. **53**(1), 124–130 (2005). [CrossRef]

*K*in the conventional algorithm [5

5. S. Rodríguez, R. Pérez-Jiménez, F. J. López-Hernández, O. González, and A. Ayala, “Reflection model for calculation of the impulse response on IR-wireless indoor channels using ray-tracing algorithm,” Microw. Opt. Technol. Lett. **32**(4), 296–300 (2002). [CrossRef]

7. D. Takase and T. Ohtsuki, “Spatial multiplexing in optical wireless MIMO communications over indoor environment,” IEICE Trans. **E 89-B**(4), 1364–1371 (2006). [CrossRef]

## 2. Models of sources, receivers, and reflectors

### 2.1 Sources and Receivers

3. J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Comm. **11**(3), 367–379 (1993). [CrossRef]

*m*is its mode number which specifies the directionality of the source.

### 2.2 Reflectors

5. S. Rodríguez, R. Pérez-Jiménez, F. J. López-Hernández, O. González, and A. Ayala, “Reflection model for calculation of the impulse response on IR-wireless indoor channels using ray-tracing algorithm,” Microw. Opt. Technol. Lett. **32**(4), 296–300 (2002). [CrossRef]

*ρ*, where

*ρ*is the reflection coefficient of the surface. As a result, in MMCA, the number of rays keeps constant, but, in the proposed method, the number of rays decreases exponentially with the times of reflection.

## 3. Description of the method

*S*represents the source,

*R*represents the receiver,

*S*and a receiver

*R*in an environment free of reflector. If the distance

*d*between them is very large relative to the receiver size, all of beams of light will arrive at the receiver at approximately the same time. Thus, the LOS impulse response can be written as Eq. (4) where

*θ*is the angle between

*ψ*is the angle between

*d*is the distance between the source and receiver

*c*is the speed of light, the rectangular function is given by Eq. (5)

*N*) of photons. Consider a differential reflector

*ρ*and Lambert’s reflection pattern. In

*k*th bounce, it is assume that the number of photons arriving at

*M*. Thus,

*θ*is the angle between

*ψ*is the angle between

*d*is the distance between

*i*th photon in the

*k*th bounce can be written as Eq. (8)

*k*th bounce is given by Eq. (9)

*k*th bounce. It is assumed that the energy of each photon keeps constant from one bounce to another, for the energy of the photon contributing to the receiver is very small relative to the energy of itself. For this reason, the proposed method is sometimes called photon tracing method.

*N*photons have been traced one by one, the process of photon tracing comes to an end.

## 4. Data processing

*i*th photon in

*k*th bounce,

## 5. Results

2. F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Ray tracing algorithms for fast calculation of the channel impulse response on diffuse IR-wireless indoor channels,” Opt. Eng. **39**(10), 1510–1512 (2000). [CrossRef]

2. F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Ray tracing algorithms for fast calculation of the channel impulse response on diffuse IR-wireless indoor channels,” Opt. Eng. **39**(10), 1510–1512 (2000). [CrossRef]

## 6. Computational complexity

**39**(10), 1510–1512 (2000). [CrossRef]

*N*photons are emitted in a simulation, then, after each of photons experiences their first impinging, there remain

*number of photons remained*is the number of photons remained after

*k*th bounce, which is exactly the number of photons to be traced. It can be seen from Table 2 that the number of photons to be traced after

*k*th bounce depends on

*N*,

*k,*and

*number of photons remained*will decrease exponentially with

*k*. In contrast, the number of rays to be traced after

*k*th bounce in MMCA is always

*N*. The contrasts are shown in Fig. 2 . The three curves marked with circle symbols are simulation results of the proposed method. The configuration parameters in each simulation come from Table 1 except that all surfaces of the room have the same reflection coefficients, which are 0.4, 0.6, and 0.8 respectively. The curves marked with plus symbols are calculation results according to Table 2. It is obvious that simulation results fit calculation results very well. The curve marked with triangle symbols is the simulation results of MMCA [4

4. O. González, S. Rodriguez, R. Perez-Jimenez, B. R. Mendoza, and A. Ayala, “Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm,” IEEE Trans. Commun. **53**(1), 124–130 (2005). [CrossRef]

*k*may keep increasing up to infinite, so the total number of photon tracing can be obtained from Eq. (14)

*k*should be a finite value

*K*in computer simulations, then the total number can be revised as Eq. (15)

4. O. González, S. Rodriguez, R. Perez-Jimenez, B. R. Mendoza, and A. Ayala, “Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm,” IEEE Trans. Commun. **53**(1), 124–130 (2005). [CrossRef]

## 7. Conclusions

## Acknowledgments

## References and links

1. | Wikipedia. “Ray_tracing_(physics)”. http://en.wikipedia.org/wiki/Ray_tracing_(physics). |

2. | F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Ray tracing algorithms for fast calculation of the channel impulse response on diffuse IR-wireless indoor channels,” Opt. Eng. |

3. | J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Comm. |

4. | O. González, S. Rodriguez, R. Perez-Jimenez, B. R. Mendoza, and A. Ayala, “Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm,” IEEE Trans. Commun. |

5. | S. Rodríguez, R. Pérez-Jiménez, F. J. López-Hernández, O. González, and A. Ayala, “Reflection model for calculation of the impulse response on IR-wireless indoor channels using ray-tracing algorithm,” Microw. Opt. Technol. Lett. |

6. | D. Takase, and T. Ohtsuki, “Optical wireless MIMO communications (OMIMO),” in |

7. | D. Takase and T. Ohtsuki, “Spatial multiplexing in optical wireless MIMO communications over indoor environment,” IEICE Trans. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(060.2605) Fiber optics and optical communications : Free-space optical communication

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 11, 2010

Revised Manuscript: August 1, 2010

Manuscript Accepted: August 2, 2010

Published: August 12, 2010

**Citation**

Minglun Zhang, Yangan Zhang, Xueguang Yuan, and Jinnan Zhang, "Mathematic models for a ray tracing method and its applications in wireless optical communications," Opt. Express **18**, 18431-18437 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18431

Sort: Year | Journal | Reset

### References

- Wikipedia. “Ray_tracing_(physics)”. http://en.wikipedia.org/wiki/Ray_tracing_(physics) .
- F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Ray tracing algorithms for fast calculation of the channel impulse response on diffuse IR-wireless indoor channels,” Opt. Eng. 39(10), 1510–1512 (2000). [CrossRef]
- J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, “Simulation of multipath impulse response for indoor wireless optical channels,” IEEE J. Sel. Areas Comm. 11(3), 367–379 (1993). [CrossRef]
- O. González, S. Rodriguez, R. Perez-Jimenez, B. R. Mendoza, and A. Ayala, “Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm,” IEEE Trans. Commun. 53(1), 124–130 (2005). [CrossRef]
- S. Rodríguez, R. Pérez-Jiménez, F. J. López-Hernández, O. González, and A. Ayala, “Reflection model for calculation of the impulse response on IR-wireless indoor channels using ray-tracing algorithm,” Microw. Opt. Technol. Lett. 32(4), 296–300 (2002). [CrossRef]
- D. Takase, and T. Ohtsuki, “Optical wireless MIMO communications (OMIMO),” in Proceedings of IEEE Grobal Telecommunications Conference (Institute of Electrical and Electronics Engineers, Dallas, 2004), pp. 928–932.
- D. Takase and T. Ohtsuki, “Spatial multiplexing in optical wireless MIMO communications over indoor environment,” IEICE Trans. E 89-B(4), 1364–1371 (2006). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.