## Comparison of Monte Carlo ray-tracing and photon-tracing methods for calculation of the impulse response on indoor wireless optical channels |

Optics Express, Vol. 19, Issue 3, pp. 1997-2005 (2011)

http://dx.doi.org/10.1364/OE.19.001997

Acrobat PDF (779 KB)

### Abstract

We present a comparison between the modified Monte Carlo algorithm (MMCA) and a recently proposed ray-tracing algorithm named as photon-tracing algorithm. Both methods are compared exhaustively according to error analysis and computational costs. We show that the new photon-tracing method offers a solution with a slightly greater error but requiring from considerable less computing time. Moreover, from a practical point of view, the solutions obtained with both algorithms are approximately equivalent, demonstrating the goodness of the new photon-tracing method.

© 2011 Optical Society of America

## 1. Introduction

1. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE **85**, 265–298 (1997). [CrossRef]

2. 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]

3. F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Modified Monte Carlo scheme for high-efficiency simulation of the impulse response on diffuse IR wireless indoor channels,” Electron. Lett. **34**(19), 1819–1820 (1998). [CrossRef]

4. 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), 2775–2780 (2000). [CrossRef]

5. C. R. Lomba, R. T. Valadas, and A. M. de Oliveira Duarate, “Experimental characterisation and modelling of the reflection of infrared signals on indoor surfaces,” IEE Proc., Optoelectron. **145**, 191–197 (1998). [CrossRef]

6. 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. O. González, S. Rodríguez, R. Pérez-Jiménez, 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]

8. M. Zhang, Y. Zhang, X. Yuan, and J. Zhang, “Mathematic models for a ray tracing method and its applications in wireless optical communications,” Opt. Express **18**(17), 18431–18437 (2010). [CrossRef] [PubMed]

*et al*. proved the reduction in computational cost of the new re-designed algorithm, but they did not comment on the accuracy of the results. In this paper, we compare in more detail the differences between the two algorithms, MMCA and PTA, according to error analysis and computational cost.

## 2. Algorithms description

### 2.1. LOS impulse response

*E*and receiver

*R*in an environment without reflectors, with a large distance

*d*between both, the received power is approximately where the emitter is modeled using a generalized Lambertian radiation pattern

*R*(

_{E}*ϕ*,

*n*).

*A*

_{eff}(

*φ*) represents the effective signal collection area of the receiver.

*n*is the

*mode number*of the radiation lobe which specifies the directionality of the emitter,

*P*the power radiated by the emitter,

_{E}*A*the physical area of the receiver, and FOV the receiver field of view (semi-angle from the surface normal).

_{r}### 2.2. Multiple-bounce impulse response

*E*and receiver

*R*in a room with reflectors. The radiation from the emitter can reach the receiver after any number of reflections (see Fig. 1). In the algorithm, many rays are generated at the emitter position with a probability distribution equal to its normalized radiation pattern

*R*(

_{E}*ϕ*,

*n*)/

*P*. The power of each generated ray is initially

_{E}*P*/

_{E}*N*, where

*N*is the number of rays used to discretize the source. In MMCA, when a ray impinges on a surface, the reflection point is converted into a new optical source, thus, a new ray is generated with a probability distribution provided by the reflection pattern of that surface,

*R*(

_{S}*θ*,

*θ′*). The process continues during the simulation time. After each reflection, the power is reduced by the reflection coefficient of the surface, and the reflected power reaching the receiver is computed by where

*d*is the distance between the reflection point and receiver, and

*R*(

_{s}*θ*,

*θ′*) is the Phong’s model, used to describe the reflection pattern of the surface [6

6. 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]

*r*and the directivity of the specular component of the reflection

_{d}*m*. This model is described by where

*ρ*is the surface reflection coefficient,

*P*represents the optical power of the incident ray,

_{i}*θ*is the observation angle, and

*θ′*represents the incidence angle.

*ρ*. Moreover, when a new ray is generated, its direction of propagation is determined according to the probability distribution of the reflection pattern as in MMCA, but now the power of the new ray keeps unalterable and equal to

*P*/

_{E}*N*. In the same way, after each reflection, the reflected power reaching the receiver is computed by using Eq. (5), but with

*P*=

_{i}*P*/

_{E}*N*in Eq. (6). Therefore, both algorithms perform similarly because in PTA only

*ρN*rays of power

_{k}*P*

_{ray}=

*P*/

_{E}*N*are reflected (being

*N*the number of rays remained after the previous

_{k}*k*th bounce) whereas in MMCA all the rays are reflected but with output power

*P*

_{ray}=

*ρP*. As we will see, the fact that, in PTA, not all the incident rays are reflected leads the number of computational operations to decrease rapidly with each new bounce with respect to MMCA.

_{i}## 3. Computational complexity

7. O. González, S. Rodríguez, R. Pérez-Jiménez, 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]

2. 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]

5. C. R. Lomba, R. T. Valadas, and A. M. de Oliveira Duarate, “Experimental characterisation and modelling of the reflection of infrared signals on indoor surfaces,” IEE Proc., Optoelectron. **145**, 191–197 (1998). [CrossRef]

*elementary*calculations that are performed:

*N*is the number of rays,

*K*is the number of reflections that are considered, and

*N*is the number of surfaces that define the room. An elementary calculation is defined as the calculation of power contribution and delay from a point source (emitter or reflection point of a ray) to the receiver, as in Eq. (5), or the assessment of the propagation of the new generated ray to determine a new point source. The previous value of

_{S}*N*− 1 remaining surfaces have to be considered as possible future reflecting surfaces (in rooms with complex geometries, some surfaces can be placed on other greater surfaces, e.g. windows or doors over walls; therefore, when a ray sets up from a surface with others superimposed to it, all of them can be discarded during the searching of the new collision point). Moreover, not always the ray contributes in the received power, because sometimes the emitting point is out of the FOV of the receiver and its contribution does not have to be computed. Similarly, not all the rays reach the maximum number of reflections

_{S}*K*during the simulation time

*t*

_{max}either. However, this value represents a good approximation to the required number of elementary calculations of MMCA.

*N*rays (photons) are initially launched, after the

*k*th bounce, only

*ρ̃*

_{k}N_{k}_{−1}rays (photons) continue their path, where

*N*

_{k}_{−1}is the

*number of photons remained*after the (

*k*− 1)-th bounce (with

*N*

_{0}=

*N*), and

*ρ̃*is an average parameter of the reflection coefficient at the

_{k}*k*th bounce which depends on the reflection coefficients of the surfaces, but also on the radiation and reflection patterns, and the position and other characteristics of the emitter. The number of elementary calculations can be computed as:

*K*− 1)-th reflection. These last rays are propagated to determine the reflecting surfaces (and the reflecting points) and then their power contributions are computed. This is the reason why the previous equation has been truncated in the

*N*

_{K}_{−1}term.

*ρ̃*depends on which bounce is being considered. However, we can assume an average reflection coefficient

_{k}*ρ̃*which allows us to represent any bounce obtained as an average over all the

*ρ̃*. Then, the previous series is reduced to

_{k}*N*× (

_{S}*N*+

*ρ̃N*+

*ρ̃*

^{2}

*N*+ ... +

*ρ̃*

^{K−1}

*N*). This numeric series converges to:

*ρ̃*= 0.69 (only taking into account the areas of the surfaces), the number of rays is

*N*= 500000, the number of surfaces is

*N*= 6 and the number of considered reflections

_{S}*K*= 10. With MMCA, the number of elementary calculations is upper bounded by

## 4. Error estimation

7. O. González, S. Rodríguez, R. Pérez-Jiménez, 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]

*P′*reaching the receiver during a small time interval Δ

_{j}*t*can be estimated from the variance of

*P′*, var (

_{j}*P′*) The biggest admissible Δ

_{j}*t*is defined as the largest interval which ensures that the same ray does not contribute twice to a receiver near the walls, being var (

*P′*) given by [7

_{j}**53**(1), 124–130 (2005). [CrossRef]

*N*is the total

_{f,j}*number of flights*during the

*j*th time interval (rays flying during that interval),

*N*the number of rays that contribute in the power reaching the receiver during Δ

_{j}*t*, and

*p*is the power contribution of the

_{i,j}*i*th ray (

*i*= 1, 2,...,

*N*) arriving during that interval obtained by using Eq. (5). Therefore,

_{j}*P′*is the power value in the

_{j}*j*th histogram interval computed with the Monte Carlo ray-tracing algorithm. In MMCA,

*N*coincides with

_{f,j}*N*, because during a certain interval

*j*there are exactly

*N*rays flying, those generated by the emitter, which are never discarded. However, in PTA, after each reflection a certain number of rays are removed according to the reflection coefficient of surfaces, then

*N*decreases exponentially along the time, being different for each time interval (which has been indicated by the sub-index

_{f,j}*j*in

*N*:

_{f,j}*N*

_{f,j1}≥

*N*

_{f,j2},

*j*

_{1}<

*j*

_{2}).

*P′*given by Eq. (9) divided by the computed contribution power defined as

_{j}*j*th time interval.

## 5. Simulation results

4. 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), 2775–2780 (2000). [CrossRef]

8. M. Zhang, Y. Zhang, X. Yuan, and J. Zhang, “Mathematic models for a ray tracing method and its applications in wireless optical communications,” Opt. Express **18**(17), 18431–18437 (2010). [CrossRef] [PubMed]

*r*= 1), but rooms with materials characterized by the Phong’s model (such as the examples described in [7

_{d}**53**(1), 124–130 (2005). [CrossRef]

*K*= 10 and the simulation time

*t*

_{max}= 120 ns.

*N*= 500000 rays (photons) are generated from the emitter: in Fig. 2(a) we have the response computed by the PTA method, whereas that obtained by MMCA is depicted in Fig. 2(b). We can observe how very similar impulse responses are obtained with both methods, only distinguished by a slight greater ripple in that provided by PTA.

*k*(see Fig. 4). One can see how in MMCA the number of contributions always presents a relative high value (> 1/2 peak value) except at extreme time instants (

*t*> 80%

*t*

_{max}) when the rays are moved apart from each other and their contributions become more and more spread. In addition, no more rays are generated after the

*k*th reflection. On the contrary, in PTA the number of contributions decreases rapidly (in an exponential way) with time or with the bounce index. Therefore, the received power (impulse response) is computed by using a lower and lower number of information samples, leading the algorithm to present a greater relative error at longer time instants.

*P′*: where

_{j}*j*= 1, 2,...,

*J*and

*J*=

*t*

_{max}/Δ

*t*. We can see how PTA presents only an approximately 58% higher mean relative error than MMCA, since its greater values for the relative error are found at time instants where the impulse response exhibits quite low values. We have checked that by using

*N*= 1000000 and

*N*= 1500000 rays, PTA presents a mean relative error of +12.5% and −8.1% with respect to MMCA with

*N*= 500000. However, MMCA continues displaying a lower relative error for

*t*> 50 ns.

*N*= 500000, the simulations performed with the PTA for

*N*= 1000000 and

*N*= 1500000 needed the 56% and 84%, respectively, of elementary calculations. However, for the latter we obtained a relative error inferior to that of MMCA when

*t*< 50 ns in spite of requiring 16% less computation time.

## 6. Conclusions

*photon-tracing algorithm*. We have established quantitative parameters to carry out this comparison according to computational cost and accuracy of the provided solution. We have stated analytically and by means of simulation results that PTA presents a lower computational cost than conventional MMCA. However, regarding the error committed by both algorithms, MMCA is more reliable than PTA, although the mean relative error of the latter can be considered acceptable taking into account the reduction in computing time. In addition, more rays can be used by the new method still requiring lower simulation run-time in order to obtain more accurate results. Therefore, we can conclude that PTA begins to appear as a good substitute to MMCA with superior performance.

## Acknowledgments

## References and links

1. | J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE |

2. | 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. |

3. | F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, “Modified Monte Carlo scheme for high-efficiency simulation of the impulse response on diffuse IR wireless indoor channels,” Electron. Lett. |

4. | 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. |

5. | C. R. Lomba, R. T. Valadas, and A. M. de Oliveira Duarate, “Experimental characterisation and modelling of the reflection of infrared signals on indoor surfaces,” IEE Proc., Optoelectron. |

6. | 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. |

7. | O. González, S. Rodríguez, R. Pérez-Jiménez, 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. |

8. | M. Zhang, Y. Zhang, X. Yuan, and J. Zhang, “Mathematic models for a ray tracing method and its applications in wireless optical communications,” Opt. Express |

**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: October 29, 2010

Revised Manuscript: December 14, 2010

Manuscript Accepted: December 16, 2010

Published: January 19, 2011

**Citation**

Oswaldo González, Silvestre Rodríguez, Rafael Pérez-Jiménez, Beatriz R. Mendoza, and Alejandro Ayala, "Comparison of Monte Carlo ray-tracing and photon-tracing methods for calculation of the impulse response on indoor wireless optical channels," Opt. Express **19**, 1997-2005 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1997

Sort: Year | Journal | Reset

### References

- J. M. Kahn, and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265-298 (1997). [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]
- F. J. López-Hernández, R. Pérez-Jiménez, and A. Santamaría, "Modified Monte Carlo scheme for high-efficiency simulation of the impulse response on diffuse IR wireless indoor channels," Electron. Lett. 34(19), 1819-1820 (1998). [CrossRef]
- 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), 2775-2780 (2000). [CrossRef]
- C. R. Lomba, R. T. Valadas, and A. M. de Oliveira Duarate, "Experimental characterisation and modelling of the reflection of infrared signals on indoor surfaces," IEE Proc., Optoelectron. 145, 191-197 (1998). [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]
- O. González, S. Rodríguez, R. Pérez-Jiménez, 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]
- M. Zhang, Y. Zhang, X. Yuan, and J. Zhang, "Mathematic models for a ray tracing method and its applications in wireless optical communications," Opt. Express 18(17), 18431-18437 (2010). [CrossRef] [PubMed]

## 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.