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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 1997–2005
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Comparison of Monte Carlo ray-tracing and photon-tracing methods for calculation of the impulse response on indoor wireless optical channels

Oswaldo González, Silvestre Rodríguez, Rafael Pérez-Jiménez, Beatriz R. Mendoza, and Alejandro Ayala  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 1997-2005 (2011)
http://dx.doi.org/10.1364/OE.19.001997


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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

Recently, a new ray-tracing method, which has been denominated photon-tracing algorithm (PTA), has been proposed [8

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]

]. This method is based on the MMCA but now the rays are not always propagated after reflection. Instead, they are only propagated in a certain proportion according to the reflection coefficient, which makes the number of considered rays decrease rapidly after each reflection. Zhang 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.

The paper is organized as follows: Section 2 describes the Monte Carlo ray-tracing algorithm, highlighting the differences between MMCA and PTA. The computational cost is evaluated in section 3. The equations that allow us to determine the error in computing the impulse response are presented in section 4. Finally, several computer-simulated results are reported in section 5 in order to compare both methods. The conclusions are summarized in section 6.

2. Algorithms description

In the modified Monte Carlo ray-tracing algorithm, ray directions are randomly generated according to the radiation pattern from the emitter. The contribution of each ray from the source or after a bounce to the receiver is computed deterministically. Consequently, the discretization error is due to the number of random rays. The line-of-sight (LOS) and multiple-bounce impulse responses are considered when calculating the total impulse.

2.1. LOS impulse response

Given an emitter E and receiver R in an environment without reflectors, with a large distance d between both, the received power is approximately
PR=1d2RE(ϕ,n)Aeff(φ)
(1)
where the emitter is modeled using a generalized Lambertian radiation pattern RE (ϕ,n). Aeff (φ) represents the effective signal collection area of the receiver.
RE(ϕ,n)=n+12πPEcosn(ϕ),0ϕπ2
(2)
Aeff=Arcosφrect(φFOV)
(3)
rect(x)={1,|x|10,|x|>1
(4)

Here n is the mode number of the radiation lobe which specifies the directionality of the emitter, PE the power radiated by the emitter, Ar the physical area of the receiver, and FOV the receiver field of view (semi-angle from the surface normal).

2.2. Multiple-bounce impulse response

We consider now an emitter 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 RE (ϕ,n)/PE. The power of each generated ray is initially PE/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, RS(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
PR=1d2RS(θ,θ)Aeff(φ)
(5)
where d is the distance between the reflection point and receiver, and Rs (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]

]. This model is able to approximate the behavior of those surfaces that present a strong specular component. It considers the reflection pattern as a sum of both diffuse and specular components. In this way, surface characteristics are defined by two new parameters, the percentage of incident signal that is reflected diffusely rd and the directivity of the specular component of the reflection m. This model is described by
RS(θ,θ)=ρPi[rdcosθπ+(1rd)m+12πcosm(θθ)]
(6)
where ρ is the surface reflection coefficient, Pi represents the optical power of the incident ray, θ is the observation angle, and θ′ represents the incidence angle.

Fig. 1 Geometry of emitter and receiver with reflectors. Reflection pattern of the surface is described by Phong’s model.

3. Computational complexity

In PTA, if we consider that N rays (photons) are initially launched, after the kth bounce, only ρ̃kNk−1 rays (photons) continue their path, where Nk−1 is the number of photons remained after the (k − 1)-th bounce (with N0 = N), and ρ̃k is an average parameter of the reflection coefficient at the kth 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:
NopPTA=NS×(N+ρ˜1NN1+ρ˜2N1N2+...+ρ˜K1NK2NK1)
(7)

4. Error estimation

Both PTA and MMCA are Monte Carlo based ray-tracing algorithms. Therefore the error analysis for MMCA given in [7

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]

] is directly applicable to PTA. There, it was demonstrated that the error in computing the power P′j reaching the receiver during a small time interval Δt can be estimated from the variance of P′j, var (P′j) The biggest admissible Δt 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′j) given by [7

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]

]:
var(Pj)=i=1Njpi,j21Nf,j(i=1Njpi,j)2
(9)

The relative error can be computed as the square root of the variance (one standard deviation) of P′j given by Eq. (9) divided by the computed contribution power defined as
Pj=i=1Njpi,j
(10)

Then
relerr(Pj)=[i=1Njpi,j2(i=1Njpi,j)21Nf,j]1/2
(11)

The above equation allows us to estimate the relative error in computing the received power in the jth time interval.

5. Simulation results

In order to compare PTA and MMCA methods, the simulation room described in [4

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

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]

] has been evaluated. The main characteristics of this room are indicated in Table 1. In this example, the reflecting surfaces are purely diffuse Lambertian (rd = 1), but rooms with materials characterized by the Phong’s model (such as the examples described in [7

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]

]) have also been evaluated, obtaining similar results to those presented below. The emitter is placed at the center of the room rested on the ceiling and pointing straight down. The receiver is located on the floor near a corner pointing straight up. The maximum number of considered reflections is K = 10 and the simulation time tmax = 120 ns.

Table 1. Parameters for simulation

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In Fig. 2 we show the impulses responses (received power) obtained by using both methods when 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.

Fig. 2 Impulse responses obtained with (a) photon-tracing (PTA) and (b) ray-tracing (MMCA) algorithms.

The accuracy of both methods can be compared quantitatively by means of the relative error committed by both algorithms during the calculation of the impulse response. Figure 3 shows the relative error for PTA and MMCA calculated by using Eq. (11). It can be observed how MMCA offers a solution (impulse response) with a relative error roughly constant along the time. However, in PTA the relative error tends to increase with time, what is logical because the number of rays (photons) that contribute in the calculated received power decreases along the time, since many of them begin to be discarded when they collide against the room surfaces. This is understood clearly if we observe the number of rays that contribute in the received power along the time or at a certain bounce 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%tmax) 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 kth 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.

Fig. 3 Relative error of both algorithms: PTA (blue) and MMCA (green).
Fig. 4 Number of contributions (a) along the time and (b) at the kth reflection.

In Table 2 we compare PTA and MMCA according to mean relative error and number of elementary calculations. The mean relative error has been weighted by the value of P′j:
meanrelativeerror=j=1JPj×relerr(Pj)j=1JPj
(12)
where j = 1, 2,...,J and J = tmaxt. 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.

Table 2. Comparison between PTA and MMCA

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The number of elementary calculations required for both algorithms (see Table 2) demonstrates that PTA is much more efficient than MMCA (∼28% out of the total of operations required by the latter), as we had stated in section 3. We can observe how the predicted values in that section are very close to those obtained during the simulations. Therefore these expressions can be used to determine in advance the number of required calculations approximately. In comparison with MMCA using 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.

Finally, we show in Table 2 the power distribution according to the bounce index. We can observe how both algorithms present very similar values, demonstrating a good behavior of the PTA during the distribution of power in the remained rays after each reflection.

6. Conclusions

In this paper, we have compared the conventional modified Monte Carlo ray-tracing algorithm with a recently proposed one, which has been called 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

This work has been funded in part by the Spanish Research Administration ( TEC2009-14059-C03-02 and TEC2009-14059-C03-01), by the Canary Government ( SolSubC200801000306) and the Plan E (Spanish Economy and Employment Stimulation Plan).

References and links

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]

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


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References

  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]

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