## Abstract

In this paper, a simulated annealing (SA) algorithm is proposed to be used in the optimization of the spot pattern for the indoor diffuse optical wireless network application. The channel response is analyzed using conventional grid-based patterns and a field of view (FOV) of 30° is found to give a good performance balance in the uniformity of the received power distribution and multipath dispersion. Using the algorithm to determine the spot pattern for the minimum standard deviation of the received power, an improvement of more than 85% is realized. To optimize the spot pattern at 30° FOV, a merit function is incorporated into the algorithm for two parameters, and the SA algorithm is run to obtain optimized spot patterns for both a 4.5m and 6m extent of the spot pattern. Various weights are used, and a performance improvement of 39% and 78% is observed for the 4.5m and 6m spot pattern sizes respectively which shows that the approach can be used to effectively optimize the spot pattern in the indoor optical wireless application.

©2005 Optical Society of America

## 1. Introduction

Currently deployed radio frequency (RF) wireless technologies give users a wide range of mobility, while allowing bandwidths of tens of megabits. However, this bandwidth may be unable to meet the bandwidth requirements of data-rate intensive applications such as video on demand and video conferencing [1]. With the broadcast and propagation properties of RF systems, network security has become a major area of concern in the use of wireless local area networks [2], as, unlike traditional wired technologies, the transport medium is easily accessible to anyone.

Optical wireless networking technologies are not new [3], but in recent years, there has been a renewal of interest in developing optical wireless as an alternative to RF networks. Compared to RF networks, optical networks are confined to the indoor environment, minimizing the access of the transport medium and hence reducing security risks. This also encourages carrier re-use [4]. Optical wireless networks have been shown through simulations and experimental measurements to be capable of large bandwidths in hundreds of megabits [5, 6] and with the availability of an unregulated spectrum, even higher bandwidths are theoretically possible.

However, one of the major limitations of optical wireless networking technologies is the inherent line of sight (LOS) nature of optical carriers. To overcome this obstacle a number of solutions have been proposed, which can be distinguished into diffuse and spot diffusing techniques. In the diffuse method [7], a diffusing material such as a lens diffuses the light onto a surface such as a ceiling, and a wide field of view (FOV) lens is used to collect the reflected signals. This enables a simple system configuration but at the expense of a higher power requirement and higher delay spread. In spot diffusing, the transmitting beams are focused onto several small spots on the ceiling, by using several laser transmitters [8] or by using a beam splitting hologram [9, 10]. By designing the receiver and transmitter in tandem, and using a typically smaller receiver FOV, a lower delay spread can be achieved by limiting the number of signals which reach the receiver. [11]

In spot diffusing, the choice of receiver and transmitter design is important in obtaining the desired performance. While some studies have been made on the analysis of the diffusing spot pattern [12, 13, 14], most of the work has concentrated on optimizing the receiver design [15]. However, the diffusing spot pattern should not be neglected as this would also affect the system performance, in particular the uniformity of the received signal power and the multipath dispersion throughout the room. In this paper, we will first present in Section 2 the model and the parameters that are used in the simulation process, as well as the metrics that are used to determine the system performance. In Section 3, the results obtained using various diffusing spot grid configurations will be presented. The optimum FOV will also be discussed. To further improve the system performance, in Section 4, a simulated annealing optimization algorithm will be used to obtain the optimum spot pattern for various performance metrics. The optimization algorithm will first be presented followed by a comparison between the simulated annealing technique and a simpler iterative minimization. Next, the simulated annealing algorithm will be used to first obtain the optimized spot pattern for a single parameter optimization and following that a double parameter optimization. Throughout the section comparisons will be made between the conventional grid designs and the optimized patterns. A discussion is presented in Section 5 and conclusions from the work performed in this paper will be made in Section 6.

## 2. Mathematical model and parameters

#### 2.1 Mathematical Model

The impulse response between a single source and a single receiver can be determined using the following equation from Barry [16].

*where n* is the Lambertian mode number of the source, *ϕ* is the angle between the source normal and the line connecting the source to the receiver, *dΩ* is the solid angle subtended by the receiver at the source, *θ* is the angle between the receiver normal and the line connecting the receiver and the source, *FOV* is the half-angle of the receiver *FOV, r* is the distance between the source and the receiver, and c is the speed of light in a vacuum. The effective result of (1) is that the impulse response is simply a scaled and delayed delta Dirac function. By multiplexing the application of (1), the impulse response between a source and a receiver off any number of reflecting surfaces can be obtained. *FOV* limits the number of paths reaching the receiver by ignoring paths whose angle at the receiver is more than *FOV*.

#### 2.2. Design Parameters

A typical room size of 6m×6m×3m was used in the model, with a ceiling and wall reflectivity of 0.7. A Lambertian mode number of 1 was used for the diffusing surfaces. No objects were simulated to be in the room. To collect the transmitted impulse response, a 1cm^{2} active area photodetector directed vertically at the ceiling is used. The transmitter was placed at the center of the room and directed vertically at the ceiling, with a total power of 1W directed into the diffusing spot patterns at the ceiling. The simulation is carried out in a ‘dark’ environment without any illumination from light sources or from windows. A metric to determine the performance of a wireless mobile network is the RMS delay spread, $\Gamma =\sqrt{{\overline{{\tau}^{2}}-\left(\overline{\tau}\right)}^{2}}$ [17], where *τ̄*=Σ*P*(*τ*)·*τ*/Σ*P*(*τ*). *τ* is the delay associated with the signal, relative to the first arriving signal, *P*(*τ*) is the associated power of the signal, *τ*̄ is the mean delay spread, and $\overline{{\tau}^{2}}$ is the second moment of the delay spread. A higher delay spread limits the data rate that can be transmitted, and a conservative estimate of the data rate can be found by calculating the inverse of 10*Γ*. This data rate does not take into account equalization and other forms of signal conditioning which could be used to increase the data rate.

## 3. Conventional grid-based diffusing spot designs

Diffusing spots are typically arranged in square grid-type designs which are easy to implement. In this section, various numbers of diffusing spots arranged in such grids will be used in the model, together with various receiver FOV values, to determine the combination of spot pattern and more importantly the receiver FOV that will give the best results. The optimum FOV obtained will be used in the optimization of the spot pattern in Section 4. In practical implementation, losses can be expected at the transmitter such as absorption and back reflection at the hologram mask. However, for the scope of this paper, it is assumed that there is no loss in power from the transmitter to the ceiling.

#### 3.1. Impulse Response Graphs

To analyze the effect of spot patterns and receiver FOV on the system response, the model was run with 8 different spot patterns and 5 different receiver FOVs. A Lambertian diffuse pattern was used for the transmitter, as well as 5 other grid-based designs as shown in Fig. 1: 2×2, with a 2.85m separation between the spots; 4×4, with a 1.35m separation; 6×6, with a 0.90m separation; 8×8, with a 0.60m separation distance and 10×10, with a 0.45m separation distance. Patterns consisting of a single spot (PT) as well as the case of uniformly illumination (uniform) were also used. Each pattern had a total of 1W which was equally distributed into all the spots in the design. Values of 10°, 20°, 30°, 45° and 90° were used.

The impulse response graphs for some of the spot patterns and FOVs are shown in Fig. 2, with the detector placed at 2 positions within the room, and the received power and delay spread results are tabulated in Table 1. At position ‘A’, the detector is placed near the room corner with coordinates (0.75, 0.75, 0.0) and at position ‘B, the detector is placed the exact room center (3.0, 3.0, 0.0). In Fig. 2(a), the received power is very low for most of the transmitting spot patterns. This can be attributed to the 10° FOV of the receiver and the placement at the corner, resulting in only 2^{nd} order bounces from the walls to be received by the detector and thus the generally lower receiver power. The Lambertian pattern results in more signal power being received as the wider coverage of the pattern allows some of the 1^{st} order signals from the ceiling to be reflected onto the detector. The highest power detected was when the ceiling was uniformly illuminated using Fig. 1(h), giving a substantially higher power as compared to the other patterns. This is due to the uniform illumination of the ceiling which increases the number of sources within the FOV. However, the tradeoff is that the impulse response is longer compared as to when other patterns are used. When the 10° FOV detector is placed at the room center, the limited FOV only allows 1^{st} order signals from the ceiling to be received. As the 2×2 and 4×4 patterns do not have spots that are within the receiver FOV, no signal is received resulting in a lost connection at the center when these 2 patterns are used. When a single spot lies within the FOV, such as in the 10×10 design, a negligible delay spread results due to the restriction of multi-paths entering the detector. The greatest amount of power is received when a single spot is used. This is since all the power is concentrated within that single spot and when the single spot falls within the receiver FOV, a large contribution to the received power is from the direct path. However, a single spot is unsuitable for use due to the inherent susceptibility of the single spot transmitting pattern to LOS blocks and the relatively large amount of power contained within that single spot.

Figure 2(c) and Fig. 2(d) show the impulse response results when a wide FOV is used for the receiver. The wide FOV allows all signals to be able to reach the detector, regardless of the point of origin of the signals from the walls or the ceiling. This leads to a much higher received power than when the narrow FOV is used, with an improvement of a factor of at least 10. In Fig. 2(d), the large peak from the 2×2 spot pattern is due to the larger concentration of power within the 4 spots of the pattern. However, as the nearest spots are further away from the centre than the other designs, the signals from the spots take a longer time to reach the detector. While the uniform pattern has the highest power at the corner i.e. at position A, it also has the least power in the centre i.e. position B, and the resulting delay spread is also the highest of the different spot patterns used. This is since the spots are uniformly distributed with equal power in each, resulting in many signals with similar amplitudes reaching the receiver. This is undesirable, as, to minimize the effect of delay spread, ideally there should only be a single strong signal, with the subsequent paths being as weak as possible and this can be seen from Fig. 2(c) and 2(d), where the uniform pattern has the longest impulse response of the various patterns considered.

#### 3.2 Survey of results

To further analyze the results, another metric, the standard deviation of the received power at various parts of the room, *σP*, is taken into consideration. From the earlier results, particularly at the narrower FOVs, certain positions may be unable to receive any signals if no sources, either 1^{st} order from the ceiling or 2^{nd} order from the walls falls, within the receiver FOV. This is particularly prevalent at the corners, which tend to suffer a sharp drop-off in power as compared to positions near the room centre. A low standard deviation relative to the average received power would imply that the signal power distribution throughout the receiving plane, in this case the floor, would be more uniform than a high relative standard deviation, and would reduce the drop-off at the corners. Furthermore, a more uniform signal distribution would simplify the design of the receiver electronics by reducing the required dynamic power range of the electronics. Also, the average of the RMS delay spread across all the detector positions, *µ _{t}* is calculated to for the overall delay spread performance.

The graphs of the average received power (*µP*), standard deviation of the average received power (*σ _{P}*), the average delay spread (

*µ*) and the standard deviation of the RMS delay spread across all the locations (

_{t}*σ*) are shown in Fig. 2(a), 2(b), 2(c) and 2(d). The receiver FOV has a large impact on the system performance. As the FOV is reduced, the number of multipaths reaching the detector is reduced, which reduces the delay spread, limiting multipath dispersion effects and increasing the possible data rates. However, the tradeoff is that with a reduction in FOV, few paths reaching the detector means that the average received signal power is also reduced accordingly. The smaller gradient in Fig. 3(a) between the 45° and 90° FOV can be attributed to the type of signals that are prevented from reaching the receiver. At 90°, as earlier mentioned, all signals reaching the detector are accepted. When the FOV is reduced to 45° however, the weaker 2

_{t}^{nd}order signals from the walls are rejected more than the stronger 1

^{st}order signals from the ceiling. For example, when the receiver is placed at the room centre, a 45° FOV describes a circle of radius 3m at the ceiling. Effectively this means only 1

^{st}order signals from the ceiling is accepted. When the FOV is increased to 90° at the same position, the additional power at the receiver is due to the acceptance of 2

^{nd}order signals from reflections off the room walls.

From Fig. 3(a) and 3(c), the variation of the spot patterns do not have a large effect on the performance of the system with a variation of within 10% when different patterns are used. However, the choice of patterns affects the *σ _{P}*, with a variation of more than 50% of the standard deviation values, particularly at narrower FOVs. At an FOV value of 10°,

*σ*is as much as

_{P}*µ*. A large variation in received power implies that some locations are not receiving any signals. Although an extremely low delay spread of less than 1ns is possible at this FOV, alignment is needed for the spot distribution with the room size to ensure that the receiver is able to receive signals, which may be difficult to ensure practically. The single point and 2×2 spot patterns are clearly unsuitable for use at narrow FOVs as

_{P}*σ*is larger than

_{P}*µ*.

_{P}From Fig. 3(c), it can be observed that use of a wide FOV of 45° and 90°, allows a low *σ _{P}* relative to

*µ*to be achieved. However, the tradeoff is a higher

_{P}*µ*due to a larger number of multipaths being allowed to reach the detector. A maximum RMS delay spread of 1.85ns is used as a cutoff value. Using a conservative estimate of 1/(10×

_{t}*σ*), this delay spread value gives a maximum data rate of approximately 54 Mbit/s. Use of equalizers and other communication techniques such as diversity can be used to increase the data rate further but this value will be used as a performance benchmark within the scope of this paper. From the graphs shown in Fig. 3(c), the maximum FOV within this benchmark is seen to be 30°.

_{τ}Although an FOV of 20° gives a better delay spread performance according to Fig. 3(c), the larger relative *σ _{t}* implies a larger variation in the delay spread performance. Furthermore, at 30°, the larger FOV results in a 100% increase in

*µ*as well as a lower

_{P}*σ*as observed from Fig. 3(a) and Fig. 3(b), with the tradeoff being a 20% increase in

_{P}*µ*. A larger FOV would also simplify the design of the spot pattern. Using a 20° FOV, for a detector placed at the corner to be able to see a spot, the spot must be less than 2.5m away from the corner. If a 30° FOV detector is used, the spot need only be less than 6.5m away from the corner, reducing the likelihood of a dropped link if the transmitter is offset from the centre. Based on the arguments given above it can be concluded that a receiver FOV of 30° would give a good balance of delay spread and signal power distribution performance.

_{t}The performance of the various spot patterns at a 30° FOV is summarized in Table 2. The high standard deviation of the single point and 2×2 spot patterns for received power and delay spread indicate a large variation in the system performance, making the pattern clearly unsuitable for use at this FOV value. Due to the large illumination area when the Lambertian and uniform spot patterns are used, they result in the highest average delay spread values. The uniform illumination pattern has the lowest *σ _{P}*, but at the expense of higher

*µ*and the lowest

_{t}*µ*. The 4×4 and 6×6 spot patterns result in the

_{P}*σ*, with the 4×4 pattern performing marginally better in terms of consistency when the ratio of the

_{P}*σ*over the

_{P}*µ*is determined.

_{P}## 4. Optimization of spot pattern

#### 4.1. Simulated Annealing Technique

Using the optimum FOV as determined in Section 3, the spot pattern was next optimized using a simulated annealing (SA) technique for this optimum FOV of 30°. The simulated annealing technique was proposed by Kirkpatrick et al [18] as a means of finding a global minimum in an optimization process. It is analogous to the annealing process of a molten metal, in which a material finds the lowest possible energy state during the cooling process. The chief advantage of using the SA technique is that it avoids being trapped in a local minimum by introducing a temperature coefficient that is slowly reduced. The governing equation is *P*=*exp*(-*df/T*), where *P* is the probability of accepting an undesired result *df*, and *T* is the current temperature. The technique uses a random search process. If a desired result is obtained, the technique accepts the change as the best obtained result. If an undesired result is obtained, that undesired result is accepted with a probability of *P*. This avoids the algorithm from working on the assumption that the first minimum is the global minimum, as can be the case when a simple iterative minimization (IM) process is utilized, which simply only accepts desirable results with no probability of accepting undesired results. After a number of iterations, *T* is reduced by a factor of 0.95 [19]. The initial temperature *T _{0}* should result in an average acceptance probability

*χ*

_{0}of about 0.8 [20], and this can be determined from -

*df/ln(χ*

_{0}).

In previous work by Yao, Chen and Lim [10], the SA approach was found to be able to optimize the hologram mask to convert a point source into an extended source, with the lowest cost function as compared to the error reduction and input output methods. In this application, the SA algorithm is used to determine the optimum distribution of spots on the room ceiling to achieve a desired performance metric. The algorithm begins with a randomly generated array of spots. A dark pixel represents no illumination by the transmitter, while a light pixel represents a diffusing spot. At each iteration, four pixels which are symmetric about the origin are inverted. If the pixels were previously dark, they are made light, and if light, are made dark. The relevant metric such as the standard deviation of the received power is then calculated, and compared with the previous result to give *df*, where an undesired result may or may not be accepted depending on *P*. The iterations are repeated till the end of one cycle, whereby the annealing temperature is reduced, and the cycle of iterations begins anew for the new temperature. The process can run till either no improvement can be observed or until a set number of cycles have been reached.

To determine the performance of the SA algorithm, a comparison is made between two runs, one using a simple IM algorithm, the other using the SA algorithm, to minimize *σ _{P}* and the plots of the results for the IM and the SA methods are shown in Fig. 4, which plots the number of cycles required against the normalized

*σ*. On the plot for the IM approach, the minimum value is obtained after about 2300 cycles. The graph also reflects the hard optimization nature of the algorithm, which only moves downwards. From the plot of the graph obtained using the SA technique, there is a higher probability of accepting undesired results in the early cycles when the temperature is high. As the number of cycles increase, the temperature drops and the probability of accepting undesired results drops. The obtained standard deviation using the SA algorithm is an improvement of 19% over the IM approach. However, the tradeoff is that the SA algorithm requires approximately twice the computing time than the simpler IM method.

_{p}To determine the minimum standard *σ _{P}* that can be achieved using the SA method, five runs were conducted using the various FOV values that were described in the earlier sections to obtain the specific pattern that gives the minimum

*σ*for each value of the receiver FOV. A comparison is made between the graphs of the conventional grid designs and the optimized results in Fig. 5, and the accompanying spot patterns are shown in Fig. 6. Using the SA technique, a reduction of the

_{P}*σ*by a factor of 7 can be achieved. It can be generally observed that this is achieved by having a higher spot density at the edges and corners rather than at the centre. However, the tradeoff is a slightly higher

_{P}*µ*than the conventional designs which is more apparent at the larger FOVs for 45° and 90°. At 30° FOV, the SA obtained pattern is still able to achieve a delay spread of 1.77ns, less than the benchmark of 1.85ns, still achieving a 93% reduction in the

_{t}*σ*below that of the conventional grid designs, and 87.5% reduction over the uniform illumination pattern.

_{P}#### 4.2. Optimization of Spot Patterns at an FOV of 30°

In the previous subsection, when the SA algorithm was set to run to minimize the *σ _{P}* only, the corresponding

*µ*was found to still be below the benchmark of 1.85ns. To determine the tradeoff between

_{t}*σ*and

_{P}*µ*at the 30° receiver FOV, another round of optimization was performed. Instead of optimizing for only one parameter, a merit function was assigned to the SA algorithm to take into account the optimization of

_{t}*µ*(W2) as well as

_{t}*σ*(W1). With a normalized total weight of 1.0, various combinations of weights for the two parameters can be run to obtain the spot patterns. The optimization was also performed for two sizes of the overall extent of the spot pattern. For the first size, the spot pattern was allowed to cover the entire ceiling surface and for the second spot size, the extent of the pattern was limited to only a 4.5m×4.5m extent, which is 56.25% of the total ceiling surface area. The 6m extent simulates an ideal condition in which the hologram pattern matches the size of the ceiling, while the 4.5m extent spot pattern simulates a hologram projection and ceiling size mismatch, as well as the extent of the conventionally sized grid holograms shown earlier. In Fig. 8, the obtained spot distributions using the SA algorithm for the various conditions are shown. Figure 7(a) shows the results for

_{P}*µ*and

_{P}*σ*on the same graph, and likewise, Fig. 7(b) shows

_{P}*µ*and

_{t}*σ*. In the graphs, the normalized weight for

_{t}*µ*is the complement of the described power standard deviation weight i.e. W1+W2=1. Results from the 4×4 and uniform spot patterns have also been included in the graphs to serve as a basis of comparison.

_{t}As W1 is increased, *µ _{P}* and

*σ*drop while

_{P}*µ*increases. This is because a larger weight is placed on the SA algorithm in optimizing

_{t}*σ*over the

_{P}*µ*, resulting in a tradeoff. By optimizing for a more uniform power distribution via emphasis on W1, the RMS delay spread is adversely affected since a greater number of multipath signals reach the detector and this is evident from the graphs. The drop in

_{t}*σ*with the increase in W1 also shows that a more consistent delay spread performance is achieved. When the system is optimized for a higher value of W2, a delay spread of less than 1ns is obtained. However, the high

_{t}*σ*relative to the low average RMS delay spread implies a large variation in the actual delay spread values, which may be undesirable. It is also observed from the graphs that the extent of the size of the spot pattern affects the obtained results. When the spot pattern is limited to an area of 4.5m×4.5m, there is a drop in the metrics for the received power by a factor of at least 2.5 as compared to the larger spot pattern. Although the smaller pattern is also able to achieve a better delay spread performance, the relatively higher

_{t}*σ*also indicates a larger variation in the delay spread performance.

_{t}The performance of the various spot patterns obtained via the SA technique should be analyzed both in terms of the absolute metric values as well as the relative ratios of the standard deviation over the average. Although a low standard deviation of either the delay spread or received power is certainly desirable, a low average of that metric compared to the standard deviation implies an inconsistent system performance and the ratio should also be considered in evaluating the performance of the spot patterns. *R _{P}* is defined as

*σ*over

_{P}*µ*, and

_{P}*R*is defined as

_{t}*σ*over

_{t}*µ*, and the graphs are shown in Fig. 9. For the 4.5m x 4.5m sized spot pattern the spot pattern obtained that optimized only

_{t}*σ*gives the best performance of the other results in terms of the ratio i.e. the consistency of the performance of the system. When compared to the 4×4 spot pattern, it is interesting to note that the absolute results are 6.5% more and 1.3% more for the

_{P}*µ*and

_{P}*µ*respectively, use of the SA algorithm has reduced the

_{t}*R*and

_{P}*R*by a factor 39% and 23% respectively. Furthermore, in comparison with the results from the uniform signal power distribution, it is observed that the performance is similar with respect to both absolute and relative values. However, while the uniform pattern required illumination of the entire ceiling, the 4.5m×4.5m pattern only required illumination of 56.25% of the room ceiling. For the 6m×6m sized pattern, a value of 0.15 for W1 reduces

_{t}*R*by more than 78% as compared to the 4×4 pattern, while still maintaining a comparable absolute and relative RMS delay spread performance. The higher uniformity of the signal power results is due to the larger extent of the spot distribution, and it can be observed that the optimized spot patterns has a high concentration of spots at the corner to compensate for the lower corner power. In comparing the optimized pattern for the 6m extent with the uniform illumination pattern, it is observed when W1=1,

_{P}*R*and

_{P}*R*are less than that of uniform illumination pattern by 79% and 14% respectively, with the tradeoff being a 4.7% increase in the average delay spread.

_{t}#### 4.3 Analysis when spot pattern is not centered on the ceiling

In the prior analysis, it was always assumed that the spot patterns are located at the centre of the ceiling. However, this may not always be the case, as the transmitter could be located in any location in a mobile transceiver. To analyse the performance of the transmitting patterns when not centred at the room ceiling, two scenarios are considered, in which the transmitting patterns are diagonally offset 1.5m and 3m respectively, as shown in Fig. 11(a) and Fig. 11(c). To simplify the analysis, the segments of the pattern directed onto the surrounding walls are not considered. The SA pattern for W1=1, the uniform pattern, a 10×10 grid and a single spot are used, and the results are shown in Fig. 11(b) and Fig 11(d).

Figure 11(b) and Fig. 11(d) show the percentage of detector locations that received a normalized power within the range on the horizontal axis. The single Point pattern results in a poor distribution of power, with more than 70% of the locations for both scenarios receiving only 0.1 of the normalized power P. The SA-obtained pattern resulted in the best spatial distribution of the various patterns studied. In both situations, SA had the lowest percentage of points with normalized power less than 0.1. At normalized powers beyond 0.3, SA has a higher percentage than the various other patterns for most of the ranges. For the 1.5m offset, the SA pattern resulted in a 11.75% improvement in the number of locations which received a higher normalized power beyond 0.3. For the 3m offset, the improvement was even better with 14% of the locations having a normalized power of 0.3 or more. The better spatial distribution of the power is due to the higher spot density at the corners of the SA pattern as can be seen in Fig. 10(c). When the pattern is offset to the corner, this translates into a higher concentration of spots in the centre which results in a better spatial distribution.

#### 4.4. Q-factor and Bit Error Rate performance

To study the communications channel performance of the obtained spot pattern, a simple receiver model is used [20]. Assuming OOK operation with equal probability for ON state and OFF state. Using the expression for the Q-factor i.e. Q=(I_{on}-I_{off})/(σ_{on}-σ_{off}), where I_{state} refers to the signal current in the ON or OFF state, and σ_{state} refers to the sum of the shot noise and thermal noise in the ON or OFF state, the corresponding bit error rate (BER) can be found from expression BER=0.5erfc(Q/√2). In this model, it is taken that there is no received power in the OFF state. Furthermore, the shot noise current due to ambient lighting is taken to be 25dB more than the corresponding signal noise current [15]. The thermal noise is considered negligible when compared to the shot noise and thus σ_{on}=σ_{off}. The results of the BER analysis for the optimized pattern for W1=1 is shown in Fig. 12. For the purposes of calculation of the noise, the data rate is taken as 155Mbits/s.

In Fig. 12(a), the BER is plotted as a function of the receiving plane. The higher BER values are found near the walls and the corners, and this is due to the lower power at these positions. From Fig. 12(b), the percentiles of the locations corresponding to the BER values are shown. About 25% of the locations have a BER value less then 10^{-6}, with more than 90% having a BER value of less than 5×10^{-6}. The highest obtained BER is 10^{-5}. The system performance can be further improved by using multiple receiver branches in a diversity scheme to increase the signal to noise ratio and the corresponding BER, though at the cost of increased system complexity. [14]

## 5. Discussion

In this paper, a simulated annealing technique was used to obtain the optimum spot pattern. However, this pattern was assumed to be centered in the room ceiling. In a practical situation, the transmitter may be anywhere in the room. When the receiver is at the sides and corners of the room, the spots may be projected onto the wall instead of the ceiling, and that would affect the system performance. Although an analysis was done in this work, the effect of the pattern being projected onto the walls was ignored and further work can be done to simulate this condition. Although a significant amount of noise can be filtered out using optical techniques such as wavelength bandpass filters and electronic frequency bandpass filters, the presence of residual noise is unavoidable in any system. By including illumination sources in the model, a more realistic model of the room environment can be achieved.

The merit function can also be further developed to include more variables. In the earlier sections, only two target parameters, the standard deviation of the received power and the average of the RMS delay spread across the receiving plane, were considered. It was observed that the relative magnitudes of the mean and the standard deviation for the metrics are also of interest, particularly when trying to obtain a consistent system performance. A low standard deviation relative to a high mean for the received power would indicate a more uniform distribution of power as compared to a high standard deviation and low mean. Further work will be explored to include these ratios as target parameters in the SA schedule.

A practical approach to obtaining a spot pattern on the ceiling while only using a single laser can be obtained by the use of holograms [9]. However, the central spot is by and large unavoidable due to the diffraction efficiency and fabrication limitations. The SA optimization can be used to design a hologram that already takes into consideration the effect of a central spot, and it may be possible to make use of the central spot as one of the spots in the spot diffusing pattern to achieve a desired result.

## 6. Conclusions

This paper presented the work done to optimize the receiver FOV and an optimum distribution of spots on the ceiling using a simulated annealing technique. A survey of the effect of various conventional grid-based designs when an FOV of 10°, 20°, 30°, 45° and 90° for the receiver was modeled in a typical indoor environment, and the performance of these spot patterns was analyzed. Considering the coverage of the room ceiling and the RMS delay spread benchmark required to obtain an unequalized bandwidth of 54Mbits/s, an FOV of 30° was chosen as it represents a good tradeoff between the performance criteria. At this FOV, the conventional 4×4 and 6×6 grid of spots gave a better performance, with both having an RMS delay spread of 1.53ns, well below the benchmark. Although the Lambertian and uniform illumination patterns have good distribution of signal power, the tradeoff is a higher delay spread.

To further optimize the performance of the system, a simulated annealing technique was used to obtain the optimum distribution of spots for a target parameter. In comparing the SA technique with a simpler iterative minimization method, it was found that the SA technique resulted in an improvement of 19% over the IM method, although the required computing time was doubled. The SA method to optimize the spot pattern to obtain the minimum delay spread for the five different FOVs, led to a 93% reduction of in the standard deviation of the received signal power below that of the conventional grid designs, and 87.5% reduction over the uniform illumination pattern, while still having an delay spread within design objectives.

At 30° FOV, the spot pattern was then optimized using a two parameter merit function, taking into consideration the standard deviation of the received power as well as the average RMS delay spread. The weights for these 2 parameters were varied as well as the extent of the size of the spot pattern distribution, and the results were plotted. Use of a weight of 1.00 for the smaller distribution of spots results in a more consistent performance by increasing the ratios of the standard deviation and average of the average power and RMS delay spread by a factor of 39% and 23% respectively over conventional spot patterns. While the performance was similar to using a uniform illumination pattern over the entire ceiling, the optimized pattern only required illumination of 56% of the ceiling. When the spot pattern was allowed to optimize over the entire ceiling area, values of W1 were used to reduce the ratio of the received power metrics by more than 78% over conventional grid and uniform illumination patterns. The optimized pattern for W1 was also shown to have a better spatial distribution of signal power over the other patterns considered when the pattern was offset

One of the challenges in the use of infrared wireless LAN is the potential loss of signal when the receiver does not see any spots, especially for the smaller FOV receivers. By using a global optimization technique together with a 30° FOV receiver, a more uniform signal distribution and system performance can be achieved without a compromise in the data rate performance of the system. By specifying other merit functions, the SA algorithm can be modified to meet the demands of other parameters such as the ratios between average and standard deviation of the metrics, and obtain an optimum spot distribution for those demands.

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