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This study investigates the sensitivity of a three-dimensional (3D) indoor ray tracing (RT) model for the use of the uniform theory of diffraction and geometrical optics in radio channel characterizations of indoor environments. Under complex indoor environments, RT-based predictions require detailed and accurate databases of indoor object layouts and the electrical characteristics of such environments. The aim of this study is to assist in selecting the appropriate level of accuracy required in indoor databases to achieve good trade-offs between database costs and prediction accuracy. This study focuses on the effects of errors in indoor environments on prediction results. In studying the effects of inaccuracies in geometry information (indoor object layout) on power coverage prediction, two types of artificial erroneous indoor maps are used. Moreover, a systematic analysis is performed by comparing the predictions with erroneous indoor maps and those with the original indoor map. Subsequently, the influence of random errors on RMS delay spread results is investigated. Given the effect of electrical parameters on the accuracy of the predicted results of the 3D RT model, the relative permittivity and conductivity of different fractions of an indoor environment are set with different values. Five types of computer simulations are considered, and for each type, the received power and RMS delay spread under the same circumstances are simulated with the RT model.
© 2016 Optical Society of America
1. Introduction
With the rapid growth of wireless communications, cell sizes continue to decrease [1]. Therefore, the accurate planning of cellular networks (especially in complex indoor environments) requires complete knowledge of propagation conditions and losses in wireless channels [2]. The wireless communication in an urban street and that in buildings are based on the same idea, and they generally use the ray tracing (RT) method [3]. This method, which provides not only path-loss information but also time delay, angle spread, and polarization information, is a good alternative in meeting these requirements [4]. Accurate RT simulations are also of interest for a broad range of applications, such as prediction of channel impulse response for wireless optical communications [5] and optical performance [6], diagnostic tools in the bio-medical field [7], and optical design tools for luminaires, displays, and photovoltaic devices [8, 9].
An indoor propagation channel differs considerably from outdoor response. The distance between transmitting and receiving antennas is relatively shorter than that in outdoor systems because of the high attenuation caused by internal walls and furniture and because of low transmitter (Tx) power. In complex indoor environments, all possible combinations of reflected, transmitted, and diffracted rays reaching a mobile should be considered. In all cases, the geometrical search for valid propagation paths is the most time-consuming and most challenging part of any implementation [10]. Thus, the acceleration of the determination of propagation paths is the kernel and should be solved first for complex environments. For this purpose, some scholars have explored acceleration methods for RT models. In [11] and [12], only rays with certain contributions to the receiver (Rx) were traced. In [13], the effective propagation area method and dominant corner extraction method were used. In [14], a new three-dimensional (3D) propagation prediction model that makes use of two-dimensional RT results was proposed. In most published works, only the main structure of a building (such as interior walls, floors, and ceilings) is considered. To the best of our knowledge, only a few studies have investigated the effects of indoor objects, such as tables and cupboards. To consider the influence caused by these indoor objects, a study recently presented a new 3D indoor RT model [15].
A detailed knowledge of the environment is required when applying any RT model. The prediction accuracy of an RT model is obviously affected by the accuracy of environment information. Thus, a thorough investigation of the influence of the accuracy of indoor environment databases on RT-based prediction is valuable in quantifying prediction errors as functions of database inaccuracies. RT models have been employed for over two decades, but published studies on this subject remain very limited. In [16], the effects of several different sets of electrical parameters for a selected street on path-loss prediction results were investigated with an RT-based model. In [17], the variation of the received power was examined for different wall characteristics and database inaccuracies (by randomly misplacing the walls) on several routes. Similarly, Rizk et al. focused on the influence of random errors in a building vector database and computed the sensitivity of the selected reflection coefficient attributed to building walls [18]. In the aforementioned references, few studies explored the influence of indoor environment (especially complex environments) database accuracy on RT-based prediction. In addition, these studies did not investigate coverage prediction for certain areas and instead discussed only the simulation results related to the received power or path loss (narrowband parameters) on several specific routes. The influence of database accuracy on RT-based RMS delay spread (a parameter characterizing wide-band multipath channels) predictions is seldom considered.
To compensate for the lack of previous studies, the current study focuses on analyzing the influence of indoor environment database accuracy on power coverage predictions. It also discusses the relationship between the prediction results for the RMS delay spread and database accuracy. Because the simulation of radio waves is conceptually analogous to the problem of light transport, the present work could provide a research method in areas of the prediction of indoor wireless optical channels and indoor optical wireless communication environment modeling. This paper is organized as follows. A general description of indoor environments and the 3D RT-based model is given in Section 2. The effects of inaccuracies in indoor object layout are analyzed in Section 3. Subsequently, the dependence of predictions on the electrical characteristics (i.e., relative permittivity and conductivity) of walls, ceiling, floor, and indoor objects is presented in Section 4. The conclusions are drawn in Section 5.
2. Description of indoor environments 3D ray tracing model
2.1 Indoor environment
Conventionally, the geometric description of an outdoor environment can be mainly extracted from digital maps [19]. By contrast, the geometric description of complex indoor environments may be obtained from AutoCAD files [20] or other electronic documents in different formats. In general, the indoor geometry database used for RT models does not feature a uniform data format. In other words, no universal format exists for indoor wireless propagation environments. Therefore, in this study, the geometric description of an indoor environment is imported as the format [15]. In typical indoor environments (such as an office), their geometric information is classified into two types. The first type mainly describes relatively large indoor objects, such as tables and cupboards. Small indoor objects, such as chairs, are not included in our model. Every indoor object can be laid out with arbitrary orientation and modeled as a polygonal cylinder with uniform height. Therefore, only the geometric information on all the vertices on the roofs of cylinders needs to be registered. All vertices on the roofs of cylinders make up a polygon and are ordered in a counterclockwise direction. The other type of information involves walls (modeled as smooth vertical surfaces) and attachments (which could be a door, a window, or even a hole in the walls) contained in these walls, and they are modeled as rectangles whose vertices are also ordered in a counterclockwise direction. This modeling technique is essential and can be expediently given full consideration for the properties of different attachments, thereby highlighting the accurate advantage of the RT model. The dielectric property of each material involved in an environment is registered with a unique identification number.
2.2 3D ray-tracing model and model setup
The 3D RT method is chosen for the prediction of the signal level in the area of interest. A good agreement is found between the simulated results obtained with the RT method and the measurements [15]. In the model, the following phenomena are considered: the direct line-of-sight (LOS) ray when it exists, up to seven reflections, up to two diffractions, up to two transmissions or all possible combinations of up to four reflections, a single diffraction or transmission per path, all possible combinations of up to a single reflection, and two diffractions or transmissions per path. The reflected and diffracted wave fields are computed by using Fresnel reflection coefficients and uniform theory of diffraction coefficients, respectively (the calculation of the two coefficients can be traced in [21]). More in detail, the direct, reflected, and refracted field are evaluated through geometrical optics, and the diffracted ray field is evaluated through uniform theory of diffraction.
To analyze the effects of indoor environments on RT-based predictions, this study investigates a practical complex indoor environment (Sections 3 and 4). The geometric model of this indoor environment is shown in Fig. 1, which includes the floor and ceiling to consider the reflections produced in both objects. The room measures 7.9 m wide, 17.9 m long, and 3.85 m high. Additional details about the geometric features can be obtained in [15].
In our calculations in the next two sections, two Tx locations are considered, as depicted in Fig. 1. In the figure, the two Txs are located at Th and Tl, and their corresponding heights are 1.8 and 0.6 m, respectively. Note that the straight line passing through the locations of Th and Tl is perpendicular to the floor. The transmitted power is 1 W at 2.6 GHz. The Tx and Rx are half-wavelength vertically polarized dipoles.
In analyzing the sensitivity of power predictions to the indoor environment, coverage predictions for two predicted planes are performed and described in Sections 3 B, 4 B, and 4 D. The locations of Rx are uniformly distributed 0.5 m apart in a predicted plane. The height of the predicted plane is denoted as Ph and is set to 0.6 and 1.5 m. Exactly 554 test points feature Ph = 1.5 m, and 463 test points feature Ph = 0.6 m.
To analyze the sensitivity of RMS delay spread predictions to the indoor environment, route predictions along two routes are performed and described in Sections 3 C, 4 C, and 4 E. As depicted in Fig. 1, routes AB and CD are chosen because they emphasize different propagation mechanisms. The heights of routes AB and CD are 1.8 and 0.6 m, respectively. Relative to the two Tx locations Th and Tl, routes AB and CD are in the LOS region and non-LOS (NLOS) region, respectively. In our calculations, up to 25 test points (Rxs) are predicted and uniformly distributed 0.5 m apart along the test streets.
The material characteristics used in Section 3 are summarized in Table 1. All the electrical properties and characteristics of the indoor environment elements, such as doors, windows, and tables, are set according to [22–25]. Moreover, these parameters can practically characterize indoor environments [Fig. 1] and were validated in [15]. Consequently, these parameters serve as the reference electrical parameters in Section 4, which discusses the influence of electrical parameters.
Table 1. Material characteristics used in this study.
3. Effects of geometric indoor environment on 3D ray tracing-based predictions
This section investigates the sensitivity of the proven 3D RT model to the inaccuracies of the geometric information of the environment by misplacing the simulated lateral faces of indoor objects randomly and analyzing the effects on the received signal levels. With all-sided propagation environments considered, power predictions are implemented for all the test points that are uniformly distributed with constant spatial steps in the studied geographical zone. Hence, loss predictions are obtained in a more universal manner than the prediction of several specific routes. The RMS delay spread is a critical parameter in characterizing wide-band multipath channels; thus, the effects of erroneous indoor maps on prediction results are analyzed in LOS and NLOS indoor channels. Consequently, this approach has more practical applications than an approach that only considers large-scale power.
3.1 Generation of geometric information with random errors and execution of error statistics
The prediction accuracy of the RT model depends on the modeling of the environment so that the complex problem of wave propagation in a mobile radio environment can be reduced to a manageable level [16]. Thus, a good modeling approach should consider all-sided environment information, and the propagation model should be rapidly implemented on the basis of the storage format of this information. In achieving the best trade-off between the appropriate level of accuracy required in geometrical databases and prediction accuracy, the rule of the influence of indoor environment information on prediction results must be analyzed systematically. Some conclusions are made with regard to the degree of dependence of the RT model on the accuracy of the geometric environment. For this purpose, random errors are artificially produced with generators of uniform distributions, as shown in Fig. 2. The generation of the errors is similar to and in accordance with [18]. In general, the length and height of a wall could be easily and accurately obtained. However, the size of indoor objects and their relative positions may be erroneously measured to varying degrees. Therefore, our study focuses on analyzing errors in indoor objects. The following two types of errors are considered for complex indoor environments in this study:
Fig. 2 Two types of errors for indoor objects in an indoor map: (a) error in edge position and (b) error in indoor object size. The term N(μ, ε2) denotes a normal distribution with μ and ε as mean and standard deviation. The term U(0, 2π) denotes a uniform distribution with 0 and 2π as lower and upper limits of the variation.
- (a) Errors in the edge position of an indoor object [Fig. 2(a)] – the vertical edges of the object are displaced (the radial displacement is d > 0) randomly and uniformly around their original location in the horizontal direction. No displacement occurs in the vertical direction, and the implementation of the artificial errors in the indoor object size is subject to this limitation.
- (b) Errors in the indoor object size while keeping the orientation of the lateral face of the object [Fig. 2(b)] – the lateral faces are displaced parallel to their original locations, and the magnitude of the displacement is chosen randomly for each lateral face from a normal distribution with zero mean and standard deviation ε.
For each type of error, the following four values of ε are considered: 0.02, 0.04, 0.05, and 0.07 m. For each of the eight cases determined by a type of error (errors in the edge position or in object size) and a value of ε, 25 maps or realizations are generated with different seeds, starting from the indoor map [Fig. 1] as the original map. Thus, 200 erroneous maps are generated.
According to the generation of the erroneous maps mentioned above, 200 erroneous indoor maps are created for two Tx locations, and all test points uniformly distributed and covering each erroneous indoor map are predicted using the RT model. Therefore, a mass of prediction results are obtained and handled for the sensitivity of power coverage predictions to erroneous geometric environments.
In this section, the prediction data are handled similarly to the proposed method in [18]. Several concepts should be defined before conducting this operation. RP (test point i) represents the predicted power or RMS delay spread (as the reference prediction) at the ith test point using the original indoor map. Similarly, RP (test point i, error type j) is the predicted power or RMS delay spread at the ith test point using the erroneous map that belongs to error type j with the radial displacement d (determined by ε). Errors in the edge position and indoor object size are denoted as error types 1 and 2, respectively. To indicate model sensitivity to the indoor object database, we let ERRrp be the error that results from the comparisons between the predictions using the erroneous maps and the predictions using the original indoor map:
For each realization, error type, and value of ε, the ERRrp values for about 554 or 463 test points (for coverage prediction) covering the area of interest or 25 test points (for route prediction) are treated as random variables characterized by their means E[ERRrp] = AERRrp and their standard deviations σERRrp. Averaging the 25 realizations provides the average values E[AERRrp] and E[σERRrp] for each error type and value of ε.
3.2 Sensitivity of power coverage predictions to erroneous geometric information
Table 2 provides the error statistics of the differences in power maps between the use of the original indoor map and the use of the erroneous indoor maps. With the increase in the magnitude of the displacement, for the two errors, the values of E[AERRrp] for the Tx location Th with Ph = 1.5 m varies slowly within the minimal interval, whereas the values of E[AERRrp] for the Tx location Tl with Ph = 0.6 m varies quickly within the maximal interval. For the Tx location Th with Ph = 1.5 m, all the test points are found nearly in the LOS area, where the direct LOS ray can provide the main contribution. Consequently, the RT predictions of the test points distributed in LOS regions are not evidently affected by surrounding walls, indoor objects, and other objects. In contrast to the values of E[AERRrp], the values of E[σERRrp] are larger and vary quickly. Only when ε > 0.05 m do the values of E[σERRrp] relative to the edge position error exceed the values of E[σERRrp] relative to the object size error. This conclusion is similar to that of an analysis of outdoor RT-based power predictions in [18]. Two reasons may explain this conclusion. On the one hand, the displacement, which is determined by the value ε within the intervals of [0.02, 0.04], could be neglected in comparison with indoor object size. In this case, the edge position error is very small and exerts little to no influence on the orientation of the lateral face. On the other hand, indoor objects may not be the important factors of radio wave propagation in an indoor space (especially in LOS cases). Such radio wave propagation is affected by each direction; hence, this situation is a full 3D problem. No propagation toward the horizontal or vertical direction prevails in complex indoor environments. Consequently, similar error statistics are derived for edge position error and object size error.
Table 2. Error statistics from the differences in power maps between the use of the original indoor map and using erroneous indoor maps.
In fully visualizing the influence of object size error and edge position error on power, the differences in the power map between using the original indoor map and erroneous indoor maps are identified and presented in Figs. 3 and 4, respectively. In obtaining the differences, the mean powers using different erroneous maps minus powers based on the original map makes the difference at each test point. For the two types of errors, the differences in power maps are limited in [−5 dBm, 5 dBm] in the predictable range, where direct rays and low-order reflected rays could arrive. The poorest agreement between the predictions using erroneous maps and reference predictions may occur in some areas where NLOS cases are produced from the original LOS cases through artificial errors. In addition, the marked differences in the power map occur at the test points in the predicted plane with Ph = 0.6 m. In this case, all the test points are well below the indoor objects, resulting in the existence of large NLOS areas.
Fig. 3 Object size error (ε = 0.07 m) for two Tx locations: (a) Th with Ph = 1.5 m, (b) Th with Ph = 0.6 m, (c) Tl with Ph = 1.5 m, and (d) Tl with Ph = 0.6 m.
Fig. 4 Edge position error (ε = 0.07 m) for two Tx locations: (a) Th with Ph = 1.5 m, (b) Th with Ph = 0.6 m, (c) Tl with Ph = 1.5 m, and (d) Tl with Ph = 0.6 m.
3.3 Sensitivity of RMS delay spread predictions to erroneous maps
The error statistics of the predictions with respect to the RMS delay spread along routes AB and CD [Fig. 1] are shown in Figs. 5 and 6, respectively. In the condition in which Tx and Rx are well below indoor objects, a small prediction error occurs when the errors in the indoor object database do not affect the orientation of the lateral face of objects. This conclusion is similar to that of the analysis of RT-based path-loss predictions in urban microcellular environments in [18]. However, in many circumstances (unlike the abovementioned conditions), the predictions are more sensitive to object size errors than to errors in the orientation of the lateral face of objects. When all the test points are distributed along the NLOS route, the predictions are increasingly sensitive to errors in the indoor geometric environment. Under NLOS conditions, the ray paths that arrive at the test points are primarily composed of high-order rays. These rays undergo multiple reflections and diffractions; the computation accuracy of the electric field is lowered several times. This rule of the influence of environmental information on prediction results is also appropriate for power coverage prediction. This rule becomes evident when Txs and Rxs are well above indoor objects. In this condition, more transformations between LOS areas and NLOS areas may occur. Consequently, the values of E[AERRrp] and E[σERRrp] along route CD (in the NLOS area) are larger than those along route AB (in the LOS area).
Fig. 5 Error statistics of the predictions with respect to RMS delay spread for different Tx locations along route AB
Fig. 6 Error statistics of the predictions with respect to RMS delay spread for different Tx locations along route CD
4. Influence of electrical parameters on predictions
In general, the electrical parameters of each object in the environment of interest comprise relative permittivity (εr) and conductivity (σ) in the simulation prediction. For outdoor urban environments, all building walls are generally characterized by the same electrical parameter. However, in typical indoor environments (such as an office), the electrical parameters of the corresponding materials are associated with each wall, ceiling, floor, or indoor object. In particular, wall properties may change in rectangular portions to reproduce actual discontinuities (doors, windows, and others) [26]. Therefore, a systematic investigation into the influence of different fractions of an indoor environment on RT-based prediction is essential to achieve good prediction accuracy for the RT method.
4.1 Simulation setup and execution of error statistics
For a systematic investigation into the influence of different fractions of an indoor environment on prediction, this study considers five types of computer simulations.
- Type I. The electrical parameter of an indoor object is changed solely.
- Type II. The electrical parameter of the ceiling is changed solely.
- Type III. The electrical parameter of the floor is changed solely.
- Type IV. The electrical parameter of the wall is changed solely.
- Type V. The electrical parameters of the ceiling, floor, and wall are changed simultaneously and equally.
In using simulation type I, the influence of indoor objects on RT-based prediction can be analyzed. On the basis of the other four types, we can obtain the effects of the ceiling, floor, and wall on RT-based prediction and determine which part of the room is the important factor that affects indoor radio wave propagation.
The studied indoor environment and the simulation results for the two Tx locations Th and Tl are shown in Fig. 1 to clarify the influence of inaccuracies in electrical parameters on coverage prediction or route prediction (along routes AB and CD). A sensitivity analysis with respect to the electrical parameters of the environment is already presented in [27]. Such analysis is completed and further investigated in the present work. Except for the study of εr or σ for each wall, ceiling, floor (or both), and indoor object, all other setups used for the simulations in this section are identical to those in Section 2 B.
In clarifying the handling process of the prediction results, several concepts must be defined. RP1 (test point i, εr, σ) represents the predicted power or RMS delay spread at the ith test point from the original indoor map with specific electrical parameters (εr and σ). To show the indication of model sensitivity to the relative permittivity of the environment, we let ERrp be the error that results from the comparisons between the predictions that use different electrical parameters and the predictions that use referenced electrical parameters:
where the referenced values for the permittivity (εr0) of the environment are summarized in Table 1.
For each realization, the values of σ and ERrp on all the test points are treated as random variables characterized by their means E[ERrp] and their standard deviations σERrp.
For convenience, when the value εr is equal to the referenced relative permittivity εr0, we let Eσ and σσ denote the values of E[ERrp] and σERrp, respectively; that is, Eσ = E[ERrp] and σσ = σERrp. Similarly, Eεr = E[ERrp] and σεr = σERrp with σ = σ0. If the value of σ is constant, then the 16 pairs of Eεr and σεr correspond to different values of εr. If the value of εr is constant, then 9 pairs of Eσ and σσ correspond to different values of εr for type I, and 11 pairs of Eσ and σσ correspond to different values of εr for types II–V.
4.2 Sensitivity of power coverage predictions to relative permittivity
Figure 7 provides the error statistics based on the comparisons between power predictions with different values of relative permittivity and power predictions, with σ = 0.015 S/m as the reference. This reference relative permittivity is very small. Hence, the values that are greater than the reference value are considered in Fig. 7. The values of εr uniformly vary from 2.5 to 10 with an invariable separation (0.5).
Fig. 7 Average error and standard deviation of the difference between predictions and the reference prediction when the relative permittivity varies for type I
As depicted in Fig. 7, as the value of relative permittivity increases, the values of Eεr and σεr increase linearly. The minimum sensitivity is found in LOS areas, where strong LOS rays exist and other high-order rays provide negligible contributions. When Tx is placed at position Th with a height Ph = 0.6 m, the values of σεr and Eεr are maximum because the majority of the test points are distributed in the NLOS region. In this condition, the high-order rays prevail, and the predicted results are intensively influenced by the relative values of permittivity of indoor objects.
Figures 8 (a) and (b) show the error statistics of the predictions with respect to power against relative permittivity for the different types of computer simulations (types II–V). In these figures, the error of relative permittivity is equal to the value obtained with the chosen relative permittivity minus the reference permittivity (see Table 1). The behavior of the RT model is examined as the values of relative permittivity range from −1.6 to 1.6 with an invariable separation (0.2). In comparing the different curves with respect to the average error and standard deviation, the increasing trend of σεr (when the values of εr are less than the reference permittivity) is basically similar to the decreasing trend of σεr (when the values of εr are greater than the reference permittivity). In general, the relative values of permittivity of the surrounding walls in the boundary (i.e., walls, ceiling, and floor) of indoor spaces have the greatest effect on RT-based predictions, whereas the influence of the ceiling on indoor radio wave propagation is more obvious than the influence of the floor because strong reflections on the floor are obstructed by indoor objects, in which case the rays propagated from the floor are seldom affected. Note that when Tx is placed at location Tl, the floor obviously influences indoor radio wave propagation. In this case, the distance between Tx and the floor is smaller than the distance between Tx and the ceiling. As a result, the ray propagated from the floor for the same multi-reflected signal allows strong signals to reach the test point. In addition, most of the test points are distributed in the NLOS region, resulting in few reflected rays on the ceiling. Furthermore, the power predictions are found to be increasingly sensitive to variations in the low values of permittivity.
Fig. 8 Error statistics of the predictions with respect to power against relative permittivity for different types of computer simulations (types II–V): (a) Eεr and (b) σεr.
4.3 Sensitivity of RMS delay spread predictions to relative permittivity
The RT-based RMS delay spreads are calculated along route AB (LOS case) and route CD (NLOS case). In addition, two Tx locations are considered, and 64 groups of simulations are obtained. The values of εr uniformly vary from 2.5 to 10 with an invariable separation (0.5). An analysis similar to that in the previous section is performed here to examine the effect of the relative permittivity of indoor objects on the predictions.
Figure 9 shows the influence of the error of the relative permittivity of indoor objects on RMS delay spread predictions. The effects of the relative permittivity of indoor objects on RMS delay spread predictions are especially obvious (particularly for the standard deviation) along the NLOS route. Conversely, the minimum values of σεr are found along the LOS route. This conclusion, as well as the corresponding reason, is similar to that of the analysis of RT-based power predictions. By contrast, the trend of the curve for errors in RMS delay spread predictions is relatively flat. In using the RT model for prediction along the LOS and NLOS routes, the increasing values of σεr become linearly proportional to the absolute value of the error of relative permittivity.
Fig. 9 Average error and standard deviation of the difference between predictions and the reference prediction when the relative permittivity varies for type I along the two routes
Figure 10 shows the error statistics of the predictions with respect to RMS delay spread against relative permittivity for the different types of computer simulations. When Tx is placed at position Tl with a height Ph = 0.6 m, the values of σεr and Eεr are maximum for the different types of computer simulations. In this condition, all test points are distributed along the NLOS route, and Tx is well below indoor objects. The ray paths that arrive at the test points are primarily composed of high-order rays. These rays undergo multiple reflections and diffractions; the computation accuracy of the electric field is reduced several times. Whether for a Tx location or a test route, the surrounding walls in the boundary of indoor spaces achieve the greatest effect on RT-based predictions, whereas the ceiling and floor do not show any obvious effect on predictions. As a result, the surrounding walls in the boundary of indoor spaces are the important factors affecting indoor radio wave propagation. When Tx is placed at position Th with a height Ph = 1.5 m, the influence of the ceiling on indoor radio wave propagation is especially obvious. In this case, the distance between Tx and the ceiling is small. As a result, the ray propagated from the ceiling for the same multi-reflected signal can carry strong energy and is evidently affected.
Fig. 10 Error statistics of the predictions with respect to RMS delay spread against relative permittivity for the different types of computer simulations (types II–V): (a) Eεr and (b) σεr.
4.4 Sensitivity of power coverage predictions to conductivity
The reference conductivity (0.015 S/m) remains relatively low. The power results (for simulation type I) are examined for indoor object conductivities: 1.5 × 10−3, 7.5 × 10−3, 1.5 × 10−2, 7.5 × 10−2, 1.5 × 10−1, 7.5 × 10−1, 1.5, 7.5, and 10 S/m. The error statistics of the difference with respect to power for type I is indicated in Fig. 11. A high number of LOS areas in a predicted plane indicate small values of Eσ and σσ. When Tx is placed at position Th with a height Ph = 1.5 m, the influence of indoor objects on indoor radio wave propagation is not obvious. The increasing trend of σσ (when the values of σ are less than the reference conductivity) is more obvious than the decreasing trend of σσ (when the values of σ are greater than the reference conductivity).
Fig. 11 Average error and standard deviation of the difference between predictions and the reference prediction when the conductivity varies for type I
For the ceiling, floor, and surrounding walls, the values of σ vary uniformly from 0.2 S/m to 1.1 S/m with an invariable separation (0.1). According to the analysis above regarding the effect of the relative permittivity of the floor on RT-based predictions, the influence of the conductivity of the floor on the predictions is not obvious and can be neglected. The conductivity of the floor is relatively low in comparison with that of the ceiling or surrounding walls. If the conductivity of the ceiling, floor, and surrounding walls are changed simultaneously and equally, then the conductivity of the floor dramatically increases. Therefore, comparing the four simulation types is not appropriate, and the prediction for type III is not evaluated.
Figure 12 shows the error statistics of the predictions with respect to power against conductivity for the three types of computer simulations. When test points are in the predicted plane and are mostly under the NLOS condition, the values of σσ are maximum. The increasing trend of σσ caused by the walls is basically similar to the increasing trend of σσ caused by the ceiling with the increase of σ. The surrounding walls in the boundary of indoor spaces demonstrate the greatest effect on RT-based predictions. This finding leads to the conclusion that the surrounding walls in the boundary of indoor spaces are the important factors affecting indoor radio wave propagation.
Fig. 12 Error statistics of the predictions with respect to power against conductivity for the three types of computer simulations: (a) Eσ and (b) σσ.
4.5. Sensitivity of RMS delay spread predictions to conductivity
The simulation analysis of the RMS delay spread predictions, the other simulation setup, and the statistical method to obtain the simulation results in this section are similar to those in described in Section 4 C. The values of conductivity are chosen and are similar to those in Section 4 D.
The average error and standard deviation of the difference between the predictions and the reference prediction when the conductivity varies for type I along the two routes are shown in Fig. 13. With the increasing conductivity along different routes, the values of Eσ are relatively low and show no obvious change. This change is limited to the range from −0.1 S/m to 0.73 S/m. When all test points are distributed in the lower predicted plane and are mostly under the NLOS condition, the values of σσ significantly increase. Furthermore, the difference between the values of σσ for predictions in the high predicted plane and the values of σσ for predictions in the low predicted plane increase significantly as conductivity increases. This conclusion is similar to that of the analysis of the effects of conductivity on RT-based power predictions.
Fig. 13 Average error and standard deviation of the difference between predictions and the reference prediction when the conductivity varies for type I along the two routes
Figure 14 shows the error statistics of the predictions with respect to RMS delay spread against conductivity for the three types of computer simulations. The influence of the surrounding walls on RT-based predictions is more evident than the influence of the ceiling on RT-based predictions. If all the test points are found in the high predicted plane, then the said phenomenon becomes increasingly obvious. A high number of LOS areas in a predicted plane indicate small values of Eσ and σσ. This conclusion is similar to that of the analysis of the influence of relative permittivity on RT-based predictions.
Fig. 14 Error statistics of the predictions with respect to RMS delay spread against conductivity for the three types of computer simulations: (a) Eσ and(b) σσ.
5. Conclusion
In this study, the sensitivity of the power and RMS delay spread predictions of the 3D indoor RT model presented in [15] is investigated. The variation of the predictions is examined for geometric and electrical database inaccuracies.
The sensitivity analysis of indoor geometric environments indicates that the minimum sensitivities are found in LOS regions and that model sensitivity increases with increasing NLOS conditions. The predictions in this study are more sensitive to object size errors than to edge position errors in most situations. In a complex indoor environment, radio wave propagation is affected from each direction (full 3D problem). When the coverage prediction for a certain area comprising direct rays and low-order reflected rays is calculated, only a low-accuracy indoor map is required. Conversely, a high-accuracy indoor map must be provided for coverage prediction in deep shadow areas.
The sensitivity of the RMS delay spread predictions to indoor geometric environments along the two routes is examined. In most circumstances, the predictions are more sensitive to object size errors than to errors in the orientation of the lateral face of objects. Therefore, in producing the indoor object layout database needed for the RT technique, special care should be taken to keep the position of the lateral faces as accurate as possible. When all the test points are distributed along the NLOS route, predictions are found to be especially sensitive to errors in the indoor geometric environment. In addition, the sensitivity of RMS delay spread predictions to environment information is higher than that of power predictions (especially in the NLOS region).
The sensitivity of the indoor model to the inaccuracies in the input electrical parameters is also studied. With the increasing LOS coverage area, the values of Eεr and σεr with respect to power and RMS delay spread decrease. In general, the electrical parameters of the surrounding walls in the boundary of indoor spaces demonstrate the greatest effect on RT-based predictions, whereas the electrical properties of the floor show minimal influence on predictions. If many indoor objects exist, the effect caused by simulation type I becomes relatively evident. Under this condition, a high accuracy with respect to the size and position of indoor objects should be achieved to obtain reliable prediction results. Therefore, small indoor objects, such as chairs, can be appropriately simplified or even excluded in indoor environment databases.
A comprehensive analysis of the sensitivity of RT-based predictions to geometric and electrical environments indicates that obtaining a high-accuracy indoor map is important. However, given that developing an indoor map for high-complexity environments is difficult, an indoor map with a suitable accuracy can be used instead. The analysis in this study can serve as a reference for indoor RT-based simulations. The used RT model is based on geometrical optics and its extensions uniform theory of diffraction. So, our work has the potential to the application in optics aspect, particularly in the area on sustainable energy, environment and green technologies given the efficiency analysis. However, the effects of the polarization of Tx and Rx, different directional antennas, and different parts of complex indoor environments on MIMO channels are excluded from the study. Future studies will concentrate on the effects of detailed environmental elements on RT-based prediction to provide a reliable theoretical basis for indoor MIMO communication.
Acknowledgments
The authors acknowledge financial support from the National Natural Science Foundation of China (Grant No. 61401345), the National Science Foundation for Distinguished Young Scholars of China (Grant No. 61225002), the Fundamental Research Funds for the Central Universities (Grant No. JB140510), the New Teacher Innovation Fund of Xidian University (Grant No. XJS14005), and the Foundation of Huawei Technologies CO., Ltd. (Contract No. YBWL2010247).
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