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An efficient two-dimensional ray-tracing algorithm for modeling of urban microcellular environments
Int. J. Electron. Commun. (AEÜ) 66 (2012) 439–447
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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue
An efficient two-dimensional ray-tracing algorithm for modeling of urban microcellular environments Sanjay Soni ∗ , Amitabha Bhattacharya Department of Electronics and Electrical Communication Engineering, IIT Kharagpur, West Bengal, India
a r t i c l e
i n f o
Article history: Received 30 April 2011 Accepted 11 October 2011 Keywords: Deterministic propagation model Microcellular scenario Ray tracing Uniform theory of diffraction (UTD)
a b s t r a c t In this work, an efficient ray-tracing approach based on decomposition of visibility-tree into sub-tree has been compared with the conventional approach. The comparison is done in the context of both the computational complexity and accuracy of the ray-tracing algorithm. Three independent urban scenarios are considered for this comparison. It is observed that the efficient approach based on use of sub-tree reduces the computational complexity significantly. Further, the proposed ray-tracing algorithm is applied to a microcellular environment to compute rms delay spread and the results are compared with available measurements. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Due to ever-demanding high-data rate by mobile customers in 3 G/4 G networks (e.g. in 4 G, significantly higher bit rate than 2 Mbps with typically 100 Mbps as maximum), there is great interest in multiple-input multiple-output (MIMO) technology which enhances channel capacity by taking advantage of the multipath radio channel [1,2]. This technology uses adaptive antenna elements whose performance strongly depends on the directional property of channel (such as delay spread, angular spread). This requires accurate modeling of the propagation channel [3]. Among different propagation models available, deterministic models are most preferable prediction tool as they can give accurate prediction in various kinds of scenarios without any need of the cumbersome measurement [4,5]. Deterministic model based on the ray approximation of electromagnetic field is most suitable when the scattering body is electrically large. Thus, ray model finds wide applications in the urban scenarios where buildings are involved. The main hurdle in successful implementation of ray-tracing tool for commercial purposes is the enormous computation time taken by the computer programme. In the literature, various approaches have been suggested to enhance the speed of the raytracing tool. For instance, in [6], it is shown that some ray paths are common between transmitter (Tx) and receiver (Rx). Thus, they can be computed by determining the ray path only once. This is done by decomposing the ray paths in four categories Tx–Rx, Tx–D, D–D, D–Rx where D stands for diffraction. The ray trajectory between Tx
∗ Corresponding author. Tel.: +91 1368 228030; fax: +91 1368 228062. E-mail addresses: [email protected], [email protected] (S. Soni). 1434-8411/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2011.10.005
and Rx is established by appropriately concatenating the elements of the above categories. In [7], the proposed method overcomes the problem of the Shooting-and-Bouncing (SBR) method’s requirement of tracing all of the rays to the receiver emitted by the transmitter by noting that most of the emitted rays do not reach the receiver. In [8], simplification of input database is proposed. In this approach, the two nearby buildings with same heights are joined. Second approach is to reduce input database by selecting “active set” of buildings and discarding rest of the buildings. In [9], Carluccio et al. uses modified Newton search algorithm for tracing the multiple diffracted paths assuming a series of arbitrarily placed wedges that are completely visible to each other. When the height of the building is much larger than the height of the transmitter and the receiver, the ray-tracing tool can be simplified to trace the ray paths only in the horizontal plane and hence input database of the buildings does not require height information of the buildings. In such scenarios, the predicted results of two-dimensional ray-tracing are shown to be adequate [10–12]. Image-based two-dimensional ray-tracing algorithm is presented in [13]. This algorithm is based on ray-tube method and it uses visibility-tree (VT) of the building scenario to construct the ray paths. It is fast in the sense that it does not require any testing to examine whether the selected ray-path in the VT of building layout is valid or not. However, this method requires redundant computation time to compute children of images and virtual sources, which is very time consuming. In [14] a more efficient approach to implement the ray-tracing tool using VT has been presented. The approach is based on VT and pre-processing of the data before actual ray-tracing tool is run. One of the main challenges faced in implementing the ray-tracing tool is the sharp rise in the computational complexity with the increase in the order of scattering [9,12].
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Fig. 1. An example of a building scenario and its visibility tree: (a) 2D view of building scenario and (b) visibility-tree.
In this paper, the comparison between an efficient ray-tracing method based on decomposition of VT into sub-tree has been done with conventional ray-tracing method. The comparison has been done in the context with computational complexity as well as accuracy of algorithm both. For this purpose, three microcellular scenarios of Ottawa city, Bern city and Fribourg city have been considered. It is observed that ray-tracing tool based on sub-tree concept results in considerable reduction in computational time. It is further observed that there is a moderate rise in the computational complexity with the increase in the scattering order. The paper is organized as follows. Section 2 describes briefly the concept of sub-tree, their construction. This also includes a brief discussion on the propagation model, the extension of twodimensional model to three-dimensional model and rms delay spread. Section 3 presents the comparison between predictions and available measurements of both path loss and rms delay spread. Section 4 concludes the work.
2. Proposed ray tracing algorithm 2.1. Construction of visibility-tree The scheme of the propagation environments considered in this work can be observed in Fig. 1. Fig. 1(a) is a two-dimensional view of a multiple building environment where Tx and Rx are considered in the same plane. To determine the valid ray path from Tx to Rx in a given scenario, the use of VT is an effective approach [13]. Fig. 1(b) illustrates an example of the construction of the VT of the building scenario shown in Fig. 1(a). There are basically two kinds of secondary sources generated: (i) image source (IS) at the surface of a wall (ii) virtual source (VS) at the corner. These are called children of Tx. Each of the secondary sources can now produce its own children that contain both image source and virtual source. For example, in Fig. 1(b), IS1 at I-order level generates k children VS1 , VS2 . . . VSk . They are called II-order sources. This recursion continues till the desired scattering order is reached. To determine the ray path, a receiver is tested if it lies in the shadow region of a given child at desired scattering order. If Rx lies in the visibility region of the child source, then, the backward ray path from Rx to Tx can be traced and valid ray-path can be determined without doing shadow-test. For instance, in Fig. 1 if Rx lies in the lit region of image source IS1 at II order, then, the II-order ray path is Rx–IS1 –VSm –Tx as shown in red colour. Therefore, we note that construction of VT for a given
scenario plays a crucial role in the determination of a ray path. In the VT, it is the determination of the valid children of a given source that takes most of the computational time. Once the children up to the desired scattering levels are obtained, determination of the ray path is done simply by moving in backward direction from Rx to Tx. 2.2. Construction of sub-tree In the ray-tracing algorithm based on sub-tree, we decompose the visibility-tree of a given scenario into a number of sub-trees which can be generated from the building database irrespective of the position of Tx in the scenario. Therefore, the required VT of a given scenario is nothing but simply a concatenation of these stored sub-trees. In order to elaborate this approach, we consider a simplified two-building scenario as shown in Fig. 2. We note that there are eight corners and eight faces of this two buildings scenario. Image source at the face and virtual source at j j the corner are denoted as ISi and VSi respectively where i is the index number of the image/virtual source and j is the scatteringorder. The VT corresponding to this building scenario is shown in Fig. 3. This VT is prepared up to third-order scattering with the root of tree as transmitter. The whole VT is divided into number of subtrees. For simplicity, only three sub-trees at scattering level-1 are shown in Fig. 3. These are sub-tree1, sub-tree2 and sub-tree3. These
Fig. 2. Simplified two-building scenario (2D view).
S. Soni, A. Bhattacharya / Int. J. Electron. Commun. (AEÜ) 66 (2012) 439–447 Table 1 Number of virtual sources with increase in the order of the scattering for a particular scenario of Fig. 2. Scattering order
VS1
VS2
VS3
VS4
VS5
I-Order II-Order III-Order IV-Order
1 3 10 21
1 2 11 34
1 3 5 35
1 4 11 30
1 3 6 37
sub-trees are independent of visibility-tree and hence they can be prepared independently using building database. Note that subtree2 contains sub-tree1 as its subset. Similarly, sub-tree3 contains sub-tree1 as its subset. Therefore, once the subtree1 is prepared and stored, it can be directly used in the construction of sub-tree2 j and sub-tree3. It is interesting to note that children of VSi and chilj VSi
obtained dren of VSki are exactly same. Therefore, children of by processing the building database can be used at any scattering order. As we go down to the higher scattering level, this situation is more often encountered where sub-tree, prepared once using building database, can be employed more frequently. Table 1 shows an example of how the number of VS increases as we go down the scattering order, establishing the above mentioned fact clearly. This results in great saving in computation time by avoiding computation of higher order sources. This also results in moderate rise in computational complexity while going from Nth scattering level to (N + 1)th scattering level as will be shown in Section 3. Considering Fig. 2 once again, there will be eight sub-trees corresponding to eight corners of the buildings. The root of the sub-tree is always the virtual source. Similarly, we can construct the subtrees with image source as the root of the sub-tree provided these image sources are the children of the virtual source. This is due to the reason that if they are the children of the virtual source at corner, only then their position will be fixed in the building scenario. These sub-trees are constructed irrespective of the position of the transmitter and the receiver in the buildings scenario. We also note from Fig. 3 that the length of the sub-tree can be as long
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Table 2 Look-up table showing the range of the children and their number. Index No.
1. 2. 3. – N
Total No. of children
Range
IS
VS
IS
VS
n1 n2 n3 – nN
m1 m2 m3 – mN
1 − S1 (S1 + 1) − S2 (S2 + 1) − S3 – (SN−1 + 1) − SN
1 − U1 (U1 + 1) − U2 (U2 + 1) − U3 – (UN−1 + 1) − UN
as we require depending on the scattering order. Fig. 5 shows the flowchart for the ray-tracing method based on sub-tree. 2.3. Preparation of look-up table We note that there are N sub-trees possible for the building scenario with N corners shown in Fig. 2. Each of the sub-tree has its root as virtual source as shown in Fig. 5. We store the first order IS children of all sub-trees in a matrix IMAGE SOURCE (i,J), i = 1, 2,. . .P;j = 1,2 where P is the total image source children of all sub-trees. Similarly, We store the first order VS children of all subtrees in a matrix VIRTUAL SOURCE (i,J), i = 1,2,. . .Q; j = 1, 2 where Q is the total virtual source children of all sub-trees. In order that the main ray-tracing program picks up suitable children of a sub-tree, we have to define corresponding range of children belonging to a particular sub-tree. Therefore, we store the information about this range in a look-up table shown in Table 2. The first column of the table shows the index number of corner points of the building. For each index number of corner point, the number of image and virtual sources are listed in columns II and III. The range of IS and VS are given in fourth and fifth columns respectively. Though this table incorporates the first order children of all sub-trees, the table can be extended to incorporate the total children of desired scattering order of all sub-trees. SN =
N
ni
i=1
Fig. 3. Decomposition of visibility-tree into sub-trees.
(1)
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Fig. 4. Multiple diffractions and reflections from building.
Fig. 5. Flowchart for proposed ray-tracing algorithm.
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UN =
N
mi
(2)
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Table 3 Comparison between conventional and new approach for prediction of path loss for different number of building walls.
i=1
where N is total number of corner points of the given map, ni is the total number of image children of ith sub-tree, mi is the total number of virtual children of ith sub-tree.
No. of walls
Available approach (computation time) (s)
New approach (computation time) (s)
20 42 72
4.9 34 167
1.86 8.3 28
2.4. Computation of field at receiver In the free space, the electric field in the direction of (, ) in the far field of transmitting antenna at a distance of r is given by [15]: exp(−jˇr) E(r, , ) = [E0, (, )ˆe + E0, (, )ˆe ] r where
E0,(,) (, ) =
(3)
PT 0 (, ) g 2 (,)
(4)
where PT is the average transmitted power, 0 is the intrinsic impedance of the free space and g(,) is the antenna gain in (, ) direction. Let E and E be the and components of the field at receiver. Then, the total field at receiver is given by Etotal = E × eˆ + E × eˆ
(5)
It may be noted that for vertically polarized antenna (in this case, the vertical field component will be parallel to wall surface), E will be much greater than E . Fig. 4 shows the multiple diffractions and reflections scenario from the buildings. Consider, for example, a multiple scattering scenario where the field reflects N times before the first diffraction, Q times between two diffractions and M times after second diffraction. The component of field at receiver is given by [15]
exp(−jˇrTR ) ⊥ Rn Ds (1) rTR N
E = E0, (, )
n=1
×
rT 1 Rq⊥ Ds (2) r12 (rT 1 + r12 )
rT 1 + r12 ⊥ Rm r2R (rT 1 + r12 + r2R )
Q
q=1
2.5. Extension of 2D approach to 3D approach Conversion of 2D ray path to 3D ray path can be done following the approach discussed in [6]. Propagation environment for single building scenario and double building scenario are shown in Fig. 6. Fig. 6 (a) depicts 2D ray model and its corresponding 3D ray model. We note that there are two possible 3D ray paths Ray-1 and ground reflected ray path called Ray-1 corresponding to one 2D ray path. These 3D ray paths are computed using approach in Fig. 7. We note that x1 + x2 = d and x1 /x2 = ht /hr . From these two equations, we can compute ground reflection point. In order the compute the height of diffraction point E for Ray-1 , we can use (ht − hr ) EF = d1 (d1 + d2 )
Fig. 6(b) shows another scenario where 2D Rays Ray-1 and Ray2 corresponds to 3D rays Ray-1 , Ray-1 and Ray-2 respectively. These rays can be determined using the approach discussed in Fig. 7. In order to include the possibility of roof top Ray-3, 2D ray-tracing tool needs to be modified to check if the Ray-3 exists or not. This can be done by comparing the height of the building with the height of the intersection point A using previous approach. If the height of the building is lesser, then, this ray will exist otherwise, this will be blocked. 2.6. Delay spread The delay spread is a measure of multipath richness of a channel. R.M.S delay spread is given by [21]
M
(6)
m=1
(10)
2 N (t − t¯ ) P i=1 i N P i=1 i
=
N
=
(ti )2 P
i=1 N
P i=1 i
2 − (t¯ )
(11)
where Ri⊥ is the reflection coefficient of ith surface with perpendicular polarization, rT1 , r12 , r2R , rTR are the distances as defined in Fig. 4. Parameter Ds is the diffraction coefficient (with soft polarization) and it is defined in [17]. After combining all the received multipath components coherently (with phase), the total power is given by
¯i =
N 2 E,i × g + E,i × g PR = 80
where N is total number of multipath for given position of Tx and Rx.
i=1
2
(7)
where wave number; 0 is instrinsic impedance of free space; N is number of multipath components received at Rx If the receiving antenna is vertically polarized, which is generally the case in mobile communication, and then the received power is given as
2 E,i × g PR = 80 N
i=1
2
(8)
and path loss is given as PL(dB) = PT (dB) − PR (dB) + GT,max (dBi) + GR,max (dBi)
(9)
where ti is the arrival time of ith multipath and it is calculated as ti = Li /c, Li is the total distance of ith multipath and c is the speed of the light. The parameter ¯i is mean delay time and is defined as
N i=1 N
t¯ × P
(12)
P i=1 i
3. Results and discussion 3.1. Application of the proposed ray-tracing algorithm to Ottawa city In this section, the overall performance of proposed ray-tracing is analyzed. As two-dimensional ray-tracing is suitable for microcellular scenario, we took microcellular environment (Ottawa city) for which measurement was carried out by Whitteker [16]. In a particular region (core) of this city, there are 15 buildings and total 72 walls. This urban scenario is presented as 2D view in Fig. 8. The measurement was carried out along the Bank St. and Tx is at 263 Laurier St. and frequency of operation was considered as 910 MHz.
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Fig. 6. Propagation environment: (a) single-building and (b) double-building.
In our calculations, reflected and diffracted fields are computed using the Fresnel reflection coefficient and the diffraction coefficient of El-Sallabi et al. [17]. In all the prediction results, conductivity of = 0.001 S/m and relative permittivity of εr = 7 were used. The value of εr is consistent with the range 5 ≤ εr ≤ 7 by direct measurement [3] and the value of is consistent with [18]. Path loss predictions were obtained using both the available approach of [13] and the proposed ray-tracing method. Table 3 shows the comparison between available two-dimensional ray-tracing and new approach for the prediction of path loss for different number of
Fig. 8. Map of Ottawa city.
walls of buildings of the scenario depicted in Fig. 8. Total time taken to process the database for constructing sub trees is 30 s. It can also be noted that using available ray-tracing approach [13], there is an exponential growth in the computation time (CT) as the number of walls increases. On the other hand, the proportion by which the CT increases for the proposed approach is much slower. It is because of more frequent use of sub-trees in the higher Table 4 Comparison between conventional and new approach for prediction of path loss at RX using conventional and new approach (Intel Core2quad CPU,[email protected] GHz,3.24 GB RAM): complete simulation (CT = computation time). Prediction order
Fig. 7. Conversion of 2D ray to 3D rays.
2 Refl, 2 Diffr 3 Refl, 2 Diffr
Conventional approach
New approach
CT (s)
CT (s)
4.5 167
No. of sources 1687 62,578
2.58 28
No. of sources 1184 23,281
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Fig. 9. Comparison between path loss obtained using proposed ray-tracing algorithm and available measurement [16].
order of scattering levels. This reduces computational load from Nth to (N + 1)th scattering level. Table 4 shows the comparison of computation time for the complete simulation of path loss prediction at a given Rx location. The comparison is shown for two cases: (i) 2 reflections and 2 diffractions, (ii) 3 reflections and 2 diffractions. It can be noted that the proposed method results in computation time reduction by more than 80 percent. This table also gives the details of total number of sources (Image source plus virtual sources) in both the cases. This gives an indication of the computational complexity. It is also quite obvious from the table that for conventional cases, there is sharp rise in the computational load (almost 35 times) whereas using proposed method, there is a moderate rise in computational load (almost 11 times). Here, the extent of improvement is not claimed to be same for all cases (as will be seen in the following example), but in any case it does represent adequate improvement. Though the comparison is shown only up to 3 reflections and 2 diffractions, it is quite obvious that proposed method will give drastic reduction in the computational load in the higher scattering order as well. Fig. 9 shows the comparison between predicted results obtained using proposed method and available measurement [16]. Simulation results obtained based on available ray-tracing method [13] is also included for comparison purpose. We note that both are exactly same. The prediction result available elsewhere [6] is also included for comparison. This result is reported to be generated using 5 reflections and 2 diffractions. We
note that both the results (i.e. proposed result and Daniela result) are almost same with some discrepancy in the lower part of the plot. That may be due to lesser number of reflections order used in the proposed algorithm. 3.2. Application of the proposed ray-tracing algorithm in Bern city We consider another example of microcellular scenario of Bern city (Switzerland) for which measurement is available in [12]. In the region of interest of this city, there are 23 buildings with 103 walls. The frequency of operation is 1.89 GHz. Conductivity and relative permittivity of the wall were considered to be = 0.001 S/m and εr = 5 respectively as reported in [12,18]. The map for Bern city is shown in Fig. 10. The transmitter is located at ‘src2’ (See Fig. 10) and the receiver movement is along Rodtmatt St. Comparison of the proposed ray-tracing algorithm with available approach [13] is presented in Tables 5 and 6. In Table 5, the comparison is made with respect to increase in the number of walls. Here, it is noted that computation time using new approach is significantly lower than that required using available approach. In ray-tracing algorithm, scattering order in ray-tracing engine was set to 3 reflections and 2 diffractions. Table 6 shows complete simulation for calculation of the field at the given receiver location. Here, comparison is done with the increase in the scattering order. We note that there is drastic reduction in computational load with the increase in scattering order. Fig. 11 shows the comparison of path loss prediction using proposed ray-tracing algorithm and available measurement [12]. In this path loss prediction result, our ray-tracing program also included the transmission through the building, scattering from tree [19, Eqs. (13) and (16)] and knife-edge model [20, Section V]. This result shows that there is no compromise on the prediction accuracy of the proposed ray-tracing algorithm. Table 5 Comparison between conventional and new approach for prediction of path loss for different number of building walls in Bern city.
Fig. 10. Map of Bern City [11,12].
No. of walls
Available approach (computation time) (s)
New approach (computation time) (s)
20 69 103
5.62 416 1132
2.76 64 118
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Table 6 Comparison between conventional and new approach for prediction of path loss at RX of Bern city (Intel Core2quad CPU,[email protected] GHz, 3.24 GB RAM): complete simulation. Prediction order
2 Refl, 2 Diffr 3 Refl, 2 Diffr
Conventional approach
New approach
CT (s)
No. of sources
CT (s)
16 1132
4048 232,545
6 118
Table 7 Comparison between conventional and new approach for the prediction of path loss in Fribourg city (Intel(R) Pentium (R) Dual CPU T 3400 @ 2.16 GHz,1.96 GB of RAM): complete simulation (no. of RX points taken along receiver route is 80). Prediction order
No. of sources 2677 66,380
CT, computation time.
3.3. Application of the proposed ray-tracing algorithm in Fribourg city Finally, the proposed ray-tracing algorithm is applied to Fribourg city [11, Fig. 3]. Measurement route is shown by red colour for which measurement was carried out by Rizk et al. [11]. Operating frequency taken is 1.8 GHz. Relative permittivity and conductivity were considered to be εr = 5 and = 0.0001 S/m respectively. Table 7 shows the comparison of computation time for prediction of path loss in the receiver route between the conventional and the proposed algorithm. Total number of Rx samples taken in the receiver route is 80. We note that almost 50% percent computation time is saved. Here, computation time reduction is not as significant as in the two cases reported in the previous two subsections. This is due to lesser number of buildings involved in the path loss predictions. From the above two examples, it is quite obvious that the advantage of the proposed algorithm is more significant when number of buildings involved in simulation is large. Fig. 12 shows the comparison of path loss obtained using proposed algorithm and available measurement [11]. Significant gap between prediction
2 Refl, 2 Diffr 3 Refl, 2 Diffr
Conventional approach
New approach
CT (s)
CT (s)
46 448
No. of sources 15,440 144,640
38 260
15,680 145,760
CT, computation time.
and measurement in the lower part of the result is possibly due to ignoring scattering from tree as shown in [19]. In the figure, prediction using available ray-tracing approach [13] is also included. This is exactly same as that obtained using proposed algorithm. Thus, we note that there is no compromise with prediction accuracy in the proposed algorithm. 3.4. Application of proposed algorithm to compute delay spread of multipath channel Delay spread is a measure of variety of multipath related effects of a channel. In this section, we present a comparison between the mean rms delay spread obtained using the proposed twodimensional ray-tracing model and available measurements. We also incorporate the prediction results presented by El-Sallabi [3] using VPL method. The propagation scenario for this analysis is shown in Fig. 13. For the prediction results, the wall conductivity and relative permittivity were chosen to be 0.001 S/m and 5 respectively. Frequency of operation is chosen as 2.154 GHz. Description of measurement scenario is presented in [3]. Figs. 14 and 15 show the predicted rms delay spread results (in ns) and available measurements for the route CD and GH route of the scenario in Fig. 13. We note that there is a wide range of fluctuations in the measurements. As explained in [3], this is due to the fact that each arriving ray is assumed to have time dependency that is delayed version of PN sequence used in the measurement. Predictions presented in [3] follow the same details of the measurements. Hence we can see the significant fluctuations in the VPL based prediction results. In the lack of these details, we followed the simple approach based on (12) to obtain the rms delay spread. It can be observed that the proposed prediction gives reasonably fair estimates of the rms delay spread using two-dimensional ray tracing in the multipath environment.
Fig. 11. Comparison of path loss obtained using proposed ray-tracing algorithm and available measurement [12].
Fig. 12. Comparison of path loss obtained using proposed ray-tracing algorithm and available measurement [11].
No. of sources
Fig. 13. Map of Helsinki city [3].
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Acknowledgment The authors are thankful to Prof. S. Sanyal, Department of Electronics and Electrical Communication Engg. IIT Kharagpur, for valuable suggestion. References
Fig. 14. Comparison of rms delay spread with available measurements [3] for the route CD.
Fig. 15. Comparison of rms delay spread with available measurements [3] for the route GH.
4. Conclusion In this paper, we have presented an efficient two-dimensional ray-tracing algorithm for the characterization of urban environment. Computation time of proposed technique was compared with available ray-tracing approaches to validate it’s computational efficiency. The accuracy of the proposed algorithm was tested by applying the algorithm to various microcellular scenarios and comparing the path loss, rms delay spread, thus obtained, with available measurements. Further, using the proposed technique, a moderate rise in the computational complexity with increase in the scattering order was observed.
[1] Kwakkernaat MRJAE, de Jong YLC, Bultitude RJC, Herben MHAJ. High-resolution angle-of-arrival measurements on physically-nonstationary mobile radio channels. IEEE Trans Antennas Propag 2008;56(August (8)):2720–9. [2] Cerasoli C. The use of ray tracing models to predict MIMO performance in urban environments. In: IEEE Military Communications Conference. 2006. [3] El-Sallabi HM, Liang G, Bertoni HL, Rekanos IT, Vainikainen P. Influence of diffraction coefficient and corner shape on ray prediction of power and delay spread in urban microcells. IEEE Trans Antennas Propag 2002;50(May (5)):703–12. [4] Coco S, Laudani A, Pollicino G. GRID-based prediction of electromagnetic fields in urban environment. IEEE Trans Magn 2009;45(March (3)). [5] Mantel OC, Oostveen JC, Popova MP. Applicability of deterministic propagation models for mobile operators. In: The Second European Conference on Antennas and Propagation. November 2007. p. 1–6. [6] Schettino DN, Moreira FJS, Rego CG. Efficient ray tracing for radio channel characterization of urban scenarios. IEEE Trans Magn 2007;43(April (4)):1305–8. [7] Mohtashami V, Shishegar AA. Accuracy and computational efficiency improvement of ray tracing using line search theory. IET Microw Antennas Propag September 2010;4(9):1290–9. [8] Degli-Esposti V, Fuschini F, Vitucci EM, Falciasecca G. Speed-up techniques for ray tracing field prediction models. IEEE Trans Antennas Propag May 2009;57(5):1469–80. [9] Carluccio G, Albani M. An efficient ray tracing algorithm for multiple straight wedge diffraction. IEEE Trans Antennas Propag Nov 2008;56(11):3534–42. [10] Erceg V, Rustako AJ, Roman RS. Diffraction around corners and its effects on the microcell coverage area in urban and suburban environments at 900 MHz, 2 GHz, and 6 GHz. IEEE Trans Veh Technol August 1994;43(3):762–6. [11] Rizk K, Wagen J-F, Gardiol F. Two-dimensional ray tracing modeling for propagation prediction in microcellular environments. IEEE Trans Veh Technol May 1997;46:508–18. [12] de Jong YLC, Koelen MHJL, Herben MHAJ. A Building-Transmission model for improved propagation prediction in urban microcells. IEEE Trans Veh Technol March 2004;53(2):490–502. [13] Son H-w, Myung N-H. A deterministic ray tube method for microcellular wave propagation prediction model. IEEE Trans Antennas Propag August 1999;47(8):1344–50. [14] Agelet FA, Formella A, Rábanos JMH, de Vicente FI, Fontán FP. Efficient raytracing acceleration techniques for radio propagation modeling. IEEE Trans Veh Technol November 2000;49(6):2089–104. [15] Remcom, Wireless Insite. Site-specific Radio Propagation Prediction Software User’s Manual Version 2.3; 2008. [16] Whitteker JH. Measurements of path loss at 910 MHz for proposed microcell urban mobile systems. IEEE Trans Veh Technol August 1988;37(3):125–9. [17] El-Sallabi HM, Rekanos IT, Vainikainen P. A new heuristic diffraction coefficient for lossy dielectric wedges at normal incidence. IEEE Antennas Wireless Propag Lett 2002;1:165–8. [18] Athanasiadou GE, Nix AR. Investigation into the sensitivity of the power predictions of a microcellular ray tracing propagation model. IEEE Trans Veh Technol July 2000;49(4):1140–51. [19] de Jong YLC, Herben MHAJ. A tree-scattering model for improved propagation prediction in urban microcells. IEEE Trans Veh Technol March 2004;53:503–13. [20] Yun Z, Zhang Z, Iskander MF. A ray-tracing method based on the triangular grid approach and application to propagation prediction in urban environments. IEEE Trans Antennas Propag May 2002;50:750–8. [21] Rappaport TS. Wireless communications—principle and practice. 2nd ed. Prentice-Hall; 2000.
Contents lists available at SciVerse ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue
An efficient two-dimensional ray-tracing algorithm for modeling of urban microcellular environments Sanjay Soni ∗ , Amitabha Bhattacharya Department of Electronics and Electrical Communication Engineering, IIT Kharagpur, West Bengal, India
a r t i c l e
i n f o
Article history: Received 30 April 2011 Accepted 11 October 2011 Keywords: Deterministic propagation model Microcellular scenario Ray tracing Uniform theory of diffraction (UTD)
a b s t r a c t In this work, an efficient ray-tracing approach based on decomposition of visibility-tree into sub-tree has been compared with the conventional approach. The comparison is done in the context of both the computational complexity and accuracy of the ray-tracing algorithm. Three independent urban scenarios are considered for this comparison. It is observed that the efficient approach based on use of sub-tree reduces the computational complexity significantly. Further, the proposed ray-tracing algorithm is applied to a microcellular environment to compute rms delay spread and the results are compared with available measurements. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Due to ever-demanding high-data rate by mobile customers in 3 G/4 G networks (e.g. in 4 G, significantly higher bit rate than 2 Mbps with typically 100 Mbps as maximum), there is great interest in multiple-input multiple-output (MIMO) technology which enhances channel capacity by taking advantage of the multipath radio channel [1,2]. This technology uses adaptive antenna elements whose performance strongly depends on the directional property of channel (such as delay spread, angular spread). This requires accurate modeling of the propagation channel [3]. Among different propagation models available, deterministic models are most preferable prediction tool as they can give accurate prediction in various kinds of scenarios without any need of the cumbersome measurement [4,5]. Deterministic model based on the ray approximation of electromagnetic field is most suitable when the scattering body is electrically large. Thus, ray model finds wide applications in the urban scenarios where buildings are involved. The main hurdle in successful implementation of ray-tracing tool for commercial purposes is the enormous computation time taken by the computer programme. In the literature, various approaches have been suggested to enhance the speed of the raytracing tool. For instance, in [6], it is shown that some ray paths are common between transmitter (Tx) and receiver (Rx). Thus, they can be computed by determining the ray path only once. This is done by decomposing the ray paths in four categories Tx–Rx, Tx–D, D–D, D–Rx where D stands for diffraction. The ray trajectory between Tx
∗ Corresponding author. Tel.: +91 1368 228030; fax: +91 1368 228062. E-mail addresses: [email protected], [email protected] (S. Soni). 1434-8411/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2011.10.005
and Rx is established by appropriately concatenating the elements of the above categories. In [7], the proposed method overcomes the problem of the Shooting-and-Bouncing (SBR) method’s requirement of tracing all of the rays to the receiver emitted by the transmitter by noting that most of the emitted rays do not reach the receiver. In [8], simplification of input database is proposed. In this approach, the two nearby buildings with same heights are joined. Second approach is to reduce input database by selecting “active set” of buildings and discarding rest of the buildings. In [9], Carluccio et al. uses modified Newton search algorithm for tracing the multiple diffracted paths assuming a series of arbitrarily placed wedges that are completely visible to each other. When the height of the building is much larger than the height of the transmitter and the receiver, the ray-tracing tool can be simplified to trace the ray paths only in the horizontal plane and hence input database of the buildings does not require height information of the buildings. In such scenarios, the predicted results of two-dimensional ray-tracing are shown to be adequate [10–12]. Image-based two-dimensional ray-tracing algorithm is presented in [13]. This algorithm is based on ray-tube method and it uses visibility-tree (VT) of the building scenario to construct the ray paths. It is fast in the sense that it does not require any testing to examine whether the selected ray-path in the VT of building layout is valid or not. However, this method requires redundant computation time to compute children of images and virtual sources, which is very time consuming. In [14] a more efficient approach to implement the ray-tracing tool using VT has been presented. The approach is based on VT and pre-processing of the data before actual ray-tracing tool is run. One of the main challenges faced in implementing the ray-tracing tool is the sharp rise in the computational complexity with the increase in the order of scattering [9,12].
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Fig. 1. An example of a building scenario and its visibility tree: (a) 2D view of building scenario and (b) visibility-tree.
In this paper, the comparison between an efficient ray-tracing method based on decomposition of VT into sub-tree has been done with conventional ray-tracing method. The comparison has been done in the context with computational complexity as well as accuracy of algorithm both. For this purpose, three microcellular scenarios of Ottawa city, Bern city and Fribourg city have been considered. It is observed that ray-tracing tool based on sub-tree concept results in considerable reduction in computational time. It is further observed that there is a moderate rise in the computational complexity with the increase in the scattering order. The paper is organized as follows. Section 2 describes briefly the concept of sub-tree, their construction. This also includes a brief discussion on the propagation model, the extension of twodimensional model to three-dimensional model and rms delay spread. Section 3 presents the comparison between predictions and available measurements of both path loss and rms delay spread. Section 4 concludes the work.
2. Proposed ray tracing algorithm 2.1. Construction of visibility-tree The scheme of the propagation environments considered in this work can be observed in Fig. 1. Fig. 1(a) is a two-dimensional view of a multiple building environment where Tx and Rx are considered in the same plane. To determine the valid ray path from Tx to Rx in a given scenario, the use of VT is an effective approach [13]. Fig. 1(b) illustrates an example of the construction of the VT of the building scenario shown in Fig. 1(a). There are basically two kinds of secondary sources generated: (i) image source (IS) at the surface of a wall (ii) virtual source (VS) at the corner. These are called children of Tx. Each of the secondary sources can now produce its own children that contain both image source and virtual source. For example, in Fig. 1(b), IS1 at I-order level generates k children VS1 , VS2 . . . VSk . They are called II-order sources. This recursion continues till the desired scattering order is reached. To determine the ray path, a receiver is tested if it lies in the shadow region of a given child at desired scattering order. If Rx lies in the visibility region of the child source, then, the backward ray path from Rx to Tx can be traced and valid ray-path can be determined without doing shadow-test. For instance, in Fig. 1 if Rx lies in the lit region of image source IS1 at II order, then, the II-order ray path is Rx–IS1 –VSm –Tx as shown in red colour. Therefore, we note that construction of VT for a given
scenario plays a crucial role in the determination of a ray path. In the VT, it is the determination of the valid children of a given source that takes most of the computational time. Once the children up to the desired scattering levels are obtained, determination of the ray path is done simply by moving in backward direction from Rx to Tx. 2.2. Construction of sub-tree In the ray-tracing algorithm based on sub-tree, we decompose the visibility-tree of a given scenario into a number of sub-trees which can be generated from the building database irrespective of the position of Tx in the scenario. Therefore, the required VT of a given scenario is nothing but simply a concatenation of these stored sub-trees. In order to elaborate this approach, we consider a simplified two-building scenario as shown in Fig. 2. We note that there are eight corners and eight faces of this two buildings scenario. Image source at the face and virtual source at j j the corner are denoted as ISi and VSi respectively where i is the index number of the image/virtual source and j is the scatteringorder. The VT corresponding to this building scenario is shown in Fig. 3. This VT is prepared up to third-order scattering with the root of tree as transmitter. The whole VT is divided into number of subtrees. For simplicity, only three sub-trees at scattering level-1 are shown in Fig. 3. These are sub-tree1, sub-tree2 and sub-tree3. These
Fig. 2. Simplified two-building scenario (2D view).
S. Soni, A. Bhattacharya / Int. J. Electron. Commun. (AEÜ) 66 (2012) 439–447 Table 1 Number of virtual sources with increase in the order of the scattering for a particular scenario of Fig. 2. Scattering order
VS1
VS2
VS3
VS4
VS5
I-Order II-Order III-Order IV-Order
1 3 10 21
1 2 11 34
1 3 5 35
1 4 11 30
1 3 6 37
sub-trees are independent of visibility-tree and hence they can be prepared independently using building database. Note that subtree2 contains sub-tree1 as its subset. Similarly, sub-tree3 contains sub-tree1 as its subset. Therefore, once the subtree1 is prepared and stored, it can be directly used in the construction of sub-tree2 j and sub-tree3. It is interesting to note that children of VSi and chilj VSi
obtained dren of VSki are exactly same. Therefore, children of by processing the building database can be used at any scattering order. As we go down to the higher scattering level, this situation is more often encountered where sub-tree, prepared once using building database, can be employed more frequently. Table 1 shows an example of how the number of VS increases as we go down the scattering order, establishing the above mentioned fact clearly. This results in great saving in computation time by avoiding computation of higher order sources. This also results in moderate rise in computational complexity while going from Nth scattering level to (N + 1)th scattering level as will be shown in Section 3. Considering Fig. 2 once again, there will be eight sub-trees corresponding to eight corners of the buildings. The root of the sub-tree is always the virtual source. Similarly, we can construct the subtrees with image source as the root of the sub-tree provided these image sources are the children of the virtual source. This is due to the reason that if they are the children of the virtual source at corner, only then their position will be fixed in the building scenario. These sub-trees are constructed irrespective of the position of the transmitter and the receiver in the buildings scenario. We also note from Fig. 3 that the length of the sub-tree can be as long
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Table 2 Look-up table showing the range of the children and their number. Index No.
1. 2. 3. – N
Total No. of children
Range
IS
VS
IS
VS
n1 n2 n3 – nN
m1 m2 m3 – mN
1 − S1 (S1 + 1) − S2 (S2 + 1) − S3 – (SN−1 + 1) − SN
1 − U1 (U1 + 1) − U2 (U2 + 1) − U3 – (UN−1 + 1) − UN
as we require depending on the scattering order. Fig. 5 shows the flowchart for the ray-tracing method based on sub-tree. 2.3. Preparation of look-up table We note that there are N sub-trees possible for the building scenario with N corners shown in Fig. 2. Each of the sub-tree has its root as virtual source as shown in Fig. 5. We store the first order IS children of all sub-trees in a matrix IMAGE SOURCE (i,J), i = 1, 2,. . .P;j = 1,2 where P is the total image source children of all sub-trees. Similarly, We store the first order VS children of all subtrees in a matrix VIRTUAL SOURCE (i,J), i = 1,2,. . .Q; j = 1, 2 where Q is the total virtual source children of all sub-trees. In order that the main ray-tracing program picks up suitable children of a sub-tree, we have to define corresponding range of children belonging to a particular sub-tree. Therefore, we store the information about this range in a look-up table shown in Table 2. The first column of the table shows the index number of corner points of the building. For each index number of corner point, the number of image and virtual sources are listed in columns II and III. The range of IS and VS are given in fourth and fifth columns respectively. Though this table incorporates the first order children of all sub-trees, the table can be extended to incorporate the total children of desired scattering order of all sub-trees. SN =
N
ni
i=1
Fig. 3. Decomposition of visibility-tree into sub-trees.
(1)
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Fig. 4. Multiple diffractions and reflections from building.
Fig. 5. Flowchart for proposed ray-tracing algorithm.
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UN =
N
mi
(2)
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Table 3 Comparison between conventional and new approach for prediction of path loss for different number of building walls.
i=1
where N is total number of corner points of the given map, ni is the total number of image children of ith sub-tree, mi is the total number of virtual children of ith sub-tree.
No. of walls
Available approach (computation time) (s)
New approach (computation time) (s)
20 42 72
4.9 34 167
1.86 8.3 28
2.4. Computation of field at receiver In the free space, the electric field in the direction of (, ) in the far field of transmitting antenna at a distance of r is given by [15]: exp(−jˇr) E(r, , ) = [E0, (, )ˆe + E0, (, )ˆe ] r where
E0,(,) (, ) =
(3)
PT 0 (, ) g 2 (,)
(4)
where PT is the average transmitted power, 0 is the intrinsic impedance of the free space and g(,) is the antenna gain in (, ) direction. Let E and E be the and components of the field at receiver. Then, the total field at receiver is given by Etotal = E × eˆ + E × eˆ
(5)
It may be noted that for vertically polarized antenna (in this case, the vertical field component will be parallel to wall surface), E will be much greater than E . Fig. 4 shows the multiple diffractions and reflections scenario from the buildings. Consider, for example, a multiple scattering scenario where the field reflects N times before the first diffraction, Q times between two diffractions and M times after second diffraction. The component of field at receiver is given by [15]
exp(−jˇrTR ) ⊥ Rn Ds (1) rTR N
E = E0, (, )
n=1
×
rT 1 Rq⊥ Ds (2) r12 (rT 1 + r12 )
rT 1 + r12 ⊥ Rm r2R (rT 1 + r12 + r2R )
Q
q=1
2.5. Extension of 2D approach to 3D approach Conversion of 2D ray path to 3D ray path can be done following the approach discussed in [6]. Propagation environment for single building scenario and double building scenario are shown in Fig. 6. Fig. 6 (a) depicts 2D ray model and its corresponding 3D ray model. We note that there are two possible 3D ray paths Ray-1 and ground reflected ray path called Ray-1 corresponding to one 2D ray path. These 3D ray paths are computed using approach in Fig. 7. We note that x1 + x2 = d and x1 /x2 = ht /hr . From these two equations, we can compute ground reflection point. In order the compute the height of diffraction point E for Ray-1 , we can use (ht − hr ) EF = d1 (d1 + d2 )
Fig. 6(b) shows another scenario where 2D Rays Ray-1 and Ray2 corresponds to 3D rays Ray-1 , Ray-1 and Ray-2 respectively. These rays can be determined using the approach discussed in Fig. 7. In order to include the possibility of roof top Ray-3, 2D ray-tracing tool needs to be modified to check if the Ray-3 exists or not. This can be done by comparing the height of the building with the height of the intersection point A using previous approach. If the height of the building is lesser, then, this ray will exist otherwise, this will be blocked. 2.6. Delay spread The delay spread is a measure of multipath richness of a channel. R.M.S delay spread is given by [21]
M
(6)
m=1
(10)
2 N (t − t¯ ) P i=1 i N P i=1 i
=
N
=
(ti )2 P
i=1 N
P i=1 i
2 − (t¯ )
(11)
where Ri⊥ is the reflection coefficient of ith surface with perpendicular polarization, rT1 , r12 , r2R , rTR are the distances as defined in Fig. 4. Parameter Ds is the diffraction coefficient (with soft polarization) and it is defined in [17]. After combining all the received multipath components coherently (with phase), the total power is given by
¯i =
N 2 E,i × g + E,i × g PR = 80
where N is total number of multipath for given position of Tx and Rx.
i=1
2
(7)
where wave number; 0 is instrinsic impedance of free space; N is number of multipath components received at Rx If the receiving antenna is vertically polarized, which is generally the case in mobile communication, and then the received power is given as
2 E,i × g PR = 80 N
i=1
2
(8)
and path loss is given as PL(dB) = PT (dB) − PR (dB) + GT,max (dBi) + GR,max (dBi)
(9)
where ti is the arrival time of ith multipath and it is calculated as ti = Li /c, Li is the total distance of ith multipath and c is the speed of the light. The parameter ¯i is mean delay time and is defined as
N i=1 N
t¯ × P
(12)
P i=1 i
3. Results and discussion 3.1. Application of the proposed ray-tracing algorithm to Ottawa city In this section, the overall performance of proposed ray-tracing is analyzed. As two-dimensional ray-tracing is suitable for microcellular scenario, we took microcellular environment (Ottawa city) for which measurement was carried out by Whitteker [16]. In a particular region (core) of this city, there are 15 buildings and total 72 walls. This urban scenario is presented as 2D view in Fig. 8. The measurement was carried out along the Bank St. and Tx is at 263 Laurier St. and frequency of operation was considered as 910 MHz.
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Fig. 6. Propagation environment: (a) single-building and (b) double-building.
In our calculations, reflected and diffracted fields are computed using the Fresnel reflection coefficient and the diffraction coefficient of El-Sallabi et al. [17]. In all the prediction results, conductivity of = 0.001 S/m and relative permittivity of εr = 7 were used. The value of εr is consistent with the range 5 ≤ εr ≤ 7 by direct measurement [3] and the value of is consistent with [18]. Path loss predictions were obtained using both the available approach of [13] and the proposed ray-tracing method. Table 3 shows the comparison between available two-dimensional ray-tracing and new approach for the prediction of path loss for different number of
Fig. 8. Map of Ottawa city.
walls of buildings of the scenario depicted in Fig. 8. Total time taken to process the database for constructing sub trees is 30 s. It can also be noted that using available ray-tracing approach [13], there is an exponential growth in the computation time (CT) as the number of walls increases. On the other hand, the proportion by which the CT increases for the proposed approach is much slower. It is because of more frequent use of sub-trees in the higher Table 4 Comparison between conventional and new approach for prediction of path loss at RX using conventional and new approach (Intel Core2quad CPU,[email protected] GHz,3.24 GB RAM): complete simulation (CT = computation time). Prediction order
Fig. 7. Conversion of 2D ray to 3D rays.
2 Refl, 2 Diffr 3 Refl, 2 Diffr
Conventional approach
New approach
CT (s)
CT (s)
4.5 167
No. of sources 1687 62,578
2.58 28
No. of sources 1184 23,281
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445
Fig. 9. Comparison between path loss obtained using proposed ray-tracing algorithm and available measurement [16].
order of scattering levels. This reduces computational load from Nth to (N + 1)th scattering level. Table 4 shows the comparison of computation time for the complete simulation of path loss prediction at a given Rx location. The comparison is shown for two cases: (i) 2 reflections and 2 diffractions, (ii) 3 reflections and 2 diffractions. It can be noted that the proposed method results in computation time reduction by more than 80 percent. This table also gives the details of total number of sources (Image source plus virtual sources) in both the cases. This gives an indication of the computational complexity. It is also quite obvious from the table that for conventional cases, there is sharp rise in the computational load (almost 35 times) whereas using proposed method, there is a moderate rise in computational load (almost 11 times). Here, the extent of improvement is not claimed to be same for all cases (as will be seen in the following example), but in any case it does represent adequate improvement. Though the comparison is shown only up to 3 reflections and 2 diffractions, it is quite obvious that proposed method will give drastic reduction in the computational load in the higher scattering order as well. Fig. 9 shows the comparison between predicted results obtained using proposed method and available measurement [16]. Simulation results obtained based on available ray-tracing method [13] is also included for comparison purpose. We note that both are exactly same. The prediction result available elsewhere [6] is also included for comparison. This result is reported to be generated using 5 reflections and 2 diffractions. We
note that both the results (i.e. proposed result and Daniela result) are almost same with some discrepancy in the lower part of the plot. That may be due to lesser number of reflections order used in the proposed algorithm. 3.2. Application of the proposed ray-tracing algorithm in Bern city We consider another example of microcellular scenario of Bern city (Switzerland) for which measurement is available in [12]. In the region of interest of this city, there are 23 buildings with 103 walls. The frequency of operation is 1.89 GHz. Conductivity and relative permittivity of the wall were considered to be = 0.001 S/m and εr = 5 respectively as reported in [12,18]. The map for Bern city is shown in Fig. 10. The transmitter is located at ‘src2’ (See Fig. 10) and the receiver movement is along Rodtmatt St. Comparison of the proposed ray-tracing algorithm with available approach [13] is presented in Tables 5 and 6. In Table 5, the comparison is made with respect to increase in the number of walls. Here, it is noted that computation time using new approach is significantly lower than that required using available approach. In ray-tracing algorithm, scattering order in ray-tracing engine was set to 3 reflections and 2 diffractions. Table 6 shows complete simulation for calculation of the field at the given receiver location. Here, comparison is done with the increase in the scattering order. We note that there is drastic reduction in computational load with the increase in scattering order. Fig. 11 shows the comparison of path loss prediction using proposed ray-tracing algorithm and available measurement [12]. In this path loss prediction result, our ray-tracing program also included the transmission through the building, scattering from tree [19, Eqs. (13) and (16)] and knife-edge model [20, Section V]. This result shows that there is no compromise on the prediction accuracy of the proposed ray-tracing algorithm. Table 5 Comparison between conventional and new approach for prediction of path loss for different number of building walls in Bern city.
Fig. 10. Map of Bern City [11,12].
No. of walls
Available approach (computation time) (s)
New approach (computation time) (s)
20 69 103
5.62 416 1132
2.76 64 118
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Table 6 Comparison between conventional and new approach for prediction of path loss at RX of Bern city (Intel Core2quad CPU,[email protected] GHz, 3.24 GB RAM): complete simulation. Prediction order
2 Refl, 2 Diffr 3 Refl, 2 Diffr
Conventional approach
New approach
CT (s)
No. of sources
CT (s)
16 1132
4048 232,545
6 118
Table 7 Comparison between conventional and new approach for the prediction of path loss in Fribourg city (Intel(R) Pentium (R) Dual CPU T 3400 @ 2.16 GHz,1.96 GB of RAM): complete simulation (no. of RX points taken along receiver route is 80). Prediction order
No. of sources 2677 66,380
CT, computation time.
3.3. Application of the proposed ray-tracing algorithm in Fribourg city Finally, the proposed ray-tracing algorithm is applied to Fribourg city [11, Fig. 3]. Measurement route is shown by red colour for which measurement was carried out by Rizk et al. [11]. Operating frequency taken is 1.8 GHz. Relative permittivity and conductivity were considered to be εr = 5 and = 0.0001 S/m respectively. Table 7 shows the comparison of computation time for prediction of path loss in the receiver route between the conventional and the proposed algorithm. Total number of Rx samples taken in the receiver route is 80. We note that almost 50% percent computation time is saved. Here, computation time reduction is not as significant as in the two cases reported in the previous two subsections. This is due to lesser number of buildings involved in the path loss predictions. From the above two examples, it is quite obvious that the advantage of the proposed algorithm is more significant when number of buildings involved in simulation is large. Fig. 12 shows the comparison of path loss obtained using proposed algorithm and available measurement [11]. Significant gap between prediction
2 Refl, 2 Diffr 3 Refl, 2 Diffr
Conventional approach
New approach
CT (s)
CT (s)
46 448
No. of sources 15,440 144,640
38 260
15,680 145,760
CT, computation time.
and measurement in the lower part of the result is possibly due to ignoring scattering from tree as shown in [19]. In the figure, prediction using available ray-tracing approach [13] is also included. This is exactly same as that obtained using proposed algorithm. Thus, we note that there is no compromise with prediction accuracy in the proposed algorithm. 3.4. Application of proposed algorithm to compute delay spread of multipath channel Delay spread is a measure of variety of multipath related effects of a channel. In this section, we present a comparison between the mean rms delay spread obtained using the proposed twodimensional ray-tracing model and available measurements. We also incorporate the prediction results presented by El-Sallabi [3] using VPL method. The propagation scenario for this analysis is shown in Fig. 13. For the prediction results, the wall conductivity and relative permittivity were chosen to be 0.001 S/m and 5 respectively. Frequency of operation is chosen as 2.154 GHz. Description of measurement scenario is presented in [3]. Figs. 14 and 15 show the predicted rms delay spread results (in ns) and available measurements for the route CD and GH route of the scenario in Fig. 13. We note that there is a wide range of fluctuations in the measurements. As explained in [3], this is due to the fact that each arriving ray is assumed to have time dependency that is delayed version of PN sequence used in the measurement. Predictions presented in [3] follow the same details of the measurements. Hence we can see the significant fluctuations in the VPL based prediction results. In the lack of these details, we followed the simple approach based on (12) to obtain the rms delay spread. It can be observed that the proposed prediction gives reasonably fair estimates of the rms delay spread using two-dimensional ray tracing in the multipath environment.
Fig. 11. Comparison of path loss obtained using proposed ray-tracing algorithm and available measurement [12].
Fig. 12. Comparison of path loss obtained using proposed ray-tracing algorithm and available measurement [11].
No. of sources
Fig. 13. Map of Helsinki city [3].
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447
Acknowledgment The authors are thankful to Prof. S. Sanyal, Department of Electronics and Electrical Communication Engg. IIT Kharagpur, for valuable suggestion. References
Fig. 14. Comparison of rms delay spread with available measurements [3] for the route CD.
Fig. 15. Comparison of rms delay spread with available measurements [3] for the route GH.
4. Conclusion In this paper, we have presented an efficient two-dimensional ray-tracing algorithm for the characterization of urban environment. Computation time of proposed technique was compared with available ray-tracing approaches to validate it’s computational efficiency. The accuracy of the proposed algorithm was tested by applying the algorithm to various microcellular scenarios and comparing the path loss, rms delay spread, thus obtained, with available measurements. Further, using the proposed technique, a moderate rise in the computational complexity with increase in the scattering order was observed.
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