{{projectinfo|Application| Modeling Polarimetric Wave Propagation In The Lower Manhattan Scene Using EM.Terrano|ART MANH Fig title.png|In this project, you we will analyze a very large urban canyon propagation scene involving hundreds of buildings with complex geometries and a very large irregular terrain.|*[[Building Geometrical Constructions in CubeCAD| CubeCAD]]
*[[EM.Terrano]]
*Impenetrable Surfaces
*Short Dipole SourceTransmitter*Field Distribution*Source PolarizationReceiver
*Received Power Coverage Map
*Pattern Rotation*Frequency Sweep|All versions|{{download|http://www.emagtech.com|EM.Terrano Lesson 8|[[EM.Cube]] 14.10}} None }}
== Introduction ==
In this application note, we will demonstrate how [[EM.Terrano]] can be used to analyze a very large urban canyon propagation scene using its ultrafast k-d tree based ray tracer. Specifically, we will construct a scene representing a portion of lower Manhattan, NY, with real-sized models of its buildings and terrain profile. The total size of the area is 1.3km × 1.2km, and it will be covered with a very large rectangular grid of receivers equally spaced with a uniform spacing of 5m. You can learn more about the basic procedure for setting up an propagation scene and running a SBR simulation in "[[EM.Terrano Tutorial Lesson 23: Analyzing An Outdoor A Multipath Outdoor Propagation Scene]]".
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The [[SBR Method|SBR method]] assumes that electric fields propagate as spherical waves with the general form:
<math> \mathbf{A}(\mathbf{r}) \frac{e^{-jkR}}{R} </math>
where R = |<strong>r-r'</strong>| is the distance between the source and observation points, <strong>r</strong> and <strong>r'</strong> are the position vectors of the observation and source points, respectively, k = √ε<sub>r</sub>k<sub>0</sub>, ε<sub>r</sub> is the relative permittivity of the propagation medium, k<sub>0</sub> = 2πf/c is the free-space propagation constant, f is the operational frequency and c is the speed of light in the free space. <strong>A</strong>(<strong>r</strong>) is complex vector function that defines the polarization and strength of the field at the observation point. Propagating spherical waves are modeled as ray tubes or beams that emanate from a field source point (or a transmitter), travel in the free space, bounce from obstacles and scatterers and are collected at a field observation point (or a receiver). As a ray tube travels in space, its footprint expands due to the beam's spatial divergence. Every scatterer in the scene is discretized into a group of conjoined triangular facets, which are created by a triangular surface mesh generator. When a propagating ray is intercepted by a facet, one of the following events takes place:
* The incident ray is reflected back into the free space.
* The incident ray is diffracted from one of the three edges of the facet.
Therefore, a ray-triangle intersection algorithm is required to test if a given ray hits a given facet of a building. Unfortunately, even very fast ray-triangle intersection algorithms are computationally very expensive. If the mesh triangles are not organized spatially, a brute-force SBR solver will have to perform N×M intersection calculations, where N is the total number of rays and M is the total number of triangular facets in the scene. This is one of the main reasons why conventional ray tracers take a significant amounts of time to solve large-scale propagation scenes that involve hundreds of buildings and thousands of rays.
The solution to the geometric problem of ray-facet intersections can be significantly accelerated by using a k-dimensional (k-d) tree to subdivide the computational domain. A k-d tree, a special case of a binary tree, is a spatial partitioning data structure that is used for organizing points in a k-dimensional space. Using a k-d tree reduces the number of ray-triangle intersection calculations that is required for each ray. In other words, the algorithm used to navigate or traverse a k-d tree is computationally less expensive than a brute-force ray-triangle intersection calculation. The k-d tree is constructed by dividing the computational domain into two partitions and then sorting all the triangles into the two partitions. This process is repeated for the two newly generated partitions until a termination criterion is met. The key to constructing a useful and computationally efficient k-d tree is intelligently deciding on when and where to split each space. A well constructed k-d tree can be many times faster than a poorly constructed one.
After constructing a k-d tree for a given propagation scene, all the triangular facets of the simulation domain are organized spatially. Each k-d tree leaf is loaded with a list of all the facets that are contained in that leaf. Each shooting ray will traverse through the k-d tree and will perform ray-triangle intersection calculations only on triangles that are contained in the traversed leaves. Many different k-d tree traversing algorithms have been developed in recent years. [[EM.CubeTerrano]]'s SBR simulation engine uses an improved version of an efficient algorithm called the Recursive Ray Traversal Algorithm [3].
==Setting Up the Manhattan Propagation Scene==
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[[EM.Terrano]]'s default transmitter is a vertically polarized half-wave dipole antenna. The default transmitter is placed at x = 1150m and y = 700m, at a height of 30m above the ground elevation at that point. A large rectangular grid of 260 × 240 isotropic receivers is placed 1.5m above the ground elevation that covers the entire scene. A total of 62,400 receivers with an equal spacing of 5m are used to collect the rays in the scene.
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== SBR Analysis of the Manhattan Scene with a Vertical Dipole Transmitter ==
The operating frequency of the SBR simulation is set to f = 2.4GHz. At this frequency, the free-space wavelength is λ<sub>0</sub> = 0.125m. Due to the very large size of the propagation scene, we choose a ray angular resolution of 0.1° for this simulation. [[EM.Terrano]] uses a special algorithm based on the tessellation of an icosahedron to generate ray tubes with uniform and equal cross sectional areas. With the specified 0.1° angular resolution, this algorithm generate generates a total of 648,000 initial rays emanating from the location of the transmitter in all directions. The reflected rays are limited to a maximum of 10 bounces. The sensitivity of the receivers is set to -100dBm. This means that if the power of any ray falls below this threshold before reaching a receiver, that ray is terminated in the scene. The figure below shows the computed receiver power coverage map of the propagation scene.
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[[Image:ART MANH Fig4.png|thumb|left|550px|The mesh view received power coverage map of the Manhattan scene.]]
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:<math> P_R = P_T G_T G_R \left( \frac{\lambda_0}{4\pi R} \right)^2 = 29.14dBm + 20log_{10}\left( \frac{0.125}{4\pi(28.5)} \right) \approx -40dBm</math>
The table below summarizes the computational statistics associated with the SBR simulation performed earlier: {| class="wikitable"|-| Operating Frequency| 2.4GHz|-| Free-Space Wavelength| λ<sub>0</sub> = 0.125m|-| Total Domain Area Size| 1.3km × 1.2km (10,400λ<sub>0</sub> × 9,600λ<sub>0</sub>)|-| Total Number of Buildings| 914|-| Total Number of Facets| 146,205|-| Total Number of Edges| 27,357|-| Total Number of Transmitted Rays| 648,000|-| Total Number of Receivers| 62,400|-| Total Number of k-d Tree Nodes| 125,038|} The SBR simulation was performed on an ASUS laptop computer with an Intel i7-4860 HQ processor with eight cores. The laptop features 32GB RAM memory. Due to the complexity of the geometry of many buildings, the mesh generation task took a sizable amount of time. The table below shows the time spent by the CPU for the mesh generation and SBR ray tracing portions of the total simulation time: {| class="wikitable"|-| Simulation Step| Computation Time|-| Facet Mesh Generation| 80s|-| SBR Ray Tracing| 160s|-| Total| 240s|}
== Using a Directional Yagi-Uda Array ==
A vertical dipole has a fairly omni-directional far-field radiation pattern along the azimuth direction. Next, we will use a directional antenna array as the transmitter. For this purpose, we choose a horizontal end-fire Yagi-Uda dipole array with a reflector, an exciter and 10 director elements. The individual dipole elements are Y-directed and the array's linear axis is oriented along the X-axis. The design of the Yagi-Uda dipole array is discussed in detail in [[EM.Libera Tutorial Lesson 2: Designing A Yagi-Uda Dipole Array]]. The figure below shows the 3D far-field radiation pattern of 12-element Yagi-Uda array at an operating frequency of 2.4GHz.
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[[Image:ART MANH Fig10.png|thumb|left|550px|The 3D far-field radiation pattern of a horizontally polarized Yagi-Uda dipole array with 10 director elements.]]
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The above radiation pattern was computed using [[EM.Libera]], [[EM.Cube]]'s MoM3D computational module. The φ and θ components of the far-zone electric fields in the standard spherical coordinate system are stored in an ASCII data file with a ".RAD" file extension. With an original angular resolution of 1°, the data file contains a total of 361 × 181 = 65,341 rows containing the real and imaginary parts of the complex-valued E<sub>θ</sub> and E<sub>φ</sub> field components. This provides a fully polarimetric representation of the source's outgoing fields, which are then picked up and traced by [[EM.Terrano]]'s fully coherent and 3D-polarimetric SBR solver. Note that the complexity of the radiation pattern data file does not add any computational overhead to the SBR simulation. The figure below shows the received power coverage map of the Manhattan scene with an X-directed horizontal Yagi-Uda array. You can visibly see the directionality of the pattern of the transmitter from the coverage map in the area below the transmitter. Also note that the maximum received power in this case is -37dBm, which is 7dB larger than the previous case with a vertical dipole transmitter. This increase is accounted for by the greater directivity (D<sub>0</sub> = 10dB) of the transmitter antenna, and hence the larger transmitted EIRP.
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[[Image:ART MANH Fig11.png|thumb|left|550px|The receiver received power coverage map of the Manhattan scene with an X-directed horizontal Yagi-Uda array.]]
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[[EM.Terrano]] allows the user to rotate an imported far-field radiation pattern about the three principal coordinate axes. Next, we rotate the radiation pattern of the Yagi-Uda array by 135° and run a new SBR simulation of the propagation scene. The results are shown in the figure below. Note the deeper wave penetration into the urban canyon as well as the strong reflection from the blocks of buildings into the open area behind the transmitter.
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[[Image:ART MANH Fig12.png|thumb|left|550px|The receiver received power coverage map of the Manhattan scene with a horizontal Yagi-Uda array rotated clockwise by 135°.]]
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==Moving the Source Transmitter into a Dense Area== Next, we moved the transmitter with the rotated Yagi-Uda array into a denser area and placed it at x = 900m and y = 450mm as shown in the figure below.
Next, open the property dialog of the short dipole source and move its coordinated to (750m, 450m, 40m), keeping it at the same height as before. Run a new SBR simulation of the scene and visualize the received coverage map.
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[[Image:PROP255ART MANH Fig15.png|thumb|550pxleft|640px|The received power coverage map new location of the transmitter (in blue circle) in the lower Manhattan scene with the transmitter moved into an area with high building density.]]
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As you can see from the above figureInstead of running a single-frequency SBR analysis, this time we ran a frequency sweep of the coverage area is still very limitedpropagation scene over the frequency range [1.5GHz, with a few streets getting some coverage2. If you open the property dialog 5GHz]. A frequency step of the short dipole source100MHz was chosen, hence, you will notice a default value total of 1A for the dipole current11 frequency samples. The dialog also shows a value geometrical optic (GO) part of 227.4mW the simulation is the same for all frequency samples except for the dipole source's radiated power. Increase fact that the dipole current to 10Areflection coefficients are at the building specular points are frequency-dependent. This will increase Diffraction coefficients are frequency-dependent, too. [[EM.Terrano]] first determines all the radiated power 100optical paths in the scene using the k-fold to 22d tree algorithms.7WThen, as it calculates the reflection and diffraction coefficients at each frequency sample and computes the field components and received power at the location of a short dipole radiator varies as |I|<sup>2</sup>each receiver. Run As a new SBR simulation and compare result, [[EM.Terrano]]'s frequency sweeps are extremely fast. The figures below show the received power coverage mapsof the propagation scene with a rotated horizontal Yagi-Uda array at three different frequencies: f = 1.5GHz, 2GHz and 2.5GHz. The color-coded intensity scale is identical for all three plots. As you would have expected, the maximum received power level drops as the operating frequency is increased.
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[[Image:PROP259ART MANH Fig16.png|thumb|350pxleft|550px|The Short Dipole Source dialogreceived power coverage map of the Manhattan scene with a rotated horizontal Yagi-Uda array at f = 1.5GHz.]]
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[[Image:PROP260ART MANH Fig17.png|thumb|left|550px|The received power coverage map of the Manhattan scene with significantly increase transmitted powera rotated horizontal Yagi-Uda array at f = 2.0GHz.]]
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Overall, increasing the transmitted power lifts the coverage map by 20dB (100-fold) as you can verify from the maximum received power values in the two plots. However, the dense array of nearby buildings still block most of the coverage into the streets even though a good number of new signal paths have appeared in the latest coverage map.
The figures below show the distribution of the three X, Y and Z components of the electric field in the Manhattan scene. Note that your short dipole source is vertically polarized. The X and Y field components are a result of multipath effects.
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[[Image:PROP256ART MANH Fig18.png|thumb|380pxleft|The distribution of the X-component of electric field due to the vertically polarized dipole source.]]</td><td>[[Image:PROP257.png|thumb|380px550px|The distribution received power coverage map of the YManhattan scene with a rotated horizontal Yagi-component of electric field due to the vertically polarized dipole sourceUda array at f = 2.]]</td><td>[[Image:PROP258.png|thumb|380px|The distribution of the Z-component of electric field due to the vertically polarized dipole source5GHz.]]
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The table below shows the computation time of different stages of the SBR sweep simulation:
{| class="wikitable"
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| Simulation Step
| Computation Time
|-
| Facet Mesh Generation
| 80s
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| SBR Ray Tracing
| 160s
|-
| Frequency Computations
| 90s
|-
| Total
| 330s
|}
== References ==
[3] M. Hapala and V. Havran. Review: Kd-tree traversal algorithms for ray tracing. Computer Graphics Forum, 30(1):199213, March 2011.
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