Changes

In a free-space line-of-sight (LOS) communication system, the signal propagates directly from the transmitter to the receiver without encountering any obstacles (scatterers). Electromagnetic waves propagate in the form of spherical waves with a functional dependence of e^j(?^t-k₀R)/R, where R is the distance between the transmitter and receiver, ? = 2pf, f is the signal frequency, k₀ = ?/c = 2p/?₀, c is the speed of light, and ?₀ is the free-space wavelength at the operational frequency. By the time the signal arrives at the location of the receiver, it undergoes two changes. It is attenuated and its power drops by a factor of 1/R², and additionally, it experiences a phase shift of 2pR/?₀, which is equivalent to a time delay of R/c. The signal attenuation from the transmitter to the receiver is usually quantified by '''Path Loss''' defined as the ratio of the received signal power (P_R) to the transmitted signal power (P_T). Assuming isotropic transmitting and receiving radiators (i.e. radiating uniformly in all directions), the Path Loss in a free-space line-of-sight communication system is given by Friis’ formula:

The above formula assumes that the receiving antenna is polarization-matched. Normally, there is a polarization mismatch between the transmitting and receiving antennas. In the case of directional transmitting and receiving antennas, Friis’ formula takes the following form:

where '''u_T''' and '''u_R''' are the unit polarization vectors of the transmitting and receiving antennas, and G_T and G_R are their gains, respectively.

=== Multipath Propagation Channel ===

Link budget analysis for a multipath channel is a challenging task due to the large size of the computational domains involved. Typical propagation scenes usually involve length scales on the order of thousands of wavelengths. To calculate the path loss between the transmitter and receiver, one must solve Maxwell's equations in an extremely large space. Full-wave numerical techniques like the Finite Difference Time Domain (FDTD) method, which require a fine discretization of the computational domain, are therefore impractical for solving large-scale propagation problems. The practical solution is to use asymptotic techniques such as SBR, which utilize analytical techniques over large distances rather than a brute force discretization of the entire computational domain. Such asymptotic techniques, of course, have to compromise modeling accuracy for practical computation feasibility.

Figure 1: A multipath propagation scene showing all the rays arriving at a particular receiver.

The incident, reflected and transmitted rays are each characterized by a triplet of unit vectors:

The Incident, Reflected and Transmitted Rays at the Interface Between Two Dielectric Media

The reflected ray is assumed to originate from a virtual image source point. The three triplets constitute three orthonormal basis systems. Below, it is assumed that the two dielectric media have permittivities ε₁ and ε₂, and permeabilities μ₁ and μ₂, respectively. A lossy medium with a conductivity σ can be modeled by a complex permittivity ε_r = ε'_r –jσ/ε₀. Assuming '''n''' to be the unit normal to the interface plane between the two media, and Z₀ = 120Ω , the incident polarization vectors as well as all the reflected and transmitted vectors are found as:

The reflected unit vectors are found as:

The transmitted unit vectors are found as:

where

The reflection coefficients at the interface are calculated for the two parallel and perpendicular polarizations as:

=== Penetration Through Thin Walls Or Surfaces ===

In "Thin Wall Approximation", we assume that an incident ray gives rise to two rays, one is reflected at the specular point, and the other is transmitted almost in the same direction as the incident ray. The reflected ray is assumed to originate from a virtual image source point. Similar to the case of reflection and transmission at the interface between two dielectric media, here too we have three triplets of unit vectors, which all form orthonormal basis systems.

The Incident and Transmitted Rays through a Thin Wall

The transmission coefficients are calculated for the two parallel and perpendicular polarizations as:

where

=== Wedge Diffraction From Edges ===

For the purpose of calculation of diffraction from building edges, we define a "Wedge" as having two faces, the 0-face and the ''n''-face. The wedge angle is a = (2-''n'')p, where the parameter ''n'' is required for the calculation of diffraction coefficients. All the diffracted rays lie on a cone with its vertex at the diffraction point and a wedge angle equal to the angle of incidence in the opposite direction. A diffracted ray is assumed to originate from a virtual image source point. Three triplets of unit vectors are defined as follows:

The three triplets constitute three orthonormal basis systems. The propagation vector '''k'''' of the diffracted ray has to be constructed based on the diffraction cone as follows:

where the resolution of the angle θ_w is chosen to be the same as the resolution of the incident ray.

The Incident Ray and Diffract Ray Cone at the Edge of a Building

The other unit vectors for the incident and diffracted rays are found as:

The diffraction coefficients are calculated in the following way:

where ''F(x)'' is the Fresnel Transition function:

In the above equations, we have

where ''N^±'' are the integers which most closely satisfy the equations 2''n''π''N^±'' - ν = ±π.

The SBR simulation engine requires a finite computational domain. All the stray rays that hit the boundaries of this finite domain are terminated during the simulation process. Such rays exit the computational domain and travel to the infinity, with no chance of ever reaching any receiver in the scene. When you define a propagation scene with various elements like buildings, walls, terrain, etc., a dynamic domain is automatically established and displayed as a wireframe box with green lines that surrounds the entire scene. Every time you create a new object, the domain is automatically adjusted and extended to enclose all the objects in the scene. You can change the size and color of the domain box through the Ray Domain Settings Dialog, which can be accessed in one of the following three ways:

# Select the '''Simulate''' > '''Computational Domain''' > '''Settings...''' item of the Simulate Menu.

# Right click on the '''Ray Domain''' item of the Navigation Tree and select '''Domain Settings...'''

You can define any arbitrary surface by entering an equation of the two variables x and y as z = f(x,y). In this case, you have to select the '''Custom Function''' option in the dropdown list labeled '''Model'''. You should enter your equation as any mathematical expression in the box labeled '''Function f(x,y)'''. You can use any of EM.Cube's mathematical functions listed in the '''Function Dialog''' or combine several of them. Note that after selecting the custom function option, the height of the surface is determined by your equation, and the '''Height''' box is disabled. You can also introduce random noise and create a rough terrain. You can do this by setting a nonzero value for '''Noise''', which represent the RMS peak-to-valley amplitude of the surface roughness. The figures below show two custom terrain surfaces modeled by the equation z = (x.y)/20 defined over the range [0, 10] in both X and Y directions. Random noise has been added to both surfaces, with the noise amplitude being 0.2 and 0.5 for the left and right figures, respectively.

Figure 3: Two noisy custom terrain surfaces both defined as z = (x.y)/20: (Left) RMS noise amplitude = 0.2, (right) RMS noise amplitude = 0.5.

Another type of terrain model that the terrain generator provides is '''XY Grid Data'''. In this case, you define a rectangular XY grid with a uniform grid cell size along the X and Y directions and manually define the Z-elevation for each grid point. This is similar to the surface generator's "2D Uniform Grid" model type in CubeCAD. Based on your input to '''Range Start''', '''Range Stop''' and '''Range Step''' along X and Y, a 2D grid is set up and displayed in a table at the bottom of the terrain generator dialog. By default, all the Z-elevations are set to zero initially. You can click on each table cell and overwrite it with a new value. At the end, click the '''Create''' button of the dialog to add the new grid-based terrain object to the Navigation Tree.

A grid-based terrain object.

You can also export all the terrain objects in the project workspace as a terrain file with a '''.TRN''' file extension. You can even import a DEM terrain model from an external file and then save and export it as a native terrain (.TRN) file. To export the terrain, select '''File''' > '''Export...''' from [[Propagation Module]]'s '''File Menu'''. The standard Windows Save Dialog opens up with the default file type set to '''.TRN'''. Type in a name for your new terrain file and click the '''Save''' button to export the terrain data.

Figure 1: An imported external terrain model.

To define a directional transmitter radiator, you need to select the "User Defined" option in the "Radiator" section of the Transmitter Dialog. You can do this either at the time of creating a transmitter set, or afterwards by opening the property dialog of the transmitter set. In the "Custom Pattern Parameters", click the '''Import Pattern''' button to set the path for the radiation data file. This opens up the standard Windows Open dialog, with the default file type or extension set to ".RAD". Browse your folders to find the right data file. A radiation pattern file usually contains the value of "Total Radiated Power" in its file header. This is used by default for power calculations in the SBR simulation. However, you can check the box labeled "'''Custom Power'''" and enter a value for the transmitter power in Watts. EM.Cube can also rotate the imported radiation pattern arbitrarily. In this case, you need to specify the '''Rotation''' angles in degrees about the X-, Y- and Z-axes. Note that these rotations are performed sequentially and in order: first a rotation about the X-axis, then a rotation about the Y-axis, and finally a rotation about the Z-axis.

Figure 1: [[Propagation Module]]'s Transmitter dialog with a user defined radiator selected.

To define a new Receiver Set, go to the Observables section of the Navigation Tree, right click on the '''Receivers''' item and select '''Insert Receiver...''' A dialog opens up that contains a default name for the new Receiver Set as well as a dropdown list labeled '''Select Radiator Set'''. In this list you will see all the available base sets that you have already define in the project workspace. Select and designate the desired base set as the receiver set. Note that if the base set contains more than one point, all of them are designated as receivers. After defining a receiver set, the points change their color to the receiver color, which is yellow by default. The first element of the set is represented by a larger ball of the same color indicating that it is the selected receiver in the scene. The Receiver Set Dialog is also used to access individual receivers of the set for data visualization at the end of a simulation. At the end of an SBR simulation, the button labeled "Show Ray Data" becomes enabled. Clicking this button opens the Ray Data Dialog, where you can see a list of all the received rays at the selected receiver and their computed characteristics.

Figure 1: [[Propagation Module]]'s Receiver dialog.

=== Viewing SBR Mesh ===

Note that unlike EM.Cube's other computational modules that express the default mesh density based on the wavelength, the resolution of the SBR mesh generator is expressed in project length units. The default edge length value of 100 units might be too large for non-flat objects. You may have to use a lower value to capture the curvature of your curved structures adequately.

Figure 1: [[Propagation Module]]'s Mesh Settings dialog.

Note: You have to make sure that the resolution of your terrain, its fluctuation scale and building dimensions are all comparable. Otherwise, on a high-resolution, rapidly varying terrain, you will have buildings whose bottoms are in contact with the terrain only at a few points and parts of them hang in the air.

A Scene with Buildings and Terrain Before and After Adjusting Elevation

To better understand why there are two separate sets of points in the scene, note that a point array (CAD object) is used to create a uniformly spaced base set. The array object always preserves its grid topology as you move it around the scene. However, the transmitters or receivers associated with this point array object are elevated above the irregular terrain and no longer follow a strictly uniform grid. If you move the base set from its original position to a new location, the base points' topology will stay intact, while the associated transmitters or receivers will be redistributed above the terrain based on their new elevations.

Figure 1: Transmitters and receivers adjusted above an uneven terrain and their associated base sets.

# Visualize the coverage map and plot other data.

[[File:PROP12.png]]

If the associated radiator set is isotropic, so will be the transmitter set. By default, an isotropic transmitter has vertical polarization. You can use the '''Polarization''' radio button to select one of the two options: '''Vertical''' or '''Horizontal'''. If the associated radiator set consists of '''Short Dipole''' or '''User Defined''' radiators, it is indicated in the transmitter property dialog. In the case of a short dipole radiator, you can set a value for the dipole current in Amperes. The radiation resistance of a short dipole of length ''dl'' is given by:

The radiated power of a short dipole carrying a current I₀ is then given by:

For isotropic and user defined radiators you can set the '''Input Power''' and '''Phase''' of a transmitter set in Watts and degrees, respectively. This can be accessed from the '''Transmitter Chain''' dialog, which will be described in detail in the next section. The radiation pattern of the associated radiator set is normalized and used in conjunction with the input power value to create a weighted distribution of transmitted rays. In certain cases like hybrid simulations, you may want to use the actual values of the far field to define the transmitter power rather than a normalized radiation pattern. Note that the pattern (.RAD) file contains the value of total radiated power in its header. In this case, check the box labeled '''"Calculate Power From Radiation Pattern"'''. This is calculated directly from the complex θ and φ components of the far field data by integrating them over the entire space (4π solid angle). Note that this option is available only when the radiator is of the User Defined type. When this box is checked, the transmitter chain button is grayed out. By default, an isotropic transmitter emanates rays uniformly in all directions at the angular resolution specified by the user. A transmitter with a user defined associated radiator may represent a highly directional radiation pattern with the main beam pointing in a certain direction. You can additionally force and limit the '''Angular Extents''' of rays to a certain solid angle around the transmitter. This is especially useful and computationally efficient when the transmitter is on one side of the scene, and all the scatterers and receivers are on the other side. In this case, there is no need to generate rays in all directions. To limit the angular extents of rays, define the Start and End values for both Theta (θ) and Phi (φ) angles. The value of the angular resolution of the rays can be changed from the Run Dialog as will be discussed later.

In a regular SBR simulation, you have a transmitter and one or more arrays of receivers in your scene. At the end of the simulation, you can visualize the coverage map of the transmitter over the receiver sets. A coverage map shows the total '''Received Power''' by each of the receivers and is visualized as a color-coded intensity plot. You can visualize the coverage maps of individual receiver sets. At the end of a SBR simulation, each Received Power Coverage Map is listed under the receiver set's name in the Navigation Tree. To display a coverage map, simply click on its entry in the Navigation Tree. The coverage map plot appears in the Main Window overlaid on the scene. A legend box on the right shows the color scale and units (dB). The 3-D coverage maps are displayed as horizontal confetti above the receivers. If the receivers are packed close to each other, you will see a continuous confetti map. If the receivers are far apart, you will see individual colored squares. You can also visualize coverage maps as colored 3-D cubes. This may be useful when you set up your receivers in a vertical arrangement or the scene has a highly uneven terrain. To change the type of coverage map visualization, open the receiver set's property dialog and select the desired option for '''Coverage Map: Confetti''' or '''Cube''' in the '''"Visualization Options"''' section of the dialog.

Received power coverage map: (Left) confetti style, and (Right) cube style.

You can change the settings of the coverage map by right clicking on its entry in the Navigation Tree and selecting '''Properties...''' or by double-clicking on the legend box. In the Output Plot Settings dialog, you can choose from one of three Color Map options: '''Default''', '''Rainbow''' and '''Grayscale'''. The visualization plot uses default values for the color scale. In the section titled "Limits", you can choose the radio button labeled '''User Defined'''. Then, you have to enter new values for the '''Lower''' and '''Upper''' Limits of the plot. You can also show or hide the Legend Box or change its '''Background''' and '''Foreground''' colors by clicking the buttons provided for this purpose.

Output Plot Settings

At the end of a SBR simulation, each receiver receives a number of rays. Some receivers may not receive any rays at all. You can visualize all the rays received by a certain receiver from the active transmitter of the scene. To do this, right click the '''Receivers''' item of the Navigation Tree. From the context menu select '''Show Received Rays'''. All the rays received by the currently selected receiver of the scene are displayed in the scene. The rays are identified by labels, are ordered by their power and have different colors for better visualization. You can display the rays for only one receiver at a time. The receiver set property dialog has a list of all the individual receivers belonging to that set. To display the rays received by another receiver, you have to change the '''Selected Receiver''' in the receiver set's property dialog. If you keep the mouse focus on this dropdown list and roll your mouse scroll wheel, you can scan the selected receivers and move the rays from one receiver to the next in the list. To remove the visualized rays from the scene, right click the Receivers item of the Navigation Tree again and from the context menu select '''Hide Received Rays'''.

Visualization of received rays at the location of the selected receiver.

Note: The rays are summed up coherently at the receiver.

Analyzing a selected ray from the ray data dialog.

Besides visualizing the coverage map and received rays in the EM.CUBE's [[Propagation Module]], you can also plot the '''Path Loss''' of all the receivers belonging to a receiver set as well as the '''Power Delay Profile''' of individual receivers. To plot these data, go the '''Observables''' section of the Navigation Tree and right click on the '''Receivers''' item. From the context menu, select '''Plot Path Loss''' or '''Plot Power Delay Profile''', respectively. The path loss data between the active transmitter and all the receivers belonging to a receiver set are plotted on a Cartesian graph. The horizontal axis of this graph represents the index of the receiver. Power Delay Profile is a bar chart that plots the power of individual rays received by the currently selected receiver versus their time delay. If there is a line of sight (LOS) between a transmitter and receiver, the LOS ray will have the smallest delay and therefore will appear first in the bar chart. Sometimes you may have several rays arriving at a receiver at the same time, i.e. all with the same delay, but with different power level. These will appear as stacked bars in the chart.

=== Output Data Files ===

The angles of arrival are the θ and φ angles of a received ray measured in degrees and are referenced in the spherical coordinate systems centered at the location of the receiver. The angles of departure for a received ray are the θ and φ angles of the originating transmitter ray, measured in degrees and referenced in the spherical coordinate systems centered at the location of the active transmitter, which eventually arrives at the receiver. The total time delay is measured in nanoseconds between t = 0 nsec at the time of launch from the transmitter location till being received at the receiver location. The last four columns show the real and imaginary parts of the received electric fields with vertical and horizontal polarizations, respectively. The complex field values are normalized in a way that when their magnitude is squared, it equals the received ray power. If the active transmitter is an isotropic radiator with either a vertical or horizontal polarization, then the field components corresponding to the other polarization will have zero entries in the output data file.

A typical SBR output data file.

By default, you run a single-frequency simulation in EM.CUBE's [[Propagation Module]]. You set the operational frequency of a SBR simulation in the project's '''Frequency Dialog''', which can be accessed in a number of ways:

# Using the keyboard shortcut '''Ctrl+F'''.

# By double clicking the frequency section (box) of the '''Status Bar'''.<br />

(Left) Project's frequency dialog and (Right) the frequency settings dialog.

You can also select the '''Frequency Sweep''' option in the '''Simulation Mode''' drop-down list of the '''Run Dialog'''. Click the '''Settings...''' button on the right side of this dropdown list to open up the Frequency Settings Dialog. Based on the original values of the project center frequency and bandwidth, the '''Start Frequency''' and '''End Frequency''' have default values. You can also change the '''Number of Samples'''. Once you click the '''Run''' button, EM.CUBE performs a frequency sweep by assigning each of the frequency samples as the current operational frequency and running the SBR simulation engine at that frequency. All the simulation data at all frequency samples are saved into the output data files including &quot;SBR_results.RTOUT&quot;. After the completion of a frequency sweep simulation, as many coverage maps as the number of frequency samples are generated and added to the Navigation Tree under the Receiver Set's entry. You can click on each of the coverage maps corresponding to each of the frequency samples and visualize it in the project workspace. You can also animate the coverage maps. To do so, right click on the receiver set's name in the Navigation Tree and select '''Animation''' from the contextual menu. The coverage maps start to animate by their order on the Navigation Tree. Once the entire list is displayed sequentially, it starts all over again from the beginning of the list. During the animation, the '''Animation Controls''' dialog appears at the lower right corner of the screen. This dialog has a number of buttons for pause/resume, step forward/backward, and step to the end/start. The title of each coverage map is shown in the box labeled '''Sample''' as it is displayed in the main window. You can also change the speed of animation. The default frame duration has a value of 300 (3x100) milliseconds. To stop the animation, simply press the keyboard's '''Esc Key'''.

Multiple coverage maps on the Navigation Tree at the end of a frequency sweep and starting an animation from the contextual menu.

Animation controls dialog in the project workspace.

In EM.CUBE, all the CAD object properties as well as certain source, material and mesh parameters can be assigned as variables. Variables are defined to control and vary the values of such parameters either for editing purposes or to run parametric sweep or optimization. Variable are defined using the '''Variables Dialog''', which can be accessed in the three ways:

# Using the keyboard shortcut '''Ctrl+B'''.

The variables dialog is initially empty. To add a new variable, click the '''Add''' button to open up the '''Add Variable/Syntax Dialog'''. In this dialog you have to type in a name for the new variable and choose a type. The default type is '''Uniformly Spaced Samples'''. You also need to specify the '''Start''', '''Stop''' and '''Step''' values for the variable. In the figure below, a variable called &quot;Tx_Height&quot; is defined that varies between 2 and 10 with equal steps of 2. This means the sample set {2,4,6,8,10}. When you return to the variables dialog, the syntax of the new variable is shown as 2:10:2. The last number in this syntax is always the variable step. In this example, this variable is going to be used to control the height of the transmitter in a propagation scene.

EM.CUBE's variable dialog and the dialog for defining a new variable.

Next, you have to attach the variable to the CAD object. In this case, the CAD object is the point object that represents the transmitter's radiator. To attach a variable to a CAD object, open the object's property dialog and type in the name of the variable as the value of a property or parameter. In this case, the variable Tx_Height is going to control the Z-Coordinate of the point object. Once the value of the object parameter is replaced by the name of an already defined variable, it is updated with the current value of that variable. In the case of a variable of &quot;Uniformly Spaced Samples&quot; type, the current value is the start value. This value will be incrementally varied during a parametric sweep simulation process. Note that a variable can take a fixed value or a discrete set of values, too. You can always open the variables dialog and change the value or syntax of any variable. To make a new or modified value effective, click the '''Apply''' button of the variables dialog. You can test the values by performing a '''Dry Run''' of the selected variable. This runs an animation of the project workspace as the value of the variable changes and all the related CAD objects are updated accordingly. Note that you can attach the same variable to more than one CAD object property or to the properties of different objects. You can also define multiple values or syntaxes to the same variable. To do so, open the '''Add Variable/Syntax Dialog''', and instead of typing in a new variable name, choose an existing variable name from the '''Name''' dropdown list. This will add a new value or syntax to the existing syntax(es) of the selected variable. When you return to the variables dialog, variables with more than one value or syntax will have a dropdown list in the '''Syntax''' column. You can choose any of these values or syntaxed at any time and make the change effective by clicking the '''Apply''' button.

Replacing the value of a CAD object parameter with a variable name.

To run a parametric sweep, open the '''Run Dialog''' and select the '''Parametric Sweep''' option in the '''Simulation Mode''' drop-down list. If you have not defined any variables in the project, the box in the '''Variables''' row before the '''View''' will be red. You have to turn it into green before you can run a simulation. By clicking the '''View''' button, you can open up the variables dialog from here. Once you click the '''Run''' button, EM.CUBE performs a parametric sweep by incrementally varying the values of all the defined variables from their start to stop values at the specified steps and updating all the related CAD objects. After the completion of a parametric sweep simulation, as many coverage maps as the total number of variable samples are generated and added to the Navigation Tree under the receiver set's entry. You can click on each of the coverage maps and visualize it in the project workspace. You can also animate the coverage maps sequentially. To do so, right click on the receiver set's name in the Navigation Tree and select '''Animation''' from the contextual menu. To stop the animation, simply press the keyboard's '''Esc Key'''.

Choosing parametric sweep as the simulation mode in the run dialog. Note that one variable has been defined and EM.CUBE is ready to run the simulation.

The coverage map of the scene at the end of a parametric sweep where the sweep variable is the transmitter height.

In the [[Propagation Module]], when you ran a sweep simulation (frequency, transmitter or parametric), you also have the option to generate two additional coverage maps: one for the mean of all the individual sample coverage maps and another for their standard deviation. To do so, in the '''Run Dialog''', check the box labeled '''&quot;Create Mean and Standard Deviation Coverage Maps&quot;'''. Note that the mean and standard deviation values displayed on the individual coverage maps correspond to the spatial statistics of the receivers in the scene, while the mean and standard deviation coverage maps correspond to frequency, transmitter or variable sets defined for the sweep simulation. Also, note that both of the mean and standard deviation coverage maps have their own spatial mean and standard deviation values expressed in dB at the bottom of their legend box.

The mean coverage map at the end of a transmitter sweep.

The standard deviation coverage map at the end of a transmitter sweep.