Cellular Network Radio Propagation Modeling with Deep Convolutional Neural Networks

There are a number of commonly used empirical models, for example, Okumura-Hata model (hata80_emp, 19), Stanford University Interim model (stanford_interim_emp, 20), standard propagation model (SPM) (spm_emp, 21) and Walfisch-Ikegami model (antenna_prop4_wireless_book, 1, 2).

The above models all use a few parameters and equations to characterize typical classes of radio links such as urban, suburb and rural. The parameters are determined based on measured and averaged losses in the aforementioned typical classes of ratio links. Thus empirical models are not site-specific, i.e., they use exactly the same parameters for two different sites as long as the sites fall in the same link category.

For example, the Okumura-Hata model is valid for microwave frequencies from 150 to 1500 MHz, and is suited for both point-to-point and broadcast communications, and covers mobile station antenna heights of 1-10 m, base station antenna heights of 30-200 m, and link distances from 1-10 km. The Okumura-Hata model uses separate sub-models for the urban, suburban and rural environments. The urban model is formulated as follows.

For small or medium-sized city,

For large cities and carrier frequency in $150\text{MHz}\leq f\leq 200\text{MHz}$,

And for large cities and carrier frequency in $200\text{MHz}\leq f\leq 1500\text{MHz}$,

$L_{U}$ is the path loss in urban areas in decibel (dB). $h_{B}$ is the height of base station antenna in meter. $h_{M}$ is the height of mobile station antenna in meter. $C_{H}$ is the antenna height correction factor. $f$ is the carrier frequency in Megahertz. $d$ is the distance between the base and mobile stations in kilometer.

The models for suburban and rural areas are just slightly modified version of the above model and are omitted here.

SPM, derived from Okumura-Hata model with a few more parameters, is another commonly used model in industry, and we will compare our proposed method with SPM in Section 4.

As we can see, the disadvantage of empirical models is that they use very limited input information and parameters to coarsely characterize radio links, so the prediction can be very inaccurate. In our experience, the root-mean-square error (RMSE) between the un-calibrated prediction and ground truth is typical around $12$-$15$dB.

Ray tracing techniques based on geometrical optics (ikegami91_ray, 3, 4) produce the most widely used deterministic path loss models. Reflection, scattering and diffraction are modeled and calculated using ray tracing, and the effects are combined as shown in Figure 1.

The input to ray tracing models contains much more information about a specific site than that of empirical models. GIS maps, often containing building layers (specifying heights, shapes and locations of buildings), terrain layers (altitude of ground) and clutter layers (categorized land use), are typically used as input to ray tracing. Examples of these layers can be found in Figure 2, where the values in each layer are typically normalized.

Antenna parameters such as location, height, azimuth, tilt and radiation pattern are also required in ray tracing.

Ray tracing can calculate deterministic prediction of path loss at any specified points. With site-specific and rich input information, it is usually more accurate than empirical models. However, many factors affecting radio propagation such as the material of building enclosure and density of vegetation are not available in typical GIS maps, resulting in prediction error (measured in RMSE) close to 10dB. Thus, to perform well for a specific site in practice, ray tracing models require calibration which is essentially over-fitting a few parameters to measurements from the same site using expensive drive tests. Moreover, in a new site planning task, these measurements are outright not available.

Ray tracing based radio propagation models are very complex and are implemented as commercial softwares by several companies implementing their proprietary technologies. They are usually very computationally intensive and are often too slow for cellular network planning, which is essentially an optimization process with thousands of steps and each step recalculates the path loss of the whole area under consideration.

Another disadvantage of ray tracing models is that unconventional input information such as satellite photos of areas cannot be easily incorporated into the models. These new sources contain even richer information than traditional GIS map layers.

We observe that the inputs to ray-tracing based propagation models include GIS map layers, which are essentially numerical matrices and can be considered as the channels of an image, as shown in Figure 2. Each pixel of the image here is corresponding to a square area patch of $m$ by $m$ meters in the physical world, where $m$ is the resolution of GIS map. Typical resolutions of an GIS map are 5 meters, 10 meters and 20 meters.

Ray-tracing based models also utilize engineering parameters such as antenna location (latitude, longitude and height), orientation and radiation pattern as input information. These parameters are not in matrix format and cannot be directly incorporated into input image tensors. There are two possible ways to include these inputs into machine learning frameworks, e.g., neural network architectures. We can keep the original format of these engineering parameters and use as scalar inputs to a neural network in parallel to the image tensor constructed with GIS map layers. By this design we are asking the neural network to learn some underlying mechanisms on these parameters which we already know with simple physics. We think a better approach is to translate these parameters into additional channels of the input image tensor so that we have a unified input in image format. By these translations, we are essentially applying domain knowledge into the input tensor design, as will be demonstrated in the next subsection.

Propagation models predict path loss values at each location in an area, which can be gridded into square patches of $l$ by $l$ meters. If we assume the path loss at any point within a patch to be the same (a common practice by field engineers), the output can be formatted into a matrix of path loss values, which can also be considered as an image with each pixel corresponding to a small area patch of $l$ by $l$ meters. The intensity of a pixel is set to the path loss value at the square patch representing the pixel. Note that in practice $l$ is typically larger than or equal to $m$.

Thus the prediction process is essentially an image-to-image regression problem, i.e., mapping from one image domain to another. Instead of using ray tracing techniques to calculate the mapping, we propose to use an alternative machine learning and data driven approach to learn the mapping. More specifically we proposed to use a deep convolutional neural network (ConvNet) as the propagation model and use simulated and real field data to train the ConvNet. Details of the network architecture and training methodology will be presented later in this section.

One key aspect to use a ConvNet as propagation model is the design of input tensor, such that all necessary information is organized in a format to be utilized effectively by the neural network.

We design the input to be a 3-dimensional tensor, with the shape of $C\times W\times H$. $C$ is the number of channels defined below. $W$ and $H$ are the width and height (in pixels) of the target area and they are design parameters. For example, if the GIS map has a resolution of $m=5$ meters and we would like to predict the path loss within an area of 1000 by 1000 meters, the width and height of the patch are both 200 pixels.

As mentioned previously, layers (building, terrain and clutter) of GIS maps are naturally image channels, while engineering parameters have to be translated into extra channels of the input image tensor.

Instead of using extra channels to encode the latitude and longitude of the antenna, we propose to use the 2-D location of the antenna as the center to cut out a map patch (with all three layers) from the whole map. Subsequent channels also use the same center.

In practice the antenna can be either installed on a small tower on top of a building, or hang on the side. We encode the antenna height information in a channel of the input tensor where each pixel has the same value representing the antenna height in meters.

With the above intuition, we design the input tensor with 8 channels and summarize each channel as follows:

• •

  Clutter channel contains the clutter layer of a GIS map, which categorizes each pixel into one of the surface feature types, such as sea, forest, urban, river, lake, village and other ground cover information. It is a categorical value and usually numericalized as integers from 1 to 21.

• •

  Building channel contains the building layer of a GIS map, which describes the general outline and height information of buildings. If there is building on a pixel, the value of the pixel is the height of building at the pixel. If there is no building on the pixel, the value is simply zero.

• •

  Terrain channel contains the terrain layer of a GIS map, describing the altitude of each pixel, which is normalized by subtracting the lowest point in the map patch.

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  Azimuth channel indicates the horizontal angle difference between the line of sight from each pixel to the antenna and direction of main lobe.

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  Tilt channel indicates the vertical angle difference between the line of sight from each pixel to the antenna and direction of main lobe.

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  Antenna height channel is designed to encode the height of antenna simply with the height in meters at each pixel of the map patch.

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  Frequency channel describes the frequency of carrier signal. Signals with different carrier frequencies usually have different attenuation, hence different path losses. We simply copy the frequency value of a sample (the carrier frequency of the base station) to each pixel.

• •

  Antenna Gain channel describes the antenna radiation pattern, i.e., how the radio energy is distributed in the space around antenna. We assume the 3-D radiation pattern is intercepted by the terrain surface below the antenna, and the gain at each pixel on the surface is filled in this channel of the tensor.

Examples of the channels are shown in Figure 2. Note that in this framework, the input tensor can be easily modified when new information is available. For example, satellite photos of an area can be incorporated as an additional channel for the input tensor.

Referring to the conventional propagation models, we observe that the path loss at a specific location is determined by both the macro condition of the environment such as the overall density of buildings in the area, and the micro condition close to the receiver such as the number of walls nearby and their specific locations.

To capture the above characteristics, a good neural network architecture for path loss prediction should assemble different levels of abstraction inside a neural network to form predictions. We propose to use a U-Net (Ronneberger15_cnn, 18) type of architecture, which is also known as encoder-decoder with skip connections. These skip connections are critical and serve as the combinations of various abstraction levels. The detailed architecture is shown in Figure 3.

In all the convolution and de-convolution layers, we use 3x3 kernel size and set stride to 2. Padding is applied such that feature map size is only altered by striding. Note that in Figure 3 the input tensor was designed to have the same width and height as the output tensor (path loss matrix), i.e., the input map layers has the same resolution as the path loss prediction. If they have different resolution, it is easy to modify the architecture to up-sample/down-sample the path loss matrix in the width and height dimensions. We dub our neural network PLNet for easy reference later in this article.

The most straight forward way to train PLNet is supervised training. The input tensors are based on engineering parameters of base stations and GIS maps, constructed according to Section 3.2. Corresponding labeled path loss matrices are collected from field measurements or alternatively from commercial path loss simulators. Details about data generation will be explained in Section 4. With the labeled path loss matrices (supervisors) for all the training samples, we can use mean absolute error (MAE) or mean square error (MSE) as the loss function for training. MAE is defined as follows and MSE can be defined similarly,

where $\mathbf{\Theta}$ is the weights of PLNet and $\mathbf{X}=\{\mathbf{X}_{i}\}_{i=1}^{N}$ are the input tensors as described in Section 3.2. $\mathbf{Y}=\{\mathbf{Y}_{i}\}_{i=1}^{N}$ and $\hat{\mathbf{Y}}=\{\hat{\mathbf{Y}}_{i}\}_{i=1}^{N}$ are the ground-truth and predicted path loss matrices respectively. $N$ is the number of samples, and $W$ and $H$ are the width and height of the path loss matrix. $\mathbf{Y}_{i}(j,k)$ refers to path loss value at the $j^{th}$ row and $k^{th}$ column of the ground truth path loss matrix of the $i^{th}$ sample.

Note that in field collected data not all pixels have valid measurements, because it is not realistic to ask road testing equipment to traverse every single pixel in the area. Actually it is not uncommon that only 5-10% of the pixels in a path loss matrix have measured values. Pixels without valid measurements should be excluded from the calculation of loss function, hence will not affect the gradient back propagation. Just clamping the error from these pixels at zero will effectively prevent them from interfering the training process.

First we clarify that the results from this subsection, i.e., results using simulated data (generated from ray tracing based models), are to demonstrate that PLNet has the capacity to capture meaningful patterns from the input tensor by ”learning from ray tracing”. In real application, PLNet will compete with ray tracing based models, and we show in the next subsection that PLNet trained by real field data outperformed ray tracing significantly.

We use the simulated data from 36 cities (728,574 samples) to train PLNet, and data from the other 4 cities (96,724 samples) to test the performance of PLNet.

It takes about 80 epochs for the training to converge, and we achieve test performance of $\text{RMSE}=7.48\text{dB}$ using data from the 4 test cities.

Samples of predicted and ground truth (calculated by Volcano in this case) path loss matrices are shown in Figure 4. The predicted path loss matrix captures the overall patterns of the ground truth, but it is a little blurred as a result of using MSE/MAE as loss function.

We also visualize the convolutional filters of the first layer in Figure 5. The original PLNet has $3\times 3$ convolution filters for each channel and it is hard to visualize such small filters, so we increase the filter size of the first layer filters to $11\times 11$ just for visualization purpose. We can see that the filters for building and terrain channels are indeed trained to discover low level patterns of the corresponding map layers.

There are two interesting observations. First, the filter patterns of the terrain channel is wider and smoother than those of the building channel. This is because buildings have much sharper edge in map compared to smooth changing terrains. Second, the filter patterns are similar to the ConvNets trained for image recognition purpose, with the exception of some patterns having distinctive highlighted pixels. When these filters convolve with the input building or terrain channels, they search for higher buildings or terrains near the current location. This is because protruding objects can reflect electromagnetic waves and affect radio propagation.

From these results, we can see that the lower layers of PLNet are trained to capture patterns of input map, even though it is trained with simulated data generated by raytracing models. Thus it can be a good pre-training in the case where real field data is limited.

Field collected data from 48 cities world-wide (319,350 samples) are utilized to train our PLNet, and data collected from another 4 cities (8,316 samples) are used to test the performance of PLNet against empirical and ray-tracing based methods. The results are shown in Table 1, where PLNet ST stands for PLNet by supervised training, and SPM is a widely applied empirical model in industry. We can see the proposed PLNet out-perform the conventional methods by a large margin in each test city.

We also plot samples of predicted path loss matrices by PLNet and conventional methods in Figure 6. Note that only the center area of the ground truth path loss matrix has valid values. PLNet predictions are more consistent with ground truth and are less dramatic in the surrounding area.

Because of the large errors, both empirical and ray-tracing based methods recommend to “calibrate” their models before actual prediction. Basically calibration is overfitting the conventional models to certain region of consideration, for example a city. It is worth pointing out that in many cases data required by calibration are not available, for example, in new construction of cellular networks, where no base station has been constructed yet. In other cases, data required by calibration has to be recorded through expensive road test. So PLNet has considerable advantage here because it can perform well without calibration.

But for the sake of completeness, we present the performance comparison after calibration. A map of City D is shown in Figure 7, where the calibration data is collected from the roads in blue color, and the data collected from roads in red color are used for test after the calibration.

Both SPM and Volcano use calibration data to fine-tune a few parameters designed for calibration (volcano_ray, 22). For PLNet, we use the calibration data to fine tune the neural network for a few epochs. The RMSE for SPM and Volcano are 9.50dB and 8.53dB respectively, improved from the un-calibrated performance. PLNet still has the best accuracy with $\text{RMSE}=7.81\text{dB}$.

PLNet also has significant computation speed advantage over ray tracing based method. To predict the path loss matrix of a single cell with $320\times 320$ pixels, Volcano needs 30 seconds (8-core Intel i7 CPU), while PLNet only needs 0.33 seconds (implemented in TensorFlow with one Nvidia Titan Xp GPU). Of course PLNet benefits from GPU computation, but at this time we are not aware of any ray tracing based path loss prediction software taking advantage of GPU acceleration.