1 Introduction↩︎

As a promising paradigm shift from conventional environment-unaware to environment-aware communication and sensing, channel knowledge map (CKM) has been recently proposed to address the challenge of channel knowledge acquisition with the prior local environment [1], [2]. As a location-specific channel knowledge database, CKM can effectively reflect the channel knowledge related to the node location and dependent on the local environment, such as channel gain, time of arrival (ToA), angle of arrival (AoA), angle of departure (DoA), etc [3]. Different from the physical environment map [4], CKM focuses on the intrinsic characteristics of wireless channels, which effectively avoids the complicated computation from physical environment to channel knowledge. Therefore, the prior local environment embedded in CKMs can greatly facilitate the performance optimization and resource allocation of future wireless communications.

Efficient construction of CKM is the key to realizing CKM-based environment-aware communication and sensing. Essentially, the construction of CKM is the process of combining limited data with prediction methods such as interpolation and inference and obtaining the location-specific channel knowledge in the region [2]. CKM construction based on existing channel knowledge can be categorized into same-AP construction and cross-AP construction. For the same AP, the CKM can be completed or the CKM resolution can be improved by using the physical environment map [5] or the channel knowledge of the nearest neighbor nodes [6]. For the cross-AP construction, the mutual information proves the existence of channel state information (CSI) dependence across APs, and CSI features such as received power at a specific location are inferred from the source CSI [7]. Learning-based channel mapping is also used for cross-antenna channel prediction at specific candidate locations [8].

Different from the CSI inference for a specific location across APs/antennas above, this letter focuses on the fundamental problem that constructing the complete CKMs of potentially new APs efficiently based on the CKMs of existing APs. Specifically, consider a dense network, such as a cell-free network, in which the existing APs are equipped with CKMs for the region. A potentially new AP is introduced into the network with only its location information known. A cross-AP CKM generation system needs to be designed whose inputs are the location information of the potentially new AP with the CKMs of other APs in the network, while the output is the complete CKM inference of the new AP. With the densification of network nodes, the generation of the cross-AP inferred CKMs of potentially new APs effectively expands the system potential. The overhead of constructing CKMs for all APs can be reduced during the initial CKM construction phase through a combination of measurement and cross-AP inference strategies. For newly introduced potential APs, traversing to generate CKMs in different locations can guide the environment-aware placement. CKM inference across APs also realizes cost-effective CKM updates during subsequent system maintenance.

The cross-AP CKM inference is built on the location diversity of distributed APs and the same physical environment they share. Specifically, the wireless environment is an outward manifestation of the physical environment, which is also embedded in the spatial variations of the wireless environment [2]. Therefore, there is a natural correlation and dependence between the CKMs of distributed APs at different locations. Although this implicit relationship is difficult to characterize concretely, it can be applied to cross-AP CKM inference with learning-based approaches. In this paper, neural network is used to learn the implicit correlation between CKMs of different APs related to the wireless environment, and ultimately to realize cross-AP CKM inference. As shown in Fig. 1, this learning-based cross-AP CKM inference avoids the complex computation from physical environment to channel knowledge. The UNet model [9] is first trained with CKM datasets from different physical environments. After training, the locations of the potentially new APs are fed into the model along with the CKMs of other existing APs to output the CKMs of the potentially new APs. This cross-AP CKM inference learns the correlation between CKMs and is capable of generating complete CKMs of potentially new APs with high accuracy. The generated CKM results are compared with other benchmarks to validate the feasibility and effectiveness of CKM inference across APs.

2 Problem Formulation↩︎

As shown in Fig. 1, consider a cell-free network with \(N\) APs.The coordinate of the \(n\)th AP is denoted as \(\mathbf{c}_n \in \mathbb{R}^{2\times 1}\). The problem to be solved is: how to construct a CKM for a potentially new AP at location \(\mathbf{c}_0\) based on the CKMs of existing APs in the cell-free network? Specifically, the problem is first analyzed with a typical kind of CKM, channel gain map (CGM), as an example. The entire network area is divided into \(W \times W\) grids, where each grid records a channel gain value. Therefore, for the \(n\)th AP, its CKM, which mainly stores the location of the AP itself and the channel gain corresponding to each grid, can be expressed as \[\label{eq:CKMstore3232} \mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},\tag{1}\] where \(\mathbf{G}_n\in \mathbb{R}^{W\times W}\) denotes the channel gain of the \(n\)th AP.

Consider a potentially new AP 0 for which only its location information \(\mathbf{c}_0\) is known. The focus of this paper is on how to reduce or even avoid actual measurements to generate the CKM for the AP 0. Based on the correlation on the physical environment, CKMs of other APs within the network area are naturally a source of information that can be mined. Therefore, the CKM construction problem for AP 0 can be formulated as a cross-AP inference problem from CKMs of other existing distributed APs to the CKM of AP 0 as \[\label{eq:mapping} f: \big\{ \mathcal{M}_n \big\}_{n=1}^{N};\mathbf{c}_0\rightarrow \mathbf{G}_0 .\tag{2}\]

The training and inference phases of cross-AP CKM inference are illustrated in Fig 2. In the training phase, UNet learns the propagation characteristics in the wireless environment by performing supervised learning with the help of the channel knowledge database and optimizing the UNet parameters. In the inference phase, the coordinate \(\mathbf{c}_0\) of the introduced AP 0 and the CKMs of other APs in the area are combined and input into the trained UNet model, and the final inferred CKM \(\mathbf{G}_0\) is output.

The proposed CKM inference method across APs is based on the fact that the physical environment shared by the APs determines the wireless channel conditions. We believe that the cross-AP CKM inference has great potential in cell-free networks. In the initial phase of the network deployment, the cross-AP CKM inference can effectively reduce the overhead of CKM construction. Specifically, CKMs are mapped for only part of the APs through actual measurements or ray tracing, while CKMs for the remaining APs are directly inferred to effectively reduce the overall construction overhead. Meanwhile, in a cell-free network already equipped with CKMs, cross-AP CKM inference can effectively guide the environment-aware deployment of potentially new APs. By inferring the corresponding CKMs, the coverage of the potentially new AP in different locations can be effectively modeled, so that the locations where the wireless environment best meets the requirements can be selected for AP deployment.

3 Model Training↩︎

In this section, the construction and training of the UNet for cross-AP CKM inference is presented. Specifically, CKMs from distributed APs within the cell-free network are first transformed and combined to generate input data for UNet training. Then, the design of the UNet architecture for cross-AP CKM inference is presented.

3.1 Input Data Generation↩︎

The physical and wireless environments interact with each other. As an example, abrupt variations in channel strength in space often imply abrupt variations in the wireless environment, which can be further inferred from variations in the physical environment due to obstacles, buildings, etc. Therefore, effective cognition of the wireless environment can be obtained by synthesizing the CKMs of distributed APs in cell-free networks. Although this cognition of the wireless environment is difficult to express directly in a concrete mathematical form, it can be learned through neural networks. Specifically, the AP location stored in each CKM is first converted into a \(W\times W\) AP location map through one-hot encoding, i.e., for any \(n\)th AP, there is \[\mathbf{M}_{\rm{AP},n}(\mathbf{c})=\left \{ \begin{align} 1, \quad \mathbf{c}=\mathbf{c}_n \\ 0, \quad \mathbf{c}\neq \mathbf{c}_n, \end{align}\right .\] where \(\mathbf{c}\) is any poss coordinate in the CKM.

Combining AP location maps with their corresponding CGMs can be a good way to help UNet learn wireless environment characteristics. In general, the grid of the AP tends to have the maximum value of channel gain in \(\mathbf{G}_n\). To highlight the AP location feature and strengthen the influence of AP location on CKM, the AP location map \(\mathbf{M}_{\rm{AP},n}\) is considered to be weighted with the corresponding channel gain matrix \(\mathbf{G}_n\) as the feature map of the \(n\)th AP \[\mathbf{M}_{n}=(1-\omega)\mathbf{G}_n +\omega \mathbf{M}_{\mathrm{AP},n},\] where increasing the weight \(\omega\) strengthens the corresponding AP location feature but weakens the details of the CKM.

Further, combining \(\mathbf{M}_{n}\) of all the \(N\) distributed APs in the cell-free network results in a feature map of \(N\)-dimensional channels, where the feature map of each AP corresponds to one dimension. Name the AP of which the CKM needs to be generated as the target AP. The next part is to combine the target AP location map with the \(N\)-dimensional feature maps of the other APs. Notice that the AP location information stored in map \(\mathbf{M}_{\rm{AP},{\rm target}}\) is sparse. To make the convolutional kernel fully capture the key AP location features and avoid feature omissions during the learning process, pre-convolution is first needed to reinforce the features of the \(\mathbf{M}_{\rm{AP},{\rm target}}\). By convolving with the kernel \(\mathbf{v}=\mathbf{1}\in \mathbb{R}^{3\times 3}\), all elements in the surrounding \(3\times 3\) grids of the AP location \(\mathbf{c}_{\rm{target}}\) are set to 1, effectively expanding the coverage of location information. The pre-convolution kernel can improve the extraction of AP location features, and make its performance more stable in the face of sparse location data. The target AP location map after pre-convolution is \[\mathbf{M}^{*}_{\rm{AP},{\rm target}}=\mathbf{M}_{\rm{AP},target}\ast \mathbf{v}.\]

Finally, the target AP location map \(\mathbf{M}^{*}_{\mathrm{AP},{\rm target}}\) is overlaid to generate the input data \(\mathbf{M}_{\mathrm{input}}\in \mathbb{R}^{W^2 \times (n+1)}\) of the UNet network. The generation process and structure of \(\mathbf{M}_{\mathrm{input}}\) is shown in Fig. 3.

3.2 UNet Design and Training↩︎

The structure of the UNet network for cross-AP inference is designed based on the CGMs and their corresponding AP location maps in RadioMapSeer [10]. For other datasets like CKMImageNet [11], parameters such as the number of UNet input channels need to be adjusted according to the characteristics of the dataset.

The details of the UNet structure are shown in Fig. 5. Note that for each physical environment map in RadioMapSeer, there are 80 AP location maps and corresponding simulated CGMs. Therefore, the input data of the UNet for cross-AP inference is a high-dimensional matrix with 80 channels. According to the process of input data generation, the first channel dimension consists of the target AP location map \(\mathbf{M}^{*}_{\rm{AP},target}\) after mask operation. All the other channels are feature maps composed of the weighted sums of CGMs and AP location maps of other APs in the same physical environment map. As a supervised learning, the output data in the UNet architecture is the inferred CGM \(\mathbf{G}^{\rm{infer}}_{\rm{target}}\) of the target AP with one channel.

The network architecture has the symmetry of the classical UNet structure and can be divided into a downsampling part and an upsampling part. To enhance the learning of wireless environment features at the edges of the building, the \(5\times5\) convolution kernel is used extensively, which helps to capture the local spatial information. For cross-AP CKM inference, the high-channel input data can lead to parameter redundancy. Dimensionality-reduction convolutions effectively reduce the parameter redundancy, significantly lowering computational overhead and improving model generalization. Further, in the process of down-sampling and up-sampling, the network extracts and recovers increasing high-level features layer by layer, while retaining low-level features with hop concatenation. Meanwhile, additional convolutions are added at the resolutions of \(64\times64\) and \(32\times32\), which helps to enrich and strengthen the local feature representation and the CKM property characterization.

During the training process, each AP randomly takes turns to play the role of target AP, while the remaining APs act as the other existing APs with CKMs. The input data \(\mathbf{M}_{\mathrm{input}}\) is generated and fed into the UNet and outputs the corresponding inferred CKM \(\mathbf{G}^{\rm{infer}}_{\rm{target}}\). The corresponding CGM \(\mathbf{G}_{\rm{target}}\) of the target AP is the ground-truth. The mean square error (MSE) between \(\mathbf{G}_{\rm{target}}\) and \(\mathbf{G}^{\rm{infer}}_{\rm{target}}\) is used as the loss function as \[e=\frac{1}{W^2} \sum_{i=1}^{W^2}( \mathbf{G}_{\rm{target}}(i)-\mathbf{G}^{\rm{infer}}_{\rm{target}}(i) )^2.\] By updating the parameters according to the loss, the trained UNet model can be finally obtained. The detailed process of the training phase is shown in Algorithm 4.

4 Inference Results↩︎

In this section, the UNet network is trained based on RadioMapSeer Dataset. The trained UNet network will perform cross-AP CKM inference for validation. The inference results for CKM will be compared with the benchmark schemes to demonstrate the feasibility of generating inferred CKM across APs in cell-free networks without physical environment.

4.1 Training Settings↩︎

To ensure the independence between the training dataset and the validation dataset, 500 of the 700 different physical environment maps in RadioMapSeer are arbitrarily selected for model training. The training dataset consists of 80 AP location maps and their CGMs corresponding to each physical environment map and does not contain the physical environment maps themselves. During the training process, each AP of each physical map acts as the target AP in turn and uses the corresponding target CGM as the ground-truth, constituting a total of 40000 sets of input-output training data. The training is performed based on Adam, where the initial learning rate is \(10^{-3}\). The UNet training is carried out for 15 epochs, where the batch size is 15. To mitigate overfitting, the model that minimized the MSE loss in the validation set over the 15 epochs is saved.

4.2 Training Results↩︎

After training, CKM inference across APs was performed on the validation dataset with the UNet model. The 100 physical environment maps in RadioMapSeer Dataset that are disjoint from the training dataset are selected as the validation dataset. Similarly, the validation dataset is composed of 80 AP location maps with CGMs for each physical environment map. Two basic CKM construction methods are chosen as benchmarks.

• Weighted cross-AP CKM inference scheme. The inferred CKM is represented as a weighted sum of the CKMs of the other \(N\) APs.

  \[\mathbf{G}^{\rm{infer}}_0=\sum_{n=1}^{N} w_{n}\mathbf{G}_n =\sum_{n=1}^{N} \frac{e^{-\beta d_{t,n}}}{\sum_{i=1}^{N} e^{-\beta d_{t,i}} }\mathbf{G}_n,\] where \(w_{n}\) denotes the weight based on the Euclidean distance, \(\beta=0.1\) denotes the weight parameter. The distance between the target AP and the AP \(i\) is \(d_{t,i}\).

• Path loss model in urban microcell proposed by 3GPP TR 38.901 [12].

The MSEs and root MSEs (RMSEs) of different cross-AP CKM inference schemes are shown in Table I. Note that the range of the channel gain from the noise floor to the maximum in RadioMapSeer Dataset is 100dB, the unit of the RMSE is dB. As shown in Table I, the accuracy of the learning-based cross-AP CKM inference is about 2.38 dB. This mean error is on the same level as that of RadioUNet(2.03dB), but without the need for the physical environment map as training data. Compared with the benchmark schemes, the accuracy of the proposed cross-AP CKM inference is 3dB higher than the distance-based weighted inference, and 33dB higher than the model-based inference. The comparisons of the cross-AP CKM inference and the benchmark schemes are presented in Fig. 6. The inference is performed in different physical environments ranging from simple to complex. The distance-based weighted inference blurs key features of the CKM such as the AP location. In contrast, a comparison with the CKM ground-truth reveals that the proposed cross-AP CKM inference well preserves the target AP location and learns the wireless environment features and wireless propagation characteristics. Even unaware of the physical environment map, the CKM inference still exhibits the attenuation and mutation characteristics under the occlusion of buildings.

5 Conclusion↩︎

This paper proposes a cross-AP CKM inference in cell-free networks. By taking advantage of the correlation of the wireless environment and the shared physical environment among APs, the trained UNet utilizes other existing APs’ CKMs to generate CKMs of potentially new APs without the need for physical environment maps or any onsite measurement. The comparisons with the CKM inference by benchmark schemes validate the feasibility and effectiveness of cross-AP CKM inference, which is significant for the construction and updating of CKMs as well as the environment-aware deployment of potentially new APs in dense networks.

References↩︎