Deep CNN based Pilot Allocation Scheme in Massive MIMO systems

Abstract

This paper introduces a pilot allocation scheme for massive MIMO systems based on deep convolutional neural network (CNN) learning. This work is an extension of a prior work on the basic deep learning framework of the pilot assignment problem, the application of which to a high-user density nature is difficult owing to the factorial increase in both input features and output layers. To solve this problem, by adopting the advantages of CNN in learning image data, we design input features that represent users' locations in all the cells as image data with a two-dimensional fixed-size matrix. Furthermore, using a sorting mechanism for applying proper rule, we construct output layers with a linear space complexity according to the number of users. We also develop a theoretical framework for the network capacity model of the massive MIMO systems and apply it to the training process. Finally, we implement the proposed deep CNN-based pilot assignment scheme using a commercial vanilla CNN, which takes into account shift invariant characteristics. Through extensive simulation, we demonstrate that the proposed work realizes about a 98% theoretical upper-bound performance and an elapsed time of 0.842 ms with low complexity in the case of a high-user-density condition.

1. Introduction

The Multi-Input Multi-Output (MIMO) technology is the most important technology in terms of network capacity in mobile communication [1,2]. The base stations (BS) for 6G technology have considered in using a number of antennas for the large capacity improvements [3,4]. This is because the use of more antennas in massive MIMO technologies than conventional MIMO technologies significantly improves the radiative energy efficiency and throughput by the elimination of intra-cell interference and concentrates power in narrow areas of space [5,6]. The Massive MIMO is generally known to be more than 100 times more efficient in radiant energy, as well as 10 times more capacity growth. In order to mitigate these negative impacts from adopting massive MIMO technology such as "contamination of pilot", which is the inter-cell interference effect [7,8], it requires the difficult task of assigning the pilots for users in mobile communication [9,10].

There are various approaches for realizing efficient pilot assignment solutions that improve the capacity of the overall systems [15]. It is challenging due to the pilot allocation problem with a large complexity, and it is difficult to formulate a mathematical model [11,12]. Firstly, Marzetta proposed a random pilot assignment scheme (R-PAS) as a basic approach that can be easily deployed in a practical system [7]. The R-PAS provides an minimum elapsed time owing to its fixed time complexity. However, the overall network capacity of the R-PAS is limited as compared to the theoretical optimal bound. This is owing to the fact that the R-PAS does not take into consideration the users’ different channel qualities. In [16], a smart pilot assignment scheme (S-PAS) was invented, which allocates the pilot sequence with the smallest inter-cell interference to the worst channel condition’s user in a sequential manner. In a multi-cell environment, all the cells are required to iteratively solve their own optimization problems until the solution is converged [13,14]. Owing to such an iterative procedure in the multi-cell environment, the S-PAS results in an increased computing delay as the number of cells increases. Here, the computing delay also depends on its convergence step size where there is a trade-off between performance and computing delay cost according to the step size. Recently, as a first approach to adopting deep learning techniques, which have recently emerged as a useful tool for solving complex and difficult problems with a large search space, in our preliminary work [17], we proposed a deep-learning-based pilot assignment scheme (DL-PAS). The DL-PAS achieves almost a 99.38% ideal performance with low complexity within a computing time of 0.92 ms on average. Even though this preliminary work suggests a novel method for adopting machine learning technology into a wireless networking problem, it is restrictively applicable only to low-density users due to the factorial increase in input features and output layers [18,19]. Thus, a performance study on high-density users was not conducted. Accordingly, the previous studies are based on heuristic approaches with sub-optimal performance or are hard to deploy in a practical system owing to their high time/space complexity [20,21]. Thus, a practical solution to the pilot assignment problem is required, which motivated the works [22].

This paper introduces a deep convolutional-neural-network-based (CNN-based) pilot assignment scheme (DC-PAS) for the massive MIMO systems. The proposed work is an extension of a prior work and adopts the advantages of CNN in learning image data [23-25]. The important contributions are summarized as follows.

• In our scheme, by applying the CNN mechanism to the pilot assignment problem, we design new input features of the users’ locations in all cells as the image data with a two-dimensional fixed-size matrix.

• Instead of using a sigmoid function, we construct output layers with linear space complexity according to the number of users. Here, in order to improve the accuracy of the proposed work, we utilize a sorting mechanism to map the pilot index into the user index in the ascending order.

• A theoretical framework for the network capacity model of massive MIMO is derived by extending the model presented in [7], and we apply it into the training process.

• We deploy the proposed DC-PAS under a commercial vanilla CNN by considering shift invariant characteristics. Using simulation, we check that the proposed work realizes about a 98 % theoretical upper-bound performance and an elapsed time of 0.842 ms with low complexity in the case of a high-userdensity condition.• Our research is not limited to this vanilla CNN. There are CNNs that have been modified recently [23, 25, 28], and it is easy to apply to reflect modified CNNs. Later, it will be the future work to apply the revised CNN to improve performance.

We can thus develop a practical learning-based pilot assignment scheme that is applicable to high-density users and suggest a method by which the deep learning mechanism could improve the network performance in the case of complex problems.

Section II presents a overview of the massive MIMO. In Section III, we formulate the problem as a deep learning model and present the novel DC-PAS. Numerical results and performance analysis are explained in Section IV. We summarize and conclude this work in Section V.

2. System Model

For the reader’s convenience, in Table 1, we present a list of the major symbols that are used in this paper with their definitions.

Table 1. List of major symbols

We consider the massive MIMO systems that consist of 𝐿𝐿 multiple cells. Each cell has a BSBS with 𝑀𝑀 omni-directional antennas, where 𝑀𝑀 is generally assumed to be very large in the massive MIMO systems such that 𝑀𝑀 → ∞. Pilot contamination occurs owing to a user assigned with the same pilot assignment from another cell. As illustrated in Fig. 1, the pilot contamination is small when assigning the same pilot to another cell’s users who are far from the cell. Within each cell, there are multiple terminals with a single antenna that conducts the uplink transmission. In order to model this in a manner similar to our preliminary work [17], we define 𝐼 as the set of cells, where |𝐼|=𝐿 denotes the total number of cells, and 𝐾_𝑖 as the set of terminals in the 𝑖-th cell for each 𝑖 ∈ 𝐼, where |𝐾_𝑖|=𝑈_𝑖 denotes the total number of terminals in the cell. For the channel estimation, a set of orthogonal pilots is assigned and transmitted to the terminals, where the total number of orthogonal pilots used in each cell is 𝑃 . Correspondingly, the BSBS can measure the channel condition for up to 𝑃 terminals, which means that the maximum number of terminals within a cell that can simultaneously transmit data is only up to 𝑃 terminals. The channel is composed of large- and small-scale propagation effects. The large-scale propagation effect comprises pathloss and shadowing. For each 𝑖,𝑗 ∈𝐼 and 𝑘 ∈ 𝐾_𝑗, we define 𝑟_𝑖,𝑘 and 𝑧_𝑖,𝑘, which denote the distance and shadowing, respectively, between the BSBS of the 𝑖𝑖-th cell and 𝑘𝑘-th terminal, which may reside in the 𝑗𝑗-th cell. Accordingly, as in [7,8], the large-scale propagation effect can be modeled using 𝛽_𝑖,𝑘 = _{\(r_{ik}^{-y}\)}, where 𝛾 is the pathloss exponent. In addition, in the case of the small-scale propagation effect, the fading coefficient between the 𝑚-th antenna of the BS in 𝑖-th cell and 𝑘-th terminal is denoted by ℎ_{𝑚,𝑖,𝑘,} where the coherence time 𝑇_𝑐 is assumed to be constant over the fading coefficient ℎ_{𝑚,𝑖,𝑘}. Finally, the overall channel gain 𝑔_{��,𝑖,𝑘} between the 𝑚-th antenna of the BS in 𝑖-th cell and 𝑘-erminal is denoted by 𝑔_{𝑚,𝑖,𝑘} = \(h_{m, i, k} \sqrt{\beta_{i, k}}\). As assumed in other literatures on massive MIMO, we assume that {𝛽_𝑖,𝑘} values are known at the BSs such that it can be learned in the network. This is because {𝛽_𝑖,𝑘} are varying much more slowly than the small-scale fading coefficients, and they are independent of frequency [8]. In contrast, the fading coefficient ℎ_{𝑚,𝑖,𝑘} should be estimated for every coherence time 𝑇_𝑐. For the estimation of ℎ_{𝑚,𝑖,𝑘}, it is assumed that a set of 𝑃 orthogonal pilot sequences is used for each cell wherein 𝑃 depends on the channel and system parameters such as the maximum delay spread, channel coherence time, etc. In addition, the orthogonality of the pilots can be implemented using a few methods: i) each pilot can be transmitted in the form of one orthogonal frequency division multiplexed (OFDM) symbol in one subcarrier or ii) each pilot can be transmitted in the form of an orthogonal sequence over the time–frequency domain. Here, the proposed work is not restricted to any specific implementation; only 𝑃 the number of pilots is important in our analysis.

Fig. 1. Importance of assigning the pilot

Moreover, as in [7], we assume that all other cells in the network use the same pilot set. We investigated the effect of using different orthogonal pilot sequences in other cells, and it was demonstrated that essentially the same performance was obtained.

The channel vector from the 𝑘-th terminal in the 𝑗-th cell to the 𝑚-th antenna of the BS in the 𝑖-th cell is given by

\(\mathrm{g}_{m, i, j, k}=\mathrm{h}_{m, i, j, k} \sqrt{\beta_{\mathrm{i}, \mathrm{j},\mathrm{k}}}=\mathrm{h}_{m, i, j, k} \sqrt{\mathrm{z}_{\mathrm{i}, \mathrm{j}, \mathrm{k}} \mathrm{r}_{\mathrm{i}, \mathrm{j}, \mathrm{k}}^{-\gamma}}\)       (1)

where ℎ_{𝑚,𝑖,𝑗,𝑘} is the small-scale fading between the 𝑚-th antenna of the BS in the 𝑖-th cell and 𝑘-th terminal in the 𝑗-th cell, 𝛽_{𝑖,𝑗,𝑘} is the large-scale fading between the 𝑖-th cell and 𝑘-th terminal in the 𝑗-th cell, 𝑧_{𝑖,𝑗,𝑘} is the shadowing between the 𝑖-th cell and 𝑘-th terminal in the 𝑗-th cell, 𝑟_{𝑖,𝑗,𝑘} is the distance between the 𝑖-th cell and 𝑘-th terminal in the 𝑗-th cell, and 𝛾 is the pathloss exponent.

The uplink SIR of the 𝑘-th terminal in the 𝑖𝑖-th cell is obtained as

\(\operatorname{SIR}_{i, k}^{u}=\frac{\left|g_{m, i, i, k}^{H} g_{m, i, i, k}\right|^{2}}{\sum_{\mathrm{j} \neq \mathrm{i}}\left|g_{m, i, j, k}^{H} g_{m, i, j, k}\right|^{2}+\frac{\left|\epsilon_{i, k}^{u}\right|^{2}}{\rho u}}\)

With 𝑀 → ∞, we obtain

       \(\underline{M \rightarrow \infty} \frac{\beta_{i, i, k}^{2}}{\sum_{j \neq i} \beta_{i, j, k}^{2}}\)       (2)

In the uplink, the capacity of the 𝑘𝑘-th terminal in the 𝑖𝑖-th cell is given by

\(C_{i, k}^{u}=B_{w}\left(\frac{1}{\alpha}\right)\left(\frac{T_{\text {block }}-T_{\text {pilot }}}{T_{\text {block }}}\right)\left(\frac{T_{u}}{T_{s}}\right) \log _{2}\left(1+\operatorname{SIR}_{i, k}^{u}\right)\left(\frac{\frac{b p s}{t e r m i n a l}}{c e l l}\right)\)        (3)

where 𝐵_𝑤 is the bandwidth, 𝛼 is the frequency reuse factor, 𝑇_b𝑙ock is the time for the OFDM symbols of one frame, 𝑇_p𝑖iot is the time for sending reverse pilots, 𝑇_𝑢 is the data transmission interval, and 𝑇_𝑠 is an OFDM symbol interval.

The sum capacity of 𝑖𝑖-th cell is given by

\(C_{i}^{u}=\sum_{k=1}^{U_{i}} C_{i, k}^{u} \quad\left(\frac{b p s}{c e l l}\right)\)       (4)

The total capacity of the entire network comprising 𝐿 cells is given by

\(C_{\text {network }}^{u}=\sum_{i=1}^{L} C_{i}^{u} \quad(b p s)\)       (5)

For achieving maximum capacity of the entire systems, the optimization over all the cells in the network should be conducted simultaneously. Here, the large-scale propagation effect 𝛽_{𝑖,𝑗,𝑘} is considered as a correlated parameter as it results in inter-cell interference.

In [7], considering only the special case of 𝑃 = 𝑈_𝑖, ∀_𝑖, the capacity of the entire systems was studied. In practice, however, the assumption of 𝑃 = 𝑈_𝑖, ∀_𝑖 is not realistic because the actual number of terminals 𝑈_𝑖 can be varied from different cells, and moreover, it can change with time. That is, the number of actual terminals 𝑈_𝑖 cannot be generalized as 𝑃. This has motivated our work.

Considering the general case of an arbitrary number of terminals 𝑈_𝑖, our objective is to improve the capacity of the massive MIMO systems. The case of 𝑈_𝑖 > 𝑃 can be considered as a high-user-density scenario; in this case, only 𝑃 terminals out of the 𝑈_𝑖 terminals must be selected for data transmission in each time slot, which will be referred to as terminal selection. In contrast, the case of 𝑈_𝑖≤ 𝑃 can be considered as a low-user-density scenario; in this case, all 𝑈_𝑖 terminals should be selected for data transmission, which means that no terminal selection is required. However, 𝑈_𝑖 pilots should be selected for channel estimation out of 𝑃 pilots. This is referred to as pilot selection. Finally, when the terminals or pilots are selected, the (selected) pilots must then be assigned to the (selected) terminals, which is referred to as pilot assignment. It should be noted that pilot assignment is required for both overloaded and underloaded cases. When there are 𝑈_𝑖 terminals and 𝑃 pilots in a cell, min(𝑈_𝑖, 𝑃) means the maximum number of pilots, which can be assigned to terminals in a cell. Specifically, min(𝑈_𝑖, 𝑃) pilots are assigned to min(𝑈_𝑖, 𝑃) terminals. We first derive the capacity as a function of the selected terminals, selected pilots, and pilot assignment. Using the capacity expression, we then propose an optimal equation.

3. Proposed DC-PAS : Deep CNN Learning Framework

The pilot assignment with our proposed work is based on a deep CNN learning that could be suitable for large neuron networks with shift-invariant characteristics such as the image recognition area [23].

3.1 Preliminaries

There is a limitation of the DL-PAS in applications to a high-user-density nature owing to the factorial increase in both output nodes and multi-layer perceptions (MLP) weight.

As shown in Fig. 2, even though the DL-PAS performs well in the case of a low-user-density, it requires a significant amount of output layers even in the case of nine users. For ten users, almost 25,401,600 output nodes are required. That is, it is impossible to implement the DL-PAS in a practical system for handling a high-user-density.

Fig. 2. Problem of high density in DL-PAS

In machine learning, the deep CNN is a image oriented neural network in many areas. The deep CNN outperforms other classification algorithms with little preprocessing by reflecting the location characteristics of pictures. Such advantages have inspired us to apply the deep CNN mechanism to the pilot assignment problem in order to solve the problem focused on in prior works. The MLP is a mesh-shaped neuron network structure that is easy to create. Even though it performs well with almost a 100 % accuracy in learning for small-sized neuron networks, it requires high complexity and a huge amount of data to learn for realizing a relevant accuracy. That is, it is only suitable for small-sized neuron networks. Conversely, deep CNN is suitable for large-sized neuron networks. For instance, the ResNet, which is a good example of a deep CNN, consists of more than 100 layers and a high number of input nodes as the number of image pixels in the picture [23]. It should be noted that deep CNN is less precise in learning in the case of a small-sized neuron network.

3.2 Proposed DC-PAS algorithm

We introduce the entire deep learning process to apply to the proposed system. Deep learning can be adopted in the pilot assignment scheme by following procedure:

3.2.1 Getting data for machine learning

In the case of 𝑈_𝑙 > 𝑃, only 𝑃 terminals must be selected out of 𝑈𝑙 for data transmission. The set of selected terminals is defined as follows.

Definition 1 (Set of Selected Terminals): Let Ω_𝑙 denote the index set of the selected terminals in the 𝑙-th cell (out of all 𝑈_𝑙 terminals) with |Ω_𝑙| = min(𝑃,𝑈_𝑙).

The next issue is pilot selection: when 𝑈_𝑙 ≤ 𝑃, only 𝑈_𝑙 pilots must be selected out of 𝑃 for the channel estimation of 𝑈_𝑙 terminals. The set of selected pilots is defined as follows.

Definition 2 (Set of Selected Pilots): Let Γ_𝑙 denote the set of selected pilots in the _𝑙-th cell (out of all 𝑃 pilots) with |Γ_𝑙| = min(𝑃,𝑈_𝑙).

Once the terminals and pilots are selected, they must be one-to-one mapped. This pilot assignment is defined by a one-to-one mapping Π_𝑙(⋅) between the terminal indices in Ω_𝑙 and the pilot indices in Γ_𝑙.

Definition 3 (Mapping between Terminals and Pilot (pilot assignment)): When 𝑝 ∈ Γ_𝑙, Π_𝑙(𝑝) denotes the index of the terminal in Ω_𝑙, which 𝑝-th pilot is allocated to. When 𝑝 ∉ Γ_𝑙, Π_𝑙(𝑝) does not give any index of the terminal in Ω_𝑙, and it is denoted as Π_𝑙(𝑝) = 0, which means that the 𝑝-th pilot is not assigned to any terminal in 𝑙-th cell.

Using the expression of \(c_{network} ^u\) in equation (5), we propose the following optimization problem in order to maximize the network capacity:

\(\left(\Omega_{l}^{\mathrm{opt}}, \Gamma_{l}^{\mathrm{opt}}, \Pi_{l}^{\mathrm{opt}}\right)=\arg \max _{\left(\Omega_{l}, \Gamma_{l}, \Pi_{l}\right)} C_{\text {network }}\left(\Omega_{l}, \Gamma_{l}, \Pi_{l}\right)\)        (6)

It should be noted that (6) is a joint optimization over all three variables (Ω_𝑙, Γ_𝑙 ,Π_𝑙). However, if we consider the cases of 𝑈_𝑙 > 𝑃 and 𝑈_𝑙 ≤ 𝑃 individually, the problem is reduced to a simpler joint optimization of (Ω_𝑙,Π_𝑙) or (Γ_𝑙,Π_𝑙).

Subsequent to extracting the equation of the joint optimization for the terminal selection and pilot assignment according to the location of each terminal, we use that equation as the learning data for the deep CNN model.1) Induce joint optimization for terminal selection and pilot assignment in an equation according to the terminal location.2) Calculate the joint optimization for the terminal selection and pilot assignment according to the random terminal location.3) Classify the data using the deep CNN model. The input value is considered as the terminal location, and the output is considered as the result of the joint optimization for the terminal selection and pilot assignment. 4) Data could be obtained from the result of the joint optimization for the terminal selection and pilot assignment equation.

3.2.2 Building the deep CNN model and the proposed pilot assignment scheme

We use 32 kernels (3x3 size) and max-pooling (2x2 size). Finally, we use two fully connected layers to distinguish the output as shown in Fig. 3.

Fig. 3. Proposed deep CNN-based pilot assignment scheme

The algorithm for designing the input feature is explained in Algorithm 1. We initialize the input data from the coordinates of all the cells and terminals by mapping two-dimensional coordinates onto a one-dimensional array. Based on this input data, we create a two-dimensional image matrix for the CNN.

Algorithm 1 Algorithm for designing the input feature based on the square matrix imagination method

Algorithm 2 Algorithm for designing the output label and assigning the pilot for each terminal based on the pilot value sorting method

The algorithm for designing the output label and assigning the pilot for eachterminal is presented in Algorithm 2. We set the pilot assignment results and called it as the “pilot value” as an output label. For instance, for four pilots, there are four pilot values for identifying those pilots with an index such that {3,2,1,4}. Each location means terminal’s order. That is, 3rd pilot is mapped to 1st terminal, 2nd pilot is 2nd terminal, 1st pilot is 3rd terminal, and 4th pilot is mapped to 4th terminal, respectively. Here, the order of the output is a combination of the assigned pilot values for each cell and the number of terminals residing in the cell. It should be noted that during the prediction, the output value might return a real number, such as {1.2,1.9,3.4,4.1}, and not an integer. Thus, the post-processing is necessary for mapping a real number value into an actual pilot value. In Algorithm 2, this post-processing is based on an ascending sorting of the real number values and then mapping them into the integer-valued pilot values in the ascending order. For example, each cell should first be found. For the first cell, we can sort the predicted values in the increasing order. Their order will be Π_1,1={1.2} <Π_1,2={1.9} < Π_1,3={3.4} < Π_1,4={4.1}. We then assign pilots to Π_1,1={1} < Π_1,2={2} <Π_1,3={3} < Π_1,4={4} for each terminal in the first cell. Similarly, for the seventh cell, the expected value is first debited. Their order will be Π_7,1={0.9} < Π_7,3={2.1} < Π_7,2={3.3} <Π_7,4={3.9}. We then assign the pilot value to Π_7,1={1} < Π_7,3={2} < Π_7,2={3} < Π_7,4={4} for each terminal in the seventh cell. The manipulation of training data sets consists of three stages: the creation of training sets, the design of key metrics, and the labeling. First, training samples shall be designed for the generation of training sets. It is going to adopt location of devices as training set in order to detect major effects such as large-scale fading. For 𝐿 cells, 𝑈_𝑙 terminals, and P pilots (𝑈_𝑙 = 𝑃 for balance), the input size is 2 (position/terminal coordination) x 𝑈_𝑙 (terminal/cell) x 𝐿 (cell). For example, suppose each of the seven cells has four terminals, the input feature vector is [8 x 8 x 1]. Second, major performance metrics should be designed. This is adopted as a key metric to maximize total uplink network capacity𝐶_n𝑒twork(Ω_𝑙, Γ_𝑙, Π_𝑙). Finally, we should label on the basis of this key performance indicator. Then the label for each set of training is calculated (e.g. pilot assignment). Common deep learning algorithms, such as CNN, choose one of the combinations of the total result values on the label. That is, the number of output-layer neurons is the number of combinations that get the total result. For DC-PAS, 𝐿 cell, 𝑈_𝑙 terminal, and 𝑃 pilot (𝑈_𝑙 = 𝑃 for balance), it is not practical to deploy as the terminal 𝑈_𝑙 number increases because the output label is (𝑈_𝑙!) x 𝐿. Thus, to address this problem, a new version of the deep learning network is proposed, with minor modifications to the conventional deep learning network, to reduce the number of labels. The key idea is to resize the output layer to the number of pilot allocation cases per cell. This is because each cell has its own pilot task. Thus, for 𝐿 cell, 𝑈_𝑙 terminal, and 𝑃 pilot, the size of the output label is (𝑈_𝑙) x 𝐿. For example, assuming that each of the seven cells has 10 terminals, the combination of the results is required to be (10!) x 7 = 25,401,600 ≈ 2.54 x 10⁷. This means that the size of the output label is [2.54 x 10⁷] and the number of neurons in the output layer is [2.54 x 10⁷]. For example, if each of the seven cells has four terminals, 10 x 7 = 70, only a combination of result values is required. That is, while the traditional method requires [2.54 x 10⁷], the number of neurons in the output layer is only [70]. Therefore, the proposed DC-PAS would reduce the number of neurons in the order of 10⁶. Afterwards, extract Π_𝑖,𝑘={Ρ} label, where Π_𝑖,k is a pilot allocation with the 𝑖-th cell and the 𝑘-th terminal.

4. Performance Evaluation

Here, the detailed performance evaluation of the suggested DC-PAS algorithm is explained. We compare the proposed DC-PAS with the traditional schemes such as a deep-learning-based pilot assignment scheme (DL-PAS), a random pilot assignment scheme (R-PAS), a smart pilot assignment scheme (S-PAS) and a exhaustive-search pilot assignment scheme (ES-PAS). The performance measures used are the deep learning accuracy, normalized capacity gain, elapsed time, and training time. The deep learning accuracy is considered as a metric for evaluation of classification models. Specifically, accuracy has the following definition: Accuracy=Number of correct predictions/Total number of predictions[26,27]. Accordingly, in this paper, we measured this learning accuracy by comparing between exhaustive based pilot assignment (theoretical upper-bound = correct predictions) and prediction based pilot assignment.

Table 2 lists the details of the deep learning system environment with the TensorFlow with the Adam optimizer, the loss function of the mean square error, the learning rate of 10⁻⁴, the activation of Elu and the dropout which are 0.5 for train and 1.0 for test. We get the simulation results for a mobile network consisting of twelve outer cells in the outer ring and nineteen hexagonal cells consisting of seven inner cells. Twelve external cells are used as boundaries of the network, and the capacity is measured only in seven internal cells in Fig. 4 and Table 3. As in [7], we suppose that the cell radius is 1600 m and that there is a hole in the center of the cell with a radius 100 m. For each cell, the terminals are created uniformly within each cell except in the central holl area. Assume that the frequency reuse factor is 1, the path-loss exponent is 4, and the shadowing standard deviation is 8.0 dB.

Table 2. Deep learning system environment

Table 3. Mobile network model environment

Fig. 4. Simulation setting

Fig. 5 shows the performance analysis for low-user-density (𝑈_𝑙 ≤ 𝑃). The accuracy of the suggested DC-PAS with various numbers of users, i.e., two users and four users, is denoted by DC-PAS(2) and DC-PAS(4), respectively, as the samples number varies from 2,000 to 20,000. The DL-PAS with various users’ numbers, i.e., two users and four users are also denoted by DL-PAS(2) and DL-PAS(4), respectively.

Fig. 5. Performance in low-user-density case (𝑈_𝑙 ≤ 𝑃) for (a) accuracy, (b) normalized capacity gain

As depicted in Fig. 5(a), the accuracy of the suggested DC-PAS increases as the sample size increases, where the achieved accuracy of the suggested DC-PAS is almost 96.09%, which is similar to the 96.82% of the DL-PAS. For the normalized capacity gain in Fig. 5(b), both the DL-PAS and the proposed DC-PAS can realize a similar performance as 99.44% and 99.22%, respectively. The proposed DC-PAS is characterized by searching for similar patterns but has few patterns for a small number of users, which indicates a poor performance. The DC-PAS is slightly less capable than the DL-PAS but does not differ significantly. That is, the normalized capacity gain of the suggested DC-PAS is similar to that of the DL-PAS.

Fig. 6 shows the comparison of the output nodes with the DL-PAS versus that of the suggested DC-PAS. As expected, as the number of terminals increases, the number of output nodes in the case of the DL-PAS has faster growth rates than those of the suggested DC-PAS, which has linear growth rates. That is, the proposed DC-PAS requires a smaller number of output nodes for a high number of terminals.

Fig. 6. Comparison of output nodes with the DL-PAS versus that with the proposed DC-PAS

Fig. 7 shows performance analysis for a high-user-density (𝑈_𝑙 > 𝑃). The proposed accuracy of DC-PAS is expressed as DC-PAS (6), DC-PAS (8), and DC-PAS (10) because each of the different sample numbers, along with six users. As Fig. 7(a) shows, the proposed accuracy of the DC-PAS increases to an average of 93.84% of the proposed work. When user densities are high, DL-PAS should be complex in implementation and excluded from evaluation. Fig. 7(b) illustrates the capacity gains for the various schemes. We define the normalized capacity gain 𝐺_𝑛 as

Fig. 7. Performance for the high-user-density case (𝑈_𝑙 > 𝑃) for (a) accuracy of the suggested DC-PAS, (b) normalized capacity gain, and (c) elapsed time

\(G_{n}=\frac{C_{\text {network }}^{u}}{C_{\text {network-upper }}^{u}}\)       (7)

where the theoretical upper-bound capacity \(C_{\text {network-upper }}^{u}\) can be calculated from ES-PAS.

In this study, various iterations of S-PAS(1000) were tested with S-PAS(10), S-PAS(100), and S-PAS(1000) repeated 10 times, 100 times, and 1000 times, respectively.

The normalized capacity gain is ES-PAS(100%) > DC-PAS(98%) > S-PAS(1000)(95%) > S-PAS(100)(93%) > S-PAS(10)(91%) > R-PAS(89%) for each scheme. The DC-PAS realizes a nearly 98.15% normalized capacity gain of in 20,000 samples and ten terminals. As shown in Fig. 7(a), the proposed normalized capacity increase in DC-PAS increases slightly with the number of samples. On the other hand, other existing schemes are independent of the number of samples because they do not adopt a learning mechanism. As a result, it is clear that the proposed work exceeded the existing pilot assignment system.Specifically, R-PAS, which randomly assigns pilots to users, shows the worst performance despite the complexity and ease of deployment. Normalised gains from S-PAS increase with the number of repetitions. It should be noted that increasing the number of iterations of the S-PAS significantly increases the elapsed time and constrains the S-PAS. Fig. 7(c) shows performance in terms of time elapsed from various pilot allocation plans. where R-PAS has the lowest elapsed time because of its low complexity. Only 0.842 ms is required for the proposed DC-PAS. It should be noted that the computational ability and training time of the in-depth learning process can be further analyzed as a practical concern. For S-PAS, the elapsed time increases with the number of repetitions such as S-PAS(1000) > S-PAS(100) > S-PAS(10). It also evaluates the elapsed time of the ES-PAS returning the theoretical capacity upper limit. Despite notable performance, the elapsed time in this case is nearly 13 hours in the same environment. This means that this evaluation is not included in the figure because the ES-PAS cannot be placed in the actual system.

Fig. 8 shows the performance in terms of the elapsed time and training time for the proposed DC-PAS. We observe that the elapsed time increases according to the number of terminals as shown in Fig. 8(a). This is because, as the number of terminals increases, the number of output nodes increases linearly, thus resulting in a larger model size and greater complexity. In addition, we further observe that the training time also increases according to the number of samples as shown in Fig. 8(b). This is because a larger number of samples requires a larger amount of data for processing in the training phases.

Fig. 8. Performance analysis (a) elapsed time of the suggested DC-PAS, (b) training time

5. Conclusion

We have proposed a DC-PAS in order to reduce the difficulty of applying a deep-learning-based scheme to a high-user-density nature because of the factorial increase in both input features and output layers. Specifically, the input and labels are considered as the users’ positions in all the cells and the pilot allocations, respectively. The pre-trained optimal pilot allocations with the given users’ positions are provided through a mathematical calculation for a high-user-density as the training data set, and a near-optimal pilot assignment is returned from analyzing the data set. The obtained results demonstrate that our proposed DC-PAS realizes an nearly 98% performance with low complexity and requiring a computing time of only 0.842 ms. In the future, in order to reduce the load changes and change the minimum pilot assignment, the current work can be expanded by utilizing reinforcement learning considering the previous state.