Performance Analysis of Hybrid Cellular and Cell-free MIMO Network

The future wireless communication networks will witness a proliferation of mobile applications and unprecedented growth in wireless data. The realization of higher spectrum and energy efficiency with superior costs remains a challenging issue in current research. As one of the most prominent wireless technologies proposed in recent years, cell-free network is considered as a promising network architecture in the beyond fifth-generation (B5G) and sixth generation (6G) mobile communication system [1]. Different from traditional cellular systems, cell-free network is a user-centric coverage architecture that discards the traditional concept of cellular boundaries [2]. The central processing unit (CPU) controls the access points (APs) to cooperate to provide services to user equipment (UE) on the same time-frequency resources, thus realizing higher spatial multiplexing [3, 4]. Cell-free network improves the energy and spectral efficiency of the system, and effectively reduces the performance gaps between UE by ensuring that there are spatially short-range APs that provide stable services to UE.

However, the deployment of cell-free systems in existing commercial mobile networks still faces serious challenges. First, the construction of cell-free systems requires the deployment of the distributed APs throughout the network and the construction of the corresponding fronthaul links, which brings heavy time costs and deployment expenses. Second, simply introducing the cell-free system without cooperation will inevitably cause mutual interference with existing cellular systems, significantly limiting the system performance. Therefore, the deployment of cell-free systems is bound to be a long-term evolutionary process, and hybrid cellular and cell-free cooperation networks are both a necessity and a desirable choice for B5G and 6G.

Some recent works have investigated the performance analysis and resource allocation of hybrid cellular and cell-free networks. A cell-free and legacy cellular coexistence system deployed on the existing system architecture, as well as the corresponding precoding, power control, etc., are outlined in [5]. With appropriate UE association criteria and coordinated beamforming, hybrid cell-free and small cell systems can provide superior downlink rates for static and dynamic UE than the single architecture [6]. However, the above works do not take into account the impact of the spatial distribution of base stations (BSs), APs, and UE on network performance.

Due to the densification and irregularity of wireless node distributions in the network, traditional grid-based deployment models are difficult to reflect the practical system performance. Stochastic geometry models the spatial distribution of wireless nodes with point processes and can effectively characterize the lower bound of the actual system performance. There have been some works using stochastic geometry to analyze the performance of cell-free networks and coordinated multiple points (CoMP) communication in terms of energy efficiency (EE) [7], power control [8] and channel hardening analysis [9], etc. However, there is still a lack of work related to the characterization of hybrid cellular and cell-free networks. In addition, the existing stochastic geometry-based heterogeneous network studies, which separate different network layers from each other [10], are not applicable to the analysis of hybrid cellular and cell-free networks.

To gain some insights of the performance limit of the hybrid network, this paper develops a stochastic geometry-based model for hybrid cellular and cell-free networks, which reveals the coupling of the signal and interference from both the cellular and cell-free networks. However, the aggregate signals from the BSs and APs with conjugate beamforming make it difficult to characterize the distribution of the signal strength and the corresponding signal-to-interference plus noise ratio (SINR). To tackle this issue, we first derive the closed-form expressions for the average signal strength and interference power from the APs, and then the aggregate signal strength distribution is approximated via moment matching. Finally, the coverage probability is characterized on the basis of the Laplace transform of the system interference power. The analysis of network coverage probability is verified by extensive simulations, and it can be used to guide the network deployment and interference management in the hybrid cellular and cell-free networks.

As shown in Fig. 1, a hybrid cellular and cell-free network is considered in this paper. The locations of BSs, cell-free APs, and single-antenna UE are modeled by independent homogeneous Poisson point processes (HPPP) $\Lambda_{B}$, $\Lambda_{A}$ and $\Lambda_{U}$, with density $\lambda_{B}$/$\mathrm{km}^{2}$, $\lambda_{A}$/$\mathrm{km}^{2}$ and $\lambda_{U}$/$\mathrm{km}^{2}$, respectively. Each BS is equipped with $N_{B}$ antennas, while each AP is equipped with $N_{A}$ antennas. Considering the different configurations that AP and BS can support, $P_{A}$ and $P_{B}$ denote the maximum downlink power of AP and BS with $P_{A}<P_{B}$. In the network, $d_{mi}$ and $l_{ji}$ denote the distance between BS $m$ and UE $i$, and that between AP $j$ and UE $i$, respectively. We consider a typical UE, referred as UE 0, which is jointly served by the closest BS, named BS 0, with the distance $d_{00}$, and all the cell-free APs in $\mathcal{A}$. The channel vector between BS $m$ and UE $i$ is denoted by $\mathbf{h}_{mi}\triangleq[h_{mi,1},...,h_{mi,N_{B}}]\in\mathbb{C}^{N_{B}\times 1}$, while the channel vector between AP $j$ and UE $i$ is $\mathbf{g}_{ji}\triangleq[g_{ji,1},...,g_{ji,N_{A}}]\in\mathbb{C}^{N_{A}\times 1}$. The channel model consisting of distance-dependent large-scale fading and random small-scale fading is considered as

where $\beta_{mi}$ and $\delta_{ji}$ are path loss of the channel with $\beta_{mi}=\beta_{0}d_{mi}^{-\alpha_{1}}$ and $\delta_{ji}=\delta_{0}l_{ji}^{-\alpha_{2}}$. $\omega_{B}$ and $\omega_{A}$ are the sets of all the BSs and APs, respectively. The small-scale fading in both $\bm{\zeta}_{mi}$ and $\bm{\xi}_{ji}$ are independent and identically distributed (i.i.d.) $\mathcal{CN}(0,1)$ random variables (r.v.s).

The entire network area is represented by $\mathcal{A}$.

During the downlink transmission, UE in the same cell will be served by the same BS. In addition, both BSs and APs select conjugate beamforming in order to obtain low computational complexity and good performance, and also to avoid channel state information (CSI) interactions between APs [1]. Therefore, the signal transmitted by BS $m$ is

where $\phi_{B,m}$ denotes the set of UE that are served by BS $m$, and $q_{n}\sim\mathcal{CN}(0,1)$ denotes the information-bearing symbols for UE $n$. $\eta_{B}$ denotes the power constraint parameter with $\mathbb{E}[\mathbf{x}_{B,m}^{H}\mathbf{x}_{B,m}]=P_{B}$. For convenience, $\eta_{B}$ is expressed as the average number of users per BS, i.e., $\eta_{B}=\frac{1}{|\bar{\phi}_{B}|}=\frac{\lambda_{B}}{\lambda_{U}}$, where $|\bar{\phi}_{B}|$ denotes the average of $|\phi_{B,m}|$ for any $m$.

Denote the set of UE in the network as $\phi_{U}$. Therefore, the corresponding downlink signal from each cell-free AP $j$ is

where $\eta_{A}$ denotes the power constraint parameter. Note that the averaged number of UE in $\phi_{U}$ is $\mathbb{E}[|\phi_{U}|]=\bar{U}=\lambda_{U}|\mathcal{A}|$, $\eta_{A}$ can be denoted as $\eta_{A}=\frac{1}{\bar{U}}$ to ensure $\mathbb{E}[\mathbf{x}_{A,j}^{H}\mathbf{x}_{A,j}]=P_{A}$.

For any UE $i$ in the network, $i^{*}$ is denoted as the index of the associated and nearest BS providing the service. With the collaboration of cell-free APs and cellular BSs, the downlink signal received by the typical UE 0 is

where the first term $S_{0A}$ and the second term $S_{0B}$ represent the desired signals from BS 0 and cell-free APs, respectively. The total interference is shown in the third term $I_{U}$. Each term in $I_{U}$ consists of the signal sent by cellular BSs and cell-free APs to any other UE. The last term $n_{0}$ denotes the additive white Gaussian noise (AWGN) with power $\sigma^{2}$.

Based on (5), the interference power caused by the signal intended to UE $i$ is given by

In interference $I_{i}$, BS channel vector $\mathbf{h}_{i^{*}0}$ is independent of channel vector $\mathbf{g}_{j0},\forall j\in\omega_{A}$. Meanwhile, the channel gains $\mathbf{g}_{i0}$ and $\mathbf{g}_{j0}$ from different APs are also independent of each other, $\forall i\neq j$. Taking into account the law of large number and the mutual independence of channels as well as beamforming vectors between different BSs and APs, $I_{i}$ can be approximated as

The approximation in (7) indicates that $I_{i}$ can be expressed in the form of the sum of the power of the signal from the associated BS $i^{*}$ and each AP. Further, by classifying the interference into intra-cell interference $I_{B0}$, inter-cell interference $I_{B}$, and interference $I_{A}$ due to the APs, the total interference $\sum_{i\in\phi_{U}\backslash\{0\}}I_{i}$ can be rewritten as

where

Therefore, the corresponding received SINR at UE 0 is approximately expressed as

where

In this section, the statistical distributions of the received signal and interference power are characterized. The signal strength is approximated via moment matching, with its first- and second-order moments derived in closed-form. The intra- and inter-cell interference caused by BSs are approximated as a Gamma r.v. and a weighted sum of Gamma r.v.s, respectively. The average interference caused by cell-free APs is derived in closed-form. In addition, the performance of the network can be obtained by analyzing a typical UE 0 according to Slivnyak’s theorem [11].

In this section, the coverage probability of hybrid cellular and cell-free network is analyzed based on the distribution of signal strength and various interference components, derived in the preceding section. In general, the coverage probability is the complementary cumulative distribution function (CCDF) of SINR over the overall network, which can be defined as

where $T$ denotes the target threshold of the SINR $\Omega$.

In this section, the analytical results of the coverage probability of the hybrid cell and cell-free network are verified by the comparison with the Monte-Carlo (MC) simulation results. Each result of the MS simulation is averaged from 1000 randomly generated wireless node distributions with 5 realizations per channel. All the wireless nodes are randomly distributed in a circular area of radius 500m and UE 0 is located at the center of the circle. The densities of BSs, APs and UE are $\lambda_{B}=40/\mathrm{km}^{2}$, $\lambda_{A}=200/\mathrm{km}^{2}$ and $\lambda_{U}=160/\mathrm{km}^{2}$, respectively. Other relevant parameters are as follows: $\alpha_{1}=2.7$, $\alpha_{2}=1.8$, $\frac{P_{B}}{\sigma^{2}}=130$dB, $P_{A}=3\times 10^{-5}P_{B}$, $N_{B}=4$, $N_{A}=2$, $C=3\times 10^{8}$m/s, $f=3.5$GHz and $\beta_{0}=\delta_{0}=(\frac{C}{4\pi f})^{2}$.

Fig. 3 shows the coverage probability in cellular networks ($P_{A}=0$), cell-free networks ($P_{B}=0$), and hybrid cellular and cell-free networks for different SINR thresholds $T$. The coverage probability is obtained from the linear weighted probability of the upper and lower integers of $k_{\theta_{S}}$, i.e.,

From Fig. 3, It is expected that the coverage probability analysis of the hybrid network is generally consistent with the results of MC simulation under different SINR threshold, and can also be applied to the special case where $P_{A}=0$. Compared with traditional cellular networks, hybrid networks effectively improve the communication performance of edge UE and reduce the performance gaps between UE. Compared with cell-free networks, such hybrid networks can achieve higher peak SINR. Therefore, by deploying low-power APs, hybrid cellular and cell-free networks can provide UE with uniformly good communication services while obtaining better peak SINR performance.

In this paper, the hybrid cellular and cell-free network is modeled by the stochastic geometry approach, revealing the coupling of the signal and interference from both the cellular and cell-free networks. Moment matching is used to approximate the aggregate signal received from the hybrid network to address the difficulty of distribution analysis due to conjugate beamforming. The coverage probability is then obtained by the Laplace transform for interference. The analysis of the coverage probability of hybrid networks is validated by MC simulation, demonstrating that hybrid networks can reduce the performance gap while improving the peak SINR performance.

This work was supported by the National Key R&D Program of China with Grant number 2019YFB1803400, and by the National Natural Science Foundation of China with grant number 62071114.