# Scalable Cell-Free Massive MIMO with Multiple CPUs

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

- Despite [18] having proved that excellent performance can be achieved when all the APs are connected to a single CPU for joint transmission, it is difficult to let a single CPU control all the APs when the number of APs is large. Moreover, joint transmission requires a single CPU to centralize the signal processing, which puts high demands on the CPU’s processing capacity.
- The traditional precoding and power control of CF-M-MIMO act on the overall APs and UEs, which are difficult to realize in practice. Although a series of studies have proved the superiority of the final results [18,24,25], the complexity of these algorithms grows polynomially with the number of APs or UEs. Moreover, each AP needs to transmit instantaneous CSIs to the CPU, which is also difficult to achieve scalability when there are a large number of APs in the CF-M-MIMO systems.

- We consider a taxonomy with four different implementations of CF-M-MIMO with multiple CPUs, which are classified by different degrees of cooperation among the CPUs. The four different levels of cooperation can be called centralized connectivity, distributed connectivity and complex processing, distributed connectivity and simple processing, and no connectivity, respectively. The difference of these levels is shown in Table 1. We derive novel SE expressions for different levels of multiple CPUs cooperation in the uplink transmission. In addition, unlike most scenarios that specify the number of CPUs participating in the service, we consider a completely user-centric way to select APs.
- We propose a novel signal processing algorithm for cooperation among multiple CPUs. Each CPU processes the local information from its APs, and then transmits these signals to a CPU for final decoding. Based on the generalized Rayleigh quotient, we use simple weighted processing to linearly combine received signals from multiple CPUs with statistical CSIs.
- We compare the performance of different cooperation levels. Monte Carlo simulation results show that our proposed distributed connectivity scheme can achieve scalability with lower backhual burden, and the performance loss is negligible compared to the centralized connectivity scheme.

#### 1.2. Paper Structure

## 2. System Model

**Notation:**Boldface lowercase letters denote column vectors and boldface uppercase letters denote matrices. The superscripts ()${}^{*},{\left(\right)}^{\mathrm{T}}$, and ()${}^{\mathrm{H}}$ indicate transpose, conjugate and transpose. Since we cannot guarantee full rank of matrices, we use ${\left(\xb7\right)}_{}^{\u2020}$ to denote the matrix pseudoinverse. The $n\times n$ identity matrix stands for ${\mathrm{I}}_{\mathrm{N}}$. The $diag\left({\mathbf{h}}_{1},\dots ,{\mathbf{h}}_{n}\right)$ is used to denote a block-diagonal matrix. $z\sim \mathcal{CN}\left(0,{\sigma}^{2}\right)$ denotes a circularly symmetric complex Gaussian random variable (RV) z with zero mean and variance ${\sigma}^{2}$. $\left|\mathcal{Z}\right|$ is applied to denote the cardinality of the set $\mathcal{Z}$.

#### 2.1. Channel Estimation

#### 2.2. Uplink Payload Transmission

#### 2.3. Dynamic Cooperation Clustering Network

## 3. Multiple CPUs Cooperative Transmission

#### 3.1. Level 4: Centralized Connectivity

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.2. Level 3: Distributed Connectivity and Complex Processing

**Proposition**

**2.**

**Proof.**

**Corollary**

**2.**

Algorithm 1 Optimization algorithm for Level 3 |

Input: Channel gain ${h}_{kl}$, MMSE combining ${\mathbf{v}}_{ku}$, noise variance ${\sigma}^{2}$, DCC matrix ${\mathbf{D}}_{ku}^{}$Output: ${\mathbf{a}}_{k}$1: Initialization: calculate ${\mathcal{U}}_{k}^{}$-dimensional vector ${\mathbf{g}}_{ki}={\left[{\mathbf{v}}_{k1}^{\mathrm{H}}{\mathbf{D}}_{k1}{\mathbf{h}}_{i1}^{},...,{\mathbf{v}}_{k|{\mathcal{U}}_{k}^{}|}^{\mathrm{H}}{\mathbf{D}}_{k|{\mathcal{U}}_{k}^{}|}{\mathbf{h}}_{i|{\mathcal{U}}_{k}^{}|}^{}\right]}^{\mathrm{H}}$ 2: for u = 1:U do3: for k = 1:K do4: if ${D}_{ku}=1$ then5: calculate diagonal matrix ${\mathbf{N}}_{k}=\mathrm{diag}\left(\mathbb{E}\left\{{\u2225{\mathbf{v}}_{k1}^{H}{\mathbf{D}}_{k1}\u2225}^{2}\right\},\cdots ,\mathbb{E}\left\{{\u2225{\mathbf{v}}_{k|{\mathcal{U}}_{k}^{}|}^{H}{\mathbf{D}}_{k|{\mathcal{U}}_{k}^{}|}\u2225}^{2}\right\}\right)$ 6: update ${\mathbf{g}}_{kk}={\left[{\mathbf{v}}_{k1}^{\mathrm{H}}{\mathbf{D}}_{k1}{\mathbf{h}}_{k1}^{},...,{\mathbf{v}}_{k|{\mathcal{U}}_{k}^{}|}^{\mathrm{H}}{\mathbf{D}}_{k|{\mathcal{U}}_{k}^{}|}{\mathbf{h}}_{k|{\mathcal{U}}_{k}^{}|}^{}\right]}^{\mathrm{H}}$ 7: calculate ${\mathbf{a}}_{k}$ by (23). 8: end if9: end for10: end for |

#### 3.3. Level 2: Distributed Connectivity and Simple Processing

**Proposition**

**3.**

**Proof.**

#### 3.4. Level 1: No Connectivity

**Proposition**

**4.**

**Proof.**

## 4. Simulation Results

#### 4.1. Uplink Transmission

#### 4.2. Power Allocation

#### 4.3. Varying Numbers of CPUs

#### 4.4. Varying Numbers of UEs

#### 4.5. Varying the Uplink Transmit Power

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Example of virtual clusters in the CF-M-MIMO with multiple CPUs. The yellow area, red area and blue area represent virtual cluster, real cluster and the intersection of virtual cluster and real cluster, respectively.

**Figure 2.**Comparison of CF-M-MIMO with different levels of multiple CPUs cooperation in the case that partial APs serve the UEs.

**Figure 3.**Comparison of CF-M-MIMO with different levels of multiple CPUs cooperation in the case that all APs serve the UEs.

**Figure 4.**In the case that partial APs serve the UEs, the SE comparison of Level 3 between full power and power allocation.

**Table 1.**Type of CSIs that needs to exchange among CPUs, and the comparison of different levels of cooperation.

Type of CSIs | Level of Computational Complexity | |
---|---|---|

Level 4 centralized connectivity | instantaneous CSIs | high |

Level 3 distributed connectivity and complex processing | statistical CSIs | medium |

Level 2 distributed connectivity and simple processing | statistical CSIs | low |

Level 1 no connectivity | − | lowest |

**Table 2.**Number of complex scalars that other CPUs send to the Master CPU through the backhaul, either in each coherence block or for each realization of the UE locations/statistics.

Each Coherence Block | Statistical Parameters | |
---|---|---|

Level 4 | ${\tau}_{\mathrm{c}}\left|{\mathcal{Z}}_{k}^{}\right|-{\tau}_{\mathrm{c}}\left|{\mathcal{A}}_{MCk}\right|$ | $\left(K\left|{\mathcal{Z}}_{k}^{}\right|\right)/2$ |

Level 3 | $K({\tau}_{\mathrm{c}}-{\tau}_{\mathrm{p}}^{})\left(\right|{\mathcal{U}}_{k}^{}|-1)$ | $K\left|{\mathcal{Z}}_{k}^{}\right|+\left(\right|{\mathcal{U}}_{k}^{}{|}^{2}{K}^{2}+K|{\mathcal{U}}_{k}^{}\left|\right)/2$ |

Level 2 | $K({\tau}_{\mathrm{c}}-{\tau}_{\mathrm{p}}^{})\left(\right|{\mathcal{U}}_{k}^{}|-1)$ | − |

Level 1 | − | − |

**Table 3.**Computational complexity of different level of cooperation per coherence block. Only complex multiplications and complex divisions are considered, and additions/subtractions are neglected.

Computing Combining Vectors | Computing Weighted Vectors | |||
---|---|---|---|---|

Multiplications | Divisions | Multiplications | Divisions | |

Level 4 | $\begin{array}{c}\hfill \frac{\left(3{\left|{\mathcal{Z}}_{k}^{}\right|}^{2}+\left|{\mathcal{Z}}_{k}^{}\right|\right)K}{2}+\frac{{\left|{\mathcal{Z}}_{k}^{}\right|}^{3}-\left|{\mathcal{Z}}_{k}^{}\right|}{3}\\ \hfill +\left|{\mathcal{Z}}_{k}^{}\right|{\tau}_{p}\left({\tau}_{p}-K\right)\end{array}$ | $|{\mathcal{Z}}_{k}^{}|$ | − | − |

Level 3 | $\begin{array}{c}\hfill {\displaystyle \sum _{u\in {\mathcal{U}}_{k}^{}}}(\frac{\left(3{\left|{\mathcal{P}}_{u}^{}\right|}^{2}+\left|{\mathcal{P}}_{u}^{}\right|\right)K}{2}+\frac{{\left|{\mathcal{P}}_{u}^{}\right|}^{3}-\left|{\mathcal{P}}_{u}^{}\right|}{3}\\ \hfill +\left|{\mathcal{P}}_{u}^{}\right|{\tau}_{p}\left({\tau}_{p}-K\right))\end{array}$ | $\sum _{u\in {\mathcal{U}}_{k}^{}}}\left|{\mathcal{P}}_{u}^{}\right|$ | $\begin{array}{c}\hfill \frac{\left({|{\mathcal{U}}_{k}^{}|}^{2}+\left|{\mathcal{U}}_{k}^{}\right|\right)K}{2}+\frac{{|{\mathcal{U}}_{k}^{}|}^{3}-\left|{\mathcal{U}}_{k}^{}\right|}{3}\\ \hfill +|{\mathcal{U}}_{k}^{}{|}^{2}\end{array}$ | $|{\mathcal{U}}_{k}^{}|$ |

Level 2 | $\begin{array}{c}\hfill {\displaystyle \sum _{u\in {\mathcal{U}}_{k}^{}}}(\frac{\left(3{\left|{\mathcal{P}}_{u}^{}\right|}^{2}+\left|{\mathcal{P}}_{u}^{}\right|\right)K}{2}+\frac{{\left|{\mathcal{P}}_{u}^{}\right|}^{3}-\left|{\mathcal{P}}_{u}^{}\right|}{3}\\ \hfill +\left|{\mathcal{P}}_{u}^{}\right|{\tau}_{p}\left({\tau}_{p}-K\right))\end{array}$ | $\sum _{u\in {\mathcal{U}}_{k}^{}}}\left|{\mathcal{P}}_{u}^{}\right|$ | − | − |

Level 1 | $\begin{array}{c}\hfill \frac{\left(3{\left|{\mathcal{A}}_{MCk}^{}\right|}^{2}+\left|{\mathcal{A}}_{MCk}^{}\right|\right)K}{2}+\frac{{\left|{\mathcal{A}}_{MCk}^{}\right|}^{3}-\left|{\mathcal{A}}_{MCk}^{}\right|}{3}\\ \hfill +\left|{\mathcal{A}}_{MCk}^{}\right|{\tau}_{p}\left({\tau}_{p}-K\right)\end{array}$ | $|{\mathcal{A}}_{MCk}^{}|$ | − | − |

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**MDPI and ACS Style**

Li, F.; Sun, Q.; Ji, X.; Chen, X. Scalable Cell-Free Massive MIMO with Multiple CPUs. *Mathematics* **2022**, *10*, 1900.
https://doi.org/10.3390/math10111900

**AMA Style**

Li F, Sun Q, Ji X, Chen X. Scalable Cell-Free Massive MIMO with Multiple CPUs. *Mathematics*. 2022; 10(11):1900.
https://doi.org/10.3390/math10111900

**Chicago/Turabian Style**

Li, Feiyang, Qiang Sun, Xiaodi Ji, and Xiaomin Chen. 2022. "Scalable Cell-Free Massive MIMO with Multiple CPUs" *Mathematics* 10, no. 11: 1900.
https://doi.org/10.3390/math10111900