# Analysis and Optimization for Downlink Cell-Free Massive MIMO System with Mixed DACs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- With the conjugate beamforming and AQNM, the tight closed-form downlink rate expression is derived based on the minimum mean square error (MMSE) channel estimation technique. It reflects the effect of the APs’ number, the downlink transmitted power, the number of quantization bits, and the proportion of perfect DACs in the mixed-DAC architecture;
- (2)
- Considering the fact that, in the future, emerging situations such as telemedicine, driverless, and real-time operations, these situations often require high priority, while the remaining are served at the same rate. Then, the weighted max–min power allocation scheme is proposed. As the objective function of the optimization problem can be effectively proved to be quasi-concave, then the critical bisection algorithm is used to solve it;
- (3)
- Numerical results are given to verify the derived results and prove that the proposed scheme can be effectively solved by the bisection algorithm.

## 2. System Model

## 3. Uplink Channel Training

**Lemma**

**1.**

#### 3.1. Downlink Data Transmission

#### 3.2. Rate Performance Analysis

**Theorem**

**1.**

## 4. Weighed Max–Min Power Allocation Scheme

Algorithm 1 Bisection Algorithm for Solving ${P}_{2}$ |

Input: $M,K,N,{N}_{1},\tau ,{\rho}_{\mathrm{d}},{\gamma}_{mk}.$ |

Output: ${\epsilon}_{k},\forall k,\forall m$ |

(1) Initialize the lower value ${r}_{\mathrm{l}}$ and the upper bound ${r}_{\mathrm{u}}$. Give the tolerance $\delta >0.$ |

(2) While ${r}_{\mathrm{u}}-{r}_{\mathrm{l}}\le \delta $ |

(3) Set $r=\left({r}_{\mathrm{l}}+{t}_{\mathrm{u}}\right)/2$ and solve the following feasible problem ${\tilde{P}}_{2}$. |

(4) If ${\tilde{P}}_{2}$ is feasible, set ${r}_{\mathrm{l}}=r$, go to step 2, else set ${r}_{\mathrm{u}}=r$. |

(5) end If |

(6) end While |

## 5. Numerical Results

#### 5.1. Simulation Parameters

#### 5.2. Results and Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem**

**1.**

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**Figure 1.**Downlink sum-rate versus the number of access points (Aps) with different numbers of users, with $M=60,N=10$.

**Figure 2.**Downlink sum-rate versus the transmitted power with different antenna configurations, with $MN=600,K=10.$.

**Figure 3.**Cumulative distribution functions (CDFs) of the downlink per-user rate for different users, with $K=10,N=10.$.

**Figure 4.**Downlink sum-rate versus the number of bits for different $\kappa $, with $M=60,N=10,K=10$.

**Figure 5.**Downlink sum-rate versus the $\kappa $ for different quantization bits, with $M=60,N=10,K=10$.

${b}_{m}$ | 1 | 2 | 3 | 4 | 5 |

${\alpha}_{m}$ | 0.6366 | 0.8825 | 0.96546 | 0.990503 | 0.997501 |

Parameter | Value |
---|---|

B | 20 MHz |

T | 200 |

${\rho}_{\mathrm{p}},{\rho}_{\mathrm{d}}$ | 100, 200 mW |

${d}_{0},{d}_{1},D$ | 10, 50, 1000 m |

**Table 3.**The Achievable Per-User Rate for proposed algorithm, for $1\u2a7dn\u2a7d3$, ${\omega}_{k}=0.5$, else ${\omega}_{k}=1$.

Achievable Per-User Rate (bits/s/Hz) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Quantization Bits | ${R}_{1}$ | ${R}_{2}$ | ${R}_{3}$ | ${R}_{4}$ | ${R}_{5}$ | ${R}_{6}$ | ${R}_{7}$ | ${R}_{8}$ | ${R}_{9}$ | ${R}_{10}$ |

DAC = 1 | 5.07 | 5.07 | 5.07 | 4.15 | 4.15 | 4.15 | 4.15 | 4.15 | 4.15 | 4.15 |

DAC = 3 | 5.91 | 5.91 | 5.91 | 4.96 | 4.96 | 4.96 | 4.96 | 4.96 | 4.96 | 4.96 |

DAC = 10 | 6.02 | 6.02 | 6.02 | 5.06 | 5.06 | 5.06 | 5.06 | 5.06 | 5.06 | 5.06 |

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**MDPI and ACS Style**

Zhou, M.; Zhang, Y.; Qiao, X.; Tan, W.; Yang, L. Analysis and Optimization for Downlink Cell-Free Massive MIMO System with Mixed DACs. *Sensors* **2021**, *21*, 2624.
https://doi.org/10.3390/s21082624

**AMA Style**

Zhou M, Zhang Y, Qiao X, Tan W, Yang L. Analysis and Optimization for Downlink Cell-Free Massive MIMO System with Mixed DACs. *Sensors*. 2021; 21(8):2624.
https://doi.org/10.3390/s21082624

**Chicago/Turabian Style**

Zhou, Meng, Yao Zhang, Xu Qiao, Weiqiang Tan, and Longxiang Yang. 2021. "Analysis and Optimization for Downlink Cell-Free Massive MIMO System with Mixed DACs" *Sensors* 21, no. 8: 2624.
https://doi.org/10.3390/s21082624