## 2 System Model
### Channel Hardening  
Channel hardening phenomenon is the phenomenon where the variance of the mutual information of the MIMO channel changes slowly as the number of antennas grows [22; 23]. Channel hardening can bring several advantages in large dimensional signal processing. As the number of antennas increases, the channel hardening phenomenon makes the multi-antenna fading channel almost deterministic. In centralized massive MIMO systems, this phenomenon is exploited to obtain high spectral efficiency and high reliability. Cell-free massive MIMO distributes AP antennas randomly in a scene, and the large-scale fading between each AP and the terminal is no longer the same. With one antenna per AP, increasing the AP density does not lead to channel hardening. Equation (3) is the channel gain variation to prove channel hardening [16; 24].  
\[\text{Var}\Bigg{[}\frac{\|\mathbf{g}_{k}\|^{2}}{\mathbb{E}[\|\mathbf{g}_{k}\|^{2}| \mathbf{r}]}\Bigg{]}=\frac{\text{Var}[\|\mathbf{g}_{k}\|^{2}|\mathbf{r}]}{(\mathbb{E}[\|\bm {g}_{k}\|^{2}|\mathbf{r}])^{2}}\to 0\text{ as }M\rightarrow\infty, \tag{3}\]  
where \(\mathbf{g}_{k}\) is the channel vector for the \(k\)th terminal, \(\mathbf{r}\) is the distance vector between APs and the terminal, and \(M\) is the number of antennas. The channel gain is  
\[\|\mathbf{g}_{k}\|^{2}=\mathbf{g}_{k}^{H}\mathbf{g}_{k}=\sum_{m=1}^{M}\beta_{mk}h_{mk}^{*} h_{mk}. \tag{4}\]  
When the positions of the antenna and the terminal are determined, \(\text{Var}\Big{[}\|\mathbf{g}_{k}\|^{2}|\mathbf{r}\Big{]}\) and \(\mathbb{E}\Big{[}\|\mathbf{g}_{k}\|^{2}|\mathbf{r}\Big{]}\) are as follows.  
\[\text{Var}\Big{[}\|\mathbf{g}_{k}\|^{2}|\mathbf{r}\Big{]}=\sum_{m=1}^{M}\beta_{mk}^{2 }\text{Var}[h_{mk}^{*}h_{mk}], \tag{5}\]  
and  
\[\mathbb{E}\Big{[}\|\mathbf{g}_{k}\|^{2}|\mathbf{r}\Big{]}=\sum_{m=1}^{M}\beta_{mk} \mathbb{E}[h_{mk}^{*}h_{mk}]. \tag{6}\]  
To analyze the difference in channel hardening between centralized massive MIMO and cell-free massive MIMO, the received power of the terminal is assumed to be normalized, \(\sum_{m=1}^{M}\beta_{mk}=1\). Substitution of (5) and (6) into (3) yields  
\[\text{Var}\Bigg{[}\frac{\|\mathbf{g}_{k}\|^{2}}{\mathbb{E}\Big{[}\|\mathbf{g}_{k}\|^{2 }|\mathbf{r}\Big{]}}\Bigg{]}=\frac{\sum_{m=1}^{M}\beta_{mk}^{2}}{(\sum_{m=1}^{M} \beta_{mk}^{2})^{2}}=\begin{cases}\frac{1}{M},&for\text{ centralized}\\ \sum_{m=1}^{M}\beta_{mk}^{2},&for\text{ cell-free}\end{cases}. \tag{7}\]  
Figure 2: In FDD system, uplink and downlink channels are different, and CSIs are estimated separately.  
When \(\beta_{1k}=\cdots=\beta_{Mk}\), \(\sum_{m=1}^{M}\beta_{mk}^{2}\) is minimum. In cell-free massive MIMO systems, the large-scale fading will not be identical. Therefore, the channel hardening is not as good as that of the centralized massive MIMO system.  
The variation in the channel gain of centralized massive MIMO and cell-free massive MIMO with the same number of antennas is shown in Table 1. As the number of antennas increases, the channel of the centralized massive MIMO system tends to be deterministic. However, this finding does not hold for the cell-free massive MIMO channel with the same number of antennas, where the channel gain is far less stable than that of the centralized massive MIMO system.