# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 4 Uplink Transmission
### Uplink Data Transmission  
The uplink received signal at each Access Point (AP) is expressed as  
\[s_{k}=\sum_{l=1}^{L}\hat{s}_{kl}=\sum_{l=1}^{L}\mathbf{v}_{k}^{H}D_{k}\mathbf{ y}_{ul}^{l}. \tag{66}\]  
where \(s_{k}\sim\mathcal{N}(0,p_{k}^{ul})\) denotes the information-bearing signal transmitted by User Equipment (UE) \(k\) with power \(p_{k}^{ul}\), and \(\mathbf{N}_{l}\sim\mathcal{N}(0,\sigma^{2})\) represents the independent receiver noise. The matrix \(D_{k}\) is employed to select the UEs served by each AP \(k\). Based on the signal in (66), the APs and the central processing unit (CPU) can decode the symbols using one of two methods: centralized or distributed. In the distributed approach, each AP independently decodes the information. Conversely, the centralized method gathers information from the APs, which is then processed at the CPU. This study employs the Least-Squares Frequency-Domain (LSFD) technique as the centralized method. The core idea of LSFD is that each AP computes local estimates of the desired data for all UEs in the first layer and forwards them to the CPU for final decoding in the second layer. Specifically, an estimate of the data symbol from UE \(k\) at AP \(l\) is determined by local linear combining using the vector \(\mathbf{v}_{kl}\in\mathbb{C}^{N\times 1}\) in the first layer. To detect the symbol transmitted by the \(k\)-th UE, the \(l\)-th AP calculates the product of the received signal \(\mathbf{y}_{l}[n]\) and the conjugate of its locally estimated channel. The resulting value, \(\hat{s}_{kl}=\hat{h}_{kl}^{T}\mathbf{y}_{l}[n]\), is then sent to the CPU via the fronthaul. The CPU utilizes the coefficients \(a_{kl}[n]\) to compute \(\hat{s}_{k}[n]\)[19] as follows:  
\[\hat{s}_{kl}=\mathbf{v}_{kl}^{H}\mathbf{y}_{l}=\mathbf{v}_{kl}^{H}\mathbf{h}_{ kl}s_{k}^{H}+\mathbf{v}_{kl}^{H}\sum_{t\in\mathbb{P}_{k}}^{K}\mathbf{h}_{tl}s_{t} +\mathbf{v}_{kl}^{H}\sum_{t\
otin\mathbb{P}_{k}}^{K}\mathbf{h}_{tl}s_{t}+ \mathbf{v}_{kl}^{H}\mathbf{n}_{l}. \tag{67}\]  
The final estimate \(\hat{s}_{k}\) is computed as  
\[\hat{s}_{k}= \sum_{l=1}^{L}a_{kl}^{*}\mathbf{v}_{kl}^{H}\mathbf{h}_{kl}s_{k}+ \sum_{l=1}^{L}a_{kl}^{*}\mathbf{v}_{kl}^{H}\sum_{t\in\mathbb{P}_{k}\setminus \{k\}}\mathbf{h}_{tl}s_{t} \tag{68}\] \[+\sum_{l=1}^{L}a_{kl}^{*}\mathbf{v}_{kl}^{H}\sum_{t\
otin\mathbb{P }_{k}}\mathbf{h}_{tl}s_{t}+\sum_{l=1}^{L}a_{kl}^{*}\mathbf{v}_{kl}^{H}\mathbf{ n}_{l}.\]  
\[\hat{s}_{k}=\underbrace{\mathbf{v}_{k}^{H}D_{k}\mathbf{b}_{hk}s_{k}}_{\text{A} }+\underbrace{\mathbf{v}_{k}^{H}D_{k}\mathbf{e}_{hk}s_{k}}_{\text{B}}+ \underbrace{\sum_{ \begin{subarray}{c}i=1\\ i\
eq k\end{subarray} }^{K}\mathbf{v}_{k}^{H}D_{k}\mathbf{h}_{i}s_{i}}_{\text{I} }+\underbrace{\mathbf{v}_{k}^{H}D_{k}\mathbf{n}}_{\text{N}}, \tag{69}\]  
where \(\mathbf{A}\) is the desired signal over the estimated channel, \(\mathbf{B}\) is the desired signal over the unknown channel, \(\mathbf{I}\) is interference, and \(\mathbf{N}\) is noise. The channel estimate \(\hat{H}_{l}=Z_{l}\Phi\in\mathbb{C}^{N\times\tau_{p}}\) is then utilized, where \(\Phi=\left[\phi_{1},\dots,\phi_{\tau_{p}}\right]\in\mathbb{C}^{\tau_{p}\times \tau_{p}}\) is defined in (38). Thus, the channel estimate \(\hat{h}_{kl}\) can be expressed as  
\[\hat{h}_{kl}=c_{kl}\hat{H}_{l}e_{i_{k}}, \tag{70}\]  
where \(e_{i_{k}}\) denotes the \(i_{k}\)-th column of the identity matrix \(I_{\tau_{p}}\). After estimating the local data, the second layer of centralized decoding is carried out using the local estimates \(\{\hat{s}_{kl}:l=1,\dots,L\}\) and the LSFD coefficients \(\{a_{kl}:l=1,\dots,L\}\) to compute \(\hat{s}_{k}=\sum_{l=1}^{L}a_{kl}^{*}\hat{s}_{kl}\). Dynamic Mode Decomposition (DMD) is employed to process \(S\) consecutive snapshots of the pilots, sequentially storing a reduced eigenstructure to estimate the channel, forming the matrix \(\Phi\). This approach proves to be more efficient than the Least Mean Square Error (LMSE) method commonly used for Channel Estimation.