# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 2 Materials and Methods  
### Simulation Scenario  
Our approach adheres to the standard massive MIMO protocol outlined in [21]. We adopt the notation from [5]; the system is illustrated in Figure 1, where a CF-MMIMO system operating under Time Division Duplexing (TDD) is used. All single-antenna User Equipment (UE) is simultaneously served by all Access Points (APs) (in this work Access Points and base stations (BSs) are exactly the same), \(L\) represents the number of APs, \(M\) is the number of antennas at each AP, and \(K\) is the number of UEs. The APs are linked to a central processing unit (CPU) through front haul connections. The communication channel undergoes continuous variations, segmented into \(\tau_{p}\) channel uses for uplink pilots and \(\tau_{u}\) for uplink data, with the total coherence interval given by \(\tau_{c}=\tau_{p}+\tau_{u}\) (Figure 2). We utilize a block-fading model, where \(h_{kl}\in\mathbb{C}^{N\times 1}\) denotes the channel response between the \(k\)-th UE and the \(l\)-th AP, for \(k=1,\ldots,K\) and \(l=1,\ldots,L\). Each fading block is independently drawn from a Rayleigh distribution, as described in [19]. The number of time-frequency snapshots considered in this study is represented by \(\tau_{s}\). 1. \(\tau_{p}\) symbols for uplink pilots;
2. \(\tau_{u}\) symbols for uplink data;
3. \(\tau_{d}\) symbols for downlink data. The channel vector between \(AP_{l}\) and \(UE_{k}\) is denoted as \(h_{l,k}\in\mathbb{C}^{M\times 1}\), incorporating both small-scale and large-scale fading characteristics. The following assumptions are made:  
* The channel is reciprocal due to the precise calibration of the hardware chains, which can be achieved through standard methods [21]. * The channel is not static within a specific time-frequency interval, known as the coherence interval. It varies continually and independently between these intervals [3]. * The channel, \(h_{l,k}\in\mathbb{C}^{N}\), follows a Rayleigh fading distribution, \(\text{CN}\sim(0,\beta_{l,k})\), where \(\beta_{l,k}\) represents the constant large-scale fading coefficient (channel variance) between \(AP_{l}\) and \(UE_{k}\). This coefficient is consistent across antenna elements, meaning it does not depend on the antenna element index \(m\), \(m=1,\ldots,M\)[5]. * The large-scale fading coefficients are known in advance using DMD prediction at each AP. They change slowly and are influenced by UE mobility. Therefore, we assume that channel variances are estimated early in the process, and these estimates are then used to gauge the current channel response [11]. * Unlimited Front-Haul Network Capacity. The capability of CF-MMIMO with front-haul capacity limitations has been explored in [4, 22]. The TDD frame consists of three phases [7]:  
Figure 2: Communication blocks in the time–frequency plane. There appear uplink data pilots (\(\tau_{p}\)) and uplink (\(\tau_{u}\)) and downlink data. 1. Uplink (UL) pilot transmission, also known as UL training;
2. Uplink Data Transmission; and
3. Downlink (DL) data transmission. We adopt the block-fading model, where \(h_{kl}\in\mathbb{C}^{N\times 1}\) represents the channel response between the \(k\)-th UE and the \(l\)-th AP, with \(k=1,\ldots,K\) and \(l=1,\ldots,L\). Within each block, an independent realization is drawn from an independent Rayleigh fading distribution:  
\[h_{l,k}=N_{c}(0,\beta_{kl}I_{N}) \tag{2}\]  
The large-scale fading coefficients \(\beta_{l,k}\) incorporate pathloss and shadow fading as follows [7]:  
\[\beta_{l,k}=PL_{l,k}.10^{\frac{c_{kl}\beta_{l,k}}{10}} \tag{3}\]  
where \(d_{l,k}\) is the distance between AP 1 and UE \(\rm k\), including the AP's and UE's heights. The shadow fading accounts for spatial correlation both between APs and between UEs. We consider an area of radius R meters (1000 m). APs and UEs are uniformly distributed at random on the circle. For the simulation, the movement of the UEs is randomly positioned within the circle of radius R, and their motion is confined within it.