The Cell-Free mMIMO Network Based on IRSs: Mathematical Analysis and Performance Evaluation

The implementation of massive multiple input multiple output systems mMIMO can greatly enhance the spectral efficiency SE performance for users. Although, the cell-edge users as well as indoor users still have a bad performance. The cell-Free network, wherein the large number of existing antennas is replaced by the same number of distributed access points APs, can enlarge the SE performance even for shadowed users. The performance can be further enhanced after application of cooperation. When a Cell-Free mMIMO system is enriched with intelligent reflecting surfaces IRS, its performance may be improved. In this manuscript, the authors will carry out the mathematical model and simulation for the Cell-Free networks employing IRSs. The performance metrics can include the SE as well as energy efficiency EE. From the simulation results, it can be observed that implementation of IRSs inside the cell-Free networks can greatly increase the SE and EE of the system.


Introduction
During the motivation from the 1 st generation mobile system to the 5 th one, there were a lot of tools. These tools can include; small cells, mMIMO, and much more. The small cells let the mobile cellular system have more frequency reuse. In addition, the cellular system becomes more power efficient. By the same way, the implementation of MIMO technology can increase the system SE. Thanks to the MIMO technology, more gain can be obtained. This gain can include; diversity gain, beam forming, or spatial multiplexing gain [1][2][3][4][5][6][7][8].

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The green communication systems are the systems that have low power consumption. The cell-Free green networks were proposed before in [19]. The authors tried to increase the system power efficiency. Others tried to increase the SE of the cell-Free networks by deployment of relays as a form of cooperative communication [20]. In fact, the relays were good solutions for the SE enhancement in cell-Free networks. Others tried to examine the adaptation process between the small cells and the cell-Free networks [21]. Their tries aimed to optimize the network performance. The authors of [22] considered the channel aging in the cell-Free networks.
The cell-Free mMIMO network was modeled and simulated in [23]. The authors considered four levels of cooperation among the APs. The SE, based on bit error rate BER, was given in a closed formula. In [24], the combination between the cell-Free operation as well as energy harvesting was proposed. The authors assumed an IOT based cell-Free system that is based on power efficient sensors. Other authors assumed the random access resource allocation among the APs. The authors of [26] studied a cell-Free mMIMO wherein multiantenna APs serve single-antenna unmanned aerial vehicles "UAVs" and ground users "GUEs". The unsupervised learning in beam-forming was handled in cell-Free networks in [27]. In our manuscript, the cell-Free mMIMO system performance will be improved by implementation of intelligent reflecting surfaces "IRSs".
IRSs are tools that can provide the cooperative communication functionality. They are analogues to relays in their operation. In fact, the IRSs can be used to focus signals in specific direction or neglect "nulling" signals in another direction. IRSs may be preferable than relays as there is no noise amplification. In order to have the highest channel gain, the IRSs should be implemented at the receiver or at the transmitter [28][29][30][31]. There are a lot of differences between relays and IRSs [32][33][34][35].
Other work handled the IRS systems that are affected by the electromagnetic interference EMI. The authors of [36] studied the cooperative communication systems that are affected by EMI and noise. They derived the suitable mathematical models. The system was simulated. Then, they tried to compensate the interference effects by applying Alamouti codes. The authors studied the EMI in the 5G propagation channels [36]. Subsequently, they modified their analysis and simulation in order to include the millimeter wave propagation channels [37]. They used 3 × 4 space time block codes. Their work is extended to repeat the study of IRSs based systems that are affected by the EMI, in the Tera Hertz frequency bands [38].
In [39], the cell-Free mMIMO security against eavesdroppers was investigated. When the central processor receives the estimated channel parameters, the central processor can optimize the preceding parameters. The previous optimization aimed to increase the SE as well as reduction of the power consumption. In [40], the authors assumed a framework to model and simulate the scalable cell-Free mMIMO network whereas the resource allocation and distribution were assumed in [41]. The authors of [41] assumed the application of the cell-Free technology in an internet of things IOT in order to provide high reliability network with low latency.
Our work, in [42], handled the cell-Free mMIMO operation when the cognitive relays were employed. The work, in our manuscript, is an extension of the previously mentioned work wherein the IRSs are employed inside a cell-Free mMIMO system. As stated before, both relays and IRSs have the same functionality. However, the IRSs can provide an improved performance due to lack of active elements that can generate noise and provide additional noise values. Therefore, it is predicted that employing the IRSs is better than relays. However, the operation mechanism itself in IRSs can differ from that exists in relays. The pilot signal groups as well as SE calculation are completely different.
Our manuscript is concerned with the IRSs implementation inside the cell-Free networks. The IRSs can provide gain and channel amplification in certain direction. The existence of the IRSs let the cell-Free systems be more power-efficient. In other words, the IRSs existence increases the cell-Free SE as well as EE. This manuscript is organized as follows; Sect. 3 provides the mathematical analysis of the Cell-Free mMIMO network which is based on IRSs. Subsequently, the proposed system is simulated and performance comparisons are held in Sect. 4. Finally, conclusions are given in Sect. 5.

IRSs Based Cell-Free mMIMO System
In this section, the system model is considered. The cell-Free mMIMO is assumed to have APs with L number, each AP has N antennas. The system can serve K number of users. The existing APs can cooperate with each other. Moreover, they are connected over a central controller over the existing front hauls. Let h k,l refers to the channel parameter which exists between the users and the serving AP. The system is modeled and simulated in the uplink operation only. The AP can receive pilots as well as data from UEs and IRSs.

Pilot Transmission
In each frame, there are pilot signals as well as data signals. The existing time, available for pilots, is divided into group of signals. The pilots can be given as; φ 1 , φ 2, φ 3, …, φ n . Assuming that the existing pilots, in general, can be divided into two groups which are; • Group 1 is the summation of pilots that are applied to estimate the channel between UEs and IRSs. • Group 2 is the summation of pilots that are applied to estimate the channel between UEs and APs.
If we assumed that the pilot signals, which are mentioned previously φ n , has a time duration value of τ p . Assume that this duration is divided into two parts; each part has duration of τ p /2. When there is a limited number of pilots or when the number of available pilots is less than the number of existing users, there may be pilot contamination [23]. The received pilot, at an IRS, can be expressed as; The channel between UE and an AP can be modeled as follow; The transmission power of the i th user can be given by p i and the channel parameter, between a UE and an IRS, is expressed as SD . The noise vectors can be stated as LSR and LSD where each of them has subscript that refers to its channel path. The received pilots, at an IRS, can be correlated with an existing replica, in such a way that, the correlation signal is; The minimum mean square error MMSE, can be applied for channel estimation, the ̂ SRkl can be given by; where; Equation 5 gives an expression for the correlation matrix of the received pilot signal at an IRS.
The AP can receive pilots from UEs for channel estimation as well as control processes. There is a correlation between the received pilot and its known replica. This correlation can be formed as;

Data Transmission
At each AP, there are two versions of the received signals. The first version comes directly from UEs whereas the other version comes from IRSs. The received AP signals can be given by; where y is the received signal, s i is the transmitted signal from a UE, while s R is the IRS reflected signal, n is the channel noise, and RD is the channel vector between an IRS and an AP, SD is the channel vector between a UE and an AP that can include the path loss as well as shadowing.

Cooperation Among the APs
A cell-Free mMIMO system is considered wherein IRSs are implemented. Each user can be served from an AP and an IRS at least. The IRSs can provide signal directivity or signal nulling in a direction. The APs can cooperate among each other with four levels of cooperation. In the following paragraphs, the SINR and SE of each level are clarified.
where BW is the bandwidth, P C is the power consumed in the circuits, and P T is the transmitted power. The transmission power, during the uplink is the mobile equipment power as well as relays. The EE may be calculated per unity BW value.

Simulation Results
In this section, the cell-Free mMIMO system, including IRSs, is simulated. The simulation parameters are concluded in Table 1. These parameters were used before in a lot of reference. The reason for the simulation parameters repetition is to have fair comparisons with already published work [23,42]. The SE and EE are the performance metrics. The Cumulative Distribution Function CDF is evaluated for SE and EE. The CDF of SE can represent the number of users that have SE less than or equal to a certain value. The same thing is valid for the CDF value of the EE.
The propagation path loss models are chosen to be the 5G models. Moreover, the fading channels are considered. The fading and shadowing models are applied before and these models can be considered as the most appropriate models for cell-Free mMIMO systems [23]. Figure 1 and 2 displays the SE and EE performance of the proposed system without and with application of IRSs. Figure 1a displays the CDF of the SE performance of the cell-Free mMIMO without IRS deployment. On the other side, Fig. 1b displays the CDF performance of the SE of a cell-Free mMIMO when the IRSs are deployed. From Fig. 1a, b, it can be observed that, the deployment of IRSs can increase the SE performance of the cell-Free networks. Figure 2a, b clarifies the EE performance of the cell-Free mMIMO network before and after the IRSs deployment. Figure 2a displays the CDF performance of the EE of the system before IRS implementation whereas Fig. 2b shows the EE performance of the cell-Free network after implementation of IRSs. From Fig. 2a  Does not exist-Fixed IRSs be at transmitters or at receivers in order obtain the best channel characteristics. In our manuscript, the IRSs positions are assumed to be at the APs. The APs are the destinations as the uplink operation is considered. The future work can include the mobile IRSs. When using the mobile IRSs, the Doppler spread compensation should be considered. The main difference between IRSs static and mobile deployment is the propagation channel characteristics. However, the concept and their impact on the cell-Free networks is the same.

Conclusions
This manuscript was concerned with the implementation of randomly distributed IRSs in cell-Free mMIMO systems in order to provide the cooperative communication services. There were mathematical analysis and simulation through this manuscript. The simulation results displayed that, implementation of IRSs could increase the SE and EE performance of the cell-Free networks. The future work can extend to include; application of the modern signal processing techniques especially in the cooperation mechanisms, among the APs, in order to raise the SE and EE performance.