Performance Analysis and Simulation of Millimeter Wave Cell-Free mMIMO Networks

Towards the 5Gnetworks and beyond, there are a lot of emerging technologies. These technologies include but not limited to; multiple input multiple output (MIMO), cell-Free networks, and millimeter wave bands. The cellular MIMO can provide a satisfied performance for users except the shadowed and the cell-edge ones. In order to overcome this disadvantage, the cell-Free networks are deployed. Through applications of distributed access points (APs), the cell-Free networks can provide a ubiquitous coverage for users as whole. Therefore, they can provide a high throughput for users. Moreover, the applications of millimeter wave bands can provide a high bandwidth and hence, a high throughput for users. In other words, the application of the cell-Free technology combined with the millimeter wave bands can extremely enlarge the users’ throughput. This is the motivation of our manuscript. The purpose of this manuscript is to provide mathematical model and simulation for the cell-Free mMIMO network, operating in the millimeter wave bands. The performance metrics can include; the spectral efficiency (SE), bit error rate (BER), and energy efficiency (EE). It is observed that the centralized cooperation among the APs can let users have a satisfied throughput even the system employs the maximal ratio combining (MRC). Furthermore, all cooperation levels, among APs, can perform well even for non-light of sight (NLOS) environment.


Introduction
The large increase in the given users' data rates is the goal of each mobile generation [1][2][3][4]. In order to achieve higher data rates, there are a lot of emerging tools. These tools may include, MIMO, beam forming, millimeter wave bands, cell-Free networks, and much more. The small cells, in general, can increase the frequency reuse. In addition, it can reduce the system consumption power. The cell-Free is a combination of the small cells and massive MIMO technology. The MIMO system implements a lot of antennas at each transmitter and at each receiver. The multiple antennas can provide; diversity gain, beam forming gain, and spatial multiplexing gain [5][6][7][8]. In fact, the MIMO systems can perform well for all users. However, they cannot provide a satisfied performance for the cell-edge users as well as the shadowed ones.
The massive MIMO (mMIMO) is an improved version of the MIMO systems in order to increase; spatial multiplexing gain, diversity gain, beam forming gain. All the fore-mentioned gains were already increased. However, the bad performance for shadowed users and the cell-edge ones is still a great challenge [5][6][7][8]. There were trials to improve the mMIMO performance in previous researches. The mMIMO performance was improved by applying signal processing techniques for interference mitigation in [9,10]. The error suppression techniques could improve the mMIMO performance. Such these systems can have a satisfied throughput; however, the shadowed users are still having a bad performance.
The cell-Free network concept depends on application of small and fully random distributed APs in order to let a user, anywhere, have a serving AP. Sometimes each user can find a lot of serving APs around it [11,12]. The concept of the cell-Free mMIMO is very close to the mMIMO, however, the cell-Free depends on application of small and fully distributed APs instead of application of a lot of centralized antennas on one base station. For more clarifications, the cell-Free mMIMO can provide a massive number of antennas but in a distributed manner throughout the existing coverage area. By application of distributed APs, each user can have a lot of nearby serving APs [11][12][13][14]. Therefore, there is a ubiquitous coverage for the existing mobile system. The cell-Free mMIMO concept is shown in Fig. 1.   Fig. 1 The cell-Free mMIMO network concept [13,14] There were a lot of researches in the cell-Free mMIMO network. In [13], there was a trial to carry out a mathematical model for a cell-Free mMIMO system. The authors assumed that there were four levels of cooperation among the APs. They tried to find out a competitive cell-Free mMIMO system design to the mMIMO system. They assumed the application of minimum mean square error (MMSE) and maximal ratio combining (MRC) in their analysis. Their research concluded that level 4 cooperation in the cell-Free mMIMO can perform well and it can provide the best performance when comparing with the other levels. The authors informed us that the performance of the cell-Free mMIMO system can be dominated by the applied wireless channel model. The performance comparison between the mMIMO and the cell-Free mMIMO was extended in [14]. The authors provided a closed mathematical formula for the throughput based on BER. In addition, they simulated the BER performance of the mMIMO and the cell-Free mMIMO. They already held comparisons between the two technologies. The authors also tried to apply an interference mitigation tools in order to improve the throughput performance. They used the cognitive radio technology as a powerful interference mitigation technique.
In our manuscript, the cell-Free mMIMO system is simulated in the millimeter wave frequency bands. These bands can provide high bandwidth, high throughput, little noise, and much more. As a result of the millimeter wave band advantages, this manuscript is concerned with the analysis and simulation of the cell-Free mMIMO in the millimeter wave bands. Our analysis and simulation is carried out assuming four level of APs cooperation. In addition, two combining schemes are applied. These schemes include; MMSE and MRC.

Related Work
A lot of neighboring base stations can cooperate in order to provide service coverage for users everywhere. Sometimes, the cooperation can waste the bandwidth due to the overhead signaling. This cooperation can be replaced by application of MIMO wherein a lot of antennas are existing at each transmitter and at each receiver. The implementation of a lot of serving antennas can increase the users' throughput. However, there are some users suffering from the bad service. These users are shadowed users and the cell-edge ones. In order to overcome this drawback, the cell-Free mMIMO can provide better services for all users even they are shadowed users or cell-edge ones. In [11,12], there was a comparison between the cellular mMIMO and the cell-Free mMIMO system. In these references, there were complicated signal processing techniques in order to improve the cell-Free network performance. Subsequently, there was a mathematical model and a simulation tool for the cell-Free mMIMO network in [13]. It was assumed that there were four cooperation levels among the APs as well as a lot of signal processing "combining" techniques. These techniques included; MMSE and MRC. The authors exerted their utmost in order to find the best cell-Free scenario that is comparable to the cellular mMIMO. The SE was only the performance metric. This work is extended in [14] wherein the comparisons, between the cellular MIMO and the cell-Free mMIMO, still exist. However, the BER was considered and its effect was simulated. The authors considered the cognitive radio as an interference mitigation tool in order to reduce the interference and hence improve the users' throughput.
In [15], the authors optimized the weighted sum energy efficiency (WSEE) for the uplink of a cell-Free mMIMO system by using deep deterministic policy gradient (DDPG) technique, which does not require labeled data. Subsequently, the authors in [16] adopt the cell-Free mMIMO to support URLLC in a smart factory. Their performance analysis was based on estimating the lower bound data rate for users. In [17], the authors studied the secrecy of the energy efficiency in multi-user for eavesdropping. They used the ergodic secrecy rate as a metric for performance evaluation. They tried to model the total consumed power especially in APs in a cell-Free mMIMO. The authors of [18] studied the orthogonal pilot assignment performance in a cell-Free mMIMO based on users' activity detection. They considered, in their research, the interference among the users. Others investigated in their letter the security of user-centric cellfree massive multiple-input multiple-output (UC-CF mMIMO) system in the presence of multiple collusive eavesdroppers (Eves) [19]. The authors of [20] analyzed the grant-free random access (GFRA) in cell-free massive multiple-input multiple-output (CF-mMIMO) systems.
Unfortunately, the previous cell-Free studies were carried out at radio frequency bands however, the cell-Free concept can be applied in 5G networks and beyond which are operating in millimeter wave frequency bands. Therefore, the study of mMIMO and cell-Free mMIMO, at millimeter wave bands, has a great attention and this study will not be clarified up till now. In this paper, the study of cellular mMIMO and the cell-Free mMIMO, which are operating at millimeter wave frequency bands, is carried out.
The main contribution of our paper can be stated as follows; • The performance of the cell-Free mMIMO and the cellular mMIMO are mathematically analyzed and simulated by applying the millimeter wave propagation models. The motivations to the operation in the millimeter wave bands are; high available bandwidth, high security, easy and simple antenna design, and much more. They are the bands that already applied in the 5G networks and beyond • The performance of the cell-Free mMIMO system, operating in the millimeter wave band, is not limited to SE only. Our performance metrics can include; SE based BER as well as the EE • The SE, EE, and BER performance of the cell-Free mMIMO operating in the millimeter wave bands is compared to the cellular mMIMO system. There is a trial to find out the most competitive cooperation mechanism of the cell-Free mMIMO to the cellular mMIMO one when both of them operate in the millimeter wave frequency bands • The SE based BER and EE performance of the cell-Free mMIMO and the cellular mMIMO is studied for LOS operation and NLOS operation when the millimeter wave bands are applied.
Our paper is organized as follows; Sect. 3 provides the mathematical analysis of the cell-Free network operating in the millimeter wave band. In Sect. 4, the cooperation mechanisms, among the APs in the millimeter wave cell-Free mMIMO network, will be explained. The millimeter wave channel models are clarified in Sect. 5. Subsequently, the simulation comparisons between the millimeter wave cell-Free mMIMO networks and the millimeter wave cellular mMIMO are held in Sect. 6. Finally, conclusions about the paper are given in Sect. 7.

Cell-Free mMIMO Mathematical Model
Consider a cell-Free network model that has L distributed APs. Each AP has N antennas. The APs are connected over a centralized controller "cloud-edge processor". These APs can serve K users. Assume that h k,l describes the channel between the lth AP and the kth user. The uplink transmission process can be divided into; pilot transmission and data transmission [14].

Pilot Transmission
Let the pilot signals be; φ 1 , φ 2, φ 3 ,…, φ τp . These pilot signals have the same length which is τ p . The orthognality of the existing pilots is assumed. These pilots are used for channel estimation as well as control processes [14]. Assume that the UEs transmit their pilots; the pilots that are received can be expressed as; where p i is the transmit power of ith user, h i is the channel vector between ith user and lth AP, and N l is the noise signal. To estimate the channel parameters, each AP should correlate the received pilot signal with a locally generated version of the pilot signal. The result of this correlation can be as follows; where τp is the time allowed for pilot transmission [14]. By using the MMSE, the channel parameter, h k,l , can be given by;  (4) gives an expression for the correlation matrix of the received signal [14].

Data Transmission
The received complex base band signal, in the uplink, can be given by; where y is the received signal, si is the transmitted signal from ith user, n is the channel noise, and h il is the channel parameter that include, path loss, fading, shadowing, and much more [14].

Cooperation Among the APs
In the cell-Free mMIMO, there are a lot of cooperating APs that can provide service for users. The cooperation, among the APs, can be carried out by four methods [14] which can be stated as follows;

Fully Centralized APs "Level 4"
During this cooperation, the AP can act as relays, in such a way that, it can relay both pilot and data to the central controller. It does not have the capability to operate any processing functions. The received signal, in this cooperation mechanism, is related to the transmitted one by the following relation; The received signal, y l , can be expressed as a function of the transmitted signal, s i , as follow; where n l is the noise signal. The SE can be calculated as follow; where τ p is the pilot length, τ c is the block length (pilot and data). SINR represents the signal to interference and noise ratio. α is a parameter depends on the BER as follow; where BER is the bit error rate. With the help of [14], 21, the energy efficiency, EE, can be expressed as; Fig. 4 a- 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. The EE can be calculated per unity BW value.

Level 3
In this level, the APs can share in the channel estimation by simple processing for the received pilots. However, the full pilot and data detection are carried out at the ntral processor. In this cooperation level, the SE can be calculated as follow; The only difference in SE calculations, from the previous cooperation level, is the SINR value.

Level 2
During this cooperation level, the APs can detect the pilot in order to have the channel estimates. Then, the central controller can receive only the average of channel estimates that will be used in data detection and extraction. The SE and EE can be calculated as in Eqs. (7) and (9).

Fully Distributed "Level 1"
In this cooperation, detection of pilots and data can be carried out at each AP. Really, the APs has the capability of data and pilot detection. Then, the central controller, in this case, can only provide the cooperation mechanism among the received data from different APs as they are considered as one base station. The spectral efficiency, SE, and the signal to interference plus noise ratio, SINR, can be calculated by the following relations; The EE can be calculated as in Eq. (9).

Millimeter Wave Channel Models
The millimeter wave channels can be modeled by a lot of methods, however, the applied model, in this manuscript, is the simplest one that is provided by Rappaport [22]. In fact, the millimeter wave bands can have a lot of advantages such as; high band width, little noise, high throughput, high security, enormous available free bands, and much more. There was a measurement model in [23]. This model considered the cellular system operating at 60 GHz for outdoors. Although, the millimeter wave can provide short range communication only, they are widely used. The authors in [24] applied the millimeter wave channels as backhauls especially in the forest communication. The millimeter channel models were applied in indoors especially in small scale applications [25]. Others considered the millimeter that operates at 60 GHz [26]. Subsequently, in [27], the characteristics of spatial channel modeling were simulated for 73 GHz mmWave bands using NYUSIM. Spatial consistency channel models for moving users and channel models for static users without spatial consistency consideration had been compared in terms of different channel parameters for both LOS and non-LOS (NLOS) environment. The authors of Ref. [28] presented an empirically based analysis of propagation characteristics in two vegetated suburban areas with different types and fractions of vegetation cover in 5G millimeter-wave "mmWave" bands. The Cumulative Distribution Function (CDF) is used in the performance metric [29]. In this paper, the millimeter wave channel models, which are in Ref. [22], was applied in the mMIMO and the cell-Free mMIMO networks in order to estimate the performance of the two strategies at millimeter wave frequency bands. The models, provided by Theodore Rappaport [22], are the easiest way to validate the performance of mMIMO and the The values of α, β, and ζ are included in Table 1.

Simulation Results
In this section, the cell-Free mMIMO system is simulated assuming that the millimeter wave channel models are applied. The simulation parameters are concluded in  Figure, it is obvious that when the BER values are low, the communication system reliability is very high. When the BER values are minimized, the SE and the EE are minimized. There is a direct proportional relationship between BER and the SE. It also can be deduced that, the Level 4 "Fully Centralized" can provide better SE and EE than the cellular mMIMO network especially for the shadowed users and cell-edge ones when the MMSE is applied. Furthermore, the MRC can let the cell-Free mMIMO have a non-satisfied performance. Then, it is not recommended to operate the cell-Free mMIMO at the millimeter wave by the MRC especially at 28 GHz band even there are good LOS operation between the APs and UEs. Figures 4 and 5 display the performance comparisons between the cellular mMIMO and the cell-Free mMIMO when applying the NLOS-28 GHz model especially for the BER values of 10 -3 and 10 -8 respectively. It can be concluded that the all cooperation levels, in a cell-Free mMIMO, can have a satisfied SE and EE performance. On the other side, the cellular mMIMO has a bad SE and EE performance. It can be noticed that, the cell-Free mMIMO can operate better than the cellular mMIMO especially at the millimeter wave channel bands. It also can be deduced that the cell-Free mMIMO can greatly solve the problems of bad performance for shadowed users. The performance of the cell-Free mMIMO is still satisfying even the existing channel model is a millimeter wave one operating at NLOS conditions. Figures 6 and 7 show SE and EE of cell-Free mMIMO system and cellular mMIMO when the LOS-73 GHz model is applied assuming that the BER is 10 -3 and 10 -8 respectively. From Figs. 6 and 7 it can be observed that the low BER value can provide a reliable communication system. Moreover, the Level 4 "fully centralized" can have a better performance than the cellular mMIMO when the MMSE or the MRC are applied. This is a unique characteristic to this propagation model. The SE and EE performance comparisons between the cellular mMIMO and the cell-Free mMIMO are held for the other millimeter wave channel models. Figures 8, 9, 10, and 11 displays the SE and EE performance of the MMSE and MRC in the cell-Free mMIMO for 73 GHz NLOS operation "Model 1 and Model 2" at different BER values. From these figures, it can be concluded that the low BER results in a high reliable communication system. However, this results in low SE and EE values. The 73-GHz NLOS model 1 and model 2 can have a low SE and EE performance. However, the cell-Free mMIMO can provide a little better SE and EE performance than the cellular mMIMO. In general, the NLOS propagation models can let the mMIMO and the cell-Free mMIMO have a non-satisfied performance.

Conclusions
During this manuscript, the cell-Free mMIMO was mathematically modeled and simulated assuming that the existing channel models are millimeter wave channels. The combination of the cell-Free concept as well as the millimeter wave channel can provide large users' throughput. The performance metrics were the SE and EE. It can be concluded that the