5G is being used all around the world due to its fast data throughput, low latency, low jitter, and great mobility. Various enabling technologies, including as massive MIMO and millimeter wave, are being used to meet the 5G needs. Massive MultipleInput MultipleOutput (MIMO) is one of the most important technologies for 5G and beyond wireless communication networks since it has the potential to offer very high improvements in Spectral Efficiency (SE) and Energy Efficiency (EE)
As mentioned above, on single cell single user/multiuser environments, several research works on hybrid precoders/combiners have been done




A Kalmanbased beamforming technique has been considered. 
Did not consider digital combiner and multicell environment. 

The extended simultaneous orthogonal matching pursuit (ESOMP) algorithm was considered. 
For performance comparison, they didn't use a Kalmanbased approach even though Kalmanbased beamformers offer reasonable spectral efficiency at a lower cost of complexity. 

To increase total beamforming gain, proposed hybrid precoding by dividing the highest feasible rate optimization problem into a series of sub rate optimization problems. 
For performance comparison, a Kalmanbased approach was not considered. 

Multiuser performance analysis was taken into consideration. 
Did not take into account the multicell, multiuser situation as we do in our research. 

In comparison to stateoftheart methods, hybrid precoding was considered for nonorthogonal multiple access transmission schemes and resulted in higher spectral efficiency. 
For performance comparison, the Kalman precoding was not used. 

The energy efficiency of hybrid precoding was examined in a multicell multiuser scenario. 
For performance comparison, a Kalmanbased approach was not considered. 

For performance analysis, they looked at multicell millimeter wave systems that used phase shifterbased analog beamforming and regularized zero forcing digital beamforming. 
As we do in this research, Kalmanbased beamforming was not considered. 
The rest of the paper is organized as follows. Section2 presents precoding schemes, Section3 presents proposed method. Section 4 presents the results and discussion, and section 5 concludes the paper.
In this section, precoding schemes: Zero Forcing (ZF) hybrid precoder, Minimum Mean Square Error (MMSE) fully digital precoder, kalman based hybrid precoder, and hybrid MMSE precoder have been described and used for performance comparison later in the simulation part.
Multiple antenna transmitters can be used in this approach to eliminate multiuser interference in a millimeter wave massive MIMO scenario. High noise reduction can be achieved since it nulls interference from the layer of other symbols. Authors in
The MMSE scheme is used to minimize error between Base Station (BS) transmitted symbols and user terminal received signals. This technique provides the best out of the Maximum Ratio Combining (MRT) and Zero Forcing (ZF) hence reasonable performance having moderate interference and noise can be achieved
In this precoder, Base Station (BS) sends a message signal s, after which the mobile station (MS) estimated signal is s_{e} = [s_{e1}, ..., s_{eA}] ^{T} with the training vector s(n) at the iteration, n. The mean squared error of the training vector between the transmitted signal, s and estimated signal, s_{e}, Es− se^{2 }is minimized by kalman filter algorithm
By considering s_{e} = [s_{e1}, ..., s_{eA}] ^{T} at the MS the vector observed at the n iteration can be expressed: as s_{eA} (n) = (w_{a}^{H} H_{a} F_{RF} F_{BB) }s(n) + n_{a}(n) then the minimization problem can be written as:
where He is the effective channel.
The Kalman filter state expression by considering FBB can be written as:
where the e(n) is nth Kalman iteration error, K(n) represents the gains of Kalman filter and E{diag[e(n)]} is the error s(n)−se(n). Then the expanded equations can be written as:
where Qn is noise n(n) covariance matrix which can be written as:
In this section, precoding schemes that exist in the literature have been discussed. We found that, precoding schemes can achieve Spectral Efficiency of Massive MIMO but with the complexity higher than kalman based precoding. We hence, proposed kalman based precoding under multicell multiuser environment which achieves spectral efficiency with affordable complexity. In the next section, we developed a system model in order to formulate the problem, defines the solution criteria and apply deep insights towards the solution.
In this section, mathematical model for kalman based beamformer that incorporates the digital combiner, hybrid zero forcing, hybrid minimum mean square error, and fully digital mean square error under multicell multiuser scenario has been developed.
Symbols: F_{RF}, F_{BB}, W_{RF} and W_{BB }denotes analog precoder, digital precoder, analog combiner, and digital combiner respectively. [.]^{H}, [.]^{T}, [.]^{1} denotes Hermitian, transpose, and inverse of matrix respectively. E [.] represents expectation operator, ._{F} denotes frobenius norm. N_{s}, N_{T}, N_{R} and N_{RF }denotes number of data streams, number of transmitting antennas, number of receiving antennas, and the number of radio frequency chains respectively. H represents channel matrix.
From the
The estimated signal at the receiver once combining vectors WRF for all MSs and analog precoder FRF_{k, l}at the BS are determined can be:
For transmission point (TP) ‘i’ and user ‘k’ in cell ‘l’ the terms can be noted as follow:
Having (6) received signal by user ‘K’ in the cell ‘L’ can be expressed as:
By substituting (6) into (7)
The symbol based filter algorithm that minimizes the sum MSE for multicell multiuser scenario can be:
Error e (n) at the nth Kalman iteration is for multicell multiuser scenario can be formulated as:
Having the baseband combining matrix W_{BB,} Kalman hybrid multicell multiuser state equation becomes:
The diagonal matrix error representation of multicell multiuser can be expressed as:
By substituting (11) in to (10):
Formulization of hybrid millimeter wave combining matrix through the Kalman hybrid multi cellbased approach that minimize error e(n) can be:
Taking the assumption WRF then (13) can be written as:
With the given error calculation (9) the minimization problem can be expressed as:
The spectral efficiency of Kalman hybrid precoding under multi cell multi user scenario can be:
The value
The channel matrix for linear precoders under multicell multiuser scenario can be:
The signal to noise ratio of MMSE precoding under multi cell multi user can be expressed as:
The spectral efficiency of MMSE precoding can then be SE=
The spectral efficiency of fully digital MSE precoding under multicell multiuser scenario can be:
The signal to noise ratio of zero forcing precoding under multi cell multi user scenario can be expressed as:
The spectral efficiency of Zero Forcing precoding can then be SE=
where
In this section we have developed mathematical formulations for precoding schemes including kalman based, fully digital MSE, hybrid MMSE and zero forcing under multicell multiuser environment. In the next section, we have presented simulation results for performance comparison.
The entire performance of beamforming for massive MIMO networks at millimeter wave frequency is evaluated and examined in this section using MATLAB simulation results. A hybrid beamforming for massive MIMO at mm Wave frequency with a Base station (BS) implemented with 16 × 16 Uniform phased array (UPA) and associated with 4 mobile stations (MSs) is considered, each mobile station is implemented with 4 × 4 Uniform phased array (UPA). Angle of departure (AoDs) and angle of arrival (AoAs) are uniformly distributed over [−π/2, π/2]. The azimuth AoAs/AoDs are expected to be uniformly distributed in [0, 2π], the elevation AoAs/AoDs are uniformly distributed in [−π/2, π/2], and perfect channel knowledge is considered for performance analysis. The simulation parameters are mentioned in the form of table in


Elevation AoA/AoD 
[−π/2, π/2] 
Azimuth AoA/AoD 
[0, 2π] 
Number of transmitting antennas 
128, 64, 256, 512 
Number of users 
4, 8 
SNR 
[10, 20] 
The multicell multiuser based kalman beamformer algorithm for performance evaluation is shown in algorithm 1.
Three important points can be raised from computed results. I) SE of all beamformers improved by increasing transmitting antennas under multicell multiuser scenario as can be seen in
In
The simulation results in
The beamformers with multi cell systems exhibit better spectral efficiency as compared to beamformers with single cell systems by considering transmission points (TPs) antennas varying from 64 (8X8) to 128(16X8) for single cell and from 256 (16X16) to 512 (16X32) for multicell.
The beamformers with multi cell systems exhibit better spectral efficiency as compared to beamformers with single cell systems even the number of users increases.
This research considered hybrid beamforming for multicell multiuser environment. It extended single cell multiuser scenario for Kalman based beamformer to multicell multiuser scenario in which intercell and intracell interferences are counted. In addition, extended work performance has been compared with other linear beamformers including zero forcing, hybrid minimum mean square error and fully digital mean square error beamformers. The simulation results show that SE of all beamformers improved by increasing transmitting antennas and decrease by increasing the number of users under multicell multiuser scenario. The spectral efficiency of Kalman based beamformer, hybrid minimum mean square error beamformer, fully digital beamformer, and zero forcing beamformer at signal to noise ratio of 20 dB for multicell containing 512 transmitting antennas is around 14.1 bps/Hz, 12.1 bps/Hz, 18.8bps/Hz, 12 bps/Hz respectively while the spectral efficiency of aforementioned beamformers for multicell containing 256 transmitting antennas is around 12 bps/Hz, 10.2 bps/Hz, 15 bps/Hz , and 10 bps/Hz respectively for the same signal to noise ratio. Similarly, the spectral efficiency of Kalman based beamformer, hybrid minimum mean square error beamformer, fully digital beamformer, and zero forcing beamformer at signal to noise ratio of 20 dB for multicell containing 4 users is around 12 bps/Hz, 10.2 bps/Hz, 15 bps/Hz, 10 bps/Hz respectively while the spectral efficiency of aforementioned beamformers for multicell containing 8 users is around 9.8 bps/Hz, 8.1 bps/Hz, 10.3 bps/Hz, and 7 bps/Hz respectively for the same signal to noise ratio. Simulation result also shows SE of beamformers under multicell multiuser scenario is better than that of single cell multiuser scenario.