Figure 1 shows the system model providing the coverage of three cells of the mobile communication system. The system is connected to a unit frequency reuse deployed at the base station and each mobile user can receive the transmitted signals from all available BSs, which are three. The signals indicated by the bold arrow are desired signals and a dotted arrow represents interference signals received from other BS.

In addition, Figure 1 demonstrates multi-cell, multi-user MIMO, where each BS is equipped with more than one antenna. A BS communicates with the group of MS users simultaneously, and each MS is equipped with an antenna. A base station uses available CSI to reduce or eliminate multi-user interference through linear precoding schemes.

The three neighboring BSs establish cooperative transmission sites connected with the high-rate optical fiber in the given system model and share information such as resources, scheduling and precoding matrix. In this study, it is assumed that all base stations share the CSI. At the same time, the user information is shared with the BSs that cooperate.

This study uses mathematical notations, such as lower-case letters for vectors and upper-case letters for matrices. (·)^H stands for the Hermitian transpose, and (·)^T represents the transpose matrix. indicates the expectation operator and indicates the identity matrix's size that is denoted by , and N represents either number of transmitting or receiving antennae.

In a Multi-Cell MU MIMO setup, each mobile station, MS is jointly served by each cooperative site of BSs that has a B number of BSs. In this system, BS has N_t a number of transmitting antennas in this system, and M mobile station has N_r a number of receiving antennas. The BSs assist each other by sharing CSI to transmit data to the available mobile station. Furthermore, each BS uses its channel to serve all the available MSs from a system, representing the channel matrix, which is used to connect j^th MS from k^th BS, whereby indicate the number of MSs and represent available BSs on the system, respectively. The value channel matrix used in the BS involves a number of sub-matrices, which are used to allow communication to the mobile stations and descriptions expressed as

From the above Equation (1) the linear precoding scheme from k^th BS to j^th MS is expressed as

where represented as precoder at BSs and indicated as data symbols transmitted to j MS. Based on the cooperative BS, the j MS receives the signal from all BSs. In the system, the transmitted signals from only one BS are considered desired signals, and others are considered interfered signals. The expression of the system expressed as

where represent the received signal and denoted as a noise vector. A noise vector is a signal modeled by zero-mean circularly symmetric complex Gaussian (ZMCSCG) with a given noise power which is equal to , hence .

Furthermore, the equation (3) was modified by adding the noise component as

where denoted as a channel matrix connecting j^th MS user with j^th BS, and denoted with noise vector. From equation (4), the expression represents the desired signals and the interference that occurred on Multi-cell Multiuser. Furthermore, the component represents the j^th MS and k^th BS channel matrix.

From the equations above, the total amount of power applied to the j^th MS user at the transmitter expressed as

whereby β represents the received data having the same power and let represent data symbols transmitted to j MS and

whereby G is normalized. In simplicity, the amount of the data received equals the number of receiving antennae equal to one [12] . The matrix satisfies the condition whereby the interference is equal to zero.

To design the precoding matrix in the MU-MIMO scenario as in a SU-MIMO, the interference in MU-MIMO is suppressed by reducing the overlap of row vector spaces in channel matrices. A row vector space spanned by different users through effective channel matrices and MIMO processing gain requires mobile users to use the available subspaces. The precoder matrix preserved complex elements and added redundancy to the input symbol streams to improve system performance [13] .

The precoder output is launched into the MIMO channel through N_t transmit antennas. The signal is received by N_r receiving antennas and processed by the linear decoder optimized by the fixed and known channel. The linear decoder also operates in the complex domain and removes any redundancy introduced by the precoders. Figure 2 below shows a flow block of a linear precoder system when the signals are coded. The signals are transmitted via the communication channel, and at the receiver, the signals are decoded for retained the original signals. The three linear precoding schemes ZF, BD and SLNR are discussed and described in detail following subsections.

This study uses the Gaussian random channel that includes two models, the Rayleigh, and Rician distributions. Both distributions are subjected to either Line of sight (LOS) or Non-line of sight (NLOS). The channel matrix of between transmitter and receiver is obtained using real, and imaginary variables at a given zero mean, and equal variance. Furthermore, we assumed the is a normalized channel of i.i.d matrix. Then the channel matrix of each distribution expressed as

Rayleigh distribution: (24)

Rician distribution: (25)

In Equation (25), K denoted as the value of the rician factor defined as the ratio of a power of channel at LOS component with the power of the scattered component. This is obtained as to the pure LOS aat [23] . Furthermore, the when value of the channel is considered as Rayleigh fading and when value pf channel is non-fading channels. The channel matrix of LOS is assumed as a deterministic matrix with rank [24] .

From the above schemes, the assumption is made that the transmitter knows CSI. This is incurred through reverse channel estimation in time-division-duplex (TDD) or feedback in frequency-division-duplex (FDD) [25] .

Fortunately, in the above consideration at the receiver, CSI is assumed to be perfect and an error-free feedback link. However, it is impossible for the channel to perfect due to errors during the transmission between the transceivers [26] .

This section addresses the impact of the errors in the Precoding Schemes. The expression of the channel state information with estimated errors expressed as

where denoted as actual channel matrix and represents as zero-mean complex Gaussian random matrix of a real channel. indicated as an additive error component that follows i.i.d complex Gaussian distribution, which has zero mean and variance of mutually independent presented as . Furthermore, from Equation (26) the error component is added and accounted as noise obtained from channel estimated error [27] . The desired and interference components remain the same, hence the linear precoding schemes that cancel the interference term.

The results were obtained by conducting the simulation through Matlab, and analyses were made by comparing the performance of the three schemes in variation of average capacity rate in bits/sec/Hz, and the signal-to-noise ratio in (dB). The parameters used in this study are summarised in Table 1.

Through the simulation, the individual scheme obtained the results and presented on the graphs. The performance per each scheme discussed is founded on the changes in average capacity rate with the signal-to-noise ratio. Additionally,

each scheme is generalized using a unique attributes such as eigenvalue, channel inversion, precoder matrix for the SLNR, the ZF, and the BD scheme respectively.

During the simulation, the value of rician factor was set to be −10 dB. The results shown in Figure 3 indicate that the SLNR precoding scheme offers better performance compared to ZF and BD schemes in both channel distribution models. Furthermore, as the value of SNR increases the performance of each scheme is almost equal for both distributions. This is because the value of the Rician factor approaches zero, which implies that the channel distribution is considered as a Rayleigh model and the radio frequency propagation in the bands is more reliable under NLOS than under LOS conditions.

The results in Figure 4 show that SLNR performs better than ZF and BD at all SNR values. Also, the standard Rayleigh distribution performs better than the

Rician distribution in all SNR values, which means that the Rician distribution is about the same as the Gaussian distribution around the mean. This implies that random multipath components arriving at different angles are superimposed on a stationary dominant signal. The two figures showed that when the SNR value is small, an average capacity rate becomes so scarce compared to the high SNRs, because the number of transmitted antennas is larger than the received signal on the system, which receives multiple signals due to multipath transmission from the transmitter.

The CSI at a small value of SNRs in the MIMO systems mostly follows the idea of multiplicative increases in the average capacity rate. On the other hand, when the SNR value is high, the capacity rate increases according to the additive idea in a system where the number of transmitting antennas is larger than that of receiving antennas. In this scenario, the optimal input is subjected to the CSI on the transmitter side.

Figure 5 shows that SLNR performance is better than ZF and BD. However, BD is also better than ZF because ZF removes all interference from the mobile user. In addition, the performance of all schemes is reduced due to the presence of estimation errors. In this case, when the estimation error variance becomes small, the average capacity rate decreases as the value of SNR increases.

Figure 6 shows the result obtained when the variance of the estimation error is equal to 0.5. The presence of the error implies that the received signals are affected and the average capacity rate decreases as the value of SNRs increases. In addition, the SLNR precoding scheme has better performance compared to ZF and BD for both distributions. In addition, the graph shows that the SLNR scheme suffered more as the estimation error variance increased. As the reference SNR increases, the interference between the mobile users becomes more significant compared to the thermal noise of the receiver.

Subsequently, mitigating such interference by a cooperative multi-cell MIMO system becomes more valuable for system performance. In addition, system performance can be improved when using the channel estimation approach. Channel estimation is used by combining the orthogonal symbols at the beginning of the frame as a pilot or using a training sequence to estimate the value of both the transmitter (BS) and receiver (MS) channel state information (CSI).