Performance analysis of downlink massive MIMO system with precoding techniques and pilot reuse factor

Abstract

Massive multiple input multiple output (Ma-MIMO) is one of the basic enabler technologies for 5 G wireless communication networks to provide higher spectral efficiency (SE) by using large number of antennas and serves many users simultaneously in the same frequency—time resources. In a time division duplex (TDD) Ma-MIMO downlink (DL) system, the base station (BS) estimates the channel using uplink (UL) training but a pilot contamination (PC) is the major challenge in multi-cell Ma-MIMO system. This challenge causes severe inter-cell interference at the home cell and then limits the performance of the system. Precoding techniques with pilot reuse factor (PRF) are used to mitigate the PC effect. The performance of DL multi cell Ma-MIMO system is analyzed under spatial correleted Rayleigh fading channel with both perfect and imperfect channel state information (CSI). Precoding is used to mitigate the intra-cell interferences and PRF reduces the PC effect by assigning orthogonal pilots to neighboring cells. SE performance of multi cell Ma-MIMO system is analyzed with maximum ratio transmitter (MRT), zero forcing (ZF) and minimum mean square error (MMSE) precoding techniques and combined with PRF. Performance comparison of precoding techniques showed that MMSE provides the best SE performance for both CSI and it has highest SE at PRF of four among all when the number of antennas are changed. When both the number of users (K) and antennas (M) are increased simultaneously and $\mathit{M}>>\mathit{K}$, the SE is improved by more than 25% and also system performance with perfect CSI is better than imperfect CSI.

Keywords:

• CSI
• Ma-mimo
• PRF
• PC
• Precoding

Public Interest Statement

Massive multi-input multi-output (MIMO) is a form of wireless communication technology in which base stations are equipped with a very high number of antenna components in order to improve and boost network performance. Massive MIMO effectively enhances the communication system’s spectrum efficiency and channel capacity while concurrently increasing connection reliability and data transmission rate. Massive MIMO systems, however, experience a bottleneck because of the pilot contamination brought on by users exchanging non-orthogonal pilots. Pilot contamination is therefore a major problem in massive MIMO systems. The strong inter-cell interference caused by this issue consequently limits the performance of the system. In order to reduce the pilot contamination effect, various precoding techniques with a pilot reuse factor are used in this work.

1. Introduction

Massive MIMO (Ma-MIMO) is a multi user-MIMO (MU-MIMO) system where a base station (BS) with a large number of antennas array serves MU terminals simultaneously, each having a single antenna or more antennas, in the same time-frequency resource (Rusek et al., 2012). It has been proposed as a solution to scalability and uses simple linear signal processing both on the uplink (UL) and downlink (DL). In Ma- MIMO (M antennas is much larger than K users), only the BS requires and learns channel and operates in time division duplex (TDD) mode to exploit channel reciprocity. Sending UL pilots and exploiting channel reciprocity are used for channel state information (CSI) acquisition. In Ma-MIMO systems, deploying large number of antennas at the BS focuses the transmit energy into smaller regions. This leads high improvement in radiated spectral efficiency (SE). The SE improvement results from spatial multiplexing gain (Asif et al., 2022; Elijah et al., 2015; Rusek et al., 2012). Thus, Ma-MIMO is expected to increase the SE in the order of $10\times$ (Asif et al., 2022; Marzetta, 2010; Quoc Ngo, 2015; Rusek et al., 2012). Ma-MIMO is one of the key technologies for 5 G and beyond wireless communication networks due to its higher SE.

Practical Ma-MIMO systems contain many cells even though pilot contamination (PC) is the major challenges in multi-cell Ma-MIMO systems. In a multi-cell scenario, user within each cell receives inter-cell interference in addition to intra-cell interference. The CSI of Ma-MIMO is estimated at the BS via UL training based on TDD. MUs send orthogonal signals which are known at the BS in the UL training. However, in TDD operation the coherence block is limited and unique orthogonal signals for all users in all cells cannot be realized (Al-Hubaishi et al., 2019). This indicates that orthogonal signals can be reused within a cell or in adjacent cells. This result intercell interference and channel estimation error. This effect is called PC. The intercell interference is because of pilot contamination and this restricts the system capacity by the work (Hoydis et al., 2013). Therefore, PC degrades the system performance and becomes one of the main reasons for the performance loss in Ma-MIMO systems. To alleviate this problem and analyze the performance of multi-cell Ma-MIMO systems various researches have been conducted using different schemes or techniques like precoding, channel estimation and pilot scheduling. The work by Zhao et al., (2017) suggested a combination of PC precoding (PCP) and pilot assignment to combat PC in multi-cell MU- Ma-MIMO systems. Authors used two heuristic pilot assignment schemes, such as a swapping-based and a greedy scheme combine with ZF-PCP and studied the system performance of the DL transmission in Ma-MIMO in terms of sum rate metrics as the number of BS antennas is increased and also studied the computational complexity of the proposed schemes. But the PCP matrix depends on the pilot assignment information and altered depending on the update in pilot assignment information. (Al-Hubaishi et al., (2019) proposed an efficient pilot assignment (EPA) scheme against PC in Multi cell Ma-MIMO Systems. Authors used this scheme combined with two UL receiver detectors, maximum ratio combiner (MRC) and zero forcing (ZF) to reduce PC problem in Multi cell UL Ma-MIMO Systems. But the work did not consider intra cell interference. Thakur & Chandra Mishra, (2019) proposed performance analysis of energy-efficient MU Ma-MIMO system. Authors investigated the EE and data rate MU Ma-MIMO system performance by assuming channel reciprocity error using precoding schemes like MRT, ZF and PRF to reduce the PC and optimize the SE for different users density per cell. Although the system is restricted to line of sight propagation. Salh & Audah, (2020) suggested a PRF technique to mitigate PC for Channel Estimation in Multi-cell Ma-MIMO Systems. Authors used a channel-estimation scheme by applying an orthogonal pilot reuse sequence to minimize PC in edge users with minimized channel quality as per the approximation of large-scale fading, and evaluated the performance of this scheme using the ZF and MRT precodings under the channel is i.i.d Rayleigh fading.

In this research work, the performance of DL multi-cell Ma-MIMO system is analyzed using PRF and linear precoding techniques. We used low complexity precoding techniques such as MRT, ZF and multi-cell minimum mean square error (M-MMSE) to minimize intra-cell interference by directing the signal to the intended user and nulls to unintended users and to reduce the inter-cell interference we used PRF values one, three, and four. M-MMSE is used to reduce inter-cell interference in addition to intra-cell interference.

This paper is organized as follows. Section 2 explains system model and section 3 describes PC and PRF. In section 4, precoding techniques and SE of both perfect and imperfect CSI are described. Simulation results and discussion are demonstrated in section 5 and finally, concluding remarks are made in section 6.

2. System model

As shown in Figure 1, we considered a Multi-Cell MU-DL TDD-Ma-MIMO system model. In this model, the system composed of L hexagonal cells, and each cell contains a BS with M-antennas and K single-antenna user terminals and in each cell, K single-antenna users communicate simultaneously to their BS.

Figure 1. Multi-Cell Multi-user Downlink TDD-Massive MIMO System Model.

Let $h_{jk}^i$ is the channel response between user k in cell j and the BS in cell i, and this channel response represents the propagation channels of the $k^{\mathit{th}}$ user located in the $j^{\mathit{th}}$ cell and the BS in $i^{\mathit{th}}$ cell and it is modeled as spatial correlated Rayleigh fading. The channel responses are all Gaussian distributed with zero mean and it is entirely defined through the correlation matrices (Marzetta, 2010). Because practical channels are generally spatially correlated, the BS antennas have non-uniform radiation patterns and the physical propagation environment makes some spatial directions more probable to carry strong signals from the transmitter to the receiver than other directions (Bjornson et al., 2017; Marzetta, 2010). By considering pathloss, shadowing, multipath fading and spatial channel correlation, a correlated Rayleigh fading channel model of a user is expressed as (Bjornson et al., 2017).

(1) ${\mathrm{h}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}},\sim {\mathrm{N}}_{\mathrm{C}}\left(0,{\mathrm{R}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}\right)$(1)

where ${\mathrm{N}}_{\mathrm{C}}$ denotes complex Gaussian distribution. In which the channel response converges to complex Gaussian distribution.

${\mathrm{R}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}\in {\mathbb{C}}^M{\times }^M$ is positive semi-definite spatial channel correlation matrix. The spatial channel correlation matrix describes the macroscopic propagation effects including the radiation patterns and antenna gains at the transmitter and receiver. Thus, the average channel gain from the ${\mathrm{i}}^{\mathrm{th}}$ BS antenna to user k in cell j is given by (Marzetta & Quoc Ngo, 2016).

(2) ${\beta }_{jk}^i=\frac{1}{M}\mathrm{tr}\left(R_{jk}^i\right)=\mathrm{{\rm Y}}-10\mathit{\alpha }{log}_{10}\frac{{\mathrm{d}}_{jk}^i}{1\mathrm{km}}+{\mathrm{F}}_{jk}^i$(2)

where ${\beta }_{jk}^i$ the large-scale fading coefficient that describes the effect of both path-loss and shadowing, ${\mathrm{d}}_{jk}^i$ is distance between the ${\mathrm{k}}^{\mathrm{th}}$ user in the j cell and the BS in ${\mathrm{i}}^{\mathrm{th}}$ cell and the receiver, $\mathit{\alpha }$ is the path-loss exponent, and $\mathrm{{\rm Y}}$ is the median channel gain at the reference distance of 1 km.

${\mathrm{F}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}$ represents a log-normal shadowing, ${\mathrm{F}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}\sim \mathrm{N}\left(0,{\sigma }_{\mathrm{s}\mathrm{h}}^2\right)$ is around the nominal value $\mathrm{{\rm Y}}+10\mathit{\alpha }{log}_{10}\frac{{\mathrm{d}}_{jk}^i}{1\mathrm{km}}$ where ${\sigma }_{\mathit{sh}}$ is the standard deviation of the shadow fading model. The shadow fading adds random correction term to obtain a model that better fits with practical channel measurements (Rong et al., 1996).

To simplify our analysis, we considered the local scattering-based spatial channel correlation model. The local scattering correlation model describes the basic characteristics of spatial channel correlation in terms of angular standard deviation (ASD) and the nominal angle. These are the basic parameters to model the spatial channel correlation matrix. The ASD describes the random deviation from the nominal angle with given standard deviation. The scenario of a Ma-MIMO system under non line of sight (NLOS) propagation with the local scattering spatial correlation model is shown in Figure 2.

Figure 2. NLOS propagation under local scattering spatial correlation model (Bjornson et al., 2017).

This approach helps to develop a model for the spatial correlation matrix for NLOS propagation between the user and the BS equipped with uniform linear array (ULA) antennas. The received signal at the BS is the superposition of N-multipath components. In Gaussian angular distribution with small ASD values, an approximate closed form expression of the spatial channel correlation matrix can be expressed as (Bjornson et al., 2017).

(3) ${\mathrm{R}}_{\mathrm{jm}}=\mathit{\beta }\mathit{e}{}^{j2\pi d_h(\mathit{j}-\mathit{m})sin({\sigma }_{\phi })}{\mathrm{e}}^{-\frac{{\sigma }_{\phi }^2}{2}\mathit{j}2\mathit{\pi }{d}_h(\mathit{j}-\mathit{m})cos{((\mathit{\phi }))}^2}$(3)

The DL transmission has two steps: Training step and DL data transmission step.

(i) Uplink Training Phase (Uplink Pilot Transmission) Based Channel Estimation

In the training step, the BS estimates the CSI from K users that depends on the received pilot sequences in the UL so as to make efficient use of the massive number of antennas. Users transmit pilots to BSs, and then the BSs will estimate the corresponding channel coefficients. Let ${\tau }_p$ samples are reserved for UL pilot signaling or training phase in each coherence block. Each UE sends a pilot sequence that spans these ${\tau }_p$ samples. All user terminals in the system send pilot sequences to BSs synchronously, and then the ${\mathrm{i}}^{\mathrm{th}}$ BS receives the pilot sequences transmitted from users and these received pilot sequences at the ${\mathrm{i}}^{\mathrm{th}}$ BS is denoted by ${\mathrm{Y}}_{\mathrm{i}}^{\mathrm{P}}\in {\mathbb{C}}^{\mathrm{M}\times {\mathit{\tau }}_{\mathrm{p}}}$ as shown in equation 4 (Bjornson et al., 2017).

(4) ${\mathrm{Y}}_{\mathrm{i}}^{\mathrm{P}}=\sum _{\mathit{k}=1}^{K_i}\sqrt{{\rho }_{\mathit{ik}}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}{{\mathit{\varphi }}_{\mathit{ik}}}^{\mathrm{H}}+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^L\sum _{\mathit{m}=1}^{K_j}\sqrt{{\rho }_{\mathit{jm}}}{\mathrm{h}}_{j\mathrm{m}}^{\mathrm{i}}{{\mathit{\varphi }}^{\mathrm{H}}}_{\mathrm{jm}}+{\mathrm{N}}_i$(4)

where the superscript P in ${\mathrm{Y}}_{\mathrm{i}}^{\mathrm{P}}$ stands for pilots. $p_{\mathit{ik}}$ is The UL transmit power which scales the elements of ${\varphi }_{\mathit{ik}}$. H is used as transpose. ${\mathrm{N}}_{\mathrm{i}}$ denotes the additive white Gaussian noise (AWGN). The first term in equation 4(4) ${\mathrm{Y}}_{\mathrm{i}}^{\mathrm{P}}=\sum _{\mathit{k}=1}^{K_i}\sqrt{{\rho }_{\mathit{ik}}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}{{\mathit{\varphi }}_{\mathit{ik}}}^{\mathrm{H}}+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^L\sum _{\mathit{m}=1}^{K_j}\sqrt{{\rho }_{\mathit{jm}}}{\mathrm{h}}_{j\mathrm{m}}^{\mathrm{i}}{{\mathit{\varphi }}^{\mathrm{H}}}_{\mathrm{jm}}+{\mathrm{N}}_i$(4) represents the received pilot signals from users in the home or target cell. The middle term express the inter-cell interference signal from the neighbor cells, which causes the pilot contamination.

The BS performs a de-spreading operation by correlating the received signals with each of the K pilot sequences. This is equivalent to right-multiplying the received signal matrix by the matrix of pilots, yielding.

(5) ${\mathrm{Y}}_{ik}^i=\sqrt{{\rho }_{\mathit{ik}}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}{\varphi }_{ik}^{\mathrm{H}}{\varphi }_{ik}^{\star }+\sum _{\mathit{m}=1,\mathit{m}\ne \mathit{k}}^{K_i}\sqrt{{\rho }_{\mathit{im}}}{\mathrm{h}}_{\mathrm{i}\mathrm{m}}^{\mathrm{i}}{\varphi }_{im}^{\mathrm{H}}{\varphi }_{ik}^{\star }+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^L\sum _{\mathit{m}=1}^{K_j}\sqrt{{\rho }_{\mathit{jm}}}{\mathrm{h}}_{\mathrm{j}\mathrm{m}}^{\mathrm{i}}{{\mathit{\varphi }}_{\mathit{jm}}}^{\mathrm{H}}{\varphi }_{ik}^{\star }+{\mathrm{N}}_i{\varphi }_{\mathrm{i}\mathrm{k}}^{\star }$(5)

The second and third terms in equation 5(5) ${\mathrm{Y}}_{ik}^i=\sqrt{{\rho }_{\mathit{ik}}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}{\varphi }_{ik}^{\mathrm{H}}{\varphi }_{ik}^{\star }+\sum _{\mathit{m}=1,\mathit{m}\ne \mathit{k}}^{K_i}\sqrt{{\rho }_{\mathit{im}}}{\mathrm{h}}_{\mathrm{i}\mathrm{m}}^{\mathrm{i}}{\varphi }_{im}^{\mathrm{H}}{\varphi }_{ik}^{\star }+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^L\sum _{\mathit{m}=1}^{K_j}\sqrt{{\rho }_{\mathit{jm}}}{\mathrm{h}}_{\mathrm{j}\mathrm{m}}^{\mathrm{i}}{{\mathit{\varphi }}_{\mathit{jm}}}^{\mathrm{H}}{\varphi }_{ik}^{\star }+{\mathrm{N}}_i{\varphi }_{\mathrm{i}\mathrm{k}}^{\star }$(5) represent interference and contain inner products of the form ${\varphi }_{ik}^H{\varphi }_{\mathit{ik}}$ between the pilot of the desired UE and the pilot of another UE i in cell j. To estimate the channel of a particular user, the BS needs to know the transmit pilot sequence of the user. so, the pilots should be deterministic sequences and pilot assignment is made when the user connects to the BS.

Ideally, all pilot sequences to be orthogonal, but practically, the pilots are limited,which, is ${\tau }_p$ -dimensional vectors due to coherence block is limited, for a given ${\tau }_p$, we can only find a set of at most ${\tau }_p$ mutually orthogonal sequences. The limited length of the coherence blocks is limited, ${\tau }_p\le {\tau }_c$. Thus, assigning mutually orthogonal pilots to all UEs is impossible in practice. Assuming that the system uses a set of ${\tau }_p$ mutually orthogonal pilot sequences. These can be gathered as the columns of the UL pilot book $\mathit{\varphi }\in {\mathbb{C}}^{{\tau }_p}\times {\tau }_p$ which satisfies ${\mathrm{\Phi }}^H\mathit{\varphi }={\tau }_p{I}_p.$

For conventional pilot reusing scheme, the ${\mathrm{k}}^{\mathrm{th}}$ users in all cells reuse the ${\mathrm{k}}^{\mathrm{th}}$ pilot sequence, i.e. The set with the indices of all UEs use the same pilot sequence as UE k in cell i (Al-Hubaishi et al., 2019).

(6) ${\rho }_{\mathit{ik}}=\left\{(\mathrm{j},\mathrm{m}):{\varphi }_{\mathit{jm}}={\varphi }_{\mathit{ik}.}\mathit{j}=1\dots \dots .\mathrm{L},\mathrm{k}=1\dots \dots {\mathrm{K}}_{\mathrm{j}}\right.$(6)

Equation 6(6) ${\rho }_{\mathit{ik}}=\left\{(\mathrm{j},\mathrm{m}):{\varphi }_{\mathit{jm}}={\varphi }_{\mathit{ik}.}\mathit{j}=1\dots \dots .\mathrm{L},\mathrm{k}=1\dots \dots {\mathrm{K}}_{\mathrm{j}}\right.$(6) indicates UE m in cell j uses the same pilot as UE k in cell i, $(\mathit{j},\mathit{m})\in P_{\mathit{ik}}$ and Using then equation in 5 is simplified as (Al-Hubaishi et al., 2019):

(7) ${\mathrm{Y}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}=\sqrt{{\rho }_{\mathit{ik}}}{\tau }_{\mathrm{p}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}+\sum _{(\mathit{j},\mathit{m})\in \mathit{pik}\mathrm{\setminus pik}}\sqrt{{\rho }_{\mathit{jm}}}{\tau }_{\mathrm{p}}{\mathrm{h}}_{jm}^{\mathrm{i}}+{\mathrm{N}}_{\mathrm{i}}{\varphi }_{\mathit{ik}}$(7)

The first term in equation 7(7) ${\mathrm{Y}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}=\sqrt{{\rho }_{\mathit{ik}}}{\tau }_{\mathrm{p}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}+\sum _{(\mathit{j},\mathit{m})\in \mathit{pik}\mathrm{\setminus pik}}\sqrt{{\rho }_{\mathit{jm}}}{\tau }_{\mathrm{p}}{\mathrm{h}}_{jm}^{\mathrm{i}}+{\mathrm{N}}_{\mathrm{i}}{\varphi }_{\mathit{ik}}$(7) represents the desired pilot signals. The second and third terms describe interfering pilot and noise respectively. Where ${\mathrm{Y}}_{\mathrm{i}\mathrm{k}}^i$ is the processed received signal. The processed received signal is a sufficient statistics to estimate ${\mathrm{h}}_{\mathrm{j}\mathrm{k}}^i$. In order to implement precoding and decoding, the home cell needs only an estimate of its own channel matrix. Assuming that the BS uses MMSE channel estimation.

The MMSE estimator of ${\mathrm{h}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}$ is the vector that minimizes the MSE $\mathit{E}{∥h_{jk}^i-{\hat{h}}_{\mathit{jk}}^{\mathit{i}}∥}^2$

where ${\hat{h}}_{\mathit{jk}}^{\mathit{i}}$ is given as (Bjornson et al., 2017).

(8) ${\hat{h}}_{\mathit{jk}}^{\mathit{i}}=\mathit{E}\left\{h_{jk}^i|Y_i^P\right\}=\sqrt{{\rho }_{\mathit{jk}}}{R}_{jk}^i{\mathrm{\Psi }}_{jk}^i{Y}_{jk}^{i,P}$(8)

where ${\mathrm{\Psi }}_{jk}^i={\sum }_{\mathit{j}{\mathit{r}}^{\prime }{k}^{\prime }\in \mathit{pjk}}{\left(\sqrt{{\mathit{\rho }}_{{\mathit{j}}^{\prime }{k}^{\prime }}}{\tau }_{\mathrm{p}}{R}_{jk}^i+I_M\right)}^{-1}$ is the inverse of normalized correlation matrix. $R_{jk}^i$ is the spatial correlation matrix of the channel to be estimated. $Y_{jk}^{i,P}$ is the received pilot sequences at the ${\mathrm{i}}^{\mathrm{th}}$ BS that are transmitted from users in the j cells. The estimation error ${\tilde{\mathrm{h}}}_{\mathrm{jk}}^{\mathrm{i}}={\mathrm{h}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}-{\hat{\mathrm{h}}}_{\mathrm{jk}}^{\mathrm{i}}$ has the correlation matrix.

3. Pilot contamination and pilot reuse factor

Figure 3 depicts the specific uplink pilot contamination in a situation involving several cells in which the BS in cell i receives pilots from the adjacent cell where PC is caused by sharing of non-orthogonality of pilot sequences among users in adjacent cells, which is a crucial problem in Ma-MIMO (Zhao et al., 2017). It exists because of the practical necessity to reuse the time-frequency resources across cells. Each user terminals send UL pilot sequences to its BS simultaneously. This helps to know the channel responses of its user terminals. The BS needs to know the channel responses of its user terminals and these are estimated in the UL by sending pilot signals. Each pilot signal is corrupted by noise and inter-cell interference when received at the BS. For example, consider the scenario illustrated below where K number of user terminals are transmitting simultaneously, so that the BS receives a correlated of their pilot sequences. Thus, the desired pilot signal is contaminated. When estimating the channel from the desired terminals in a home cell(served cell), the BS cannot easily separate the transmitted pilot sequences from all terminals. This has two basic indications: First, the interfering pilot sequences reduce the channel estimation quality. Second, the BS unintentionally estimates a superposition of the channel from the required terminals and from the interferes. As a result, the estimated channel at the BS is contaminated. This degrades the estimation quality. Later, the desired terminals transmit payload data and the BS wishes to coherently combine the received signals, using the channel estimate. It will then unintentionally and coherently combine part of the interfering signal as well. This is particularly poisonous when the BS has M antennas, since the array gain from the receive combining increases both the signal and the interference power proportionally to M. Similarly, in DL transmission when the BS transmits a beam formed DL signal towards its terminal, it will unintentionally direct some of the signals towards interferes. At the end, the system performance is degraded (Zhao et al., 2017).

Figure 3. Uplink Pilot Contamination in a multi cell scenario where BS i in cell i receive pilots from adjacent cell (Zhao et al., 2017).

In order to mitigate the PC, consider arbitrary pilot allocation with the only requirement of ${\tau }_p\ge \mathit{K}$ (Rappaport et al., 1996). The parameter $\mathit{N}=\frac{{\tau }_p}{\mathit{K}}\ge 1$ is called the PRF. Full pilot reuse results in high inter-cell interference during channel estimation, that can be minimized using pilot reuse factor. PRF is designed to have each cell within a cluster to use unique orthogonal signals. This helps to mitigate inter-pilot contamination effect. It is calculated in a similar manner to the cellular frequency reuse factor (Rappaport et al., 1996) given as: $\mathit{N}=i^2+\mathit{ij}+j^2$ where $\mathit{i}$ and $\mathit{j}$ are positive integers. The possible PRF values are:1,3,4,7,9,12.

It is analogous to traditional frequency reuse. terminals within the pilot reuse area are confined to utilize only a fraction of the time-frequency resources during the channel estimation phase. However, with pilot reuse, each terminal is free to use all the available resources for data transmission during the rest of the coherence interval. The PRF 1/N is the rate at which pilot resources may be reused in the network.

Where N is the number of cells that are assigned orthogonal pilots. A factor N reduces the pilot contamination effect by assigning orthogonal pilots to neighboring cells, the next-neighbor cells and so on. The total number of pilots reserved for pilot transmission are ${\tau }_p=\mathit{KN},$

where K is the number of users per cell.

We assumed that every cell served exactly K terminals; the minimum required duration of the pilot sequences is ${\tau }_p$, would be equal to KN. For each of the N groups of pilots, i = 1, … , N, denoted by $K_{\mathit{max}}$, i the maximum number of terminals served by any of the cells in that group. Then the required pilot duration is equal to the total number of mutually orthogonal pilots in the system, ${\mathrm{T}}_{\mathrm{P}}={\sum }_{i=1}^N\mathit{\hspace{0.17em}}{\mathrm{K}}_{max}\cdot \mathrm{i}$. Each terminal in a cell is randomly assigned one of the pilot sequences allocated to that cell. The reuse factor should divide the total number of cells. The cells are enumerated such that the ${\mathrm{n}}^{\mathrm{th}}$ cell and the $(\mathrm{n}\pm \mathrm{N}{)}^{\mathrm{th}}$ cell use the same set of orthogonal pilots.

(ii) Signal Model of Downlink Data Transmission

The BS has to ensure that each terminal receives only the signal intended for it. DL data transmission uses a linear precoding operation that combines data symbols with the channel estimates to create the actual signals that the array transmits. The ${\mathrm{j}}^{\mathrm{th}}$ BS in cell j independently transmitting a signal, $X_j$ is ${\mathrm{X}}_{\mathrm{j}}={\sum }_{\mathit{m}=1}^{K_j}\mathit{\hspace{0.17em}}{\mathrm{A}}_{\mathrm{jm}}{\mathrm{S}}_{\mathrm{jm}}$ (Wang et al., 2018). where ${\mathrm{A}}_{\mathrm{jk}}$ is the precoding vector that determines the spatial directivity of the transmission. ${\mathrm{S}}_{\mathrm{jk}}\sim {\mathrm{C}}_{\mathrm{N}}\left(0,{\rho }_{\mathit{jk}}\right)$ is a vector of K symbols intended for the K terminals in the cell. ${\rho }_{\mathit{jk}}$ is the signal power. The average precoding vector is assumed to be unity. that is: $\mathrm{E}\left\{{∥{\mathrm{A}}_{\mathrm{jk}}∥}^2\right\}=1$, Such that $\mathrm{E}\left\{{∥{\mathrm{A}}_{\mathrm{jk}}{\mathrm{s}}_{\mathrm{jk}}∥}^2\right\}={\rho }_{\mathit{jk}}$, is the transmit power allocated to this UE. Users in the ${\mathrm{i}}^{\mathrm{th}}$ cell received signal is defined as (Marzetta & Quoc Ngo, 2016):

(9) ${\mathrm{Y}}_{\mathrm{i}}=\sum _{\mathit{j}=1}^L{\left({\mathrm{H}}_{\mathrm{i}}^{\mathrm{j}}\right)}^{\mathrm{H}}{\mathrm{X}}_j+{\mathrm{N}}_{\mathrm{i}}$(9)

where ${\mathbf{N}}_{\mathrm{i}}={\left[{\mathrm{n}}_{\mathrm{i}1},{\mathrm{n}}_{\mathrm{i}2}\dots {\mathrm{n}}_{\mathrm{ik}}\right]}^{\mathrm{T}}\in {\mathbb{C}}^K$ is the noise vector of the k user in cell i. Each users received DL signal in cell i can be written as

(10) ${\mathrm{Y}}_{\mathit{i}\mathrm{k}}={\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}{\mathrm{S}}_{\mathrm{ik}}+\sum _{\mathit{m}=1,\mathit{k}\ne \mathit{m}}^{\mathit{Ki}}{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{im}}{\mathrm{S}}_{\mathrm{im}}+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}\sum _{\mathit{m}=1}^{\mathit{Kj}}{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{j}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{jm}}{\mathrm{S}}_{\mathrm{jm}}+{\mathrm{N}}_{\mathrm{ik}}$(10)

Where ${\mathrm{A}}_{\mathrm{jm}}$ is the ${\mathrm{m}}^{\mathrm{th}}$ column of precoding matrix ${\mathrm{A}}_{\mathrm{j}}$.

Equation 10(10) ${\mathrm{Y}}_{\mathit{i}\mathrm{k}}={\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}{\mathrm{S}}_{\mathrm{ik}}+\sum _{\mathit{m}=1,\mathit{k}\ne \mathit{m}}^{\mathit{Ki}}{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{im}}{\mathrm{S}}_{\mathrm{im}}+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}\sum _{\mathit{m}=1}^{\mathit{Kj}}{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{j}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{jm}}{\mathrm{S}}_{\mathrm{jm}}+{\mathrm{N}}_{\mathrm{ik}}$(10) represents the desired signal for the user k in cell i, intra-cell interference signal, and inter-cell interference signal, respectively.

4. Precoding techniques and spectral efficiency with perfect and imperfect CSI

Transmit precoding is a generalization of beamforming to support multi-stream transmission in multi-antenna wireless communications. It is a versatile technique for signal transmission from an array of M antennas to one or multiple users. In wireless communications, its goal is to increase the signal power at the intended user and reduce interference to non-intended users. Figure 4 represents a block diagram of a multi-cell DL Ma-MIMO system with linear precoding; the specifics of each precoding will be discussed in the following subsections.

Figure 4. Block diagram representation of a multi-cell DL Ma-MIMO system with linear precoding (Marzetta & Quoc Ngo, 2016).

4.1. Precoding techniques

The three linear precoding techniques like MRT, MMSE and ZF are used to reduce the inter-user interference or intra-cell interference for both perfect and imperfect CSI in this research work. In perfect CSI, channel estimation is not required. Hence, the pilot sequences received from the users aren’t required because the BS knows the propagation channel matrix ${\mathrm{H}}_{\mathrm{i}}^{\mathrm{i}}$ of multi cell Ma-MIMO system. The precoding matrix is described in terms of propagation channel matrix ${\mathrm{H}}_{\mathrm{i}}^{\mathrm{i}}$ (Qasim Jabbar & Li, 2016). Assume the scalling factor of precoding vector is unity. Then by considering the expression in equation 10 above, the SINR of ${\mathrm{k}}^{\mathrm{th}}$ user in the cell i for perfect CSI can be evaluated as follows (Qasim Jabbar & Li, 2016):

(11) ${\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{ik}^{\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}=\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|{\mathit{A}}_{\mathit{ik}}{\left({h^i}_{\mathit{ik}}\right)}^{\mathit{H}}\right|}^2}{{\sum }_{{\mathit{m}}^{\prime }=1,\mathit{k}\ne \mathit{m}}^{\mathit{K}}{\rho }_{\mathit{im}}{\left|{\mathit{A}}_{\mathit{im}}{\left({h^i}_{\mathit{ik}}\right)}^{\mathit{H}}\right|}^2+{\sigma }_n^2}$(11)

The SE for ${\mathrm{k}}^{\mathrm{th}}$ user, k in i cell is given as shown below in equation 12.

(12) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}={log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(12)

The SE of the system for perfect CSI is calculated using the three precoding techniques. (a) MRT precoding

MRT precoding is the DL version of maximum ratio combining of the UL. MRT is a pre-processing technique at the transmitter. It maximizes signal gain at the intended user and its precoding matrix ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{M}\mathrm{R}\mathrm{T}}$ is expressed as follows (Qasim Jabbar & Li, 2016).

(13) ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{M}\mathrm{R}\mathrm{T}}={\left({{\mathrm{H}}_{\mathrm{i}}}^{\mathrm{i}}\right)}^{\mathrm{H}}$(13)

Then the ${\mathrm{k}}^{\mathrm{th}}$ user SINR and SE of MRT are expressed respectively as follows

(14) ${\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{M}\mathrm{R}\mathrm{T}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}=\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|\mathit{\hspace{0.17em}}{{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}\right|}^4}{{\sum }_{{\mathit{m}}^{\prime }=1,\mathit{k}\ne \mathit{m}}^{\mathit{K}}{\rho }_{\mathit{im}}{\left|\mathit{\hspace{0.17em}}{{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{im}}{\left({{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}\right)}^{\mathrm{H}}\right|}^2+{{\sigma }_n}^2}$(14)

(15) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{M}\mathrm{R}\mathrm{T}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}={log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{M}\mathrm{R}\mathrm{T}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}\right)$(15)

(b) ZF precoding

ZF precoding is a type of precoding technique, which eliminates the interference by transmitting the signal toward the intended user while nulling in the directions of other users. It is the pseudo-inverse of the CSI matrix. The ZF precoding matrix ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{Z}\mathrm{F}}$ is:

(16) ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{Z}\mathrm{F}}={\left({{\mathrm{H}}_{\mathrm{i}}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\left({\mathrm{H}}_{\mathrm{i}}\mathrm{i}{\left({{\mathrm{H}}_{\mathrm{i}}}^{\mathrm{i}}\right)}^{\mathrm{H}}\right)}^{-1}$(16)

Then the ${\mathrm{k}}^{\mathrm{th}}$ user SINR and SE of ZF are expressed respectively as follow:

(17) ${\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}=\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|\mathit{\hspace{0.17em}}{{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}{\mathrm{A}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}}\right|}^2}{{{\sigma }_n}^2}$(17)

(18) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}={log}_2\left(1+\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}{\mathrm{A}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}}\right|}^2}{{{\sigma }_n}^2}\right)$(18)

(c) MMSE Precoding

MMSE tries to strike a balance between getting the most signal amplification and reducing the interference. It reduces the interference and improves the performance by minimizing the mean square error. This technique is generated by the mean square error method and the precoding matrix ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}$ is expressed as follows (Qasim Jabbar & Li, 2016).

(19) ${\mathrm{A}}_{\mathrm{i}}^{\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}={\left({{\mathrm{H}}_{\mathrm{i}}}^{\mathrm{i}}\right)}^{\mathrm{H}}\left[{\left({\mathrm{H}}_{\mathrm{i}}\mathrm{i}{\left({{\mathrm{H}}_{\mathrm{i}}}^{\mathrm{i}}\right)}^{\mathrm{H}}+\mathrm{K}{\sigma }^2{\mathrm{p}}^{-1}\right)}^{-1}\right]$(19)

(20) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}={log}_2\left(1+\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|\mathit{\hspace{0.17em}}{{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}{{\mathrm{A}}_{\mathrm{ik}}}^{\mathrm{MMSE}}\right|}^2}{{\sum }_{m=1,k\ne m}^K{\rho }_{\mathit{im}}{\left|\mathit{\hspace{0.17em}}{\mathrm{A}}_{\mathrm{im}}{\left({{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}\right)}^{\mathrm{H}}\right|}^2+{{\sigma }_n}^2}\right)$(20)

Where

${\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}\mathrm{Perf},\mathrm{C}\mathrm{S}\mathrm{I}}=\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|\mathit{\hspace{0.17em}}{{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}{{\mathrm{A}}_{\mathrm{ik}}}^{\mathrm{MMSE}}\right|}^2}{{\sum }_{m=1,k\ne m}^K{\rho }_{\mathit{im}}{\left|\mathit{\hspace{0.17em}}{\mathrm{A}}_{\mathrm{im}}{\left({{\mathrm{h}}^{\mathrm{i}}}_{\mathrm{ik}}\right)}^{\mathrm{H}}\right|}^2+{{\sigma }_n}^2}$

4.2. Spectral efficiency with imperfect CSI

In order to utilize the advantages of Ma-Mu-MIMO, accurate CSI is required at the BS. In multi-cell system, there is reused of pilots by users while user terminals send UL pilots to the BS to estimate the channel response within cell due to the coherence block is limited.

In imperfect CSI, there is estimation error because the transmitter not knows all the components of ${\mathrm{H}}_{\mathrm{j}}^{\mathrm{i}}$. That is $\left[\mathrm{tr}\left(C_j^i\right)]\ne 0\right.$, i.e $\mathrm{E}\left\{h_{jk}^i\right\}\ne \mathrm{E}\left\{{\hat{h}}_{\mathit{jk}}^{\mathit{i}}\right\}$

The channel estimation error, ${\tilde{h}}_{\text{jk}}^{\text{i}}={\text{h}}_{\text{jk}}^{\text{i}}-{\widehat{h}}_{\text{jk}}^{\text{i}}$ and the channel vector, ${\mathrm{h}}_{\mathrm{j}\mathrm{k}}^{\mathrm{i}}$ is expressed as ${\text{h}}_{\text{jk}}^{\text{i}}={\tilde{h}}_{\text{jk}}^{\text{i}}+{\widehat{h}}_{\text{jk}}^{\text{i}}$ (Bjornson et al., 2017)

Since we use TDD mode (UL pilot) that limits the pilot overhead, no DL pilots and we depend on instead on channel hardening. Thus, the users do not know their instantaneous channel realizations. However, they can learn their average equivalent channels, $\mathrm{E}\left\{{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}\right\}$. (i.e.without beamforming training, the users detect the signals based on the statistical CSI).

The DL signal vector received by user k in cell i for imperfect CSI can be expressed as (Marzetta & Quoc Ngo, 2016):

(21) $\begin{array}{c}{\mathrm{Y}}_{\mathrm{ik}}=\mathrm{E}\left\{{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}\right\}{\mathrm{S}}_{\mathrm{ik}}+\left({\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}-\mathrm{E}\left\{{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}\right\}\right)\\ {\mathrm{S}}_{\mathrm{ik}}+\sum _{\mathit{m}=1,\mathit{k}\ne \mathit{m}}^K{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{im}}\mathit{\hspace{0.17em}}{\mathrm{S}}_{\mathrm{im}}+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^L\sum _{\mathit{m}=1\mathit{k}\ne \mathit{m}}^K{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{j}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{jm}}\mathit{\hspace{0.17em}}{\mathrm{S}}_{\mathrm{jm}}+{\mathrm{N}}_{\mathrm{ik}}\end{array}$(21)

The second term describes receiver’s lack of knowledge about H. The first term in equation 21(21) $\begin{array}{c}{\mathrm{Y}}_{\mathrm{ik}}=\mathrm{E}\left\{{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}\right\}{\mathrm{S}}_{\mathrm{ik}}+\left({\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}-\mathrm{E}\left\{{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{ik}}\right\}\right)\\ {\mathrm{S}}_{\mathrm{ik}}+\sum _{\mathit{m}=1,\mathit{k}\ne \mathit{m}}^K{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{im}}\mathit{\hspace{0.17em}}{\mathrm{S}}_{\mathrm{im}}+\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^L\sum _{\mathit{m}=1\mathit{k}\ne \mathit{m}}^K{\left({\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{j}}\right)}^{\mathrm{H}}{\mathrm{A}}_{\mathrm{jm}}\mathit{\hspace{0.17em}}{\mathrm{S}}_{\mathrm{jm}}+{\mathrm{N}}_{\mathrm{ik}}\end{array}$(21) is the desired signal received over the deterministic average precoded channel ${\hat{\mathrm{h}}}_{\mathit{jk}}^{\mathit{i}}{A}_{\mathit{ik}}$, where as the remaining terms are random variables with realizations that are unknown to the UE.

The SINR and SE for ${\mathrm{k}}^{\mathrm{th}}$ user in i cell for imperfect CSI are expressed in equation 22 and equation 23, respectively.

(22) ${\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}=\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|\mathrm{E}\left\{{\left({\mathit{A}}_{\mathrm{i}k}\right)}^{\mathrm{H}}\right\}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right|}^2}{{\left.{\sum }_{j=1}^L{\sum }_{m=1}^K{\rho }_{\mathit{jm}}\left|\mathrm{E}{\left\{{\left({\mathit{A}}_{\mathrm{jm}}\right)}^{\mathrm{H}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{j}}\right\}}^2-{\rho }_{\mathit{ik}}\right|\mathrm{E}\left\{{\left({\mathit{A}}_{\mathrm{ik}}\right)}^{\mathrm{H}}\right\}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right|}^2+{\sigma }_n^2}$(22)

(23) ${\mathrm{S}\mathrm{E}}_{i\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}=\left(\frac{{\mathit{\tau }}_{\mathrm{d}}}{{\tau }_{\mathrm{c}}}\right){log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(23)

Equation 22(22) ${\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}=\frac{{\mathit{\rho }}_{\mathit{ik}}{\left|\mathrm{E}\left\{{\left({\mathit{A}}_{\mathrm{i}k}\right)}^{\mathrm{H}}\right\}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right|}^2}{{\left.{\sum }_{j=1}^L{\sum }_{m=1}^K{\rho }_{\mathit{jm}}\left|\mathrm{E}{\left\{{\left({\mathit{A}}_{\mathrm{jm}}\right)}^{\mathrm{H}}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{j}}\right\}}^2-{\rho }_{\mathit{ik}}\right|\mathrm{E}\left\{{\left({\mathit{A}}_{\mathrm{ik}}\right)}^{\mathrm{H}}\right\}{\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}\right|}^2+{\sigma }_n^2}$(22) is the lower bounded DL ergodic channel capacity of UE k in cell i

The SE expression can be expressed for any channel model and precoding scheme.

Where the expectations are with respect to the channel realizations.

$h_{jk}^i={\tilde{\mathrm{h}}}_{\mathit{jk}}^{\mathit{i}}+{\hat{\mathrm{h}}}_{\mathit{jk}}^{\mathit{i}}$ and $\mathrm{E}({\tilde{\mathrm{h}}}_{\mathit{jk}}^{\mathit{i}}{\hat{\mathrm{h}}}_{\mathit{jk}}^{\mathit{i}})=0$. Due to the estimate and the estimation error are independent and have zero mean.

Where $\frac{{\tau }_d}{{\tau }_c}$ is the prelog factor, which is fraction of all samples used for DL data. It also represents the ratio of the time needed in sending data to the coherence interval and describes the pilot overhead. It is equivalent to $1-(\frac{{\tau }_p}{{\tau }_c})$.

Equation 23(23) ${\mathrm{S}\mathrm{E}}_{i\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}=\left(\frac{{\mathit{\tau }}_{\mathrm{d}}}{{\tau }_{\mathrm{c}}}\right){log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(23) is re-write equivalent to in equation 24

(24) ${\mathrm{S}\mathrm{E}}_{i\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}=\left(1-(\frac{{\mathit{\tau }}_{\mathrm{p}}}{{\tau }_{\mathrm{c}}})\right){log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(24)

The sum SE for imperfect CSI of ${\mathrm{i}}^{\mathrm{th}}$ cell is

(25) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}=\mathrm{E}(\sum _{\mathit{k}=1}^K{\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Imperf},\mathrm{C}\mathrm{S}\mathrm{I}}[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}/\mathrm{cell}])$(25)

We also used Linear precoding techniques such as MRT, ZF and M-MMSE to analyze the performance of multicell Ma-MIMO systems with imperfect CSI. These Linear precoding techniques are selected using the UL-DL duality precoding design principle. The UL-DL duality theorem describes that the SE achieved in the UL can be achieved also in the DL, if the UL combining vectors are used as DL precoding vectors. i.e. if the DL power is allocated in a particular way based on the UL powers and the precoding vectors are selected based on the detection vectors (Bjornson et al., 2014). Hence, under imperfect CSI, the DL precoding vectors are based on the UL receive combining vectors are expressed as follow (Bjornson et al., 2017):

(26) $A_{\mathit{ik}}=\frac{{\mathit{V}}_{\mathit{ik}}}{\sqrt{Y_{\mathit{ik}}}}$(26)

where $Y_{\mathrm{ik}}=\sqrt{\mathrm{E}\left\{{∥{\mathrm{V}}_{\mathrm{ik}}∥}^2\right\}}$

$V_i=\left[V_{\mathit{i}1}\dots .V_{\mathit{ik}}\right]=\left\{\begin{array}{cc}{V}_i^{\mathrm{M}\mathrm{R}\mathrm{C}}& \mathrm{with}\mathrm{MRT}\mathrm{precoding}\\ V_{\mathrm{i}}^{ZF}& \mathrm{with}\mathrm{ZF}\mathrm{precoding}\\ V_i^{\mathrm{M}-\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}& \mathrm{with}\mathrm{M}-\mathrm{MMSE}\mathrm{precoding}\end{array}\right.$

MRT precoding techniques

The MRT precoding for UE k in cell i,${\mathbf{A}}_{\mathrm{ik}}$ is determined based on the channel estimate, ${\mathrm{h}}_{\mathrm{i}\mathrm{k}}^{\mathrm{i}}$ or the UL receive combining vectors, ${\mathrm{V}}_{\mathrm{i}\mathrm{k}}^{MRC}={{\hat{\mathrm{h}}}^{\mathrm{i}}}_{\mathrm{ik}}$, ${\mathbf{A}}_{\mathrm{ik}}=\frac{{\hat{\mathrm{h}}}_{\mathrm{ik}}^{\mathrm{i}}}{\sqrt{{\mathrm{{\rm Y}}}_{\mathrm{ik}}}}$ (Bjornson et al., 2017).

(27) $\mathrm{S}{{\mathrm{E}}^{\mathrm{MRT}}}_{\mathrm{ik}}==\left(\frac{{\mathit{\tau }}_{\mathrm{d}}}{{\tau }_{\mathrm{c}}}\right){log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{I}\mathrm{m}\mathrm{p}\mathrm{C}\mathrm{S}\mathrm{I},\mathrm{M}\mathrm{R}\mathrm{T}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(27)

ZF precoding techniques

ZF precoding is linear precoding technique that is used to cancel the intra-cell interference. This precoding is a normalization of ZF combining vector. The ZF combining vector is assumed to implement a pseudo-inverse of the estimated channel matrix (Bjornson et al., 2014). The ZF precoding for UE k in cell i is expressed as ${\mathbf{A}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}}=\frac{{\mathrm{V}}_{\mathrm{i}\mathrm{K}}^{\mathrm{Z}\mathrm{F}}}{∥{\mathrm{V}}_{\mathrm{i}\mathrm{K}}^{\mathrm{Z}\mathrm{F}}∥}$ where ${\mathbf{V}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}}={\hat{\mathrm{H}}}_{\mathrm{i}}^{\mathrm{i}}{\left(\left({\hat{\mathrm{H}}}_{\mathrm{i}}^{\mathrm{iH}}\right){\hat{\mathrm{H}}}_{\mathrm{i}}^{\mathrm{i}}\right)}^{-1}$, the ZF combining vector. The SE expression of average-normalized ZF precoding

(28) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{Z}\mathrm{F}}=\left(\frac{{\mathit{\tau }}_{\mathrm{d}}}{{\tau }_{\mathrm{c}}}\right){log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{I}\mathrm{m}\mathrm{p}\mathrm{C}\mathrm{S}\mathrm{I},\mathrm{Z}\mathrm{F}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(28)

M- MMSE precoding techniques

The multi-cell MMSE precoding matrix of cell i is expressed based on the multi-cell MMSE receive detector matrix of cell i, ${\mathbf{A}}_{\mathbf{i}}^{\mathrm{M}-\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}=\frac{{\mathbf{V}}_{\mathbf{i}}^{\mathrm{M}-\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}}{\parallel {\mathbf{V}}_{\mathbf{i}}^{\mathrm{M}-\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}\parallel }$ (Bjornson et al., 2014, 2017).

(29) ${\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{k}}^{\mathrm{M}-\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}=\left(\frac{{\mathit{\tau }}_{\mathrm{d}}}{{\tau }_{\mathrm{c}}}\right){log}_2\left(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{k}}^{\mathrm{I}\mathrm{m}\mathrm{p}\mathrm{C}\mathrm{S}\mathrm{I},\mathrm{M}-\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}}\right)[\mathrm{bit}/\mathrm{s}/\mathrm{Hz}]$(29)

M-MMSE combining maximizes the instantaneous SINR and also minimizes the MSE in the data detection; which is, the average squared distance between the desired signal and the processed received signal.

5. Simulation results and discussion

The parameters and their values for matlab simulation are listed under Table 1.

Table 1. Parameters and values

parameters	values (considerations)
Number of cells	Multi-cells ((Marzetta, 2010; Quoc Ngo, 2015))
Cell radius	1000m
Channel state information	Perfect and imperfect CSI at the BS
Channel model	Correlated Rayleigh fading
Number of antennas per BS (Fixed)	100 (Salh & Audah, 2020)
Range of antennas per BS (When varied)	10–110
Number of users (Fixed)	10 (Salh & Audah, 2020)
Number of users(Varied)	0–40 (Thakur & Chandra Mishra, 2019)
UL and DL power	20dBm (Bjornson et al., 2017)
PRF	1,3,4
Channel gain at 1 km $(\mathrm{{\rm Y}})$	148.1 dB (Bjornson et al., 2017)
Path-loss exponent $(\mathit{\alpha })$	3.76 (Marzetta, 2010)
Receiver noise power	94 dBm (Bjornson et al., 2017)
Bandwidth	20MHz (Marzetta & Quoc Ngo, 2016)
Carrier frequency	2GHz (Marzetta & Quoc Ngo, 2016)
Coherence interval	300 (Marzetta & Quoc Ngo, 2016)
Number of channel realizations	100
Shadow fading standard deviation $({\sigma }_{\mathit{sf}})$	10dB (Marzetta, 2010)

The simulation parameters were selected for the simulation based on the operation of Massive MIMO technology that can help for the performance of wireless communication systems (particularly, for 5 G, Massive MIMO is one of the enabler technologies for 5 G). For example, 2 GHz carriers frequency was selected because Massive MIMO technology can operate at this frequency and enhance the spectral efficiency of 5 G without increasing bandwidth (for example at 20 MHz).So, the parameters that are described in Table 1 are operate at 2 GHz carriers and communication channel bandwidth of 20 MHz. $\mathit{\gamma }=-148.1\mathit{dB}$ is the median channel gain at reference distance of 1 km, the path loss exponent, $\mathit{\alpha }=3.76$ and the standard deviation, ${\sigma }_{\mathit{sf}}=10$ are inspired by the NLoS macro cell 3GPP model for 2 GHz carriers (Bjornson et al., 2017). At 20 MHz communication channel bandwidth, UL and DL transmit power are 20dBm per UE(assume both are the same) and receiver noise power is $-94\mathit{dBm}$ (including thermal noise and a noise figure of $7\mathit{dB}$ in the receiver hardware in case of imperfect CSI).

5.1. Spectral efficiency for different number of BS antennas with imperfect CSI

Using the above parameters and Rayleigh fading channel under local scattering spatial channel correlation model with given nominal arrival angle, 30^°and Gaussian angular standard deviation, ${\sigma }_{\phi }$ = 10^° are considered for SE evaluation of Ma-MIMO system.

Figure 5 shows the average sum SE versus number of BS antennas (M) for the three precoding techniques and PRF of one, three and four. The figure illustrates the average sum SE increases as increase number of antennas for 3 precoding techniques. As we can see from the result the M-MMSE provides the best performance than ZF and MRT precoding for the given PRFs. Because M-MMSE can reduce the inter cell interference in addition to intra cell interference. ZF shows better performance than MRT Precoding For PRF one, the length of required pilot sequence ${\tau }_p$ is the same as number of users. Thus each cell reuses the same k pilots. Similarly, for 3 and 4 reuse factors, the figure shows that M-MMSE provides the best system performance. ZF shows better performance than MRT Precoding and MRT gives the lowest SE. The result also shows that average sum SE of M-MMSE Precoding with PRF of four are higher than this Precoding with PRF of three and average sum SE of M-MMSE Precoding with PRF of three are higher than this Precoding with PRF of one due to it can better suppress the interference from user terminals in the neighboring cells when these user terminals use other pilots. For 3 and 4 reuse factors the SE of MRT slightly reduces since the improved estimation quality does not outweigh the reduced pre-log factor when the estimate is only used to coherently combine the desired signal and not to cancel interference. likewise, the SE of ZF reduces for 3 and 4 reuse factors. Thus, ZF and MRT provide lower average sum SE. M-MMSE Precoder achieves 12.5% higher SE than ZF and 75% higher SE MRT for all taken PRFs and it has highest SE at PRFs of four among all because it reduces the level of PC by using different pilot for user terminals in surrounding cells. ZF achieves more than 50% better performance than MRT precoding.

Figure 5. Average sum SE with respect to the number of BS antennas for M-MMSE, ZF and MRT precoding techniques (for a PRF of one, three and four and K=10).

5.2. Spectral efficiency versus different number of users with imperfect CSI

Figure 6 shows average sum SE increases versus number of users for the three precoding techniques with PRF = 1. As it can be seen MR (i.e. MRT) gives the lowest SE than all because MRT only increases the signal gain but it doesn’t suppress the interference. and M-MMSE provides the best performance. The PRF is one; the number of users is the same as pilot sequence length,${\tau }_p$. The sum SE increases as number of user increases until its SE reaches saturation point. (Hint: MR in Figures 6, 7 and 8 is represent MRT precoding).

Figure 6. Average sum SE with respect to the number of BS antennas for M-MMSE, ZF and MRT precoding techniques (for a PRF of one and M=100).

Figure 7. Average sum SE with respect to the number of users for M-MMSE, ZF and MRT. Precoding techniques (for a PRF of three and M=100).

Figure 8. Average sum SE with respect to the number of users for M-MMSE, ZF and MRT. Precoding techniques (for a PRF of four and M=100).

To look at the SE of the system for larger PRF, Figures 7 and 8 are simulated for PRF 3 and 4 respectively. Figure 7 illustrates average sum SE versus of users’ increases from 0 to 30 for all linear precoding techniques with PRF is three. The SE increases as number of users increase then reaches maximum point(at K = 30) and finally decreases as number of users’ increases from 30 to 40. Similar to Figure 6, MRT gives the lowest SE and M-MMSE provides the best performance. ZF shows better performance than MR Precoding. The PRF is three; the required pilot sequence length is three times of number of users.

Figure 8 shows first average sum SE increases as number of users increases for all linear precoding techniques with PRF is four, then reaches maximum point and keeps constant (from K = 20 to 30 for M-MMSE and MRT) and finally decreases as number of users increases from 30 to 40. The result shows that MRT and ZF provide lower performance than M-MMSE. MRT gives the lowest SE. The pilot group is divided into four and the length of pilot sequence is four times of K. Average sum SE decreases as number of users’ increases from 0 to 40 for all linear precoding techniques with PRF changes from one to three and four as shown Figure 8 because PRF 3 and 4 are effective at small number of users. The SE of three precodings reaches saturation level (maximum point) for different number of users due to the precodings performance depends the coherence block.

5.3. Spectral efficiency for different number of BS antennas with perfect and imperfect CSI

Figure 9 shows average sum SE with respect to the number of Antennas for three precoding techniques with perfect and imperfect CSI (PRF of one) and K = 10 The figure illustrates the SE of the system using all linear precoding techniques (MRT, ZF and MMSE) with both imperfect and perfect CSI, the SE of the system using linear precoding techniques with perfect CSI achieve better performance than their corresponding precoding techniques under imperfect CSI during changing the number of BS antennas from 10 to 110 and fixed number of users, K = 10.

Figure 9. Average sum SE with respect to the number of Antennas for MMSE, ZF and MRT precoding techniques with perfect and imperfect CSI (PRF of one) and K = 10.

5.4. Spectral efficiency for different number of BS antennas and users with perfect and imperfect CSI

B Average sum SE with respect to the number of Antennas for MMSE, ZF and MRT precoding techniques with perfect and imperfect CSI (PRF of four) and K = 10, k = 20

Figure 10 shows that the SE increases as the number of BS antennas increases and users per cell is increased from 10 to 20 in case of perfect CSI and also the same PRF is used in case of imperfect CSI. The result also shows that the sum SE is better when the number of users per cell is changed from k = 10 to 20 and the number of BS antennas is also changed simultaneously and $\mathit{M}>>\mathit{K}$ for both CSI. The SE increases slowly with small number of BS antennas and the number of users is changed from 10 to 20. This is because the BS does not have enough spatial degrees of freedom to separate the users. When we compare the SE of the system using all linear precoding techniques (MRT, ZF and MMSE) with both imperfect and perfect CSI, the SE of the system using linear precoding techniques with perfect CSI achieve better performance than their corresponding precoding techniques under imperfect CSI during changing the number of BS antennas from 10 to 110 and fixed number of users, K = 10.

Figure 10. Average sum SE with respect to the number of Antennas for MMSE,ZF and MRT precoding techniques with perfect and imperfect act CSI (PRF of four) and K = 10, K = 20.

6. Conclusion

A Ma-MIMO system gives higher SE without increasing frequency spectrum. This system uses linear precoding techniques like MRT, ZF, MMSE and M-MMSE at the BS. The known channel is estimated from the UL via TDD mode and MMSE estimation is used. The SINR and SE are derived for both perfect and imperfect CSI. In this thesis, we have analyzed and evaluated the performance of multicell DL Ma- MIMO system using linear precoding techniques and PRF under spatially correlated Raleigh fading channel model with both perfect CSI and Imperfect CSI for different BS number of antennas and number of users per cell. M-MMSE Precoder achieves higher SE than ZF and MRT for all taken pilot reuse factors due to it can better suppress the inter cell interference of users in neighboring cells for using fixed number of users and for different BS number of antennas when these user terminals use other pilots and also it has highest average sum SE at PRF of four among all because it reduces the level of PC. ZF achieves better performance than MRT Precoding. The result also depicts that the sum SE is increased initially, then reach at optimal value and decreased finally when the number of users per cell is changed and fixed number of antennas is used even the PRF is increased. This indicates that large PRF is effective at small number of users and small pilot reuse factor is effective at large number of users. The sum SE is increased when both the number of users (K) and antennas (M) are changed at the same time. The sum SE of all precoder with perfect CSIT better than the sum SE of all precoder with imperfect CSIT. It exceeds the sum SE of all precoder with imperfect CSIT by more than 25% for$\mathit{M}>>\mathit{K}.$

In this study, we used PRF and precoding techniques to examine the SE performance of a multicell Ma-MIMO system. Even though different scenarios and assumptions are used to evaluate performance, there are still some situations and factors that require more research.

• In our work, we evaluated performance under the assumption that there is only one user-side antenna (single antenna users), and that this number is constrained by SE criteria. But by assuming multi-antenna users, further research may be done.

• Analysis of performance evaluation in terms of EE and complexity indicators is also possible for future work.

• In this research, we assumed equal transmit power allocation, however it can be expanded to an optimization technique employing power allocation algorithms.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Fikreselam Gared Mengistu

Fikreselam Gared Mengistu received his PhD from Taiwan National University of Science and Technology (NTUST), Taiwan, in communication engineering. He received his MSc from Addis Ababa University, Ethiopia, in communication engineering and BSc in electrical engineering from Bahir Dar University, Ethiopia.

Gebey Admassu Worku

Gebey Admassu Worku received his MSc and BSc from Bahir Dar University, Ethiopia. Gebey is currently working in Debre Markos University, Ethiopia, as a lecturer.