Performance evaluation of frequency division duplex (FDD) massive multiple input multiple output (MIMO) under different correlation models

Introduction

Global wireless data-traffic has grown dramatically in the last years. According to recent statistics, monthly data-traffic would exceed 607 Exabytes (EBs) by 2025 (IMT, 2015). To this end, sixth-generation (6G) wireless communications networks are being developed to accommodate the substantial growth in mobile data-traffic (Chen et al., 2020; Chowdhury et al., 2020). Specifically, 6G wireless communication networks aim to increase the data rate, support 4K video streaming, exploit massive Machine-to-Machine (M2M) devices, increase link reliability, and reduce latency (Dang et al., 2020; Agiwal, Roy & Saxena, 2016; Aljiznawi et al., 2017). However, in order to meet these requirements, advanced physical layer technologies are still required. Massive multiple-input multiple-output(massive-MIMO) is to date considered as a promising technology to meet the huge requirements for high data rates in the future the 6G networks (Han, Jornet & Akyildiz, 2018; De Carvalho et al., 2020). Unlike traditional MIMO techniques, massive-MIMO systems allow the base station (BS) to deploy a large number of antennas in order to increase the data rate, enhance link reliability, and achieve energy efficiency (Rusek et al., 2013; Ngo, Larsson & Marzetta, 2013; Malkowsky et al., 2017; Han, Jornet & Akyildiz, 2018; De Carvalho et al., 2020). Furthermore, massive-MIMO systems have the potential to improve the array gain and link budget. To this end, sufficiently accurate channel state information (CSI) for downlink (DL) precoding is need to achieve the full potential gain of massive-MIMO systems. However, deploying large arrays at the BS incurs an unacceptably large training overhead especially when the coherence time (CT) is short. Therefore, finding a feasible solution for achieving sufficiently accurate CSI estimation under short CT is essential. The majority of massive-MIMO works have forced on time division duplex (TDD) transmission due to the technical challenge of CSI estimation, see e.g., Jose et al. (2011), Ngo, Larsson & Marzetta (2013), Hoydis, Ten Brink & Debbah (2013), Yang & Marzetta (2013), Zuo et al. (2016), Li et al. (2019). The uplink (UL) and DL channels are considered to be reciprocal in TDD operation mode, hence allowing the UL CSI estimation to be exploited for DL precoding without requirements for the DL CSI estimation. However, the calibration error and hardware impairments are to date considered as severe limitations for the TDD systems in practice (Kaltenberger et al., 2010; Bjrnson et al., 2013; Mi et al., 2017; Vieira et al., 2017). Furthermore, most of current mobile systems operate in the frequency division duplex (FDD) mode. Thus, there is a crucial requirement for enabling FDD operation mode. However, the DL and UL channels in FDD systems are non-reciprocal, and hence, CSI estimation using UL training sequence can not be achieved in FDD systems (Tse & Viswanath, 2005). Specifically, to achieve a minimum mean-square-error (MSE) in the FDD systems, users must estimate the DL CSI of each of the N BS antennas (Hassibi & Hochwald, 2003; Rusek et al., 2013; Björnson, Larsson & Marzetta, 2016). In massive-MIMO systems with enormous antenna arrays at the BS, the training overhead becomes excessive. As such, CSI estimation becomes extremely difficult, especially when CT is short (Björnson, Larsson & Marzetta, 2016; Alsabah, Vehkapera & O’Farrell, 2020; Naser et al., 2020). This is because the DL training sequence for CSI estimation would take up most of the CT allocated period, leaving no time for delivering useful data to the users. Therefore, finding a practical CSI estimate design with a shorter training sequence length is critical for FDD systems with short CT.

Several works have been proposed to address the challenge of CSI estimation overhead in FDD massive-MIMO systems, see e.g., Adhikary et al. (2013), Nam, Caire & Ha (2017), Rao & Lau (2014), Gao et al. (2015), Gao et al. (2016), Han, Lee & Love (2017), Choi, Love & Bidigare (2014), Noh et al. (2014), So et al. (2015). In particular, the works in Adhikary et al. (2013) and Nam, Caire & Ha (2017) develop a two-stage precoding method for DL CSI estimation, termed as joint-spatial-division-and-multiplexing (JSDM). In these works, the users can be divided into groups where the BS can select a specific group to be served. Then, a spatial multiplexing technique can be used to server the selected groups. The authors found that the pilot length for CSI estimation can be increased proportionally with the number of groups and not with N. As such, the overhead in FDD systems can be reduced. While the works in Adhikary et al. (2013) and Nam, Caire & Ha (2017) try to reduce the overhead of CSI estimation, advanced scheduling algorithms become essential for grouping the users. Furthermore, the works in Adhikary et al. (2013) and Nam, Caire & Ha (2017) are not considered the challenge of DL training with single-stage precoding. As such, these works are not able to predict the FDD massive-MIMO performance when a short CT is considered. The works in Rao & Lau (2014), Gao et al. (2015), Gao et al. (2016) and Han, Lee & Love (2017) consider the compressed sensing (CS) based techniques to reduce the CSI estimation overhead in FDD systems; while the works in Choi, Love & Bidigare (2014) and Noh et al. (2014); So et al. (2015) exploit the temporal correlation based on Kalman filter and spatial correlation to reduce the CSI estimation overhead in FDD systems. Although, the aforementioned works try to address the challenge of training sequence design for CSI estimation, the algorithms used are still computationally complex. In addition, future wireless communications networks aim to be operated in scenarios with high frequencies and mobilities, where both required a short CT. As such, a sufficiently accurate CSI estimation for FDD massive-MIMO systems with limited CT is essential. The authors in Wagner et al. (2012), Zheng et al. (2019) and Mostafa, Newagy & Hafez (2021) investigate UL CSI estimation with TDD systems using ZF precoding. They found that the number of UL sequences is proportional to the number of users and independent of the number of BS antennas N that can be made as large as required. However, there is a need to investigate the ZF precoding in FDD systems. The authors in Amadid et al. (2022) and Boulouird et al. (2022) consider the spatially correlated channel in TDD protocol. Therefore, there is an essential requirement to investigate the FDD massive-MIMO systems with limited CT considering different spatial correlation models. Furthermore, the achievable sum rate (ASR) performance in the typical wireless networks is greatly influenced by the received signal and by the level of correlation. While much focus is dedicated to examine the received signal effect on ASR, little consideration is given to examine the correlation effect on ASR. Furthermore, the uncorrelated channel cases have been investigated in the majority of massive-MIMO studies, see e.g., Jose et al. (2011), Marzetta et al. (2016), Chen, Guevara & Pollin (2017), Liu et al. (2017), Ngo, Larsson & Marzetta (2013). A common conclusion arising from the aforementioned works is that a necessary condition for channel hardening can be achieved with massive-MIMO over which the variance goes to zero. However, this is not the case when the channel is highly correlated. Spatial channel correlation model is essential for evaluating the performance of any wireless communications systems. This is because it can reflect the propagation characteristics of the signals in practical radio environments. Therefore, this article aims to investigate the massive-MIMO performance using different channel correlation models namely the P-DoF, the Weichselberger model (Weichselberger et al., 2006), the one ring (OR) developed by Jakes in Jakes & Cox (1994), Laplacian (Lap) (Molisch, 2012), and exponential  (Loyka, 2001; Albdran et al., 2016; Albdran, Alshammari & Matin, 2017; Croisfelt Rodrigues, Marinello & Abrão, 2019) channel models.

System Model

This present article assumes a single-cell wireless communication system. A Rayleigh fading channel is considered in this article. However, finding an efficient training sequence design in Rician fading channel (Amadid, Belhabib & Zeroual, 2022) can be considered in the future. We would like to emphasis that our article discusses a uniform linear array and not three-dimensional array. However, three-dimensional array, e.g., Amadid, Boulouird & Riadi (2022), can be considered in the future. A time-varying channel is considered, which is partitioned into several coherence blocks. To this end, a block-fading model with time and frequency resources is used, over which the channels are assumed to be frequency flat (Alsabah, 2020). The time slot, however, which is enumerated in symbols per transmission block, corresponds to channel CT τ_c. The channel CT is partitioned into the training phase to enable DL CSI estimation, and the data phase for useful data transmission as shown in Fig. 1. The transmit power is equally distributed between the training and data phases. We assume massive-MIMO system with a base station (BS) that uses N antennas. The BS is communicated with K users over the same time and frequency resources assuming N ≫ K. Furthermore, this article considers a single-stage precoding. Fig. 2 shows the system model considered in this article with FDD protocol.

Downlink CSI estimation and training sequence design

In the typical FDD systems, the BS needs to perform a precoding based on DL CSI estimation to serve multiple users, simultaneously. During the CSI estimation phase, the BS transmits DL sequences of length τ_p with training power denoted by ρ_p. The rest of the time is spent on data transmission, which is given as τ_d = τ_c − τ_p. This article focuses on minimizing the training sequence length τ_p over a limited CT τ_c. To this end, this article constructs the training sequences in the DL based on the dominant eigenvectors of the transmit correlation matrix. To obtain the CSI estimation of the DL channel, the BS sends training sequences of length τ_p. The training sequence matrix is denoted by S_p ∈ ℂ^N×τ_p. This training sequence must meet the energy constraint. This implies that $\mathrm{tr}\left(\right.{\mathbf{S}}_{\mathrm{p}}^{\text{H}}{\mathbf{S}}_{\mathrm{p}}\left)\right.={\rho }_{\mathrm{p}}{\tau }_{\mathrm{p}}$. This article considers a scenario where the users have common spatial correlation matrices. In addition, this article assumes an equal power allocation between the training and data phases, i.e., ρ_p = ρ_d. However, non-uniform power allocations across the users and optimization with respect to different chooses of ρ_p and ρ_d could be considered in the future. Therefore, the k-th user received training signal, y_k ∈ ℂ^τ_p, can be expressed as (1)${\displaystyle {\mathbf{y}}_k={\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{h}}_k+{\mathbf{n}}_k,}$ where n_k ∈ ℂ^τ_p represents the receiver noise. The received noise is assumed to be complex Gaussian with $\mathcal{CN}$$\left(\right.{\mathbf{0}}_{\mathrm{}},{\mathbf{I}}_{{\tau }_{\mathrm{p}}}\left)\right.$ and h_k ∈ ℂ^N ∼ $\mathcal{CN}$$\left(\mathbf{0},{\mathbf{R}}_{\mathrm{k}}^{\mathrm{}}\right)$ denotes the complex baseband DL channel vector, which can be represented using Karhunen–Loeve method as (2)${\displaystyle {\mathbf{h}}_k={\mathbf{R}}_k^{1/2}{\tilde{\mathbf{h}}}_k,}$ where ${\tilde{\mathbf{h}}}_k\sim$$\mathcal{CN}$(0, I_N) denotes the DL channel and satisfies ${\mathbf{R}}_k=E\left[{\mathbf{h}}_k{\mathbf{h}}_k^{\text{H}}\right]\in {\mathbb{C}}^{N\times N}$. The spatial correlation matrix R_k has an eigenvalue-decomposition of (3)${\displaystyle {\mathbf{R}}_k={\mathbf{T}}_k{\Delta }_k{\mathbf{T}}_k^{\mathrm{H}},}$ where T_k = [t_k,1, …, t_k,N] ∈ ℂ^N×N represents the eigenvectors of R_k and Δ_k denotes the eigenvalues of R_k, which are arranged as δ_k,1 ≥ δ_k,2 ≥ ⋯ ≥ δ_k,N. The correlation matrix is considered as a large-scale channel statistics, which are assumed to be frequency invariant, and hence, can be efficiently obtained at both transceivers sides (Xie et al., June-2018). Since the channel statistics are assumed to be known, linear filter that exploits the channel statistics is considered. Hence, an optimized CSI estimation in the DL with τ_p < N can be obtained using Bayesian estimation. An example of Bayesian estimation is the minimum mean square error (MMSE) CSI estimation G_k. The MMSE filter uses the statistics of channel and the statistics of noise (Kay, 1993). (4)${\displaystyle {\mathbf{G}}_k={\mathbf{R}}_k{\mathbf{S}}_{\mathrm{p}}\left(\right.{\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{R}}_k{\mathbf{S}}_{\mathrm{p}}+{\mathbf{I}}_{{\tau }_{\mathrm{p}}}{\left)\right.}^{-1}.}$ Accordingly, the resulting channel estimate ${\hat{\mathbf{h}}}_k\sim$ $\mathcal{CN}\left(\mathbf{0},{\Theta }_k\right)$ is obtained as (5)${\displaystyle {\hat{\mathbf{h}}}_k={\mathbf{G}}_k{\mathbf{y}}_k.}$ where y_k is the received training signal in Eq. (1). In what follows, the MSE of the channel estimation that can be calculated using Monte Carlo simulation is provided as (6)${\displaystyle {\text{MSE}}_{\text{sim}}=\sum _{k=1}^{K}E\left|\right|{\mathbf{h}}_k-{\hat{\mathbf{h}}}_k|{|}^2,}$ where (sim) refers to Monte Carlo simulation that is used to calculate the simulated MSE. The expression of the MSE in Eq. (6) is valid due to the orthogonality principle between the channel estimation error ${\hat{\mathbf{h}}}_k$ and the channel estimation ${\tilde{\mathbf{h}}}_k$, where both channels are typically considered to be uncorrelated. Accordingly, the per user channel estimation error vector, which is independent of ${\hat{\mathbf{h}}}_k$, can be expressed as (7)${\displaystyle {\mathbf{e}}_k={\mathbf{h}}_k-{\hat{\mathbf{h}}}_k.}$ Let define the per user error covariance matrix as C_e,k ∈ ℂ^N×N, which can be expressed as

(8)${\displaystyle {\mathbf{C}}_{e,k}=\mathbb{E}\left\{\right.{\mathbf{e}}_k{\mathbf{e}}_k^{\mathrm{H}}\left\}\right.,}$ (9)${\displaystyle =\mathbb{E}\left\{\right.\left(\right.{\mathbf{h}}_k-{\hat{\mathbf{h}}}_k\left)\right.\left(\right.{\mathbf{h}}_k-{\hat{\mathbf{h}}}_k{\left)\right.}^{\mathrm{H}}\left\}\right.,}$ (10)${\displaystyle =\left(\right.{\mathbf{R}}_k-{\mathbf{R}}_k{\mathbf{S}}_{\mathrm{p}}\left(\right.{\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{R}}_k{\mathbf{S}}_{\mathrm{p}}+{\mathbf{I}}_{{\tau }_{\mathrm{p}}}{\left)\right.}^{-1}{\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{R}}_k\left)\right..}$

The MMSE of the CSI estimation should satisfy $\Theta =\mathbb{E}\left\{\right.{\hat{\mathbf{h}}}_k{\hat{\mathbf{h}}}_k^{\mathrm{H}}\left\}\right.$. As such, the MMSE of the CSI estimation can be expressed as (11)${\displaystyle {\mathbf{\Theta }}_k={\mathbf{R}}_k{\mathbf{S}}_{\mathrm{p}}\left(\right.{\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{R}}_k{\mathbf{S}}_{\mathrm{p}}+{\mathbf{I}}_{{\tau }_{\mathrm{p}}}{\left)\right.}^{-1}{\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{R}}_k.}$

The expression in Eq. (10) is minimized by maximizing Eq. (11), which represents the MMSE of the CSI channel estimation. To this end, the MSE can be analytically expressed as (12)${\displaystyle {\text{MSE}}_{\text{an}}=\sum _{k=1}^K\mathrm{tr}\left({\mathbf{C}}_{e,k}\right),}$ where tr stands for the trace of a matrix, which represents the sum of its eigenvalues. The subscript (an) stands for the analytical form of the MSE. The expression in Eq. (12) provides the output of the CSI estimator, which minimizes the MSE. The expression in Eq. (12) implies that the overall MSE performance relies on the eigenvalues of the expression in Eq. (10), which characterizes the overall MSE. Using the well known matrix inversion Lemma, the formulation of the error covariance matrix in Eq. (10) can be simplified to (13)${\displaystyle {\mathbf{C}}_{e,k}=\left(\right.{\mathbf{R}}_k^{-1}+{\mathbf{S}}_{\mathrm{p}}{\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\left)\right.}^{-1},}$ where S_p should satisfy the energy constraint, which is given as $\text{tr}\left({\mathbf{S}}_{\mathrm{p}}^{\mathrm{H}}{\mathbf{S}}_{\mathrm{p}}\right)={\tau }_{\mathrm{p}}{\rho }_{\mathrm{p}}$. Clearly, the expression in Eq. (13) relies on the correlation matrix R_k, the sequence matrix S_p, the training sequence length τ_p, and the training power ρ_p. The expression in Eq. (13) motivates the use of the structure of the correlation matrix in the DL training design.

Data transmission with linear precoder

This section presents the data transmission using linear precoding technique. To this end, the k-th user received data signal can be expressed as (23)${\displaystyle y_k=\sqrt{{\rho }_{\mathrm{d}}}{\mathbf{h}}_k^{\mathrm{T}}\mathbf{V}\mathbf{s}+n_k,}$ where the vector h_k ∈ ℂ^N is the DL instantaneous channel and s = [s₁, …, s_K]^T ∈ ℂ^K denotes the vector of data symbols, which is considered to be an independently and identically distributed and satisfies 𝔼[ss^H] = I_K, and V = [v₁, …, v_K] ∈ ℂ^N×K is the precoding matrix at the BS. Parameter n_k denotes the received noise, which is also considered to be an independently and identically distributed. Therefore, the received SINR, denoted as γ_k, can be expressed as (24)${\displaystyle {\mathrm{\gamma }}_k=\frac{\mid {\mathbf{h}}_k^{\mathrm{T}}{\mathbf{v}}_k{\mid }^2}{\frac{1}{{\rho }_{\mathrm{d}}}+\sum _{l\ne k}^K|{\mathbf{h}}_k^{\mathrm{T}}{\mathbf{v}}_l{|}^2}.}$ where ρ_d denotes the SNR, which denotes the transmit power in the data transmission. To this end, zero forcing (ZF) precoder is considered, and hence, the normalized ZF precoding is given as (25)${\displaystyle {\mathbf{v}}_k=\sqrt{\frac{p_k}{\left|\right.\left|\right.\left(\right.{\hat{\mathbf{H}}}^{†}{\left)\right.}^{:,k}\left|\right.{\left|\right.}_2^2}}\left(\right.{\hat{\mathbf{H}}}^{†}{\left)\right.}^{:,k},}$ where $p_k=\left|\right|{\mathbf{v}}_k|{|}_2^2$ is the power allocated to user k, which satisfies the transmit power constrain ${\sum }_{k=1}^K{p}_k=1$. This normalization is used to guarantee that the per user transmit power to be constant in the data phase. The term ${\hat{\mathbf{H}}}^{\mathrm{T}}=\left[{\hat{\mathbf{h}}}_1,\dots ,{\hat{\mathbf{h}}}_K\right]$ considers the estimate of the downlink true channel, which is named in this article as (imperfect CSI), where (†) corresponds to the Moore–Penrose pseudo-inverse. With perfect channel estimation, which is named in this article as (perfect CSI), the precoder can be written as (26)${\displaystyle {\mathbf{v}}_{\text{per},k}=\sqrt{\frac{p_k}{\left|\right.\left|\right.\left(\right.{\mathbf{H}}^{†}{\left)\right.}^{:,k}\left|\right.{\left|\right.}_2^2}}\left(\right.{\mathbf{H}}^{†}{\left)\right.}^{:,k},}$ where the notation (per) stands for perfect CSI estimation. As such, the SINR with perfect CSI can be written as (27)${\displaystyle {\mathrm{\gamma }}_{\text{per},k}=\frac{\mid {\mathbf{h}}_k^{\mathrm{T}}{\mathbf{v}}_{\text{per},k}{\mid }^2}{\frac{1}{{\rho }_{\mathrm{d}}}}.}$

Accordingly, the DL ASR with imperfect CSI estimation, denoted by Rate, is expressed as (28)${\displaystyle \text{Rate}=\left(\right.\frac{{\tau }_{\mathrm{c}}-{\tau }_{\mathrm{p}}}{{\tau }_{\mathrm{c}}}\left)\right.\sum _{k=1}^K\mathbb{E}\left\{\right.{\log }_2\left(\right.1+{\mathrm{\gamma }}_k\left)\right.\left\}\right.\text{[bit/s/Hz]}.}$

In addition, the DL ASR, with perfect CSI estimation, denoted by Rate_per, is written as (29)${\displaystyle {\text{Rate}}_{\text{per}}=\sum _{k=1}^K\mathbb{E}\left\{\right.{\log }_2\left(\right.1+{\mathrm{\gamma }}_{\text{per},k}\left)\right.\left\}\right.\text{[bit/s/Hz]}.}$

The above expectation operation is obtained by taking into account all sources of randomness. A Monte Carlo simulation is used to determine the SINR. As previously stated, this work considers a practical correlation models. To this goal, the user channel’s elements are correlated rather than isotropically dispersed with Rayleigh fading model. This indicates that h_k has a strong spatial directionality. The received signal in Eq. (24) is dependent on the BS’s correlation, CSI estimation, and linear precoding strategy. However, due to the estimation error as well as the pre-log term, the CSI estimate imperfection lowers the ASR. The ASR maximizing will be discussed more in the following section.

Physical spatial correlation models

For modelling channel variations in conventional wireless communications systems, a random distribution is used. The channel statistics, which correspond to the random variable distributions, are commonly assumed to be available throughout the network (Björnson et al., 2017). As a result, Knowing the channel statistics is sufficient to design efficient communications systems (Stein, 1987; Björnson et al., 2017).

The uncorrelated Rayleigh fading model is a typical approach for modeling the covariance matrix (Marzetta et al., 2016). In Rayleigh fading model, the channel coefficients are considered to be independently and uniformly distributed. Besides, the signal Rayleigh fading model can be propagated across a wide range of scattering objects. Furthermore, in the Rayleigh fading model, the energy should be evenly distributed in all directions (Tse & Viswanath, 2005). However, the requirement for channel to be uncorrelated appears to be quite stringent. The channel coefficients in massive-MIMO systems are correlated because to the usage of a large arrays at the BS, which are closely spaced. In practice, the arrays face non uniform emission patterns and varying polarization. This makes the massive-MIMO channels correlated (Yu et al., 2004; Wallace & Jensen, 2001; Shepard et al., 2012; Gao et al., 2012). Besides, practical experiments in outdoor and indoor scenarios demonstrate that the MIMO channels are correlated, see, e.g., Yu et al. (2004), Wallace & Jensen (2001), Shepard et al. (2012), Gao et al. (2012). As a result, considering more realistic correlation models are necessary to give a a closed to practice performance indication of massive-MIMO systems. Because of the high size of the array in massive-MIMO systems, the antenna spacing at the BS should be reduced, resulting in significant correlations. Due to this correlation, most of the eigenvalues become closed to zero. In the high correlation scenario, the channel covariance matrix would exhibit a large eigenvalue spread. In this case a large portion of the energy in the channel can be concentrated into a few directions, rather than being spread isotropically in all directions. Furthermore, antenna correlations are advantageous when it comes to channel estimation because of the possibility of rejecting the contamination in the estimates. If the contamination is reduced, then the MSE denoted can be reduced. For further discussion about this issue please refer to Heath Jr & Lozano (2018). A measure of the spatial correlation of channel is the eigenvalue distribution or what known as the eigenstructure of the channel covariance matrix. This indicates which spatial directions are statistically more likely to contain strong signal components than others. Strong spatial correlation is characterized by large eigenvalue variations, where the channel is confined to a small eigensubspace. Unlike the uncorrelated channels, with strong spatial correlation, a few eigenvalues dominate and hence the uncertainty in the channels can be significantly reduced.

To this end, spatial correlation models that reflect the practical propagation signals in radio environments should be considered. This is very essential for evaluating the performances of wireless communication systems. Therefore, this article aims to investigate the FDD massive-MIMO systems using the state-of-the-art spatial correlation models.

To reduce the systems overhead, signals can be propagated in the dominated angular domain that contains a fewer dimensions in compared to the number of BS antennas N. In order to reflect the spatial correlation in the massive-MIMO channels, different correlation models will be discussed in the following subsections. These channel correlation models will also be evaluated later in ‘Numerical Results and Performance Evaluation’.

Exponential model

The exponential model discussed here provides a full rank R. As such, the (m, n)th element of R with an exponential model can be written as Loyka (2001) and Croisfelt Rodrigues, Marinello & Abrão (2019) (35)${\displaystyle \mathbf{R}=\left\{\begin{array}{cc}{r}^{n-m},\hfill & m\le n,\hfill \\ {\left(r^{m-n}\right)}^{\ast },\hfill & m>n,\hfill \end{array}\right.}$ where r(0 ≤ |r| ≤ 1) represents the correlation coefficient. The eigenvalue dispersion of the channel covariance matrix is represented by the correlation factor r. As such, in the exponential correlation model, raising the factor r leads to stronger spatial correlations.

The channel covariance matrix’s eigenvalue distribution can be considered as a metric for quantifying the level of correlation. For instance, a very weak or no correlation denotes an identical eigenvalue distribution for all users, but large correlations suggest a small fraction of eigenvalues, which can be dominated and the remainder of the eigenvalues are almost zero. Furthermore, substantial eigenvalue changes are explained by strong correlations. Figures 3, 4, 5, 6, and 7 are provided to illustrate the effect of spatial channel correlation. These figures present the eigenvalues distribution in the P-DoF, Weichselberger, OR, Lap, and Exponential models, respectively. Different level of correlations are presented in each correlation model. The results show that even with a very strong correlation, Exponential model provide a full-rank covariance matrix since the normalized eigenvalues are not closed to zero. In what follows, we investigate the effect of channel hardening on the massive-MIMO systems.

Channel hardening in massive-MIMO systems

Small-scale channel fading is one of the most common flaws in wireless communication systems. Random fluctuations in the channels are generated by microscopic changes in the propagation conditions. The oscillations render the channel gain unstable, because the channel gain might be very small, causing the broadcast data to be received incorrectly. To prevent small-scale fading, signal diversity can be performed. In particular, the diversity is achieved by delivering a signal over many channels with independent realizations. Besides, the spatial diversity can be simply achieved by deploying many antennas at the BS and/or at the receivers. Spatial diversity leads to channel hardening in massive-MIMO systems with a massive number of BS antennas. In simple terms, channel hardening means that a fading channel behaves as if it were not fading, implying that the channel gain after beamforming is nearly constant (Hochwald, Marzetta & Tarokh, 2004). As such, we may have a highly deterministic channel gain because the channel hardening could increase the reliability of communication networks. A propagation channel provides h_k asymptotic channel hardening if (36)${\displaystyle \frac{\left|\right|{\mathbf{h}}_k|{|}^2}{\mathbb{E}\left\{\left|\right|{\mathbf{h}}_k|{|}^2\right\}}\to \left[N\to \infty \right]1}$

The expression in Eq. (36) implies that the channel gain approaches its mean value when the number of antennas at the BS grows very large. In this article, we consider the variance to illustrates how the channel and its means is closed to channel hardening so that (37)${\displaystyle \mathbb{V}\left\{\right.\frac{\left|\right|{\mathbf{h}}_k|{|}^2}{\mathbb{E}\left\{\left|\right|{\mathbf{h}}_k|{|}^2\right\}}\left\}\right.=\frac{\mathbb{V}\left\{\left|\right|{\mathbf{h}}_k|{|}^2\right\}}{{\left(\mathbb{E}\left\{\left|\right|{\mathbf{h}}_k|{|}^2\right\}\right)}^2}=\frac{\left(\right.\text{tr}{\left(\mathbf{R}\right)}^2\left)\right.}{\left(\right.\text{tr}\left(\mathbf{R}\right){\left)\right.}^2},}$ where the variance expression in Eq. (37) should be close to zero as N goes to infinity. The expression in Eq. (37) implies that the correlation, denoted by the eigenvalue variation of R, increases the variance and thus reduces the level of channel hardening.

Figures 8, 9, 10, 11, and 12 are provided to illustrate the effect of channel hardening in the presence of correlation with different correlation models. The variance is obtained based on the expression in Eq. (37) as a function of number of BS antennas. The uncorrelated Rayleigh fading channel is compared with different correlation models. As can be observed, the smaller the variance, the more the channel is hardened. As expected, the lowest variance is achieved with uncorrelated Rayleigh fading channel. The results show that high correlation leads to a large loss in the channel hardening in all the correlation models considered.

Numerical results and performance evaluation

This section presents some of numerical results, which examine the performance of massive-MIMO using FDD protocol. Specifically, the system performance is evaluated in terms of the ASR considering ZF precoding and the normalized MSE. In this present article, the results are provided for the uniform linear array configuration in different correlation models. Table 1 summaries the simulation parameters, which are considered in the performance evaluation. The MSE and ASR performances are averaged over 10000 independent channel realizations using a Monte-Carlo simulation. The number of training sequence length is selected based on the effective eigenvalues that are dominated. These effective eigenvalues for different correlation models are demonstrated in Figs. 3, 4, 5, 6, and 7, respectively. A short CT is considered with τ = 100 symbols. In this article, the training power ρ_p is assumed to be equal to the data power. However, optimization between the training and data powers are left to the future work, as stated earlier.

Figures 13, 14, 15, 16, and 17 demonstrate plots of the normalized MSE and DL ASR versus the SNR ρ_d in dB under different correlation models. The solid lines demonstrate the numerical analysis, while the markers represent simulation. The results demonstrate that an excellent matching between the simulated and the analytical results with various correlation models are obtained. The results show that the MSE of CSI estimation is reduced when the spatial correlation is increased. Besides, increasing the SNR improve the CSI estimation significantly. This is due to the fact that increasing the transmit power reduces the MSE performance. This is due to that fact that the error variance would approach zero in the high power regions. In addition, the error variance is reduced with the increasing of the spatial correlation. Specifically, when the level of correlation is relatively increased, the power in the channel is increased and the eigendirections denoted by the dominated eigenvalues is reduced. Consequently, this would provide a sufficient improvement in the CSI estimation accuracy even when a reduced training sequence length is used. Furthermore, the results indicate clearly the effect of carrying out DL CSI estimation. This is clearly observed in the plots when comparing the perfect CSI and imperfect CSI estimation. To this end, the results show the loss in the ASR is due the pre-log fraction, which effects the ASR considerably. The highest ASR with imperfect CSI estimation is obtained using the P-DoF and exponential channel models with c = o.1 and r = 0.9, respectively. Considering the perfect CSI estimation, the results clearly show that increasing the level of correlation reduces the ASR considerably under all the correlation models considered. In summary, the results indicate that minimizing the MSE of the DL CSI estimation does not maximizing the ASR in the limited CT. Using large arrays may enhance the spatial resolution and provide a sufficiently accurate CSI estimation. However, this is not necessary would enhance the ASR. Finally, the results provide an essential insight into the performances of the massive-MIMO systems with FDD transmission protocol and considering different spatial correlation models.

Concluding remarks and future research directions

This article investigated the FDD massive-MIMO systems performance using different correlation models. To this end, statistical structure of massive-MIMO channel captured by the physical correlation is explored to find sufficiently accurate DL CSI estimation considering a short CT. The training sequence in DL is constructed based on the eigenvectors of the transmit correlation matrix. In this case, only the effective dominated eigenvectors is exploited to reduce the DL CSI estimation overhead. In addition, this article also examined the channel hardening phenomenon in the massive-MIMO systems considering FDD transmission protocol. The results showed that with strong correlation, a large loss in the channel hardening is obtained. Furthermore, the MSE minimization of CSI estimation and ASR maximization with short CT are evaluated under different physical correlation models. The results demonstrated that the spatial correlation minimizes the MSE of CSI estimation and can help in maximizing the ASR when short CT is considered. Future work may consider the investigation of the FDD massive-MIMO systems with high frequencies such as Millimeter waves and terahertz (THz) frequencies. Besides, exploiting the deep learning approach for CSI estimation could be possible in the future, see e.g., Abdulwahhab & Jergees (2016), Ali & Taha (2021). Further, A resource allocation mechanism for future wireless communications networks (Fakhri et al., 2017) with massive MIMO systems could also be considered in future.

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