Uplink Achievable Rate analysis of Massive MIMO Systems in Transmit-correlated Ricean Fading Environments

Abstract

In this article, the uplink achievable rate is investigated for massive multiple-input multiple-output (MIMO) under correlated Ricean fading channel, where each base station (BS) and user are both deployed multiple antennas. Considering the availability of prior knowledge at BS, two different channel estimation approaches are adopted with and without prior knowledge. Based on these channel estimations, a two-layer decoding scheme is adopted with maximum ratio precoding as the first layer decoder and optimal second layer precoding in the second layer. Based on two aforementioned channel estimations and two-layer decoding scheme, the exact closed form expressions for uplink achievable rates are computed with and without prior knowledge, respectively. These derived expressions enable us to analyze the impacts of line-of-sight (LoS) component, two-layer decoding, data transmit power, pilot contamination, and spatially correlated Ricean fading. Then, numerical results illustrate that the system with spatially correlated Ricean fading channel is superior in terms of uplink achievable rate. Besides, it reveals that compared with the single-layer decoding, the two-layer decoding scheme can significantly improve the uplink achievable rate performance.

1. Introduction

Massive multiple-input multiple-output (MIMO) is an important technique for beyond-5G (B5G) communication networks, since its ability to enhance system performance (e.g. spectral and energy efficiency) with large antenna arrays [1]. In practice, most of the studies mainly focus on analyzing massive MIMO performance for single-antenna user scenario, e.g. [2-4]. On this basis, the system performance will be significantly improved by using deep learning which has many advantages in massive training samples, noise influence, and high-dimensional feature data [5, 6]. However, many modern user devices such as smart vehicles, tablets, and laptops are equipped with multi-antennas to enhance the achievable rate performance [7, 8]. Thus, as one of performance metrics for massive MIMO, the achievable rate of multi-antenna users is necessary to be analyzed for understanding the effects of additional antennas at users [9, 10].

To this end, for a single-cell scenario with multi-antenna users, both uplink and downlink achievable rates are derived under spatially correlated Rayleigh fading channel [11]. It shows that the additional user antennas can significantly improve the achievable rate via increasing spatial multiplexing. However, a novel insight is presented in [12], which reveals that the inner-user interference seriously affects the achievable rate when the users are deployed multiple antennas. For mitigating interference, a hybrid beamformer is proposed by using the channel covariance matrix [13]. Moreover, an improved maximum ratio (MR) precoding scheme is presented for multi-antenna users to mitigate the interference in [14, 15]. Compared with the conventional precoder, this precoder can significantly enhance the downlink achievable rate. In a multi-cell scenario, employing zero forcing (ZF) and regularized ZF (RZF) precoding scheme, the closed-form downlink achievable rates are derived [16]. It validates that the achievable rates are significantly affected by pilot contamination (PC). Based on the interference alignment theory, a novel approach is presented to reduce PC via designing precoder and combiner matrices [17]. In addition, combining the benefits of the spatial modulation and massive MIMO techniques, the closed form achievable rate expressions are derived with the MR and ZF combining precoders, respectively [18]. This expression facilitated the system-level analysis, in which the system-level achievable rates are maximized by optimizing the user antennas.

The aforementioned research focuses on investigating the system performance under Rayleigh fading channel. However, the practical channel in densely deployed massive MIMO system is always composed of line-of-sight (LoS) components, which should be modeled as spatially correlated Ricean fading [19]. To this end, for a single-cell scenario with multi-antenna users, the closed-form uplink achievable rate expression is computed under correlated Ricean fading in [20]. However, this study assumes that the full knowledge of channel covariance matrices is known at the BS as prior information, while practical BS sometimes does not obtain any prior information. Moreover, the impact of PC is not considered in this study. In fact, PC is still challenging and cannot be ignored [21]. To mitigate PC, a two-layer decoding scheme is studied for massive MIMO with single-antenna users in [22]. However, this work focuses on the system with single-antenna users, and only considers the case that the full knowledge of channel covariance matrices is known at the BS.

Motivated by the aforementioned observations, the uplink achievable rate for multi-cell scenario with multi-antenna users is investigated in this article under correlated Ricean fading channel by capturing the effects of LoS components, PC, two-layer decoding, and data transmit power. The specific contributions of this article are outlined as follows:

(1) A multi-cell multi-antenna users massive MIMO network is proposed under spatially correlated Ricean fading that captures the effect of LoS and non-line-of-sight (NLoS) components in real communication systems. Particularly, the Kronecker correlation channel model is adopted for describing LoS propagation and spatially correlated multipath fading.

(2) By employing two different channel estimations and two-layer decoding scheme, the closed-form achievable rate expressions are computed under correlated Ricean fading channel. These derived expressions provide generalized expression for analyzing the impacts of multi-antenna users, inter-cell interference, channel estimation errors and spatially correlated Ricean fading. Simulation results illustrate that the uplink achievable rate is significantly improved by employing two-layer decoding scheme under correlated Ricean fading channel.

2. System Model

In this work, an uplink massive MIMO network with L cells is consider, where each cell contains that one BS simultaneously serves K users. In particular, each user and BS are deployed with N and M antennas, respectively. For the sake of brevity, the sets of BS, users, and user antenna are denoted as 𝒧 ∈ {1,..., L}, 𝒦 ∈ {1,..., K}, and 𝒩 ∈ {1,..., N}, respectively. Moreover, the kth user in the ith cell is defined as u_ik, where k ∈ 𝒦 and i ∈ 𝒧. The channel matrix between u_ik and the lth BS is defined as H_ik^l = [h_ik^l1,..., h_ik^lN] ∈ ℂ^MxN, where h_ik^ln ∈ ℂ^Mx1 represents the channel vector between nth antenna of u_ik and the lth BS. Furthermore, the length of coherence interval is denoted by τ_c. Each coherence interval consists of two parts: τ_p(τ_p ≥ KN) and τ_c - τ_p symbols are utilized for uplink training and data transmission, respectively.

In practical scenario, the LoS path can be obstructed due to the presence of blockages as shown in Fig. 1. Thus, the channel is needed to be characterized as a combination of both LoS and NLoS components. For this purpose, based on Kronecg channel H_ik^l is given by

\(\begin{aligned}\boldsymbol{H}_{i k}^{l}=\overline{\boldsymbol{H}}_{i k}^{l}+\underbrace{\left(\boldsymbol{R}_{i k}^{l}\right)^{1 / 2} \boldsymbol{G}_{i k}^{l}\left(\boldsymbol{U}_{i k}^{l}\right)^{1 / 2}}_{\widetilde{G}_{i k}^{l}}\end{aligned}\)       (1)

Fig. 1. Illustration for massive MIMO network under Ricean fading.

where \(\begin{aligned}\overline{\boldsymbol{H}}_{i k}^{l}=\left[\overline{\boldsymbol{h}}_{i k}^{l 1}, \ldots, \overline{\boldsymbol{h}}_{i k}^{l N}\right] \in \mathbb{C}^{M \times N}\end{aligned}\) is an matrix corresponds to the deterministic LoS component and \(\begin{aligned}\tilde{\boldsymbol{G}}_{i k}^{l}=\left[\tilde{\boldsymbol{g}}_{i k}^{l 1}, \ldots, \tilde{\boldsymbol{g}}_{i k}^{I N}\right] \in \mathbb{C}^{M \times N}\end{aligned}\) represents the stochastic NLoS component. In (1), R_ik^l ∈ ℂ^MxM stands for a covariance matrix at lth BS. G_ik^l represents the M × N small-scale fading matrix, and the elements of G_ik^l obey distribution with C𝒩 (0,1). Moreover, U_ik^l ∈ ℂ^NxN is a covariance matrix capturing the spatial correlation at u_ik. Let \(\begin{aligned}\boldsymbol{U}_{i k}^{l}=\tilde{\boldsymbol{U}}_{i k}^{l} \boldsymbol{\Lambda}_{i k}^{l} \tilde{\boldsymbol{U}}_{i k}^{l}\end{aligned}\) be the eigenvalue decomposition of U_ik^l, where \(\begin{aligned}\tilde{\boldsymbol{U}}_{i k}^{l} \in \mathbb{C}^{N \times N}\end{aligned}\) is a unitary matrix and Λ_ik^l = diag{λ_ik^l1,⋯λ_ik^lN} contains the eigenvalues of U_ik^l.

2.1 Uplink Pilot Training

In pilot-based channel estimation phase, all users are required to simultaneously send pilot sequences to the BS for channel estimation. Let Φ_lk = [ϕ_lk,1^l,..., ϕ_lk,N^l] ∈ ℂ^τ_pxN be the pilot sequence matrix of u_lk, satisfying that

       (2)

The pilot signals Y_l^p ∈ ℂ^Mxτ_p received at lthBS is expressed as

\(\begin{aligned}\boldsymbol{Y}_{l}^{p}=\sum_{l \in \mathcal{L}} \sum_{k \in \mathcal{K}} \sqrt{\rho_{u}} \boldsymbol{H}_{l k}^{l}\left(\boldsymbol{\Phi}_{l k}\right)^{\mathrm{H}}+\boldsymbol{N}_{l}\end{aligned}\),       (3)

where ρ_u represents the transmitted pilot power. N_l ∈ ℂ^Mxτ_p denotes the additive noise matrix, and the elements of N_l obey distribution with C𝒩 (0,1).

To obtain the channel estimation of h_lk^l, Y_l^p is projecting onto Φ_lk as follows:

\(\begin{aligned}\boldsymbol{Y}_{l k}^{p}=\boldsymbol{Y}_{l}^{p} \boldsymbol{\Phi}_{l k}=\sqrt{\rho_{u}} \tau_{p} \boldsymbol{H}_{l k}^{l}+\sum_{l^{\prime} \neq l} \sqrt{\rho_{u}} \tau_{p} \boldsymbol{H}_{l^{\prime} k}^{l}+\boldsymbol{N}_{l} \boldsymbol{\Phi}_{l k}\end{aligned}\).       (4)

Let \(\begin{aligned}\hat{\boldsymbol{H}}_{l k}^{l, \text { MMSE }}=\left[\hat{\boldsymbol{h}}_{l k}^{l l, \mathrm{MMSE}}, \ldots, \hat{\boldsymbol{h}}_{l k}^{l N, \text { MMSE }}\right] \in \mathbb{C}^{M \times N}\end{aligned}\) denote the minimum mean-squared error (MMSE) estimation of H_lk^l, which is expressed as [24]

\(\begin{aligned}\operatorname{vec}\left(\hat{\boldsymbol{H}}_{l k}^{l, \mathrm{MMSE}}\right)=\operatorname{vec}\left(\overline{\boldsymbol{H}}_{l k}^{l}\right)+\left(\boldsymbol{\Lambda}_{l k}^{l} \otimes \boldsymbol{R}_{l k}^{l}\right)\left(\sum_{l^{\prime}=1}^{L} \boldsymbol{\Lambda}_{l^{\prime} k}^{l} \otimes \boldsymbol{R}_{l^{\prime} k}^{l}+\left(\tau_{p} \sqrt{\rho_{u}}\right)^{-1} \boldsymbol{I}_{N} \otimes \boldsymbol{I}_{M}\right)^{-1} \operatorname{vec}\left(\boldsymbol{Y}_{l k}^{p}-\tilde{\boldsymbol{Y}}_{l k}^{p}\right)\end{aligned}\),       (5)

where \(\begin{aligned}\tilde{\boldsymbol{Y}}_{l k}^{p}=\sum_{l^{\prime} \in \mathcal{L}} \tau_{p} \sqrt{\rho_{u}} \overline{\boldsymbol{H}}_{l^{\prime} k}^{l}\end{aligned}\).

Moreover, let \(\begin{aligned}\overline{\boldsymbol{h}}_{l k}^{l n}\end{aligned}\) and \(\begin{aligned}\tilde{\boldsymbol{g}}_{l k}^{l n}\end{aligned}\) be nth column of \(\begin{aligned}\overline{\boldsymbol{H}}_{l k}^{l}\end{aligned}\) and \(\begin{aligned}\tilde{\boldsymbol{G}}_{l k}^{l}\end{aligned}\), respectively. Then, the MMSE estimation in (5) is represented as

\(\begin{aligned}\left[\begin{array}{c}\hat{\boldsymbol{h}}_{l k}^{l, \mathrm{MMSE}} \\ \vdots \\ \hat{\boldsymbol{h}}_{l k}^{l N, \mathrm{MMSE}}\end{array}\right]^{\mathrm{T}}=\left[\begin{array}{c}\overline{\boldsymbol{h}}_{l k}^{l 1} \\ \vdots \\ \overline{\boldsymbol{h}}_{l k}^{l N}\end{array}\right]^{\mathrm{T}}+\boldsymbol{\Xi}\left[\begin{array}{c}\sum_{l^{\prime} \in \mathcal{L}} \tau_{p} \sqrt{\rho_{u}} \tilde{\boldsymbol{g}}_{l^{\prime} k}^{l 1}+\boldsymbol{n}_{l k} \\ \vdots \\ \sum_{l^{\prime} \in \mathcal{L}} \tau_{p} \sqrt{\rho_{u}} \tilde{\boldsymbol{g}}_{l^{\prime} k}^{l N}+\boldsymbol{n}_{l k}\end{array}\right]^{\mathrm{T}}\end{aligned}\),       (6)

where \(\begin{aligned}\boldsymbol{\Xi}=\operatorname{diag}\left\{\lambda_{l k}^{l 1}, \boldsymbol{R}_{l k}^{l}\left(\boldsymbol{Q}_{l k}^{1}\right)^{-1}, \cdots, \lambda_{l k}^{l N} \boldsymbol{R}_{l k}^{l}\left(\boldsymbol{Q}_{l k}^{N}\right)^{-1}\right\}\end{aligned}\) and n_lk = vec(N_lΦ_lk) . Here, \(\begin{aligned}\boldsymbol{Q}_{l k}^{n}=\sum_{l^{\prime} \in \mathcal{L}} \lambda_{l^{\prime} k}^{l n} \boldsymbol{R}_{l^{\prime} k}^{l}+\left(\tau_{p} \rho_{u}\right)^{-1} \boldsymbol{I}_{M}\end{aligned}\).

Based on (6), the MMSE estimation \(\begin{aligned}\hat{\boldsymbol{h}}_{l k}^{l n, \mathrm{MMSE}}\end{aligned}\) and the estimation error \(\begin{aligned}\tilde{\boldsymbol{h}}_{l k}^{l n, \mathrm{MMSE}}=\boldsymbol{h}_{l k}^{l n}-\hat{\boldsymbol{h}}_{l k}^{l n, \mathrm{MMSE}}\end{aligned}\) are distributed as

\(\begin{aligned}\hat{\boldsymbol{h}}_{l k}^{l n, \mathrm{MMSE}} \sim C \mathcal{N}\left(\overline{\boldsymbol{h}}_{l k}^{l n}, \boldsymbol{\Omega}_{l k}^{l n}\right)\end{aligned}\),       (7)

\(\begin{aligned}\tilde{\boldsymbol{h}}_{l k}^{l n, \mathrm{MMSE}} \sim C \mathcal{N}\left(0, \boldsymbol{C}_{l k}^{l n}\right)\end{aligned}\),       (8)

where Ω_lk^ln = (λ_lk^ln)²R_lk^l(Q_lkⁿ)^-1R_lk^l and C_lk^ln = λ_lk^lnR_lk^l - Ω_lk^l.

Note that the full knowledge of covariance matrices R_lk^l is required for MMSE channel estimation as prior information. For another practical scenario, if there is no prior information of R_lk^l, we can use the least-square (LS) approach to achieve the estimation of \(\begin{aligned}\hat{\boldsymbol{h}}_{l k, n}^{l}\end{aligned}\) [25]. Based on this method, the LS estimation \(\begin{aligned}\hat{\boldsymbol{h}}_{l k}^{l n}\end{aligned}\) is given by \(\begin{aligned}\hat{\boldsymbol{h}}_{l k}^{l n, \mathrm{LS}}=\left(1 / \tau_{p} \sqrt{\rho_{u}}\right) \hat{\boldsymbol{y}}_{l k, n}\end{aligned}\), where \(\begin{aligned}\hat{\boldsymbol{y}}_{l k, n}=\sum_{l^{\prime} \in \mathcal{L}} \boldsymbol{h}_{l^{\prime} k}^{l n}+\boldsymbol{n}_{l^{\prime} k}\end{aligned}\) denotes the nth column of Y_lk^p . Moreover, the LS estimation \(\begin{aligned}\hat{\boldsymbol{h}}_{l k}^{l n}\end{aligned}\) and estimation error \(\begin{aligned}\tilde{\boldsymbol{h}}_{l k}^{l n}\end{aligned}\) can be distributed as

\(\begin{aligned}\hat{\boldsymbol{h}}_{l k}^{l n, \mathrm{LS}} \sim C \mathcal{N}\left(\frac{1}{\tau_{p} \sqrt{\rho_{u}}} \overline{\boldsymbol{y}}_{l k, n}, \frac{1}{\tau_{p} \rho_{u}} \boldsymbol{Q}_{l k}^{l n}\right)\end{aligned}\),       (9)

\(\begin{aligned}\tilde{\boldsymbol{h}}_{l k}^{l n, \mathrm{LS}} \sim C \mathcal{N}\left(\overline{\boldsymbol{h}}_{l k}^{l n}-\frac{1}{\tau_{p} \sqrt{\rho_{u}}} \overline{\boldsymbol{y}}_{l k, n}, \frac{1}{\tau_{p} \rho_{u}} \boldsymbol{Q}_{l k}^{l n}-\lambda_{l k}^{l n} \boldsymbol{R}_{l k}^{l}\right)\end{aligned}\),       (10)

where \(\begin{aligned}\overline{\boldsymbol{y}}_{l k, n}=\sum_{l^{\prime} \in \mathcal{L}} \tau_{p} \sqrt{\rho_{u}} \overline{\boldsymbol{h}}_{l^{l k}}^{l n}\end{aligned}\).

It is clear that the LS channel estimation is more complicated than the MMSE channel estimation. Specifically, the estimation error \(\begin{aligned}\tilde{\boldsymbol{h}}_{l k}^{l n}\end{aligned}\) has non-zero mean, which is required to be considered for analyzing the system performance.

2.2 Uplink Data Transmission

The received data signal y_l ∈ ℂ^Mx1 at the lth BS is given by

\(\begin{aligned}\boldsymbol{y}_{l}=\sum_{l^{\prime \prime} \in \mathcal{L}} \sum_{i \in \mathcal{K}} \boldsymbol{H}_{l^{\prime \prime} i}^{l} \boldsymbol{F}_{l^{\prime \prime} i} \boldsymbol{x}_{l^{\prime \prime} i}+\tilde{\boldsymbol{n}}_{l}\end{aligned}\).       (11)

where x_l"i = [x_l"i¹,..., x_l"i^N]^T ∈ ℂ^Nx1 is the uplink data signal transmitted from u_l"i to the BS, satisfying 𝔼{x_l"i(x_l"i)^H} = I_N. Moreover, F_l"i = diag[p_l"i¹,..., p_l"i^N] ∈ ℂ^NxN denotes the uplink transmit power of u_l"i and \(\begin{aligned}\tilde{\boldsymbol{n}}_{l} \in \mathbb{C}^{M \times 1}\end{aligned}\) represents the additive noise at lth BS, satisfying \(\begin{aligned}\mathbb{E}\left\{\tilde{\boldsymbol{n}}_{l}\left(\tilde{\boldsymbol{n}}_{l}\right)^{\mathrm{H}}\right\}=\boldsymbol{I}_{M}\end{aligned}\).

3. Uplink Achievable Rate Analysis

The exact closed form expressions in terms of uplink achievable rate are computed in this section by employing two-layer decoding scheme based on MMSE and LS channel estimation, respectively. Particularly, the main idea of two-layer decoding scheme is that the received data signal is first decoded by local decoding (i.e. MR precoding) at the BS to suppress intra-cell interference. Then, the decoded signals of users that shared the same pilot are collected by a central station for jointly processing in the second layer to mitigate PC and inter-cell interference.

3.1 Uplink Achievable Rate

Let W_lk = [w_lk¹,..., w_lk^N] ∈ ℂ^MxN denote the MR combining matrix in the first layer, of which the nth column is defined as \(\begin{aligned}\boldsymbol{w}_{l k}^{n}=\hat{\boldsymbol{h}}_{l k}^{l n}\end{aligned}\). Then, the local estimate of y_l is written as s_lk = (W_lk)^Hy_l. Extracting the nth row of s_lk, the nth data-stream received at the BS is given by

\(\begin{aligned} \boldsymbol{s}_{l k}^{n}= & \underbrace{p_{l k}^{n}\left(\boldsymbol{w}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{h}_{l k}^{n} x_{l k}^{n}}_{\text {Desired signal }}+\underbrace{\sum_{m \in \mathcal{N} \backslash\{n\}} p_{l k}^{m}\left(\boldsymbol{w}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{h}_{l k}^{l m} x_{l k}^{m}}_{\text {Inner-user interference }}+\underbrace{\sum_{l^{\prime \prime} \in \mathcal{L} \backslash\{l\}} \sum_{k^{\prime} \in \mathcal{K}} \sum_{m \in \mathcal{N}} p_{l^{\prime \prime} k^{\prime}}^{m}\left(\boldsymbol{w}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{h}_{l^{\prime \prime} k^{\prime}}^{l m} x_{l^{\prime \prime} k^{\prime}}^{m}}_{\text {Inter-cell interference }} \\ & +\underbrace{\sum_{k^{\prime} \in \mathcal{K} \backslash\{k\}} \sum_{m \in \mathcal{N}} p_{l k^{\prime}}^{m}\left(\boldsymbol{w}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{h}_{l k^{\prime}}^{l m} x_{l k^{\prime}}^{m}}_{\text {Intra-cell interference }}+\underbrace{\left(\boldsymbol{w}_{l k}^{n}\right)^{\mathrm{H}} \overline{\boldsymbol{n}}_{l}}_{\text {Noise }}\end{aligned}\)        (12)

where \(\begin{aligned}\overline{\boldsymbol{n}}_{l} \sim C \mathcal{N}(0,1)\end{aligned}\) denotes the additive noise component.

After that, the first-layer decoding signals in (12) are converged to a central station for the second-layer decoding as

\(\begin{aligned}\hat{\boldsymbol{s}}_{l k}^{n}=\sum_{l^{\prime} \in \mathcal{L}}\left(a_{l k, n}^{l^{\prime}}\right)^{\mathrm{H}} \boldsymbol{s}_{l^{\prime} k}^{n}\end{aligned}\)       (13)

where a_lk,n^l' is the second-layer decoding weight.

Define that b_lkk',nm = [(w_lkⁿ)^Hh_lk'^1m,..., (w_Lkⁿ)^Hh_lk'^Lm]^T ∈ ℂ^L and a_lkⁿ = [a_lk,n¹,..., a_lk,n^L]^T ∈ ℂ^L _, then the signal \(\begin{aligned}\hat{\boldsymbol{S}}_{l k}^{n}\end{aligned}\) is represented by

\(\begin{aligned} \hat{\boldsymbol{s}}_{l k}^{n}= & p_{l k}^{n}\left(\mathbf{a}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{b}_{l k k, n n} x_{l k}^{n}+\sum_{m \in \mathcal{N} \backslash\{n\}} p_{l k}^{m}\left(\mathbf{a}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{b}_{l k k, n m} x_{l k}^{m}+\sum_{k^{\prime} \in \mathcal{K} \backslash\{k\}} \sum_{m \in \mathcal{N}} p_{l k^{\prime}}^{m}\left(\mathbf{a}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{b}_{l k k^{\prime}, n m} x_{l k^{\prime}}^{m}+\hat{n}_{l k} \\ & +\sum_{l^{\prime \prime} \in \mathcal{L} \backslash\{l\}} \sum_{k^{\prime} \in \mathcal{K}} \sum_{m \in \mathcal{N}} p_{l^{\prime \prime} k^{\prime}}^{m}\left(\mathbf{a}_{l k}^{n}\right)^{\mathrm{H}} \boldsymbol{b}_{l^{\prime \prime} k k^{\prime}, n m} x_{l^{\prime \prime} k^{\prime}}^{m}\end{aligned}\).       (14)

where \(\begin{aligned}\hat{n}_{l k}=\sum_{l^{\prime} \in \mathcal{L}}\left(a_{l k, n}^{l^{\prime}}\right)^{\mathrm{H}}\left(\boldsymbol{w}_{l k}^{n}\right)^{\mathrm{H}} \overline{\boldsymbol{n}}_{l} \cdot\end{aligned}\).

Based on (14), employing the use-and-then-forget (UatF) technique [26], the uplink achievable rate of u_lk with two-layer decoding scheme is given by

\(\begin{aligned}\tilde{\mathcal{R}}_{l k}=\left(1-\frac{\tau_{p}}{\tau_{c}}\right) \sum_{n \in \mathcal{N}} \log _{2}\left(1+\gamma_{l k, n}\right)\end{aligned}\),       (15)

where γ_lk,n denotes signal to interference plus noise ratio (SINR) for the nth data-stream of u_lk, which is given by

\(\begin{aligned}\gamma_{l k, n}=\frac{\left|p_{l k}^{n}\left(\mathbf{a}_{l k}^{n}\right)^{\mathrm{H}} \mathbb{E}\left\{\boldsymbol{b}_{l k k, n n}\right\}\right|^{2}}{\left(\mathbf{a}_{l k}^{n}\right)^{\mathrm{H}}\left(\boldsymbol{D}_{l k}\right) \mathbf{a}_{l k}^{n}}\end{aligned}\),       (16)

where \(\begin{aligned}\boldsymbol{D}_{l k}=\sum_{l^{\prime} \in \mathcal{L} k} \sum_{k^{\prime} \in \mathcal{K}} \sum_{m \in \mathcal{N}}\left(p_{l^{\prime} k^{\prime}}^{m}\right)^{2} \boldsymbol{\Gamma}_{l l^{\prime} k^{\prime}}-\left(p_{l k}^{n}\right)^{2} \mathbb{E}\left\{\boldsymbol{b}_{l k, n n}\right\} \mathbb{E}\left\{\left(\boldsymbol{b}_{l k k, n n}\right)^{\mathrm{H}}\right\}+\boldsymbol{Z}_{k n}\end{aligned}\). Here, Z_kn = diag(𝔼{||w_lkⁿ||²},..., 𝔼{||w_Lkⁿ||²}) and Γ_l"kk'= 𝔼{b_lkk,nm(b_lkk,nm)^H} ∈ ℂ^LxL .

The expression (15) provides a generalized uplink achievable rate expression for a multi-cell scenario with multi-antenna users. Note that the expression (15) can be applied to practical spatially correlated Ricean fading channel with arbitrary channel estimation and first-layer decoder. However, the expression in (15) is not in closed-form, which cannot effectively quantify the effects of LoS component, PC, and channel estimation error. Moreover, for enhancing the performance of system, the closed-form expressions are desired to formulate the optimization scheme such as data power control and resource allocation. To this end, using the MR combining as the first-layer decoder, the closed-form expressions of (15) are derived with and without prior knowledge in the next subsection.

In addition, it can be observed that the second-layer decoding vector a_lk,n seriously affects the performance expression. Thus, to enhance system performance, the second-layer decoding vector a_lk,n can be optimized by the following corollary.

Corollary 1: For a given set of first-layer decoder (i.e. MR precoding) and data transmit power matrix, the uplink achievable rate in (15) can be maximized by

a_lk^n* = (D_k)^-1𝔼{b_lkk,nm}       (17)

and the maximum value of \(\begin{aligned}\tilde{\mathcal{R}}_{l k}\end{aligned}\) can be given by

\(\begin{aligned}\tilde{\mathcal{R}}_{l k}^{*}=\left(1-\frac{\tau_{p}}{\tau_{c}}\right) \sum_{n \in \mathcal{N}} \log _{2}\left(1+\left(\mathbb{E}\left\{\boldsymbol{b}_{l k k, n n}\right\}\right)^{\mathrm{H}}\left(\boldsymbol{D}_{k}\right)^{-1} \mathbb{E}\left\{\boldsymbol{b}_{l k k, n n}\right\}\right)\end{aligned}\).      (18)

Proof: The SINR expression γ_lk,n in (16) can be recognized as a generalized Rayleigh quotient with respect to a_lkⁿ . Thus, according to [26, Lemma B.10], the maximum uplink achievable rate in \(\begin{aligned}\tilde{\mathcal{R}}_{l k}^{*}\end{aligned}\) (18) and optimal a_lk^n* in (17) can be obtained.

Corollary 1 reveals that the second-layer decoding matrix can be designed to maximize the uplink achievable rate when the first-layer decoder and data transmit power coefficients are given.

3.2 Closed-form Uplink Achievable Rate for Prior Knowledge Case

Employing MR combining as the first-layer decoder, the closed-form expression of (15) is computed for prior knowledge case as follows.

Theorem 1: If the MMSE channel estimation and MR combining \(\begin{aligned}\boldsymbol{w}_{l k, n}=\hat{\boldsymbol{h}}_{l k, n}^{l}\end{aligned}\) are adopted, the uplink achievable rate of u_lk with two-layer decoding scheme can be derived in closed-form as

\(\begin{aligned}\tilde{\mathcal{R}}_{l k}^{\mathrm{MMSE}}=\left(1-\frac{\tau_{p}}{\tau_{c}}\right) \sum_{n \in \mathcal{N}} \log _{2}\left(1+\frac{\left|p_{l k}^{n}\left(\mathbf{a}_{l k}^{n, \mathrm{MMSE}}\right)^{\mathrm{H}} \mathbb{E}\left\{\boldsymbol{b}_{l k k, n n}^{\mathrm{MMSE}}\right\}\right|^{2}}{\left(\mathbf{a}_{l k}^{n, \mathrm{MMSE}}\right)^{\mathrm{H}}\left(\boldsymbol{D}_{l k}^{\mathrm{MMSE}}\right) \mathbf{a}_{l k}^{n, \mathrm{MMSE}}}\right)\\\end{aligned}\),       (19)

where 𝔼{b_lkk,nm^MMSE} = diag(Z_kn) and

\(\begin{aligned}\boldsymbol{D}_{l k}^{\mathrm{MMSE}}=\sum_{l^{\prime \prime} \in \mathcal{L}} \sum_{k^{\prime} \in \mathcal{K}} \sum_{m \in \mathcal{N}}\left(p_{l^{\prime} k^{\prime}}^{m}\right)^{2} \overline{\boldsymbol{\Gamma}}_{l^{\prime \prime} k^{\prime}}^{\mathrm{MME}}+\sum_{l^{\prime \prime} \in \mathcal{L} \backslash\{l\}}\left(p_{l^{\prime \prime} k}^{n}\right)^{2} \tilde{\boldsymbol{\Gamma}}_{l^{\prime \prime} k}^{\mathrm{MMSE}}-\left(\boldsymbol{\Lambda}_{k, n}\right)^{2}+\boldsymbol{Z}_{k n}^{\mathrm{MMSE}}\end{aligned}\).       (20)

In (20), [Z_kn^MMSE]_ll is the (l,l)th element of Z_kn^MMSE , which can be given by \(\begin{aligned}\left[\boldsymbol{Z}_{k n}^{\mathrm{MMSE}}\right]_{l l}=\operatorname{tr}\left(\boldsymbol{\Omega}_{l k}^{l n}+\overline{\boldsymbol{h}}_{l k}^{l n}\left(\overline{\boldsymbol{h}}_{l k}^{l n}\right)^{\mathrm{H}}\right)\end{aligned}\). Moreover, \(\begin{aligned}\overline{\boldsymbol{\Gamma}}_{l^{\prime} k k^{\prime}}^{\mathrm{MMSE}} \in \mathbb{C}^{L \times L}\end{aligned}\) is a diagonal matrix with (l,l)th element given by

(21)

Moreover, \(\begin{aligned}\boldsymbol{\Lambda}_{k, n}=\operatorname{diag}\left(\left|\overline{\boldsymbol{h}}_{1 k}^{1 n}\right|^{2}, \ldots,\left|\overline{\boldsymbol{h}}_{L k}^{L n}\right|^{2}\right)\end{aligned}\) and \(\begin{aligned}\tilde{\boldsymbol{\Gamma}}_{l^{\prime \prime} k}^{\mathrm{MMSE}}=\boldsymbol{z}_{l^{\prime \prime} k}\left(\boldsymbol{z}_{l^{\prime \prime} k}\right)^{\mathrm{H}}\end{aligned}\), where z_l"k = [tr(λ_l"k¹ⁿR_l"k¹(Q_lkⁿ)^-1λ_lk¹ⁿR_lk¹),..., tr(λ_l"k^LnR_l"k^L(Q_Lkⁿ)^-1λ_Lk^LnR_Lk^L)]^T.

Proof: Refer to Appendix A.

From Theorem 1, the following important insights into the behaviors of a multi-cell scenario with multi-antenna users over correlated Ricean fading channel is obtained.

(1) Theorem 1 is a generalized expression for analyzing the impacts of multi-antenna users, inter-cell interference, channel estimation errors and spatially correlated Ricean fading. In particular, through accordingly adjusting the parameters, this expression contains many special cases. For example, when N=1 , the expression (19) denotes a uplink achievable rate expression with single-antenna users over spatially correlated Ricean fading. Moreover, if \(\begin{aligned}\overline{\boldsymbol{h}}_{l k, n}^{l}=0\end{aligned}\) , then the impacts of LoS components is neglected. Under such circumstance, the expression (19) stands for an uplink achievable rate expression over spatially correlated Rayleigh fading.

(2) It is clear that the desired signal term in the numerator depends on the channel estimation accuracy and the LoS component \(\begin{aligned}\overline{\boldsymbol{h}}_{l k, n}^{l}\end{aligned}\) . In fact, a larger \(\begin{aligned}\overline{\boldsymbol{h}}_{l k, n}^{l}\end{aligned}\) can significantly improve the channel estimation accuracy. Thus, a stronger LoS component makes an important contribution to enhancing the performance of system. Numerical results also validate this analysis.

(3) Moreover, Theorem 1 shows that the uplink achievable rate is seriously affected by PC term \(\begin{aligned}\tilde{\boldsymbol{\Gamma}}_{l^{\prime \prime} k k^{\prime}}\end{aligned}\) in the denominator. Thus, to enhance the performance of system, the pilot decontamination schemes such as the effective power control and the optimization of second-layer decoding can be formulated by using the expression (19).

3.3 Closed-form Uplink Achievable Rate for No Prior Knowledge Case

Employing MR combining as the first-layer decoder, the closed-form expression of (15) is computed for no prior knowledge case as follows.

Theorem 2: If the LS channel estimation and MR combining \(\begin{aligned}\boldsymbol{w}_{l k, n}=\hat{\boldsymbol{h}}_{l k, n}^{l}\end{aligned}\) are adopted, the uplink achievable rate of u_lk with two-layer decoding scheme is derived in closed-form as

\(\begin{aligned}\tilde{\mathcal{R}}_{l k}^{\mathrm{LS}}=\left(1-\frac{\tau_{p}}{\tau_{c}}\right) \sum_{n \in \mathcal{N}} \log _{2}\left(1+\frac{\left|p_{l k}^{n}\left(\mathbf{a}_{l k}^{n, \mathrm{LS}}\right)^{\mathrm{H}} \mathbb{E}\left\{\boldsymbol{b}_{l k k, n, \mathrm{~L}}^{\mathrm{LS}}\right\}\right|^{2}}{\left(\mathbf{a}_{l k}^{n, \mathrm{LS}}\right)^{\mathrm{H}}\left(\boldsymbol{D}_{l k}^{\mathrm{LS}}\right) \mathbf{a}_{l k}^{n, \mathrm{LS}}}\right)\end{aligned}\),       (22)

where

\(\begin{aligned}\boldsymbol{D}_{l k}^{\mathrm{LS}}=\sum_{l^{\prime} \in \mathcal{L}} \sum_{k^{\prime} \in \mathcal{K}} \sum_{m \in \mathcal{N}}\left(p_{l^{\prime} k^{\prime}}^{m}\right)^{2} \overline{\boldsymbol{\Gamma}}_{l^{\prime} k k^{\prime}}^{\mathrm{LS}}-\left(p_{l k}^{n}\right)^{2} \mathbb{E}\left\{z_{l k}\right\} \mathbb{E}\left\{\left(z_{l k}\right)^{\mathrm{H}}\right\}+\boldsymbol{Z}_{k n}^{\mathrm{LS}}\end{aligned}\).       (23)

In (23), \(\begin{aligned}\overline{\boldsymbol{\Gamma}}_{l^{\prime} k k^{\prime}}^{L L} \in \mathbb{C}^{L \times L}\end{aligned}\) is a diagonal matrix with (l,l)th l l element given by

(24)

where \(\begin{aligned}\overline{\boldsymbol{c}}_{l k}^{n}=\overline{\boldsymbol{y}}_{l k, n}-\overline{\boldsymbol{h}}_{l^{\prime \prime} k}^{l n}\\\end{aligned}\) . In (22), 𝔼{b_lkk,nm^LS} = 𝔼{z_lk} , where \(\begin{aligned}\boldsymbol{z}_{l k}=\left[\operatorname{tr}\left(\lambda_{l k}^{1 n} \boldsymbol{R}_{l k}^{1 n}+\sum_{l^{\prime} \in \mathcal{L}} \overline{\boldsymbol{h}}_{l^{\prime} k}^{1 n}\left(\overline{\boldsymbol{h}}_{l k}^{1 n}\right)^{\mathrm{H}}\right), \ldots, \operatorname{tr}\left(\lambda_{l k}^{L n} \boldsymbol{R}_{l k}^{L n}+\sum_{l^{\prime \prime} \in \mathcal{L}} \overline{\boldsymbol{h}}_{l^{\prime} k}^{L n}\left(\overline{\boldsymbol{h}}_{l k}^{L n}\right)^{\mathrm{H}}\right)\right]^{\mathrm{H}}\end{aligned}\) . In (23), [Z_kn^LS]_ll is the (l,l)th element of Z_kn , which can be given by \(\begin{aligned}\left[\boldsymbol{Z}_{k n}^{\mathrm{LS}}\right]_{l l}=1 / \tau_{p} \rho_{u}\left(\operatorname{tr}\left(\boldsymbol{Q}_{l k}^{l n}\right)+\left\|\overline{\boldsymbol{y}}_{l k, n}\right\|^{2}\right)\end{aligned}\).

Proof: Refer to Appendix B.

The closed-form expression of (15) for no prior knowledge case is derived in Theorem 2, where the same insights as the observations in Theorem 1 can be easily obtained. Particularly, Theorem 2 shows that the interference terms of SINR with LS channel estimation are larger than that using MMSE channel estimation. This is because that the mean values for LS channel estimation are not used as prior information.

4. Numerical Results

In simulation, the numerical result is generated for verifying the correctness of theoretical analysis in Section 3. For a wrap-around cellular network with 4 cells, the users are uniformly and independently distributed in each cell [26]. Besides, each coherence block contains τ_c = 200 samples and τ_p = KN (except Figure 7) in each cell. The LoS component from u_lk to the lth BS can be modeled as [27]

\(\begin{aligned}\overline{\boldsymbol{H}}_{i k}^{l}=\sqrt{\frac{\kappa_{i k}^{l}}{\kappa_{i k}^{l}+1}} \boldsymbol{\delta}_{r}\left(\theta_{i k}^{l, r}\right) \cdot\left(\boldsymbol{\delta}_{t}\left(\theta_{i k}^{l, t}\right)\right)^{\mathrm{T}}\end{aligned}\)       (25)

where κ_lk^l = 13 - 0.03d_lk^l [dB] refers to Ricean factor. d_lk^l stands for the distance between u_lk and the lth BS [28]. Besides, θ_ik^l,r and θ_ik^l,t denote the angle of arrival and departure. Then, δ_t(θ_ik^l,t) and δ_r(θ_ik^l,r) represent the transmit and receive array response vector, which can be given by [20]

δ_t(θ_ik^l,t) = [1, e^{jπsin(θ_ikl,t)},..., e^{jπ(M-1)sin(θ_ikl,t)}]^T       (26)

δ_t(θ_ik^l,t) = [1, e^{jπsin(θ_ikl,r)},..., e^{jπ(N-1)sin(θ_ikl,r)}]^T       (27)

Moreover, the correlation matrices R_ik^l and U_ik^l are modeled by employing the exponential correlation model from [29]. Thus, the (x, y)th and (a, b)th elements of R_ik^l and U_ik^l are given by [R_ik^l]_x,y = ρ_te^{j(x-y)θ_ikl,t}and [U_ik^l]_a,b = ρ_re^{j(a-b)θ_ikl,r} , where ρ_t and ρ_r are correlation coefficients. In simulations, the pilot and data power is assumed to be 200mW for each user.

Fig. 2 shows the sum uplink achievable rates with the two-layer decoding when the number of BSs antennas M is increasing. The curves stand for the sum uplink achievable rate obtained by using the expressions in (19) and (22), while the “W” markers represent the sum uplink achievable rates obtained by using Monte Carlo method. It is clearly that the markers and the curves as well as simulation results are nearly overlap, which validates the correctness of our derived expressions. Particularly, spatially correlated Ricean fading channels achieve higher uplink achievable rate than the spatially correlated Rayleigh fading channels for both MMSE and LS estimation. This is because the existence of LoS component in Ricean fading channel can significantly improve achievable rate of the proposed system. Moreover, for MMSE channel estimation, the highest sum uplink achievable rates can be obtained for both the curves of analytical results and the markers. In other words, the MMSE channel estimation can achieve higher uplink achievable rate than that of LS channel estimation since the knowledge of covariance matrices is known and utilized. It can be seen from Fig. 2, for all cases, the sum achievable rates are steadily increasing as M grows largely.

Fig. 2. Achievable rate with two-layer decoding versus the number of BS antennas (K = 10 and N = 2).

The impact of different channel estimation for uncorrelated Ricean fading is displayed in Fig. 3. It can be observed that the markers and the curves as well as simulation results are nearly overlap. Similar to Fig. 2, for the MMSE channel estimation, the highest sum uplink achievable rates can be obtained for both the curves of analytical results and the markers. This is because the mean vector of MMSE channel estimation is utilized for enhancing the accuracy of estimation. In addition, the sum achievable rate is steadily increasing for MMSE and LS channel estimators as M grows largely.

Fig. 3. Achievable rate with two-layer decoding versus the number of BS antennas for uncorrelated Ricean fading with K = 10 and N = 2 .

Fig. 4 and Fig. 5 illustrate the achievable rate comparison of the proposed two-layer precoding scheme and other precoding method for MMSE and LS channel estimation, respectively, where K = 20 and N = 2. To study the influence of two-layer decoding scheme on the uplink achievable rate, the case of approximate second-layer detector is considered, in which the second-layer detector is only computed by using the diagonal elements of the channel correlation matrices. It is clear that the two-layer decoding scheme enhance the achievable rate with MMSE and LS channel estimation. More importantly, the LS channel estimation obtains more benefits than MMSE channel estimation does from employing the two-layer decoding scheme. This is because the two-layer decoding scheme can effectively suppress the interference caused by LS channel estimation.

Fig. 4. Achievable rate versus M with MMSE channel estimation (K = 20 and N = 2).

Fig. 5. Achievable rate versus M with LS channel estimation (K = 20 and N = 2).

In Fig. 6, the cumulative distribution function (CDF) curves for the per-user uplink achievable rate are plotted with M = 200 , N = 2 and K = 20 . It can be observed that the two-layer decoding scheme can improve the uplink achievable rate for both MMSE and LS channel estimation cases. Fig. 6 reveals that whether the two-layer decoding scheme is adopted or not, the MMSE channel estimation can achieve higher uplink achievable rate than the LS channel estimation. In particular, LS channel estimation can achieve more benefits from two-layer decoding scheme.

Fig. 6. CDF of per-user achievable rate for single-layer and two-layer decoding.

Fig. 7 depicts the CDF curves of per-user achievable rate for different pilot lengths, where case 1 and case 2 denote τ_p = 40 and τ_p = 10, respectively. For both case, the MMSE channel estimation can obtain higher uplink achievable rate than LS channel estimation. However, the CDF curve of MMSE become closer to the LS channel estimation when τ_p = 40. This is because the larger pilot length can enhance the channel estimation quality, thereby mitigating the effect of PC.

Fig. 7. CDF of per-user achievable rate with M = 200, N = 2 and K = 20.

In Fig. 8, the CDF curves of per-user achievable rate are plotted for different number of users, in which M = 200 , N = 2. It reveals that the user obtains lower uplink achievable rate when the number of users increases. Moreover, as expected, the MMSE channel estimation achieves higher achievable rate than that of LS method for different number of user.

Fig. 8. CDF of the per-user achievable rate.

5. Conclusions

In this work, we investigate the uplink achievable rate of the proposed system. In such system, the channel is modeled as correlated Ricean fading. By employing two-layer decoding scheme, closed-form uplink achievable rate expressions are derived with and without prior knowledge case, respectively. These expressions can be utilized for formulating the uplink power control and other resource allocation schemes. Simulation results illustrate that the uplink achievable rate is effectively enhanced by employing two-layer decoding scheme under correlated Ricean fading channel.

Appendix A

Appendix B