NOMA-Based Coexistence of Near-Field and Far-Field Massive MIMO Communications

The $m$-th legacy near-field user’s observation is given by $y_{m}=\mathbf{h}_{m}^{H}\mathbf{x}+n_{m}$, where $\mathbf{x}$ denotes the signal vector sent by the base station, $n_{m}$ denotes the additive Gaussian noise with its power denoted by $\sigma^{2}$, $\mathbf{h}_{m}$ is based on the spherical-wave propagation model [5, 6, 12]:

where $\alpha_{m}=\frac{c}{4\pi f_{c}\left|{\bm{\psi}}^{\rm NF}_{m}-{\bm{\psi}}_{0}\right|}$, $c$, and $f_{c}$ denote the free-space path loss, the speed of light, and the carrier frequency, respectively. We note that the line of sight (LoS) path is assumed to be available for the near-field users, as they are within the Rayleigh distance from the base station [6, 12].

The $k$-th far-field user receives the following signal: $z_{k}=\mathbf{g}_{k}^{H}\mathbf{x}+w_{k}$, where $w_{k}$ denotes the additive Gaussian noise having the same power as $n_{m}$, the conventional beamsteering vector is used to model the far-field user’s channel vector, $\mathbf{g}_{k}$, as follows: [2]

and $\theta_{k}$ denotes the conventional angle of departure. Scheduling is assumed to be carried out to ensure that each participating far-field user has an LoS connection to the base station. Because the LoS link is typically $20$ dB stronger than the non-LoS links, only the LoS link is considered in (1) and (2) [13].

Remark 1: Comparing (1) to (2), one can observe that the near-field channel model is fundamentally different from the far-field one. In particular, the channel vector in (2) is mainly parameterized by the angle of departure, $\theta_{k}$, but the elements of the vector in (1) depend on the near-field user’s specific location.

For illustrative purposes, full-digital near-field beamforming based on the zero-forcing principle is adopted in this letter: $\mathbf{P}\triangleq\begin{bmatrix}\mathbf{p}_{1}&\cdots&\mathbf{p}_{M}\end{bmatrix}=\mathbf{H}\left(\mathbf{H}^{H}\mathbf{H}\right)^{-1}\mathbf{Q}$, where $\mathbf{H}=\begin{bmatrix}\mathbf{h}_{1}&\cdots&\mathbf{h}_{M}\end{bmatrix}$, and $\mathbf{Q}$ is an $M\times M$ diagonal matrix to ensure power normalization. In particular, the $i$-th element on the main diagonal of $\mathbf{Q}$ is given by $\left[\mathbf{Q}\right]_{i,i}=\left[\left(\mathbf{H}^{H}\mathbf{H}\right)^{-1}\right]_{i,i}^{-\frac{1}{2}}$, which ensures that the beamforming vectors are normalized, i.e., $\mathbf{p}_{m}^{H}\mathbf{p}_{m}=1$, for all $m\in\{1,\cdots,M\}$.

The NOMA principle is applied to ensure that each preconfigured spatial beam is used as a type of bandwidth resource for serving additional far-field users, which means that the signal vector sent by the base station is given by

where $P_{m}$ is the transmit power allocated to the $m$-th near-field user’s signal, $f_{m,k}$ denotes the coefficient assigned to the $k$-th far-field user on beam $\mathbf{p}_{m}$, and ${s}^{\rm NF}_{m}$ and ${s}^{\rm FF}_{k}$ denote the signals for the near-field and far-field users, respectively. If the $k$-th far-field user uses only a single beam, $f_{m,k}$ can be viewed as a power allocation coefficient. If multiple beams are used, $f_{m,k}$ can be interpreted as a beamforming coefficient.

Therefore, the observation at the $m$-th near-field user can be written as follows:

For the considered NOMA network, the near-field users have better channel conditions than the far-field users, and hence have the capability to carry out SIC. To reduce the system complexity, assume that at most a single far-field user is scheduled on a near-field user’s beam, and each of the $K$ far-field users utilize $D_{x}$ beams, where $KD_{x}\leq M$. Denote the subset collecting the indices of the beams used by the $k$-th far-field user by $\mathcal{S}_{k}$, where $|\mathcal{S}_{k}|=D_{x}$.

Assume that the $k$-th far-field user is active on $\mathbf{p}_{m}$. On the one hand, this far-field user’s signal can be decoded by the $m$-th near-field user with the following data rate: $R_{m,k}^{\rm FF-NF}=\log\left(1+\frac{|f_{m,k}|^{2}h_{m}}{\sigma^{2}+P_{m}h_{m}}\right)$, where $h_{m}=|\mathbf{h}_{m}^{H}\mathbf{p}_{m}|^{2}$. If the first stage of SIC is successful, the near-field user can remove the far-field user’s signal and decode its own signal with the following data rate: $R_{m}^{\rm NF}=\log\left(1+\frac{P_{m}}{\sigma^{2}}h_{m}\right)$.

On the other hand, the $k$-th far-field user directly decodes its own signal with the following data rate:

where $\gamma_{k}=\sigma^{2}+\sum^{M}_{m=1}P_{m}g_{m,k}+\underset{i\neq k}{\sum}|\tilde{\mathbf{g}}^{H}_{k}\mathbf{f}_{i}|^{2}$ $\tilde{\mathbf{g}}_{k}=\mathbf{P}^{H}\mathbf{g}_{k}$, $g_{m,k}=|\mathbf{g}_{k}^{H}\mathbf{p}_{m}|^{2}$, and $\mathbf{f}_{k}=\begin{bmatrix}f_{1,k}&\cdots f_{M,k}\end{bmatrix}^{T}$.

The aim of this letter is to maximize the far-field users’ sum data rate while guaranteeing the near-field users’ QoS requirements, based on the following optimization problem:

where $\mathbf{1}_{x\neq 0}$ denotes an indicator function, i.e., $\mathbf{1}_{x\neq 0}=1$ if $x\neq 0$, otherwise $\mathbf{1}_{x\neq 0}=0$, $R$ denotes the near-field users’ target data rate, and $P$ denotes the transmit budget per beam. We note that the resource allocation problem in (P1) requires the base station to have access to the users’ channel state information (CSI). This CSI assumption can be realized by asking each user to first carry out channel estimation based on the pilots broadcast by the base station and then feed back its CSI to the base station via a reliable control channel.

Remark 2: The objective function in (P1a) indicates that the $k$-th far-field user prefers to use a beam on which both $h_{m}$ and $g_{m,k}$ are strong. Therefore, to find $\mathcal{S}_{k}$ and remove the indicator function in (P1a), a simple sub-optimal scheduling scheme can be used first, where the far-field users are successively asked to select the best $D_{x}$ beams based on the following criterion: ${\arg}~{}\underset{m}{\max}~{}\min\left\{\frac{h_{m}}{\max\{h_{1},\cdots,h_{M}\}},\frac{g_{m,k}}{\max\{g_{1,k},\cdots,g_{M,k}\}}\right\}$. Here, the channel gains, $h_{m}$ and $g_{m,k}$, are normalized to ensure that they are in the same order of magnitude. As a result, $f_{m,k}=0$ for $m\not\in\mathcal{S}_{k}$, and only $f_{m,k}$, $m\in\mathcal{S}_{k}$, need to be optimized. We note that optimal scheduling is possible by applying integer programming as in [8]. However, this is out of the scope of this paper due to the space limitations.

In this section, the general case with $K\geq 1$ and $D_{x}\geq 1$ is considered first. Problem (P1) can be first recast as follows:

where $\epsilon=2^{R}-1$, and constraint (P2d) is due to the use of the scheduling scheme discussed in Remark 3. Because (P2b) and (P2c) are decreasing functions of $P_{m}$, it is straightforward to show that the optimal solution of $P_{m}$ is $P_{m}^{*}=\min\left\{\frac{\sigma^{2}\epsilon}{h_{m}},P\right\}$. We note that the main challenges involving problem (P2) are the non-convex constraints in (P2b) and (P2c), which motivates the use of SCA. To facilitate the application of SCA, problem (P2) can be first recast as follows:

where $\eta_{k}=\sigma^{2}+\sum^{M}_{m=1}P_{m}^{*}g_{m,k}$ and $\mu_{m}=\frac{\sigma^{2}+P_{m}^{*}h_{m}}{h_{m}},m\in\mathcal{S}_{k}$. The term $\frac{|\tilde{\mathbf{g}}^{H}_{k}\mathbf{f}_{k}|^{2}}{x_{k}}$ can be approximated as an affine function via the Taylor expansion. In particular, first express $|\tilde{\mathbf{g}}^{H}_{k}\mathbf{f}_{k}|^{2}$ as follows:

where $(\cdot)^{T}$ denote the transpose, $\bar{\mathbf{f}}_{k}=\begin{bmatrix}{\rm Re}({\mathbf{f}}_{k})^{T}&{\rm Im}({\mathbf{f}}_{k})^{T}\end{bmatrix}^{T}$, $\hat{\mathbf{g}}_{k}=\begin{bmatrix}{\rm Re}(\hat{\mathbf{g}}_{k})^{T}&{\rm Im}(\hat{\mathbf{g}}_{k})^{T}\end{bmatrix}^{T}$, $\check{\mathbf{g}}_{k}=\begin{bmatrix}-{\rm Im}(\hat{\mathbf{g}}_{k})^{T}&{\rm Re}(\hat{\mathbf{g}}_{k})^{T}\end{bmatrix}^{T}$. By building the real-valued vector, $\tilde{\mathbf{f}}_{k}=\begin{bmatrix}\bar{\mathbf{f}}_{k}^{T}&x_{k}\end{bmatrix}^{T}$, and applying the first order Taylor expansion, $\frac{|\tilde{\mathbf{g}}^{H}_{k}\mathbf{f}_{k}|^{2}}{x_{k}}$ can be approximated as follows:

where $\tilde{\mathbf{f}}^{0}_{k}=\begin{bmatrix}{\rm Re}({\mathbf{f}}_{k}^{0})^{T}&{\rm Im}({\mathbf{f}}_{k}^{0})^{T}&x_{k}^{0}\end{bmatrix}^{T}$ denotes the initial value of $\tilde{\mathbf{f}}_{k}$, and $\triangledown_{k}$ is given by

Constraint (P3c) can be similarly approximated by using an initial value $f_{m,k}^{0}$.

Therefore, problem (P3) can be approximated as follows:

which is a convex optimization problem and can be straightforwardly solved by convex solvers.

The implementation of SCA requires that $\tilde{\mathbf{f}}_{k}^{0}$ is a feasible solution of problem (P3), and $\tilde{\mathbf{f}}_{k}^{0}$ can be obtained as follows. First, by using (P3d), we choose $f_{m,k}^{0}=0$ for $m\not\in\mathcal{S}_{k}$, and $f_{m,k}^{0}=(P-P^{*}_{m})$ for $m\in\mathcal{S}_{k}$, which means that the following choices of $x_{k}^{0}$, $1\leq k\leq K$, are feasible:

Based on $\tilde{\mathbf{f}}_{k}^{0}$, SCA can be applied in an iterative manner, i.e., by using $\tilde{\mathbf{f}}_{k}^{0}$ and solving problem (P4), a new estimate of $\tilde{\mathbf{f}}_{k}$ can be generated and used to replace $\tilde{\mathbf{f}}_{k}^{0}$ in the next iteration. In general, SCA can only converge to a stationary point, which cannot be guaranteed to be the optimal solution. Motivated by this, the optimal performance of NOMA transmission is studied for the two special cases in the following subsection.