1. Introduction
The wireless traffic demand of grouporiented services and applications such as mobile TV, satellite communications, etc., is predicted to increase significantly in future wireless networks, and physicallayer wireless multicast transmission is a promising approach for such rapidly increasing demand. Recently, physicallayer multicasting has been incorporated in the 3GPP release known as eMBMS [
1] and has received extensive research interest [
2,
3,
4]. In massive multipleinput multipleoutput (MIMO) systems [
5,
6,
7], large numbers of antennas are equipped at the base station (BS) so that a smaller number (compared with the number of BS antennas) of user terminals (UTs) can be simultaneously served in the same time and frequency resources. Compared with conventional smallscale MIMO systems [
8], massive MIMO systems can significantly improve the spectral efficiency and transmission reliability. Due to the ability of massive MIMO in shaping the multicast transmission signals, physicallayer multicasting combined with massive MIMO promises to improve the multicast transmission quality of service in the evolution of future wireless networks [
9,
10].
Energy efficiency (EE) is an important performance metric in wireless transmission design and has received growing attention from both academia and industry [
11,
12]. Energyefficient transmission design for MIMO unicasting was investigated in some existing works, e.g., [
13,
14,
15]. Meanwhile, energyefficient multicasting transmission was also studied in several previous works. For example, energyefficient coordinated multicast transmission design for multicell scenarios was studied in e.g., [
16,
17]. In addition, energyefficient joint unicast and multicast transmission design for multicell MIMO systems was investigated in [
18]. Please note that in most of the existing works on energyefficient physicallayer multicast transmission, e.g., [
16,
17,
18], instantaneous channel state information (CSI) is assumed to be known at the BS and exploited for transmission design.
For physicallayer massive MIMO multicast transmission, the transmission performance will be highly dependant on the quality of the available CSI at the BS. CSI acquisition at the BS is a challenging task in massive MIMO for both frequencydivision duplex (FDD) and timedivision duplex (TDD) protocols. For FDD massive MIMO systems, acquisition of instantaneous CSI at the BS will lead to a huge pilot and feedback overhead. Meanwhile, the hardware limitations might destroy the reciprocity between the uplink and the downlink channels for TDD massive MIMO systems, and thus instantaneous CSI acquisition is still not an easy task [
19,
20,
21,
22]. Compared with the instantaneous CSI, the statistical CSI varies over much larger time scales and thus can be efficiently and accurately obtained. In addition, massive MIMO channels exhibit new statistical properties with a large number of antennas [
23], which, then, can be exploited in energyefficient massive MIMO multicast transmission design.
In this paper, we investigate physicallayer multicast precoding for energyefficient massive MIMO transmission. Our work differs from previous works in that only statistical downlink CSI of the UTs is available at the BS. Our major contributions are summarized as follows:
We identify the eigenvectors of the optimal multicast transmit covariance matrix in terms of maximizing the system EE in closed form, which reveals that optimal energyefficient multicast transmission should be performed in the beam domain in massive MIMO and simplifies the largedimensional matrixvalued energyefficient massive MIMO multicast transmission design into a power allocation problem in the beam domain with significantly reduced optimization variables.
We propose a sequential beam domain power allocation approach with guaranteed convergence to the global optimum via exploiting Dinkelbach’s transform.
We used the largedimensional random matrix theory to deduce the deterministic equivalent (DE) of the optimization objective function to further reduce the computational complexity involved in sample averaging.
The rest of this paper is organized as follows. We present the massive MIMO beam domain channel model in
Section 2. In
Section 3, energyefficient multicast precoding for singlecell massive MIMO transmission is investigated. The eigenvectors of the optimal energyefficient multicast transmit covariance matrix is identified in closedform and a sequential beam domain power allocation algorithm is proposed. Numerical results are presented in
Section 4. Finally, we conclude the paper in
Section 5.
The notations adopted throughout the paper are listed as follows: We denote by ${\mathbb{R}}^{M\times N}$ the $M\times N$ dimensional realvalued vector space and ${\mathbb{C}}^{M\times N}$ the $M\times N$ dimensional complexvalued vector space, respectively. We adopt uppercase boldface letters to denote matrices and lowercase boldface letters to denote column vectors, respectively. We adopt $\mathbf{X}\succcurlyeq \mathbf{0}$ to denote that $\mathbf{X}$ is a positive semidefinite matrix. We use $\mathrm{tr}\left\{\xb7\right\}$ to denote the matrix trace operation and $\mathrm{det}\left\{\xb7\right\}$ to denote the matrix determinant operation, respectively. We use $\mathsf{E}\left\{\xb7\right\}$ to denote the expectation operation. We adopt $\mathcal{CN}\left(\mathbf{a},\mathbf{B}\right)$ to denote the circular symmetric complexvalued Gaussian distribution with mean $\mathbf{a}$ and covariance $\mathbf{B}$. We adopt ${\left(\xb7\right)}^{H}$, ${\left(\xb7\right)}^{T}$, and ${\left(\xb7\right)}^{*}$ to denote the conjugatetranspose, transpose, and conjugate operations, respectively. We adopt ⊙ to denote the Hadamard product. We adopt ${\left[\mathbf{A}\right]}_{m,n}$ to denote the $(m,n)$th element of matrix $\mathbf{A}$. We adopt ≜ and ∼ to denote “be defined as” and “be distributed as”, respectively.
2. Massive MIMO Beam Domain Channel Model
We consider a singlecell massive MIMO system consisting of one BS which is equipped with M antennas, and K UTs, where UT k is equipped with ${N}_{k}$ antennas. In this paper, we focus on the single group multicast transmission case where the BS sends one common message to all the UTs in the multicast group.
We denote by
$\mathbf{x}\in {\mathbb{C}}^{M\times 1}$ the multicast signal intended for the UTs in the multicast group, then the signal received at UT
k at a time instant can be written as
where
${\mathbf{H}}_{k}$ represents the downlink channel matrix from the BS to UT
k at the given time instant, and
${\mathbf{n}}_{k}\sim \mathcal{CN}\left(\mathbf{0},{\sigma}^{2}{\mathbf{I}}_{{N}_{k}}\right)$ denotes the additive Gaussian noise with
${\sigma}^{2}$ being the noise power. We assume that the multicast signal
$\mathbf{x}$ satisfies
$\mathsf{E}\left\{\mathbf{x}\right\}=\mathbf{0}$ and
$\mathsf{E}\left\{\mathbf{x}{\mathbf{x}}^{\mathrm{H}}\right\}=\mathbf{Q}\in {\mathbb{C}}^{M\times M}$ where
$\mathbf{Q}$ is the multicast signal transmit covariance matrix.
Before investigating the energyefficient massive MIMO multicast transmission design, we first introduce the channel model adopted in our work which has a great impact on the considered multicast design. Specifically, Weichselberger’s channel model [
24,
25] is adopted in our work. Please note that in the adopted Weichselberger’s channel model, the channel correlation properties between the transmitter (BS) and the receiver (UT) are jointly taken into account, which is different from the Kronecker channel model where only the correlations at both ends are considered. With Weichselberger’s jointly correlated channel model, the channel matrix
${\mathbf{H}}_{k}$ in (1) can be written as follows
where
${\mathbf{U}}_{k}\in {\mathbb{C}}^{{N}_{k}\times {N}_{k}}$ and
${\mathbf{V}}_{k}\in {\mathbb{C}}^{M\times M}$ are both deterministic unitary matrices, and
${\mathbf{G}}_{k}\in {\mathbb{C}}^{{N}_{k}\times M}$ is a random matrix where the elements are all zeromean independently distributed. Please note that in massive MIMO related literature,
${\mathbf{G}}_{k}$ is usually referred to as the beam domain channel matrix [
19,
21,
23]. The secondorder channel statistics of
${\mathbf{G}}_{k}$ can be fully characterized by the following matrix
As the
$\left(i,j\right)$th element of
${\mathsf{\Omega}}_{k}$ corresponds to the average power of the
$\left(i,j\right)$th element of the beam domain channel matrix,
${\mathsf{\Omega}}_{k}$ is usually referred to the beam domain channel power matrix. Please note that
${\mathsf{\Omega}}_{k}$ varies in a larger time scale than instantaneous CSI. In addition, the channel statistics are approximately constant for a wide range of frequencies [
23,
26]. Therefore, statistical CSI can be obtained via averaging the samples over time and frequency in a realistic wideband wireless transmission system with guaranteed accuracy. In this paper, we assume that the statistical CSI of the multicast UTs can be perfectly known at the BS.
In massive MIMO systems with a sufficiently large number of BS antennas, the channels usually exhibit new properties. One particular property of the massive MIMO channels is that the eigenvector matrices of the correlation matrices at the BS of different transmission links (UTs) tend to be the same and can be uniquely determined by the array topology equipped at the BS asymptotically [
19,
21,
23]. Specifically, we denote by
$\mathbf{V}$ this unique matrix, then the massive MIMO channel adopted in this paper can be modeled as
It is worth noting that the above channel model has been verified in typical scenarios [
19,
21,
23] and has been extensively adopted in previous works. The massive MIMO channel model in (4) will be adopted throughout this paper.
3. EnergyEfficient Multicast Precoding for Massive MIMO
In this section, we investigate energyefficient multicast precoding for massive MIMO where the BS only has knowledge of the statistical CSI of all multicast UTs and all the UTs know their own instantaneous CSI perfectly. In particular, with multicast transmit covariance matrix
$\mathbf{Q}$, the achievable ergodic multicast can be written as follows
where
${R}_{k}$ is UT
k’s achievable ergodic rate which can be written as
where (a) follows from the massive MIMO beam domain channel model in (4), and the determinant identity
$\mathrm{det}\left\{\mathbf{I}+\mathbf{A}\mathbf{B}\right\}=\mathrm{det}\left\{\mathbf{I}+\mathbf{B}\mathbf{A}\right\}$.
Before proceeding, we present the power consumption model. In massive MIMO systems, the BS consumes most of the power and we focus on the power consumption at the BS. Specifically, we adopt the power consumption model as follows [
11,
13]
where
$\mathrm{tr}\left\{\mathbf{Q}\right\}$ is the multicast transmit power, which is scaled by a coefficient
$\zeta \ge 1$ modeling the reciprocal of the transmit amplifier drain efficiency,
${P}_{\mathrm{c}}$ denotes the circuit power consumption of each BS antenna, and
${P}_{\mathrm{s}}$ denotes the static power consumption at the BS. Then, the system EE of multicast transmission with bandwidth
W is given by
In the following, we investigate energyefficient multicast precoding with statistical CSI. We aim to design the optimal transmit covariance
$\mathbf{Q}$ that can maximize the multicast EE in (8). The problem to be dealt with can be formulated as follows
where
${P}_{\mathrm{max}}$ denotes the multicast power budget at the BS.
We denote by $\mathbf{Q}=\mathsf{\Phi}\mathsf{\Lambda}{\mathsf{\Phi}}^{H}$ the eigenvalue decomposition of the transmit covariance matrix $\mathbf{Q}$. Please note that the columns of $\mathsf{\Phi}$ denote the eigenvectors of $\mathbf{Q}$ and can represent the multicast signaling directions. Meanwhile, diagonal elements of $\mathsf{\Lambda}$ denote the eigenvalues of $\mathbf{Q}$ and the powers allocated to each beam can be reflected in $\mathsf{\Lambda}$. We first investigate the eigenvectors of the optimal multicast transmit covariance matrix in the following theorem.
Theorem 1. The eigenvectors of the optimal transmit covariance matrix ${\mathbf{Q}}^{\mathrm{opt}}$ in terms of maximizing the systems EE are constituted by the columns of the matrix $\mathbf{V}$ in (4)
, i.e., The result presented in Theorem 1 shows that the optimal energyefficient multicast signaling directions should align with the eigenvectors of the transmit correlation matrices at the BS. Thus, the optimal multicast precoding in terms of maximizing the systems EE should be performed in the beam domain. With Theorem 1, the matrixvalued energyefficient transmit covariance matrix design can be simplified into a vectorvalued beam domain power allocation problem with significantly reduced optimization variables. In particular, without loss of optimality, the energyefficient multicast transmit precoding optimization problem in (9) can be simplified to the following problem
in which
In the objective function of the beam domain power allocation problem (11),
${R}_{\mathrm{mc}}\left(\mathsf{\Lambda}\right)$ and
$P\left(\mathsf{\Lambda}\right)$ are concave and linear functions with respect to
$\mathsf{\Lambda}$, respectively. Thus, the optimization problem in (11) is a concavelinear fractional program. Dinkelbach’s transform and CharnesCooper transform are two classic approaches to solve the concavelinear fractional program [
11]. In this work, we adopt Dinkelbach’s transform to solve this optimization problem. Compared to CharnesCooper transform, Dinkelbach’s transform has an advantage that no extra constraints must be included. Via invoking Dinkelbach’s transform, the energyefficient precoding optimization problem in (11) can be solved via iteratively solving the following sequence of convex optimization problems
where
${\eta}^{\left(\ell \right)}$ is an introduced auxiliary variable in Dinkelbach’s transform, which is iteratively updated as follows
with
ℓ denoting the iteration index. It can be shown that the above Dinkelbach’s transformbased iteration approach is guaranteed to converge to the global optimum of the original optimization problem in (11) with a superlinear convergence rate [
11].
Although each transformed subproblem in (15) is a convex program, the optimization complexity might still be high in practice due to the sample average in calculating the expectation operation. To reduce the implementation complexity in MonteCarlo averaging over channel realizations, we further employ the largedimensional random matrix theory [
27,
28] to calculate the DE of the ergodic rate. By replacing the multicast rate with its DE, the sequence of the convex optimization problems in (15) can be rewritten as follows
In (17),
${\overline{R}}_{k}\left(\mathsf{\Lambda}\right)$ is the DE of
${R}_{k}\left(\mathsf{\Lambda}\right)$ given by
where
${\mathsf{\Gamma}}_{k}\in {\mathbb{C}}^{M\times M}$,
${\tilde{\mathsf{\Gamma}}}_{k}\in {\mathbb{C}}^{{N}_{k}\times {N}_{k}}$, and
${\tilde{\mathsf{\Phi}}}_{k}\in {\mathbb{C}}^{{N}_{k}\times {N}_{k}}$ can be efficiently calculated via solving the following fixedpoint equations
It is worth noting that
${\mathbf{B}}_{k}\left(\mathbf{X}\right)\triangleq \frac{1}{{\sigma}^{2}}\mathsf{E}\left\{{\mathbf{G}}_{k}^{H}\mathbf{X}{\mathbf{G}}_{k}\right\}\in {\mathbb{C}}^{M\times M}$ and
${\mathbf{C}}_{k}\left(\mathbf{X}\right)\triangleq \frac{1}{{\sigma}^{2}}\mathsf{E}\left\{{\mathbf{G}}_{k}\mathbf{X}{\mathbf{G}}_{k}^{H}\right\}\in {\mathbb{C}}^{{N}_{k}\times {N}_{k}}$ appeared in (19) are both matrixvalued functions which both output diagonal matrices with the corresponding
ith output diagonal elements given by
respectively. The DE
${\overline{R}}_{k}\left(\mathsf{\Lambda}\right)$ can be efficiently calculated using the channel statistics
${\mathsf{\Omega}}_{k}$ in a few iterations without exhaustive averaging involved in the MonteCarlo approach, and then the computational complexity of the optimization problem in (15) can be further reduced.
Please note that the DE
${\overline{R}}_{k}\left(\mathsf{\Lambda}\right)$ is a quite tight approximation of
${R}_{k}\left(\mathsf{\Lambda}\right)$ for massive MIMO channels in typical settings [
27,
28]. In addition, as
${\overline{R}}_{k}\left(\mathsf{\Lambda}\right)$ in (18) is a concave function with respect to the power allocation matrix
$\mathsf{\Lambda}$, each subproblem in (17) is a convex program and can be efficiently solved using standard techniques. Formally, we present the description of our proposed energyefficient multicast beam domain power allocation algorithm for massive MIMO transmission using Dinkelbach’s transform and largedimensional random matrix theory in Algorithm 1. The computational complexity of Algorithm 1 is presented as follows. From [
11], we can obtain that the convergence rate of Dinkelbach’s transformbased iteration will exhibit a superlinear trend in the subproblem sequence (15). In addition, each subproblem in (15) is a convex program and thus the computational complexity of each subproblem is polynomial in the numbers of variables and constraints [
29]. Moreover, the complexity of calculating DE involved in the objective of each subproblem is relatively low due to the quick convergence [
28] in solving (19).
Algorithm 1 Beam Domain EnergyEfficient Multicast Power Allocation Algorithm 
Require: Initialization power allocation ${\mathsf{\Lambda}}^{\left(0\right)}$, beam domain statistical CSI ${\mathsf{\Omega}}_{k}$, threshold $\u03f5$ Ensure: Beam domain energyefficient multicast power allocation pattern $\mathsf{\Lambda}$ 1: Initialization: $\ell =0$, calculate ${\eta}^{\left(\ell \right)}$ using (16) 2: while$\underset{k}{\mathrm{min}}\phantom{\rule{4pt}{0ex}}W{\overline{R}}_{k}\left({\mathsf{\Lambda}}^{\left(\ell \right)}\right){\eta}^{\left(\ell \right)}P\left({\mathsf{\Lambda}}^{\left(\ell \right)}\right)\ge \u03f5$ do 3: Update $\ell \leftarrow \ell +1$ 4: Calculate ${\mathsf{\Lambda}}^{\left(\ell \right)}$ via solving (17) with ${\eta}^{\left(\ell 1\right)}$ 5: Calculate ${\eta}^{\left(\ell \right)}$ using (16) 6: end while 7: Return $\mathsf{\Lambda}={\mathsf{\Lambda}}^{\left(\ell \right)}$

4. Numerical Results
Extensive numerical results are provided in this section to corroborate the performance of the proposed energyefficient multicast precoding for massive MIMO transmission with only statistical CSI available at the BS. In the simulation, we adopt the 3GPP SCM channel model and the suburban macrocell propagation scenario [
30]. In addition, both the BS and the UTs are equipped with the uniform linear arrays (ULAs) where the antennas are spaced with half wavelength. We list the major simulation parameters in
Table 1.
Firstly, the convergence performance of the proposed energyefficient multicast power allocation algorithm is evaluated in
Figure 1. We can observe from
Figure 1 that the proposed energyefficient multicast power allocation algorithm converges very rapidly for different values of multicast power budget
${P}_{\mathrm{max}}$. Usually, the proposed algorithm converges after very few iterations. In addition, we can observe that the proposed multicast power allocation algorithm achieves nearoptimal EE performance.
We then compare the EE performance of the proposed EE maximization approach with that of the rate maximization approach which aims to maximize the achievable ergodic multicast rate in
Figure 2. We can observe from
Figure 2 that the proposed EE maximization approach and the conventional rate maximization approach achieve similar EE performance in the low transmit power regime, which indicates that using the maximum power budget is almost EE optimal. Meanwhile, in the high transmit power regime, our proposed EE maximization approach significantly outperforms the conventional rate maximization approach. In addition, we can also observe that the proposed DE result is quite accurate compared with the MonteCarlo results in a wide range of power budget values.
Finally, the EE performance of our proposed approach versus the number of BS antennas is evaluated in
Figure 3. We can observe from
Figure 3 that the EE performance of our proposed approach will reduce as the number of BS antennas
M increases, which indicates that power consumption dominates the EE performance in the case with large numbers of antennas.