Sum-Rate Maximization for a Hybrid Precoding-Based Massive MIMO NOMA System with Simultaneous Wireless Information and Power Transmission

Abstract

Non-orthogonal multiple access (NOMA) has emerged as a key enabling technology in the realm of millimeter-wave (mmWave) massive MIMO (mMIMO) systems for enhancing spectral efficiency (SE). Furthermore, it is believed that simultaneous wireless information and power transmission (SWIPT) will allow for the system’s energy efficiency (EE) to be maximised. The effectiveness of the mmWave mMIMO-NOMA system along with SWIPT has been examined in this article under multi-user (MU) scenarios. This paper’s major goal is to construct a low-complexity hybrid-precoder (HP) while taking into account the sub-connected (SC) architecture. The linear precoder is a computationally demanding technique as a result of the matrix inversion. The authors of this paper have suggested a symmetric sequential over-relaxation (SSOR) complex regularised zero-forcing (CRZF) linear precoder. The power distribution for the mmWave mMIMO-NOMA system and power splitting factors for SWIPT are jointly tuned to maximize the sum rate along with the suggested SSOR-CRZF precoder. In regards to complexity, SE, and EE, the SSOR-CRZF-HP surpasses conventional linear precoders.

1. Introduction

High-speed and low-latency features are much sought after in communication systems of the future. Massive multiple-input multiple-output (mMIMO) technology in conjunction with millimeter-wave (mmWave) communication and non-orthogonal multiple access (NOMA) is crucial for achieving this goal [1]. By utilising the extensive, accessible wide bandwidth offered in the mmWave band while adding the benefit of high multiplexing gains, the mmWave-mMIMO system greatly improved the spectral efficiency (SE) as well as energy efficiency (EE) in comparison with traditional systems [2]. Recent research works indicate that NOMA in an mmWave-mMIMO system has tremendous potential to improve SE [3,4,5].

A substantial amount of research has focused on simultaneous wireless information and power transmission (SWIPT), which has the capacity to transmit both information (through information decoding (ID)) as well as energy (via energy harvesting (EH)), as a potential remedy regarding the problem with energy scarcity in power-constrained wireless networks [6,7,8]. Although SWIPT significantly improves battery life and EE, its practical application is severely constrained by the ineffectiveness of the wireless transmission of power caused by scattering as well as attenuation in route loss. Integrating SWIPT with massive multiple-input multiple-out (mMIMO) technology serves as a technically doable way to get around these restrictions, and it enables SWIPT to reach its full potential [9,10,11]. The combination of NOMA and SWIPT has garnered significant interest lately. Successive interference cancellation (SIC) demonstrates a sizeable gain for SWIPT systems when taking into account a bipolar ad hoc network [12]. The author in [13] introduces a mmWave mMIMO-NOMA SWIPT system with hybrid precoding, user grouping, power splitting (PS), and power allocation (PA) for enhancing the SE as well as the EE. The authors in [14] propose a novel cooperative SWIPT-NOMA transmission method to assist cell-edge users. The authors in [4] introduce an alternating optimization (AO) algorithm to decouple the joint PA and PS nonconvex optimization problem for enhancing the SE as well as the EE. In [15], the performance of cooperative MIMO-NOMA networks is examined by incorporating broadcast antenna selection and SWIPT protocols. The authors in [16] propose a low-complexity optimisation technique linked to PS control and timeslot allocation in order to maximise the weighted sum throughput for each downlink (DL) as well as uplink (UL) for MIMO-NOMA systems, along with SWIPT. In an effort to improve the SE and EE, the authors in [17] introduce a novel affinity propagating clustering technique to support the grouping of users and a Lagrangian duality approach to decouple the joint PA as well as transmitting power. The authors in [18] have enhanced the NOMA-SWIPT system by simultaneously designing the transmit beamforming and PS coefficients to seek high EE for information transmission.

HPs have undergone a great deal of effort in their design. The maximum ratio transmission (MRT) system has been thoroughly explored, in addition to zero-forcing (ZF), regularised ZF (RZF) [19], and minimum mean square error (MMSE) [20]. There have been numerous proposals over the years for designing effective HPs, including the Kalman filter-reliant strategy [21,22], block diagonalization mechanism [23], modified block diagonalization mechanism [24], and the singular-value-decomposition (SVD) reliant strategy [25], as well as the lattice reduction (LR) reliant strategy [26,27,28,29]. The authors of  [30] have presented an outline enabling LR-assisted precoding via vector perturbation (VP) to improve the performance of mMIMO systems. Here, the authors have looked into the possibilities of utilising the approximate message passing (AMP) method, LR-ZF, and LR-SIC, as well as LR-ZF to enhance the symbol error rate (SER) efficacy for the mMIMO network while reducing complexity. In mMIMO, network complexity remains a significant dilemma. For building low-complexity HPs, artificial neural network [31,32], deep learning [33,34]-reliant strategies have been presented and thoroughly examined.

An upsurge of complexity within HP is a serious dilemma as it severely impacts the implementation cost for mMIMO systems, and this is because of the complexity brought on by the matrix inversion procedure. To make the precoders less complex, certain efforts have been made. Regarding this matter, there have been proposals as well as thorough studies of truncated polynomial expansion (TPE) [35] as well as Neumann series (NS) [36] precodings. With a few iterations, the NS precoder is advantageous. However, when there are many iterations, they display a comparable level of complexity that is on par with that of ZF. Diverse precoding strategies, notably the Jacobi iteration-reliant [37] precoder, Gauss-Seidel (GS) [38] precoding, successive over-relaxation (SOR) [39] precoding, symmetric SOR (SSOR) [40] precoding, weighted SSOR (WSSOR) [41] precoding, and modified SOR (MSOR) [42] precoding have also been put forward and assessed in order to further reduce computational complexity. These techniques can deliver close to ideal ZF performance by successfully iteratively converting the inversion of a matrix onto an equation of linear form.

However, the efficacy of the system can be increased to an even greater extent by employing a more advanced HP architecture as well as a mechanism for grouping users. The aforementioned motivated the authors to streamline the CRZF precoder using the SSOR approach in order to improve the functioning associated with the mmWave-mMIMO-NOMA-SWIPT mechanism. An SSOR-CRZF precoder that is sub-connected (SC) has been proposed by the authors of this work. The MRT, ZF, RZF, Kalman, SSOR, and CRZF precoders are compared to the recommended SSOR-CRZF precoder in this work. Computational complexity, SE, and EE have all been assessed in relation to the performances. The presented SSOR-SRZF precoder outperforms with less computational complexity than the MRT, ZF, RZF, Kalman, and SSOR precoders.

This work’s primary contribution may be summarized succinctly as follows:

• We examine the optimisation of both HP and PS techniques to enrich SWIPT-enabled mmWave-mMIMO-NOMA systems. Based on the channel correlation, a modified version of the K-Means (MKM) user grouping algorithm was used. Here, we examine the proposed HP and optimization of the PA and PS factors to improve the sum rate of the proposed system.
• We have constructed a hybrid/joint mmWave MIMO-NOMA precoding technique/scheme. Depending on the user groupings, a scheme for analog precoding (AP) has been designed to guarantee that every beam receives equivalent channel gain at its maximum. We introduce the SSOR-CRZF digital precoder (DP) for eradicating inter-user interference. We define the problem as a combined optimization of power distribution and PS variables in order to make the overall power as well as the minimal rate limits for every UE more straightforward.
• In comparison to traditional precoders, the presented SSOR-CRZF precoder is evaluated in this study for its ability to improve both SS as well as EE within the mmWave-mMIMO-NOMA-SWIPT network while having less complexity. In terms of SE, EE, and computational complexity, its performance is contrasted with that of MRT, ZF, RZF, Kalman, SSOR, and CRZF-based precoders. It is shown that the recommended strategy outperforms the system significantly more than the current state-of-the-art algorithms.
• The performance of the proposed system is evaluated with respect to variations in the signal-to-noise ratio, number of users, number of BS antennas, number of phase quantization bits, fading parameters, and under imperfect CSI conditions.
• Given the limitations of transmit power and EH need, the combined PA and PS control is mathematically formulated to maximise the sum rate.

The remainder of the article is organized as follows. A system model is described in Section 2. Problem formulation and proposed solutions related to joint optimization are presented in Section 3. Section 4 presents a comparison associated with the complexity of the suggested as well as traditional precoders. In Section 5, multiple algorithms are simulated/evaluated, and a comparison analysis is done. Section 6 highlights the related challenges and future research direction. Section 7 has the conclusions.

2. System Model

Here, a DL-MU scenario with a SWIPT-enabled mmWave-MIMO-NOMA network is depicted within Figure 1 and is taken into consideration for the performance assessment. In this case, the base station (BS) has $N_T$ transmit antennas. Here, the performance evaluation focuses primarily on the SC structure. As in Figure 1, $N_{RF}$ chains of RF have been utilized for extending the backing to K single antenna UEs [3,4] ($N_{RF}\le K<N_T$). As far as the SC structure is concerned, every RF chain has been coupled to M ($=N_T/N_{RF}$) antennas. Additionally, Ref. [3] states that the number of beams (D) created cannot be greater than $N_{RF}$. For the purpose of simplicity, it is presumed that $D=N_{RF}$. Under the proposed system model, the core concept of NOMA is exploited to support UEs through each beam. Thus, in other words, the K UEs can be supported through D clusters corresponding to each beam.

As in this proposed model, all UEs data streams have been coded by utilizing baseband DP ${\mathbf{f}}_d^{BB}\left[N_{RF}\times 1\right]$, $d\in \left[1,D\right]$ regarding the $d^{th}$ beam and [$N_T\times N_{RF}$] RF AP ${\mathbf{F}}^{RF}$ exploiting the analogue phase shifter. In the case of AP, all of its components must be present to meet ${\left({\left[{\mathbf{F}}^{RF}\right]}_{:,j}{\left[{\mathbf{F}}^{RF}\right]}_{:,j}^H\right)}_{l,l}=N_T^{-1}$ with $\left|{\left[{\mathbf{F}}^{RF}\right]}_{i,j}\right|=N_T^{-1/2},1\le i\le N_T,1\le j\le N_{RF}$ and quantized phases: ${\left[{\mathbf{F}}^{RF}\right]}_{i,j}={\left(N_T\right)}^{-1/2}exp\left(j{\left[\mathsf{\Phi }\right]}_{i,j}\right)$, and ${\left[\mathsf{\Phi }\right]}_{i,j}$ can be quantized as ${\left[\mathsf{\Phi }\right]}_{i,j}\in :=\left\{{\displaystyle \frac{2\pi n}{2^B}}:n=0,\dots ,2^B-1\right\}$. In addition to the aforementioned constraints, the overall power constraint has been enforced via normalising DP ${\mathbf{f}}_d^{BB},d=1,\dots ,D$ for the $d^{th}$ beam so that ${∥{\mathbf{F}}^{RF}{f}_d^{BB}∥}_2=1$ for $d=1,\dots ,D$. No other constraints are imposed on DP. Let ${\mathsf{\Xi }}_d$ for $d=1,\dots ,D$ represents the user set corresponding to the $d^{th}$ beam with $\left|{\mathsf{\Xi }}_d\right|\ge 1$, and ${\mathsf{\Xi }}_i\cap {\mathsf{\Xi }}_j=$ for $i\ne j$; thus, ${\sum }_{d=1}^D\left|{\mathsf{\Xi }}_d\right|=K$.

In the case of a fully hybrid precoder, the AP matrix can be defined as

$${{\mathbf{F}}^{RF}}^{\left(full\right)}=\left[{{\overline{f}}_1^{RF}}^{\left(full\right)},{{\overline{f}}_2^{RF}}^{\left(full\right)}\dots {{\overline{f}}_{N_{RF}}^{RF}}^{\left(full\right)}\right],$$

(1)

where ${{\overline{f}}_m^{RF}}^{\left(full\right)}\in {\mathbb{C}}^{N_T\times 1}\forall m=1,2,\dots ,N_{RF}$ is the steering vector, having identical amplitude $1/\sqrt{N_T}$, nevertheless with distinct phases [43].

For SC architecture, the AP matrix can be defined as

$${{\mathbf{F}}^{RF}}^{\left(sub\right)}=\left[\begin{array}{ccc}{{\overline{f}}_1^{RF}}^{\left(sub\right)}& 0\dots & 0\\ 0& {{\overline{f}}_2^{RF}}^{\left(sub\right)}& 0\\ :& :& :\\ 0& 0\dots & {{\overline{f}}_{N_{RF}}^{RF}}^{\left(sub\right)}\end{array}\right],$$

(2)

wherein ${{\overline{f}}_m^{RF}}^{\left(sub\right)}\in {\mathbb{C}}^{M\times 1}\forall m=1,2,\dots ,N_{RF}$ is the steering vector, having identical amplitude $1/\sqrt{M}$, nevertheless with distinct phases [44].

Let $x_{d,k}$$(\mathbb{E}\left[{\left|x_{d,k}\right|}^2\right]=1)$ represent the signal delivered to the $k^{th}$ UE i the $d^{th}$ beam. Although the system exploits the NOMA scheme, the BS utilizes superposition code to support multiple UEs simultaneously. Additionally, the entire power (P) has been divided evenly among all K UEs. At the receiver side, every UE exploits successive interference cancellation (SIC) to retrieve the desired information signal.

The $k^{th}$ UE within the $d^{th}$ beam receives the baseband signal $y_{d,k}$$[k=1,\dots ,\left|{\mathsf{\Xi }}_d\right|]$, which can be expressed as

$$y_{d,k}={\mathbf{h}}_{d,k}^H{\mathbf{F}}^{RF}\sum _{i=1}^D\sum _{j=1}^{\left|{\mathsf{\Xi }}_i\right|}{\mathbf{f}}_i^{BB}\sqrt{p_{i,j}}{x}_{i,j}+n_{d,k}.$$

(3)

$$\begin{array}{c}\hfill \underset{\mathrm{Desired}\mathrm{signal}}{\underbrace{={\mathbf{h}}_{d,k}^H{\mathbf{F}}^{RF}{\mathbf{f}}_d^{BB}\sqrt{p_{d,k}}{x}_{d,k}}}\\ \hfill \underset{\mathrm{Intra}-\mathrm{beam}\mathrm{Interference}}{\underbrace{+{\mathbf{h}}_{d,k}^H{\mathbf{F}}^{RF}{\mathbf{f}}_d^{BB}\left(\sum _{j=1}^{k-1}\sqrt{p_{d,j}}{x}_{d,j}+\sum _{j=k+1}^{\left|{\mathsf{\Xi }}_d\right|}\sqrt{p_{d,j}}{x}_{d,j}\right)}}\\ \hfill \underset{\mathrm{Inter}-\mathrm{beam}\mathrm{Interference}}{\underbrace{+{\mathbf{h}}_{d,k}^H{\mathbf{F}}^{RF}\sum _{i\ne b}\sum _{j=1}^{\left|{\mathsf{\Xi }}_i\right|}{\mathbf{f}}_i^{BB}\sqrt{p_{i,j}}{x}_{i,j}}}\underset{\mathrm{Noise}\mathrm{Signal}}{\underbrace{+n_{d,k}}},\\ \hfill ={\mathbf{h}}_{d,k}^H{\mathbf{F}}^{RF}{\mathbf{F}}^{BB}\mathbf{P}\mathbf{x}+n_{d,k},\end{array}$$

(4)

wherein  ${\mathbf{F}}^{RF}\in {\mathbb{C}}^{N_T\times N_{RF}}$ and  ${\mathbf{F}}^{BB}=[{\mathbf{f}}_1^{BB},\dots ,{\mathbf{f}}_D^{BB}]\in {\mathbb{C}}^{N_{RF}\times D}$ are the precoding matrix corresponding to AP and DP. Here, the channel vector on behalf of every $k^{th}$ UE within the $d^{th}$ beam is represented by ${\mathbf{h}}_{d,k}^H\in {\mathbb{C}}^{1\times N_T}$. $\mathbf{P}=diag\left\{{\mathbf{p}}_1,\dots ,{\mathbf{p}}_D\right\}$ symbolizes the PA matrix. The signal vector that gets transmitted has been defined via $\mathbf{x}={\left[x_{1,1},\dots ,x_{1,\left|{\mathsf{\Xi }}_d\right|},\dots ,x_{D,1},\dots ,x_{D,\left|{\mathsf{\Xi }}_d\right|}\right]}^T\in {\mathbb{C}}^{K\times 1}$; the $p_{d,k}$ denotes the average power received for the $k^{th}$ UE within the $d^{th}$ beam, and $n_{b,k}\sim \mathcal{C}\mathcal{N}(0,{\sigma }_n^2)$ represents the AWGN having zero mean along with ${\sigma }_n^2$ variance introduced across the $k^{th}$ UE within the $d^{th}$ beam. As in (4), there is a high possibility that the received signal can be greatly affected by the interference. Therefore, SIC will be very effective at the UE in detecting the desired signal free from interference.

2.1. Channel

In this paper, the authors have considered the geometrically expanded Saleh–Valenzuela model [45]. The channel ${\mathbf{h}}_{d,k}$ ($\in {\mathbb{C}}^{N_T\times 1}$) associated with the $k^{th}$ UE within the $d^{th}$ is considered to be the cumulative effect from ${\mathcal{L}}_{b,k}$ and can be expressed as

$${\mathbf{h}}_{d,k}=\sqrt{{\displaystyle \frac{N_T}{{\mathcal{L}}_{d,k}}}}\sum _{l=1}^{{\mathcal{L}}_{d,k}}{\alpha }_{d,k}^{\left(l\right)}\mathbf{a}\left({\mathsf{\Theta }}_{d,k}^{\left(l\right)},{\mathsf{\Phi }}_{d,k}^{\left(l\right)}\right).$$

(5)

where ${\alpha }_{d,k}^{\left(l\right)}$ reflects the complex gain within the $l^{th}$ path and corresponds to the $k^{th}$ UE within the $d^{th}$ beam. Furthermore, it has been presumed to have complex Gaussian characteristics with a mean of 0 as well as a variance of 1. $\sqrt{{\displaystyle \frac{N_T}{{\mathcal{L}}_{d,k}}}}$ symbolizes the normalization factor. Herein, ${\mathsf{\Theta }}_{d,k}^{\left(l\right)}$ as well as ${\mathsf{\Phi }}_{d,k}^{\left(l\right)}$ symbolize the azimuth as well as the elevation angle of departure (AOD) pertaining to the $l^{th}$ path. The normalized array response vector is introduced by the factors $\mathbf{a}\left({\mathsf{\Theta }}_{d,k}^{\left(l\right)},{\mathsf{\Phi }}_{d,k}^{\left(l\right)}\right)$. Consider the $N_T$ elements uniform linear array (ULA), wherein ${\mathcal{N}}_1$ elements are arranged in a horizontal direction as well as ${\mathcal{N}}_2$ elements in a vertical direction [$N_T={\mathcal{N}}_1{\mathcal{N}}_2$].

Under such element distribution, the array response vector $\mathbf{a}\left({\mathsf{\Theta }}_{d,k}^{\left(l\right)},{\mathsf{\Phi }}_{d,k}^{\left(l\right)}\right)$ can be conveyed as

$$\mathbf{a}\left({\mathsf{\Theta }}_{d,k}^{\left(l\right)},{\mathsf{\Phi }}_{d,k}^{\left(l\right)}\right)={\mathbf{a}}_{az}\left({\mathsf{\Theta }}_{d,k}^{\left(l\right)}\right)\otimes {\mathbf{a}}_{el}\left({\mathsf{\Phi }}_{d,k}^{\left(l\right)}\right)$$

(6)

where

$${\mathbf{a}}_{az}\left({\mathsf{\Theta }}_{d,k}^{\left(l\right)}\right)={\displaystyle \frac{1}{\sqrt{{\mathcal{N}}_1}}}{\left(1,e^{j\varrho dsin\left(\mathsf{\Theta }\right)},\dots ,e^{j({\mathcal{N}}_1-1)\varrho dsin\left(\mathsf{\Theta }\right)}\right)}^T.$$

(7)

and

$${\mathbf{a}}_{el}\left({\mathsf{\Phi }}_{d,k}^{\left(l\right)}\right)={\displaystyle \frac{1}{\sqrt{{\mathcal{N}}_2}}}{\left(1,e^{j\varrho dsin\left(\mathsf{\Theta }\right)},\dots ,e^{j({\mathcal{N}}_2-1)\varrho dsin\left(\mathsf{\Theta }\right)}\right)}^T,$$

(8)

with the inter-element distance $d=\lambda /2$ as well as $\varrho =2\pi /\lambda$. Additionally, it is assumed that the receivers are completely synchronised with respect to time and frequency and that the BS as well as the UEs have perfect knowledge of CSI. The authors have also tested the proposed system under the imperfect CSI condition. As a part of the channel modeling and further exploration, the authors have considered that all the arrivals are Nakagami-m distributed and have tested the proposed system under the expanded Saleh–Valenzuela channel with the Nakagami-m distributed small-scale fading model.

2.2. Power Splitting Receiver and Sum Rate

The received signal corresponding to each UE will be separated into two components as part of the SWIPT modelling. One component will be employed for information decoding (ID), while the other module it utilised for energy harvesting (EH). The signal intended for EH can be articulated as follows:

$$y_{d,k}^{EH}=\sqrt{1-{\rho }_{d,k}}{y}_{d,k},$$

(9)

where ${\rho }_{d,k}\in [0,1]$ denotes the PS factor associated with the $k^{th}$ UE within the $d^{th}$ beam. As in [17], the harvested energy can be expressed as

$$P_{d,k}^{EH}=\eta \left(1-{\rho }_{d,k}\right)\left(\sum _{i=1}^D\sum _{j=1}^{\left|{\mathsf{\Xi }}_i\right|}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_i^{BB}∥}_2^2{p}_{i,j}+{\sigma }_n^2\right),$$

(10)

where ${\overline{\mathbf{h}}}_{d,k}^H(={\mathbf{h}}_{d,k}^H{\mathbf{F}}^{RF})$ represents the effective channel vector and $0\le \eta \le 1$ symbolizes the energy conversion efficiency. Similar to (9), the signal corresponding to ID can be conveyed as

$$y_{d,k}^{ID}=\sqrt{{\rho }_{d,k}}{y}_{d,k}+u_{d,k},$$

(11)

where $u_{d,k}\sim \mathcal{C}\mathcal{N}(0,{\sigma }_u^2)$ denotes the noise introduced because of the PS. Based on [3], the estimated received signal for ID associated with the $k^{th}$ UE within the $d^{th}$ beam can be conveyed as

$$\begin{array}{c}\hfill {\widehat{y}}_{d,k}^{ID}=\sqrt{{\rho }_{d,k}}\left({\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}\sqrt{p_{d,k}}{x}_{d,k}+{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}\sum _{j=1}^{k-1}\sqrt{p_{d,j}}{x}_{d,j}+\right.\\ \hfill \left.{\overline{\mathbf{h}}}_{d,k}^H\sum _{i\ne d}\sum _{j=1}^{\left|{\mathsf{\Xi }}_i\right|}{\mathbf{f}}_i^{BB}\sqrt{p_{i,j}}{x}_{i,j}+n_{d,k}\right)+u_{d,k}.\end{array}$$

(12)

Following [3,17], the SINR at the $k^{th}$ UE within the $d^{th}$ beam can be conveyed as

$${\gamma }_{d,k}={\displaystyle \frac{{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2{p}_{d,k}}{{\varsigma }_{b,k}}},$$

(13)

where

$$\begin{array}{c}\hfill {\varsigma }_{d,k}={∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2\sum _{j=1}^{k-1}{p}_{d,j}+\\ \hfill \sum _{i\ne b}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2\sum _{j=1}^{\left|{\mathsf{\Xi }}_i\right|}{p}_{i,j}{\sigma }_n^2+{\displaystyle \frac{{\sigma }_u}{{\rho }_{d,k}}}.\end{array}$$

(14)

Thus, the achievable data transmission rate at the $k^{th}$ UE within the $d^{th}$ directional beam is meant to be portrayed as

$$R_{d,k}=lo{g}_2\left(1+{\gamma }_{d,k}\right).$$

(15)

The system’s achievable sum rate can be obtained by    

$$R_{sum}=\sum _{d=1}^D\sum _{k=1}^{\left|{\mathsf{\Xi }}_d\right|}{R}_{d,k}.$$

(16)

2.3. User Grouping

It is convenient to divide the UEs into D groups ($D=N_{RF}$) because there are more UEs $\left(K\right)$ than $N_{RF}$. Additionally, there is the composite channel matrix  $\mathbf{H}=[{\mathbf{h}}_1,\dots ,{\mathbf{h}}_K]$ that needs to be taken into account. For the proposed mmWave-NOMA-SWIPT system, an improved K-Means user grouping algorithm is taken into consideration with an initial correlation threshold $\zeta (0<\zeta <1)$. Here, the crucial component for the grouping is the normalised correlation of channels between user channels. By minimising the normalised correlation of channels between the beam-picked representatives, one representative UE has been initially chosen for every beam. Based on the correlation of channels, UEs are divided into various beams in order to reduce inter-beam interference. The MKM algorithm is presented in Algorithm 1. The distinctive feature associated with this Algorithm 1 has been essentially cluster head picking, as in Steps 10–15. In the following method, selecting the best representative (Step 12) takes into account the minimal correlation of channels between the picked representatives. The UEs from strongly correlated channels are then placed on an identical beam to reduce intra-beam noise/interference. On the other hand, UEs with lower correlated channels have been assigned to distinct beams in order to reduce inter-beam interference.

Algorithm 1 MKM User Grouping Algorithm

3. Problem Formulation and Solutions

3.1. Problem Formulation

For maximizing the sum rate (as in (16)) within the proposed network, here, the authors have addressed the issue of joint optimization of the PS, PA, and digital as well as analog RF precoders. The statement of the problem can be formulated as

$$\begin{array}{c}\hfill \underset{p_{d,k},{\rho }_{d,k},{\mathbf{f}}_d^{BB},{\mathbf{F}}^{RF}}{max}{R}_{sum}\left(R_{sum}=\sum _{d=1}^D\sum _{k=1}^{\left|{\mathsf{\Xi }}_d\right|}{R}_{d,k}\right)\end{array}$$

(17a)

$$\begin{array}{c}\hfill s.t.R_{d,k}\ge R_{d,k}^{min},\forall d,k,\end{array}$$

(17b)

$$\begin{array}{c}\hfill p_{d,k}\ge 0,\forall d,k,\end{array}$$

(17c)

$$\begin{array}{c}\hfill \sum _{d=1}^D\sum _{k=1}^{\left|{\mathsf{\Xi }}_d\right|}{p}_{d,k}\le P,\end{array}$$

(17d)

$$\begin{array}{c}\hfill P_{d,k}^{EH}\ge P_{d,k}^{min}\forall d,k,\end{array}$$

(17e)

$$\begin{array}{c}\hfill {\left|{\mathbf{F}}_{i,j}^{RF}\right|}^2={\displaystyle \frac{1}{N_T}},1\le i\le N_T,1\le j\le N_{RF},\end{array}$$

(17f)

$$\begin{array}{c}\hfill {∥{\mathbf{F}}^{RF}{\mathbf{f}}_d^{BB}∥}_2=1\forall d,\end{array}$$

(17g)

Here, constraint (17b) ensures the minimum desirable data rate for the $k^{th}$ UE within the $d^{th}$ beam. The constraints (17c) and (17d) pertain to the transmitted power. The transmitted power at the BS associated with every UE ought to be positive yet should not surpass the permitted level P. Constraint (17e) represents the fact that each $k^{th}$ UE within the $d^{th}$ beam is required to harvest minimum energy of $P_{d,k}^{min}$. Here, (17f) represents the constant-modulus constraint related to the AP. The HP matrix’s unit power restriction is given by (17g). It turns out that the optimization problem as given in  (17a)–(17g) is a non-convex optimization dilemma, making it challenging to find a globally optimal solution.

3.2. Solution: Hybrid Precoder

The maximization of sum-rate can be achieved by lowering inter-beam interference as well as by increasing effective channel gain. With fewer hardware restrictions, the HP scheme can utilise the mmWave-mMIMO system to its full potential. Motivated by  [3,4,46], the hybrid precoder designing step is divided into two segments, and the analogue RF precoder as well as the digital baseband precoder are discussed separately. Here, the AP algorithm (for ${\mathbf{F}}^{RF}$) is inspired by the work in  [4]. As part of the low-dimensional DP design, the authors have introduced a low complexity algorithm (for ${\mathbf{F}}^{BB}$) and compared its performance with those of current digital precoders.

3.2.1. Analog Precoder (AP)

The purpose of the AP is to produce significant array gain through the leverage of a multitude of antennas within the mMIMO system. This is possible to accomplish by aligning the phases of the aggregate DL channel $\mathbf{H}$ = $[{\mathbf{h}}_1,\dots ,{\mathbf{h}}_K]$. Following  [3], quantized phase shifters are utilized for the design of the AP. Both FC and SC architectures are taken into account and compared. The pseudo-code for the FC AP is presented in Algorithm 2.

In the case of the SC structure, step (11) can be realized by exploiting ${\overline{ap}}_d^{sub}\left(n\right)={\displaystyle \frac{1}{\sqrt{M}}}exp\left(j{\displaystyle \frac{2\pi \theta }{2^B}}\right)$, where $n=(d-1)M+1,(d-1)M+2,\dots ,dM$. Subsequently, once the formulation of the AP is completed, the DP design is carried out with the goal of maximizing the achievable sum rate while drastically suppressing noise/interference.

Algorithm 2 Analog Precoder

3.2.2. Digital Precoder (DP)

In contrast to traditional ZF [47] and RZF [47] precoders, as studied in Mostafa, 2021, the CRZF [48] greatly enhances system performance. Nevertheless, similar to ZF as well as RZF, it ultimately incorporates an inversion of the matrix. Such computation becomes practically hard to realise in the case of the MIMO system. The authors of this work suggest the SSOR-CRZF DP for the system shown in (1). Without resorting to matrix inversion, the suggested approach, as in Algorithm 3, creates a CRZF matrix using the iterative SSOR technique. In the proposed HP approach, a lower-dimensional baseband DP (SSOR-CRZF) has been developed while taking the effective channel $\left(\overline{\mathbf{H}}\right)$ into consideration following the construction of the AP (${\mathbf{F}}^{RF}$). In Algorithm 3, as an initial step, the CRZF filtering matrix (${\mathbf{P}}_{crzf}$) is first calculated, which may be conveyed as follows:

$${\mathbf{P}}_{crzf}=\left(\breve{\mathbf{H}}{\breve{\mathbf{H}}}^H+\alpha {\mathbf{I}}_{N_{RF}\times N_{RF}}\right)$$

(18)

wherein $\alpha$ constitutes a parameter of complex valued regularization [48] related to the complex AWGN noise.

However, because it is necessary to take the pseudo-inverse, the computing cost rises. The authors in [40,42] have advocated SSOR-reliant precoding via making use of the wireless channel’s asymptotically orthogonality characteristic in mMIMO with the goal of mitigating complexity. As stated in [40], for the CRZF precoder, the transmitted signal ($\mathbf{x}$),

$$\begin{array}{c}\hfill \mathbf{s}={\mathbf{F}}_{CRZF}^{BB}\mathbf{x}={\beta }_{CRZF}{\mathbf{H}}^H{\left(\breve{\mathbf{H}}{\breve{\mathbf{H}}}^H+\alpha {\mathbf{I}}_{N_{RF}\times N_{RF}}\right)}^{-1}\mathbf{x}\\ \hfill ={\beta }_{CRZF}{\breve{\mathbf{H}}}^H{\mathbf{P}}_{crzf}^{-1}\mathbf{x}={\beta }_{CRZF}{\breve{\mathbf{H}}}^H\mathbf{t}.\end{array}$$

(19)

where the power normalisation factor is denoted by ${\beta }_{CRZF}$. The key purpose of utilising the SSOR technique in this approach aims to build the precoder matrix $\mathbf{t}$, avoiding any inversion of the matrix (${\mathbf{P}}_{crzf}^{-1}$). The SSOR method begins by decomposing the matrix ${\mathbf{P}}_{crzf}$, which can then be articulated as ${\mathbf{P}}_{crzf}=\mathbf{D}+\mathbf{L}+{\mathbf{L}}^H$, wherein $\mathbf{D}$, $\mathbf{L}$, and ${\mathbf{L}}^H$ symbolise the matrix of a diagonal, matrix of lower triangular, and the matrix of upper triangular for ${\mathbf{P}}_{crzf}$, respectively. The proposed DP algorithm is illustrated in Algorithm 3. As presented in the algorithm, the desired vector $\mathbf{t}$ can be obtained after a number of iterations. Here, $\omega$ is the relaxation factor. The ideal $\omega$ can be found using the formula $\omega =2/(1+sqrt\left((1-a^2)\right))$, where $a={[1+\sqrt{(Nr/Nt)}]}^2-1]$ and $[Nr,Nt]=size\left({\mathbf{P}}_{crzf}\right)$. It is obvious that the relaxation parameter $\omega$ also becomes fixed once the mMIMO configuration is fixed. To obtain the appropriate precoding matrix, vector $\mathbf{t}$ and ${\beta }_{CRZF}{\breve{\mathbf{H}}}^H$ must be multiplied. Thus, it is evident that an iterative techniques can be used to accomplish the computationally intensive matrix inversion.    

Algorithm 3 Proposed SSOR-CRZF DP

The CRZF filtering matrix is generated inStep 4, and Steps 5 to 18 have been utilised in the SSOR procedure for obtaining the required SSOR-CRZF precoder (${\widehat{\mathbf{F}}}^{BB}$). After $N_{RF}$ iterations, the baseband DP (${\mathbf{F}}^{BB}$) is constructed using Steps 23 to 27. As a result, adhering to the design for a hybrid precoder, an AP (${\mathbf{F}}^{RF}$) has been created first, and the effective channel matrix ($\overline{\mathbf{H}}$) has then been used to build a low-dimensional DP (SSOR-ZF).

3.3. Solution: Power Allocation (PA) and Power Splitting (PS)

The sum rate can be maximized by exploiting the joint optimization of PA and PS. For addressing the above optimization dilemma, the problem definition (17) can be transformed and can be defined as

$$\underset{p_{d,k},{\rho }_{d,k}}{max}=\sum _{d=1}^D\sum _{k=1}^{\left|{\mathsf{\Xi }}_d\right|}{R}_{d,k}$$

(20)

s.t. (17b)–(17e)

It is very much clear that (20) is a non-convex optimization problem and, considering the stated constraints, it is still a challenging task.

$$\begin{array}{c}\hfill {\check{y}}_{d,k}^{ID}=\left({\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}\sqrt{p_{d,k}}{x}_{d,k}+{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}\sum _{j=1}^{k-1}\sqrt{p_{d,j}}{x}_{d,j}+\right.\\ \hfill \left.{\overline{\mathbf{h}}}_{d,k}^H\sum _{i\ne d}\sum _{j=1}^{\left|{\mathsf{\Xi }}_i\right|}{\mathbf{f}}_i^{BB}\sqrt{p_{i,j}}{x}_{i,j}+n_{d,k}\right)+{\displaystyle \frac{u_{d,k}}{\sqrt{{\rho }_{d,k}}}},\end{array}$$

(21)

Following [3], the mean square error (MSE) related to the estimation of $x_{d,k}$ can be defined as

$$\begin{array}{c}\hfill e_{d,k}=1-2\mathbb{R}\left(c_{d,k}\sqrt{p_{d,k}}{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}\right)+\\ \hfill {\left|c_{d,k}\right|}^2\left(p_{d,k}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2+{\chi }_{d,k}\right),\end{array}$$

(22)

where $c_{d,k}$ represents the equivalent channel estimation coefficient. It is proven that optimal $c_{d,k}$ leads to the minimization of the $e_{d,k}$. Based on [3], the optimized $c_{d,k}$ can be articulated as

$$c_{d,k}^o={\left(\sqrt{p_{d,k}}{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}\right)}^H{\left(p_{d,k}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2+{\chi }_{d,k}\right)}^{-1},$$

(23)

and the minimum $e_{d,k}$ can be obtained as

$$e_{d,k}^o=1-\left(p_{d,k}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2\right){\left(p_{d,k}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2+{\chi }_{d,k}\right)}^{-1},$$

(24)

Using the Sherman–Morrison–Woodbury formula for the matrix inversion, we can have

$${\left(1+{\gamma }_{d,k}\right)}^{-1}=\underset{c_{d,k}}{min}{e}_{d,k}$$

(25)

As in [3], the optimization dilemma/problem can be reformulated as

$$\underset{p_{d,k},{\rho }_{d,k}}{max}\sum _{d=1}^D\sum _{k=1}^{{\mathsf{\Xi }}_d}\underset{a_{d,k}>0}{max}\underset{c_{d,k}}{max}\left(-{\displaystyle \frac{a_{d,k}{e}_{d,k}}{ln2}}+lo{g}_2{a}_{d,k}+{\displaystyle \frac{1}{ln2}}\right)$$

(26)

s.t. (17b)–(17e)

Here, $a_{d,k}>0$ is introduced as a slack variable. The optimization problem, problem (26), can be iteratively resolved through exploiting the alternating optimization (AO) algorithm after dividing (26) into four sub-problems related to ($c_{d,k}$), ($a_{d,k}$), ($p_{d,k}$), (${\rho }_{d,k}$). Particularly, for a given solutions related to $p_{d,k}^{i-1}$ and ${\rho }_{d,k}^{i-1}$ in the ($i-1$)th iteration, the optimal solution at the (i)th iteration for $c_{d,k}^i$ leads to the solution of the optimization problem,

$$\underset{c_{d,k}}{min}{e}_{d,k}$$

(27)

After achieving the optimal solution for $c_{d,k}$, the next step is to solve the optimization problem as stated below:

$$\underset{a_{d,k}>0}{max}\left(-{\displaystyle \frac{a_{d,k}{e}_{d,k}}{ln2}}+lo{g}_2{a}_{d,k}+{\displaystyle \frac{1}{ln2}}\right)$$

(28)

This will lead to the generation of an optimal solution for $a_{d,k}$. As in [3], the optimal solution $a_{d,k}^i$ at the ith iteration can be articulated as

$$a_{d,k}^i={\displaystyle \frac{1}{e_{d,k}^o}}$$

(29)

Following achieving the optimal solution for $c_{d,k}$ and $a_{d,k}$, the optimization dilemma/ problem can be reformulated as

$$\underset{{\rho }_{d,k}}{max}\sum _{d=1}^D\sum _{k=1}^{{\mathsf{\Xi }}_d}\left(-{\displaystyle \frac{a_{d,k}{e}_{d,k}}{ln2}}+lo{g}_2{a}_{d,k}+{\displaystyle \frac{1}{ln2}}\right)$$

(30)

s.t. (17e)

Now, to obtain to obtain the optimal ${\rho }_{d,k}$, a new slack variable ${\tau }_{d,k}\ge {\displaystyle \frac{1}{{\rho }_{d,k}}}$ is introduced, and the optimization dilemma/problem (30) can be equivalently represented as

$$\underset{{\rho }_{d,k}}{max}\sum _{d=1}^D\sum _{k=1}^{{\mathsf{\Xi }}_d}{\tau }_{d,k}$$

(31)

s.t. $\left(-{\displaystyle \frac{a_{d,k}{e}_{d,k}}{ln2}}+lo{g}_2{a}_{d,k}+{\displaystyle \frac{1}{ln2}}\right)\ge {\tau }_{d,k}$.

To make problem (31) solvable, another slack variable ${\vartheta }_{d,k}={\displaystyle \frac{P_{d,k}^{min}}{\left(\eta (1-{\rho }_{d,k})\right)}}$ is introduced, and constraint (17e) in (31) can be conveyed as

$$\sum _{d=1}^D\sum _{k=1}^{{\mathsf{\Xi }}_d}{∥{\overline{\mathbf{h}}}_{d,k}^H{\mathbf{f}}_d^{BB}∥}_2^2{p}_{d,k}+{\sigma }_u^2\ge {\vartheta }_{d,k}$$

(32)

Now, to deal with the constraints relate to ${\tau }_{d,k}$ and ${\vartheta }_{d,k}$, we transform them into matrix form as in [3]

$$\left[\begin{array}{cc}{\tau }_{d,k}& 1\\ 1& {\rho }_{d,k}\end{array}\right]\ge 0,\forall d,k$$

(33)

and

$$\left[\begin{array}{cc}{\vartheta }_{d,k}& \sqrt{P_{d,k}^{min}/\eta }\\ \sqrt{P_{d,k}^{min}/\eta }& 1-{\rho }_{d,k}\end{array}\right]\ge 0,\forall d,k$$

(34)

Finally, after having the optimal values corresponding to $a_{d,k}$, $c_{d,k}$, and ${\rho }_{d,k}$, we move towards solving the optimization problem as follows:

$$\underset{p_{d,k}}{max}\sum _{d=1}^D\sum _{k=1}^{{\mathsf{\Xi }}_d}{\tau }_{d,k}$$

(35)

s.t. (17c), (31), (33) and (34).

The solution of (35) produces an optimal value of $p_{d,k}$ in the ith iteration. Thus, through the iterative AO algorithm, the resulted optimal values for $a_{d,k}$, $c_{d,k}$, $p_{d,k}$, and ${\rho }_{d,k}$ will preserve or increase the solution for the objective function in (26).

4. COMPUTATIONAL COMPLEXITY

This section analyses the SSOR-CRZF precoder’s computational complexity as well as that of a few other precoders that are already in use. The comparative analysis of the computational complexity is presented in

Table 1. Computational complexity comparison.

Precoding Scheme	Computational Complexity
ZF	$\mathcal{O}(2{\left(N_{RF}\right)}^3+{\left(N_{RF}\right)}^2)$
RZF	$\mathcal{O}(5{\left(N_{RF}\right)}^3+{\left(N_{RF}\right)}^2)$
CRZF	$\mathcal{O}(2{\left(N_{RF}\right)}^3+3{\left(N_{RF}\right)}^2)$
KALMAN	$\mathcal{O}(2{\left(N_{RF}\right)}^3+3{\left(N_{RF}\right)}^2)$
SSOR-CRZF	$\mathcal{O}(N_{RF}+{\left(N_{RF}\right)}^2+i2{\left(N_{RF}\right)}^2)$

. For ZF, RZF, CRZF, and SSOR-ZF, it is necessary for computing $\mathbf{H}{\mathbf{H}}^H$, and it is additionally feasible for computing the complexity beyond $\mathbf{H}{\mathbf{H}}^H$. Therefore, the complexity of the computation for ZF can be written as $\mathcal{O}(2{\left(N_{RF}\right)}^3+{\left(N_{RF}\right)}^2)$. Similarly, the complexity of RZF can be written as $\mathcal{O}(2{\left(N_{RF}\right)}^3+3{\left(N_{RF}\right)}^2)$. The Kalman filter (KF)-based HP algorithm is also taken into account for the comparison. As far as the KF is concerned, the calculation of the KF gain at the $n^{th}$ iteration, ${\mathbf{K}}_{\mathbf{kf}}\left(n\right)$$(=\mathbf{R}\left(n|n-1\right){\breve{\mathbf{H}}}^H{\left[\breve{\mathbf{H}}\mathbf{R}\left(n|n-1\right){\breve{\mathbf{H}}}^H+{\mathbf{Q}}_n\right]}^{-1}$), results in the computational complexity $\mathcal{O}(2{\left(N_{RF}\right)}^3+3{\left(N_{RF}\right)}^2)$. Here, the covariance matrix $\mathbf{R}\left(n|n\right)=E\left[{\mathbf{F}}^{BB}\left(n\right){\left({\mathbf{F}}^{BB}\left(n\right)\right)}^{*}\right]$ as well as ${\mathbf{Q}}_n$ has been the covariance matrix corresponding to the noise vector ($\mathbf{n}$).

In the proposed SSOR-CRZF precoder, ${\mathbf{P}}_{crzf}$, and the iteration can be written as (as in [40])

$$t_n^{i+1/2}=t_n^i+{\displaystyle \frac{\omega }{p_{ii}}}\left(s_i-\sum _{j=1}^{n-1}{p}_{nj}{t}_j^{n+1/2}+\sum _{j=n}^{N_{RF}}{p}_{nj}{t}_j^{\left(n\right)}\right),$$

(36)

where the subscript n designates the $n^{th}$ element in a vector, and $P_{nn}(n=1,\dots ,N_{RF})$ represents the diagonal elements of ${\mathbf{P}}_{crzf}$. The initial section of the complexity is mostly caused by Equation (36), and after i iterations, the computational complexity is $i2{\left(N_{RF}\right)}^2$ [42]. Additionally, the multiplication ${\mathbf{H}}^H\mathbf{t}$ cost an additional level of computational complexity of ${\left(N_{RF}\right)}^2$. The final part produces a computational complexity of $\left(N_{RF}\right)$ when ${\beta }_{CRZF}$ is multiplied by ${\mathbf{H}}^H\mathbf{t}$. Therefore, it can be inferred from the above study that the proposed SSOR-CRZF precoder has an overall computational complexity of $\mathcal{O}(N_{RF}+{\left(N_{RF}\right)}^2+i2{\left(N_{RF}\right)}^2)$.

Furthermore, to strengthen the arguments, we explore the computational time corresponding to each of the precoding schemes. For this exploration, we consider the following parameters: $N_T=16$, $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=4$. For SSOR and SSOR-CRZF, we consider the number of iterations to be 4. From the perspective of the system configuration, the simulation is carried out using an HP laptop with the following specifications: processor: 11th Gen Intel(R) Core(TM) i5-1135G7 @ 2.40GHz;RAM: 8GB; 64-bit operating system. Based on the simulation, the computational times corresponding to the RZ, CRZ, SSOR-ZF, and Kalman schemes are 1.732 ms, 1.347 ms, 1.113 ms, and 2.134 ms, respectively. This shows that the SSOR-ZF scheme produces a relatively small computational load compared to the other schemes.

5. NUMERICAL AND SIMULATION RESULTS

This section compares the proposed SSOR-CRZFHP precoder with the traditional ZF, RZF, MRT, SSOR, Kalman, and CRZF precoders to illustrate the superiority of the suggested algorithm. At the end of this section, we have included a comparative analysis of the proposed scheme with the following methods: (a) “SWIPT-Hybrid Precoding NOMA as in [3]”, (b) “SWIPT-Hybrid Precoding NOMA as in [17]”, along with (c) “SWIPT-Hybrid Precoding OMA”. In this case, a SC structure is implemented for HPs, with the transmitter side assumed to have complete access to all channel state information (CSI). The aim is to conduct a thorough quantitative evaluation of the proposed HP’s performance within an mMIMO-NOMA-SWIPT system. This evaluation focuses on three key performance metrics: SE, EE, and the complexity of computation. Utilizing the MATLAB platform, simulations are carried out across 1000 randomly generated channel implementations. The simulation parameters are as follows: BS antennas, $N_T$ = [8, 16, 32, 48, 64, 96, 128]; UE Antenna, 1; Number of RF chains, $N_{RF}$ = [2:8]; Number of Users, $K=$ [4:12]; Number of propagation paths per cluster, $\mathcal{L}=$10; Phase quantization, $B=$ [2:4]; Transmitted power, $P=30$ dB; ULA at the BS with $\lambda /2$ antenna spacing.

The evaluation and the comparison of the performance of several precoding techniques via assessing their SE relative to SNR is shown in Figure 2. The results are obtained with the following parameters: $N_T=64$, $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=4$. According to the simulation findings, the suggested SSOR-CRZF outperforms previous precoders built on the SC architecture, like MRT, ZF, RZF, Kalman, SSOR, and CRZF, by a large margin. For example at SNR = 20 dB SEs (in bps/Hz) corresponding to MRT, ZF, RZF, Kalman, SSOR, CRZF, and SSOR-CRZF are 5.392, 6.375, 11.68, 8.823, 11.08, 12.27, and 15.62, respectively. Even though it is obvious that the FD and FC schemes outperform the SC scheme in terms of SE, the SC-SSOR-CRZF-based HP method considerably enhances the SE performance in the high SNR region, as shown in Figure 2.

The achievable SE comparison regarding the alteration of $N_{RF}$ is shown in Figure 3. In the above outcome, the mmWave mMIMO-NOMA-SWIPT network is achieved considering $N_T=80$, $N_{RF}=[8,10]$, $K=12$, $\mathcal{L}=10$, and $B=4$. There is a notable improvement in SE obtained with the increase in the total quantity of $N_{RF}$. For example, 4.129 bps/Hz is the potential SE at SNR = 15 dB with $Nrf=8$ and the SC-SSOR-CRZF precoder. Under the same conditions, the possible SE with $Nrf=10$ is 9.795 bps/Hz. This clearly demonstrates the impact of the $N_{RF}$ over the SE of the system.

The Figure 4 demonstrates the implications of the SNR and $N_{RF}$ upon the SE for the proposed network.

The SE comparison between several existing precoders and the proposed SSOR-CRZF with regard to the alterations in user numbers is presented in Figure 5. The proposed HP exhibits significantly improved performance compared to the conventional precoders.

The authors of this work have evaluated the suggested lower-resolution HPs’ performance within the mmWave-mMIMO-NOMA-SWIPT network, and Figure 6 represents their findings. The mmWave-mMIMO-NOMA-SWIPT network has been taken into account for this analysis, with $N_T=32$, $N_{RF}=4$, $K=6$, $\mathcal{L}=10$, and $B=[2,3,4]$. According to Figure 6, the suggested SSOR-CRZF approach having 4-bit resolution exhibits superior effectiveness when contrasted with competitors. Although it is clearly true that system performance increases as the resolution for PS increases, PSs with high resolution were not necessary due to their added complexity and cost.

Figure 7 shows a relative comparison to demonstrate the impact of iteration number associated with the proposed SSOR-CRZF algorithm. As in the figure, for SNR = 15 dB, the SEs (in bps/Hz) for the proposed algorithm with iteration 2, 3, and 4 can be specified as 4.52, 5.11, and 5.63, respectively. It is clear that the increase in the iteration number results in improvement in the SE but at the cost of only a little increase in the complexity.

Figure 8 shows the SE of the system as a function of iteration with the following parameters: $N_T=64$, $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=4$. Basically, it depicts the relative convergence analysis of the proposed SSOR-CRZF HP-assisted system with its counterparts. This analysis is particularly important as it addresses the core issue of the problem of joint optimization of PA and PS for a system with fixed users (Here, $K=10$). Specifically, it depicts that, for the mMIMO-NOMA-SWIPT system, the SC-ZF HP Scheme [3] converges with 7 iterations, while the proposed scheme converges in 8 iterations. It is also important to note that our proposed system converges at a significantly high SE.

A comparative analysis of the proposed system with existing work is presented in Figure 9. The results are obtained with the following parameters: $N_T=64$, $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=2$. According to the simulation findings, the proposed system outperforms the existing works. Thus, one can observe that the proposed scheme has the potential to significantly improve the SE of the system.

Furthermore, the performance of the proposed SSOR-CRZF scheme is explored under the Nakagami-m fading channel and with imperfect CSI.

Figure 10 demonstrates the performance comparison of different sub-connected HP schemes with the variation in the Nakagami-m channel fading parameter (m). For this simulation, we have assumed that each arrival/path is Nakagami-m distributed, contrary to what is assumed in the expanded Saleh–Valenzuela model discussed in Section 2.1. The results are obtained with the following parameters: $N_T=16$, $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=4$. This indicates that an increase in the value of m has a significant impact on the system’s SE. As shown, the SE of the system improves as the value of m increases. This is because, with $m=0.5$, the channel can be characterized by a one-sided Gaussian distribution, whereas for $m=2.0$, it can be described by a Rician distribution.

Figure 11 depicts the impact of the imperfect channel on the SE of the system. Following [49], the estimated channel ($\widehat{\mathbf{H}}$) with CSI error can be modelled as $\widehat{\mathbf{H}}=t\mathbf{H}+\sqrt{1-t^2}\mathbf{E}$, where t$(0\le t\le 1)$ is the CSI estimation accuracy and $\mathbf{E}$ represents error matrix. The results are obtained with the following parameters: $N_T=16$, $N_{RF}=4$, $K=6$, $\mathcal{L}=10$, $t=0.8$, and $B=4$. As indicated, the imperfect CSI impacts the performance of the system. More specifically, the SSOR-CRZF scheme provides more stable performance than its counterparts.

The EE ((bps/Hz/W)) for the mmWave-mMIMO-NOMA-SWIPT network is displayed in Figure 12. As in [3], the EE can be calculated by $EE={\displaystyle \frac{R_{sum}}{P_T+N_{RF}{P}_{rf}+N_{ps}{P}_{ps}+P_{bb}}}$, where $N_{ps}$ implies the total number of employed phase shifters. On account of FC, $N_{ps}=N_T{N}_{RF}$, and for SC, $N_{ps}=N_T$. Under such analysis, the maximum transmitted power, $P_T$ = 30 mW. For all precoder algorithms, it is taken for granted to be the same. Every RF chain uses $P_{rf}$ = 300 mW of power. Similarly, power consumption associated with the phase shifter and baseband processing are $P_{ps}$ = 40 mW and $P_{bb}$ = 200 mW, respectively.

The EE comparison of several precoders versus the SNR variation is shown in Figure 12. For this performance analysis, the following parameters are considered: $N_T=64$, $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=4$. As seen in Figure 12, it is clear that the EE is worse in an FD system when compared to other systems. This is due to the FD system’s energy consumption, which is caused by the fact that the number of RF chains and base station antennas are equal. On the other hand, in an SC system, there are far fewer RF chains. Because fewer PS are used in SC-HPs than in FC-HPs, they are more energy efficient. The suggested SSOR-CRZF beats the existing schemes. For example, at SNR = 15 dB, the EE for SC-SSOR-CRZF, SC-CRZF, SC-SSOR, SC-Kalman, SC-RZF, and SC-ZF have 2.077 bps/Hz/W, 1.935 bps/Hz/W, 1.870 bps/Hz/W, 0.810 bps/Hz/W, 1.711 bps/Hz/W, and 0.572 bps/Hz/W, respectively.

Figure 13 presents a comparison of EE among different precoders as the number of BS antennas ($N_T$) varies. For this analysis, the following parameters are considered: $N_{RF}=8$, $K=10$, $\mathcal{L}=10$, and $B=4$. Figure 13 indicates that there is an optimal antenna size that maximizes the EE of the system, given a fixed $N_{RF}$ chain.

6. Challenges and Future Research Directions

The implementation of power splitting (PS) techniques in practical SWIPT systems presents significant challenges, including power loss due to signal degradation in wireless channels and reduced performance as the number of devices increases. These factors complicate the design of power optimization algorithms, which are crucial for balancing spectrum efficiency (SE) and energy efficiency (EE) in the system. In this paper, we address these issues by proposing a complex regularized zero-forcing (CRZF) precoding scheme, integrated with a power splitting optimization method through a joint optimization approach. Nonetheless, further experimental studies are necessary to evaluate the real-world effectiveness of PS techniques. There are several challenges related to SWIPT implementation in mMIMO-NOMA systems. This section highlights the challenges and future research directions related to the implementation of SWIPT in mMIMO-NOMA systems.

6.1. Challenges

The integration of SWIPT within mMIMO and NOMA systems presents a promising solution to enhance both spectrum and energy efficiency. However, the practical implementation of SWIPT in mMIMO-NOMA systems is fraught with several challenges that necessitate further investigation.

6.1.1. Power Splitting (PS) Efficiency

One of the foremost challenges in implementing SWIPT is optimizing the power splitting (PS) technique. In practical scenarios, the efficiency of PS can be severely impacted by signal degradation due to fading and other wireless channel impairments. This can lead to substantial power loss, especially as the number of users in the system increases. As a result, maintaining a balance between harvested energy and information decoding rates becomes increasingly complex, particularly in dense networks with multiple users.

6.1.2. Complexity in Algorithm Design

The joint optimization of spectrum efficiency (SE) and energy efficiency (EE) in SWIPT systems requires sophisticated algorithms that can adapt to varying channel conditions and user demands. The complexity of designing such algorithms is further compounded in mMIMO-NOMA systems due to the large number of antennas and users involved. This increases the computational burden and can lead to latency issues, which are critical in real-time applications.

6.1.3. Channel Estimation and Feedback Overhead

Accurate channel estimation is crucial for the effective implementation of SWIPT in mMIMO-NOMA systems. However, the large-scale nature of mMIMO systems results in significant feedback overhead, which can strain system resources and degrade performance. Moreover, the non-orthogonal nature of NOMA adds another layer of complexity to channel estimation, making it challenging to achieve high accuracy in practical deployments.

6.1.4. Interference Management

In mMIMO-NOMA systems, managing interference is a key challenge, particularly when multiple users share the same frequency band. The addition of SWIPT introduces new sources of interference, as the power transfer process can interfere with information decoding. Developing robust interference management techniques that can operate efficiently in such environments is essential to ensure the reliable performance of the system.

6.1.5. Energy Harvesting Efficiency

The efficiency of energy harvesting in SWIPT-enabled systems is another area that requires attention. In practical deployments, the amount of energy that can be harvested is often limited by various factors, including the distance between the transmitter and the receiver, the quality of the channel, and the level of interference. Enhancing energy harvesting efficiency without compromising the information decoding process is a significant challenge that needs to be addressed.

6.2. Future Research Directions

To address these challenges, several future research directions can be explored:

Advanced Precoding Techniques: Developing more sophisticated precoding techniques, such as those that can adapt to dynamic channel conditions, will be crucial in improving the performance of SWIPT in mMIMO-NOMA systems. Hybrid PS Schemes: Exploring hybrid power splitting schemes that can dynamically adjust the PS ratio based on real-time channel conditions and user requirements could enhance the efficiency of SWIPT systems. Machine-Learning Integration: The integration of machine-learning techniques to optimize resource allocation and manage interference in SWIPT-enabled mMIMO-NOMA systems could provide significant performance gains. Experimental Validation: Conducting real-world experiments and field trials to validate the theoretical models and algorithms developed for SWIPT in mMIMO-NOMA systems will be essential to assess their practicality and effectiveness. Energy Harvesting Technologies: Investigating new energy harvesting technologies and materials that can operate efficiently in a broader range of conditions could improve the overall viability of SWIPT in practical systems. Furthermore, one can explore the possibilities of interplay with emerging technologies like intelligent reflecting surface (IRS), cell-free MIMO, and many more.

7. Conclusions

This study investigated the effectiveness offered by an mmWave-mMIMO-NOMA-SWIPT system armed with a sub-connected architecture as well as in multi-user situations. The MKM user grouping technique has been used by the authors in this instance, followed by the hybrid precoder design. The optimization of both the power allocation and the power splitting has been carried out, and the convergence of the proposed system has been demonstrated. This paper presents and illustrates the SE and EE, as well as the computational complexity and performance of the SSOR-CRZF-based hybrid precoder. Comparing the proposed SSOR-CRZF to traditional precoders, for instance, ZF, MRT, RZF, Kalman, and SSOR, a significant performance boost is achieved along with an immense reduction in complexity. As part of future works, the authors would like to explore the proposed SSOR-CRZF hybrid precoding scheme in terms of optimization of the resource allocation, and the authors would also like to design a more sophisticated hybrid precoding design algorithm to further improve the performance.