Enhancing the Energy Efficiency of mmWave Massive MIMO by Modifying the RF Circuit Configuration

Abstract

Hybrid architectures are used in the Millimeter wave (mmWave) Massive MIMO systems, which use a smaller number of RF chains and reduces the power and energy consumption of the mmWave Massive MIMO systems. However, the majority of the hybrid architectures employs the conventional circuit configuration by connecting each of the RF chains with all the transmitting antennas at the base station. As a result, the conventional circuit configuration requires a large number of phase shifters, combiners, and low-end amplifiers. In this paper, we modify the RF circuit configuration by connecting each of the RF chains with some of the transmitting antennas of mmWave Massive MIMO. Furthermore, the hybrid analogue/digital precoders and decoders along with the overall circuit power consumptions are modelled for the modified RF circuit configuration. In addition, we propose the alternating optimization algorithm to enhance the optimal energy efficiency and compute the optimal system parameters of the mmWave Massive MIMO system. The proposed framework provides deeper insights of the optimal system parameters in terms of throughput, consumed power and the corresponding energy efficiency. Finally, the simulation results validate the proposed framework, where it can be seen that the proposed algorithm significantly reduces the power and energy consumptions, with a little compromise on the system spectral gain.

1. Introduction

The consumption of data traffic is exponentially increasing in wireless networks and a large number of advanced user terminals are getting connected to the internet with every passing day, which in turn leads to the ever-increasing demand for higher data rates, as seen in the last couple of decades [1]. This rapid proliferation of internet data traffic along with the continuous increment in the number of new advanced user terminals have made the congestion in the lower frequency band spectrum [2]. In order to handle this issue, various techniques have been proposed by the researchers, like installing the massive antennas at the base station [3], spectrum coding and other network densification algorithms [4]. These spectrum enhancing techniques alone may not be enough to fulfill the current and future demand of users, which in turn shifts the focus of researchers to explore the low congested frequency bands.

The communication by using the mmWave carrier frequencies (30–300 GHz) is deemed the vital candidate to address the above-mentioned situation because it can provide higher data rates required by the next generation of networks [5]. In addition, the mmWave carrier frequencies offer low latency, which in turn makes it a perfect candidate for various applications of current and future generation of networks such as intelligent robots, auto-driven vehicles [6] and the wearable networks [7,8]. However, there is a tradeoff in mmWave carrier frequencies between the offered bandwidth and path loss, which can be compensated by adding the large antenna array at the base station [9,10]. Thanks to the small wavelengths of mmWave carrier frequencies, the large antenna array can be efficiently packed in order to overcome the huge path losses [11]. Massive Multiple Input and Multiple Output, also called as large scale MIMO, is mostly associated with the mmWave systems because it can provide the substantial increment in the overall throughput, due to its high bandwidth, and this integration between the Massive MIMO and mmWave systems makes them the perfect candidates for the future generation of networks [12,13].

In order to further enhance the performance of the system, various precoding schemes are used along with the multiple chains of Massive MIMO. Under the scenario of low frequency spectrum, various digital linear and nonlinear precoding schemes are used such as Zero Forcing (ZL), Matched Filter (ML) and the dirty paper coding [14,15,16]. However, the signal processing in mmWave systems possesses a condition of non-trivial system constraint. For example, in the low frequency systems, where each of the transmitting antenna at the base station requires a separate dedicated RF chain and each of the RF chain consists of Digital to Analogue convertor (DAC), Analogue to Digital Convertor (ADC), mixers, oscillators and the Power Amplifiers (PA), which would definitely be a costly option to use in mmWave systems [17]. The above-mentioned problem was addressed by the researches by using the antenna selection techniques in order to reduce the number of RF chains [18,19]. However, the system performance gets reduced by utilizing the antenna selection techniques at the base station as compared to the performance achieved by using the digital precoding. The researchers also proposed the analogue-only beamforming to address the above-mentioned problem. Actually, the analogue beamforming consists of the network of analogue phase shifters, where the phase shifters are used to steer the transmitting and receiving beams in the intended directions by changing the phases of the signals [20,21,22]. However, the analogue beamforming is dependent on the phase angles quantization of signals and cannot support the multi-stream communication.

It has been seen that the hybrid architectures, where the signal processing is performed by using the combination of analogue/digital precoders and decoders, significantly improves the performance of the system as compared to using the only analogue beamforming, and thus can support multi-stream communication from the multiple end users [23,24,25,26,27]. In the conventional hybrid approaches, the researches connect the output of each RF chain with all the transmitting antennas at the base station, which in turn results in the increment in the consumed power and the corresponding energy efficiency [28,29,30,31,32,33,34,35,36]. The conventionally used hybrid approaches, where each of the RF chains is connected with all the transmitting antennas, are also used in [37,38,39,40], where the authors use the Orthogonal Matching Pursuit (OMP) to solve the problem of designing the optimal hybrid precoders and decoders. In [41], the authors propose the low complexity algorithm to design the hybrid precoders for the conventional circuit configuration with the aim of maximizing the spectral efficiency over different sub-carriers. In [42], the authors propose the beam forming algorithm to reduce the overhead of the channel feedback by considering the fully digital Massive MIMO system. Motivated by the above-mentioned research, the authors in [43] propose the designing of hybrid precoders and decoders for the multiple users with a small training and feedback by considering the conventional circuit configuration. In [44,45], the authors design the optimal hybrid precoders and decoders by using the predefined codebooks for the conventionally used hybrid circuit configuration. However, conventionally used hybrid circuit configuration is not feasible to use in the Massive MIMO systems because Massive MIMO is based on the theory of having a lot of transmitting antennas at the BS. Therefore, conventionally used circuit configuration will require a large number of phase shifters to connect each of the RF chain with all the transmitting antennas in the Massive MIMO system [46], which in turn requires further energy for excitation and compensation of the insertion losses of phase shifters [47]. Furthermore, when each of the RF chains is connected with all the transmitting antennas at the base station, then the computation complexity of the system also is increased [48].

An alternative solution, to reduce the number of phase shifters and the corresponding energy efficiency, is to modify the circuit configuration. The RF circuit configuration is modified by connecting each of the RF chain with part of the transmitting antennas, which in turn significantly reduces the complexity of the system. The idea of the above-mentioned RF circuit modification is presented in [49,50,51], where the authors design the optimal hybrid precoders and decoders by solving the optimization problem in terms of the maximization of the system spectrum efficiency. However, optimization problem should be solved in terms of the minimization of the system energy efficiency rather than the spectral efficiency. In [52], the authors utilize the only analogue precoders and decoders to precode and decode the transmitting and receiving signals for the modified RF circuit configuration, which actually contradicts the hybrid approach for the mmWave systems. In [53], the authors design the optimal precoders and decoders for the mmWave large antenna array systems by solving the optimization problem with respect to the maximization of the system energy efficiency. However, the proposed optimal solution is so complex in [53], and the power consumption for the large antenna arrays is not correctly modelled.

Thus far, there is no research available in the existing literature to design the optimal hybrid precoders and decoders for the mmWave Massive MIMO systems, by using the modified RF circuit configuration, with the aim of maximizing the overall system energy efficiency without much complexity, and by utilizing the correct modelling of the system power consumptions. Therefore, the above-mentioned missing context in the literature is explored in this paper. The optimal hybrid precoders and decoders are designed for the mmWave Massive MIMO system by utilizing the modified RF circuit configuration, where each of the RF chain is connected to some of the transmitting antennas. When the modified RF configuration is utilized, then the designing of the analogue precoders becomes challenging due to some extra circuit constraints. Therefore, the solution is derived and discussed for designing the analogue precoders in this article by taking all the system and circuit constraints into account. In the existing research, the mathematical expressions of the spectral efficiency and the corresponding system throughput are derived, without taking the overhead of pilots into account. In this paper, we derive the mathematical expressions of the spectral efficiency and the corresponding system throughput by considering the overhead of pilots. Contrary to existing research, the circuit power consumption is accurately and completely modelled by deriving the mathematical expressions of the transmitting and consumed power for the modified RF circuit configuration. In addition, the comparison is presented and discussed in terms of power consumptions between the conventional and modified RF mapping circuit configurations, where it can be seen that the modified RF circuit configuration deploys a significantly smaller number of the phase shifters and the corresponding low-end amplifiers. Furthermore, we propose the alternating optimizing algorithm in this paper, and the proposed algorithm works by maximizing the overall optimal energy efficiency of the system by considering the modified RF circuit configuration. The simulation results unveil the significance of the proposed algorithm and the modified RF circuit configuration, where it can be seen that the optimal energy efficiency of the mmWave Massive MIMO can be significantly enhanced with a little compromise on the spectral gain of the system. The main contributions of this article are summarized as follows:

• Contrary to the existing research [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,49,50,51,52,53], where the authors formulate the spectral efficiency of mmWave Massive MIMO without considering the pilot overhead factor, the mathematical expressions of the spectral efficiency of mmWave Massive MIMO are derived and formulated by considering the pilot overhead factor for the modified RF circuit configuration in this paper.
• Contrary to the existing research [49,50,51,52,53], the circuit power consumption is accurately and completely modelled by deriving the mathematical expressions of the transmitting and consumed power for the modified RF circuit configuration in this paper.
• The detailed comparison is presented and discussed in terms of power consumptions between the conventional and the modified RF mapping circuit configurations in order to highlight the significance of the proposed circuit configuration.
• Contrary to the relevant research [49,50,51,52,53], the extended version of the Saleh-Valenzuela is used to model the mmWave channel in order to embody the low rank and spatial correlation characteristics of mmWave channels in the simulations.
• Compared to the existing research [37,38,39,40,43,49,50,51,52,53], where the existing algorithms are so complex and each of the existing algorithms has some limitations, as explained in detail in the literature review, we propose the alternating optimizing algorithm in this paper, and the proposed algorithm works by maximizing the overall optimal energy efficiency of the system by considering the modified RF circuit configuration without much complexity.
• The simulation results unveil the significance of the proposed alternative optimization algorithm and the modified RF circuit configuration, where it can be seen that the proposed algorithm significantly enhances the overall optimal energy efficiency of the system as compared to the conventionally used circuit configuration.

The remainder of the article is organized as follows. Section 2 describes the system model and the derivation of the mathematical expression of the system spectral efficiency. The extended version of the Saleh-Valenzuela to model the mmWave channel is discussed in Section 3. Section 4 describes the modelling of the hybrid precoders and decoders, and the overall transmitted and consumed power for the RF modified circuit configuration is formulated in Section 5. Section 6 presents the detailed comparison between the traditional and the RF modified circuit configurations in terms of consumed power. The optimization problem is formulated and solved in Section 7. In Section 8, we numerically evaluate the proposed framework and discuss the simulation results. Conclusions are summarized in Section 9.

We use the following notation throughout this paper; $AP$ is a matix, $A{P}_k$ is a kth column of $AP$, ${\Vert AP\Vert }_F$ means the frobenius norm, ${(.)}^H,e^{(.)}$ means the Hermitian and exponential operation respectively, $\otimes$ means the kronecker product, $trace(.)$ means the trace of a matrix, ${\log }_2(.)$ means the logarithm operation, ${\left(AP\right)}^T$ means the transpose of $AP$, $\left|AP\right|$ means the modulus of $AP$, $Z_{+}$ means the positive integer and $\frac{d}{dx}(.)$,$\frac{d^2}{d{x}^2}(.)$ means the first and second order derivative with respect to $x$.

2. System Modelling and the Spectral Efficiency

Consider the system model of mmWave Massive MIMO shown in Figure 1. The base station is comprised of $T_x$ transmitting antennas, and transmitting $x=\left[x_1,x_2,x_3,\dots ,x_K\right]$ information symbols to the $K$ number of single antenna users. As it can be seen in Figure 1, the base station contains $C_t$ transmitting chains with the constraint of $x<C_t<T_x$, and the corresponding receiver contains $C_r$ receiving chains with the constraint of $x<C_r<K$, with the aim of reducing the hardware complexity of the system. The hybrid precoding brings to the theory of using the combination of analogue and digital precoders in order to reduce the hardware cost. Therefore, the base station is comprised of the hybrid combination of digital procoder $D{P}_t$ and the analogue precoder $A{P}_t$. Similarly, the single antenna end users are comprised of the hybrid combination of digital combiner $D{P}_r$ and the analogue combiner $A{P}_r$. The dimension of the digital and analogue precoder is $C_t\times x$ and $T_x\times C_t$ respectively, and the dimension of the digital and analogue combiner at the receiving end is $C_r\times x$ and $K\times C_r$ respectively. The information symbols following the process of analogue and digital precoder can be written as:

$$A=D{P}_{t}A{P}_{t}x$$

Furthermore, the received information stream by assuming the narrow-band block fading channel can be written as:

$$Y=\sqrt{p}\times H\times D{P}_t\times A{P}_t\times x+n$$

(1)

where $Y$ is the received signal and $n$ is the Additive White Gaussian Noise (AWGN). The AWGN has the unity variance and zero mean. $H$ is the discrete mmWave narrow band channel, and it is modelled in the next section.

Furthermore, the Figure 1 unveils the conventionally used transmitter circuit, where each of the RF chains is connected to all the transmitting antennas at the base station via phase shifters. The conventionally used circuit requires a large number of phase shifters, which in turn increases the power and energy consumption of the system. Therefore, the conventionally used RF signal mapping configuration is not feasible under the scenario of energy efficient systems. The Figure 2 unveils the modified RF signal mapping scheme, where the output of each RF chain is connected with some of the transmitting antennas instead of connecting with all the transmitting antennas. The modified RF signal mapping scheme requires less number of phase shifters, and the required number of phase shifters in the modified RF signal mapping scheme comes out to be $T_x$ and $K$ at the transmitting and receiving end respectively. Whereas, in the conventionally used RF signal mapping scheme, the required number of phase shifters comes out to be $T_x\times C_r$ and $K\times C_r$ at the transmitting and receiving end respectively. Due to the substantial decrement in the number of required phase shifters, the modified RF signal mapping scheme is highly suitable under the scenario of energy efficient systems. However, the modelling of the digital and analogue precoders and decoders at the base station and receiving ends is different for both of these above-mentioned configurations, and it is discussed in Section 4. The signal received at the $k$th user can be written as:

$$Y_k=\sqrt{p}\times H_k\times D{P}_{t,k}\times A{P}_{t,k}\times x_k+\sqrt{p}{\displaystyle \sum _{i=1,i\ne k}^K\left(H_i\times D{P}_{t,i}\times A{P}_{t,i}\times x_i\right)}+n$$

(2)

The received signal $Y_k$ at the $k$th user can be retrieved by multiplying the received signal with the Hermitian of the analogue and digital decoder. Therefore, the retrieved signal $Y_k^{\prime }$ can be written as:

$$Y_k^{\prime }=D{P}_{r,k}^H\times A{P}_{r,k}^H\times \left(\begin{array}{l}\sqrt{p}\times H_k\times D{P}_{t,k}\times A{P}_{t,k}\times x_k\\ +\sqrt{p}{\displaystyle \sum _{i=1,i\ne k}^K\left(H_i\times D{P}_{t,i}\times A{P}_{t,i}\times x_i\right)}+n\end{array}\right)$$

(3)

By utilizing the properties of Massive MIMO, the interference term can be neglected because the propagation channel between two different signals becomes orthogonal due to the law of large numbers. Therefore, the Equation (3) can be written as:

$$Y_k^{\prime }=D{P}_{r,k}^H\times A{P}_{r,k}^H\times \left(\sqrt{p}\times H_k\times D{P}_{t,k}\times A{P}_{t,k}\times x_k+n\right)$$

(4)

The corresponding Signal to Noise Ratio $S.N.R_k$ can be computed as:

$$S.N.R_k=\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}$$

(5)

Whereas, the sum rate at the $k$th user can be expressed as:

$$R_k={\log }_2\left(1+S.N.R_k\right)={\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)$$

(6)

The overall spectral efficiency of the system can be expressed as:

$$R_K=K\times {\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)$$

(7)

Time Division Duplex (TDD) is mostly used in Massive MIMO systems because the overhead factor is not dependent on the number of transmitting antennas as compared to Frequency Division Duplex (FDD) where the overhead factor is so huge due to this dependence on the number of transmitting antennas at the base station. Therefore, the pilot overhead factor is the same in both the circuit configurations because of using the same duplexing scheme (TDD). By considering the pilot overhead, the Equation (7) can be written as:

$$R_K=K\times \left(\begin{array}{l}{\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)-\frac{T_{sum}K}{C}\times \\ {\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)\end{array}\right)$$

(8)

The second term in Equation (8) is the overall reduction in the spectral efficiency due to pilot overhead. Furthermore, the Equation (8) can be simplified as:

$$R_K=K\left(1-\frac{T_{sum}K}{C}\right){\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)$$

(9)

where $C$ is the coherence block, and $T_{sum}$ is the relative length of the pilot.

3. Extended Saleh-Valenzuela mmWave Channel Modelling

The mmWave channels allow the dense packing of antennas at the base station due to the limited number of scattering, and the antenna correlation is very high in the mmWave transceivers due to the closely spaced antennas. Therefore, the rich scattering models, used in the low frequencies MIMO channels, are not feasible to use in the mmWave systems. In this paper, we use the extended version of the Saleh-Valenzuela model to model the mmWave channel in order to embody the low rank and spatial correlation characteristics of mmWave channels [54]:

$$H=\gamma {\displaystyle \sum _{l=1}^{N_{ray}}{\alpha }_l{G}_r\left({\varphi }_l^r,{\theta }_l^r\right)G_t\left({\varphi }_l^t,{\theta }_l^t\right)f_r\left({\varphi }_l^r,{\theta }_l^r\right)f_t^H\left({\varphi }_l^t,{\theta }_l^t\right)}$$

(10)

where $\gamma =\sqrt{\frac{T_{x}K{C}_t}{N_{ray}}}$ is the normalization factor, and $N_{ray}$ is the total propagation paths. The ${\alpha }_l$ is the path fading coefficient of the lth ray, which follows $ℂ\mathrm{N}\left(0,1\right)$. Furthermore, ${\varphi }_l^r,{\theta }_l^r$,${\varphi }_l^t,{\theta }_l^t$ represents the elevation (azimuth) angle of arrival and departure at the base station and end users respectively, which follows the uniform distribution from 0 to 2pi. The functions $f_r\left({\varphi }_l^r,{\theta }_l^r\right)$ and $f_t^H\left({\varphi }_l^t,{\theta }_l^t\right)$ are independent of the antenna elements, and denotes the receiving and transmitting response of antenna array at a specific angle of arrival and departure at the base station and end users respectively. The receiving and transmitting response of $U$th element antenna array can be modelled as [47]:

$$f\left(\varphi \right)=\frac{1}{\sqrt{U}}{\left[1,e^{j\frac{2\pi }{\lambda }d\times sin\left(\varphi \right)},\dots \dots ,e^{j\frac{2\pi }{\lambda }\left(U-1\right)d\times sin\left(\varphi \right)}\right]}^T$$

(11)

where $d$ is the antenna spacing, and $\lambda$ is the wavelength of the signal. The angle of elevation $\theta$ is not included in the above equation because the response of the antenna array is not dependent on the angle of elevation. Furthermore, the receiving and transmitting response of $U$th element antenna array on the y and z axes, where the $V$ and $B_1$ antenna elements are located, can be written as:

$$f\left(\varphi ,\theta \right)=\frac{1}{\sqrt{U}}{\left[1,\dots \dots ,e^{j\frac{2\pi }{\lambda }d\left(k_{1}sin\left(\varphi \right)sin\left(\theta \right)+k_{2}cos\left(\theta \right)\right)},\dots \dots ,e^{j\frac{2\pi }{\lambda }d\left(\left(k_1-1\right)sin\left(\varphi \right)sin\left(\theta \right)+\left(k_2-1\right)cos\left(\theta \right)\right)}\right]}^T$$

(12)

where $K_1$ and $K_1$ are constrained as $0\le k_1\le \left(V-1\right)$ and $0\le k_2\le \left(B_1-1\right)$ respectively. The functions $G_r\left({\varphi }_l^r,{\theta }_l^r\right)$ and $G_t\left({\varphi }_l^t,{\theta }_l^t\right)$ denotes the receiving and transmitting antenna gain at a specific angle of arrival and departure at the base station and end users respectively, and the receiving and transmitting antenna gains are dependent on the type of antenna elements in the array. In this paper, we assume the half wave dipole antennas at the base station, and the transmitting and receiving gain of the half wave dipole antenna is assumed to be unity over the sectors ${\varphi }_l^t\in \left[{\varphi }_{min}^t,{\varphi }_{max}^t\right],{\theta }_l^t\in \left[{\theta }_{min}^t,{\theta }_{max}^t\right]$, ${\varphi }_l^r\in \left[{\varphi }_{min}^r,{\varphi }_{max}^r\right],{\theta }_l^t\in \left[{\theta }_{min}^r,{\theta }_{max}^r\right]$:

$$G_t\left({\varphi }_l^t,{\theta }_l^t\right)=\{\begin{array}{cc}1& \forall {\varphi }_l^t\in \left[{\varphi }_{min}^t,{\varphi }_{max}^t\right],{\theta }_l^t\in \left[{\theta }_{min}^t,{\theta }_{max}^t\right]\\ 0& \mathrm{otherwise}\end{array}$$

$$G_r\left({\varphi }_l^r,{\theta }_l^r\right)=\{\begin{array}{cc}1& \forall {\varphi }_l^r\in \left[{\varphi }_{min}^r,{\varphi }_{max}^r\right],{\theta }_l^t\in \left[{\theta }_{min}^r,{\theta }_{max}^r\right]\\ 0& \mathrm{otherwise}\end{array}$$

4. Computation of the Analogue and Digital Precoder for the Modified RF Configuration

The hybrid precoders and decoders are modelled in this section by considering the scenario of RF mapping given in Figure 2. Under the perfect scenario in terms of hybrid design, the optimal hybrid precoders and decoders at the base station and end users should follow:

$$D{P}_t\times A{P}_t=P_{opt}$$

(13)

$$D{P}_t\times A{P}_t=D_{opt}$$

(14)

where $P_{opt}$ and $D_{opt}$ represents the fully optimal precoders and decoders at the base station and end user respectively. However, it is not possible to achieve the optimal situation given in Equation (13) and Equation (14), respectively, due to the unit modulus constraint on each of the entry of hybrid precoder and decoder along with the design constraints shown in Figure 2. In this paper, we focus on the designing of hybrid precoders by assuming that the hybrid decoders can be designed with the similar way as hybrid precoders. By setting the product of analogue and digital precoders close to the fully optimal precoder, the design problem can be illustrated as:

$$\underset{D{P}_t,AP}{minimize}{\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2$$

(15)

$$\mathrm{Constraint} \mathrm{to} \begin{array}{l}A{P}_t\in \mathbb{Q}\\ {\Vert D{P}_t\times A{P}_t\Vert }_F^2\le P_{max}\end{array}$$

where $\mathbb{Q}$ is the feasible set of constraints on the analogue precoder by considering the scenario given in Figure 2. Furthermore, the $A{P}_t$ represents the block diagonal matrices, and it can be written as:

$$A{P}_t=\left[\begin{array}{lllll}A{P}_1& 0& 0& \dots & 0\\ 0& A{P}_2& & & 0\\ 0& 0& A{P}_2& & 0\\ ⋮& & & \ddots & ⋮\\ 0& 0& 0& \dots & A{P}_{C_t}\end{array}\right]$$

where each of the block diagonal matric in the above matrix can be written as:

$$\mathbb{Q}=\left\{diag\left(A{P}_t\right),1\le t\le C_t|\left|A{P}_t\right|=1\&A{P}_t\left(cl\right)\in S, \mathrm{where} 1\le cl\le \left(T_x/C_t\right)\right\}$$

It is difficult to solve the optimization problem in Equation (15) by jointly considering the $A{P}_t$ and $D{P}_t$. Therefore, the original optimization problem is subdivided into two problems, where the $A{P}_t$ and $D{P}_t$ are modelled by keeping the other one constant as can be seen in the following optimization problems:

$$\begin{array}{l}\underset{A{P}_t}{minimize}{\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2\\ \mathrm{Constraint} \mathrm{to} A{P}_t\in \mathbb{Q}\\ {\Vert D{P}_t\times A{P}_t\Vert }_F^2=\left(\frac{T_x}{C_t}\right)\left(\frac{C_t}{T_x}x\right)=x\end{array}$$

(16)

$$\begin{array}{l}\underset{D{P}_t}{minimize}{\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2\\ \mathrm{Constraint} \mathrm{to} {\Vert D{P}_t\times A{P}_t\Vert }_F^2=\left(\frac{T_x}{C_t}\right)\left(\frac{C_t}{T_x}x\right)=x\end{array}$$

(17)

By relaxing the constraint on the optimization problem in Equation (16), each of the entries of $A{P}_t$ can be written as:

$$A{P}_1=\left[\begin{array}{l}{e}^{j{\theta }_1}\\ e^{j{\theta }_{\frac{T_x}{C_t}}}\end{array}\right], A{P}_2=\left[\begin{array}{l}{e}^{j{\theta }_{\frac{T_x}{C_t}+1}}\\ e^{j{\theta }_{2\frac{T_x}{C_t}}}\end{array}\right], A{P}_3=\left[\begin{array}{l}{e}^{j{\theta }_{2\frac{T_x}{C_t}+1}}\\ e^{j{\theta }_{3\frac{T_x}{C_t}}}\end{array}\right]$$

Furthermore, the matrices of analogue precoder can be generalized as:

$$A{P}_{C_t}=\left[\begin{array}{l}{e}^{j{\theta }_{\left(C_t-1\right)\frac{T_x}{C_t}+1}}\\ e^{j{\theta }_{T_x}}\end{array}\right]$$

(18)

The phase angles $\left[{\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_{{}_{\frac{T_x}{C_t}}},{\theta }_{{}_{2\frac{T_x}{C_t}}},{\theta }_{{}_{3\frac{T_x}{C_t}}},\dots ,{\theta }_{T_x}\right]$ are required to construct the overall matrix $A{P}_K$, and the set of phase angles can be extracted from the overall optimal hybrid precoder and the digital precoder by using the following approach:

For $A{P}_1$:

$$\theta \left(A{P}_{{}_{1,\frac{T_x}{C_t}}}\right)=\theta \left({\left(P_{opt}\right)}_{1:}\times {\left({\left(DP\right)}_{\frac{C_t}{T_x}:}\right)}^H\right)$$

For $A{P}_2$:

$$\theta \left(A{P}_{{}_{2,2\frac{T_x}{C_t}}}\right)=\theta \left({\left(P_{opt}\right)}_{2:}\times {\left({\left(DP\right)}_{2\frac{C_t}{T_x}:}\right)}^H\right)$$

For $A{P}_3$:

$$\theta \left(A{P}_{3,3\frac{T_x}{C_t}}\right)=\theta \left({\left(P_{opt}\right)}_{3:}\times {\left({\left(DP\right)}_{3\frac{C_t}{T_x}:}\right)}^H\right)$$

Similarly, the $A{P}_{C_t}$th entry of $A{P}_t$ can be written as:

$$\theta \left(A{P}_{C_t,T_x}\right)=\theta \left({\left(P_{opt}\right)}_{{\mathrm{C}}_t:}\times {\left({\left(DP\right)}_{T_x:}\right)}^H\right)$$

(19)

It is difficult to solve the optimization problem in Equation (17) in its present form. Therefore, the objective function of the optimization problem in Equation (17) needs to be simplified. By using the properties of trace, the objective function of the optimization problem in Equation (17) can be written as:

$${\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2=trace\left[\left(P_{opt}-D{P}_t\times AP\right){\left(P_{opt}-D{P}_t\times AP\right)}^H\right]$$

$${\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2=trace\left[\left(P_{opt}-D{P}_t\times A{P}_t\right)\left(P_{opt}{}^H-D{P}_t{}^H\times A{P}_t{}^H\right)\right]$$

(20)

As, $A{P}_t$ consists of block diagonal matrices so the Equation (20) can be written as:

$${\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2=trace\left[\left(P_{opt}-D{P}_t\times \left(I_x\otimes A{P}_t\right)\right)\left(P_{opt}{}^H-D{P}_t{}^H\times \left(I_x\otimes A{P}_t{}^H\right)\right)\right]$$

$${\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2=trace\left[\begin{array}{l}{P}_{opt}{P}_{opt}{}^H-P_{opt}\times D{P}_t{}^H\times \left(I_x\otimes A{P}_t{}^H\right)\\ +D{P}_t\times \left(I_x\otimes A{P}_t\right)\times D{P}_t{}^H\times \left(I_x\otimes A{P}_t{}^H\right)\\ -D{P}_t\times \left(I_x\otimes A{P}_t\right)\times P_{opt}{}^H\end{array}\right]$$

(21)

Let:

$$I_x\otimes A{P}_t=d_1$$

$$I_x\otimes A{P}_t{}^H=d_2$$

Therefore, the Equation (21) can be simplified as:

$${\Vert P_{opt}-D{P}_t\times A{P}_t\Vert }_F^2=trace\left[\begin{array}{l}{P}_{opt}{P}_{opt}{}^H-P_{opt}\times D{P}_t{}^H\times \left(d_2\right)\\ +D{P}_t\times \left(d_1\right)\times D{P}_t{}^H\times \left(d_2\right)-D{P}_t\times \left(d_1\right)\times P_{opt}{}^H\end{array}\right]$$

Following the above-mentioned simplification, the objective function in the optimization problem (Equation (17)) can be written as:

$$\begin{array}{l}\underset{D{P}_t}{minimize} trace\left[\begin{array}{l}{P}_{opt}{P}_{opt}{}^H-P_{opt}\times D{P}_t{}^H\times \left(d_2\right)\\ +D{P}_t\times \left(d_1\right)\times D{P}_t{}^H\times \left(d_2\right)-D{P}_t\times \left(d_1\right)\times P_{opt}{}^H\end{array}\right]\\ \mathrm{Constraint} \mathrm{to} {\Vert D{P}_t\times A{P}_t\Vert }_F^2=\left(\frac{T_x}{C_t}\right)\left(\frac{C_t}{T_x}x\right)=x\end{array}$$

(22)

5. Computation of the Consumed Power

The overall power consumed, in the case of mmWave Massive MIMO by considering the RF modified circuit configuration, is modelled in this section. The overall consumed power is equal to the transmitted power required to transmit the signal, power consumed in the circuitry of Massive MIMO, power losses due to hybrid architecture and the power required for the compensation of losses by using the low-end amplifiers:

$$P_{T/P}=P_T+P_C+P_{Loss}+P_{COM}$$

(23)

where $P_{T/P}$ is the overall consumed power, $P_T$ is the transmission power, $P_{Loss}$ is the power reduction due to the hybrid architecture, $P_{COM}$ is the power required for the low end amplifiers to compensate the losses, and $P_C$ is the overall consumed power. The $P_C$ represents the overall consumed power in the circuitry of Massive MIMO i.e., power consumption in each transceiver chain $P_{TR}$, power consumption during the coding and decoding of the transmitted and received signal $P_{C/D}$, fixed power consumption $P_{FiX}$ i.e.,

$$P_C=P_{TR}+P_{FiX}+P_{C/D}$$

(24)

The transmitted power required to transmit the signal by assuming the uniform distribution of users can be written as:

$$P_T=\frac{KBp{\alpha }^2{\Vert D{P}_t\times A{P}_t\Vert }_F^2}{{\eta }_{AM}}\times \left(\frac{d_{Mx}^{\alpha +2}-d_{Mn}^{\alpha +2}}{\left(1+\frac{\alpha }{2}\right)\left(d_{Mx}^2-d_{Mn}^2\right)}\right)$$

(25)

where ${\eta }_{AM}$ is the efficiency of the power amplifier, $d_{Mx}$, $d_{Mn}$ represents the maximum and minimum distance between the base station and end users.

As, each RF chain is equipped with the filter, mixer and the oscillator so the power consumed in each RF chain can be modelled as:

$$P_{R/F}=P_F+P_m+P_o$$

(26)

where, $P_F$ is the power consumption during the filtering of the signal, $P_m$ is the mixer power consumption, $P_o$ is the oscillator power consumption. The overall power consumption of transceiver chains by employing $C_t$ and $C_r$ transmitter and receiver chains at the base station and end users respectively can be written as:

$$P_{TR}=\left(C_t+C_r\right)P_{R/F}$$

(27)

The $P_{C/D}$ is the consumed power during the coding and decoding of the signal and it is dependent on the system spectral efficiency. The $P_{C/D}$ in terms of overall system spectral efficiency can be written as:

$$P_{C/D}=R_K\left(P_{cod}+P_{dec}\right)$$

(28)

where, $P_{cod}$ is the power consumption during the coding of the signal and $P_{dec}$ is the power consumption during the decoding of the signal.

The power losses gets significantly reduced by modifying the circuit configuration because the signal of each RF chain is divided into $T_x/C_t$ equal power outputs without the need of using any combiners as can be seen in Figure 2. Therefore, the power loses at the transmitter side can be written as:

$$P_{Loss}=\frac{T_x{L}_{ST}{L}_{P.S}}{C_t}$$

(29)

where $L_{ST}{L}_{P.S}$ represents the static power losses of splitters and the phase shifters respectively. On the receiving end, the above-mentioned power losses are compensated by using the low-end amplifiers, and the power consumption during this process can be written as:

$$P_{COM}=K{P}_{LNA}$$

(30)

The modified version of RF mapping circuit configuration requires $\left(T_x+K\right)P_{PS}$ number of phase shifters. Therefore, total consumption of phase shifters $P_{TPS}$ at the transmitter and receiving end can be written as:

$$P_{TPS}=\left(T_x+K\right)P_{PS}$$

(31)

The overall consumed power (Equation (23)) by using the Equations (24) to (31) can be written as:

$$\begin{array}{l}{P}_{T/P}=\frac{KBp{\alpha }^2{\Vert D{P}_t\times A{P}_t\Vert }_F^2}{{\eta }_{AM}}\times \left(\frac{d_{Mx}^{\alpha +2}-d_{Mn}^{\alpha +2}}{\left(1+\frac{\alpha }{2}\right)\left(d_{Mx}^2-d_{Mn}^2\right)}\right)\\ +\left(C_t+C_r\right)P_{R/F}+P_{FiX}+R_K\left(P_{cod}+P_{dec}\right)+\frac{T_x{L}_{ST}{L}_{P.S}}{C_t}\\ +\left(T_x+K\right)P_{PS}+K{P}_{LNA}\end{array}$$

(32)

6. Analytical Comparison in Terms of Consumed Power

In this section, the comparison is presented and discussed in terms of consumed power between the conventional and the modified RF mapping circuit configurations. The power losses gets significantly reduced by modifying the circuit configuration because the signal of each RF chain is divided into $T_x/C_t$ equal power outputs without the need of using any combiners as can be seen in

Table 1. Analytical Comparison in terms of Consumed Power.

Circuit Configurations	Total Power Reduction Due to the Hybrid Architecture (${\mathit{P}}_{\mathit{L}\mathit{o}\mathit{s}\mathit{s}}$)	Total Power Required to Compensate the Loses by Using Low End Amplifiers (${\mathit{P}}_{\mathit{C}\mathit{O}\mathit{M}}$)	Total Power Consumption of Phase Shifters (${\mathit{P}}_{\mathit{T}\mathit{P}\mathit{S}}$)
Figure 1	$C_t{L}_c{T}_x{L}_{ST}{L}_{P.S}$	$K\left(C_r+1\right)P_{LNA}$	$\left(T_x{C}_t+K{C}_r\right)P_{PS}$
Figure 2	$\frac{T_x{L}_{ST}{L}_{P.S}}{C_t}$	$K{P}_{LNA}$	$\left(T_x+K\right)P_{PS}$

. Whereas, the conventional version of RF mapping divides the output of each RF chain into $T_x$ equal power outputs, which in turn results into the increment in the consumed power. Furthermore, the traditional configuration of RF mapping requires $\left(T_x{C}_t+K{C}_r\right)P_{PS}$ number of phase shifters as compared to the modified version of circuit configuration where $\left(T_x+K\right)P_{PS}$ number of phase shifters are required. Due to the significant reduction of phase shifters and combiners, the smaller the number of low end amplifiers is required to compensate the losses induced at the transmitter side.

7. Alternating Optimization Algorithm and the Computation of Optimal Parameters

The expression of the energy efficiency of mmWave Massive MIMO is derived, and the optimization problem for the maximization of energy efficiency is presented in this section. The energy efficiency can be written as the overall area throughput divided by all the transmitted and consumed power:

$$E.E=\frac{\mathrm{Area} \mathrm{Throughput}}{\mathrm{Overall} \mathrm{tranmitted} \mathrm{and} \mathrm{consumed} \mathrm{power}}$$

The overall area throughput by using the Equation (9) can be written as:

$$\begin{array}{l}\mathrm{Area} \mathrm{Throughput}=B\times R_K=BK\left(1-\frac{T_{sum}K}{C}\right)\\ \times {\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)\end{array}$$

(33)

The overall transmitted and consumed power is modelled in the previous section, and it can be written as:

$$\begin{array}{l}{P}_{T/P}=\frac{KBp{\alpha }^2{\Vert D{P}_t\times A{P}_t\Vert }_F^2}{{\eta }_{AM}}\times \left(\frac{d_{Mx}^{\alpha +2}-d_{Mn}^{\alpha +2}}{\left(1+\frac{\alpha }{2}\right)\left(d_{Mx}^2-d_{Mn}^2\right)}\right)+\left(C_t+C_r\right)P_{R/F}\\ +P_{FiX}+P_{C/D}+R_K\left(P_{cod}+P_{dec}\right)+\frac{T_x{L}_{ST}{L}_{P.S}}{C_t}+\left(T_x+K\right)P_{PS}+K{P}_{LNA}\end{array}$$

Therefore, the energy efficiency can be written as:

$$E.E=\frac{BK\left(1-\frac{T_{sum}K}{C}\right){\log }_2\left(1+\frac{p{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\right)}{\begin{array}{l}\frac{KBp{\alpha }^2{\Vert D{P}_t\times A{P}_t\Vert }_F^2}{{\eta }_{AM}}\times \left(\frac{d_{Mx}^{\alpha +2}-d_{Mn}^{\alpha +2}}{\left(1+\frac{\alpha }{2}\right)\left(d_{Mx}^2-d_{Mn}^2\right)}\right)+\left(C_t+C_r\right)P_{R/F}\\ +P_{FiX}+R_K\left(P_{cod}+P_{dec}\right)+\frac{T_x{L}_{ST}{L}_{P.S}}{C_t}+\left(T_x+K\right)P_{PS}+K{P}_{LNA}\end{array}}$$

(34)

The optimization problem of the energy efficiency maximization can be illustrated as:

$$\begin{array}{cc}\mathrm{Maximize}\hfill & \hfill E.E\hfill \\ \mathrm{Constraint} \mathrm{to}\hfill & \hfill T_x\in Z_{+},K\in Z_{+}\hfill \\ & \hfill T_x>K,p>0\hfill \\ & \hfill A{P}_t\in \mathbb{Q}\hfill \\ & \hfill {\Vert D{P}_t\times A{P}_t\Vert }_F^2\le P_{max}\hfill \end{array}$$

(35)

The number of transmitting and receiving antennas cannot be negative so that is why they have been set positive in the first two constraints of optimization problem. The last two constraints in the above optimization problem deals with the designing of the hybrid analogue and digital precoders, and they have been discussed in details in Section 4. Furthermore, the expression of energy efficiency is simplified by using the following substitutions:

$$\begin{array}{l}{r}_1=BK\left(1-\frac{T_{sum}K}{C}\right),{ r}_2=\frac{{\left|A{P}_{r,k}^H\times D{P}_{r,k}^H\times H_k\times A{P}_{t,k}\times D{P}_{t,k}\right|}^2}{{\Vert A{P}_{r,k}^H\times D{P}_{r,k}^H\Vert }_F^2}\\ r_3=\frac{KB{\alpha }^2{\Vert D{P}_t\times A{P}_t\Vert }_F^2}{{\eta }_{AM}}\times \left(\frac{d_{Mx}^{\alpha +2}-d_{Mn}^{\alpha +2}}{\left(1+\frac{\alpha }{2}\right)\left(d_{Mx}^2-d_{Mn}^2\right)}\right), \\ r_4=\left(C_t+C_r\right)P_{R/F}+P_{FiX}+\frac{T_x{L}_{ST}{L}_{P.S}}{C_t}\\ +\left(T_x+K\right)P_{PS}+K{P}_{LNA},r_5=\left(P_{cod}+P_{dec}\right)\end{array}$$

Following the above-mentioned substitutions, the Equation (35) can be rewritten as:

$$E.E\left(p\right)=\frac{r_1\times {\log }_2\left(1+p{r}_2\right)}{r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)}$$

As it can be seen in the above equation, the optimization of the energy efficiency is equivalent to the computation of transmitted power that results into the maximization of energy efficiency. Therefore, the optimization problem in Equation (35) can be written as:

$$\begin{array}{cc}\hfill p=& \arg \mathrm{maximize}\left(E.E\left(p\right)\right)\hfill \\ \hfill \mathrm{Constraint} \mathrm{to} & \hfill T_x\in Z_{+},K\in Z_{+}\hfill \\ & \hfill T_x>K,p>0\hfill \\ & \hfill A{P}_t\in \mathbb{Q}\hfill \\ & \hfill {\Vert D{P}_t\times A{P}_t\Vert }_F^2\le P_{max}\hfill \end{array}$$

(36)

The energy efficiency undergoes the quasi concave response with respect to $p$, and it is explained at the end of this section. Furthermore, the solution of the above optimization problem is obtained by searching for the optimal transmitted power where the corresponding energy efficiency comes out to be maximum:

$$d\frac{\left(E.E\left(p\right)\right)}{dp}=\frac{d}{dp}\left(\frac{r_1\times {\log }_2\left(1+p{r}_2\right)}{r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)}\right)$$

$$d\frac{\left(E.E\left(p\right)\right)}{dp}=\frac{d}{dp}\left(\frac{\begin{array}{l}\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)\frac{d}{dp}\left(r_1\times {\log }_2\left(1+p{r}_2\right)\right)\\ -\left(r_1\times {\log }_2\left(1+p{r}_2\right)\right)\\ \times \frac{d}{dp}\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)\end{array}}{{\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)}^2}\right)$$

$$d\frac{\left(E.E\left(p\right)\right)}{dp}=\frac{d}{dp}\left(\frac{\begin{array}{l}\frac{r_1{r}_2\times \left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)}{\left(1+p{r}_2\right)}\\ -\left(\frac{r_5{r}_2}{1+p{r}_2}+r_3\right)\left(r_1\times {\log }_2\left(1+p{r}_2\right)\right)\end{array}}{{\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)}^2}\right)$$

$$d\frac{\left(E.E\left(p\right)\right)}{dp}=\left(\frac{\frac{r_1}{1+p{r}_2}\left[\begin{array}{l}\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)r_2\\ -\left[\left(r_3\left(1+p{r}_2\right)+r_5{r}_2\right)\times \left({\log }_2\left(1+p{r}_2\right)\right)\right]\end{array}\right]}{{\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)}^2}\right)$$

$$d\frac{\left(E.E\left(p\right)\right)}{dp}=\left(\frac{\frac{r_1}{1+p{r}_2}\left[\begin{array}{l}{r}_4{r}_2+r_{3}p{r}_2+r_5{r}_2\times {\log }_2\left(1+p{r}_2\right)-p{r}_2{r}_3\left({\log }_2\left(1+p{r}_2\right)\right)\\ -r_3\left({\log }_2\left(1+p{r}_2\right)\right)-r_5{r}_2\times {\log }_2\left(1+p{r}_2\right)\end{array}\right]}{{\left(r_{3}p+r_4+r_5\times {\log }_2\left(1+p{r}_2\right)\right)}^2}\right)$$

(37)

Equate the Equation (37) to zero in order to compute the optimal transmitted power:

$$\frac{r_1}{1+p{r}_2}\left[\begin{array}{l}{r}_4{r}_2+r_{3}p{r}_2+r_5{r}_2\times {\log }_2\left(1+p{r}_2\right)-r_3\left({\log }_2\left(1+p{r}_2\right)\right)\\ -r_5{r}_2\times {\log }_2\left(1+p{r}_2\right)-p{r}_2{r}_3\left({\log }_2\left(1+p{r}_2\right)\right)\end{array}\right]=0$$

$$\frac{r_1}{1+p{r}_2}\left[r_4{r}_2+r_{3}p{r}_2+\left[r_5{r}_2-r_3-p{r}_2{r}_3-r_5{r}_2\right]{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\frac{r_1}{1+p{r}_2}\left[r_4{r}_2+r_{3}p{r}_2-\left[r_3+p{r}_2{r}_3\right]{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\left[\frac{r_4{r}_2+r_{3}p{r}_2}{1+p{r}_2}-r_3{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\left[\frac{r_{3}p{r}_2+r_4{r}_2}{1+p{r}_2}+r_3-r_3-r_3{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\left[\frac{r_4{r}_2+r_{3}p{r}_2-r_3\left(1+p{r}_2\right)}{1+p{r}_2}+r_3-r_3{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\left[\frac{r_4{r}_2+r_{3}p{r}_2-r_3-p{r}_2{r}_3}{1+p{r}_2}+r_2-r_3{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\left[\frac{r_4{r}_2-r_3}{1+p{r}_2}+r_3-r_3{\log }_2\left(1+p{r}_2\right)\right]=0$$

$$\left[\frac{r_4{r}_2-r_3}{1+p{r}_2}-r_3\left[{\log }_2\left(1+p{r}_2\right)-1\right]\right]=0$$

(38)

Let

$$\left[{\log }_2\left(1+p{r}_2\right)-1\right]=q_1$$

(39)

Taking exponential on both sides:

$$e^{\left[{\log }_2\left(1+p{r}_2\right)-1\right]}=e^{q_1}$$

$$p{r}_2-1=e^{q_1}e$$

(40)

By utilizing the Equation (39) and Equation (40), the Equation (38) can be written as:

$$\frac{r_4{r}_2-r_3}{e^{q_1}e}=r_3{q}_1$$

$$q_1{e}^{q_1}=\frac{r_4{r}_2-r_3}{r_{3}e}$$

(41)

As we know:

$$u=f^{-1}\left(u{e}^u\right)=\mathrm{W}\left(u{e}^u\right)$$

where $W(.)$ is the Lambert Omega function, so Equation (40) can be written as:

$$q_1=\mathrm{W}\left(\frac{r_4{r}_2-r_3}{r_{3}e}\right)$$

(42)

Substitute the value of $q_1$ from the Equation (42) to Equation (39):

$$\left[{\log }_2\left(1+p{r}_2\right)-1\right]=\mathrm{W}\left(\frac{r_4{r}_2-r_3}{r_{3}e}\right)$$

Taking exponential on both sides:

$$e^{\left[{\log }_2\left(1+p{r}_2\right)-1\right]}=e^{\mathrm{W}\left(\frac{r_4{r}_2-r_3}{r_{3}e}\right)}$$

$$e^{-1}\times \left(1+p{r}_2\right)=e^{\mathrm{W}\left(\frac{r_4{r}_2-r_3}{r_{3}e}\right)}$$

$$\left(1+p{r}_2\right)=e^{\mathrm{W}\left(\frac{r_4{r}_2-r_3}{r_{3}e}\right)-1}$$

Finally, the optimal value of transmitted power that results into the maximization of energy efficiency can be computed by using the following:

$$p=\frac{e^{\mathrm{W}\left(\frac{r_4{r}_2-r_3}{r_{3}e}\right)-1}-1}{r_2}$$

(43)

The response of $E.E\left(p\right)$ with respect to $p$ is first increased and then decreased when we equate Equation (37) to zero. Furthermore, the $E.E\left(p\right)$ is negative when $E.E\left(p\right)<\frac{r_1}{r_5}$ and positive when $p>\frac{-1}{r_2}$, so the $d^2\frac{\left(E.E\left(p\right)\right)}{d{p}^2}$ comes out to be less than zero. Therefore, the $E.E\left(p\right)$ follows the quasi concave response. In the proposed alternating optimization algorithm, the first step is to construct the matrix $A{P}_t$. The phase angles are estimated with the help of Equation (19), and the estimated phase angles are used to construct the entries of $A{P}_t$ with the help of Equation (20). The $D{P}_t$ is estimated by using the entries of $A{P}_t$ with the help of Equation (22). All the entries of $A{P}_t$ are updated by using the entries of $D{P}_t$. Once we get the hybrid design, the power consumed in the circuit of mmWave Massive MIMO is computed with the help of Equations (24) to (31). Once we get the consumed power, the optimal transmitted power where the energy efficiency of the system comes out to be maximum is computed with the help of Equation (43). Furthermore, the overall transmitted and the consumed power $P_{T/P}$ along with the area throughput is calculated with the help of Equation (32) and Equation (33), respectively. Finally, the optimal energy efficiency of the system is calculated by using the $P_{T/P}$ and area throughput with the help of Equation (34). The steps of the proposed algorithm (Algorithm 1) are summarized as follows:

Algorithm 1. Proposed Alternating Optimization Algorithm

Input $T_x$, $C_t$, $K$, $C_r$, $H$, $d_{Mx}$, $d_{Mn}$. Estimate the phase angles $\left[{\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_{{}_{\frac{T_x}{C_t}}},{\theta }_{{}_{2\frac{T_x}{C_t}}},{\theta }_{{}_{3\frac{T_x}{C_t}}},\dots ,{\theta }_{T_x}\right]$ with the help of Equation (19). Utilize the estimated phase angles to calculate the entries of the analogue precoder $A{P}_t$ with the help of Equation (18). Utilize the calculated entries of the $A{P}_t$ to construct $D{P}_t$ with the help of Equation (22). Update all the entries of the $A{P}_t$ by utilizing the $D{P}_t$ constructed in the previous step. Calculate the overall consumed power of the system with the help of Equations (24) to (31). Calculate the optimal transmitted power where the energy efficiency of the system comes out to be maximum with the help of Equation (43). Utilize the optimal transmitted power to calculate the $P_{T/P}$ and Area throughput with the help of Equation (32) and Equation (33) respectively. Finally, utilize the $P_{T/P}$, Area throughput to calculate the $E.E$ with the help of Equation (34).

8. Simulation Results

The simulation results are shown and discussed in this section. The simulation parameters used for the simulations are shown in

Table 2. Simulation Parameters.

Parameter	Value
$C$	1850
$N_{ray}$	10 rays
Azimuthal sector angle	${50}^{\circ }$ wide
Elevation sector angle	${22}^{\circ }$ wide
$d$	$\lambda /2$
Channel realization	5000
$B$	19 MHz
$P_{cod}$	0.8
$P_{dec}$	0.2
$P_m$	19 mW
$P_F$	14 mW
$L_{ST}$	30 mW
${\eta }_{AM}$	32%
$T_{sum}$	22 m
$P_o$	4 mW
$L_{P.S}$	0.45 dB
$P_{LNA}$	14 mW
$d_{Mn}$	30 m
$\alpha$	3.8

.

The Figure 3 shows the overall spectral efficiency with respect to different SNRs, where the spectral efficiencies are computed for the fully digital precoding circuits, conventionally used circuit configuration and the modified RF configuration, respectively. It can be seen that the modified RF configuration approaches closer to the conventionally used circuit configuration. Furthermore, Figure 4 shows the optimized area throughput computed by using the proposed alternative algorithm for different circuit configurations with respect to different maximum distances between the base station and end users.

As it can be seen in Figure 4, the proposed RF mapping approaches closer to the traditionally used circuit configuration in terms of throughput at different number of transceiver chains along with the transmitting antennas and users. The Figure 5 unveils the optimal transmitted power computed by using the proposed alternative algorithm for different circuit configurations, where the x axis shows the maximum distances between the base station and end users and the y axis shows the optimal transmitted power required to transmit the signals in log scale.

As it can be seen in Figure 5, that when the base station needs to transmit the signals at the distant users, then the transmit power required to transmit the signals also is increased. Furthermore, the RF modified circuit requires less power to transmit the signals as compared to the conventionally used circuit configuration. In addition, when the number of transceiver chains along with the number of transmitting antennas and end users increases, then the performance of the modified configuration also is further improved in terms of transmission power as can be seen in Figure 5.

Figure 6 unveils the optimal energy efficiency computed by using the proposed alternative algorithm for different circuit configurations by setting different number of maximum distances between the base station and end users, different number of transceiver chains, different number of transmitting and receiving chains respectively. The optimal energy efficiency of the system gets reduced when the base station needs to transmit the signal at the distant users because the system needs more transmission power in order to transmit the signal at the distant user as can be seen in Figure 5.

Due to the more transmission power and the reduction in the overall system throughput, the overall optimal energy efficiency of the system gets reduced. In addition, the proposed RF circuit configuration significantly improves the overall energy efficiency of the system as it can be seen in Figure 7. The energy efficiency of the proposed RF circuit configuration is compared with respect to the energy efficiency computed by using the feedback and the OMP algorithm for the conventional circuit configuration in Figure 7. The number of transmitting antennas, users and the transmitter and receiver chains are set to be 148, 42 and 11, respectively.

Figure 8 unveils the effects of the number of transceiver chains on the overall optimal energy efficiency of the system, where the x axis is showing the number of transceiver chains deployed in the system, and the y axis is showing the optimal energy efficiency of the system. The optimal energy efficiencies are computed by setting the different number of distances and the number of transmitting antennas and end users. As it can be seen in Figure 8, the optimal energy efficiency of the proposed RF modified circuit performs better when the number of deployed transceiver chains is increased. Furthermore, the optimal energy efficiency of the conventional circuit configuration gets reduced when the number of deployed transceiver chains is increased as can be seen in Figure 8.

9. Conclusions

This paper mainly focused on enhancing the energy efficiency of mmWave Massive MIMO by modifying the RF circuit configurations. The RF circuit configuration was modified by connecting each of the RF chain to some of the transmitting antennas of mmWave Massive MIMO. As a result, the modified RF circuit configuration required a significantly smaller number of the phase shifters and the corresponding low-end amplifiers as can be seen in Table 1. Furthermore, the analogue precoders and decoders were also designed by taking all the circuit and system configurations into the account for the modified RF configuration. Contrary to the existing researches, the realistic modelling of the circuit power consumptions was presented and used in the simulations. In addition, the proposed alternating optimization algorithm worked well to compute the optimal system papers, and to enhance the performance of the system in terms of energy efficient approach. Contrary to the existing optimizing algorithms, where the researches computed the optimal solution of the hybrid architectures by maximizing the spectral efficiency of the system, the proposed framework provided deeper insights of the optimal system parameters in terms of throughput, consumed power and the corresponding energy efficiency. The simulation results showed that the modified RF circuit configuration significantly reduced the power consumption of the system, with a little compromise on the spectral gain. Due to the significant reduction in the system power and energy consumptions, the modified RF circuit configuration of the mmWave Massive MIMO can be a fascinating option for the future of wireless systems.