1 Introduction

Currently, the mobile broadband which is operating in the microwave frequency spectrum at sub 6GHz range is unable to meet the ever growing consumer requirements of data rate. Mobile communication in millimeter wave (mmWave) domain not only enables to meet the ever increasing demands of capacity and bandwidth [1, 2] but also fulfills the performance requirements of fifth generation (5G) communication systems [3, 4]. However, the consequences of operating at higher frequencies arise in terms of increased path loss [5], which can be compensated by adopting the massive MIMO architecture [6, 7]. Owing to reduced wavelength associated with high frequencies, the deployment of large scale antennas within a compact space is possible in order to implement the massive MIMO architecture. The conventional multi antenna systems are equipped with a fully digital architecture where a dedicated RF chain is connected to each antenna element [6]. However, this approach is not feasible for massive MIMO systems due to higher power constraints and reduced energy efficiency. An alternative approach is to use the analog architecture, but the analog structure cannot provide the flexibility of digital processing and does not support multi stream transmission.

The benefits of mmWave massive MIMO systems is achieved by employing hybrid architectures consisting of both the analog and digital architectures having less RF chains with respect to antennas. In the existing literature [8], two configurations of hybrid architectures are considered: (a) a fully connected architecture where an individual RF chain is linked with all the antenna elements present in the system and (b) a sub connected or a partially connected architecture where an individual RF chain is linked with limited number of antenna elements. It is shown in [9] that in terms of energy efficiency the sub connected or partially connected architecture outperforms the fully connected architecture. Most of the work in existing literature is focused on the precoder design for hybrid architecture. In [10] authors have proposed a spectral efficient orthogonal matching pursuit (OMP) based precoder design for the fully connected hybrid structure. However, the design does not account for the energy or power consumption. Authors in [11] have designed precoders separately for partially and fully connected architectures and discussed the energy efficiency of both architectures. Inspired by the evolutionary algorithms, in [12], a particle swarm optimization (PSO) based precoder design and in [13], an artificial bee colony (ABC) based precoder design for partially connected structure is presented. The analog and digital precoders in [14] were constructed using the signal to leakage noise ratio (SLNR) and zero forcing (ZF) methods, respectively. [15] applied successive interference cancellation (SIC) based scheme to design the hybrid precoder for a configuration where a limited number of antenna elements can be linked to each RF chain. Precoding using singular value decomposition (SVD) for hybrid structure is presented in [16]. However, in most of the available literature, the focus of researchers is on the spectrally efficient precoder design without emphasizing on the efficient resource allocation.

By employing massive MIMO architecture, spatial multiplexing gains can be achieved due to its higher dimensions. However, beyond a certain limit the multiplexing gain do not show any improvement which results in reduction of energy efficiency. Authors in [17], have shown that for the massive MIMO system, \(90\%\) of ergodic rate can be achieved using a subset of antennas. Recently, researchers have tilted towards efficient resource management of a massive MIMO architecture in order to maximize the energy efficiency. In [18], for a partially connected massive MIMO structure, authors have followed a hybrid heuristic approach to jointly perform the transmit antenna selection and after-wards applied SIC based algorithm for precoding to maximize the energy efficiency. Authors in [19] have performed antenna selection using the estimation of distribution (EDA) algorithm and precoding using SIC algorithm for improvement in energy efficiency. Naeem and Lee [20] and [21] have applied binary particle swarm optimization (BPSO) algorithm for the joint transmitter antenna and user selection for digital architecture. For a multi-cell, multi-user massive MIMO system, employing the adaptive markov chain monte carlo procedure, authors in [22] have carried out joint transmitter antenna selection and receiver selection with scheduling. Although above mentioned techniques helped in improving the energy efficiency of massive MIMO system, but the gain is not significant.

The power consumption and energy efficiency of a hybrid massive MIMO system depends on the number of RF chains. Hence, the optimal utilization of RF chains can contribute greatly towards improving the energy efficiency of massive MIMO systems. Kaushik et al. [23] provides a framework for RF chain selection with OMP precoding for a fully connected architecture but the said architecture is energy inefficient. Recently [24, 25] have adopted dinkelbach method-based framework to optimize the number of active transmit antennas. The Dinkelbach method is suitable for continuous fractional programming problem only and not suitable for combinatorial nature of the problem. Motivated by the efficient hybrid precoding schemes and effective utilization of RF chains, in this paper, we present configurable RF chains for a sub connected architecture, where the RF chains can be selected to optimize energy efficiency together with precoding. For a sub connected hybrid structure, first the RF chain selection is performed using low complex BPSO based heuristic algorithm and afterward precoding is performed using SIC based algorithm. Although the exhaustive search algorithm (ESA) can optimally select the best subset of available resources, but the computational complexity of ESA makes them inappropriate for practical systems. The major contributions of the research is as follows:

  • For a partially connected hybrid massive MIMO architecture, we investigate an adaptive approach for configuring the active RF chains out of total available RF chains based on current channel conditions.

  • For energy efficiency maximization of a partially connected hybrid massive MIMO system, we introduce a low complex hybrid heuristic approach for RF chain selection and precoding, where heuristic algorithm i.e., Binary Particle Swarm Optimization or Estimation of Distribution Algorithm is used for RF chain selection and SIC is used for precoding.

  • It is shown that hybrid heuristic algorithm outclasses the ESA algorithm in terms of computational complexity and achieves near optimal performance in terms of achievable rate.

  • For high SNR regime, beyond a certain number of optimal RF chains, the improvement in spectral efficiency is negligible. Hence, activating a large number of RF chains is not feasible in terms of energy efficiency.

  • On the contrary for low SNR regime, only single RF chain achieves the optimum performance in terms of spectral efficiency and energy efficiency. Hence, the hybrid precoding architecture can be configured as analog precoder while operating in low SNR regimes for optimal performance.

The paper is organized as follows. The system architecture adopted for the paper is explained in Sect. 2. Section 3 explains the proposed RF chain selection algorithms. Section 4 discusses the precoding using the SIC based approach. The results and simulations are presented in Sect. 5, whereas the conclusions are drawn in Sect. 6.

Notation: \({\mathbf {a}}\) and \({\mathbf {A}}\) boldface letters denote vectors and matrices, respectively; the transpose, conjugate transpose, inversion, determinant of a matrix and the Frobenius norm of a matrix are denoted as \(\left( .\right) ^{T}\), \(\left( .\right) ^{H}\), \(\left( .\right) ^{-1}\), \(\left| .\right| \), and \(\left\| .\right\| _{F}\) respectively.

Fig. 1
figure 1

System architecture for partially connected hybrid massive MIMO system with RF chain selection and precoding

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2 System architecture

We have considered a massive MIMO architecture with a sub connected structure. The aim is to optimize the energy efficiency of the system by selecting the RF chains that experiences good channel conditions out of the total available RF chains. Figure 1, represents the base station equipped with \(N_{T}\) transmit antennas with \(N_{S}\) data streams serving the data to \(N_{RF}\) total available RF chains. Each RF chain in linked with limited number of antenna elements (\(M_{T}\) = \(N_{T}/N_{RF}\)). The base station is serving a single user having \(N_{R}\) antennas with fully digital architecture. The channel state information (CSI) is represented as \({\mathbf {H}} \in C^{ N_{R} \times N_{T} }\) and is considered available for simulations. Using the available CSI, \(N_{rf}\) RF chains experiencing the best channel characteristics are selected out of total \(N_{RF}\) RF chains. Hence the system will be reduced to \(N_{t}\) = \(M_{T} \times N_{rf}\) transmit antennas connected to selected RF chains.

The mmWave propagation path has limited scatterers, so we cannot model the channel for mmWave frequencies using the model estimated for below 6GHz frequencies. For our current work the geometric Saleh-Valenzuela model [26] is adopted to represent the mmWave channel as shown in Eq. (1).

$$\begin{aligned} {\mathbf {H}} = \sqrt{\left( \dfrac{N_{T} N_{R}}{\epsilon L}\right) } \sum _{l=0}^{L}\eta _{l}\mathbf{a }_{R}(\mu _{l})\mathbf{a }^{H}_{T}(\theta _{l}) \end{aligned}$$
(1)

where the mmWave channel between the transmitter and receiver is represented as \({\mathbf {H}}\), the number of effective mmWave channel paths is represented as L, path loss between transmitter and receiver is represented as \(\epsilon \), the path gain associated with the lth path is \(\eta _{l}\), receiver and the transmitter spatial signatures are represented as \(\mathbf{a }_{R}\) and \(\mathbf{a }_{T}\) respectively, whereas \(\mu _{l}\) and \(\theta _{l}\) represents the angle of arrival (AoA) and angle of departure (AoD) at the receiver and the transmitter respectively. For our present work, we have adopted a uniform linear array (ULA) structure, however, the algorithms work equally well for other array structures.

The receiver and transmitter spatial signatures with \(\lambda \) being the wavelength and d being the antenna spacing for a ULA structure are expressed as Eqs. (2) and (3) respectively.

$$\begin{aligned} \mathbf{a }_{R}(\mu )= & {} \frac{1}{\sqrt{N_{R}}} \left[ 1,\exp ^{j\frac{2\pi }{\lambda }d\sin (\mu )},......,\exp ^{j(N_{R}-1)\frac{2\pi }{\lambda }d\sin (\mu )}\right] ^T\nonumber \\ \end{aligned}$$
(2)
$$\begin{aligned} \mathbf{a }_{T}(\theta )= & {} \frac{1}{\sqrt{N_{T}}} \left[ 1,\exp ^{j\frac{2\pi }{\lambda }d\sin (\theta )},......,\exp ^{j(N_{T}-1)\frac{2\pi }{\lambda }d\sin (\theta )}\right] ^T \nonumber \\ \end{aligned}$$
(3)

For the conventional system without any RF chain selection and precoding, the spectral efficiency is expressed as Eq. (4)

$$\begin{aligned} R = \log \left| \left( {\mathbf {I}}_{N_{R}} + \frac{\rho }{N_{s}} \mathbf {H (H)^\textit{H}}\right) \right| \end{aligned}$$
(4)

Let \(\varvec{\Phi }\) represents the set of all combinations of selecting the required RF chains and \(N_{RF} \atopwithdelims ()N_{rf}\) represents the number of possible combinations for RF chain selection. A specific selected set of RF chains is represented as \(\varvec{\phi }\). The reduced channel as a result of RF chains selection is represented as \(\mathbf {H}^{\phi } \in C^{ N_{R} \times N_{t}}\), where \(N_{t}\) represents the number of antennas connected to selected RF chains only. The channel spectral efficiency with reduced RF chains and transmit antennas is expressed as Eq. (5)

$$\begin{aligned} R = \log \left| \left( {\mathbf {I}}_{N_{R}} + \frac{\rho }{N_{s}} \mathbf {H}^\phi (\mathbf {H}^\phi )^\textit{H}\right) \right| \end{aligned}$$
(5)

where \(\rho \) is the average signal to noise ratio (SNR) and \({\mathbf {I}}_{N_{R}}\) is a \(N_{R} \times N_{R}\) identity matrix. The selection procedure can be modeled as an optimization problem as shown in Eq. (6)

$$\begin{aligned} \underset{\phi \in \varvec{\Phi }}{\arg \max }\ R = \underset{\phi \in \varvec{\Phi }}{\arg \max }\ \log \left| \left( {\mathbf {I}}_{N_{R}} + \frac{\rho }{N_{s}} \mathbf {H}^\phi (\mathbf {H}^\phi )^\textit{H}\right) \right| \end{aligned}$$
(6)

After the antenna selection, precoding at the transmitter for the selected antennas is performed. For a mmW architecture an appropriate precoding scheme is considered essential in order to direct the transmitted signal in desired direction and to overcome path-loss. In this paper, we have employed SIC based technique for precoding after antenna selection in order to increase the system efficiency. The complete system flow chart is shown in Fig. 2.

3 RF chain selection algorithms

Fig. 2
figure 2

Flow chart for proposed joint RF chain selection and precoding algorithm for partially connected hybrid massive MIMO architecture

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This section presents the algorithms adopted for RF chain selection. The algorithm that can achieve the optimal performance is termed as exhaustive search algorithm (ESA). In order to select the optimal RF chains for energy efficiency (EE) maximization, ESA for every channel realization (current channel conditions) searches for the best RF chains out to total available RF chains that gives the optimal spectral and energy efficiency. Searching over a large number of RF chains gets computationally inefficient. In our work, we have mitigated the need to search over all possible combinations and finding the optimal solution. Hence our work presents an energy efficient solution with low computational complexity. The ESA is summarized in Algorithm 1. Contrary to ESA, random search algorithm (RSA) randomly selects the RF chains. However, due to a high degree of randomness, such selection is not efficient.

figure f

3.1 BPSO algorithm based proposed solution

Evolutionary algorithms provides several options to address optimization problems [27]. One of the evolutionary algorithms namely particle swarm optimization (PSO) has proven efficient for optimizing the selection problems [28, 29]. PSO is an acclaimed meta-heuristic optimization method that drives its inspiration from investigating actions of individuals in a swarm like birds in a flock. We have utilized the discretized PSO known as binary particle swarm optimization (BPSO) [30] and applied for RF chain selection. Implementation of BPSO starts with modeling the particle position in accordance with the problem statement. The particle position represents a possible solution. The position and velocity of every particle is randomly initialized. Next the position and velocity of each particle is updated using the governing equations of the algorithm. The fitness of every solution is determined by an objective function. Afterwards, the personal best position of each particle global best position of the swarm are updated.

The BPSO algorithm is characterized by following notations throughout the article (\(I_{s}\), \(O_{F}\), \(N_{pop}\), \(N_{iter}\), \(D_{M}\), \(\chi ^{i}\), \(P_{B}^{i}\), \(G_{B}^{i}\), \(U^{i}\)). The algorithm is summarized in Algorithm. 2.

  • The solution space of the algorithm is represented as \(I_{s}\).

  • The objective function is denoted as \(O_{F}\). In our case the objective function is represented by sum rate equation. The objective of BPSO algorithm is to maximize the objective function.

  • \(N_{pop}\) represents the population size of the swarm.

  • \(N_{iter}\) represents the maximum number of generations of the BPSO algorithm.

  • \(D_{N_{RF}}\) represents the particle position dimension within the swarm

  • At the ith iteration, the particle position within the swarm is represented as \(\chi ^{i}\) = \([X_{1}^{i}, X_{2}^{i}, ....., X_{N_{pop}}^{i}]\), where \(X_{k}^{i}\), \(k = 1, 2, ...., N_{pop}\) represents the position of particles within the swarm indexed by k. The vector \(X_{k}^{i}\) contains \((x_{k,1}^{i}, x_{k,2}^{i}, ....., x_{k,D_{M}}^{i})\), where \(D_{N_{RF}}\) represents the dimensions of particle position within the swarm, also the elements of vector \(X_{k}^{i}\) are either ’0’ or ’1’.

  • The fitness function is evaluated for every vector \(X_{k}^{i}\) and \(X_{bk}^{i}\) = \(\arg \max X_{k}^{i}\) denotes the best position the kth particle has experienced up to ith iteration. For all the \(N_{pop}\) particles within the swarm their respective best positions is denoted as \(P_{B}^{i}\) = \([X_{b1}^{i}, X_{b2}^{i}, ....., X_{bN_{pop}}^{i}]\). The global best position ever experienced by any particle within the swarm is denoted as \(G_{B}^{i}\)

  • The velocity for \(N_{pop}\) particles within the swarm is represented as \(U^{i}\) = \([U_{1}^{i}, U_{2}^{i}, ....., U_{N_{pop}}^{i}]\).

figure g

4 Precoding algorithm

For communications in mmWave domain, precoding is considered essential to overcome the signal attenuation. Existing literature [8,9,10,11,12,13,14,15,16] proposes many schemes for precoding and beamforming keeping in view both partially and fully connected architectures. Since our work focuses on a energy efficient solution for a joint RF chain selection and precoding problem, hence we have opted for the sub connected architecture and adopted the SIC based precoding approach proposed in [9] which is proved to be energy efficient with less computational complexity.

The precoding is performed after the RF chain selection only for selected RF chains. Hence our system is reduced to \(N_{rf}\) active RF chains out of total \(N_{RF}\) available RF chains. For the partially connected architecture as shown in Fig. 1, the digital precoder is represented as \({\mathbf {D}}\) = diag [\(d_{1}\), \(d_{2}\), . . . , \(d_{N}\)] having dimensions of \(N_{rf} \times N_{s}\), where \(d_{n}\) \(\in \) R. Only those data streams will be activated which are corresponding to selected RF chains and are represented as \(N_{s}\) and is assumed to be equal to \(N_{rf}\) (for simplicity \(N_{s}\) = \(N_{rf}\) = N). Analog precoding is performed on data streams passing through \(N_{rf}\) RF chains. Each selected data stream is precoded by an \(M_{T} \times 1\) RF precoder \({\mathbf {a}}_{n}\) (n = 1, ...., N) \(\in \) \({\mathcal {C}}^{M_{T}\times 1}\) realized by phase shifters. After the hybrid (digital with analog) precoding, data is transmitted through antennas connected to selected RF chains. Thus, the signal vector \({\mathbf {y}}\) having dimensions \(M_{R} \times 1\) received at the user is expressed as Eq. (10)

$$\begin{aligned} {\mathbf {y}} = \sqrt{P_{av}} \mathbf {H}^\phi \mathbf {ADs} + {\mathbf {z}} \end{aligned}$$
(10)

where the average received power is denoted as \(P_{av}\), the transmitted signal vector is denoted as \({\mathbf {s}}\). \(\mathbf {F = AD}\) is the hybrid precoding matrix, comprising of digital precoding matrix for power allocation \({\mathbf {D}}\) and RF precoding matrix \({\mathbf {A}}\) = [\({\mathbf {a}}_{1}\), \({\mathbf {a}}_{2}\), . . . , \({\mathbf {a}}_{N}\)], which satisfies \(\left\| F\right\| ^2_{F} \le N_{s}\) which is the power constraint for transmitter. \({\mathbf {z}}\) = \([z_{1}, z_{2}, . . . , z_{N}]^T\) is Gaussian noise which is complex i.i.d in nature, \({{\mathcal {C}}}{{\mathcal {N}}}(0,\sigma ^2)\). Lastly, \(\mathbf {H}^\phi \) is the channel matrix between transmitter antennas connected to selected RF chains and receiver. The spectral efficiency of system is given as Eq. (11).

$$\begin{aligned} R = \log \left| \left( {\mathbf {I}}_{N_{R}} + \frac{\rho }{N_{s}} {\mathbf {H}}^\phi {\mathbf {F}}{\mathbf {F}}^\textit{H} ({\mathbf {H}}^\phi )^\textit{H}\right) \right| \end{aligned}$$
(11)

Adopting the SIC algorithm for each RF chain, the maximum achievable rate R is decomposed. The hybrid precoding matrix \({\mathbf {F}}\) is decomposed for each RF chain as \({\mathbf {F}} = [{\mathbf {f}}_{1}, {\mathbf {f}}_{2}, ..., {\mathbf {f}}_{n}]\), where \({\mathbf {f}}_{n}\) is the nth column of \({\mathbf {F}}\), hence the total decomposed achievable rate is expressed as Eq. (12)

$$\begin{aligned} R = \sum _{n=1}^{N} \log \left( 1 + \frac{\rho }{N_{s}}{\mathbf {f}}_{n}^\textit{H} ({\mathbf {H}}^\phi )^\textit{H} {\mathbf {T}}_{n-1}^{-1} {\mathbf {H}}^\phi {\mathbf {f}}_{n}\right) \end{aligned}$$
(12)

where \({\mathbf {T}}_{n} = {\mathbf {I}}_{n} + \frac{\rho }{N_{s}} {\mathbf {H}}^\phi {\mathbf {F}}_{n}{\mathbf {F}}_{n}^\textit{H} ({\mathbf {H}}^\phi )^\textit{H}\) and \({\mathbf {T}}_{0} = {\mathbf {I}}_{N}\). Hence the total capacity is decomposed into sub-capacity for selected RF chains. After-wards SIC algorithm optimizes the sub-capacity of first RF chain and updates the matrix \({\mathbf {T}}_{1}\), similarly the sub-capacity of second RF chain is optimized and same procedure is carried out for all selected RF chains. The precoder for the nth RF chain \({\mathbf {f}}_{n}^{opt}\) to maximize the spectral efficiency is expressed as Eq. (13)

$$\begin{aligned} {\mathbf {f}}_{n}^{opt} = \log \left( 1 + \frac{\rho }{N_{s}}{\mathbf {f}}_{n}^\textit{H} {\mathbf {G}}_{n-1} {\mathbf {f}}_{n}\right) \end{aligned}$$
(13)

Here \({\mathbf {G}}_{n-1} = ({\mathbf {H}}^\phi )^\textit{H} {\mathbf {T}}_{n-1}^{-1} {\mathbf {H}}^\phi \). It is worth mentioning that the nth precoding vector \({\mathbf {f}}_{n}\) has only got \(M_{T}\) non zero elements from \((M_{T}(n - 1) + 1)\) th to \((M_{T}n)\) th. Matrix \({\mathbf {G}}_{n-1}\) having dimensions \(M_{T} \times M_{T}\) is obtained using \({\mathbf {G}}_{n-1} = {\mathbf {R}} ({\mathbf {H}}^\phi )^\textit{H} {\mathbf {T}}_{n-1}^{-1} {\mathbf {H}}^\phi {\mathbf {R}}\). The selection matrix \({\mathbf {R}}\) is used to select corresponding \((M_{T}(n - 1) + 1)\)th to \((M_{T}n)\)th rows and columns for nth sub array. SIC precoding algorithm for partially connected structure is summarized in Algorithm. 3.

figure h

5 Simulations and results

This section presents the performance results of the proposed algorithm compared with the exhaustive search and random search algorithms. We also identify the SNR regimes where the hybrid precoding or analog precoding can be adopted. The analog precoding solution can be implemented using the single RF chain whereas the hybrid precoding will be implemented using the selected RF chains \(N_{rf}\) which maximizes the energy efficiency. The channel realizations are generated using [26], we have generated 10,000 realizations for our simulations. The transmit antennas are set as \(N_{T} = 1024\) and 512, receive antennas are set as \(N_{R} = 16\). The system is equipped with \(N_{RF} = 16\) RF chains and the selection of RF chains is performed on the basis of energy efficiency maximization. Number of paths for the simulations is kept as \(L = 10\).

5.1 Energy efficiency maximization and selection of optimal number of rf chains \(N_{rf}\)

In this subsection we will discuss the proposed approach formulated to obtain the optimal number of RF chains required for energy efficiency maximization. Also we will discuss the SNR regimes where the hybrid or analog precoding can be adopted. Adopting the energy consumption model as discussed in [11], the energy efficiency \(\eta \) is defined as Eq. (14)

$$\begin{aligned} \eta = \frac{R}{P_{total}} = \frac{R}{P_{t} + N_{RF}P_{RF} + N_{PS}P_{PS}} \end{aligned}$$
(14)

where \(P_{total} = P_{t} + N_{RF}P_{RF} + N_{PS}P_{PS}\) represents total power consumption of the hybrid sub connected structure. \(P_{t}\), \(P_{RF}\), \(P_{PS}\) are the transmitted power, RF chain power consumption and phase shifter power consumption respectively. \(N_{RF}\) and \(N_{PS}\) are number of RF chains and phase shifters respectively. Keeping in view the small cell mmWave communication scenario, we have considered \(P_{RF}\) as 250mW, \(P_{PS}\) as 1mW and \(P_{t}\) as 1W [11]. Using the above equation we will determine the optimal number of RF chains which will maximize the system energy efficiency.

Fig. 3
figure 3

Energy efficiency analysis with variable RF chains at low SNR for massive MIMO architecture with \(N_{T} = 1024\)

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Fig. 4
figure 4

Energy efficiency analysis with variable RF chains at high SNR for massive MIMO architecture with \(N_{T} = 1024\)

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Figures 3 and 4 shows the results for low and high SNR respectively. The analysis are performed for \(N_{T} = 1024\). It can be seen that in low SNR regime the energy efficiency shows the decreasing trend as the number of RF chains are increased. Hence for low SNR regime, we can restrict our system to adopt analog only precoding for energy efficiency maximization.

For high SNR regime we can see that the energy efficiency increases till a certain point and start decreasing afterwards. Hence we can adopt hybrid precoding when operation at high SNR regime. Also, we can obtain the optimal number of RF chains required for energy efficiency maximization. From the simulation results it is evident that \(N_{rf}\) can be kept as 6 for optimal results at high SNR.

5.2 Convergence and accuracy of BPSO algorithm

In this section, we discuss the convergence and accuracy of BPSO based scheme opted for RF chain selection. The tuning parameters required for BPSO algorithm are number of iterations \(N_{iter}\) and population size \(N_{pop}\). These parameters can be selected based on required efficiency and computational complexity. For simulations, we have kept \(N_{rf}\) as 6 to achieve maximum energy efficiency. Analysis are performed for \(SNR = 0\) dB. It can be seen in Fig. 5 that the algorithm is converging in around 40 iterations. The performance with \(N_{pop}\) = 30 is higher than \(N_{pop} = 10\). The \(N_{pop}\) and \(N_{iter}\) parameters can be selected keeping in view the computational complexity.

Fig. 5
figure 5

Convergence analysis of proposed BPSO based selection algorithm for \(SNR = 0 dB\)

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5.3 Spectral efficiency analysis

The spectral efficiency analysis of proposed algorithm is performed with ESA, RSA and estimation of distribution algorithm (EDA). EDA follows a Bayesian learning approach, where a new population is generated on the basis of probability distributions experienced from previous iterations. Figure 6 shows that the proposed algorithm approaches ESA in terms of spectral efficiency. However, ESA is not suitable for implementation due to high computational complexity. BPSO tuning parameters are set as \(N_{iter} = 40\) and \(N_{pop} = 30\). Transmit antennas are set as \(N_{T} = 1024\) and receive antennas are set as \(N_{R} = 16\). Available RF chains are \(N_{RF} = 16\) and selected RF chains are \(N_{rf} = 6\). The analysis of BPSO with EDA algorithm is shown in Fig. 7. It is evident form the analysis that BPSO not only outperforms EDA in terms of spectral efficiency, but also BPSO converges in lower number of iterations as compared to EDA algorithm.

Fig. 6
figure 6

Spectral efficiency analysis of joint RF chain selection and precoding for partially connected hybrid massive MIMO architecture

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Fig. 7
figure 7

Spectral efficiency analysis of selection algorithms (BPSO and EDA) for partially connected hybrid massive MIMO architecture

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5.4 Complexity analysis

In this section, we have presented the computational complexity analysis of the proposed algorithm with that of the exhaustive search in terms of the number of complex multiplications and additions. Additionally, the comparison of SIC based precoding scheme with OMP based precoding scheme is presented. Generally \(O(n^{3})\) is the computational complexity of a determinant and for a matrix of dimensions \(n \times n\), the total complex operations require to compute each determinant is \((1/3) \times n^{3}\). First we determine the computational complexity of selection algorithms. To evaluate spectral efficiency equation \(({\mathbf {I}}_{N_{R}} + \frac{\rho }{N_{s}} \mathbf {H}^\phi (\mathbf {H}^\phi )^\textit{H})\) approximately \((N_{R}^{2}N_{t})\) complex operations are required. The term det \(({\mathbf {I}}_{N_{R}} + \frac{\rho }{N_{s}} \mathbf {H}^\phi (\mathbf {H}^\phi )^\textit{H})\) needs \((N_{R}^{3}/3)+(N_{R}^{2}N_{t})\) determinants [20]. The ESA computes \(N_{T} \atopwithdelims ()N_{t}\) complex determinants and the computational complexity comes out to be \(N_{T} \atopwithdelims ()N_{t} \times (N_{R}^{3}/3)+(N_{R}^{2}N_{t})\). Since the massive MIMO systems are designed with large antenna arrays, the computational complexity experiences exponential growth. So comparing with ESA, the proposed algorithm only require to compute \(N_{pop} \times N_{Iter}\) determinants resulting in only \(N_{pop} \times N_{iter}\) \(\times \) \((N_{R}^{3}/3)+(N_{R}^{2}N_{t})\) complex operations. Secondly we have compared the computational complexity of OMP and SIC based precoding. The computational complexity of SIC and OMP based schemes is \(O(2M_{T}N_{s}(N^{2}_{t} (K + N_{s}M_{T})))\) and \(O(2M^{3}_{T}(M^{4}_{T}N_{t} + M^{2}_{T}L^{2} + M^{2}_{T}N_{t}^{2}L))\), respectively [31]. Hence the SIC algorithm enjoys advantage over OMP algorithm in terms of computational complexity.

As a result for a partially connected Massive MIMO system, the combination of BPSO for RF chain election and SIC for precoding gives optimal results in-terms of computational complexity and efficient in-terms of spectral and energy efficiency.

6 Conclusions

This paper focuses on the energy efficiency maximization of a partially connected mmWave massive MIMO system by employing the joint RF chain selection and precoding frame work. The power consumption and energy efficiency of a hybrid massive MIMO system depends on the number of active RF chains as RF chains constitute the most power consuming element part of the architecture. Hence, the optimal utilization of RF chains contributes greatly toward energy efficiency maximization. We have adopted BPSO algorithm for RF chain selection and performed precoding using SIC algorithm. We have identified the SNR regimes where we can adopt either analog precoding approach or hybrid precoding. We have shown that for SNR regime of -5dB or below, the analog precoding gives the best results in terms of energy efficiency. Moreover, the computational complexity of the proposed algorithm is optimal when compared with ESA. The computational complexity of ESA is of the order of \(N_{T} \atopwithdelims ()N_{t}\), whereas, our proposed algorithm only requires \(N_{pop} \times N_{Iter}\) computations. Hence, for massive MIMO systems where \(N_{T}\) is huge, ESA is not suitable for practical implementation. In future we will extend the work for multi user massive MIMO systems coupled with both precoding and beamforming.