Radio-frequency chain selection for energy and spectral efficiency maximization in hybrid beamforming under hardware imperfections

The next-generation wireless communications require reduced energy consumption, increased data rates and better signal coverage. The millimetre-wave frequency spectrum above 30 GHz can help fulfil the performance requirements of the next-generation mobile broadband systems. Multiple-input multiple-output technology can provide performance gains to help mitigate the increased path loss experienced at millimetre-wave frequencies compared with microwave bands. Emerging hybrid beamforming architectures can reduce the energy consumption and hardware complexity with the use of fewer radio-frequency (RF) chains. Energy efficiency is identified as a key fifth-generation metric and will have a major impact on the hybrid beamforming system design. In terms of transceiver power consumption, deactivating parts of the beamformer structure to reduce power typically leads to significant loss of spectral efficiency. Our aim is to achieve the highest energy efficiency for the millimetre-wave communications system while mitigating the resulting loss in spectral efficiency. To achieve this, we propose an optimal selection framework which activates specific RF chains that amplify the digitally beamformed signals with the analogue beamforming network. Practical precoding is considered by including the effects of user interference, noise and hardware impairments in the system modelling.

The next-generation wireless communications require reduced energy consumption, increased data rates and better signal coverage. The millimetre-wave frequency spectrum above 30 GHz can help fulfil the performance requirements of the next-generation mobile broadband systems. Multiple-input multipleoutput technology can provide performance gains to help mitigate the increased path loss experienced at millimetre-wave frequencies compared with microwave bands. Emerging hybrid beamforming architectures can reduce the energy consumption and hardware complexity with the use of fewer radiofrequency (RF) chains. Energy efficiency is identified as a key fifth-generation metric and will have a major impact on the hybrid beamforming system design. In terms of transceiver power consumption, deactivating parts of the beamformer structure to reduce power typically leads to significant loss of spectral efficiency. Our aim is to achieve the highest energy efficiency for the millimetre-wave communications system while mitigating the resulting loss in spectral efficiency. To achieve this, we propose an optimal selection framework which activates specific RF chains that amplify the digitally beamformed signals 2020 The Author(s) Published by the Royal Society. All rights reserved. 1

. Introduction
There is currently a strong requirement to identify the technical needs and possible solutions that will transform the wireless connectivity ecosystem for a better connected society in the future. Fifth-generation (5G) wireless standards will start to address the consumer demands and performance enhancements for mobile communication in the next 5 years [1]. Looking forward, the Cisco data traffic report indicates that in 2023 video applications will generate up to 82% of the total mobile data traffic, with up to 29.3 billion networked devices [2]. The Ericsson mobility report [3] also forecasts that there will be 2.6 billion mobile connections by the end of 2025 and at least 45% of the world's population should be able to access 5G services in the same year. The nextgeneration services are expected to be commercially implemented on a large scale in the next few years, e.g. in North America and Northeast Asia significant 5G subscriptions are expected to grow rapidly [3]. Future wireless systems require high data rates/throughput, improved coverage, lower latency, high mobility and reliability, and lower infrastructure costs [4,5].
One of the building blocks for fulfilling the requirements of the next-generation mobile communications is the use of multiple-input multiple-output (MIMO) techniques in the form of large-scale antenna arrays. Growth in spectrum availability will enhance overall network capacity in order to accommodate a large number of mobile users worldwide. Most current mobile broadband systems operate at frequencies below 6 GHz, but this spectrum is becoming increasingly crowded, especially in major cities and areas of high population density. The demand for additional spectrum can be fulfilled by the use of higher millimetre-wave bands at carrier frequencies around 30-300 GHz [6,7]. There are many potential applications associated with millimetre-wave communications, including fixed broadband access to the home, smallcell communications in dense urban areas and vehicle-to-vehicle communications. However, moving up in carrier frequency leads to new challenges of higher path loss, more significant blocking effects and unconventional channel characteristics [8]. Consideration is now also being given to using the terahertz spectrum at 300 GHz and above, where the channel conditions are often more severe, typically limiting the potential use cases to high-data-rate but very-shortrange communications applications [9]. The use of MIMO technology can provide performance gains to help to mitigate the adverse channel effects, but these systems have hardware and power consumption constraints. The very name millimetre wave highlights the very small wavelengths associated with these frequencies, which allow a large number of antennas to be placed on a compact space. Using a dedicated radio-frequency (RF) chain for each antenna element would lead to the best data-rate performance. However, this solution is difficult to implement in practice because of the excessive power consumption and hardware complexity that results. Also, using wide bandwidth analogue-to-digital converters (ADCs) and digital-toanalogue converters (DACs) as part of the RF chains at millimetre-wave frequencies becomes a further source of hardware complexity and high power consumption. A parsimonious and energy-efficient transceiver architecture is thus desired. Figure 1 shows the multiuser hybrid beamforming architecture that is often studied in the literature for use at millimetre-wave frequencies. The transmitter wishes to send K spatial streams to K receivers using spatial multiplexing techniques. Digital precoding is applied to the signals to direct these streams to the receiver using directional beamforming concepts. The digital precoder outputs are then converted into analogue form and amplified using L T RF chains. These waveforms are then directed to the transmitting antennas using an analogue precoder network. Various designs for the precoder network are possible, but may comprise a Butler matrix set-up to generate fixed beam patterns or controllable phase shifters that allow dynamic beam patterns to be created. Typically the number of RF chains L T is much less than the number of transmitting   antennas N T because of their inherent broadband operation and high power consumption. The transmitting signal travels through the wireless channel, which is different for each one of the K receiving terminals. We consider that each user equipment has N R antennas and applies analogue RF combining to the received signal, before downconversion by a single RF chain. The basic ideas of this architecture were discussed in [10], where the number of transmit (TX) and receive (RX) RF chains could be selected according to the number of spatial streams that should be transmitted and received. Techniques for optimizing signal reception in this architecture are further explored in [11]. The concept of using sparse signal-processing techniques to determine the best precoding and reception weights for maximizing data-throughput performance is described in [12]. A heuristic approach to hybrid beamforming that can achieve performance very close to the ideal case of using one RF chain is also described in [13]. Detailed survey articles that discuss issues around millimetre-wave communications and hybrid beamforming can be found in [14,15].
In recent years, optimizing the performance of the hybrid architecture has become the subject of intense study. The fully connected architecture studied in [12] connects all of the antennas to each RF chain, while the alternative partially connected structure connects each RF chain to only a subset of all antennas, which requires fewer phase shifters [16]. A detailed study in [17] also highlights that using the partially connected set-up can reduce RF losses in the system, improving performance. The partially connected set-up is therefore able to achieve a lower power consumption than the fully connected approach [18]. However, the partially connected approach may suffer from increased co-channel interference, so a low-complexity interference cancellation precoding approach is proposed in [19]. Vlachos et al. [20] studied the energy efficiency (EE) of a partially connected hybrid beamformer system, where each RF chain is connected to only a subset of the available antennas. Other authors have tried to reduce the complexity through replacing some of the phase shifters with a network of RF switches, such as described in [21,22]. It is shown in these two papers that switches can operate at lower power consumption than phase shifters, enabling further energy savings with minimal impact on throughput performance. Energy-efficient baseband signal-processing methods to mitigate interference are also studied in [23]. A simple approach to receiver design is to make use of lens antennas in place of phase shifters or switches and low-complexity hardware implementations of this approach are reported in [24][25][26]. Moghadam et al. [27] studied the impact of nonlinear effects in transmitter power amplifiers and concluded that using only one RF chain can be preferable in some scenarios to achieve the most energy-efficient operating point of the system.
The power consumption can be reduced further through making use of low-resolution quantization of waveforms in MIMO transceivers. Jointly selecting low-resolution quantization at both the TX and the RX, and optimizing bit resolution with the precoding and combining designs can provide a highly energy-efficient communication solution. This is due to the fact that royalsocietypublishing.org/journal/rspa Proc. R. Soc. A the power consumption of DACs/ADCs scales exponentially with the number of bits used [28]. Orhan et al. [29] studied how the sampling bit resolution affects the achieved data rates using an additive quantization noise model (AQNM) for the quantization process. The AQNM approach is also used in [30], which shows the impact of low-resolution sampling on the achieved data rate. The combination of a fully digital precoding TX with joint RF and baseband combining using low-resolution sampling at the RX is studied in [21]. Zhang et al. [31] proposed the idea of using a mixture of high-and low-resolution ADCs, which can achieve a higher EE than systems that use fixed resolution ADCs at all receivers. The millimetre-wave channel estimation problem when using low-resolution sampling at the RX is also discussed in [32]. Care is needed when selecting the bit resolutions to be employed as the total power consumed may be dominated by a few ADCs or DACs operating at high resolution.
As noted above, the RF chains can consume considerable power and increase the costs of the radio system [33], so another way to reduce energy is to optimize the number of activated RF chains. A brute-force technique has been used in [34] to identify the most energy-efficient hybrid precoder by designing the complete precoding solution for all of the choices for the number of RF chains. A simpler alternative approach to optimizing the number of activated RF chains was proposed in [35], which makes use of the Dinkelbach technique for optimizing the EE metric of data rate divided by power consumed.
This paper builds on the existing literature on optimal hybrid beamformer design [11] and the sparse solutions developed in [12]. More specifically, the paper builds on the Dinkelbach technique for energy-efficient RF-chain selection developed in [35] for single-user MIMO channels. We extend this work in two aspects. First, we show how this method can be implemented in a multiuser broadcast channel where one transmitter uses beamforming to send multiple data streams simultaneously to multiple user terminals. Second, we show how our approach can take into account the impact of hardware imperfections within the precoding network. This requires a completely new approach based on the mathematical technique of convex relaxation, as compared with the simpler methods used in [35]. This is necessary to handle the increased complexity of selecting RF chains for the multi-user scenario. Further modifications are required to model hardware imperfections in the system and to account for the potential co-channel interference between multiple users. Table 1 provides a list of notations used in this paper along with their descriptions.

(i) Notations and organization
The remainder of the paper is structured as follows: §2 reviews the literature on dynamic hybrid beamforming architectures. Section 3 describes the system and channel models. Section 4 discuss the EE maximization problem where spectral efficiency (SE) and power consumption models are defined. In §5, we introduce the proposed RF selection algorithm. Section 6 presents simulation results to show the performance improvements and finally §7 presents conclusions to the paper.

System and channel model (a) Millimetre-wave channel
Making use of the very wide bandwidth channels available in the millimetre-wave frequency bands is an important way to meet the needs of mobile broadband users in the next decade [36][37][38]. The higher path losses associated with the millimetre-wave spectrum compared with microwave bands can be mitigated through beamforming gains. These arise from using directional transmission and reception with large-scale antenna arrays, i.e. MIMO systems. In addition, millimetre-wave signal propagation is significantly affected by blockage effects, e.g. from the human body (attenuation of 20-35 dB [39]) and building materials such as brick (attenuation of 40-80 dB [40,41]).
a v e c t o r A m a t r i      which are important attributes when considering millimetre-wave frequency channels for nextgeneration wireless standards. One very important property of a typical millimetre-wave frequency channel is the high sparsity, i.e. there are only a few significant propagation paths in the angle and delay domains [7,42].
Assuming that orthogonal frequency division multiplexing (OFDM) is being used, we make use of the flat fading channel model to model one subcarrier of a millimetre-wave communication system. We consider P k propagation paths for the kth user, where k = 1, . .
N R RX antennas. The channel response matrix is given by (2.1) The scalar α k,p denotes the gain for the pth multi-path component (MPC) for the kth user. The scalar P k is the total number of MPCs for the kth user and the vectors b(·) ∈ C N T ×1 and a(·) ∈ C N R ×1 are the steering vectors for the TX and RX, respectively. Both TX and RX are assumed to use a uniform linear array (ULA) with antenna array spacing d = λ/2, so that the RX steering vector is defined as a(N R , θ ) = [1, e jπ cos(θ) , e j2π cos(θ) , . . . , e j(N−1)π cos(θ) ] T . (2. 2) The TX steering vector b(N T , φ) ∈ C N T ×1 is defined similarly for the TX, with θ, φ ∈ [−π/2, π/2] representing the steering angle of arrival and departure, respectively.

(b) System model
We consider a millimetre-wave downlink multi-user scenario, where the base station (BS) is equipped with an N T -element ULA and serves K users. As shown in figure 1, the BS employs analogue precoding represented by the matrix F RF ∈ C N T ×L T . To capture the hardware imperfections of the analogue beamformer, we adopt the following linear model: where x ∈ C L T ×1 is the baseband signal for amplification and upconversion by the L T RF chains, -F RF ∈ C L T ×N T is the analogue beamforming matrix, which is considered to contain noise due to hardware imperfections and fluctuations in circuit behaviour, -P TX ∈ R L T ×L T is a diagonal matrix whose entries contain the power amplification factors of the L T RF chains, -H k ∈ C N T ×N R is the millimetre-wave channel response array between the kth user and the BS, w k ∈ C N R ×1 is the combiner beamforming vector applied for the kth user: it is computed as the steering vector b that maximizes the received signal power, w H k H k Φx is the noise due to hardware imperfections, w H k K−1 =1, =k H F RF x is the inter-user interference, the vector x ∈ C K×1 represents the broadcast signal, and -n k ∈ C represents the additive white Gaussian noise (AWGN), which is complex Gaussian distributed with zero mean and variance σ 2 n , i.e. n k ∼ CN (0, σ 2 n ).

Energy efficiency maximization
The EE is defined as the ratio of the SE R (bit s −1 Hz −1 ) and the power P (watts) [43], Essentially, maximization of the EE aims for simultaneous maximization of the SE and minimization of the required power. This problem can be expressed by the following constrained optimization: where R min and P max are the predefined lower and upper bounds for the SE and power, respectively. The SE R and power P can be expressed as functions of several parameters, e.g. the analogue/digital beamforming matrices or the number of RF chains, for the TX and the RX, respectively. Thus, equation (3.2) represents a fractional optimization problem, where in general there may be no closed-form expression for the solution [43]. Depending on the optimization variable we choose to focus on, the SE R and the power P can be non-convex functions.
Let us consider the case where the optimization variable is the power consumed by L T RF chains. Mathematically, the required energy for the operation of the ith RF chain can be represented by the ith entry of the vector p [p 1 , . . . , p L T ] T , where p i is a positive real number, i.e. p i ∈ R + . Furthermore, the overall consumed power P for the hybrid beamformer MIMO system is composed by the terms P = P amp + P RFchains + P circuit (watts), (3.3) where P amp is the power required by the amplifiers for signal transmission, P RFchains is the consumed power at all the RF chains and P circuit is the consumed power at the digital and analogue circuit components. The consumed power for the RF chains contributes significantly to the overall power P. In this work, we focus on the minimization of the power of the RF chains, which is expressed as Note that (3.4) is a convex function of p since L T i=1 p i = p 1 . For hybrid beamforming, we begin with a vector s ∈ C K×1 , which contains the K data streams for transmission: s obeys the property E[ss H ] = I K . The vector s is then pre-multiplied by the matrix F BB ∈ C L T ×K , which represents the digital beamforming matrix at the transmitter to generate the baseband signal vector x. The precoder is therefore decomposed as where P TX diag(p TX ) is the diagonal matrices representing the power consumption of the RF chains at the TX, p TX . For the hybrid combiner that represents the signals processed at the receivers, we define the N R × K matrix The SE of the whole hybrid system is given by [12] R(P TX ) = log 2 I K + 1 where the matrix Q ∈ C K×K is defined as Maximizing the EE ratio in equation (3.1) is in general a difficult mathematical problem. In order to make progress, we use the following theorem to study the mathematical properties of equation (3.9).
Theorem 3.1. The SE given by (3.9) is a concave function of the diagonal matrix P TX , assuming Proof. Since F BB F H BB = I K , the expression in (3.9) is written as where λ k is the kth eigenvalue of the matrix (F H RF H H WW H HF RF ). Recall that the sum of concave functions log 2 (1 + x) is also concave. Thus, R(P TX ) is a concave function of P TX .
Therefore, in our case, the fractional problem (3.2) is given as the ratio of a concave and a convex function. A common approach for solving fractional concave-convex problems is the Dinkelbach method [44], which replaces the fractional optimization by an iterative sequence of simple problems based on the difference of the numerator and denominator. Specifically, the solution to problem (3.2) is given successively by solving the problem max(R d − κ d P k ) subject to R d ≥ R min and P d ≤ P max , (3.11) for d = 1, . . . D max , where D max is the maximum number of iterations of the method. In the above equation, R d and P d are the SE and power for the dth Dinkelbach method iteration and κ d is the calculated EE ratio based on the previous estimation of R d−1 and P d−1 . Moreover, for concave-convex problems the Dinkelbach method also provides convergence guarantees to find the globally best solution.

(a) Energy efficiency maximization via radio-frequency-chain subset selection
Assigning a zero value to the power of the ith RF chain, p i =0 represents the option of de-activating the corresponding RF chain, so that it does not contribute to the overall power expenditure. However, owing to the use of the zero value, the problem becomes a combinatorial one, where all possible combinations for the zero values that maximize the EE have to be exhaustively searched [35]. This means that the complexity of such an 'exhaustive search' solution scales exponentially with the number of RF chains L T .
To overcome the issue of the non-tractability of the exhaustive search, we consider the case where all RF chains have equal power requirements, i.e. p i = p. Then, the problem (3.9) can be formulated as a sparse subset selection one, by introducing a sparse RF-chain selection vector, s, with entries in the set {0, 1}. Incorporating this selection procedure into the expressions of rate in (3.9) and power in (3.4)

expressed as
where P fix is a fixed power consumed by the system that does not vary with the number of activated RF chains. The matrix S is a diagonal matrix where the value s i = 1 denotes that the ith RF chain is activated, or set to zero otherwise. The introduction of the selection variable permits us the approximation of the combinatorial problem (3.11) with s i ∈ {0, 1}, i = 1, . . . , L T , into an approximated convex problem with s i ∈ (0, 1). Therefore, the dth convex optimization problem in (3.11) is transformed into a sparse subset selection, i.e. min s (κ d s 1 − R d (s)) subject to R d (s) ≥ R min and P d (s) ≤ P max , (3.14) In [35], we provide an iterative algorithm that solves (3.14) via thresholding. Next, we will describe major modifications to the proposed approach that are required to optimize performance in the presence of hardware imperfections and multi-user co-channel interference. It will be seen that it can significantly reduce the overall complexity, as discussed in more detail in the following section.

Proposed technique
The RF selection process can be seen as an additional block of the hybrid beamformer structure that activates specific parts of the analogue beamformer. Thus, instead of finding the optimal phased-array matrix F RF , we choose only a subset of columns of this matrix in order to achieve the highest EE. This selection module is added between the digital and analogue parts, and can be implemented by a network of switches. Essentially, it selects a subset of the L T columns from a fixed codebook matrix which represents how the digital signals are forwarded to the analogue signal-processing network.
Let us describe the selection mechanism that represents the active/inactive RF chains at the BS, which is based on the approaches described in §3. For this, we use the binary matrix S ∈ {0, 1} N T ×N T defined in §3. Specifically, S is a diagonal matrix whose entries are either zero or 1. A physical interpretation of S is possible by considering that this matrix represents a switching network. This network activates only a maximum of L T outputs of an extended analogue combiner F e RF ∈ F N T ×N T . This extension ensures that the analogue front-end has the same number of inputs and outputs; thus, the selection is possible from the entire analogue, noisy, codebook given by where F e,ideal RF ∈ F N T ×N T is the extended ideal analogue beamformer and Φ e ∈ C N T ×N T is the extended noise matrix that captures hardware imperfections in the analogue circuitry.
These hardware imperfections may arise from one or more of the following sources [45].
- Incorporating the selection matrix, the system model of the proposed framework is expressed as where y k ∈ C is the received signal of the kth user, the hardware noise component is given by while η k (S) is the interference that affects the kth user, given by For each user, the theoretical average SE is given by where the expectation E{·} is performed over the joint space of {x, ζ k , η k }. Note that we assume that these noise sources are statistically independent; thus, there are no cross-correlation terms among them. Next, we provide an upper bound for R k (S), which is expressed based on the known covariance matrices of ζ k (S) and η k (S).
In this work, we focus on the power that is consumed by each RF chain, P RF . Each RF chain has a number of power-consuming components, such as the ADC/DAC and power amplifiers. Thus, by activating only a subset of the RF chains, the required power decreases significantly. This power level is computed using the model described in equation (3.13).
The EE problem for the multi-user downlink scenario can be expressed as for all combinations |C| with P(S (i) ) ≤ P max andR(S (i) ) ≥ R min , and select the one with the highest EE. The exhaustive search method is summarized in algorithm 1, where the remaining parameters are defined in proposition 4. 1. Note that the number of combinations |C| increases exponentially with L T . Before proceeding with the proposed technique, we would like to describe a sub-optimal but computationally affordable and straightforward approach of solving (4.6). This could be implemented by selecting the minimum number of RF chains, where the selection is performed by a naive technique, e.g. randomly or consecutively [35]. This approach can be implemented via a simple iterative search. Specifically, at each iteration the number of RF chains L T would increase by 1 until it reaches the maximum value, L T = 1, 2, . . . , L T . At each iteration, the EE will 11 royalsocietypublishing.org/journal/rspa Proc. R

Algorithm 1 Exhaustive search algorithm
Obtain S (i) by solving (4.11) 4: CalculateR (i) and P (i) 5: end for be computed and at the end of the iterations we select the L T which provides the maximum EE. This iterative search is summarized in algorithm 2.
In order to extend [35] to handle hardware imperfections and multi-user scenarios, the complexity of selecting the correct RF chains in (4.6) grows exponentially with the number of RF chains L T . In this paper, we adopt the convex relaxation strategy, where the integer values are replaced by the set of real numbers [S] , ∈ (0, 1) [48]. This approach reflects the actual system hardware and permits study of scenarios where the impairments that affect different RF chains are not symmetric. Moreover, to deal with a fractional cost function, we employ Dinkelbach iterations [49]. This method is an iterative and parametric algorithm, where a sequence of simpler problems can be shown to converge to the global solution of the overall fractional problem. Let κ (i) ∈ R, for i = 1, 2, . . . , I max , then one iteration of the Dinkelbach method can be written as (4.8) The notation S denotes the set of diagonal matrices that satisfy both constraints P(S) ≤ P max and R(S) ≥ R min . The Dinkelbach iteration steps are summarized in algorithm 3. Parameter κ (m) is defined as the previous iteration EE computation [43], i.e.
To provide a computationally tractable solution for (3.14), we use a completely different approach to [35] and derive a novel lower bound approximation for the SE expression k R k .   . . , L T , the achievable average SE for the kth user, which is given by (4.5) can be lower bounded by where σ 2 ζ k ξ H k Mξ k , σ 2 η k p =k ξ H p Nξ p and σ 2 n is the variance of the AWGN. Proof. The proof is presented briefly in appendix A.
Thus, problem (4.6) becomes where the denominator is a function of the selection matrix S, namely and (4.14) Using the lower bound of (4.14) in (4.11), and employing the Dinkelbach approach, we can replace the fractional cost function with where ω k w H k H k F e,ideal RF . Since (4.15) is convex over S, standard interior-point methods and publicly available software packages can be used to solve (4.15). Note that, even if problems (4.6) and (4.15) result in different solution matrices S, their EE performance is almost identical, as shown through the simulations in the next section.

Simulation results
In this section, we use Matlab TM computer simulation results to evaluate the performance of the proposed method. All the results are averaged over 500 Monte Carlo realizations. Let us first define the parameters and the system characteristics. We assume that the transmitter employs hybrid analogue/digital TX beamforming with N T antennas, while the number of RF chains is L T ≤ N T . Each transmission broadcasts a zero-mean random Gaussian vector with x ∈ C N R ×1 and E{xx H } = I N R . We assume ULAs at both TX and RX sides and operating over a 28 GHz outdoor millimetre-wave channel [50]. The K users are distributed uniformly random around the BS with maximum distance 5 m. Also, the MPC P k for the kth user is selected uniformly random over the set [1,15]. To focus on the TX performance, we assume that digital combining is performed at the user equipment, i.e. w k is defined as the kth column of the left orthonormal matrix, obtained by the singular value decomposition of the channel matrix H k . Default channel parameter settings are shown in table 3. For the evaluation of the proposed technique in terms of EE and SE performance, we have considered the following cases for the TX.      First, to perform a sanity check, we compare the results of the proposed technique and the exhaustive hybrid beamforming for N T = L T = N R = 8. We keep the antenna arrays to small sizes because of the computational complexity of the exhaustive hybrid beamforming. In figure 2, we plot the SE, power and EE with respect to the instantaneous signal-to-noise ratio (SNR), defined as SNR = 10 log 10 1 σ 2 n , (5.1) as well as the EE versus SE. It can be verified that the proposed technique maximizes the EE, following closely the performance of the optimum exhaustive hybrid beamforming algorithm. The iterative hybrid beamforming cannot reach the EE of the proposed technique, since it minimizes the number of the RF chains (e.g. the used codebook beams) and it does not consider the best subset of RF chains to use. Note that the hybrid beamforming and digital beamforming have the same EE performance, since N T = L T . The analogue beamforming method has only one RF chain and thus it has a minimal number of digital components; although it has low power consumption, it exhibits the lowest EE. This comes from the fact that the transmitted signal needs to be multiplexed in time or frequency between the different user terminals, as the transmitter cannot achieve spatial multiplexing. The power of the proposed and exhaustive hybrid beamforming techniques is around 2.5 J s −1 for all SNRs. This is half of the power required by the iterative hybrid beamforming and one-quarter of that required by digital and hybrid beamforming. Next, we increase the antenna array size of the BS to N T = 32, while the number of active RF chains that connect the analogue and digital parts at the BS remains L T = 8. Recall that, in order to focus on the performance at the BS, each user employs a digital combiner with N R = 8.
In figure 3, we show the SE and EE with respect to the SNR for N TX = 32 and σ φ = 0.01. The proposed technique outperforms the other baselines, even the exhaustive hybrid beamforming.   we consider that the power consumption of each phase-shifter is negligible compared with the power consumption of each RF chain [25]. This structure can be realized by using very energyefficient elements, e.g. passive phase-shifters or a Butler matrix [51]. The required power for the proposed and the exhaustive hybrid beamforming techniques remains below 5 J s −1 . The iterative hybrid beamforming requires twice as much energy per second, while the hybrid beamforming requires four times more. The results for this case indicate the high power consumption for the digital beamforming, which is over 30 J s −1 .
In figure 4, we plot the achievable SE, power and EE over the number of RF chains L T . The proposed design is able to achieve superior EE performance when compared with the other hybrid beamforming techniques. The achievable SE of iterative hybrid beamforming and exhaustive hybrid beamforming is very similar. Recall that the iterative hybrid beamforming selects the minimum number of RF chains that achieves the best EE, while exhaustive hybrid beamforming searches for the best overall subset of RF chains. However, the search space of the latter is constrained to eight codebook beams, owing to the very high computational burden.
In figure 5, we show the achievable SE, power and EE with respect to the number of users K. The proposed technique achieves high SE, following the hybrid beamforming and digital beamforming curves. The power consumption of the proposed technique remains at the same level as the analogue beamforming and exhaustive hybrid beamforming, thus it achieves the highest EE compared with the other approaches. This indicates that the proper design of the beamformer via RF-chain selection focuses the beams to different locations.
It is important to note that the proposed design has very similar SE performance to the hybrid beamforming approach [12], as shown in figures 3 and 4. Indeed, a connection between the proposed technique and the greedy algorithm introduced in [12] exists. Specifically, the algorithm of [12] estimates a sparse vector which corresponds to the digital part of the beamformer. In the proposed design, we seek a sparse binary vector which is also part of the digital beamformer. Additionally, the analogue parts are designed using static analogue codebooks in both approaches. However, the analogue part of [12] is assumed to be drawn from an idealized discrete codebook, while in the proposed technique we explicitly model the introduced noise due to hardware imperfections. Moreover, the proposed technique outperforms [12] in terms of EE, royalsocietypublishing.org/journal/rspa Proc. R  since, via the switches, it deactivates parts of the analogue beamformer that do not contribute significantly to the overall SE performance.

Conclusion
This paper discusses the advantages of hybrid beamforming architectures for millimetre-wave wireless communications systems. By using a small number of RF chains compared with the number of antennas, it is possible to improve the EE of communication. A novel RF chain selection architecture is described to allocate the best predefined analogue codebook that maximizes the EE performance of the transmitter. Via simulation results, we showed that it is beneficial in terms of EE to activate a subset of RF chains, rather than always using the maximum number that are available. The proposed algorithm outperforms all the baselines in terms of both EE and SE, when the transmitter has a large number of antenna terminals.  (A3) Given that the hardware noise, the user interference noise and the white Gaussian noise are mutually independent, the denominator of (A 2) is given by E{|ζ k (S) + η k (S) + n k | 2 } = E{|ζ k (S)| 2 } + E{|η k (S)| 2 } + E{|n k | 2 }, ( where E{|n k | 2 } = σ 2 n . This expectation can be easily calculated as S and P TX are both diagonal matrices,  Here Φ e l denotes the lth row vector and Φ e l,n denotes the lth row and nth column of Φ e , respectively. The notation S(l) is the lth diagonal entry of S and P TX (l) is the lth diagonal entry of P TX   where F e RF (l) denotes the lth row vector and F e RF (l, n) is the lth row and nth column entry of F e RF , respectively. Using equations (A 3), (A 5) and (A 7) leads to the desired result.