Energy Efficiency Optimization of 5G Radio Frequency Chain Systems

With the massive multi-input multi-output (MIMO) antennas technology adopted for the fifth generation (5G) wireless communication systems, a large number of radio frequency (RF) chains have to be employed for RF circuits. However, a large number of RF chains not only increase the cost of RF circuits but also consume additional energy in 5G wireless communication systems. In this paper, we investigate energy and cost efficiency optimization solutions for 5G wireless communication systems with a large number of antennas and RF chains. An energy efficiency optimization problem is formulated for 5G wireless communication systems using massive MIMO antennas and millimeter wave technology. Considering the nonconcave feature of the objective function, a suboptimal iterative algorithm, i.e., the energy efficient hybrid precoding (EEHP) algorithm is developed for maximizing the energy efficiency of 5G wireless communication systems. To reduce the cost of RF circuits, the energy efficient hybrid precoding with the minimum number of RF chains (EEHP-MRFC) algorithm is also proposed. Moreover, the critical number of antennas searching (CNAS) and user equipment number optimization (UENO) algorithms are further developed to optimize the energy efficiency of 5G wireless communication systems by the number of transmit antennas and UEs. Compared with the maximum energy efficiency of conventional zero-forcing (ZF) precoding algorithm, numerical results indicate that the maximum energy efficiency of the proposed EEHP and EEHP-MRFC algorithms are improved by 220% and 171%, respectively.


I. INTRODUCTION
The massive multi-input multi-output (MIMO) antennas and the millimeter wave communication technologies have been widely known as two key technologies for the fifth generation (5G) wireless communication systems [1]- [6]. Compared with conventional MIMO antenna technology, massive MIMO can improve more than 10 times spectrum efficiency in wireless communication systems [7]. Moreover, the beamforming gain based on the massive MIMO antenna technology helps to overcome the path loss fading in millimeter wave channels. For MIMO communication systems with traditional radio frequency (RF) chains and baseband processing, one antenna corresponds to one RF chain [8], [9]. In this case, a large number of RF chains has to be employed for massive MIMO communication systems. These RF chains not only consume a large amount of energy in wireless transmission systems but also increase the cost of wireless communication systems [10]. Therefore, it is an important problem to find energy efficient solutions for 5G wireless communication systems with a large number of antennas and RF chains.
To improve the performance of multiple antenna transmission systems, hybrid precoding technology combining digital baseband precoding with analog RF precoding was investigated in [9], [11]- [17]. Based on the joint design of RF chains and baseband processing, a soft antenna subset selection scheme was proposed for multiple antenna channels [9]. When a single data stream is transmitted, the soft antenna subset selection scheme was able to achieve the same signal-to-noise ratio (SNR) gain as a full-complexity scheme involving all antennas in MIMO wireless communication systems. By formulating the problem of millimeter wave precoder design as a sparsity-constrained signal recovery problem, algorithms were developed to approximate optimal unconstrained precoders and combiners in millimeter wave communication systems with large antenna arrays [11]. To maximize the sum rate of MIMO communication systems, a hybrid beamforming approach was designed indirectly by considering a weighted sum mean square error minimization problem incorporating the solution of digital beamforming systems [12]. Based on a lowcomplexity channel estimation algorithm, a hybrid precoding algorithm was proposed to achieve a near-optimal performance relative to the unconstrained digital solutions in the single user millimeter wave communication system [13]. To reduce the feedback overhead in millimeter wave MIMO communication systems, a low complexity hybrid analog/digital precoding scheme was developed for the downlink of the multi-user communication systems [14]. To maximize the minimum average data rate of users subject to a limited RF chain constraint and a phase-only constraint, a two stage precoding scheme was proposed to exploit the large spatial degree of freedom gain in massive MIMO systems with reduced channel state information (CSI) signaling overhead [15]. Based on phase-only constraints in the RF domain and a low-dimensional baseband zero-forcing (ZF) precoding, a low complexity hybrid precoding scheme was presented to approach the performance of the traditional baseband ZF precoding scheme [16]. Compared with the digital beamforming scheme, the hybrid beamforming scheme was shown to achieve the same performance with the minimum RF chains and phase shifters in multi-user massive MIMO communication systems [17]. In the above studies, hybrid precoding schemes have rarely been investigated to optimize the energy efficiency of massive MIMO communication systems. While in cellular communication systems, energy efficiency has been considered as a critical performance metric [18].
To improve the energy efficiency of MIMO communication systems, some digital precoding schemes have been studied in [19]- [21]. Based on static and fast-fading MIMO channels, an energy efficient precoding scheme was investigated when the terminals are equipped with multiple antennas [19]. Jointly considering the transmit power, power allocation among date streams and beamforming matrices, a power control and beamforming algorithm was developed for MIMO interference channels to maximize the energy efficiency of communication systems [20]. Transforming the energy efficiency of MIMO broadcast channel into a concave fractional program, an optimization approach with transmit covariance optimization and active transmit antenna selection was proposed to improve the energy efficiency of MIMO systems over broadcast channels [21]. With the massive MIMO concept emerging as a key technology in 5G communication systems, the energy efficiency of massive MIMO has been studied in several papers [10], [22]- [26]. Based on a new power consumption model for multi-user massive MIMO, closed-form expressions involving the number of antennas, number of active users and gross rate were derived for maximizing the energy efficiency of massive MIMO systems with ZF processing [10]. When linear precoding schemes are adopted at the BS, it is proved that massive MIMO can improve the energy efficiency by three orders of magnitude [22]. Considering the transmit power and the circuit power in massive MIMO systems, a power consumption model has been proposed to help optimize the energy efficiency of multi-cell mobile communication systems by selecting the optimal number of active antennas [23]. When the sum spectrum efficiency was fixed, the impact of transceiver power consumption on the energy efficiency of ZF detector was investigated for the uplinks of massive MIMO systems [24]. Considering the power cost by RF generation, baseband computing, and the circuits associated with each antenna, simulation results in [25] illustrated that massive MIMO macro cells outperforms LTE macro cells in both spectrum and energy efficiency. To maximize the energy efficiency of the massive MIMO OFDMA systems, an energy efficient iterative algorithm was proposed by optimizing the power allocation, data rate, antenna number, and subcarrier allocation in [26].
However, in all the aforementioned studies, only the transmission rate and the hardware complexity of hybrid precoding systems were analyzed for MIMO or massive MIMO communication systems. Moreover, energy efficient solutions for the RF chains and the baseband processing of 5G wireless communication systems is surprisingly rare in the open literature. On the other hand, the energy and cost increase through using a large number of RF chains is an inevitable problem for 5G wireless communication systems. Motivated by the above gaps, in this paper we propose energy and cost efficient optimization solutions for 5G wireless communication systems with a large number of antennas and RF chains. The contributions and novelties of this paper are summarized as follows.
1) The BS energy efficiency including the energy consumption of RF chains and baseband processing is formulated as an optimization function for 5G wireless communication systems adopting massive MIMO antennas and millimeter wave technologies. 2) Considering the non-concave feature of the optimization objective function, a suboptimal solution is proposed to maximize the BS energy efficiency using the energy efficient hybrid precoding (EEHP) algorithm. 3) To reduce the cost of RF circuits, the energy efficient hybrid precoding with the minimum number of RF chains (EEHP-MRFC) algorithm is developed to tradeoff the energy and cost efficiency for 5G wireless communication systems. 4) To utilize the user scheduling and resource management schemes, the critical number of antennas searching (CNAS) and user equipment number optimization (UENO) algorithms are developed to maximize the energy efficiency of 5G wireless communication systems. The remainder of this paper is outlined as follows. Section II describes the system model of 5G wirelss communication systems and the energy efficiency optimization problem is formulated. In Section III, an iteration algorithm, i.e., the EEHP algorithm is developed to maximize the energy efficiency of 5G wireless communication systems. To save the cost of RF circuits, the EEHP-MRFC algorithm is developed to tradeoff the energy and cost efficiency in Section IV. Moreover, the CNAS and UENO algorithms are developed to optimize the energy efficiency of 5G wireless communication systems considering the number of transmit antennas and user equipments (UEs). Detailed numerical simulations are presented in Section V. Finally, conclusions are drawn in Section VI.

II. SYSTEM MODEL
Considering the impact of massive MIMO antennas on the RF chains and the baseband processing, the energy efficiency of 5G wireless communication systems has to be rethought. In the following, we describe the system configuration of 5G wireless communication systems, including massive MIMO antennas, RF chains and the baseband processing. Moreover, the energy efficiency optimization problem of 5G wireless communication systems is formulated.

A. System Configuration
Without loss of generality, a single cell scenario is illustrated in Fig. 1, where a BS and K UEs are located in the 5G wireless communication system. The BS is assumed to be equipped with N Tx antennas. And based on the massive MIMO configuration, we assume N Tx 100 in our system [7], [22]. There are K active UEs, each with a single antenna, that are associated with the BS. Moreover, the transmission system of BS is equipped with N RF RF chains. One baseband data stream is assumed to be associated with one UE in 5G wireless communication systems. In this paper, our studies focuses on the downlinks of 5G wireless communication systems.
The received signal at the kth UE is expressed as Note that the consumed power for transmitting signals in (2) is expended for a given bandwidth W , which is set to be 20MHz in this paper [10]. The millimeter wave communication technology is adopted for 5G wireless communication systems. Considering the propagation characteristic of millimeter waves in wireless communications, a geometry-based stochastic modeling (GBSM) is used to express the millimeter wave channel as follows [11], [12], [27], [28] where N ray is the number of the multipath between the BS and K UEs. β k = ζ/l γ k is the large scale fading coefficient over the wireless link between the BS and the kth UE. ζ is the lognormal random variable with the zero mean and the variance of 9.2 dB. l k is the distance between the BS and the kth UE. γ is the path loss exponent. ρ ki is the complex gain of the ith multipath over the kth UE link, which denotes the small-scale fading in wireless channels and is governed by a complex Gaussian distribution. Moreover, ρ ki is i.i.d. for different values of k(k = 1, ..., K) and i(i = 1, ..., N ray ). ψ i and ϑ i are the azimuth and the elevation angle of the ith multipath at the BS antenna array, respectively. u (ψ i , ϑ i ) is the response vector of BS antenna array with the azimuth ψ i and the elevation angle ϑ i . Without loss of generality, the BS antenna array is assumed as the uniform planar antenna array in this paper. Therefore, the response vector of BS antenna array with the azimuth ψ i and the elevation angle ϑ i is expressed as [29] u (ψ i , ϑ i ) = where d is the distance between adjacent antennas, λ is the carrier wave length, M and N are the row and column number of the BS antenna array, respectively. m is denoted as the mth antenna in the row of the BS antenna array, 1 m < M ; n is denoted as the nth antenna in the column of the BS antenna array, 1 n < N .

B. Problem Formulation
Based on the system model in Fig. 1, the link spectrum efficiency of the kth UE is expressed as (5) Note that from (5) we get the instantaneous spectrum efficiency. And in this paper, we assume that the BS transmitter has perfect CSI, i.e. channel vectors h k , k = 1, ..., K are known at the BS. This assumption is widely adopted for the investigation of precoding problems in massive MIMO systems and millimeter wave transmission systems [11], [12], [22], [23]. In practical wireless communication systems, the CSI can be obtained through uplink channel estimation then applied to downlink precoding based on the channel reciprocity in the time division duplex (TDD) mode [7], [30]. Moreover, the millimeter wave multipath channel estimation utilizing compressed channel sensing was investigated in [13] and [31]. Furthermore, considering all the UEs, the sum spectrum efficiency is expressed by The total BS power is expressed as where α is the efficiency of the power amplifier, and the term K k=1 B RF b BB,k 2 is the power consumed to transmit signals for K UEs over the given bandwidth W , P RF is the power consumed at every RF chain which is comprised by converters, mixers, filters, phase shifters, etc. Considering the number of antennas is fixed in a wireless communication system, the number of phase shifters is also fixed since each phase shifter is associated with one antenna. P C is the power consumed for site-cooling, baseband processing and synchronization in the BS. To simplify the derivation, P C is fixed as a constant.
In this paper, we focus on how to maximize the BS energy efficiency (bits per Joule) by optimizing the baseband precoding matrix B BB , the RF precoding matrix B RF and the number of RF chains N RF . This optimization problem is formed by where W is the transmission bandwidth, Γ k is the minimum spectrum efficiency required by UE k , and P max is the maximum transmit power required for wireless downlinks. In general, the RF precoding is performed by phase shifters, which can change the signal phase but can not change the signal amplitude. Therefore, the amplitude of RF precoding matrix is fixed as a constant, which is added as a constraint NTx . Furthermore, there exists a minimum data rate required by each UE in practical applications. The minimum data rate is obtained by multiplying the corresponding minimum spectrum efficiency by the bandwidth. Since the bandwidth is fixed in this paper, an equivalent minimum spectrum efficiency constraint is used to satisfy the minimum data rate requirement, which is expressed as is the maximum transmit power constraint.

III. ENERGY EFFICIENT HYBRID PRECODING DESIGN
To maximize the energy efficiency in (8), an EEHP algorithm is developed to jointly optimize the baseband precoding matrix, the RF precoding matrix and the number of RF chains in the following.

A. Upper Bound of Energy Efficiency
Based on definitions of the baseband precoding matrix and the RF precoding matrix, the size of B RF ∈ C NTx×NRF and B BB ∈ C NRF×K is related with the number of RF chains N RF . To simplify the derivation, we first fix the number of RF chains. Based on the optimization objective function in the Section II, (8) is a non-concave function with regard to B RF and B BB . In general, there does not exist an analytical solution for such non-concave functions. To tackle this problem, B = B RF B BB is configured as a digital precoding matrix B ∈ C NTx×K , B = [b 1 , ..., b k , ..., b K ] whose size does not depend on the number of RF chains N RF . Furthermore, the NTx . Substituting the digital precoding matrix B into (8) and NTx , the optimization

Algorithm 1 Energy Efficient Hybrid Precoding-A (EEHP-A) algorithm.
Begin: 1) Assuming B (0) to be the initial digital precoding matrix, Ω (0) k and Ξ (0) k are calculated by (10) for the UE UE k , k = 1, ..., K; 2) At the nth, n = 1, 2, ..., iteration step, the iterative step length µ 3) Based on µ and (10), Ω end Begin problem in the Section II is transformed as Based on (9), the energy efficient is maximized by optimizing the digital precoding matrix B, where B is only constrained by the maximum transmit power P max and the minimum spectrum efficiency Γ k . Assume that the maximum energy efficiency in (8) is denoted as η max . Similarly, assume that the maximum energy efficiency in (9) is denoted asη max . Compared (8) with (9), two optimization problems have the same objective function. But (9) has less constraints than that of (8). Therefore, the solution of (9), i.e., the maximum energy efficiency of (9) is larger than or equal to the solution of (8), i.e., the maximum energy efficiency of (8). Therefore, the maximum energy efficiency η max is upper bounded by the maximum energy efficiencyη max , i.e., η max η max .

B. Energy Efficiency Local Optimization
Considering (9) is a non-concave function, it is difficult to find a global optimization solution for this optimization problem. A local optimization solutionη opt with the optimal digital precoding matrix opt K is first derived for the energy efficiency optimization in (9).
Denoting the energy efficiency and spectrum efficiency in with where δ i is the sum of the received interference power and the noise power for the UE UE i , i = 1, ..., K.R i is obtained from (9b) by replacing k with i.
When the zero-gradient condition ∂η(b k ) ∂b k = 0 is applied, the local optimization solution for the UE UE k , k = 1, ..., K is derived as To obtain the optimal digital precoding matrix B opt of the local optimization solution, an iterative algorithm is developed and called the energy efficiency hybrid precoding-A (EEHP-A) algorithm on the top of the previous page.
Based on the EEHP-A algorithm, the b opt k is obtained by the converged b (n) k . After obtaining b opt k , k = 1, · · · , K for all K UEs, the local optimization solutionη opt is achieved. To ensure the convergence of EEHP-A algorithm, the corresponding proof is given as follows. (12) and (13) are rewritten as where Z is a symmetric positive definite matrix. Furthermore, the following result is derived as (16): Based on (16) and the proposition in [33],η b are non-decreasing sequences for the UE UE k , k = 1, ..., K. Moreover,η (b k ) is upperbounded according to the proposition in [20]. It is easily known that the upper-bounded non-decreasing sequence approaches to a convergence value. Therefore,η (b k ) in EEHP-A is proved to converge.

C. Hybrid Precoding Matrices Optimization
Based on (9) and the EEHP-A algorithm, the local optimization solutionη opt is obtained. When the hybrid precoding matrices B RF B BB approach the optimal digital precoding matrix B opt with the constraint [B RF ] i,j 2 = 1 NTx , the energy efficiency η will approaches the local optimization solution η opt . Therefore, the optimal hybrid precoding matrices B opt RF and B opt BB can be solved by minimizing the Euclidean distance between B RF B BB and B opt [11], [12], [34] B opt RF , B opt BB = arg min Considering the non-convex constraint , it is not tractable to analytically solve the optimization problem in (17). Based on the millimeter wave channel in (3), the entries of BS antenna array steering matrix U = u (ψ 1 , ϑ 1 ) , ..., u (ψ i , ϑ i ) , ..., u ψ Nray , ϑ Nray ∈ C NTx×Nray are constant-amplitude which can be implemented by phase shifters in BS RF circuits. Meanwhile, as pointed out in [11], the columns vectors of steering matrix U are independent from each other in millimeter wave channels. Moreover, U ∈ C NTx×Nray and B RF ∈ C NTx×NRF have the same row numbers. To simplify the engineering application, N RF column vectors are selected from U to form the column vectors of B RF . The detailed vector selection method is described in the EEHP-B algorithm. Furthermore, the baseband precoding matrix B BB is optimized by approaching B RF B BB to B opt . As a consequence, the optimization problem in (17) is transformed as follows where ⌢ B BB ∈ C Nray×K is a digital precoding matrix con-

Begin:
1) Preset B RF as an N Tx × N RF empty matrix, and set B temp = B opt ; 2) For i = 1 : 1 : RF is solved by the step 2. The optimal baseband precoding matrix is calculated by

end Begin
Since the EEHP-B algorithm has the specified cycle number, i.e., N RF , the EEHP-B algorithm is guaranteed to converge.
When the optimal hybrid precoding matrices B opt RF and B opt BB are submitted into (8), an optimal energy efficiency η opt can be obtained. Based on the EEHP-B algorithm, the value of η opt approaches to the value ofη opt , i.e., η opt η opt . Considerinḡ η opt is a locally optimal solution for the energy efficiency optimization in (9), η opt is also a locally optimal solution for the energy efficiency optimization in (8).

D. number of RF chains Optimization
Based on EEHP-A and EEHP-B algorithms, a local optimization solution η opt with optimized hybrid precoding matrices B opt RF and B opt BB is available for the energy efficiency optimization in (8). However, the number of RF chains is fixed for the solution η opt . To maximize the energy efficiency, the number of RF chains is further optimized based on the locally optimal solution η opt .
By analyzing (9) and (17), the number of RF chains N RF is related not only to the size of the optimized hybrid precoding matrices B opt RF and B opt BB but also to the entries of B opt RF and B opt BB . Therefore, it is difficulty to derive an analytical solution for the optimal number of RF chains. However, the number of RF chains N RF is an positive integer and is limited in the specific range [K, N Tx ]. In this case, we can utilize the ergodic searching method to find the optimal number of RF chains maximizing the energy efficiency in (8). Therefore, the EEHP algorithm is developed to achieve the global optimization solution for the energy efficiency optimization in (8). η max , an suboptimal energy efficiency optimization solution is found by the EEHP algorithm in this section.
Furthermore, according to the computational complexity of matrix calculation and iterative algorithms in [35] and [36], the computational complexity of the proposed algorithm is explained as follows: The complexity of Algorithm 1 is calculated as O (N Tx K) + O N 3 Tx floating point operations (flops); the complexity of Algorithm 2 is calculated as O N 2 Tx K + N 3 RF + N 2 RF N Tx + N Tx N RF K flops; combining Algorithm 1 and 2, the complexity of the EEHP algorithm, i.e. Algorithm 3 is calculated as

IV. ENERGY EFFICIENT OPTIMIZATION WITH THE MINIMUM NUMBER OF RF CHAINS
In general, the cost of RF chain is very high in wireless communication systems. To reduce the cost of massive MIMO systems, an energy efficient solution with the minimum number of RF chains is investigated in the following.

A. Energy Efficiency Hybrid Precoding with the Minimum number of RF chains
Based on the function of RF chains and the hybrid precoding scheme, the number of RF chains is larger than or equal to the number of baseband data streams in massive MIMO communication systems [9], [39]. In this paper the number of baseband data streams is assumed to be equal to the number of active UEs. Without loss of generality, the minimum number of RF chains is configured as N min RF = K, where K is the number of active UEs in Fig. 1.
When the minimum number of RF chains is configured, the RF precoding matrix B RF has a size of N Tx × K, which is exactly the size of the conjugate transpose of the downlink channel matrix. Therefore, the entry of RF precoding matrix B RF is directly configured as [16] [ where [B RF ] i,j denotes the (i, j) th entry of the RF precoding matrix B RF , and θ i,j is the phase of the (i, j) th entry of the conjugate transpose of the downlink channel matrix. When the downlink channel matrix H H and the RF chain precoding matrix are combined together, an equivalent downlink channel matrix for all K UEs is given by Substitute (20) into (8), the energy efficiency optimization problem is transformed as (21b) Similar to the optimization problem in (8), the energy efficiencyη in (20) is maximized by the optimal baseband precoding matrixB opt BB . Considering the objective function in (21) is non-concave, it is intractable to solve for the global optimum ofη. Therefore, a local optimum solution is developed for (21) as follow.
Given thatη is a function of the baseband precoding vector of the kth UE, i.e.η (b BB,k ), the gradient ofη (b BB,k ) is derived by with [22b -22e].
whereδ i is the sum of the received interference and noise power for the ith UE,R k,eq is obtained by replacing k with i in (21b). Replacing the results of (10) by the results of (22), the baseband precoding matrixB opt BB is solved by the EEHP-A algorithm. Substituting the baseband precoding matrixB opt BB into the EEHP algorithm, a local optimum of η is solved. To differentiate this approach from the EEHP algorithm, this algorithm is denoted as the energy efficient hybrid precoding with the minimum number of RF chains (EEHP-MRFC) algorithm.
To analyze the performance of EEHP-MRFC algorithm, numerical simulation results are illustrated in Fig. 2. In generally, the optimization result of EEHP-MRFC algorithm depends on the initial values of the baseband precoding matrix, i.e. B where 1 K×K denotes the K × K matrix whose entries are equal to 1, the coefficient Pmax K is due to the maximum transmit power constraint in the BS. The detailed simulation parameters are list in Table I at the top of this page. Based on the results in Fig. 2, the EEHP-MRFC algorithm converges after a limited iteration number. Besides, the convergence rate decreases the number of UEs increases. Moreover, the converged value of the energy efficiency also decreases with the increase of the number of UEs. This result indicates that the energy efficiency of massive MIMO system is inversely proportional to the number of active UEs.

B. Energy Efficiency Optimization Considering the Numbers of Antennas and UEs
In practical engineering applications, the optimal number of transmit antennas and the optimal number of UEs can be performed by the resource management and the user schedule schemes for maximizing the energy efficiency of 5G wireless communication systems. However, the EEHP algorithm can not directly derive the analytical number of transmit antennas and UEs for maximizing the energy efficiency of 5G wireless communication systems. Therefore, we try to derive the optimal number of transmit antennas and UEs for maximizing the energy efficiency of 5G massive MIMO communication systems. To simplify the derivation, a specified scenario with rich scattering and multipaths propagation in a millimeter wave wireless channels is considered for the ergodic capacity calculation in this paper. Based on measurement results of millimeter wave channels in [28], [40], [41], the Rayleigh fading model can be adopted to describe the considered millimeter wave wireless channels.
Assume that the transmitter, i.e., the BS has the perfect CSI. When the minimum number of RF chains is configured and the ZF precoding is adopted in the system model [8], based on (19) and (20), the baseband precoding matrix is given by where D is a K ×K diagonal matrix, which aims to normalize B BB . Assume that P out is the total BS downlink transmit power consumed by K active UEs. To simplify the derivation, the equal power allocation scheme is assumed to be adopted for all K active UEs. Substituting (24) into (5), the normalized link capacity, i.e. the link spectral efficiency of UE k is expressed as [22] R k,ZF = log 2 Based on Jensen inequality, the upper-bound of the ergodic link capacity of UE k is derived by where E ( ) is the expectation operation taken over the Rayleigh fading channel H within H eq = B H RF H. Diagonal and off-diagonal entries of H eq are given by where b RF,k is the kth column of B RF . With Rayleigh fading channel considered and based on results in [16], the diagonal entry of H eq is governed by a normal distribution, i.e., and the off-diagonal entry of H eq is governed by a standard normal distribution, i.e., [H eq ] j,k ∼ CN (0, 1). Considering massive MIMO antennas are equipped at the BS, without loss of generality, the number of BS antennas is assumed to be larger than or equal to 100, i.e. N Tx 100. In this case the expected value of diagonal entries is much larger than the expected value of off-diagonal entries in H eq . Therefore, the expected value of the off-diagonal entries can be set to zero in H eq , and H eq is approximated as a diagonal matrix [16]. Based on (26), the upper-bound of the ergodic link capacity of UE k is approximated aŝ When the ergodic link capacity is replaced by the upperbound of the ergodic link capacity, the upper-bound of the BS energy efficiency is derived bŷ where P BB is the power consumed by the baseband processing for the baseband data stream, P ′ C is the fixed BS power consumption without the power consumed for the downlink transmit, the RF chains and the baseband processing.
For the upper-bound of the BS energy efficiency in (29), the following proposition is given.

Proposition:
Considering the impact of the number of UEs and transmit antennas on the BS energy efficiency, a function G (K, N T x ) is formed as (30) When G (1, 100) ≥ 0, there exists a critical number of antennas N Cri Tx . The critical number of antennas N Cri Tx is the smallest integer which is larger than or equal to the root of the equation G (1, N Tx ) = 0. When N Tx ≥ N Cri Tx , there exists an optimal number of UEs K opt ≥ 1 which maximizes the BS energy efficiency asη Max ZF . K opt is solved by the integer closest to the root of the equation G (K, N Tx ) = 0. When N Tx < N Cri Tx , the maximum BS energy efficiencyη Max ZF is achieved with the number of UEs K = 1. Moreover, the BS energy efficiencŷ η ZF decreases with the increase of the number of UEs K.
When G (1, 100) < 0, there exists an optimal number of UEs K opt which maximizes the BS energy efficiency aŝ η Max ZF . When the number of antennas N Tx is given, the optimal number of UEs K opt is solved by the integer closest to the root of the equation G (K, N Tx ) = 0.
Proof: The proposition is proved in Appendix.
Note that in the above proposition and the corresponding appendix, the minimum number of antennas is set as 100 considering the massive MIMO scenario. Based on the results in [16], the approximation in (28) is guaranteed to hold when the number of antennas is larger than or equal to 100. Based on the Appendix, G (1, N Tx ) is a monotonous decreasing function with respect to N Tx . Utilizing the bisection method, the critical number of antennas searching (CNAS) algorithm is developed to solve the critical number of antennas N Cri Tx .

end Begin
When the critical number of antennas N Cri Tx is obtained by the CNAS algorithm, the optimal number of UEs K opt is solved by G (K, N Tx ) = 0 which maximizes the BS energy efficiencyη Max ZF . Considering G (K, N Tx ) monotonously decreases with the increase of the number of UEs K, the UE number optimization (UENO) algorithm is developed to solve the optimal number of UEs K opt by the bisection method.  Table I.
The EEHP algorithm is proposed in Section III to maximize the BS energy efficiency. To tradeoff the energy and cost efficiency of the BS RF circuits, the EEHP-MRFC algorithm is developed in Section IV. To analyze the proposed EEHP and EEHP-MRFC algorithms, the energy efficient digital precoding (EEDP) algorithm and sparse precoding algorithm are simulated for performance comparisons. In the EEDP algorithm, the energy efficiency of 5G wireless communication systems is obtained by substituting the optimal digital precoding vectors b opt k derived from the EEHP-A algorithm into (9). The sparse precoding algorithm in [11] is also simulated, which implements a spectrum efficiency maximization hybrid precoding scheme.
The two proposed algorithms along with the EEDP algorithm, the sparse precoding algorithm and the traditional ZF precoding algorithm are compared in Fig. 3. In order to better compare the energy efficiency performance of the algorithms,  the total transmit power for all the UEs is used as the comparison standard [19], [43]. For each algorithm, the energy efficiency first increases with the increasing total transmit power. When the total transmit power exceeds a given threshold, the energy efficiency starts to decrease. The variation tendencies of the curves coincide with the previously published results [19], [43]. When the total transmit power is fixed, the EEDP algorithm has the highest energy efficiency which is consistent with the analysis result in Section III. The ZF precoding algorithm has the lowest energy efficiency because it uses the same number of RF chains as antennas, which leads to the highest power consumption for the RF chains. Besides, the sparse precoding algorithm performs more poorly than the EEHP algorithm but outperforms the EEHP-MRFC algorithm in terms of energy efficiency for 5G wireless communication systems. Although the energy efficiency of the EEHP-MRFC algorithm is not the best result, this algorithm is valuable for reducing the transmitter cost and the design complexity by minimizing the number of RF chains. In practical applications, the selection of the EEHP algorithm or the EEHP-MRFC algorithm depends on the desired tradeoff between the energy and cost efficiency for 5G wireless communication systems. Fig. 4 illustrates the BS energy efficiency with respect to the number of transmit antennas considering different numbers of multipath components. The sparse precoding and EEDP algorithms are also simulated to compare with the two proposed EEHP and EEHP-MRFC algorithms. It can be seen that the BS energy efficiency increases with increasing numbers of the transmit antennas. Meanwhile, increasing the number of multipath components yields the highest BS energy efficiency when the number of transmit antennas is fixed. Decreasing energy efficiency results are yielded by the algorithms in the order EEDP, EEHP, sparse precoding and finally the EEHP-MRFC algorithm, respectively.
The BS energy efficiency with respect to the number of RF chains is shown in Fig. 5. As for the EEDP and EEHP algorithms, the BS energy efficiency first increases then decreases with increasing numbers of the RF chains. For the sparse precoding and EEHP-MRFC algorithms, the BS energy efficiency always decreases when increasing the number of RF chains. When the number of multipaths is fixed as 30 and the number of RF chains is less than or equal to 26, the energy efficiency of the EEHP-MRFC algorithm is larger than that of the sparse precoding algorithm. Further, when the number of RF chains is larger than 26, the energy efficiency of the EEHP-MRFC algorithm is always less than that of the sparse precoding algorithm.
In Fig. 6, the spectrum efficiency as a function of the number of RF chains and the number of multipaths is illustrated. It can be seen that sparse precoding algorithm has the highest spectrum efficiency. As for EEDP and EEHP algorithms, the spectrum efficiency increases with increasing the numbers of the RF chains. Considering that the number  of RF chains always equals the number of UEs in the EEHP-MRFC algorithm, the spectrum efficiency of the EEHP-MRFC algorithm decreases with the increase of the number of RF chains. When the number of the RF chains is fixed, decreasing spectrum efficiency results are yielded by the algorithms in the order sparse precoding, EEDP, EEHP, and finally the EEHP-MRFC algorithm, respectively. The impact of the number of transmit antennas and the number of UEs on the BS energy efficiency is investigated in Section IV.B. Based on the CNAS and UENO algorithms, numerical simulations are shown in Fig. 7. Without loss of generality, the number of transmit antennas are configured as 100, 150 and 200, respectively. The power consumed by other parts of the BS is configured as P C = 20W in Fig. 7. The BS energy efficiency first increases then decreases with increasing of the number of UEs. The maximums of the BS energy efficiency correspond to the optimal numbers of UEs 35, 50 and 55, respectively, which is consistent with the optimal number of UEs obtained from the proposition.
VI. CONCLUSION In this paper, the BS energy efficiency considering the energy consumption of RF chains and baseband processing is formulated as an optimization problem for 5G wireless communication systems. Considering the non-concave feature of the objective function, an available suboptimal solution is proposed by the EEHP algorithm. To tradeoff the energy and cost efficiency in RF chain circuits, the EEHP-MRFC algorithm is developed for 5G wireless communication systems. Based on the CNAS and UENO algorithms, the energy efficiency of 5G wireless communication systems can be maximized by optimizing the number of UEs and BS antennas, which is easily employed in the user scheduling and resource management schemes. Compared with the maximum energy efficiency of conventional ZF precoding algorithm, numerical results indicate that the maximum energy efficiency of the proposed EEHP and EEHP-MRFC algorithms are improved by 220% and 171%, respectively. Moreover, the difference between the EEHP algorithm and the EEHP-MRFC algorithm is illustrated by numerical simulation results. Furthermore, our results provide some available suboptimal energy efficiency solutions and insights into the energy and cost efficiency of RF chain circuits for 5G wireless communication systems. For the future study, we will try to investigate the energy and spectral efficiency optimization of 5G radio frequency chain systems in the multi-cell scenario.

Proof to Proposition:
To simplify the derivation, some variables are defined as follows: z = 1 K , z ∈ ( 0, 1] , a = P out NTxπ 2 −π+4 4 , b = 1 α P out + P ′ C and c = (P RF + P BB ). The derivative ofη ZF with respect to z is given by If f (1, a) 0, then f (z, a) 0, z ∈ ( 0, 1] . Furthermore, we have the result dηZF dz 0. Therefore,η ZF monotonously increases with increase of z ∈ ( 0, 1] . In other words,η ZF monotonously decreases with the increase of K. In this case, the BS energy efficiencyη ZF is maximized by K = 1. If f (1, a) < 0, there must exist an optimal value z opt , z opt ∈ ( 0, 1] , which ensures f (z opt , a) = 0. The integer closest to 1/z opt is the optimal number of UEs K opt which maximizes the BS energy efficiencyη ZF .
The differential of f (1, a) with respect to a is derived by (34) implies that ∂f (1,a) ∂a decreases with increase of a. When the number of transmit antenna is configured as N Tx = 1, i.e. a is minimized as a min , the corresponding differential result is ∂f (1,a) ∂a a=amin < 0 considering the practical value range of P out , P ′ C , P RF and P BB [10], [25], [44], [45]. Therefore, we have the result ∂f (1,a) ∂a < 0 for all available values of a. As a consequence, f (1, a) monotonously decreases with increase of a ∈ [a min , ∞).
When f (1, a min ) 0, i.e. G (1, 100) 0, there exist a critical number of antennas N Cri Tx which ensures f (1, a) NTx=N Cri Tx = 0. When N Tx N Cri Tx , there exists an optimal value z opt which ensures f (z opt , a) = 0 and maximizes the BS energy efficiencyη ZF . As a consequence, the integer closest to 1/z opt is the corresponding optimal number of UEs K opt . When N Tx < N Cri Tx ,η ZF monotonously decreases with the increase of K. In this case, the BS energy efficiencyη ZF is maximized by K = 1.