On Ergodic Energy Efficiency of mmWave Heterogeneous Cell-Free Systems With ZF Hybrid Precoders

Energy efficiency (EE) takes a significant place in the upcoming sixth generation (6G) mobile networks. Cell-free systems have demonstrated impressive performance and attracted much attention in academia and industry. Various types and numbers of access points (APs) can be deployed in heterogeneous cell-free systems, operating at millimeter-wave (mmWave) frequencies. But, the instantaneous EE cannot give a full insight into these systems. To tackle this issue, in this article, we study the ergodic EE of mmWave heterogeneous cell-free systems. Since the derivation of a general closed-form expression for the ergodic EE is intractable, a semi-closed form expression was achieved via discretization, and its effects were investigated. Based on this expression, two specific cases of gain were obtained to describe the EE performance of the mmWave heterogeneous cell-free systems: the first determines the necessity to employ mmWave heterogeneous cell-free systems, and the second guides their deployment. Our simulation results validate the effectiveness of the two gain cases.


On Ergodic Energy Efficiency of mmWave
Heterogeneous Cell-Free Systems With ZF Hybrid Precoders Jichong Guo, Member, IEEE, Dekun Zhang, Chen Cui , Member, IEEE, and Xiqing Liu , Member, IEEE Abstract-Energy efficiency (EE) takes a significant place in the upcoming sixth generation (6G) mobile networks.Cell-free systems have demonstrated impressive performance and attracted much attention in academia and industry.Various types and numbers of access points (APs) can be deployed in heterogeneous cell-free systems, operating at millimeter-wave (mmWave) frequencies.But, the instantaneous EE cannot give a full insight into these systems.To tackle this issue, in this article, we study the ergodic EE of mmWave heterogeneous cell-free systems.Since the derivation of a general closed-form expression for the ergodic EE is intractable, a semi-closed form expression was achieved via discretization, and its effects were investigated.Based on this expression, two specific cases of gain were obtained to describe the EE performance of the mmWave heterogeneous cell-free systems: the first determines the necessity to employ mmWave heterogeneous cell-free systems, and the second guides their deployment.Our simulation results validate the effectiveness of the two gain cases.
Index Terms-Cell-free systems, energy efficiency, heterogeneous systems, millimeter-wave communications.

I. INTRODUCTION
M ILLIMETER-wave (mmWave) wireless communica- tions can allow us to dramatically improve the transmission rate by exploring the spectra that are not overexploited [1].However, the double-edged transmission characteristics of mmWave frequency bands, like its path loss, call for new networks and technologies to ensure reliable wireless communication.Among these emerging network frameworks, the cell-free system -a key enabling technology for the sixth generation (6G) mobile networks [2] -has been gaining attraction for its impressive ability to improve system performances, such as spectrum efficiency (SE) and bit error rate (BER).
Cell-free systems can be viewed as a modified version of distributed systems, first reported in papers [3], [4].In these works, all the access points (APs) were connected to a central processing unit (CPU) and served all users over the same time-frequency resources.However, this canonical form leads to impractical implementation, and many effective solutions have been proposed to circumvent this problem, such as the user-centric [5], [6] and scalable cell-free framework [7], [8], and limited fronthaul [9], [10].Owing to its excellent compatibility, many critical technologies have been imported into the cell-free systems, e.g., non-orthogonal multiple access (NOMA) [11], [12], reconfigurable intelligent surfaces (RIS) [13], [14], and unmanned aerial vehicles (UAVs) [15], [16].
The evolution of communication generations and the diversification of services cause significant heterogeneity for the current wireless communication systems.In the heterogeneous cellular networks, various classes of low power nodes (LPNs) are distributed throughout the macro cell network [17], [18].Similar to the heterogeneous cellular case, diverse types of wireless APs are randomly distributed in the heterogeneous (also called X-assisted) cell-free networks [19], [20], [21].Unlike the traditional cell-free systems, these wireless APs are not mandatory to change real-time information via a CPU.For example, it is easy to cooperate between the macrocell and femtocell base stations (BSs) [19], while intractable with UAVs worked as mobile BSs [21].The ongoing research on the heterogeneous cell-free system is at a very early stage.
Energy efficiency (EE) is a critical concern in the sustainable development of wireless communications [22], [23].Over the past few years, a lot of research has been conducted to design energy-efficient schemes and to evaluate EE performances [24], [25], [26].High EE is a significant design target in cell-free systems, with continued research providing us with several fruitful results in this direction.Specifically, several representative works are listed in Table I.As shown in Table I, most studies considered instantaneous EE as the design target, while only a few of them have focused on average EE [19], [27].For example, to best utilize the energy harvested at each AP, Hamdi and Qaraqe investigated the issues of online energy cooperation and management problem [27].They proposed an efficient online algorithm based on energy prediction to minimize power consumption in a given time period.Kim and Shim studied the maximum EE of mmWave heterogeneous cell-free systems under limited feedback [19].An energy-efficient dominating path selection algorithm is proposed to achieve more than 80% EE improvement over the conventional channel state information (CSI) feedback-based schemes.
The main reason for the focus on instantaneous EE in the aforementioned studies is that in traditional communication systems, wireless access is often through BSs with wide coverage, excellent computation capability, and complex structure, which are hard to deploy in real-time.Also, energy-efficient schemes are usually designed on the instantaneous CSI to maximize EE.However, the instantaneous EE is insufficient to characterize the randomness of the EE performance.As an alternative method of analysis, several studies applied the average EE performance in their simulation results.
Development of theory and industry have changed the concept of network with the introduction of the ultra-dense network (UDN) [31], [32], [33] and cell-free systems [3], [4], [34] then the wireless accesses widely take LPNs, such as micro BSs, pico evolved node Bs (eNBs), Femtocells, relays and APs.The miniaturized and intensive wireless accesses have allowed real-time deployment.This situation necessitates the study of the ergodic EE, instead of instantaneous EE, in communication systems, and to the best of our knowledge, the ergodic EE of heterogeneous cell-free systems is still an open issue.The ergodic EE of a centralized mmWave system with a hybrid pre-coding scheme is studied in our previous work [35].Following it, here, we discuss the more complicated ergodic EE of the mmWave heterogeneous cell-free systems.The major contributions of this work are three-fold: r Firstly, we give the instantaneous EE by modeling an mmWave heterogeneous cell-free system in which various types and numbers of APs can be deployed.Based on this expression, a generic ergodic EE with the statistical information is achieved; r The ergodic EE with the statistical information is quite complicated to calculate.Several relaxing methods are employed to derive a semi-closed ergodic EE, aiming to provide a complete insight into the systems.The effect of the main relaxing method on the ergodic EE is theoretically analyzed; r Two gain cases are derived to demonstrate the importance of heterogeneous cell-free systems and to guide their deployment.The effectiveness of these two gain cases is verified through simulations.The remainder of the article is organized as follows: Section II describes a generic mmWave heterogeneous cell-free system model and its corresponding transmit signals.Based on the system model, Section III formulates a generic instantaneous and ergodic EE, and Section IV takes several simplification steps to get a semi-closed expression.The two gain cases and their analysis is also shown in Section IV.Simulation results are presented in Section V, followed by the conclusions in Section VI.
Notation: a, a, A, A, R and C stand for a variable, a column vector, a matrix, a set, and the real and complex fields, respectively.A H , A * , and A T represent the conjugate transpose, conjugate, and transpose of A, respectively, while I refers to the unit matrix.|A| is the cardinality of the set A. Denoted by Ones(x, y) is the x × y matrix with each element being one.diag(•) is the diagonalizing function, and angle(•) is the angular function.And • F is the Frobenius norm.CN (•, •) and N (•, •) denote a circularly-symmetric complex Gaussian distribution and a real Gaussian distribution, respectively.

II. SYSTEM DESCRIPTION
A generic illustration of the deployable heterogeneous cellfree system is shown in Fig. 1.The type of APs includes both hardware and software aspects.In other words, various types of APs may be equipped with different transmitter structures and take diverse technologies, resulting in heterogeneous gains.Considering that hardware is difficult to adjust, here we mainly focus on the heterogeneous gain achieved by hardware and set each type of APs to take the same hybrid pre-coding algorithm.
The numbers of APs and their types are denoted as M and T , where T ≤ M .The special case T = 1 indicates the traditional isomorphic cell-free systems.S t is the tth type set of APs, and it corresponds to a complete configuration: working frequency f t , number of transmit antennas N t , number of radio frequency (RF) chains l t , maximum transmit power P t and so on.If the mth AP belongs to the tth type set of APs, i.e., m ∈ S t , it means that the mth AP is equipped with the configuration of S t .K users are served in an area through this deployable heterogeneous cell-free system, and each user is equipped with a single receiver antenna.The number of served users meets K ≤ t |S t |l t , where the equal sign establishes when all the APs serve different users.APs are assigned to the kth user by rules, which are contained in set A k as explained in many previous works [5], [6].Therefore, the users served by the mth AP are contained in a set A −1  m such that |A −1 m | ≤ l t .The modulated signal of the kth user is denoted by Hence, the total modulated signal at the mth AP is denoted as . A hybrid transmit architecture always bridges the gap between the digital and analog architectures, and hence, its pre-coding matrix is generic.The pre-coding matrix of the analog transmit architecture is obtained via the digital pre-coding part of the hybrid transmit architecture as a unit matrix.Hence, the mth

AP takes a local hybrid precoder, consisting of an analog part
Considering that the antenna gain just affects the value of EE, instead of the trend.In other words, the antenna gain will not affect the theoretical analysis.Then gain of each antenna element is assumed to be normalized.After processed by the power allocation matrix P m ∈ R |A −1 m |×|A −1 m | and hybrid pre-coding matrices F m and W m , the transmitted signal at the mth AP is expressed as ( Suppose that fading is constant in a block, and the channel between the mth m ∈ S t AP and the kth user is denoted as . Since the fading is composed of large-scale and small-scale fadings, we have , where the diagonal matrix I NB indicates the existence of rays.L m,k ∈ C N cl,m,k ×1 is the large-scale fading matrix.Also, H small m,k ∈ C N t ×N cl,m,k represents the small-scale fading matrix, in which N cl,m,k is the number of distinguishable paths.Without loss of generality, a column of H small m,k is the product of a small-scale fading coefficient and an array steering vector, which is decided by the transmit antenna array and the direction of arrival (DoA) of the ray.In the mmWave communication systems, when the distance between transmitter and receiver is smaller than a threshold value, such as 50 meters, the outage probability of line-of-sight (LoS) path is less than 30% [36].With the user-centric cell-free systems, we can select the APs owning the LoS path to the kth user, which forms a set A k .So when the number of APs is large, it is able to ensure the situation that mmWave LoS path exists with a high probability.Therefore, it is reasonable to set one diagonal element of I NB corresponding to the LoS path being one.
Generally, L m,k can be simplified to Lm,k Ones(N cl,m,k , 1) by taking the centroid of all the fadings, with Lm,k being the mean large-scale fading coefficient.According to the practical measurements in the mmWave communication systems [36], [37], the number of distinguishable paths is small (1 with approximately 48% probability for 28 GHz) and the power difference between the LoS path and the non-line-of-sight (NLoS) path is quite large (at least 5 dB).Then Lm,k can be approximated by the LoS large-scale fading coefficient, denoted by L m,k .This gives us where X σ represents the logarithmically Gaussian-distributed shadow fading term, with a zero-mean and variance σ 2 .The path loss exponent n 0 is always decided by the specific scenarios.In (2), b is an optimization parameter that captures the slope and f 0 is a fixed reference frequency.Besides, c and λ are the speed of light and the carrier wavelength, respectively.The distance of LoS path between the mth AP and the kth user is denoted as r m,k .
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Then the received signal of the kth user can be written as, where W m,k ∈ C l t ×1 is the kth column of W m and P m,k ∈ R is the kth diagonal element of P m .And z k ∈ C represents the additive white Gaussian noise (AWGN) for the kth user, with z k ∼ CN (0, σ 2 noise ) with σ noise being the standard noise deviation.

III. ENERGY EFFICIENCY UNDER MMWAVE HETEROGENEOUS CELL-FREE SYSTEMS
In this section, we derive the EE expression for mmWave heterogeneous cell-free systems.

A. Instantaneous Energy Efficiency Formulation
In heterogeneous cell-free systems, the instantaneous sum rate and power consumption are achieved on the premise that the APs states are provided due to the fact that various types of APs are deployable.Let S and H denote the set of types of APs and channel matrices, respectively, such that S = {S 1 , . . ., S T } and H = {H 1 , . . ., H M }.Based on the received signal in (3), the achievable rate of the kth user can be expressed as where B is the bandwidth (Hz) and Moreover, the power of interference and noise is given as The number of users is far less than the number of APs, and the receiver architectures for users are miniaturized, i.e., there is only minimal power consumption at the user equipment (UE) side, which can be neglected.Therefore, focusing on the active components of the transmitter, the power consumption of the mth AP is found to be ) where N PS,t is the number of phase shifters.This is a function of l t and N t , which is related to the transmitter architecture.For example, we have N PS,t = l t N t under the fully connected architecture, and N PS,t = N t under the subarray architecture.Besides, P BB , P RF and P PS are power costs of the baseband, an RF chain, and a phase shifter, respectively.The function P PA (•) represents the power consumption of a power amplifier (PA) at the nth transmit antenna, and it is given as the product of the output power (P out n ) and the PA efficiency.Generally, the PA efficiency model can be divided into linear and nonlinear components.Furthermore, the linear PA efficiency often exists in the lower-frequency communication systems, while the nonlinear PA efficiency exists in the mmWave and sub-Terahertz bands.Therefore, the instantaneous EE of the heterogeneous cellfree systems can be written as where R(H|S) is the sum achievable rate (bits/s) with R(H|S) = K k=1 R k (H|S).Similarly, P C all (H|S) is the total power consumption (Watts) and As illustrated in Introduction, the instantaneous EE helps to design the instantaneous signal process schemes like hybrid precoder and power allocation, etc.However, considering that the heterogeneous cell-free systems are deployable, i.e., immobile only for a short period of time, it is necessary to measure the EE performance from the statistical aspect.

B. Ergodic Energy Efficiency Formulation
According to the analyses in Section III-A, the sum rate and power consumption are functions of deployable APs.Generally, the hybrid pre-coding and power allocation matrices are designed according to the channel matrix.Furthermore, the output power at each transmit antenna is decided by the hybrid pre-coding and power allocation matrices, which then affects the power consumption of a PA.So the sum rate and power consumption are also functions of channel matrices.
Based on the instantaneous EE in (7), the ergodic EE of the deployable heterogeneous cell-free systems is defined as where E H [•] represents the expectation of " • " over H. Considering that the channel model consists of mutually independent large-scale fading and small-scale fading, let L = {L 1 , . . ., L M } denote the set of large-scale fading matrices, and = {h 1 , . . ., h M } the set of small-scale fading matrices.Then, (8) can be transformed to in which E L [.] and E [.] are the expectation functions of L and , respectively.Through integration, ( 9) becomes (10) shown at the bottom of the next page, where f (L 1 , . . ., L M ) and f (h 1 , . . ., h M ) are the joint probability density functions (PDFs) of {L 1 , . . ., L M } and {h 1 , . . ., h M }, respectively.As shown in Table I, one feasible way to achieve high EE consists of two steps.The normalized hybrid pre-coding matrices are designed to eliminate the interference between multiple users and data streams, and the optimal power allocation is calculated to maximize EE.The unitary process is followed when APs and users are both equipped with multiple antennas, and the zero-forcing (ZF) process is taken when each AP or user is equipped with a single antenna.
Compared with large-scale fading, small-scale fading plays a significant role in interference.In order to eliminate interference, the digital part of the ZF hybrid precoder at the mth AP is given as The analog part of the ZF hybrid precoder is often set to be the matched filter (MF), given as F m = angle(h * m ).This allows us to transform the sum rate as, (12) with the signal-to-interference plus noise power ratio (SINR) The interference part P I k exists for the ZF hybrid precoder locally at each AP.As m ∈ S t , we can see this interference as 6), we can see that the sum power consumption can be divided into the constant and the variable parts of the channel matrix.The former can be written as while the latter can be written as where F is the power output of the nth antenna at the mth AP, i.e., P out n in (6).When the spatial conditions are satisfied (i.e., when APs and users are dispersedly distributed), we see that the fading is independent between the various APs and users.This means that the joint probability density functions f (L 1 , . . ., L M ) and f (h 1 , . . ., h M ) can now be written as Therefore, (10) can be rewritten into (16) shown at the bottom of this page, where m=M m=1,k∈A −1 m is the successive multiplication operation.(16) gives the expression for the ergodic EE of the deployable heterogeneous cell-free systems with ZF hybrid precoders.

IV. CLOSED FORM OF ERGODIC ENERGY EFFICIENCY
In this section, a semi-closed expression of the ergodic EE is given first after several simplification methods.Based on it, two cases of heterogeneous gain are also derived in order to have direct insights into the ergodic EE.Further, we theoretically analyze the effects of the main simplification method on the ergodic EE performance.

A. Semi-Closed Expression of Ergodic EE
Due to the difficulty in directly obtaining instructive conclusions from (16), it is acceptable to make a few mathematical simplifications to the expression of the ergodic EE.
Considering a fully-connected architecture, here, we assume N PS,t = l t N t .Since each user has a singular data stream, the number of working RF chains is equal to that of serving users.Hence, the constant part of the sum power consumption becomes Following from the workings in [26], using the traditional system parameters, the nonlinear PA efficiency can be relaxed to a linear function within an acceptable error.Specially, the maximum total error caused by the linearization operation of PA efficiency is (N A P PA max )/(8π), where P PA max is the maximum output power of a PA and N A is the number of total transmit antennas.With the typical parameters, i.e., N A = 128 and P PA max = 100 mW, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
power consumptions of the baseband, RF chain, phase shifter are 200 mW, 120 mW, 20 mW, respectively, we can get P C (S) of ( 14) and P V (H|S) of ( 15) in the manuscript.Then it is able to calculate that (N A P PA max )/(8π) accounts for a very small part of the total power consumption (P C (S) + P V (H|S)), which approximately equals to 3.52%.So the linearization of a PA efficiency is acceptable.
Assuming ρ t to be the approximate PA efficiency coefficient taken by the APs in S t , the variable part of the sum power consumption in (15) can be simplified to (18) As the power allocation matrix is a joint function of the smallscale and large-scale fadings, it leads to difficulty in analyzing ξEE (S).Taking the power to be discrete, with m ∈ S t and k ∈ A −1  m , P m,k falls in the range [0, P t ], and [0, P t ] is equispaced divided into I segments.
When P m,k is located in the ith segment, we set P m,k to be the sum of its small initial value (P i m,k ) and the error (Δ i m,k ), i.e., P m,k = P i m,k + Δ i m,k , where 0 ≤ Δ i m,k ≤ P t /I.The error between P m,k and P i m,k decreases with increasing of I.When I is large enough, with In a similar manner, we are able to arrive at that Denoting the number of segments by J and L, let η j m,k (or μ l m ,k ,k ) be the left starting point of the j(or l)th segment and Effectiveness of discretization will be analyzed in Section IV-C and verified by simulation results in Section V-A.
With large enough I, O, J, and L, we are able to derive an approximate expression of γ k , that is ), which represents the interference.Hence, the approximate power consumption becomes The probability of the case where , i , j, l, o, o ).Then discrete ergodic EE can be therefore expressed as, where We know that a logarithmic function can be approximated by a piecewise function, allowing us to express the sum rate by a piecewise linear function of SINR, i.e., log 2 (1 + γ k ) ≈ β s γ k + C s .We have β s (1 ≤ s ≤ S) as the slope of the line in the sth segment and C s as the constant in the sth segment, with the total number of segments denoted as S. When the number of segments is large enough, a logarithmic function can be replaced by a piecewise function with a negligible error.The partition of segments can be uniform and non-uniform, which is decided upon by a design rule [39].The non-uniform partition can focus on the value range of SINR with high probability at the cost of complexity, while the uniform partition is simple.Following this, (21) becomes where δ k,s = 1 when γ k lies in the sth segment.Otherwise, δ k,s = 0.This semi-closed expression is useful for analyzing the ergodic EE of heterogeneous cell-free systems.Based on it, heterogeneous gain will be discussed in Section IV-B.And several insights will be given in Sections V-B, V-C, and V-D, where a part of easily configured variables are taken as examples, i.e., the transmit power, and the numbers of APs and users, which can guide the selection of system parameters.
The complexity of discretization of the integral is measured by the time complexity, which is related to the number of cyclic operation.According to the semi-closed ergodic EE in (22), calculation of both the instantaneous EE and the probability of discretized variables are required to realize discretization of the integral.And the probability of discretized variables can be given through realizing instantaneous EE of discretized variables for many times.Then it is necessary to calculate the number of cyclic operation required to achieve instantaneous EE of discretized variables.Since each AP independently designs the hybrid precoding and power allocation matrix, which are supposed to be achieved by the closed formulas, it can be considered to circulate M times.Then the corresponding time complexity can be written as O(NM), where N is the number of channel realizations.The method of bisection is used to find the order number of segment where a discretized variable lies.Then the maximum time complexity of discretization of variables is expressed as Therefore, the whole time complexity to achieve instantaneous EE of discretized variables is which is also the whole time complexity of discretization of the integral.Generally, the number of channel realizations (N ) plays a main role in the whole time complexity.As shown in the simulation results of Figs. 2 and 3, when the number of channel realizations exceeds a threshold, such as 100, the fluctuation caused by discretization tends to be stable.Therefore, the complexity of the discretization of the integral for evaluating the ergodic EE is acceptable.

B. Heterogeneous Gain
Generally, the AP with sophisticated architecture performs powerful capability, and the parameters, such as P t , N t , l t and etc, are often larger than those of APs with simpler architecture.Thus, this tends to have an increased manufacturing complexity and cost, leading to fewer number of APs in heterogeneous cell-free systems.Therefore, the ergodic EEs caused by the APs with the most sophisticated architectures and the APs with the simplest architectures represent two extremes of ergodic EEs in heterogeneous cell-free systems.The larger the difference between the two extremes, the more ergodic EE gain is achieved by the heterogeneous cell-free systems.A heterogeneous gain is used to describe the difference between two extremes.Since the APs with the most sophisticated architectures own the maximum parameters, such as P t .Let S max = S t * and S min = S t , where t * = max t {P 1 , . . ., P t , . . ., P T } and t = min t {P 1 , . . ., P t , . . ., P T }, respectively.In this manner, we have max{a, b} = a (min{a, b} = b) when a ≥ b; otherwise, we will have max{a, b} = b (min{a, b} = a).Here, we define a heterogeneous gain as which indicates whether a heterogeneous system is necessary with respect to ergodic EE.If ξ G > 1, it means that a heterogeneous system yields ergodic EE gain; otherwise, an isomorphic system does.
In fact, it is intractable to calculate the accurate ξ G .From an engineering perspective, a simple and feasible solution to calculate ξ G is desired.So, it is reasonable to calculate ξ G using (22), called ξ G,D .This still proves to be quite difficult, and several simplification methods are described below.
In the low SINR region, it is easy to derive that log 2 (1 + γ k ) ≈ γ k , which means β s = 1 and C s = 0. Here, the interference term can be ignored.Since β s and C s are both given, and the change in system parameters and signal processing mainly affects the useful signals, we chose to study the heterogeneous gain in the low SINR region.Now, (22) becomes Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Since the power allocation is used to maximize the ergodic EE, equal power allocation is a lower bound.When equal power allocation is employed, the ergodic EE is further simplified to ξD,LW where Since only a few APs own large and sophisticated architectures, longer distances between APs and users have a larger probability.That is, in comparison to S max , the advantage of using S min lies in the smaller value of large-scale fading, experienced by the transmitted signals from APs belonging to S min .If the large-scale fadings are set to be the same for S max and S min , this situation is beneficial for S max .Compared to small-scale fadings, large-scale fadings perform larger orders of magnitudes, which subsequently affects the received power.Thereafter, if the large-scale and small-scale fadings are set to be the same for S max and S min , this situation is still beneficial for S max .
Therefore, by neglecting L o m,k and η j m,k , we can have the first case of ξ G,D LW , that is where P C all (S max ) = m |N max P PS + P max ρ max ).P max , N max , and ρ max are the maximum output power, number of transmit antennas, and the PA efficiency coefficient corresponding to S max , respectively.Similarly, it is easy to derive P C all (S min ).Without the complicated CSI and statistical information, this version of the heterogeneous gain can help to determine whether a heterogeneous system is beneficial.The details are as follows, Remark 1 (for the first case of the gain): When ξ G,D,1 LW ≈ 1, it is better to employ an isomorphic system, whereas for ξ G,D,1 LW << 1 or ξ G,D,1 LW >> 1, it is reasonable to employ a heterogeneous system to achieve higher ergodic EE.
It is noted that ξ G,D,1 LW is useful but not accurate enough for determining the necessity of a heterogeneous system.Since the large-scale and small-scale fadings are mutually independent, we have p(j, o) = p( j Based on ξD,LW EE (S) in ( 27), we can obtain the second case of the heterogenous gain ξ G as, When N J and N O are set appropriately, the value of ξ G,D,2 LW approaches ξ G,D LW , thus we can get a relatively accurate value of ξ G .Therefore, ξ G,D,2 LW is a stronger determinant for the necessity of a heterogeneous system.
Moreover, with the help of ξ G,D,2 LW , we present a heuristic method to guide the deployment of heterogeneous cell-free systems: Divide the users into T clusters according to a clustering algorithm.Hence, the whole coverage area is divided into corresponding T parts.In each part of the area needed to be deployed, successively change S max = S t according to S while keeping S min constant, and further calculate the corresponding ξ G,D,2 LW .The S max with the maximum ξ G,D,2 LW is considered the recommended deployment in the corresponding part of the area.Thereafter, update S = S/S max and S min (one set in S with the minimum transmit power).The remaining parts of the area are also to be updated by removing the already deployed parts.Repeat the forward steps until each part of the area owns a serviced AP set.APs in a serviced AP set are uniformly distributed in the corresponding part of the area.
Remark 2 (for the second case of the gain): If ξ G,D,2 LW > 1, it is necessary to employ a heterogeneous system to achieve higher ergodic EE.Larger the ξ G,D,2 LW , the better EE performance.Besides, ξ G,D,2 LW is useful for guiding the deployment of heterogeneous cell-free systems.
Although Remark 1 and Remark 2 were derived in the low SINR regions, they remain valid in the high SINR regions, as verified by the simulation results.

C. Effect of Discretization
It is necessary to analyze the effects of the simplification methods on the system performance.As shown in Section III-A, the main simplification method was to discretize the variates and statistical information in the ergodic EE.Since the heterogeneous gain indicates the ergodic EE of the heterogeneous cell-free systems, we focus on the effect of discretization on it.
Based on ( 21) and ( 23), the heterogeneous gain with discretization can be further expressed as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
To have a direct insight into the effect of discretization on the heterogeneous gain, here, we use (27) to calculate ξ G,D .Since the changes in the small-scale fading are faster than that in the large-scale fading, it is reasonable to only consider the discretization of variates influenced by the former.In other words, the large-scale fading is assumed to be constant.Then we are able to arrive at Recalling that The ergodic EE with statistical information can then be written as (31) shown at the bottom of this page.
The error in Repeating this process, we obtain Further, ( 29) can be expressed as Although ξ G,D can be larger than ξ G , (35) does not indicate that the discretization method yields a gain.Factually, only the equality relationship between ξ G,D and ξ G is considered in our analysis.In particular, when J and N J approach infinity, we can arrive that J j=1 ( max + σ 2 max and μ 2 min + σ 2 min , respectively.Thus, we can see ξ G,D = ξ G .In other words, the discretization method makes sense only if both J and N J are large enough.

V. SIMULATION RESULTS AND DISCUSSIONS
In this section, we study the ergodic EE performance of the deployable heterogeneous cell-free systems.Consider a square area with a side length of 10000 meters.The channel model is set according to [38].The approximate PA efficiency coefficient is given as ρ t = 4/π (∀t).The simulation parameters are detailed in Table II.
Furthermore, the numbers of users and types of APs are set as K = 16 and T = 4, respectively.For fairness, the relationships of parameters between various types of APs are Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II SIMULATION PARAMETERS
2)N t , and P t−1 = (1/2)P t .The ZF hybrid precoders with S 1 , S 1 , S 1 , S 1 and S 3 , S 3 correspond to isomorphic systems, while those with S 1 , S 2 , S 3 , S 4 and S 3 , S 4 correspond to heterogeneous systems.Each user is served by all the APs, and each AP employs equal power allocation.A heterogeneous system is then deployed according to the proposed heuristic method in Section IV-B.
Simulation results in Fig. 2 are achieved by ξ G and ξ G,D , where ξ G is achieved by (23) with ξEE (S) showed in (16), and ξ G,D is achieved by (23) with ξD EE (S) showed in (22).   in (7), with the number of channel realization being 1000.With given S max and S min , the value of ξ G,D,1 LW is achieved according to (26) and value of ξ G,D,2 LW is achieved according to (28).Then Fig. 13 is obtained by using ξ G,D,2 LW .

A. Feasibility of Discretization
The feasibility of discretization is analyzed in Fig. 2 to investigate heterogeneous gain.Since the value range of a variable is given, the effect of the discrete step on the heterogeneous gain is the same as that of the number of segments.As shown in Fig. 2, ratio between ξ G,D and ξ G fluctuates around one.When the discrete step equals 0.1 or 0.01, the ratio between ξ G,D and ξ G approaches one with the increase of the number of channel realizations, indicating that ξ G,D can approach ξ G .It keeps in line with the analysis in IV-C.However, when the discrete step equals to 1, the ratio between ξ G,D and ξ G is always smaller than one, no matter what value of the number of channel realization is.The fundamental reason is that the loss on the sum rate caused by the large discrete step cannot be neglected.The similar phenomenon can be observed in Fig. 3, which compares  values of the semi-closed expression from (22) and average EE (denoted as 'Discrete Step = 0').With the enough small discrete step, when the number of channel realization increases, the value of semi-closed ergodic EE expression approaches that of the average EE, which verifies the effectiveness of the semi-closed expression.Besides, when the number of channel realizations exceeds the threshold, such as 100, the fluctuation tends to be stable.Therefore, we select representative {0.1, 50}, {0.01, 50}, and {0.01, 100} as the discrete steps and realization numbers in the following simulations.

B. Effect of Transmit Power
Based on the semi-closed form approximation, it is able to obtain insight on the transmit power.When the transmit power is small, SINR is mainly affected by the useful signal power and linearly increases with the transmit power.The whole power consumption equals to the sum of the variable part and the constant part.The variable part is a linear function of the transmit power.Since the transmit power is small, the whole power consumption approximately equals to the constant part.So the ergodic EE performance increases in the low transmit power  region.However, when the transmit power is large, SINR cannot linearly increase with the increasing of transmit power, but whole power consumption linearly increases with the increasing of transmit power.Then the ergodic EE decreases in the high transmit power region.Therefore, based on the semi-closed form, it is able to get that the ergodic EE performance first increases and then decreases with the increasing of transmit power, which is verified by Fig. 4.
Furthermore, as shown in Fig. 4, the average EE of the heterogeneous systems is higher than that of isomorphic systems.Besides, the performance gap between the ZF hybrid precoders with S 1 , S 1 , S 1 , S 1 and S 1 , S 2 , S 3 , S 4 is larger than that between those with S 3 , S 3 and S 3 , S 4 .For the former group, i.e., S 1 , S 1 , S 1 , S 1 and S 1 , S 2 , S 3 , S 4 , S max = S 4 and S min = S 1 , we calculated ξ G,D,1 LW using (26) to be [0.2311,0.2311,0.2305,0.2253]with various transmit powers.For the latter group, i.e., S 3 , S 3 and S 3 , S 4 , S max = S 4 and S min = S 3 , then ξ G,D,1 LW is calculated to ξ G,D,1 LW = [0.5606,0.5606, 0.5603, 0.5573].Hence, we found ξ G,D,1 LW of the latter group to be larger than that of the former group.The average EE gap in the latter group was smaller, consistent with Remark 1.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.However, in both groups, we have ξ G,D,1 LW < 1, which does not accurately reflect the necessity of a heterogeneous system.By setting the discrete step and realization number, we can compute ξD,LW LW > 1 case, implying that a heterogeneous system outperforms an isomorphic system, consistent with Remark 2. And ξ G,D,2 LW of the former group is larger than that of the latter group, the heterogeneous gain is bigger in the former group, as shown in simulation results.In the following, the discrete step and realization number are set as {0.01, 100}.
Considering the heterogeneous cell-free system with S 1 , S 2 , S 3 , S 4 , it is able to get average EEs with various hybrid pre-codings, such as ZF, maximum ratio transmission (MRT), minimum mean square error (MMSE) based hybrid pre-coding schemes.As show in Fig. 4, the MMSE hybrid pre-coding performs the best, then the ZF hybrid pre-coding, and finally the MRT hybrid pre-coding.Moreover, according to Appendix A, based on the semi-closed expressions of ergodic EEs for the MMSE hybrid pre-coding, the second version of heterogeneous gain is calculated for two groups, i.e., ξ is larger than the gain brought by the second group S 3 , S 3 and S 3 , S 4 , as shown in Fig. 5.By the same way, it is able to get the second version of heterogeneous gain for the MRT hybrid pre-coding.Here will not repeat it.

C. Effect of the Numbers of APs
Then it is able to obtain insight on the number of APs according to the semi-closed form approximation.It is noted that the increasing number of APs is reflected by the increasing values of M and |A k |.Based on the semi-closed form, when the number of APs increases, it is able to find that the whole power consumption linearly increases, while the useful signal power increases in a second power function.When the transmit power is small, SINR is mainly affected by the useful signal power, and it is able to get that SINR increases with the number of APs in a second power function.Then the ergodic EE increases in this case.When the transmit power is large, SINR is both affected by the useful signal power and interference power.It means that SINR cannot increase with the increasing number of APs in a second power function, but the whole power consumption still linearly increases with the increasing number of APs.Then the ergodic EE decreases in this case.Therefore, based on the semi-closed form, it is able to get that the ergodic EE performance first increases and then decreases with the increasing number of APs, which are verified by results of Figs. 6 and 7.
Similarly, in the low SINR region, ξ small in the high SINR region, which does not agree well with the simulation results.This indicates that an accurate heterogeneous gain considering multiple interferences is required in future studies.

D. Effect of the Numbers of Users
Based on the semi-closed form approximation, it is able to obtain insight on the number of users.When the number of users increases, limited by the number of RF chains in each AP and maximum transmit power of each AP, both the number of APs serving a user and the power allocated to a user decrease under the fair rule.Based on the semi-closed form, it is able to find that the whole power consumption linearly increases with the number of users.Besides, the useful signal power increases with the number of users in a linear function, but decreases in a second power function.Then the ergodic EE performance decreases in this case.Therefore, based on the semi-closed form, it is able to get that the ergodic EE decreases with the increasing number of users, as shown in Figs. 8 and 9.
In the low SINR region, we obtain the value of ξ

E. Effect of the Numbers of Antennas
The effects of varying the numbers of transmit antennas on the ergodic EE in the low and high SINR regions are analyzed, respectively.As shown in Figs. 10 and 11, with the increasing number of transmit antennas, the average EE performances partly increase in the low SINR region while monotonously decreasing in the high SINR region.When the number of transmit antennas increases, the increasing interference in the high SINR region is larger than that in the low SINR region.This limits the sum rate performance and results in a monotonous decrease in the high SINR region.
In the low SINR regions, we computed ξ G,D,1 LW as [0.2311,0.1815,0.1542,0.1398]for the former group and [0.5606,0.5323,0.5167,0.5085]for the latter group, whereas, in the high SINR region, we computed it to be [0.2253,0.1798,0.1537,0.1397]for the former group and [0.5573,0.5313,0.5164,0.5084]for the latter group.This analysis data explains why the average EE gap of the former group is larger than that of the latter group, in line with Remark 1. Further, we computed ξ G,D,2 LW for the former and latter groups to be [6.8397, 3.3760, 4.2131, 2.6523] and [1.1002,1.6632,1.7187,1.4321] in the low SINR region, respectively, whereas, in the high SINR region, these were computed to be [4.9308,2.1846,4.0879,2.1530]and [1.1536,1.0218,1.0089,1.0214]for the former and latter groups, respectively.In line with Remark 2, the heterogeneous gain of the former group is found to be larger than that of the latter group, verified by the simulation results.

F. Effect of the Imperfect CSI
Finally, we research the effect of imperfect CSI on the ergodic EE of heterogeneous cell-free systems.In this case, the actual channel matrix between the mth AP and the kth user is expressed as , where H m,k is the estimated channel matrix.Each element of the estimation error matrix H m,k is modeled as a complex Gaussian random variable, with mean value being zero and normalized variance being σ 2 .σ 2 = 0 corresponds to the perfect CSI.The number of channel realization is set as 1000, and discrete step equals to 0.01.Simulation results are shown in Fig. 12, with the increasing value of σ 2 , which corresponds to serious imperfect CSI, the system performance becomes worse.So the semi-closed ergodic EE expression with the perfect CSI can be seen as an upper bound.Besides, when the CSI is perfect, ZF based hybrid pre-coding performs better than MRT based hybrid pre-coding, and the situation is opposite if the CSI is seriously imperfect, like σ 2 = 10.Compared to the ZF based hybrid pre-coding, the MRT based hybrid pre-coding is insensitive to the imperfect CSI.
However, a heterogeneous gain is a ratio of two ergodic EEs under various system parameters.Especially, the first case of heterogeneous gain is achieved without the complicated CSI and statistical information, which means ξ G,D,1 LW is not affected by the imperfect CSI.The second case of heterogeneous gain is achieved by the semi-closed ergodic EE expression.With the same changed CSI, the changement of two ergodic EEs are similar.So the effect of imperfect CSI on ξ G,D,2 LW is weak.As shown in the simulation results of Fig. 13, the second cases of heterogeneous gain ξ G,D,2 LW under both ZF and MRT based hybrid pre-codings change mildly, which indicates the weak effect of imperfect CSI on the heterogeneous gain.

VI. CONCLUSION
In this article, we analyzed the ergodic EE of mmWave heterogeneous cell-free systems.Based on the system model, a general ergodic EE expression of mmWave heterogeneous cell-free systems was obtained.However, as this expression is not a closed form, it is quite difficult to extract useful information from it.We followed with a few relaxation steps to obtain a special semi-closed ergodic EE expression and analyzed the effect of discretization, verifying the validity of this semi-closed ergodic EE expression.Based on this expression, we derived two cases of heterogeneous gain, describing the ergodic EE performance of mmWave heterogeneous cell-free systems: the first one describes the necessity of mmWave heterogeneous cell-free systems, and the second can accurately measure the heterogeneous gain and provide theoretical support for deploying Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
heterogeneous networks.The simulation results demonstrated the effectiveness of the two cases of heterogeneous gain.Future work will focus on the ergodic EE of mmWave heterogeneous cell-free systems with multiple receive antennas at the UE side.

APPENDIX A ERGODIC EE WITH A GENERAL HYBRID PRECODER
A generic ergodic EE with the statistical information is given in the manuscript.Based on it, a semi-closed ergodic EE is derived with several relaxing methods.Then two cases of heterogeneous gains are discussed.ZF based hybrid pre-coding works as an example to concretely illustrate corresponding processes.And these results can also apply to a more sophisticated hybrid pre-coding proposed in the literature.The reason lies in two aspects.
The first one is that a local ZF based hybrid pre-coding is taken for each AP, in order to trade off complexity and performance.This means that interferences between APs are not completely eliminated.So SINR still exists for each user in the downlink transmission.For a hybrid pre-coding proposed in the literature, such as MMSE and MRT based hybrid pre-codings, a similar SINR is achieved.
For a generic hybrid pre-coding scheme with F M and W M being the analog and digital parts, respectively, SINR of the kth user can be written as Recall the SINR expression under the ZF based hybrid pre-coding, the main difference lies in the molecule, that is The second one is that a semi-closed expression of ergodic EE is achieved by three simplified steps, i.e., linearization of a PA efficiency, discretization of the integral, segmentation of a logarithmic function, which can apply to a situation taking a more sophisticated hybrid pre-coding.Here gives detailed explanations.
a) Linearization of a PA efficiency relaxes the nonlinear PA efficiency to a linear one.Since the power of a hybrid pre-coding scheme is normalized, the variable part of the sum power consumption, caused by PAs, is decided by the power allocation matrix.So for various hybrid pre-coding schemes, the expression of the variable part of the sum power consumption is the same, i.e., where P m,k is the power allocated to the kth user at the mth AP. b) Discretization of the integral relaxes the calculation of SINR's PDF.Since the SINR expressions are similar under various hybrid pre-coding schemes, discretization steps and results are also similar.Then discrete ergodic EE can be expressed as, with The interference item P I k equals to where δ k,s = 1 when γ k lies in the sth segment.Otherwise, δ k,s = 0. Considering that two versions of heterogeneous gains are achieved based on the semi-closed expression of ergodic EE, they also exist for various hybrid pre-coding schemes.Especially, the first case of heterogeneous gain (ξ G,D,1 LW ) is unrelated to hybrid pre-coding and stays the same under various hybrid pre-coding schemes.The second case of heterogeneous gain is the ratio of two lower bounds of semi-closed expressions, that is ξ G,D,2 LW = ξD,LW is replaced by LoS fading.When the channel gain of LoS path is obviously larger than that of NLoS path, the LoS approximation is tenable.It is noted that when the number of paths is larger than one, or the power difference between various paths is small, our work still comes into existence and similar analyses can be achieved.To elaborate these, two processing methods are considered as follows.a) In the first case, the channel matrix between the mth AP and kth user is modeled as where γ is the normalized factor.N cl is the number of scattering clusters, each of which is consisted of N ray,i propagation paths.α i,l ∼ CN (0, σ 2 i,l ) and L i,l are the small-scale fading and the large-scale fading, respectively.a(θ i,l ) represents the normalized transmit array response vector evaluated at the corresponding angles of departure, that is θ i,l .
Since the small-scale fading follows the complex Gaussian distribution with zero mean value, it is reasonable to treat the mean value of the sum fading of multiple paths as one large-scale fading.Then the multiple paths can be seen as one equivalent path, with equivalent large-scale fading being the mean value of the sum fading and equivalent small-scale fading being a complex Gaussian random variable.Hybrid pre-coding scheme is conducted under this equivalent path, then the proposed analyses in the manuscript can be directly employed.
b) In the second case, the channel matrix is modeled as where H S m,k is the path with the maximum channel gain, and H NS m,k is the sum of other paths.Hybrid pre-coding scheme is still only conducted under the path with the maximum channel gain.Under this situation, SINR of the kth user is (43) shown at the top of this page, where Compared to the SINR γ k in (13) with only LoS, the main differences lie in the additional items in the molecule and denominator of γ MP k .Based on SINR expression of γ MP k , it is able to get the semiclosed expression of ergodic EE by three main simplified steps, i.e., linearization of a PA efficiency, discretization of the integral, segmentation of a logarithmic function.Since linearization of a PA efficiency is decided by the power allocation matrix, and segmentation of a logarithmic function is decided by value of SINR, these two simplified steps are same to the case with only the LoS.For the discretization of the integral, it is necessary to discrete the added items, that is P m,k (

Fig. 1 .
Fig. 1.Illustrations of heterogeneous cell-free systems.S t represents the tth type set of APs.As an example, each color corresponds to a set of APs assigned to a user by a rule.
)p(o).Recalling the definition of probability, p(j) = N j /N J and p(o) = N o /N O , where N j enumerates δ j m,k = 1 and N J enumerates small-scale fading realizations.Similarly, N o enumerates δ o m,k = 1 and N O enumerates large-scale fading realizations.Now, (25) can be converted into ξD,LW EE
Fig. 3 is achieved by the discrete ergodic EE ξD EE showed in (22) with various discrete steps.Average EE performances in Figs.3-Fig.12 are achieved by averaging the instantaneous EE shown

p
) 2 ( O o =1 δ o m ,k L o m ,k ) 2 ( L l=1 δ l m ,k ,k μ l m ,k ,k ).L o m,k (L o m ,k ), P i m,k (P i m ,k ), η j m,k , and μ l m ,k ,k are the discrete L m,k (L m ,k ), P m,k (P m ,k ), h T m,k F M m W M m,k , and h T m ,k F M m W M m ,k, respectively.c) Segmentation of a logarithmic function relaxes the sum of achievable rates.Since the segmentation of a logarithmic function is given, i.e., log 2 (1 + γ k ) ≈ β s γ k + C s , it is the same for various hybrid pre-coding schemes.Therefore, the semi-closed expression of ergodic EE can still come into existence with a minor adjustment under various hybrid pre-codings, which isξD,M EE (S) ≈ B (i, i , j, l, o, o ) P i m,k η j m,k | 2 P I k +σ 2 noise +C s δ k,s P C all (H|S)

EE(SI k + σ 2 noisep
max )/ ξD,LW EE (S min ).Once the semi-closed expression of ergodic EE under a general hybrid pre-coding is given, it is easy to have ξ G,D,2 LW .APPENDIX B ERGODIC EE UNDER CHANNELS WITH MULTIPLE PATHSThe front analyses on ergodic EE are conducted on the LoS approximation, where the sum of LoS fading and NLoS fadingγ MP k = | m∈A k P m,k L m,k / F m W m,k F + P m,k (H NS m,k ) T F m W m,k | 2 P (i,i , j, j , l, l , o, o )

TABLE I SEVERAL
REPRESENTATIVE PAPERS ON EE IN THE CELL-FREE SYSTEMS For the sake of simplicity, we assume that each user is served by all the APs and each AP set only owns one element, which means |A−1 m | = K and |S t | = 1.Additionally, μ m,k , σ ), which are [4.9126,3.9939, 2.5362, 3.0455], and [2.1158, 1.9204, 1.6926, 1.5813], respectively.This indicates the gain brought by the first group S 1 , S 1 , S 1 , S 1 and S 1 , S 2 , S 3 , S 4 LWcan be used to explain the simulation results in line with Remark 2. However, due to the relaxation of interference in the derivation of ξ G,D,2 LW , the difference between the former group and the latter group becomes Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
G,D,1 LW , i.e., [0.2311, 0.2267, 0.2245, 0.2233] for the former group and [0.5606, 0.5581, 0.5568, 0.5562] for the latter group, respectively.In the high SINR region, it was computed to be [0.2253,0.2238, 0.2230, 0.2226] for the former group and [0.5573, 0.5565, 0.5560, 0.5558] for the latter group.This phenomenon is found to be consistent with Remark 1, indicating the average EE gap of the former group is larger than that of the latter group.For the former and latter groups, ξ G,D,2 H NS m,k ) T F m W m ,k and P m ,k (H NS m ,k ) T F m W m ,k .Then the semi-closed expression of ergodic EE under channels with multiple paths becomes (44) shown at the top of this page, whereP C all (H|S) = M m=1,m∈S t (|A −1 m | − 1) × (P BB + |A −1 m |P RF + |A −1 m |N t P PS + k∈A −1 m P i m,k ρ t ) and P I k is equal to m / ).η j m,k and μ l m ,k ,k are the discretized values of (H NS m,k ) T F m W m ,k and (H NS m ,k ) T F m W m ,k .Based on the semi-closed expression of ergodic EE, two cases of heterogeneous gains are achieved.With ξD,MP EE (S), it is able to get the corresponding two versions of heterogeneous gains by the same way to utilize ξD EE (S) with only LoS path.