CN105450275A - Optimal energy efficiency-based antenna selection method for multi-user and large-scale antenna relay system - Google Patents

Abstract

The present invention discloses an optimal energy efficiency-based antenna selection method for a multi-user and large-scale antenna relay system. The system comprises a plurality of information source users, a plurality of information sink users and a relay station, wherein the number of the information source users is equal to the number of the information sink users. The information source users and the information sink users are pairwise coupled and the information transmission between the information source users and the information sink users is realized via the relay station within two time slots. All information source users and information sink users in the system are respectively provided with a single antenna. The relay station is provided with an antenna array of a large-scale number illustrated in the drawings of the abstract. According to the technical scheme of the invention, in order to realize that the energy efficiency of the system is maximal, the antenna number of the relay station is adopted as an optimization variable for the establishment of a mathematical model. Since no clear analytical expression is available for a target function of the above optimization problem, an approximately accurate analytical expression for the target function of the optimization problem is figured out firstly based on the law of large numbers in the large dimensional random matrix theory. After that, based the quasi-concave characteristics of the optimization variable in the analytical expression, an optimal antenna number closed-form solution for realizing the optimal energy efficiency is finally solved out by means of the Lambert W function at the same time.

Description

Multi-user large-scale antenna relay system antenna selection method based on optimal energy efficiency

Technical Field

The invention belongs to the technical field of wireless communication, and particularly relates to an antenna selection method of a multi-user large-scale antenna relay system based on optimal energy efficiency.

Background

Since 2010, the american patent scientific professor Marzetta, the united states bell laboratories, has proposed a massive multiple input multiple output (massive MIMO for short), in recent years, the technology has received extensive attention from the industry and academia in the field of wireless communication with its novel characteristics, and various well-known research institutes and groups of subjects around the world have conducted intensive research on the technology. The massive MIMO technology is to configure a massive number of antenna arrays at a base station end to simultaneously serve multiple users, and the order of the antenna is much larger than that of the served users. Researches of researchers indicate that by using a large-scale antenna array to mine available space domain resources at a base station end, many new characteristics relative to a traditional MIMO system can be obtained, such as the fact that a simple linear precoding/detection method can be adopted at the base station end to effectively eliminate multi-user interference, the transmitting power of the base station end and a user end is obviously reduced, meanwhile, the achievable rate requirement of the system is not influenced, the frequency spectrum efficiency and the energy efficiency of the system are improved in a multiplied mode on the premise of not additionally increasing the time frequency resource overhead, and rich freedom is used for advanced beam forming and the like. These characteristics of massive MIMO technology also make it one of the key technologies of 5 th generation mobile communication systems.

Meanwhile, the paired-user multi-antenna relay system has been receiving general attention from the industry in the last decade. By introducing the multi-antenna relay station, the coverage area of a user can be greatly improved, the transmission rate of edge users is improved, and the reliability of a transmission link is enhanced. However, in the multi-user relay system, the inter-user interference has been a bottleneck limiting the multi-antenna relay system. To solve this problem, many scholars propose different solutions for eliminating multi-user interference, which are mainly classified into two categories: one is that orthogonal time-frequency resources are distributed among different users, and interference among the users is inhibited through resource division; and the other type is to achieve the aim of resisting the interference between users by jointly designing precoding and a receiver algorithm. However, although the first method can better eliminate the inter-user interference, it brings overhead of additional time-frequency resources, which causes a decrease in the overall spectrum efficiency of the system. The second method can greatly increase the complexity of the algorithm, and puts higher requirements on the computing resource overhead of the relay station and the sink user. Clearly, both types of solutions have serious drawbacks. Based on this, himal a. suraweera et al propose to introduce the large-scale MIMO technology into the multi-user multi-antenna relay system for the first time in 2013, and utilize the good interference suppression capability provided by the large-scale MIMO in the multi-user transmission process to solve the problem of the interference between the users of the paired-user multi-antenna relay system, and at the same time, do not need to occupy additional time-frequency resources, thereby greatly improving the spectrum efficiency performance of the system.

It should be noted that the introduction of a large-scale antenna array into a relay station also inevitably brings about some problems. The most direct problem is that the total power consumption of the radio frequency channel fixed circuit caused by the use of a large number of antennas is increased in multiples, and the increase of the total power consumption of the fixed circuit will certainly affect the overall energy efficiency performance of the relay system. It is clear that high power consumption faces a serious challenge in a wireless communication system in which green communication is mainstream in the future. Therefore, on the premise of meeting the energy efficiency performance, determining the number of the antennas required by the relay system has very important practical significance and application background, and no researcher is involved in the problem. In order to solve the problem of the optimal number of the relay antennas, an optimal antenna number optimization model based on energy efficiency maximization is provided, because an objective function in the model is too complex and inconvenient to solve, a closed form solution of the optimal number of the antennas is difficult to obtain, and the closed form solution has important guiding significance for researching influence factors of the optimal number of the antennas and action mechanisms of the factors.

The invention discloses an antenna selection method of a multi-user large-scale antenna relay system based on optimal energy efficiency. The system consists of a plurality of information source users, a plurality of information sink users and a relay station, wherein the number of the information source users is the same as that of the information sink users, and the information source users and the information sink users are paired pairwise to finish information transmission in two time slots through the relay station. In the system, all information source users and information sink users are configured with single antennas, and the relay station is configured with a large-scale number of antenna arrays, as shown in the abstract attached drawing. The method of the invention aims at maximizing the system energy efficiency and establishes a mathematical model by taking the number of the relay station antennas as an optimization variable. Because the objective function in the optimization problem has no clear analytical expression, an accurate approximate analytical expression of the objective function in the optimization problem is firstly obtained by means of a large number law in a large-dimensional random matrix theory. And finally solving to obtain an optimal antenna number closed form solution meeting the energy efficiency maximization target by utilizing the pseudo-concave characteristic of the analytical expression on the optimization variable and by means of a LambertW function.

Disclosure of Invention

The invention provides an optimal energy efficiency-based antenna selection method for a multi-user large-scale antenna relay system for enabling a paired-user large-scale antenna relay system to obtain higher energy efficiency performance, and obtains a closed form solution of the optimal number of antennas.

The invention discloses an antenna selection method of a multi-user large-scale antenna relay system based on optimal energy efficiency, which is characterized by comprising the following steps of:

1) the relay station obtains ideal channel state information from the relay station to all source users and all sink users through channel estimation, namely a channel matrixAndwherein h is_kRepresenting the channel vector from the kth source user to the relay station and obeying a complex Gaussian distributionRepresenting the channel vector for the repeater to the kth sink and obeying a complex Gaussian distributionIt is assumed that the system adopts a time division duplex system, and the channel is subject to flat block fading, i.e. the channel coefficient remains unchanged during the channel coherence time.

2) In the first time slot, K source users simultaneously transmit information symbols to the relay node, and the received signal vector at the relay is r, as shown at the end of the first time slot in fig. 1, r is expressed in the following form,

$r=\sqrt{{\mathrm{\ρ}}_s}Hx+n_r$

wherein x is [ x ]₁,x₂,…,x_K]^T，x_k(K-1, 2, …, K) denotes the transmission symbol of the kth source user andn_rwhite noise per unit power at the relay station representing the first slot and satisfying a complex Gaussian distribution

3) In the second time slot, the relay station transmits the precoding matrix by adopting the maximum ratio combination and the maximum ratioThe received signal r is amplified to form a forwarded signal vector t, as shown at the beginning of the second time slot in fig. 1, which may be represented in the form,

$t=Vr=\sqrt{\mathrm{\ξ}}{\mathrm{GH}}^{H}r$

wherein ξ is a power normalization factor to satisfy an average total transmit power constraint ρ at the relay_rThat is to say that,

then the process of the first step is carried out, $\mathrm{\ξ}=\frac{{\mathrm{\ρ}}_r}{\mathrm{\θ}}=\frac{{\mathrm{\ρ}}_r}{Tr({\mathrm{\ρ}}_s{\left(H^{H}H\right)}^2{G}^{H}G+H^H{\mathrm{HG}}^{H}G)}.$ then, the relay station forwards the signal t to all sink users, and the signal received by the k-th sink user is y_kAs indicated at the end of the second slot in FIG. 1, y_kCan be expressed in the form of,

$y_k=\sqrt{{\mathrm{\ρ}}_s}{g}_k^H{\mathrm{Vh}}_k{x}_k+\sqrt{{\mathrm{\ρ}}_s}\underset{i=1,i\≠k}{\overset{K}{\Σ}}{g}_k^H{\mathrm{Vh}}_i{x}_i+g_k^H{\mathrm{Vn}}_r+n_k$

wherein n is_kRepresents white noise of unit power at k-th sink user and satisfies complex Gaussian distribution

4) Based on the received signal expression of the sink user in step 3), the receiving drying ratio SINR expression of the kth sink user can be obtained as follows,

${\mathrm{\γ}}_k=\frac{A_k}{B_k+C_k+\mathrm{\θ}/{\mathrm{\ρ}}_r{\mathrm{\ρ}}_s}$

wherein, $A_k\stackrel{\mathrm{\Δ}}{=}|g_k^H{\mathrm{GH}}^H{h}_k{|}^2,B_k\stackrel{\mathrm{\Δ}}{=}\underset{i=1,i\≠k}{\overset{K}{\Σ}}|g_k^H{\mathrm{GH}}^H{h}_i{|}^2,C_k\stackrel{\mathrm{\Δ}}{=}\frac{{\mathrm{\σ}}_r^2}{{\mathrm{\ρ}}_s}\left|\right|g_k^H{\mathrm{GH}}^H|{|}^2.$ so that the average spectral efficiency of the kth sink user can be obtained as shown in the following formula,

wherein,indicating the spectral efficiency loss that would result from taking into account the two slot resources occupied.

5) Establishing a mathematical optimization model with the number of relay station antennas as a variable and with the goal of maximizing the total energy efficiency function eta (N) of the system at the relay station based on the average spectral efficiency expression in the step 4), as shown below,

wherein η (N) represents the energy efficiency function, S_ΣRepresenting the total spectral efficiency, P, of all users_ΣRepresents the total power consumption, μ, of the system_sMore than or equal to 1 represents the constant factor of the efficiency loss of each power amplifier device of the source user transmitter, mu_rThe constant factor P of the efficiency loss of the power amplifier device of the repeater station transmitter is more than or equal to 1_sTo representConstant fixed power consumption, P, per source user transmitter_rRepresenting a constant fixed power consumption on each antenna of the repeater station transceiver.

6) Since S is contained in the objective function in step 5)_kAnd the accurate analytical expression is difficult to obtain, which is not beneficial to solving the subsequent optimization problem. Here, according to the law of large numbers (see formula (44) in document 1): S.jin, X.Liang, K. -KWong, X.Gao, and Q.Zhu, "Ergonocatamaran for multiproirmative MIMOtwo-wayrelaynetworks," IEEETransactionson Wirelesscommunication, vol.14, No.3, pp.1488, Mar.2015.), as shown below,

law of large numbers:

let the N-dimensional vectors p and q be independent and identically distributed complex Gaussian random vectors, i.e.AndthenThe following characteristics are satisfied,

for gamma in step 4)_kThe terms contained in the expression are approximated, resulting in the following expression,

$A_k\≈\tilde{A}k=\underset{j=1}{\overset{K}{\Σ}}\left|g_k^H{g}_j{|}^2\right|h_j^H{h}_k{|}^2$

$B_k\≈{\tilde{B}}_k=\underset{i=1,i\≠k}{\overset{K}{\Σ}}\underset{j=1}{\overset{K}{\Σ}}\left|g_k^H{g}_j{|}^2\right|h_j^H{h}_i{|}^2$

$C_k\≈{\tilde{C}}_k=\frac{{\mathrm{\σ}}_r^2}{{\mathrm{\ρ}}_s}\underset{j=1}{\overset{K}{\Σ}}\left|g_k^H{g}_j{|}^2\right|\left|h_j\right|{|}^2$

$\mathrm{\θ}\≈\widetilde{\mathrm{\θ}}=\underset{i=1}{\overset{K}{\Σ}}\left({\mathrm{\ρ}}_s\underset{j=1}{\overset{K}{\Σ}}\right|h_i^H{h}_j{|}^2+{\mathrm{\σ}}_r^2\left|\right|h_i\left|{|}^2\right)\left|\right|g_i|{|}^2$

then, S_kCan be approximately expressed as follows,

fromAndit can be seen in the expression of (1) that these four terms are composed of the summation of several non-negative random variables, using the following theorem 1 (see Lemma 1: q. zhang, s. jin, k. wong, and h. b. zhu in document 2, "power scaling of upper linking in silicoms with partition-random processes," ieee journal of selectivity in signaling processes, vol.8, No.5, pp.969, oct.2014.),

theorem 1:

let two random variables P and Q satisfyAndwherein, P_nAnd Q_mAll are non-negative random variables, then the following approximate expression can be obtained

Meanwhile, the approximation accuracy of the above formula can be ensured to be higher and higher when N and M are gradually increased.

Further will beIs approximated toAs will be shown below, in the following,

the statistical property of complex Gaussian random vector product can be directly calculatedThe analytical expression of (a) is as follows,

$S_k\≈{\stackrel{\‾}{S}}_k=\frac{1}{2}{\log }_2\left(1+\frac{{\stackrel{\‾}{A}}_k}{{\stackrel{\‾}{B}}_k+{\stackrel{\‾}{C}}_k+{\stackrel{\‾}{F}}_k}\right)$

wherein,

8) consider that the number of large-scale antennas deployed at a relay station is usually much larger than the number of users, i.e., N > K, and utilize the high signal-to-noise ratio condition, i.e., ρ_r> 1 and ρ_s> 1, using the analytical expression S obtained in step 7)_kThe approximation is simplified to the form that,

${\stackrel{\‾}{S}}_k\≈\frac{1}{2}{\log }_2(1+\frac{{\mathrm{\ρ}}_r{\mathrm{\ρ}}_s(N+2)}{2(K-1){\mathrm{\ρ}}_r{\mathrm{\ρ}}_s+{\mathrm{\ρ}}_r+{\mathrm{K\ρ}}_s})$

9) based on the analytical expression in step 8)Approximately expressing the objective function η (N) of the optimization problem in step 5) asIn combination withInstead of the objective function of the optimization problem in step 5), to approximately translate into an optimization problem of the form,

10) since the optimization variable N in step 9) belongs to the set of positive integers, the optimization problem belongs to the non-convex integer programming. In order to facilitate the problem solving, the variable N is released as a continuous real variable, so that the approximate expression in the step 9) can be directly provedWith respect to N is pseudo-concave. At the same time, the first and second derivatives can be used to proveTrend about variable N being increasing first and then decreasing. Furthermore, the following theorem 2 is utilized (see Lemma 2: E.Bjornson, L.Sanguinetti, J.Hoydiand M.Debbah, "designing Multiuser MIMOfenergyeffort

Theorem 2:

the optimization problem with respect to the variable z is as follows,

$\underset{z}{max}\frac{f{\log }_2(a+bz)}{c+dz}$

wherein a and c are more than or equal to 0, and b, d and f are more than 0. The objective function is then strictly concave-like with respect to z, and has a unique optimal solution as shown below,

wherein e is a natural constant, and when z > z^optWhen the objective function is monotonically decreasing, when z < z^optThe objective function is monotonically increasing.

And a closed form solution of the optimal antenna number can be directly obtained by means of the LambertW function, as shown in the following formula,

wherein, $\mathrm{\&alpha;}\stackrel{\mathrm{\&Delta;}}{=}\frac{{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s}{2(K-1){\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s+{\mathrm{\&rho;}}_r+{\mathrm{K\&rho;}}_s},\mathrm{\&beta;}\stackrel{\mathrm{\&Delta;}}{=}K({\mathrm{\&mu;}}_s{\mathrm{\&rho;}}_s+P_s)+{\mathrm{\&mu;}}_r{\mathrm{\&rho;}}_r,$ e represents a natural constant, and e represents a natural constant,represents the LambertW function, which is defined as: equation θ e for variable x^υThe solution for v can then be represented by a lambert w function, i.e.

11) Due to the number N of the optimal antennas found in step 10)^optUsually not an integer, according to step 10) energy efficiency functionRegarding the variation relation of N, the optimal number of antennas can be finally obtained to be round { N^opt}。

Wherein, (.)^H-representing a conjugate transpose operation of the matrix,-representing a set of positive integers,-for the mathematical expectation operation of a random quantity (vector), Tr {. -the trace of the matrix, round { x } -represents taking the integer closest to the real number x,-indicating that the convergence is almost certain,-mean value μ variance σ²The complex Gaussian random distribution, | | · | -_s-average transmission power per source user, p_r-average total transmit power of the relay station.

The invention provides an antenna selection method of a multi-user large-scale antenna relay system based on optimal energy efficiency, which can directly obtain the number of optimal relay station antennas meeting the maximization of the system energy efficiency through a closed analytical expression. By selecting the optimal number of the antennas, the large-scale antenna relay system can obtain the benefits brought by the large-scale antenna array, and simultaneously reduce the overhigh circuit power consumption expense generated by the huge number of the antennas as much as possible, so that the total energy efficiency of the system can reach the optimal level. Compared with the traditional Lagrangian dual problem solving method, the method provided by the patent does not need an alternate iteration solving process, and greatly reduces algorithm complexity.

Drawings

FIG. 1 is a system model of the method of the present invention;

FIG. 2 is a basic flow chart of the algorithm of the present invention;

fig. 3 is a diagram comparing a spectrum efficiency analytical expression and a monte carlo simulation result provided in the present invention under different user pair number K scenarios;

FIG. 4 shows the fixed power consumption P at different repeater antennas_rIn a scene, the energy efficiency performance achieved by the optimal antenna number method provided by the patent is compared with the Monte Carlo numerical simulation performance.

The specific implementation mode is as follows:

the method for allocating power to a multi-user large-scale antenna relay system based on optimal energy efficiency according to the present invention is specifically described with reference to the algorithm flow chart shown in fig. 2, and includes the following steps:

1) the relay station obtains ideal channel state information from the relay station to all source users and all sink users through channel estimation, namely a channel matrixAndwherein h is_kRepresenting the channel vector from the kth source user to the relay station and obeying a complex Gaussian distributionRepresenting the channel vector for the repeater to the kth sink and obeying a complex Gaussian distributionIt is assumed that the system adopts a time division duplex system, and the channel is subject to flat block fading, i.e. the channel coefficient remains unchanged during the channel coherence time.

2) A mathematical optimization model is established at the relay station with the objective of maximizing the total energy efficiency function η (N) of the system and with the number of relay station antennas as variables, as shown below,

wherein η (N) represents the energy efficiency function, S_kRepresents the average spectral efficiency, S, of the kth sink user_ΣRepresenting the total spectral efficiency, P, of all sink users_ΣRepresents the total power consumption, mu, of the entire system_sMore than or equal to 1 represents the constant factor of the efficiency loss of each power amplifier device of the source user transmitter, mu_rThe constant factor P of the efficiency loss of the power amplifier device of the repeater station transmitter is more than or equal to 1_sRepresenting a constant fixed power consumption, P, of each source user transmitter_rRepresenting constant fixed power consumption, gamma, on each antenna of the repeater transceiver_kIndicating the reception drying ratio SINR of the kth sink user, as shown below,

${\mathrm{\&gamma;}}_k=\frac{A_k}{B_k+C_k+\mathrm{\&theta;}/{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s}$

wherein, $A_k\stackrel{\mathrm{\&Delta;}}{=}|g_k^H{\mathrm{GH}}^H{h}_k{|}^2,B_k\stackrel{\mathrm{\&Delta;}}{=}\underset{i=1,i\&NotEqual;k}{\overset{K}{\&Sigma;}}|g_k^H{\mathrm{GH}}^H{h}_i{|}^2,C_k\stackrel{\mathrm{\&Delta;}}{=}\frac{{\mathrm{\&sigma;}}_r^2}{{\mathrm{\&rho;}}_s}\left|\right|g_k^H{\mathrm{GH}}^H|{|}^2.$

3) combining the law of large numbers and theorem 1 in the specification, and considering the number of large-scale antennas and the interval of high signal-to-noise ratio, namely N > K and rho_r> 1 and ρ_s> 1, the spectral efficiency S in step 2) can be adjusted_kThe approximation is simplified to the form that,

$S_k\&ap;{\stackrel{\&OverBar;}{S}}_k=\frac{1}{2}{\log }_2(1+\frac{{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s(N+2)}{2(K-1){\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s+{\mathrm{\&rho;}}_r+{\mathrm{K\&rho;}}_s})$

4) based on the approximate expression of spectral efficiency in step 3)Replacing the objective function of the optimization problem in the step 2), approximately converting the objective function into the following form of optimization problem,

$\underset{N\&GreaterEqual;1}{max}\mathrm{\&eta;}\left(N\right)\&ap;\stackrel{\&OverBar;}{\mathrm{\&eta;}}\left(N\right)=\frac{\frac{K}{2}{\log }_2(1+\frac{{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s(N+2)}{2(K-1){\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s+{\mathrm{\&rho;}}_r+{\mathrm{K\&rho;}}_s})}{K({\mathrm{\&mu;}}_s{\mathrm{\&rho;}}_s+P_s)+{\mathrm{\&mu;}}_r{\mathrm{\&rho;}}_r+{\mathrm{NP}}_r}$

5) based on the optimization problem in step 4), a closed expression for directly obtaining the optimal number of antennas is shown as follows,

6) substituting the system parameters into the closed expression of the optimal antenna number in the step 6) to obtain a numerical solution, and then carrying out rounding { N } operation^optAnd obtaining the final integer solution of the optimal antenna number. The algorithm ends.

Wherein, (.)^H-representing a conjugate transpose operation of the matrix,-representing a set of positive integers,-for the mathematical expectation operation of a random quantity (vector), Tr {. -the trace of the matrix, round { x } -represents taking the integer closest to the real number x,mean is μ and variance is σ²The complex Gaussian random distribution, | | | | | represents the vector 2 norm operation, N-the number of relay station antennas, K-the total number of user pairs, ρ_s-average transmission power per source user, p_rInThe average total transmit power of the relay station.

FIG. 3 shows the transmission power ρ in different user pair number scenarios_r＝ρ_sWhen the number of the relay station antennas is 10dB, the spectrum efficiency approximate analysis expression provided by the patent is compared with a comparison curve of a Monte Carlo numerical simulation result along with the increase of the number of the relay station antennas. As can be seen from the figure, the analytic approximation expression provided by the patent has a very good approximation effect, and the difference between the analytic approximation expression and the Monte Carlo numerical simulation curve is almost negligible, which shows that the analytic approximation expression provided by the patent has a good effect. FIG. 4 shows different fixed power consumption P of the relay station antenna_rThe number of user pairs K is 16, the fixed power consumption transmitting power P of the source user antenna_s0dB and transmission power ρ_r＝ρ_sWhen the antenna number is 10dB, the performance comparison graph of the optimal antenna number solving method is provided. As can be seen from the figure, the total energy efficiency of the system shows a trend of increasing first and then decreasing with the number of antennas, and the optimal antenna number closed solution provided by the scheme accurately matches the optimal energy efficiency value on the energy efficiency change curve.

Claims (1)

1. A multi-user large-scale antenna relay system antenna selection method based on optimal energy efficiency is characterized by comprising the following steps:

1) the relay station obtains ideal channel state information from the relay station to all source users and all sink users through channel estimation, namely a channel matrixAndwherein h is_kRepresenting the channel vector from the kth source user to the relay station and obeying a complex Gaussian distribution Representing the channel vector of the repeater to the kth sink user and obeying a complex Gaussian distributionAssuming that a system adopts a time division duplex system, and a channel obeys flat block fading, namely a channel coefficient is kept unchanged in channel coherence time;

2) in the first slot, K source users transmit information symbols to the relay node at the same time, then the received signal vector r at the relay can be represented in the form,

$r=\sqrt{{\mathrm{\&rho;}}_s}Hx+n_r$

wherein x is [ x ]₁,x₂,...,x_K]^T，x_k(K ═ 1, 2.. times, K) denotes the transmission symbol of the kth source user andn_rwhite noise per unit power at the relay station representing the first slot and satisfying a complex Gaussian distribution

3) In the second time slot, the relay station transmits the precoding matrix by adopting the maximum ratio combination and the maximum ratioThe received signal r is amplified to form a forwarded signal vector t as follows,

$t=Vr=\sqrt{\mathrm{\&xi;}}{\mathrm{GH}}^{H}r$

wherein ξ is a power normalization factor to satisfy an average total transmit power constraint ρ at the relay_rThat is to say that,

then the process of the first step is carried out, $\mathrm{\&xi;}=\frac{{\mathrm{\&rho;}}_r}{\mathrm{\&theta;}}=\frac{{\mathrm{\&rho;}}_r}{Tr({\mathrm{\&rho;}}_s{\left(H^{H}H\right)}^2{G}^{H}G+H^H{\mathrm{HG}}^{H}G)};$ the relay station then forwards the signal t to all sink users, the signal received by the kth sink user can be represented in the form,

$y_k=\sqrt{{\mathrm{\&rho;}}_s}{g}_k^H{\mathrm{Vh}}_k{x}_k+\sqrt{{\mathrm{\&rho;}}_s}\underset{i=1,i\&NotEqual;k}{\overset{K}{\&Sigma;}}{g}_k^H{\mathrm{Vh}}_i{x}_i+g_k^H{\mathrm{Vn}}_r+n_k$

wherein n is_kRepresents white noise of unit power at k-th sink user and satisfies complex Gaussian distribution

4) Based on the received signal expression of the sink user in step 3), the receiving drying ratio SINR expression of the kth sink user can be obtained as follows,

${\mathrm{\&gamma;}}_k=\frac{A_k}{B_k+C_k+\mathrm{\&theta;}/{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s}$

wherein, $A_k\stackrel{\mathrm{\&Delta;}}{=}|g_k^H{\mathrm{GH}}^H{h}_k{|}^2,$ $B_k\stackrel{\mathrm{\&Delta;}}{=}\underset{i=1,i\&NotEqual;k}{\overset{K}{\&Sigma;}}|g_k^H{\mathrm{GH}}^H{h}_i{|}^2,$ $C_k\stackrel{\mathrm{\&Delta;}}{=}\frac{{\mathrm{\&sigma;}}_r^2}{{\mathrm{\&rho;}}_s}\left|\right|g_k^H{\mathrm{GH}}^H|{|}^2,$ so that the average spectral efficiency of the kth sink user can be obtained as shown in the following formula,

wherein,represents the spectral efficiency loss generated by taking the occupied two time slot resources into account;

5) establishing a mathematical optimization model with the number of relay station antennas as a variable and with the goal of maximizing the total energy efficiency function eta (N) of the system at the relay station based on the average spectral efficiency expression in the step 4), as shown below,

wherein η (N) represents the energy efficiency function, S_ΣRepresenting the total spectral efficiency, P, of all users_ΣRepresents the total power consumption, μ, of the system_sMore than or equal to 1 represents the constant factor of the efficiency loss of each power amplifier device of the source user transmitter, mu_rThe constant factor P of the efficiency loss of the power amplifier device of the repeater station transmitter is more than or equal to 1_sRepresenting a constant fixed power consumption, P, of each source user transmitter_rRepresents a constant fixed power consumption on each antenna of the repeater transceiver;

6) applying law of large numbers to γ in step 4)_kThe terms contained in the expression are approximated, resulting in the following expression,

$A_k\&ap;{\tilde{A}}_k=\underset{j=1}{\overset{K}{\&Sigma;}}\left|g_k^H{g}_j{|}^2\right|h_j^H{h}_k{|}^2$

$B_k\&ap;{\tilde{B}}_k=\underset{i=1,i\&NotEqual;k}{\overset{K}{\&Sigma;}}\underset{j=1}{\overset{K}{\&Sigma;}}\left|g_k^H{g}_j{|}^2\right|h_j^H{h}_i{|}^2$

$C_k\&ap;{\tilde{C}}_k=\frac{{\mathrm{\&sigma;}}_r^2}{{\mathrm{\&rho;}}_s}\underset{j=1}{\overset{K}{\&Sigma;}}\left|g_k^H{g}_j{|}^2\right|\left|h_j\right|{|}^2$

$\mathrm{\&theta;}\&ap;\widetilde{\mathrm{\&theta;}}=\underset{i=1}{\overset{K}{\&Sigma;}}\left({\mathrm{\&rho;}}_s\underset{j=1}{\overset{K}{\&Sigma;}}\right|h_i^H{h}_j{|}^2+{\mathrm{\&sigma;}}_r^2\left|\right|h_i\left|{|}^2\right)\left|\right|g_i|{|}^2$

then, S_kCan be approximately expressed asAs will be shown below, in the following,

then, toBy approximation, can obtainAs will be shown below, in the following,

the statistical property of complex Gaussian random vector product can be directly calculatedThe analytical expression of (a) is as follows,

$S_k\&ap;{\stackrel{\&OverBar;}{S}}_k=\frac{1}{2}{\log }_2\left(1+\frac{{\stackrel{\&OverBar;}{A}}_k}{{\stackrel{\&OverBar;}{B}}_k+{\stackrel{\&OverBar;}{C}}_k+{\stackrel{\&OverBar;}{F}}_k}\right)$

wherein,

8) consider that the number of large-scale antennas deployed at a relay station is usually much larger than the number of users, i.e., N > K, and utilize the high signal-to-noise ratio condition, i.e., ρ_r> 1 and ρ_s> 1, using the analytical expression obtained in step 7)The approximation is simplified to the form that,

${\stackrel{\&OverBar;}{S}}_k\&ap;\frac{1}{2}{\log }_2(1+\frac{{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s(N+2)}{2(K-1){\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s+{\mathrm{\&rho;}}_r+{\mathrm{K\&rho;}}_s})$

9) based on the analytical expression in step 8)Approximately expressing the objective function η (N) of the optimization problem in step 5) asIn combination withInstead of the objective function of the optimization problem in step 5), to approximately translate into an optimization problem of the form,

10) step 9) the optimization variable N belongs to a set of positive integers, the optimization problem belongs to a non-convex integer program; firstly, the variable N is released into a continuous real variable, and a closed form solution of the optimal antenna number can be directly obtained by means of a LambertW function, as shown in the following formula,

wherein, $\mathrm{\&alpha;}\stackrel{\mathrm{\&Delta;}}{=}\frac{{\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s}{2(K-1){\mathrm{\&rho;}}_r{\mathrm{\&rho;}}_s+{\mathrm{\&rho;}}_r+{\mathrm{K\&rho;}}_s},$ $\mathrm{\&beta;}\stackrel{\mathrm{\&Delta;}}{=}K({\mathrm{\&mu;}}_s{\mathrm{\&rho;}}_s+P_s)+{\mathrm{\&mu;}}_r{\mathrm{\&rho;}}_r,$ e represents a natural constant, and e represents a natural constant,represents the LambertW function, which is defined as: equation θ e for variable x^υThe solution for v can then be represented by a lambert w function, i.e.

11) Due to the number N of the optimal antennas found in step 10)^optUsually not an integer, according to step 10) energy efficiency functionRegarding the variation relation of N, the optimal number of antennas can be finally obtained to be round { N^opt}；

Wherein, (.)^H-representing a conjugate transpose operation of the matrix,-representing a set of positive integers,-for the mathematical expectation operation of a random quantity (vector), Tr {. -the trace of the matrix, round { x } -represents taking the integer closest to the real number x,-mean value μ variance σ²The complex Gaussian random distribution, | | · | -,ρ_s-average transmission power per source user, p_r-average total transmit power of the relay station.