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

The system of selection of multi-user based on efficiency optimum extensive antenna relay system antenna

Technical field

The invention belongs to wireless communication technology field, be specifically related to the multi-user's extensive antenna relay system antenna system of selection based on efficiency optimum.

Background technology

After U.S. Bell Laboratory scientist professor Marzetta in 2010 proposes extensive multiple-input and multiple-output (being called for short extensive MIMO) technology, this technology received the extensive concern of wireless communication field industrial quarters and academia with the characteristic of its novelty in recent years, and each well-known research institution of the whole world and seminar conduct in-depth research for this technology.So-called extensive MIMO technology, refer to that the aerial array configuring extensive quantity in base station end serves multiple user simultaneously, and antenna number magnitude will much larger than the number of users magnitude of service.Scholar's research is had to point out, by using large-scale antenna array to excavate spatial domain available resources in base station end, many new features relative to conventional MIMO system can be obtained, such as, simple linear predictive coding/detection method can be adopted effectively to eliminate multi-user interference in base station end, the achievable rate requirement of the transmitting power of remarkable reduction base station end and user side not influential system simultaneously, make the lifting at double of system spectral efficiency and energy efficiency under additionally not increasing the prerequisite of running time-frequency resource expense, the abundant degree of freedom is used for advanced wave beam forming etc.These characteristics of extensive MIMO technology, also make it become one of key technology of the 5th Generation Mobile Communication System.

Meanwhile, paired user's multiple antennas relay system is in the common concern being also subject to insider nearly ten years always.By introducing multiple antennas relay station, can greatly promote user's coverage, improving the transmission rate of edge customer, strengthen the reliability of transmission link.But in multi-user's relay system, inter-user interference is the bottleneck place of restriction multiple antennas relay system always.For this problem, numerous scholar proposes different solutions in order to eliminate multi-user interference, is mainly divided into two classes: a class is by distributing orthogonal resource between different user, suppressing inter-user interference by resource division; Another kind of is the object being reached antagonism inter-user interference by co-design precoding and receiver algorithm.But although first method can eliminate inter-user interference preferably, what bring is the expense of extra running time-frequency resource, causes the decline of entire system spectrum efficiency.Second method then can increase algorithm complex greatly, has higher requirement to the computational resource expense of relay station and stay of two nights user.Obviously, all there is serious defect in two class schemes.Just based on this, the people such as HimalA.Suraweera proposed extensive MIMO technology to introduce multi-user multi-antenna relay system first in 2013, the good suppression interference performance utilizing extensive MIMO to provide in multi-user transmission process solves the inter-user interference problem of paired user's multiple antennas relay system, simultaneously also without the need to the running time-frequency resource outside occupying volume, thus the spectrum efficiency performance of elevator system greatly.

It should be noted that while large-scale antenna array is introduced relay station, also inevitably problem served by band.The most direct problem is exactly that the radio-frequency channel permanent circuit total power consumption that causes of the use of a large amount of antenna promotes at double, and the lifting of permanent circuit total power consumption will certainly impact the overall performance efficiency of relay system.Obviously, be that in the wireless communication system of main flow, high power consumption is faced with stern challenge at following green communications.Thus, under the prerequisite meeting performance efficiency, determine that the required antenna number used of relay system has very important practical significance and application background, and this problem not yet there is researcher to set foot in.In order to solve the problem of relay station optimal antenna number, we have proposed based on efficiency maximized optimal antenna number Optimized model, because the too complicated inconvenience of target function in this model solves, closed-form solution for optimal antenna number is difficult to obtain especially, and closed-form solution has important directive significance for the mechanism of action of the influencing factor He these factors of probing into optimal antenna number.

The invention discloses the system of selection of a kind of multi-user based on efficiency optimum extensive antenna relay system antenna.This system formed by having multiple originating user of identical number and multiple stay of two nights user and a relay station, and originating user and stay of two nights user match between two and in two time slots, complete information transmission by relay station.In system, all originating user and the stay of two nights are with configuring single antenna per family, and relay station configures the aerial array of extensive quantity, as shown in Figure of abstract.The inventive method to maximize system energy efficiency for target, with relay station antenna number for optimized variable founding mathematical models.Because target function in this optimization problem is without clear and definite analytical expression, therefore, by means of the law of large numbers in large dimension Random Matrices Theory, the accurate approximate analysis expression formula of the one of target function in optimization problem is first tried to achieve.Recycle the plan recessed characteristic of this analytical expression about optimized variable, simultaneously by means of LambertW function, finally solve the optimal antenna number closed-form solution drawing and meet efficiency maximization target.

Summary of the invention

The present invention makes paired user extensive antenna relay system obtain higher performance efficiency to propose the system of selection of a kind of multi-user based on efficiency optimum extensive antenna relay system antenna, and has tried to achieve the closed-form solution of optimal antenna number.

The system of selection of multi-user based on efficiency optimum of the present invention extensive antenna relay system antenna, is characterized in that, said method comprising the steps of:

1). relay station obtains it to the imperfect channel state information between all originating user and stay of two nights user by channel estimating, i.e. channel matrix with wherein, h _krepresent that a kth originating user is to the channel vector of relay station and obedience answers Gaussian Profile represent relay station to a kth stay of two nights channel vector and obedience answer Gaussian Profile supposing the system adopts time division duplex standard, and channel obeys flat block decline, and also namely in channel coherency time, channel coefficients remains unchanged.

2). in the first time slot, K originating user sends information symbol to relay station node simultaneously, then the received signal vector at relay station place is r, and as shown in the first time slot end in accompanying drawing 1, r is expressed as form,

$r=\sqrt{{\mathrm{\ρ}}_s}Hx+n_r$

Wherein, x=[x ₁, x ₂..., x _k] ^t, x _k(k=1,2 ..., K) represent a kth originating user transmitting symbol and n _rrepresent that the first time slot meets multiple Gaussian Profile at the unit power additive white noise at relay station place

3). in the second time slot, relay station adopts maximum-ratio combing and high specific to send pre-coding matrix r amplifies to the received signal, and form forward signal vector t, as shown when in accompanying drawing 1, the second time slot is initial, t can be expressed as form,

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

Wherein, ξ is that the power normalization factor is in order to meet average total transmit power constraint ρ at relay station place _r, that is,

Then, $\mathrm{\ξ}=\frac{{\mathrm{\ρ}}_r}{\mathrm{\θ}}=\frac{{\mathrm{\ρ}}_r}{Tr({\mathrm{\ρ}}_s{\left(H^{H}H\right)}^2{G}^{H}G+H^H{\mathrm{HG}}^{H}G)}.$ Then, signal t is forwarded to all stay of two nights users by relay station, then a kth signal that stay of two nights user receives is y _k, as shown in the second time slot end in accompanying drawing 1, y _kcan form be expressed as,

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

Wherein, n _krepresent the unit power additive white noise at a kth stay of two nights user place and meet multiple Gaussian Profile

4). based on step 3) in the Received signal strength expression formula of stay of two nights user, the reception letter drying that can obtain a kth stay of two nights user is more as follows than SINR expression formula,

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

Wherein, $A_k\stackrel{\mathrm{\Δ}}{=}|g_k^H{\mathrm{GH}}^H{h}_k{|}^2,B_k\stackrel{\mathrm{\Δ}}{=}\underset{i=1,i\≠k}{\overset{K}{\Σ}}|g_k^H{\mathrm{GH}}^H{h}_i{|}^2,C_k\stackrel{\mathrm{\Δ}}{=}\frac{{\mathrm{\σ}}_r^2}{{\mathrm{\ρ}}_s}\left|\right|g_k^H{\mathrm{GH}}^H|{|}^2.$ Thus the average spectral efficiency (ase) that can obtain a kth stay of two nights user is shown below,

Wherein, represent and take take two time interval resources into account produced loss of spectral efficiency.

5). based on step 4) middle average spectral efficiency (ase) expression formula, set up to maximize system total energy effect function η (N) for target, the mathematic optimal model being variable with relay station antenna number at relay station place, as follows,

Wherein, η (N) represents efficiency function, S _Σrepresent the total frequency spectrum efficiency of all users, P _Σthe total power consumption of expression system, μ _sthe efficiency losses constant factor of each originating user power amplifier in transmitter device of>=1 expression, μ _rthe efficiency losses constant factor of>=1 expression relay station power amplifier in transmitter device, P _srepresent the constant constant power consumption of each originating user transmitter, P _rrepresent that the constant constant power on the every root antenna of relay station transceiver consumes.

6). due to step 5) in comprise S in target function _k, its accurate and analytical expression is difficult to obtain, and is unfavorable for the solution of follow-up optimization problem.Herein, according to the law of large numbers (see formula (44) in document 1: S.Jin, X.Liang, K.-KWong, X.Gao, andQ.Zhu, " ErgodicrateanalysisformultipairmassiveMIMOtwo-wayrelayne tworks; " IEEETransactionsonWirelessCommunication, vol.14, no.3, pp.1488, Mar.2015.), as follows

The law of large numbers:

If N dimensional vector p and q is independent identically distributed multiple Gaussian random vector, namely with then meet following characteristic,

To step 4) middle γ _kwhat expression comprised is everyly similar to, and can obtain 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 approximate representation be as follows,

From with expression formula in can see, all sued for peace by some non-negative stochastic variables for these four and form, utilize following theorem 1 (see the Lemma1:Q.Zhang in document 2, S.Jin, K.K.Wong, andH.B.Zhu, " PowerscalingofuplinkmassiveMIMOsystemswitharbitrary-rank channelmeans; " IEEEJournalOfSelectedTopicsInSignalProcess., vol.8, no.5, pp.969, Oct.2014.)

Theorem 1:

If two stochastic variable P and Q meet with wherein, P _nand Q _mbe non-negative stochastic variable, then, following approximate expression can be obtained

Meanwhile, can ensure that, when N and M increases gradually, above formula approximation accuracy will be more and more higher.

Further will be approximately it is as follows,

The statistical property of multiple Gaussian random vector product is utilized directly to calculate analytical expression 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 extensive antenna number that relay station disposes is usually much larger than number of users, i.e. N > > K, and utilize high s/n ratio condition, i.e. ρ _r> > 1 and ρ _s> > 1, by step 7) in the analytical expression S that obtains _kapproximate abbreviation is following form,

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

9). based on step 8) in analytical expression by step 5) in target function η (N) approximate expression of optimization problem be and use replace step 5) in the target function of optimization problem, thus the approximate optimization problem being converted into following form,

10). due to step 9) in optimized variable N belong to Positive Integer Set, this optimization problem belongs to non-convex integer programming.For the ease of problem solving, variable N is first released to continuous real variable, then directly can proves step 9) in approximate expression about N be intend recessed.Meanwhile, first derivative and second dervative is utilized to prove first increase the variation tendency subtracted afterwards about variable N.And then, utilize following theorem 2 (see the Lemma2:E.Bjornson in document 3, L.Sanguinetti, J.HoydisandM.Debbah, " DesigningmultiuserMIMOforenergyefficiency:WhenismassiveM IMOtheanswer?, " ProceedingsofIEEEWirelessCommunicationsandNetworkingConf erence, Istanbul, Apr.2014, pp.244.)

Theorem 2:

Optimization problem about variable z is as follows,

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

Wherein, a, c >=0, b, d, f > 0.Then, target function is that strict plan is recessed about z, and it is as follows to have unique optimal solution,

Wherein, e is natural constant, and as z > z ^opttime, target function is dull reduction, as z < z ^opttime, target function is monotone increasing.

And the closed-form solution of optimal antenna number directly can be obtained by means of LambertW function, be shown below,

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 natural constant, represent LambertW function, it is defined as: about equation θ=υ e of variable x ^υ, then the solution about υ can use LambertW function representation, namely

11). due to step 10) in the optimal antenna number N that obtains ^optusually be not integer, according to step 10) middle efficiency function about the variation relation of N, finally can obtain optimal antenna number is round{N ^opt.

Wherein, () ^hthe conjugate transpose operation of-representing matrix, -represent Positive Integer Set, -for the mathematic expectaion computing of random quantity (vector), Tr{}-matrix trace, the integer nearest with real number x is got in round{x}-expression, -represent and almost determine convergence, -expression average is μ variance is σ ²the distribution of multiple gaussian random, || ||-represent vectorial 2 norm computings, N-relay station antenna number, K-user to sum, ρ _sthe average transmit power of-each originating user, ρ _rthe average emitted gross power of-relay station.

The present invention proposes the system of selection of a kind of multi-user based on efficiency optimum extensive antenna relay system antenna, directly can be tried to achieve by closed analytical expression and meet the maximized optimum relay station antenna number of system energy efficiency.By selecting optimal antenna number, make extensive antenna relay system while obtaining the benefit brought of large-scale antenna array, reduce the too high circuit power consumption expense because huge antenna number produces as far as possible, thus make system total energy effect reach optimum level.Relative to traditional Lagrangian dual problem method for solving, this patent institute extracting method does not need alternating iteration solution procedure, greatly reduces algorithm complex.

Accompanying drawing explanation

Fig. 1 is the system model of the inventive method;

Fig. 2 is algorithm basic flow sheet of the present invention;

Fig. 3 is different users under number K scene, the spectrum efficiency analytical expression that this patent proposes and Monte Carlo simulation Comparative result figure;

Fig. 4 fixes power consumption P at different relay station antenna _runder scene, the performance efficiency that the optimal antenna counting method that this patent proposes reaches and Monte Carlo numerical simulation performance comparison figure.

Embodiment:

Algorithm flow chart shown in composition graphs 2 illustrates a kind of multi-user based on efficiency optimum of the present invention extensive antenna relay system power distribution method, comprises the steps:

1). relay station obtains it to the imperfect channel state information between all originating user and stay of two nights user by channel estimating, i.e. channel matrix with wherein, h _krepresent that a kth originating user is to the channel vector of relay station and obedience answers Gaussian Profile represent relay station to a kth stay of two nights channel vector and obedience answer Gaussian Profile supposing the system adopts time division duplex standard, and channel obeys flat block decline, and also namely in channel coherency time, channel coefficients remains unchanged.

2). set up to maximize system total energy effect function η (N) for target, the mathematic optimal model being variable with relay station antenna number at relay station place, as follows,

Wherein, η (N) represents efficiency function, S _krepresent the average spectral efficiency (ase) of a kth stay of two nights user, S _Σrepresent the total frequency spectrum efficiency of all stay of two nights users, P _Σrepresent the total power consumption of whole system, μ _sthe efficiency losses constant factor of each originating user power amplifier in transmitter device of>=1 expression, μ _rthe efficiency losses constant factor of>=1 expression relay station power amplifier in transmitter device, P _srepresent the constant constant power consumption of each originating user transmitter, P _rrepresent that the constant constant power on the every root antenna of relay station transceiver consumes, γ _krepresent that the reception letter of a kth stay of two nights user is dry than SINR, as follows,

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

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

3). in conjunction with theorem 1 in the law of large numbers and specification, and consider extensive antenna number and high s/n ratio interval, i.e. N > > K, ρ _r> > 1 and ρ _s> > 1, can by step 2) intermediate frequency spectrum efficiency S _kapproximate abbreviation is following form,

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

4). based on step 3) in spectrum efficiency approximate expression by step 2) in the target function of optimization problem replace, approximate transform is the optimization problem of following form,

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

5). based on step 4) middle optimization problem, directly show that the closed expression formula of optimal antenna number is as follows,

6). system parameters is substituted into step 6) in optimal antenna number close expression formula and obtain numerical solution, then carry out rounding operation round{N ^opt, final optimal antenna number integer solution can be obtained.Algorithm terminates.

Wherein, () ^hthe conjugate transpose operation of-representing matrix, -represent Positive Integer Set, -for the mathematic expectaion computing of random quantity (vector), Tr{}-matrix trace, the integer nearest with real number x is got in round{x}-expression, expression average is μ variance is σ ²the distribution of multiple gaussian random, || || represent vectorial 2 norm computings, N-relay station antenna number, K-user to sum, ρ _sthe average transmit power of-each originating user, ρ _rthe average emitted gross power of-relay station.

Fig. 3 gives different users under number scene, transmitting power ρ _r=ρ _sduring=10dB, along with the growth of relay station antenna number, the spectrum efficiency approximate analysis expression formula given by this patent and the correlation curve of Monte Carlo Numerical Simulation Results.As we can see from the figure, the analytic approximation expression formula that this patent proposes has extraordinary propinquity effect, and the difference between Monte Carlo numerical simulation curve is almost negligible, indicates the approximate analysis expression formula that this patent proposes and there is effect well.Fig. 4 gives different relay station antenna and fixes power consumption P _r, user is to number K=16, and power consumption transmitting power P fixed by originating user antenna _s=0dB and transmitting power ρ _r=ρ _sduring=10dB, the optimal antenna number method for solving performance comparison figure that this patent proposes.As can be seen from the figure, system total energy is imitated along with antenna number presents the trend first increasing and subtract afterwards, and the optimal antenna number closed solution given by this programme have matched the optimum energy valid value on efficiency change curve accurately.

Claims (1)

1., based on multi-user's extensive antenna relay system antenna system of selection of efficiency optimum, it is characterized in that, said method comprising the steps of:

1). relay station obtains it to the imperfect channel state information between all originating user and stay of two nights user by channel estimating, i.e. channel matrix with wherein, h _krepresent that a kth originating user is to the channel vector of relay station and obedience answers Gaussian Profile represent relay station to a kth stay of two nights user channel vector and obedience answer Gaussian Profile supposing the system adopts time division duplex standard, and channel obeys flat block decline, and also namely in channel coherency time, channel coefficients remains unchanged;

2). in the first time slot, K originating user sends information symbol to relay station node simultaneously, then can be expressed as form at the received signal vector r at relay station place,

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

Wherein, x=[x ₁, x ₂..., x _k] ^t, x _k(k=1,2 ..., K) represent a kth originating user transmitting symbol and n _rrepresent that the first time slot meets multiple Gaussian Profile at the unit power additive white noise at relay station place

3). in the second time slot, relay station adopts maximum-ratio combing and high specific to send pre-coding matrix r amplifies to the received signal, forms forward signal vector t as follows,

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

Wherein, ξ is that the power normalization factor is in order to meet average total transmit power constraint ρ at relay station place _r, that is,

Then, $\mathrm{\&xi;}=\frac{{\mathrm{\&rho;}}_r}{\mathrm{\&theta;}}=\frac{{\mathrm{\&rho;}}_r}{Tr({\mathrm{\&rho;}}_s{\left(H^{H}H\right)}^2{G}^{H}G+H^H{\mathrm{HG}}^{H}G)};$ Then, signal t is forwarded to all stay of two nights users by relay station, then a kth signal that stay of two nights user receives can be expressed as 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 _krepresent the unit power additive white noise at a kth stay of two nights user place and meet multiple Gaussian Profile

4). based on step 3) in the Received signal strength expression formula of stay of two nights user, the reception letter drying that can obtain a kth stay of two nights user is more as follows than SINR expression formula,

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

Wherein, $A_k\stackrel{\mathrm{\&Delta;}}{=}|g_k^H{\mathrm{GH}}^H{h}_k{|}^2,$ $B_k\stackrel{\mathrm{\&Delta;}}{=}\underset{i=1,i\&NotEqual;k}{\overset{K}{\&Sigma;}}|g_k^H{\mathrm{GH}}^H{h}_i{|}^2,$ $C_k\stackrel{\mathrm{\&Delta;}}{=}\frac{{\mathrm{\&sigma;}}_r^2}{{\mathrm{\&rho;}}_s}\left|\right|g_k^H{\mathrm{GH}}^H|{|}^2,$ Thus the average spectral efficiency (ase) that can obtain a kth stay of two nights user is shown below,

Wherein, represent and take take two time interval resources into account produced loss of spectral efficiency;

5). based on step 4) middle average spectral efficiency (ase) expression formula, set up to maximize system total energy effect function η (N) for target, the mathematic optimal model being variable with relay station antenna number at relay station place, as follows,

Wherein, η (N) represents efficiency function, S _Σrepresent the total frequency spectrum efficiency of all users, P _Σthe total power consumption of expression system, μ _sthe efficiency losses constant factor of each originating user power amplifier in transmitter device of>=1 expression, μ _rthe efficiency losses constant factor of>=1 expression relay station power amplifier in transmitter device, P _srepresent the constant constant power consumption of each originating user transmitter, P _rrepresent that the constant constant power on the every root antenna of relay station transceiver consumes;

6). utilize the law of large numbers to step 4) middle γ _kwhat expression comprised is everyly similar to, and can obtain 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 approximate representation be it is as follows,

And then, more right be similar to, can obtain it is as follows,

The statistical property of multiple Gaussian random vector product is utilized directly to calculate analytical expression 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 extensive antenna number that relay station disposes is usually much larger than number of users, i.e. N > > K, and utilize high s/n ratio condition, i.e. ρ _r> > 1 and ρ _s> > 1, by step 7) in the analytical expression that obtains approximate abbreviation is following form,

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

9). based on step 8) in analytical expression by step 5) in target function η (N) approximate expression of optimization problem be and use replace step 5) in the target function of optimization problem, thus the approximate optimization problem being converted into following form,

10). step 9) in optimized variable N belong to Positive Integer Set, this optimization problem belongs to non-convex integer programming; First variable N is first released to continuous real variable, and directly can obtains the closed-form solution of optimal antenna number by means of LambertW function, be shown below,

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 natural constant, represent LambertW function, it is defined as: about equation θ=υ e of variable x ^υ, then the solution about υ can use LambertW function representation, namely

11). due to step 10) in the optimal antenna number N that obtains ^optusually be not integer, according to step 10) middle efficiency function about the variation relation of N, finally can obtain optimal antenna number is round{N ^opt;

Wherein, () ^hthe conjugate transpose operation of-representing matrix, -represent Positive Integer Set, -for the mathematic expectaion computing of random quantity (vector), Tr{}-matrix trace, the integer nearest with real number x is got in round{x}-expression, -expression average is μ variance is σ ²the distribution of multiple gaussian random, || ||-represent vectorial 2 norm computings, N-relay station antenna number, K-user to sum, ρ _sthe average transmit power of-each originating user, ρ _rthe average emitted gross power of-relay station.