CN108521290B - A power allocation method in wireless relay cooperative network based on spatial modulation - Google Patents

Abstract

The present invention provides a power distribution method in a wireless relay cooperative network based on spatial modulation. For the cooperative spatial modulation system, using the probability density function and moment generating function of the effective signal-to-noise ratio, the theoretical expression and asymptotic approximate expression of the system bit error rate are given. According to the asymptotic approximation expression, a suboptimal power allocation scheme is proposed to minimize the bit error rate. It is verified by simulation that the power allocation method proposed by the present invention can effectively reduce the system bit error rate and improve the system performance compared with the equal power allocation scheme.

Description

A power allocation method in wireless relay cooperative network based on spatial modulation

Technical field:

The invention belongs to the field of mobile communication, and relates to a resource allocation method of a mobile communication system, in particular to a power allocation method in a wireless relay cooperative network based on spatial modulation.

Background technique:

As one of the research hotspots in the field of wireless communication in recent years, cooperative communication uses the antennas of other users in the wireless communication network as the transmission of cooperative signals of relay nodes, which solves the traditional multi-input multi-output (MIMO, Multi-Input Multi-output). In the system, the mobile terminal cannot place multiple antennas due to the limitation of volume and power. While improving the spectrum utilization rate, it effectively reduces the large cost brought by the base station. According to the different ways of processing the received signal by the relay node, it can be divided into different cooperative protocols. Compared with other protocols, the Amplify-and-Forward (AF, Amplify-and-Forward) collaboration protocol is simpler and easier to implement, so it is widely used in collaboration systems. The basic idea is that the relay node directly amplifies the received signal and then forwards it to the destination. Spatial Modulation (SM, Spatial Modulation) technology can realize single-link transceiver design by activating only one antenna in each time slot to transmit symbols, effectively overcoming inter-channel interference and synchronization problems; at the same time, using the transmission antenna serial number and transmission information bit one-to-one Mapping is the "invisible" transmission of information by the antenna serial number, with high speed and large capacity. Combining SM technology with cooperative communication, on the one hand, can reflect the advantages of SM technology and effectively avoid the disadvantages of cooperative communication system. On the other hand, cooperative relay nodes can be used to help source nodes transmit information, reflecting the advantages of cooperative communication. .

In a cooperative communication system, the power allocated by the source node and the relay node will have a certain impact on the system performance. Therefore, a reasonable power allocation scheme according to the relay location and path loss can effectively improve the system performance. Document 1 (J.Luo,R.S.Blum,L J Cimini,L.J.Greenstein.Decode-and-forward cooperative diversity with power allocation in wireless networks[J].IEEETransactions on Wireless Communications,2007,6(3):793-799.) proposed A suboptimal power allocation method is proposed to minimize the outage probability of cooperative DF systems through power control. Reference 2 (C.L.Wang and J.Y.Chen.Power allocation and relay selection for AF cooperative relaysystems with imperfect channel estimation.IEEE Transactions on VehicularTechnology, 2016,65(9):7809-7813.) for incomplete channel state information In the AF multi-relay cooperative system, an optimal closed power allocation scheme is proposed by maximizing the capacity. Document 3 (Miaowen Wen, Xiang Cheng, H. Vincent Poor and Bingli Jiao. Use of SSK modulation in two-way amplify-and-forward relaying. IEEE Transactions on Vehicular Technology, 2014, 63(4): 1498-1504) is The Space Shift Keying (SSK, Space Shift Keying) technology is introduced into the dual-relay cooperative system, and the power allocation coefficient that minimizes the bit error rate is given. The above literatures are all based on the research of power allocation scheme based on the cooperative communication system or the cooperative space shift keying system, and most of the cooperative spatial modulation systems currently use equal power allocation. Therefore, in order to improve the system performance, it is very necessary to study the adaptive power allocation of the cooperative spatial modulation system.

Invention content:

In order to improve the performance of the cooperative spatial modulation system, the present invention proposes a power allocation method in a wireless relay cooperative network based on spatial modulation based on the theoretical expression of the bit error rate.

The technical solutions adopted in the present invention include: a method for power allocation in a wireless relay cooperative network based on spatial modulation, comprising the following steps:

(1) Firstly, the physical model of the cooperative spatial modulation system is given. The system consists of a source node with N _t transmit antennas, a relay node with a single antenna, and a destination node with N _r receive antennas. Based on spatial modulation The basic idea is that the source node only activates one antenna in each time slot to send the modulated signal, and the number of bits transmitted in each time slot is log ₂ (N _t M), where the log ₂ N _t bits are used to determine the activated transmit antenna serial number, log ₂ M bits are used for constellation symbol modulation of M-QAM;

(2) The information transmission process of the cooperative system is divided into two stages. In stage 1, the source node sends the modulated constellation symbols to the relay node and the destination node through the activated transmit antenna. In stage 2, the relay node sends the modulated constellation symbol to the relay node and the destination node. Amplify and forward the signal received in phase one to the destination node;

(3) Assuming that the destination node can obtain complete channel state information, based on the received signal sent by the source node and the signal sent by the relay node, the destination node uses the maximum likelihood detection algorithm to demodulate the activated transmit antenna number and constellation at the same time. modulation symbol;

(4) According to the probability density function (PDF) and cumulative distribution function (CDF) of the effective signal-to-noise ratio of the source-to-relay, source-to-destination, and relay-to-destination links under the Rayleigh fading channel, the moment generating function (MGF) is obtained. , thus the system bit error rate _Pe is obtained as

P _e ≈P _a +P _d -P _a P _d

Wherein P _a is the detection error probability of the transmitting antenna sequence number assuming that the constellation symbol is correctly detected, and P _d is the detection error probability of the constellation symbol assuming that the transmitting antenna sequence number is correctly detected;

(5) Using the approximate expression of CDF under high signal-to-noise ratio, the approximate expression of MGF is obtained, and then the asymptotic approximate expression of Pe is obtained;

(6) Use the approximate expression of _Pe obtained in step (5) to obtain a derivative with respect to the power distribution coefficient r ₁ of the source node, and obtain the power distribution coefficient that minimizes the approximate value of _Pe according to the derivative.

Further, when N _r =1, according to the derivative of the approximate expression of P _e with respect to the power distribution coefficient r ₁ of the source node, the gradient descent method is used to obtain the sub-optimal power distribution coefficient that minimizes the approximate value of Pe _; when N _r ≥ 2, let the derivative of the approximate expression of _Pe with respect to the power distribution coefficient r1 of the source node be 0, and solve the equation to obtain the closed-form solution of the suboptimal power distribution coefficient.

The invention has the following beneficial effects: the invention provides a performance analysis scheme of a cooperative spatial modulation system, gives a theoretical expression of the bit error rate, and obtains a suboptimal power allocation scheme according to the theoretical expression. Compared with the equal power allocation scheme, this allocation scheme can effectively improve the system performance and has low computational complexity.

Description of drawings:

FIG. 1 is a step diagram of a power allocation method in a wireless relay cooperative network based on spatial modulation according to the present invention.

FIG. 2 is a schematic block diagram of a cooperative spatial modulation system in an embodiment of the present invention.

FIG. 3 is P _d of the cooperative spatial modulation system according to the embodiment of the present invention under different modulation modes.

FIG. 4 is P _a of the cooperative spatial modulation system according to the embodiment of the present invention under different modulation modes.

FIG. 5 is _Pe of the cooperative spatial modulation system according to the embodiment of the present invention under different modulation modes.

FIG. 6 is a comparison diagram of _Pe when N _r =1 between the power allocation scheme and the equal power allocation scheme according to the embodiment of the present invention.

FIG. 7 is a comparison diagram of _Pe when N _r =2 between the power allocation scheme and the equal power allocation scheme according to the embodiment of the present invention.

Detailed ways:

The present invention will be further described below in conjunction with the accompanying drawings.

The power distribution method in a wireless relay cooperative network based on spatial modulation of the present invention includes the following steps:

(1) Firstly, the physical model of the cooperative spatial modulation system is given. The system consists of a source node with N _t transmit antennas, a relay node with a single antenna, and a destination node with N _r receive antennas. Based on spatial modulation The basic idea is that the source node only activates one antenna in each time slot to send the modulated signal, and the number of bits transmitted in each time slot is log ₂ (N _t M), where the log ₂ N _t bits are used to determine the activated transmit antenna serial number, log ₂ M bits are used for constellation symbol modulation of M-QAM;

(2) The information transmission process of the cooperative system is divided into two stages. In stage 1, the source node sends the modulated constellation symbols to the relay node and the destination node through the activated transmit antenna. In stage 2, the relay node sends the modulated constellation symbol to the relay node and the destination node. Amplify and forward the signal received in phase one to the destination node;

(3) Assuming that the destination node can obtain complete channel state information, based on the received signal sent by the source node and the signal sent by the relay node, the destination node uses the maximum likelihood detection algorithm to demodulate the activated transmit antenna number and constellation at the same time. modulation symbol;

(4) According to the probability density function (PDF) and cumulative distribution function (CDF) of the effective signal-to-noise ratio of the source-to-relay, source-to-destination, and relay-to-destination links under the Rayleigh fading channel, the moment generating function (MGF) is obtained. , thus the system bit error rate _Pe is obtained as

P _e ≈P _a +P _d -P _a P _d

Wherein P _a is the detection error probability of the transmitting antenna sequence number assuming that the constellation symbol is correctly detected, and P _d is the detection error probability of the constellation symbol assuming that the transmitting antenna sequence number is correctly detected;

(5) Using the approximate expression of CDF under high signal-to-noise ratio, the approximate expression of MGF is obtained, and then the asymptotic approximate expression of Pe is obtained;

(6) Use the approximate expression of _Pe obtained in step (5) to obtain a derivative with respect to the power distribution coefficient r ₁ of the source node, and obtain the power distribution coefficient that minimizes the approximate value of _Pe according to the derivative.

When N _r =1, according to the derivative of the approximate expression of P _e with respect to the power distribution coefficient r ₁ of the source node, the gradient descent method is used to obtain the sub-optimal power distribution coefficient that minimizes the approximate value of Pe _; when N _r ≥ 2, Let the derivative of the approximate expression of _Pe with respect to the power distribution coefficient r1 of the source node be 0, and solve this equation to obtain the closed-form solution of the suboptimal power distribution coefficient.

The cooperative spatial modulation system model involved in the present invention is shown in Fig. 2. The system consists of a source node including N _t transmitting antennas, a relay node including a single antenna, and a destination node including N _r receiving antennas. The successor node adopts the AF protocol. The source node activates only one antenna to send signals in each time slot, and the other antennas do not send signals. The total number of bits transmitted in each slot is log ₂ (N _t M), where log ₂ N _t bits are used to determine the activated transmit antenna number i, i∈[1,N _t ], and log ₂ M bits are used for M -QAM constellation symbol modulation, then the transmitted symbol vector can be expressed as x _iq =[00...x _q ...0] ^T , where x _q is the ith element of x _jq , representing the qth symbol in the constellation diagram. The signal transmission process of the cooperative system is divided into two stages. In the first stage, the source node sends signals to the relay and the destination, and the signals received by the relay and the destination are expressed as

In the second stage, according to the AF protocol, the relay node amplifies the signal received in the first stage and then forwards it to the destination. The signal received at stage two is expressed as

y _rd =h _rd (Ay _sr )+n _rd (3)

和

的复高斯分布。方差δ²＝d^-α，d为两节点之间的距离，α为信道衰落系数。n_sr，n_sd和n_rd为均值为0，方差为N₀的复高斯噪声。A为放大系数，

P_s和P_r分别为源和中继的发射功率，P_s+P_r＝P_t，P_t为发射总功率，平均信噪比SNR表示为

经归一化后y_rd可表示为where h _sr , H _sd and h _rd are the source-to-relay, source-to-destination, and relay-to-destination channel matrices, respectively, and their elements obey the mean value of 0, respectively, and the variance is

and

complex Gaussian distribution. The variance δ ² =d ^−α , d is the distance between two nodes, and α is the channel fading coefficient. n _sr , n _sd and n _rd are complex Gaussian noise with mean 0 and variance N ₀ . A is the magnification factor,

P _s and P _r are the transmit powers of the source and the relay, respectively, P _s + P _r =P _t , P _t is the total transmit power, and the average signal-to-noise ratio SNR is expressed as

After normalization, y _rd can be expressed as

in

After receiving the signal sent by the source and the relay, the destination node performs joint demodulation on the transmitting antenna serial number and constellation symbol according to the maximum likelihood method. The demodulation algorithm is expressed as

和

分别表示天线序号和星座符号的估计值。in,

and

are the estimated values of the antenna sequence number and constellation symbol, respectively.

1) The present invention is directed to the calculation method of the average bit error rate of the cooperative spatial modulation system

Using P _a as the detection error probability of the constellation symbol assuming that the constellation symbol is detected correctly, P _d is the detection error probability of the constellation symbol assuming that the constellation symbol is detected correctly, then the average bit error rate _Pe of the system is expressed as

P _e ≈P _a +P _d -P _a P _d (6)

1.1) Constellation symbol detection error probability P _d

Assuming that the transmitting antenna serial number is detected correctly, the constellation symbol detection error probability P _d when MQAM modulation is used is expressed as

where BER(γ) is the bit error rate expression of MQAM under Gaussian channel

where erfc( ) complementary error function;

According to (2) and (4), the effective output SNR of the target is expressed as

在瑞利衰落信道下，γ_sd，γ_rd的PDF和γ_sr的CDF可分别表示为where γ _srd =γ _sr γ _rd /(γ _rd +C),

Under Rayleigh fading channel, the PDF of _γsd , _γrd and CDF of _γsr can be expressed as

和

in

and

According to (10), the MGF of γ _sd can be obtained

表示拉普拉斯变换。根据(11)和(12)，可得到γ_srd的CDFin

represents the Laplace transform. According to (11) and (12), the CDF of γ _srd can be obtained

where K _v (·) is a Bessel function of order v of the second kind. Then the MGF of γ _srd is expressed as

W_λ,μ(z)为Whitakker函数。in

W _λ,μ (z) is the Whitakker function.

Substituting (8) and (9) into (7), we get

where φ _u =cos((2u-1)π/(2N _p )), and N _p is the order of the Chebyshev polynomial expansion.

Substituting (13) and (15) into (16), the expression of P _d can be obtained

为了说明本发明相对于现有技术的技术效果，本实施例中通过MATLAB平台模拟仿真。图3给出了4QAM、16QAM和64QAM在Nr＝2和Nr＝4时的P_d仿真值和理论值。从图中可以看出理论值和仿真值吻合，说明所推导的P_d理论表达式是有效的。in

In order to illustrate the technical effect of the present invention relative to the prior art, in this embodiment, simulation is performed on a MATLAB platform. Figure 3 shows the simulated and theoretical values of P _d for 4QAM, 16QAM and 64QAM at Nr=2 and Nr=4. It can be seen from the figure that the theoretical value and the simulated value are consistent, indicating that the derived theoretical expression of P _d is valid.

1.2) The detection error probability _Pa of the transmitting antenna serial number

Assuming that the constellation symbols are correctly detected, the detection error probability P _a of the transmitting antenna serial number can be approximated as an upper bound formula

为天线序号估计值

和实际值i之间相差的比特数，

为成对错误概率(PEP,pairwise error probability)。根据联合解调算法，PEP可表示为in

is the estimated value of the antenna serial number

the number of bits that differ from the actual value i,

is the pairwise error probability (PEP, pairwise error probability). According to the joint demodulation algorithm, PEP can be expressed as

和

分别为源到目的、源到中继和中继到目的的有效信噪比，可分别表示为

由此可得到

和

的PDF以及

的CDFin

and

are the effective signal-to-noise ratios from source to destination, source to relay and relay to destination, respectively, which can be expressed as

It can be obtained from this

and

pdf as well

CDF

的PDF以及拉普拉斯变换，

的MGF可表示为according to

the PDF and the Laplace transform,

The MGF can be expressed as

的MGF，首先得到

的CDFfor derivation

MGF, first get

CDF

的MGF可表示为but

The MGF can be expressed as

in

Substitute (23) and (25) into (19), and then substitute (19) into (18), then P _a can be expressed as

图4为协作空间调制系统在4QAM、16QAM和64QAM调制方式下接收天线数目Nr＝2和Nr＝4时的P_a仿真值和理论值，图中仿真值和理论值能较好的吻合，说明所推导P_a的理论表达式是正确的。in

Figure 4 shows the simulated and theoretical values of Pa when the number of receiving antennas Nr=2 and Nr ₌ 4 in the 4QAM, 16QAM and 64QAM modulation modes of the cooperative spatial modulation system. The simulated and theoretical values in the figure are in good agreement. The theoretical expression of _Pa derived is correct.

Substituting (17) and (26) into P _e ≈ P _a +P _d -P _a P _d , the expression of the system average bit error rate P _e can be obtained. Fig. 5 shows the simulated and theoretical values of _Pe when the number of receiving antennas Nr=2 and Nr=4 in the 4QAM, 16QAM and 64QAM modulation modes of the cooperative spatial modulation system. In the figure, the theoretical curve and the simulation curve are consistent under different modulation modes, indicating that the theoretical expression of _Pe is valid.

2) The power distribution method proposed by the present invention based on the approximate expression of the average bit error rate.

2.1) Approximate expression of average bit error rate

Approximate closed-form expressions in terms of Bessel functions

where ψ(·) is the double gamma function. Substituting the above formula into (14), the approximate expression of γ _srd CDF can be obtained

From this, the approximate expression of MGF can be obtained

Substituting (29) into (16), the approximate expression of P _d can be obtained

Similarly, the approximate expression of P _a can be obtained

When N _r =1, when the average SNR is high, P _a of (30) and P _d of (31) can be further approximated as

Since the product term of P _a and P _d in formula (6) is compared with P _a , the value of P _d is too small and can be ignored, and P _e can be expressed as P _e ≈P _a +P _d . According to equations (32) and (33), the approximate expression of P _e can be obtained

When N _r ≥ 2, P _a of (30) and P _d of (31) can be further approximated as

Therefore, _Pe can be approximated as

2.2) Suboptimal power allocation scheme

Let P _s =r ₁ P _t , P _r =r ₂ P _t , r ₁ and r ₂ be the power distribution coefficients of the source and the relay, respectively, satisfying r ₁ +r ₂ =1, r ₁ ,r ₂ ∈[0 ,1], when N _r =1, the approximate expression (34) of _Pe can be expressed as

对该式关于r₁求导in

Derivative of this formula with respect to r ₁

时，Υ(r₁)＜0，则P_e在该区间内单调减。由此说明P_e的最小值在

区间内。根据(39)，利用梯度下降法可得到令P_e最小的r₁值From this derivative it can be seen that when r ₁ ≤ 0.5 or

When , Υ(r ₁ )<0, then P _e decreases monotonically in this interval. This shows that the minimum value of _Pe is at

within the interval. According to (39), the gradient descent method can be used to obtain the value of r ₁ that minimizes _Pe

r ₁ ^(k+1) =r ₁ ^(k) -τΥ(r ₁ ^(k) ) (40)

Among them, r ₁ ^(k+1) , r ₁ ^(k) are the r ₁ values of the k+1th and kth iterations, respectively, the initial value of the iteration is 0.5+ε, ε is a small value, and τ is the iteration step size . r ₂ can be calculated according to r ₂ =1-r ₁ , and thus the sub-optimal power coefficient when N _r =1 can be obtained. Fig. 6 shows P _e using different power allocation schemes when N _r =1, wherein the optimal power allocation scheme coefficient is to use the fminbnd function in MATLAB to minimize P formed by (17), (26) and (6). The exact expression of _e is obtained. It can be seen from the figure that the approximate values of _Pe under different power allocation schemes agree well with the simulated values under high SNR, proving that the approximate expression (34) is correct. Compared with the equal power allocation scheme (r ₁ =r ₂ =0.5), the suboptimal power allocation scheme and the optimal power allocation scheme can effectively improve the system bit error rate performance. And the suboptimal power allocation scheme can obtain almost the same performance as the optimal power allocation scheme, but the complexity is lower. This illustrates the effectiveness of the suboptimal power allocation scheme obtained by (40).

When N _r ≥ 2, the approximate expression (37) of _Pe can be expressed as

对该式关于r₁分别求一次导和二次导，则可得到in

Taking the first derivative and second derivative of this formula with respect to r ₁ , we can get

且Υ′(r₁)＞0，由此说明P_e在r₁∈[0,1]范围内有唯一最小值。令Υ(r₁)＝0，可得到一元二次方程As can be seen

And Υ'(r ₁ )>0, which means that _Pe has a unique minimum value in the range of r ₁ ∈[0,1]. Let Υ(r ₁ )=0, the quadratic equation in one variable can be obtained

Solving the above equation, the suboptimal power distribution coefficient can be obtained

(45) and (46) are closed-form expressions for the suboptimal power distribution coefficient when N _r ≥ 2. Figure 7 shows the simulated value, theoretical value and approximate value of P _e under different power distribution schemes when N _r = 2. The approximate value of Pe tends to be consistent with the simulated value under high signal-to-noise ratio _, indicating that the approximate expression of Pe is given _. Equation (37) is correct. Compared with the equal power allocation scheme, the suboptimal power allocation scheme and the optimal power allocation scheme can effectively improve the system bit error rate performance, and the performance of the suboptimal power allocation scheme is similar to the optimal power allocation scheme, but the former provides a power allocation coefficient. The closed-form solution of , with low computational complexity.

To sum up, the power allocation method proposed in the present invention has better performance than the equal power allocation scheme, and can obtain performance gains similar to the optimal power allocation scheme, but with lower complexity. This fully demonstrates the effectiveness of the power allocation method in a wireless relay cooperative network based on spatial modulation proposed by the present invention.

The foregoing has shown and described the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above-mentioned embodiments, and the descriptions in the above-mentioned embodiments and the description are only to illustrate the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have Various changes and improvements, the claimed scope of the present invention is defined by the appended claims, description and their equivalents.

The above are only the preferred embodiments of the present invention. It should be pointed out that for those skilled in the art, several improvements can be made without departing from the principles of the present invention, and these improvements should also be regarded as the invention. protected range.

Claims (2)

1. A power distribution method in a wireless relay cooperative network based on spatial modulation is characterized in that: the method comprises the following steps:

(1) firstly, a physical model of a cooperative spatial modulation system is given, wherein the system comprises N_tSource node of root transmitting antenna, relay node comprising single antenna and relay node comprising N_rThe method is characterized in that the target node of the receiving antenna is formed, a wireless channel is modeled as a Rayleigh channel, based on the basic idea of space modulation, a source node only activates one antenna to send a modulated signal in each time slot, and the number of bits transmitted in each time slot is log₂(N_tM) where log₂N_tThe bits are used to determine the activated transmit antenna sequence number, log₂M bits are used for constellation symbol modulation of M-QAM;

(2) the information transmission process of the cooperative system is divided into two stages, in the first stage, a source node sends modulated constellation symbols to a relay node and a destination node through an activated sending antenna, and in the second stage, the relay node amplifies and forwards signals received in the first stage to the destination node, namely the signals sent by the source node reach the destination node through a direct link and a relay amplifying and forwarding link;

(3) assuming that a destination node can obtain complete channel state information, based on a received signal sent by a source node and a signal sent by a relay node, the destination node simultaneously demodulates an activated sending antenna serial number and a constellation modulation symbol by adopting a maximum likelihood detection algorithm;

(4) for the above system, a Moment Generating Function (MGF) is obtained according to a Probability Density Function (PDF) and a Cumulative Distribution Function (CDF) of effective snr of a source-to-relay, a source-to-destination, and a relay-to-destination link under a rayleigh fading channel, thereby obtaining a system bit error rate P_eIs P_e≈P_a+P_d-P_aP_d

Wherein P is_aFor the detection error probability, P, of the serial number of the transmitting antenna on the assumption that the constellation symbol detection is correct_dThe probability of detection error of the constellation symbol when the sequence number of the transmitting antenna is detected correctly is assumed;

(5) obtaining an approximate expression of a moment generating function by utilizing an approximate expression of the accumulative distribution function under a high signal-to-noise ratio, and further obtaining a progressive approximate expression of a bit error rate Pe;

the approximate expression for CDF is:

where ψ (-) is a double gamma function, the approximate expression of MGF is:

P_dan approximate expression of (a) is,

P_aapproximate expression of

(6) Utilizing P obtained in step (5)_eApproximating an expression for the power distribution coefficient r of a source node₁Derivation from which P is derived_eThe smallest power division factor is approximated.

2. The method for allocating power in a wireless relay cooperative network based on spatial modulation according to claim 1, wherein: when N is present_rWhen 1, according to P_eIs expressed with respect to the power distribution coefficient r of the source node₁Is obtained by gradient descent such that P_eThe sub-optimal power distribution coefficient with the minimum approximation value; when Nr is larger than or equal to 2, derivation is carried out on the approximate expression of Pe with respect to r1, then the derivation is made to be equal to zero, and a closed-form solution of the suboptimal power distribution coefficient can be obtained after the solution.

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