CN108924067B - Time division method for training sequence and data symbol in interference alignment network - Google Patents

Abstract

The present invention provides a time division method for training sequences and data symbols in an interference alignment network. The optimal design is carried out under the premise of imperfect CSI. Given the coherence time, when considering the actual channel estimation, the transmission training sequence and data symbols are optimally designed. The time division factor α between , seeks a trade-off between channel estimation accuracy and data symbol transmission time, and maximizes the lower bound of the user achievable rate.

Description

A Temporal Segmentation Method for Training Sequence and Data Symbols in Interference Alignment Networks

technical field

The present invention relates to a time division optimization method between transmission training sequences and data symbols.

Background technique

In a multiple-input multiple-output (Multiple-Input Multiple-Output, MIMO) interference network, when multiple users communicate wirelessly, there will be interference with each other, which will affect the signal reception quality and reduce the channel capacity of the receiver. Interference Alignment (IA) is a promising interference management technique in wireless communication. Different from the interference avoidance scheme of the traditional orthogonalized channel, the IA technology can limit the total interference to one half of the signal subspace, and the other half can be used to transmit the desired signal without interference. IA technology has attracted extensive attention due to its excellent performance in increasing system capacity.

The application of the IA technology requires accurate channel state information (Channel Statement Information, CSI), but in practical applications, the obtained CSI is always imperfect due to estimation errors, quantization errors or feedback errors. In particular, when CSI is obtained by channel estimation through the training sequence, for a given coherence time, if the training time is too short, more errors will be generated in the channel estimation, thereby reducing the achievable rate of the system; on the contrary, if the training time is too long If it is long, the data symbol transmission time will be shortened, and the system rate will also be reduced. Therefore, it is of great significance to study the time division optimization scheme between the transmission training sequence and the data symbol under imperfect CSI to improve the system rate.

Reference 1 "Iterative algorithm for interference alignment in multiuserMIMO interfe-rence channels [Signal Processing Advances in WirelessCommunications IEEE Eleven-th International Work(2010):1-5]." For multiuser MIMO interference networks, in a perfect CSI scenario, A linear programming-based interference alignment method is proposed and its convergence is demonstrated. And under the actual system configuration, the proposed method is faster and simpler than other traditional algorithms.

Reference 2 "Maximum sum-rate interference alignment algorithms for MIMO channel-els [IEEE Global Telecommunications Conference (GLOBECOM), 2010, pp.1–6]." For the multi-user MIMO interference network, under the perfect CSI scenario, a A new iterative algorithm that optimizes the precoding and receiving filters alternately aims to find an IA solution that maximizes the average total rate. Simulation results show that the proposed algorithm has higher throughput than the traditional IA algorithm.

Document 3 "A limited feedback-based bit allocation method for interference alignmentm-ent [International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM), 2016, pp.1-6]." The limited feedback leads to imperfect CSI, which in turn causes the system Rate loss problem, based on fixed training time, a novel bit allocation algorithm in interference alignment is proposed. The simulation results show that when the interference alignment condition is satisfied, the proposed algorithm can greatly improve the total system rate compared with the traditional average bit allocation algorithm.

Reference 4 "On the overhead of interference alignment: Training, feedback, and coop-eration [IEEE Transactions on Wireless Communications, vol.11, no.11, pp.4192-4203, 2012]." Time and feedback to obtain the performance of the IA technique for CSI, for the imperfect CSI case, under the premise of given error power, deduce the average IA achievable rate.

Reference 5 "Capacity analysis of interference alignment with bounded CSIuncertain-ty [IEEE Wireless Communications Letters, vol.3, no.5, pp.505-508, 2014]." For the bounded error of imperfect CSI, use IA A lower bound on the system capacity is derived, and a new metric called modification degrees of freedom is introduced to describe the multiplexing performance of IA under imperfect CSI in a finite Signal-to-Noise Ratio (SNR).

Most of the existing research on IA is based on perfect CSI (such as literature 1, 2), and there are few studies on IA under imperfect CSI. In addition, the current research on IA is mainly for system design under ideal CSI conditions, or to obtain imperfect CSI through fixed training time for channel estimation (such as literatures 3, 4, 5), and then to carry out optimal design.

SUMMARY OF THE INVENTION

In order to overcome the deficiencies of the prior art, the present invention provides a time division method for training sequences and data symbols in an interference alignment network. The optimal design is carried out under the premise of imperfect CSI. Given the coherence time, the actual channel estimation algorithm and process are considered. , the time division factor between the training sequence and the data symbol is introduced, and the lower bound of the user achievable rate is maximized by optimizing the time division factor.

The technical scheme adopted by the present invention to solve its technical problem comprises the following steps:

是信道估计矩阵，ΔH_kl表示信道估计误差；

为第k个接收用户处的加性高斯白噪声，N₀表示噪声功率谱密度；令α∈(0，1)为时间分割因子，训练时间是αT个符号周期，数据符号传输时间是(1-α)T个符号周期；基于LS估计算法，得到ΔH_kl的最大误差界

其中，

(ΔH_kl)_i表示ΔH_kl的第i行，ε_t表示训练序列的发射功率，

表示传输训练序列SNR；Step 1, for an interference alignment network with K user pairs sharing the same frequency band, each user pair consists of a transmitting user and a receiving user, the transmitting user is configured with m antennas, and the receiving user is configured with n antennas; T symbol periods of the coherence time are all valid; H _kl ∈ C ^n×m is the channel matrix from the lth transmitting user to the kth receiving user,

is the channel estimation matrix, and ΔH _kl represents the channel estimation error;

is the additive white Gaussian noise at the kth receiving user, and N ₀ represents the noise power spectral density; let α∈(0,1) be the time division factor, the training time is αT symbol periods, and the data symbol transmission time is (1 -α)T symbol periods; based on the LS estimation algorithm, the maximum error bound of ΔH _kl is obtained

in,

(ΔH _kl ) _i denotes the i-th row of ΔH _kl , ε _t denotes the transmit power of the training sequence,

Indicates the transmission training sequence SNR;

其中，

是第l个发送用户基于IA技术设计的预编码矩阵，

是第k个接收用户基于IA技术设计的接收滤波矩阵，

为发送数据符号向量，假设所有的发送用户具有相同的功率约束：

其中，ε表示数据符号的最大发射功率，

表示传输数据符号SNR；Step 2, each sending user only transmits one data stream d=1 to all receiving users within the coherence time, then the signal received by the kth receiving user is:

in,

is the precoding matrix designed by the lth sending user based on the IA technology,

is the receiving filter matrix designed by the kth receiving user based on IA technology,

To transmit a vector of data symbols, it is assumed that all transmitting users have the same power constraint:

where ε represents the maximum transmit power of the data symbol,

Indicates the transmission data symbol SNR;

其中，

表示第k个用户的期望信号功率，

B＝m²n²ρ[(K-1)²-1]，C＝Tρ_tD＝m²n²ρ(K-1)²；Step 3, the lower bound of the reachable rate of the kth user pair

in,

represents the expected signal power of the kth user,

B=m ² n ² ρ[(K-1) ² -1], C=Tρ _t D=m ² n ² ρ(K-1) ² ;

实现可达速率下界最大化；By solving for the optimal time division factor

Maximize the lower bound of the achievable rate;

求关于α一阶导，并令其为0，定义

G＝n²m²ρ，得到关于α的近似表达式a₁α³+a₂α²+a₃α+a₄＝0，其中，right

Find the first derivative with respect to α and let it be 0, define

G=n ² m ² ρ, the approximate expression a ₁ α ³ +a ₂ α ² +a ₃ α+a ₄ =0 for α is obtained, where,

a ₁ =-2ACF,

a ₂ =2ACG-2(AD+BC)F+(BC-AD)(2A-F),

a ₃ =2(AD+BC)G-2BDF-(BC-AD)(2A-2B-FG),

a ₄ =2BDG-(BC-AD)(2B+G).

The approximate optimal time division factor α _opt is obtained from the approximate expression.

The beneficial effects of the present invention are: in the interference alignment network, time division technology is introduced, and a convenient optimization algorithm is studied to calculate the optimal time division factor in the compromise between the channel estimation accuracy and the data symbol transmission time. , and the optimal time division formula in the transmission process is given to maximize the lower bound of the user's reachable rate.

Description of drawings

Figure 1 is a diagram of an interference-aligned network communication model;

随α的变化图；Figure 2 shows the lower bound of the achievable rate under different ρ

The graph of the change with α;

Fig. 3 is the simulation optimal time division factor α _opt under different ρ;

随ρ的变化图；Figure 4 is the lower bound of the maximum achievable rate

Variation graph with ρ;

对比。Figure 5 shows the lower bound of the achievable rate under different time division schemes

Compared.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings and embodiments, and the present invention includes but is not limited to the following embodiments.

The invention provides a time division scheme for training sequences and data symbols in an interference alignment network. Given the coherence time, when considering the actual channel estimation, the time division factor α between the transmission training sequence and the data symbols is optimally designed, and the time division factor α between the transmission training sequence and the data symbols is optimally designed, and the channel estimation The trade-off between accuracy and data symbol transmission time maximizes the lower bound on the achievable rate of the user.

l，k＝1，…，K是第l个发送用户到第k个接收用户的信道矩阵。其中，

是信道估计矩阵，ΔH_kl表示信道估计误差。

为第k个接收用户处的加性高斯白噪声，N₀表示噪声功率谱密度。令α∈(0，1)为时间分割因子，那么训练时间是αT个符号周期，数据符号传输时间是(1-α)T个符号周期。The present invention considers an interference alignment network, assuming that there are K (K>1) user pairs sharing the same frequency band, each user pair is composed of a transmitting user and a receiving user, the transmitting user is configured with m antennas, and the receiving user is configured with n antennas . The channel model is a block slow fading channel model, that is, the channel estimation is valid during the entire coherence time (assumed to be T symbol periods).

l,k=1,...,K is the channel matrix from the lth transmitting user to the kth receiving user. in,

is the channel estimation matrix, and ΔH _kl represents the channel estimation error.

is the additive white Gaussian noise at the kth receiving user, and N ₀ represents the noise power spectral density. Let α∈(0,1) be the time division factor, then the training time is αT symbol periods, and the data symbol transmission time is (1-α)T symbol periods.

The technical scheme adopted by the present invention to solve its technical problem comprises the following steps:

其中，

(ΔH_kl)_i表示ΔH_kl的第i行，ε_t表示训练序列的发射功率，

表示传输训练序列SNR。Step 1. In practical applications, the Least Square (LS) channel estimation algorithm is widely used in wireless communications. Based on the LS estimation algorithm, the present invention obtains the maximum error bound of ΔH _kl

in,

(ΔH _kl ) _i denotes the i-th row of ΔH _kl , ε _t denotes the transmit power of the training sequence,

Indicates the transmission training sequence SNR.

l，k＝1，…，K。其中，

是第l个发送用户基于IA技术设计的预编码矩阵，

是第k个接收用户基于IA技术设计的接收滤波矩阵，

为发送数据符号向量，假设所有的发送用户具有相同的功率约束：

其中，ε表示数据符号的最大发射功率。

表示传输数据符号SNR。Step 2, each sending user only transmits one data stream (d=1) to all receiving users within the coherence time, then the signal received by the kth receiving user is:

l,k=1,...,K. in,

is the precoding matrix designed by the lth sending user based on the IA technology,

is the receiving filter matrix designed by the kth receiving user based on IA technology,

To transmit a vector of data symbols, it is assumed that all transmitting users have the same power constraint:

Among them, ε represents the maximum transmit power of the data symbol.

Indicates the transmitted data symbol SNR.

Step 3, the lower bound of the reachable rate of the kth user pair can be defined as:

表示第k个用户的期望信号功率，

B＝m²n²ρ[(K-1)²-1]，C＝Tρ_t，D＝m²n²ρ(K-1)²。in,

represents the expected signal power of the kth user,

B=m ² n ² ρ[(K-1) ² -1], C=Tρ _t , D=m ² n ² ρ(K-1) ² .

Build the optimal model as

s.t.α∈(0,1)

The achievable rate lower bound is maximized by solving the optimal time division factor α _opt .

求关于α一阶导，并令其为0，得到关于α的公式如下right

Find the first derivative with respect to α and set it to 0, and get the formula about α as follows

G＝n²m²ρ，可以得到上式的近似表达式为The theoretical optimal time division factor α _opt is determined by the root of the above formula, which can be easily solved by Matlab, but it is very difficult in practical application. We propose another simplified algorithm to solve the above equation, and introduce Taylor expansion to approximate the logarithmic function in the above equation. definition

G=n ² m ² ρ, the approximate expression of the above formula can be obtained as

a ₁ α ³ +a ₂ α ² +a ₃ α+a ₄ =0

in,

a ₁ =-2ACF,

a ₂ =2ACG-2(AD+BC)F+(BC-AD)(2A-F),

a ₃ =2(AD+BC)G-2BDF-(BC-AD)(2A-2B-FG),

a ₄ =2BDG-(BC-AD)(2B+G).

The approximate optimal time division factor α _opt can be obtained by an approximate expression.

Embodiments of the present invention consider an interference alignment network, as shown in FIG. 1 . The system includes K (K>1) pairs of users sharing the same frequency band. Each user pair consists of a transmitting user and a receiving user, the transmitting user is configured with m antennas, and the receiving user is configured with n antennas. A slow block fading channel model is used in the present invention, in which the channel estimation is valid for the entire coherence time (assumed to be T symbol periods). Let α∈(0,1) be the time division factor, then the training time is αT symbol periods, and the data transmission time is (1-α)T symbol periods.

The invention first describes the interference alignment network communication model, then deduces the lower bound of the user's reachable rate based on the LS channel estimation algorithm, and finally performs the optimal design and solution of the time division factor to maximize the lower bound of the user's reachable rate.

I. Interference Alignment Network Communication Model

为It can be seen from Figure 1 that each sending user only transmits one data stream (d=1) to all receiving users within the coherence time, then the received signal of the kth receiving user is

for

l，k＝1，…，K是第l个发送用户到第k个接收用户的信道矩阵，

代表第k个接收用户处的加性高斯白噪声(Additive White GaussianNoise，AWGN)，N₀表示噪声功率谱密度，

表示发送信号向量。令

为发送数据符号向量，则in,

l,k=1,...,K is the channel matrix from the lth transmitting user to the kth receiving user,

represents the additive white Gaussian noise (AWGN) at the kth receiving user, N ₀ represents the noise power spectral density,

Represents the transmit signal vector. make

is the transmitted data symbol vector, then

x _l =V _l s _l (2)

是第l个发送用户基于IA技术设计的预编码矩阵。假设所有的发送用户具有相同的功率约束，

其中，ε表示数据符号的最大发射功率。

表示传输数据符号SNR。in,

is the precoding matrix designed by the lth sending user based on the IA technology. Assuming that all transmitting users have the same power constraint,

Among them, ε represents the maximum transmit power of the data symbol.

Indicates the transmitted data symbol SNR.

Considering practical applications, CSI is obtained by channel estimation, quantization and feedback. Therefore, there will be some errors in the system design, that is, the CSI is not perfect. In the present invention, the imperfect CSI model is

是信道估计矩阵，ΔH_kl表示信道估计误差。in

is the channel estimation matrix, and ΔH _kl represents the channel estimation error.

式(1)中的接收信号模型可以改写为Bring equations (2) and (3) into equation (1), and consider the receive filter matrix in IA

The received signal model in equation (1) can be rewritten as

可以基于

由IA设计消除。In the second equation in (4), all signals from other user pairs

can be based on

Eliminated by IA design.

即In order to better analyze the influence of channel estimation error, the present invention introduces an error bound about ΔH _kl

which is

where (ΔH _kl ) _i represents the i-th row of ΔH _kl .

II. Lower bound of user reachable rate

Due to the randomness of the channel estimation error, it is difficult to estimate the instantaneous system reachable rate. The present invention uses the LS channel estimation algorithm of the optimal training sequence to deduce the detailed expression of the time division factor for the lower bound of the reachable rate. Then, by optimizing the time division factor, the lower bound of the achievable rate is maximized. The specific method is as follows:

The present invention adopts the LS channel estimation algorithm of the optimal training sequence, and the channel estimation error can be expressed as

和

分别表示LS估计算法下的信道估计误差和信道估计矩阵。ε_t表示训练序列的发射功率。in,

and

Represent the channel estimation error and channel estimation matrix under the LS estimation algorithm, respectively. _εt represents the transmit power of the training sequence.

Combining equations (3) and (6), we can get

Then the maximum error bound of ΔH _kl is

Based on equations (4) and (8), define the lower bound of the reachable rate of the kth user pair

表示第k个用户的期望信号功率，

B＝m²n²ρ[(K-1)²-1]，C＝Tρ_t，D＝m²n²ρ(K-1)²。in,

represents the expected signal power of the kth user,

B=m ² n ² ρ[(K-1) ² -1], C=Tρ _t , D=m ² n ² ρ(K-1) ² .

III. Time division factor optimization design and solution

The purpose of the present invention is to maximize the lower bound of the user achievable rate by optimizing the time division factor between the transmission training sequence and the data symbol when considering the actual channel estimation given the coherence time. Based on equation (9), the optimal model is constructed as

s.t.α∈(0, 1) (10)

关于α的二阶导数，即Before solving an optimization problem, we must first prove the existence of an optimal solution. Therefore, we calculated

The second derivative with respect to α, i.e.

α∈(0，1)，这表明

存在最大值。为了找到这个最大值以及对应的最优时间分割因子，本发明对

求关于α一阶导，并令其为0，得到关于α的等式如下It can be proved by simple mathematical derivation

α∈(0,1), which shows that

There is a maximum value. In order to find this maximum value and the corresponding optimal time division factor, the present invention

Find the first derivative with respect to α and set it to 0, and get the equation about α as follows

By solving equation (12), the optimal time division factor α _opt can be obtained. However, a complex logarithmic operation is included in equation (12), making the equation more difficult to solve. Furthermore, the W-Lambert function cannot be used to solve this problem due to the inclusion of α ² in equation (12). The theoretical optimal time division factor α _opt is determined by the root of equation (12), which can be easily solved with Matlab, but it is very difficult in practical application. In fact, in the simulation, under a given SNR, we can use the traversal search method for α with a small step size (generally, setting the step size to 0.001 to ensure the accuracy) in the (0, 1) interval Find the maximum α of formula (12), which is the optimal time division factor α _opt . In addition, the present invention adopts another simplified algorithm to solve equation (12). Here, we introduce Taylor expansion to approximate the logarithmic function in (12), namely

且对所有的α和ρ有

in,

and for all α and ρ we have

这一项随n的增加而迅速减小。因此，在接下来的分析中ln(k)可以只由式(13)中的第一项来近似，仅存在一些可接受的误差。Obviously, in formula (13)

This term decreases rapidly as n increases. Therefore, ln(k) can be approximated by only the first term in Eq. (13) in the following analysis, with only some acceptable error.

Then the logarithmic term in equation (12) can be approximately expressed as

G＝n²m²ρ，可以得到式(12)的详细近似表达式为Substituting equation (14) into equation (12), the definition

G=n ² m ² ρ, the detailed approximate expression of formula (12) can be obtained as

a ₁ α ³ +a ₂ α ² +a ₃ α+a ₄ =0 (15)

in,

a ₁ =-2ACF,

a ₂ =2ACG-2(AD+BC)F+(BC-AD)(2A-F),

a ₃ =2(AD+BC)G-2BDF-(BC-AD)(2A-2B-FG),

a ₄ =2BDG-(BC-AD)(2B+G).

The approximate optimal time division factor α _opt can be obtained by an approximate expression.

复高斯分布随机变量，设置m＝2，n＝2，K＝3，T＝100，

定义传输数据符号和训练序列的SNR的比值为μ＝ρ/ρ_t，简单起见，假设μ＝1。所有的推导和仿真都可以直接扩展到其它m，n和K的配置中。The present invention carries out numerical simulation and comparison on the proposed time division optimization scheme. In the simulation, it is assumed that all elements in H _kl obey the

Complex Gaussian distributed random variable, set m=2, n=2, K=3, T=100,

The ratio of the SNR of the transmitted data symbol to the training sequence is defined as μ=ρ/ρ _t , and it is assumed that μ=1 for simplicity. All derivations and simulations can be directly extended to other m, n and K configurations.

随时间分割因子α的变化。由图2可知，相同α下，

随着ρ的增大而增大。这与我们期望的结果一致。对一个确定的ρ，

随α的增大先增大后减小。当α很较低时，随着α的增大，信道估计矩阵

将变得越好，由信道估计产生的干扰泄露也会减小。因此，可达速率下界

先增大。然而，当α增大到一定程度时，α的增大将不会带来更多的速率提升。这种情况下，可达速率下界

将主要由有效信号传输时间(1-α)T。因此

随α的增大先增大后减小。由图2，我们还可以发现，对一个给定的ρ，可达速率下界

的最大值及其对应的α是存在的。而且，随着ρ的增大，信道将会变好，产生精确信道估计所需的训练时间越短。这表明ρ越大，α_opt越小。Figure 2 shows the lower bound of the achievable rate for different transmitted data symbols SNR ρ

Variation of the division factor α over time. It can be seen from Figure 2 that under the same α,

increases with the increase of ρ. This is consistent with our expected result. For a certain ρ,

It first increases and then decreases with the increase of α. When α is very low, as α increases, the channel estimation matrix

The better it gets, the less interference leakage due to channel estimation. Therefore, the lower bound on the achievable rate

Increase first. However, when α increases to a certain extent, the increase of α will not bring more rate improvement. In this case, the lower bound on the achievable rate

Will be dominated by the effective signal transit time (1-α)T. therefore

It first increases and then decreases with the increase of α. From Figure 2, we can also find that for a given ρ, the lower bound of the achievable rate

The maximum value of and its corresponding α exist. Also, as ρ increases, the channel will get better and the training time required to generate an accurate channel estimate will be shorter. This shows that the larger the ρ, the smaller the α _opt .

Figure 3 shows the simulated optimal time division factor α _opt as a function of the transmitted data symbol SNRρ. It can be seen from Figure 3 that α _opt decreases with the increase of ρ. As expected, the smaller ρ, the longer the training time will be required to provide a sufficiently accurate channel estimate. As ρ increases, the required training time is shorter.

随发送数据符号SNR ρ的变化。为了对比，我们分别将图2中得到的仿真α_opt和由式(15)求得的近似α_opt带入式(9)，给出了仿真和近似最优可达速率下界。由图4可以看出，仿真和近似速率几乎一致，只有一些小的可接受的差距。因此，考虑到从式(12)中计算理论α_opt的复杂性，在实际中，我们可以用从式(15)计算得到的近似α_opt来最大化可达速率下界。Figure 4 shows the optimal achievable rate lower bound

Variation of SNR ρ with transmitted data symbols. For comparison, we put the simulation α _opt obtained in Fig. 2 and the approximate α _opt obtained by Eq. (15) into Eq. (9), respectively, and give the simulation and approximate optimal achievable rate lower bounds. As can be seen from Figure 4, the simulated and approximate rates are almost identical, with only some small acceptable gaps. Therefore, considering the complexity of computing the theoretical α _opt from equation (12), in practice we can maximize the achievable rate lower bound with the approximate α _opt computed from equation (15).

Figure 5 compares the reachable rate lower bounds under the optimal time division factor α _opt and some fixed time division factors. We can find that the lower bound of the achievable rate under α _opt is higher than the lower bound of the achievable rate under other fixed time division factors because of the adaptability of α _opt to the channel characteristics.

Conclusion: The present invention studies the time division scheme of training sequences and data symbols in the interference alignment network. When considering the actual channel estimation at a given coherence time, based on the LS channel estimation algorithm, the statistical characteristics of the channel estimation error are analyzed, and a detailed deduction is obtained. Regarding the lower bound expression of the user's reachable rate for the time division factor, the lower bound of the user's reachable rate is maximized by optimizing the time division factor. Numerical results demonstrate the correctness and effectiveness of the proposed time division scheme.

Claims (1)

1. the time division method of training sequence and data symbol in a kind of interference alignment network, it is characterized in that comprising the following steps: Step 1, for an interference alignment network with K user pairs sharing the same frequency band, each user pair consists of a transmitting user and a receiving user, the transmitting user is configured with m antennas, and the receiving user is configured with n antennas; T symbol periods of the coherence time are all valid; H _kl ∈ C ^n×m is the channel matrix from the lth transmitting user to the kth receiving user,

is the channel estimation matrix, and ΔH _kl represents the channel estimation error;

is the additive white Gaussian noise at the kth receiving user, and N ₀ represents the noise power spectral density; let α∈(0,1) be the time division factor, the training time is αT symbol periods, and the data symbol transmission time is (1 -α)T symbol periods; based on the LS estimation algorithm, the maximum error bound of ΔH _kl is obtained

in,

(ΔH _kl ) _i denotes the i-th row of ΔH _kl , ε _t denotes the transmit power of the training sequence,

Indicates the transmission training sequence SNR; Step 2, each sending user only transmits one data stream d=1 to all receiving users within the coherence time, then the signal received by the kth receiving user is:

in,

is the precoding matrix designed by the lth sending user based on the IA technology,

is the receiving filter matrix designed by the kth receiving user based on IA technology,

To transmit a vector of data symbols, it is assumed that all transmitting users have the same power constraint:

where ε represents the maximum transmit power of the data symbol,

Indicates the transmission data symbol SNR; Step 3, the lower bound of the reachable rate of the kth user pair

in,

represents the expected signal power of the kth user,

B=m ² n ² ρ[(K-1) ² -1], C=Tρ _t , D=m ² n ² ρ(K-1) ² ; By solving for the optimal time division factor

Maximize the lower bound of the achievable rate; right

Find the first derivative with respect to α and let it be 0, define

G=n ² m ² ρ, the approximate expression for α is α ₁ α ³ +α ₂ α ² +α ₃ α+α ₄ =0, where, α ₁ =-2ACF, α ₂ =2ACG-2(AD+BC)F+(BC-AD)(2A-F), α ₃ =2(AD+BC)G-2BDF-(BC-AD)(2A-2B-FG), α ₄ =2BDG-(BC-AD)(2B+G). The approximate optimal time division factor α _opt is obtained from the approximate expression.

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