Optimal soft decision detection method for th precoding system

Abstract

The application provides an optimal soft decision detection method of a TH precoding system, considers the correlation between a signal c after channel coding and a quantized signal r, obtains accurate posterior probability (APP) through a total probability formula and a Bayes formula, inputs the APP into an LDPC decoder symbol by symbol, and can restore the original information sequence, and the complexity is basically consistent with the coding complexity under a single user non-interference system. Through simulation verification, when a single stream signal is considered and the interference signal is Gaussian distribution, the soft decision method can realize about 1dB error rate performance difference compared with a non-interference system. The soft decision detection method is applicable to different interference signal distributions, and can guarantee good performance, and has applicability for improving the overall performance of the TH precoding system.

Description

[Technical Field]

[0001] For communication systems employing Tomlinson-Harashima (TH) precoding, this invention provides an optimal soft-decision detection method. This method maximizes the efficiency of channel coding and significantly reduces the system's bit error rate. This invention belongs to the fields of communication and information systems, and signal and information processing. [Background Technology]

[0002] To meet the demands of large access numbers and high capacity, current and future wireless communications are based on multiple-input multiple-output (MIMO) systems and multiple-access (MA) systems. One of the difficulties in transmission over MIMO channels is interference among multiple users. Although interference can be suppressed by linear precoding such as zero-forcing (ZF) and minimum mean square error (MMSE) precoding, their sum-rate performance is usually far from the channel capacity limit. Gelfand and Pinsker proposed dirtypaper coding (DPC) to achieve channel capacity, but the multidimensional processing of DPC is difficult to implement in practice. Tomlinson-Harashima precoding (THP), as a simple but suboptimal symbol-by-symbol implementation of DPC, has received widespread attention. However, prior to this invention, most works only conducted theoretical analyses, and the few bit error rate performance analyses mostly used hard-decision detection. Research on soft-decision detection methods with low bit error rates is very scarce. Professor E. Peh from Singapore and Professor Y. Hayoun from Israel published an improved TH precoding receiver scheme based on soft decision, but it did not consider the correlation between TH precoding symbols, making it difficult to achieve good bit error rate performance.

[0003] Currently, the theoretical performance of TH precoding is widely used to characterize the upper limit of downlink spectral efficiency, but research on its bit error rate performance in practical applications is very limited. [Summary of the Invention]

[0004] (I) Purpose of the Invention

[0005] This invention provides an optimal soft-decision detection method that, in a system combining channel coding and TH precoding, obtains an accurate posterior probability and log-likelihood ratio (LLR), maximizing the efficiency of channel coding and significantly reducing the bit error rate of the TH precoding system.

[0006] In multi-user massive MIMO systems using TH precoding, the interference term *s* for each user is unknown at the receiver, but the interference signals for all users are known to the base station. In practical applications, the interference term is the superimposed signal from one or more users in multi-user communication, and the base station can pre-compensate for the interference during signal transmission using TH precoding.

[0007] The problem addressed by this invention is how to design a soft-decision decoder for TH precoding that can maximize the integration of channel coding and leverage the performance of the TH precoding system to improve bit error rate performance as much as possible.

[0008] (II) Technical Solution

[0009] This invention provides an optimal soft-decision detection method for a TH precoding system, comprising the following steps:

[0010] Step 1: Transmitter channel coding and modulation

[0011] First, consider the baseband signal model with real numbers. Consider a base station transmitting an information sequence w of length k to a user equipment, where c represents a data stream sequence of length n after channel coding, which can be represented as:

[0012]

[0013] in, Let G represent modular multiplication (modulo q), where G is an (n,k)-dimensional encoding generator matrix. Through multi-path binary encoding, the elements in c are encoded into base q, with each element belonging to {0,,q-1}.

[0014] After channel coding, the data stream sequence c is symbolically mapped into a q-PAM modulated signal sequence through bit-interleaved coded modulation (BICM) and Gray code mapping. When there is no interference, the modulated signal can be represented as:

[0015]

[0016] Where γ is the average energy used to normalize x. Here, equations (1) and (2) map the message sequence to an n-length q-PAM symbol sequence with an information rate of... Bit / symbol.

[0017] Step 2: TH precoding

[0018] The mathematical model of the TH precoder is described as follows: c represents a data stream sequence of length n after channel coding, s is the user's interference signal sequence, and x is the output signal of the channel-coded sequence c after passing through the TH precoder, or simply the precoded signal, which can be represented as:

[0019]

[0020] Among them, P s The average energy of the interference signal is represented by Enc, which stands for TH precoder. Its specific structure and operation are as follows: Figure 1 As shown. For the case where there is an interfering signal sequence s, the TH-precoded signal x can be specifically represented as:

[0021]

[0022] in, This represents the integer quantization of the interference signal sequence s, where Q... Z Distractor expressions Symbol-by-symbol quantization to the nearest integer value s Q , representing modulo-q subtraction, γ is used to normalize the energy of the precoded signal x. α is a performance optimization coefficient, which can be estimated using MMSE to reduce the system's bit error rate; its value is...

[0023] The sum of the TH-precoded signal x and the interference signal sequence s is called the quantized signal r, which can be expressed as:

[0024]

[0025] Step 3: Channel Transmission

[0026] The signal sequence received by the receiving end through wireless channel transmission is as follows:

[0027]

[0028] Here, the average energy of the precoded signal x and the interference signal sequence s is normalized, P x and P s Let z represent their energies, and z represent a value with a mean of 0 and a variance of σ. 2 Additive white Gaussian noise.

[0029] Step 4: Soft Decision Detection Method

[0030] This section is the core content of the invention, detailing how the receiving end calculates the corresponding LLR from the received signal using soft decision calculation.

[0031] Based on the characteristics of TH precoding, the sequence is processed symbol by symbol. Let c[t], y[t] and r[t] represent the data stream sequence c of length n after channel coding, the received signal sequence y, and the t-th symbol of the quantized signal r, respectively, t = 1, ..., n.

[0032] For the t-th symbol, its codeword value is c[t] = i, i = 0, ..., q-1. Solving for the corresponding posterior probability is the key and challenging aspect of this invention, requiring calculation based on the correlation between symbols. It can be expressed as:

[0033]

[0034] The calculation of the above formula is the core innovation of this invention. It makes full use of the correlation between symbols for calculation. The calculation of its prior probability p(r[t]) and the values ​​of other probability functions will be detailed in the specific implementation scheme section later. Then, according to the calculation formula of LLR, the value of LLR is obtained by substituting into it.

[0035] Step 5: Decoding the channel code (see details below) Figure 2 )

[0036] Based on the LLR obtained in step four, and inputting it into a standard binary LDPC iterative decoder, the soft-decision decoding result of the data stream sequence c of length n after channel coding can be obtained.

[0037] Step Six: Simulation Verification and Performance Evaluation

[0038] Under the TH precoding system model, simulations of the system bit error rate (BER) and signal-to-noise ratio (SNR) are conducted for the soft-decision detection method proposed in this invention. The user signal is considered to be a discrete q-PAM distribution with equal probability, while the interference signal is either a discrete q-PAM distribution with equal probability or a continuous Gaussian distribution.

[0039] To evaluate the performance of a single user in a TH precoding system, the bit error rate performance of the interference-free system and the TH precoding system were compared based on different coding rates, modulation schemes, and interference signal distributions.

[0040] In the absence of interference, soft-decision detection methods can achieve the best false detection rate. However, in practical applications, it is difficult to completely eliminate interference. Therefore, when evaluating the performance of soft-decision detection methods, it is necessary to consider the performance difference compared to the interference-free case.

[0041] (III) Advantages and Efficacy

[0042] This invention proposes a soft-decision detection method based on the TH precoding system. It considers the correlation between the channel-coded signal c and the quantized signal r, and obtains the accurate posterior probability (APP) using the total probability formula and Bayes' theorem. The APP is then input symbol-by-symbol into the LDPC decoder to recover the original information sequence. Its complexity is essentially consistent with the coding complexity of a single-user interference-free system. Simulation results show that when considering a single-stream signal and the interference signal has a Gaussian distribution, the proposed soft-decision method achieves a bit error rate performance difference of approximately 1 dB compared to the interference-free system. In a large-scale MIMO system, with the same system and data rate, this method achieves a power consumption difference of less than 1 dB compared to the interference-free system, and a difference of approximately 3.8 dB from the theoretical value. In terms of system and data rate performance, it can approach the theoretical limit of information in downlink multi-user channels (with a difference of approximately 1 dB). It significantly improves the system's spectral efficiency and throughput at low cost. This soft-decision detection method is applicable to different interference signal distributions and can guarantee good performance in all cases, making it suitable for improving the overall performance of TH precoding systems. [Attached Image Description]

[0043] Figure 1 This is a block diagram of the TH precoder encoder.

[0044] Figure 2 This is a schematic diagram of the proposed TH precoder decoder process.

[0045] Figure 3 This is a schematic diagram of an extended constellation with q=5 (different shapes of symbols correspond to different message values).

[0046] Figure 4 The bit error rate performance curve is shown for a single-user model when the interference signal has a Gaussian distribution.

[0047] Figure 5 For a 3×3 MIMO scenario, the system and rate are... The power performance curve at that time.

Detailed Implementation Methods

[0048] To gain a better understanding of the principles, methods, features, and performance advantages of this invention, it is described in detail below.

[0049] Step 1: Transmitter channel coding and modulation

[0050] First, consider the baseband signal model with real numbers. Consider a base station transmitting an information sequence w of length k to a user equipment, where c represents a data stream sequence of length n after channel coding, which can be represented as:

[0051]

[0052] in, Let G represent modular multiplication (modulo q), where G is an (n,k)-dimensional encoding generator matrix. Through multi-path binary encoding, the elements in c are encoded into base q, with each element belonging to {0,,q-1}.

[0053] The channel-coded data stream sequence c is symbol-by-symbol mapped into a q-PAM modulated signal sequence through bit-interleaved coded modulation (BICM) and Gray code mapping. When there is no interference, the modulated signal can be represented as:

[0054]

[0055] Where γ is the average energy used to normalize x. Here, equations (8) and (9) map the message sequence to an n-length q-PAM symbol sequence with an information rate of... Bit / symbol.

[0056] Different values ​​of a q-PAM sequence can be represented by constellation points. A primitive q-PAM constellation (set of constellation points) can be represented as follows:

[0057]

[0058] Considering the transmit power issue during TH precoding, the q constellation points are repeated in the integer field to obtain a value related to... The expanded constellation is represented as follows:

[0059]

[0060] Grouping this extended constellation according to its corresponding message value yields the following set of constellation points corresponding to different message values:

[0061]

[0062]

[0063] For the original constellation set, To expand the constellation set, To expand the subset of the constellation set corresponding to message value c, Represents a set of integers. Let q be a finite set of integers.

[0064] To illustrate the concept of extended constellation points more vividly, consider the case where q = 5. Figure 3 The corresponding graphical representation is provided. The original constellation is... Extended constellations For a given message value, such as c=3, in the original constellation... In extended constellations, it can be represented as Figure 3 The elements of this subset are represented by a “” shape.

[0065] Step 2: TH precoding

[0066] The mathematical model of the TH precoder is described as follows: c represents data of length n after channel coding, s is the user's interference signal sequence, and x is the output signal of the channel-coded sequence c after passing through the TH precoder, or simply the precoded signal, which can be represented as:

[0067]

[0068] Among them, P s The average energy of the interference signal is represented by Enc, which stands for TH precoder. Its structure is as follows: Figure 1 As shown, for the case where there is an interfering signal sequence s, the TH-precoded signal x can be specifically represented as:

[0069]

[0070] in Q is the integer quantization of the interference signal sequence s, where Q Z Quantize the interference sequence s symbol by symbol to the nearest integer value s Q , indicating in Modulo-q subtraction is performed within the finite cyclic domain, and γ is used to normalize the energy of the precoded signal x. α is a performance optimization coefficient, which can be estimated using MMSE to reduce the system's bit error rate; its value is...

[0071] The sum of the TH-precoded signal x and the interference sequence s is called the quantized signal r, which can be expressed as:

[0072]

[0073] Here, all possible values ​​of r constitute the extended constellation points. Each symbol in r is an extended constellation. One character.

[0074] Step 3: Channel Transmission

[0075] The signal sequence received by the receiving end through wireless channel transmission is as follows:

[0076]

[0077] Here, the average energy of the precoded signal x and the interference signal sequence s is normalized, P x and P s Let z represent their energies, and z represent a value with a mean of 0 and a variance of σ. 2 Additive white Gaussian noise.

[0078] Step 4: Soft Decision Detection Method (Core Content of this Invention)

[0079] Based on the characteristics of TH precoding, the sequence is processed symbol by symbol. Let c[t], y[t] and r[t] represent the data stream sequence c of length n after channel coding, the received signal sequence y, and the t-th symbol of the quantized signal r, respectively, t = 1, ..., n.

[0080] For the t-th symbol, its codeword value is c[t] = i, i = 0, ..., q-1. Solving for the corresponding posterior probability is the key and challenging aspect of this invention, requiring calculation based on the correlation between symbols, and is expressed as:

[0081]

[0082] Since r[t] and c[t] have a one-to-one correspondence, p(c[t]=i|r[t])=1, and formula (17) can be expressed in the following form:

[0083]

[0084] The following section presents the methods for calculating the prior probability p(r[t]) and the conditional probability p(y[t]|r[t]).

[0085] Since the quantized signal r is related to the signal x and the interference sequence s, the value of p(r[t]) is analyzed according to different distributions of the interference sequence s. Based on practical application scenarios, it is mainly divided into two distributions: Gaussian distribution and q-PAM distribution, which are calculated separately.

[0086] (1) When the interference sequence s follows a Gaussian distribution:

[0087] For ease of description, let s[t] denote the t-th symbol of the interference sequence s, t = 1, ..., n, whose distribution has a mean of 0 and a variance of ... Gaussian distribution Based on the properties of the Gaussian distribution, we can obtain

[0088] Right now

[0089] From the property of modulus reduction in formula (4), we can know that Therefore, for the extended constellation points in formula (11) For each value of , its prior probability p(r[t]) is equal to The probability of obtaining the corresponding interval of r[t], i.e. Based on the Gaussian distribution of s, we can obtain therefore,

[0090]

[0091] (2) When the interference sequence s is a q-PAM distribution:

[0092] Based on the properties of the q-PAM distribution, we can obtain that the probability of the interference signal s[t] for any possible value s in its original constellation is 1. Right now Let c[t] and s Q Substituting different values ​​of [t] The extended constellation corresponding to formula (13) can be obtained. And the corresponding probabilities. Let c[t] and s Q [t] conforms to the distributions of q1-PAM and q2-PAM respectively, therefore

[0093] Since the posterior probability is calculated for each y[t], p(y[t]) = 1.

[0094] From formulas (4) and (6), it can be seen that, Therefore, p(y[t]|r[t]) can be obtained in the following form:

[0095]

[0096] in,

[0097] Substituting the above probability values ​​into formula (18), we can obtain the posterior probability values ​​of each y[t] corresponding to different constellation points. By accumulating the probabilities of constellation points corresponding to the same message value, we can obtain the posterior probability of y[t] corresponding to different message values.

[0098] Regarding the calculation method of LLR, only the calculation formula for 4PAM is given here. For the posterior probability corresponding to the information element {0,1,2,3}, it can be expressed as the sum of the corresponding extended constellation points, which can be represented as:

[0099] PP(i = c(i)). Let And LLR1 and LLR2 are used as the odd-numbered and even-numbered elements of LLR, respectively, that is:

[0100]

[0101] Step 5: Decoding the channel code (see...) Figure 2 )

[0102] Based on the LLR obtained in step four, and inputting it into a standard binary LDPC iterative decoder, the soft-decision decoding result of the data stream sequence c of length n after channel coding can be obtained.

[0103] Step Six: Simulation Verification and Performance Evaluation

[0104] Within the system model of the TH precoding problem, Monte Carlo simulations of the system bit error rate (BER) versus signal-to-noise ratio (SNR) are conducted for the soft-decision detection method proposed in this invention. Interference signals are considered as both equally probable discrete q-PAM and continuous Gaussian random variables. Simulations are stopped after collecting 100 frame errors at each SNR point, and the BER and BER results are recorded. The code length during simulation is 1920, and the coding rate is...

[0105] In a single-user model, the proposed soft-decision detection method is used to simulate the bit error rate performance under Gaussian distribution interference signals and interference-free conditions. The performance curves are shown below. Figure 4 As shown, the proposed soft-decision detection method has a bit error rate that is only about 1 dB lower than that of the system in the absence of interference.

[0106] The following is a specific example:

[0107] Application of TH precoding soft decision method in 3×3 MIMO downlink channel

[0108] This invention solves the optimal soft-decision detection problem in downlink channel TH precoding systems for multi-user massive MIMO. This example presents a transceiver design combining TH precoding and MMSE precoding for a multi-user massive MIMO scenario with K users and N=K base station antennas. The values ​​of N and K can be changed according to system requirements.

[0109] The received signal of user k can be represented as:

[0110]

[0111] in, Let h be the t-th symbol in the user k signal sequence, where t = 1, ..., n. k It is the signal vector from the base station to user k, u k w is the precoding vector of user k signal. k [t] represents Gaussian white noise. For ease of explanation, let... Substituting into formula (20), The expression for the received signal of user k at this time can be obtained as follows:

[0112]

[0113] in, Since TH precoding combined with MMSE cannot eliminate interference signals, the SINR of user k can be expressed as:

[0114]

[0115] From formula (18), it can be seen that at the base station transmitter, the superimposed signals of all users j>k are considered as interference, and TH precoding is used to precode the message of user k. At the user k receiver, the optimal soft-decision detection method of this invention is used to solve for the message of user k. Since For q-PAM distributions with different parameters, when the number of users superimposed in s[t] is greater than 4, according to the Lyapunov central limit theorem, s[t] has a mean of 0 and a variance of . The distribution is a Gaussian distribution, where n is the number of users j > k. This represents the variance of the corresponding signal.

[0116] Using the case where the interference signal in step one is Gaussian distributed, the corresponding APP can be calculated. When the number of superimposed users in s[t] is less than 4, using the case where the interference signal in step one is q-PAM distributed, the corresponding APP can be calculated.

[0117] To demonstrate specific performance, simulation results and performance analysis are presented with K=3. With K=N=3, the code length is 1920, and the system and rate are... In the case of system power performance such as Figure 5 As shown. For ease of explanation, the system noise power is set to 1. In the simulation, the user can choose between BPSK and 4-PAM modulation schemes, and the selectable coding rates are [0, 1 / 4, 1 / 3, 1 / 2, 2 / 3]. The four points in the figure, from top to bottom, represent: TH precoding system combined with MMSE, all users BER < 10 -4 The average transmit power of the system at that time; BER < 10 for all users in the interference-free system. -4 The figure shows the average transmit power of the system under various conditions: the theoretical average transmit power when the user rate is a discrete value conforming to the coding rate, and the theoretical average transmit power when the user rate is continuous. All figures are averages based on 500 channel implementations. As can be seen from the figure, the performance difference between the TH precoding system combined with MMSE and the interference-free system is only about 0.6 dB; compared with the theoretically calculated performance, the difference is about 3 dB.

[0118] The simulation of this invention will use Matlab scientific computing and call the C language-based programs of the encoder, decoder, and optimizer of binary LDPC through the mex function. The simulation verification method of single-user TH precoding will be applied to the above-mentioned multi-user large-scale MIMO serial TH precoding system to obtain numerical simulation results.

[0119] Although the present invention has been disclosed above by way of embodiments, these are merely examples and are not intended to limit the invention. Anyone skilled in the art can make various equivalent changes or substitutions without departing from the spirit and scope of the invention. Therefore, the scope of protection of the present invention shall be determined by the scope defined in the appended claims.

Claims

1. An optimal soft-decision detection method for a TH precoding system, characterized in that, Includes the following steps: Step 1: Channel coding and modulation at the transmitting end; First, consider the baseband signal model with real numbers; consider that the base station needs to transmit a signal of length... Information sequence To a user device, This indicates that the length after channel coding is The data stream sequence is represented as: (1) in, Modulus multiplication, It is A dimensional encoding generator matrix; through multi-path binary encoding, ... The elements are encoded into The number system, where each element belongs to ; After channel coding, the data stream sequence The signal sequence, symbol-by-symbol mapped to q-PAM modulation using bit-interleaved coded modulation (BICM) and Gray code mapping, is represented as follows when there is no interference: (2) in, It is used for normalization The average energy; here, formulas (1) and (2) map the message sequence to long -PAM symbol sequence, its information rate is Bit / Symbol; Step 2: TH precoding; The mathematical model of the TH precoder encoder is: This indicates that the length after channel coding is Data stream sequence, It is the user's interference signal sequence. It is a channel-coded sequence The output signal after passing through the TH precoder, or simply the precoded signal, is represented as: (3) in, This represents the average energy of the interference signal. This indicates a TH precoder for sequences containing interference signals. In the case of TH precoded signal Represented as: (4) in, Indicates the interference signal sequence Integer quantization, here Distractor expressions Symbol-by-symbol quantization to the nearest integer value , Modulus Subtraction Used for pre-coded signals Perform energy normalization; These are performance optimization coefficients, estimated using MMSE to reduce the system's bit error rate; their values ​​are... ; The TH pre-encoded signal With interference signal sequence The sum of these is called the quantized signal. , represented as: （5） Step 3: Channel transmission; The signal sequence received by the receiving end through wireless channel transmission is as follows: (6) The pre-encoded signal here and interference signal sequence The average energy was normalized. and They represent their energies, This indicates that the mean is 0 and the variance is... Additive white Gaussian noise; Step 4: Soft decision detection method; Based on the characteristics of TH precoding, the sequence is processed symbol by symbol; let , and These represent the lengths after channel coding as follows: Data stream sequence Received signal sequence and quantization signal The A symbol, ; For the Each symbol has a codeword value of 1. The corresponding posterior probability needs to be calculated based on the correlation between the symbols, and is expressed as: （7） Then, according to the LLR calculation formula, substitute the values ​​to obtain the LLR value; Step 5: Decoding the channel code; The result from step four is input into a standard binary LDPC iterative decoder to obtain the channel-coded length of... Data stream sequence The soft-decision decoding result; Step Six: Simulation Verification and Performance Evaluation; Simulations were conducted on the system bit error rate (BER) and signal-to-noise ratio (SNR); the user signal was a discrete q-PAM distribution with equal probability, and the interference signal was a discrete q-PAM distribution with equal probability or a continuous Gaussian distribution; The performance of a single user in the TH precoding system is evaluated by comparing the bit error rate performance of the interference-free system and the TH precoding system under different coding rates, modulation schemes and interference signal distributions.

2. The optimal soft-decision detection method for the TH precoding system according to claim 1, characterized in that: In step one, the different values ​​of the q-PAM sequence are represented by constellation points. A primitive q-PAM constellation is represented as follows: ；（8）。 3. The optimal soft-decision detection method for the TH precoding system according to claim 2, characterized in that: Regarding Repeating the constellation points in the integer field yields a result about... The extended constellations are represented as follows: ， （9） The extended constellation is grouped according to its corresponding message value, resulting in the following set of constellation points corresponding to different message values: ， （10） ； For the original constellation set, To expand the constellation set, To expand the corresponding message value in the constellation set a subset of Represents a set of integers. for A finite set of integers containing elements.

4. The optimal soft-decision detection method for the TH precoding system according to claim 1, characterized in that: In step two, formula (5) is further expressed as: （11） in, All possible values ​​constitute the extended constellation points , Each symbol in the diagram represents an extended constellation. One character.

5. The optimal soft-decision detection method for the TH precoding system according to claim 1, characterized in that: In step four, in formula (7), since and It is a one-to-one correspondence, therefore It can be represented in the following form: （12）。 6. The optimal soft-decision detection method for the TH precoding system according to claim 5, characterized in that: Due to quantization signal With signal Interference sequences Related, therefore The value depends on the interference sequence The different distributions are analyzed; based on the actual application scenarios, they are divided into two distributions: Gaussian distribution and q-PAM distribution.

7. The optimal soft-decision detection method for the TH precoding system according to claim 6, characterized in that: When interference sequence It follows a Gaussian distribution: make Indicates interference sequence The A symbol, Its distribution satisfies a mean of 0 and a variance of . Gaussian distribution Based on the properties of the Gaussian distribution, we obtain ; Right now ; in, For extended constellation points For each value, its prior probability equal Received The probability of the corresponding interval, i.e. Based on the Gaussian distribution of s, we obtain ;therefore, 。 8. The optimal soft-decision detection method for the TH precoding system according to claim 6, characterized in that: When interference sequence For q-PAM distribution: Based on the properties of the q-PAM distribution, the interference signal is obtained. Regarding its original constellation any possible value in The probability is 1. ,Right now ;Will and Substituting different values To obtain the corresponding extended constellation And the corresponding probability; let and They conform to the distributions of q1-PAM and q2-PAM respectively, therefore ; Because it is for each Solve for the posterior probability, therefore .

9. The optimal soft-decision detection method for the TH precoding system according to claim 8, characterized in that: From formulas (4) and (6), ,therefore The result is in the following form: ； in, ; Substituting the above probability values ​​into formula (12), we obtain each The posterior probability values ​​for different constellation points are obtained by summing the probabilities of constellation points corresponding to the same message value. The posterior probability of the corresponding different message values.

10. The optimal soft-decision detection method for the TH precoding system according to claim 1, characterized in that: Regarding the calculation method of LLR, the calculation formula for 4PAM is given here; for information elements... The corresponding posterior probability, expressed as the sum of the probabilities of the corresponding extended constellation points, is as follows: ;make And LLR1 and LLR2 are used as the odd-numbered and even-numbered elements of LLR, respectively: 。

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