Signal reconstruction based on empirical Bayesian method to deal with unknown information distribution

Abstract

The present invention discloses a signal reconstruction method based on the empirical Bayesian method for processing unknown transition probabilities or unknown signal priors and transition probabilities. For the problem of unknown transition probabilities in generalized linear models, a transition probability model based on empirical Bayesian estimation is established to obtain the transition probability of the generalized linear model. According to the estimated transition probability and the state update equation, a scalar model is obtained, and a probability model based on empirical Bayesian estimation priors can be established to obtain the prior distribution of unknown signals. The transition probability model based on empirical Bayesian estimation and the probability model based on empirical Bayesian estimation priors are added to the algorithm to perform signal recovery. The algorithm continuously updates the signal estimation value through iterative calculation until the convergence condition is reached, thereby realizing accurate recovery of the original signal. The present invention solves the problem of unknown transition probabilities or unknown signal priors and transition probabilities through empirical Bayesian estimation, and can also reconstruct signals.

Description

Technical Field

[0001] The present invention belongs to the field of wireless communications and relates to a signal reconstruction technology for processing unknown information distribution based on an empirical Bayesian method. Background Art

[0002] Signal processing and information transmission are core technologies in wireless communication systems. However, due to the complexities of real-world environments, such as multipath, interference, and noise, received signals are often subject to various interferences and distortions. These factors significantly impact the performance of communication systems, especially in environments with weak signals or high noise levels.

[0003] Traditional signal estimation methods for the generalized linear model y = Hx + w include the more common generalized approximate message passing (GAMP) and generalized expectation consistent signal recovery (GEC-SR). GAMP and GEC-SR are two advanced algorithms widely used for signal recovery in large-scale linear systems. These algorithms offer significant advantages in solving problems such as sparse signal recovery and compressed sensing. However, the performance of GAMP and GEC in practical applications depends heavily on the signal's prior information and the characteristics of transition probabilities. Transition probabilities characterize the statistical properties of the intermediate variables Hx and the observation vector y. If the prior information and transition probabilities are unknown or inaccurate, the algorithm's recovery performance will be significantly reduced. These algorithms typically rely on pre-determined signal models and the distribution of transition probabilities. However, in practical applications, the true characteristics of the signal and transition probability distributions may not be fully understood, resulting in reduced estimation accuracy or even inability to estimate the signal. Therefore, accurately estimating signals under unknown prior information and transition probabilities has become an important research topic.

[0004] The empirical Bayesian approach provides an adaptive signal estimation method by using observed data to estimate the signal's prior distribution and transition probability distribution. Compared to traditional Bayesian methods, the empirical Bayesian approach does not require prior knowledge of the signal's prior distribution. Instead, it infers the prior distribution and transition probabilities from the data itself, making it more adaptable to the complex environments of practical applications.

[0005] This paper, based on the empirical Bayesian approach, adaptively estimates the signal's prior distribution and transition probabilities and applies this method to the GAMP and GEC-SR algorithms. This approach significantly improves signal recovery accuracy and system robustness without relying on pre-set prior information and transition probabilities. This provides an efficient solution for signal processing in complex wireless communication environments. Summary of the Invention

[0006] The present invention aims to address the problem in wireless communication systems where the GAMP and GEC-SR algorithms cannot recover signals when the signal prior distribution and transition probability characteristics are unknown. By introducing an adaptive estimation method based on empirical Bayesian methods, the present invention solves the problem of signal recovery even in the presence of unknown signal prior distribution and transition probability, thereby improving the accuracy of signal estimation and the robustness of the system.

[0007] The present invention proposes a signal prior distribution and transition probability estimation method based on the empirical Bayesian method, and applies it to the GAMP and GEC algorithms. The present invention provides the following technical solutions:

[0008] 1. For the problem of unknown transition probability in the generalized linear model, a transition probability model based on empirical Bayesian estimation is established to obtain the transition probability of the generalized linear model.

[0009] 2. Based on the transition probability estimated in step 1 and the state update equations of GAMP and GEC-SR, a scalar model is obtained. A probability model based on the empirical Bayesian estimation prior can be established to obtain the prior distribution of the unknown signal.

[0010] 3. Incorporating the Empirical Bayesian Estimation transition probability model and the Empirical Bayesian Estimation prior probability model into the GAMP or GEC algorithm for signal recovery. The algorithm iteratively calculates and continuously updates the signal estimate until convergence conditions are reached, accurately recovering the original signal.

[0011] Furthermore, the step 1 is specifically as follows:

[0012] 1.1 Empirical Bayesian estimation is an approach that combines Bayesian and frequentist statistical methods. In Bayesian statistics, parameters are typically considered random variables with a prior distribution. The core idea of ​​the empirical Bayesian approach is to infer or estimate the prior distribution from observed data without fully knowing it.

[0013] The empirical Bayes model is as follows:

[0014] X i =αΘ i +Ω i,i=1,…,K

[0015] Where α is the unknown channel coefficient, which combines the effects of channel fading and residual phase offset. Since the role of α is to rotate and scale the signal constellation, it is assumed that α = 1 without loss of generality. i} is a sequence composed of observation values, {Θ i} is a sequence of signals that are independent and identically distributed, {Ω i} is an independent and identically distributed sample sequence.

[0016] In the empirical Bayes model, for the unknown probability density g(θ) (which may be a discrete distribution), it has generated an unobservable random sample of independent realizations.

[0017]

[0018] Each Θ i From p i (X i |Θ i ) generates independent observations X i ,

[0019]

[0020] Edge density f i (X i ) is the previous g and p i The Bayesian deconvolution problem is to recover the probability density of g from these data.

[0021] f(x)=∫ T p(x|θ)g(θ)dθ

[0022] The prior density g(θ) may usually be a mixture of discrete and continuous distributions. For ease of explanation and calculation, assume that the sample space T of parameter Θ is finite and discrete T = (θ (1) ,θ (2) ,…,θ (t) )

[0023] where Θ i The sample set T is T, and g(θ) is estimated from the observations X. The process of selecting T can be determined through numerical experiments. Specifically, the range of T can be varied or the degree of refinement between samples can be adjusted to see whether these changes significantly affect the estimated prior distribution g(θ). If a wider range or finer samples are found to significantly improve the accuracy of the prior estimate, then it makes sense to adjust these parameters. For the number of samples t in T, it is found that increasing t does not bring significant benefits. Therefore, when choosing the number of support points t, it is generally not necessary to pursue extremely high precision.

[0024] Same X i The observation sample space φ is also assumed to be finite and discrete,

[0025] φ=(x (1) ,x (2) ,…,x (α) )

[0026] For X i The discretization of the observation sample space does not pose a restriction, because φ can be regarded as X i All order statistics of the values. Whether for discrete or continuous data, the sample space φ and the range T can be defined by appropriate methods without being troubled by practical limitations. This flexibility allows the model settings to be adjusted as needed in practical applications.

[0027] The priori g(θ) is represented by a vector g=(g1,g2,…,g t ) T , similarly, the marginal probability f(x) vector is represented as f=(f1,f2,…,f α ) T In vector g and vector f, the sum of their elements is 1.

[0028] Let p kj =P{X=x (k) |Θ=θ (j)}, where p kj represents the elements in row k and column j, which usually represent the transition from discrete state θ (j) to the observed value x (k) The convolution relationship between g(θ) and f(x) can be expressed as

[0029]

[0030] Define a matrix P of dimension α×t = (p kj ), according to the above convolution relationship, it can be transformed into matrix multiplication

[0031] f=Pg

[0032] Empirical Bayesian modeling is divided into g-model and f-model. This article mainly discusses the g-model, assuming that the prior distribution g belongs to a p-parameter exponential family on T, which can be expressed in discrete form as:

[0033]

[0034] Where g(θ) is the prior distribution to be estimated, which represents the probability distribution of the parameter θ. p) is the natural parameter vector of the model, which contains p parameters, which determine the specific form of the exponential family. j (θ) is a sufficient statistic defined on θ. A(η) is a normalization constant. Further rewriting the above equation into the following vector form,

[0035] g(θ)=e Qη / c(η)

[0036] Q is a t×p matrix, where t is the number of discrete points in the sample space T and p is the number of model parameters. This matrix is ​​often used to describe the relationship between the sample space T and the parameters θ. c(θ) is a normalization constant.

[0037] 1.2 In a massive MIMO (Multiple Input Multiple Output) system, assuming that the base station is equipped with m receiving antennas and the user end is equipped with n transmitting antennas, its generalized linear model can be expressed as

[0038] y=Hx+w

[0039] Let z = Hx, where is the m-dimensional observation vector, is the observation matrix, H is the independent and identically distributed channel matrix N(0,1 / n) obeys independent synchronous distribution, For the n-dimensional target signal to be estimated, the probability distribution of w mainly describes the statistical characteristics of the intermediate variable z and the observation vector y. The transition probability model based on empirical Bayesian estimation is as follows:

[0040] y=z+w

[0041] Where w is the transition probability to be estimated. By processing the signal at the sending end so that its mean is 0 and its variance is 1, the distribution of z is a known Gaussian distribution. Based on the above model combined with empirical Bayesian estimation, the transition probability can be estimated.

[0042] Furthermore, the step 2 is specifically as follows:

[0043] The state evolution equation (SE) is an analytical tool that can be used to predict the performance of an algorithm. It is used to understand and predict how iterative algorithms (such as GAMP and GEC-SR) converge in high-dimensional data. The SE equation provides a description of the algorithm's asymptotic behavior, especially when the data dimension tends to infinity. In the literature related to GAMP and GEC-SR, SE is used to track the evolution of the estimation error to analyze the performance of the algorithm. From the perspective of understanding, here is an analysis of SE from the matching case. In the GAMP and GEC-SR algorithms, a probability distribution similar to the following will appear

[0044]

[0045] where p X (x) is the prior distribution of the signal, is a Gaussian distribution, whose expectation x - and variance is v - This is the information obtained in the previous stage of the algorithm.

[0046] The mean square error (MSE) of the above estimate can be expressed as

[0047]

[0048] where the expectation is the probability p(x|x - ). The minimum mean square error (MMSE) estimated signal is expressed as

[0049]

[0050] In large-scale systems, replacing some variables with equivalent scalar models yields

[0051]

[0052] Where X~p X (x).

[0053] Based on the scalar model derived from the state update equation above, a probability model based on empirical Bayesian estimation priors is established in combination with empirical Bayesian estimation to obtain the prior distribution of the unknown signal. The established empirical Bayesian estimation transition probability model and the probability model based on empirical Bayesian estimation priors are incorporated into the GAMP or GEC-SR algorithm for signal recovery.

[0054] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0055] Traditional GAMP and GEC-SR algorithms typically rely on preset signal prior distribution and transition probability models. However, in practical applications, the assumptions of these models are often inaccurate, resulting in unsatisfactory signal recovery. The present invention adaptively estimates the signal prior distribution and transition probability through the empirical Bayesian method. Prior information can be dynamically acquired, and more realistic prior information can be dynamically acquired without the need for a preset model, significantly improving the accuracy of signal recovery.

[0056] By introducing the empirical Bayesian method, the present invention realizes adaptive estimation of signal prior distribution and transition probability without the need for a preset model, significantly improving the signal recovery accuracy and system robustness of the GAMP and GEC-SR algorithms in complex environments, reducing dependence on prior models, optimizing computational efficiency, and providing effective technical support for performance improvement of wireless communication systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 Schematic diagram of the signal reconstruction system model based on empirical Bayeux estimation of unknown signal priors and transition probabilities.

[0058] Figure 2 To estimate the unknown transition probabilities of the generalized approximate message passing algorithm based on empirical Bayesian estimation, the actual transition probability distribution is a mixture of Gaussian distributions. This experiment uses two Gaussian mixtures: the first Gaussian distribution has weight l1 = 0.5, mean mean1 = -2, and signal-to-noise ratio (SNR1) = 13dB. The second Gaussian distribution has weight l2 = 0.5, mean mean2 = 2, and signal-to-noise ratio (SNR2) = 13dB. The figure shows the actual probability density of the mixture of Gaussian distributions and the estimated probability density.

[0059] Figure 3 In order to estimate the prior distribution of unknown signals for the generalized approximate message passing algorithm based on empirical Bayesian estimation, real numbers are used in the experiment, which is also valid in the complex number field. The prior is four real numbers, which is equivalent to the prior for 16QAM distribution.

[0060] Figure 4 The following diagram illustrates the performance of the generalized approximate message passing algorithm in four different scenarios. The figure shows the performance of GAMP-eb when both the signal prior and transition probabilities are unknown based on empirical Bayesian estimation. The figure shows the performance of GAMP-eb-n when the transition probabilities are unknown but the prior is known based on empirical Bayesian estimation. The figure shows the performance of GAMP when both the signal prior and transition probabilities are known. LS shows the performance when using least squares. The multi-input multi-output model in this experiment has m = 1500 and n = 500.

[0061] Figure 5 To estimate the unknown transition probabilities of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, the actual transition probability distribution is a mixture of Gaussian distributions. This experiment uses two Gaussian mixtures: the first Gaussian distribution has a weight of l1 = 0.5, a mean of mean1 = -2, and a signal-to-noise ratio (SNR1) of 12dB. The second Gaussian distribution has a weight of l2 = 0.5, a mean of mean2 = 2, and a signal-to-noise ratio (SNR2) of 12dB. The figure shows the actual probability density of the mixture of Gaussian distributions and the estimated probability density.

[0062] Figure 6In order to estimate the prior distribution of the unknown signal of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, real numbers are used in the experiment, which is also valid in the complex number field. The prior is four real numbers, which is equivalent to the prior for 16QAM distribution.

[0063] Figure 7 The following diagram illustrates the performance of the generalized expectation consistent signal reconstruction algorithm in four different scenarios. GEC-eb shows the performance based on empirical Bayesian estimation when both the signal prior and transition probabilities are unknown. GEC-eb-n shows the performance based on empirical Bayesian estimation when the transition probabilities are unknown but the priors are known. GEC shows the performance when both the signal prior and transition probabilities are known. LS shows the performance when using least squares. The multi-input multi-output model used in this experiment has m = 1500 and n = 500. DETAILED DESCRIPTION

[0064] To facilitate understanding of the present invention, the present invention will be described more fully below with reference to the accompanying drawings. Preferred embodiments of the present invention are shown in the accompanying drawings. However, the present invention may be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a more thorough and comprehensive understanding of the present disclosure.

[0065] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which the present invention pertains. The terms used in the specification of the present invention are intended solely for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" as used herein includes any and all combinations of one or more of the related listed items. The technical solution of the present invention will be further described below with reference to the accompanying drawings and examples.

[0066] I hereby declare that the content described in this patent is different from the patent entitled "Iterative Signal Detection Method and System in a Scenario of Missing Prior Information" and is a further improvement on that patent. The main differences are: First, the multi-input and multi-output models are different. The model used in this patent is a generalized linear model, while the patent entitled "Iterative Signal Detection Method and System in a Scenario of Missing Prior Information" uses a standard linear model. This patent further expands the scope of the model. Second, the application scenario of this patent is to recover the signal when the transition probability is unknown or when both the signal's prior information and the transition probability are unknown, while the application scenario of the patent entitled "Iterative Signal Detection Method and System in a Scenario of Missing Prior Information" is only to recover the signal when the prior information is unknown, which is different from the application scenario of this patent. This patent focuses on recovering the signal when two important information are missing, and this patent also further improves the application scope of signal recovery.

[0067] The overall process of this program refers to Figure 1 .

[0068] Implementation Case 1

[0069] This embodiment provides a signal prior and transition probability based on empirical Bayesian estimation of generalized approximate message passing (GAMP).

[0070] S1: Get the channel matrix H with m rows and n columns and the observation signal y, and set the intermediate variable z = Hx

[0071] S2: Based on the model results of step S1, establish an empirical Bayesian estimation transition probability model y=z+n, and use empirical Bayesian to estimate the transition probability p(y|z), and obtain the estimated transition probability result as follows Figure 2 As shown, the true results are very close to the actual results.

[0072] S3: Obtain the transition probability p(y|z) in step S2, and further establish a signal prior model based on the empirical Bayesian estimation according to the state update equation SE. The prior results are estimated based on the empirical Bayesian estimation. Figure 3 shown.

[0073] S4: Combining the resulting Empirical Bayesian Estimation transition probability model and the Empirical Bayesian Estimation prior probability model with the GAMP algorithm allows for signal recovery even when signal priors and transition probabilities are missing. The algorithm iteratively updates the signal estimate until convergence is achieved, accurately recovering the original signal.

[0074] For the generalized approximate message passing algorithm based on empirical Bayesian estimation (EB-GAMP), its iterative formula is as follows:

[0075] (1) Initialization:

[0076] (2) Definition:

[0077]

[0078] (3) Iteration (t=1:T):

[0079]

[0080] (4) Output estimation signal:

[0081] Where a=1…m,i=1…n,p Y|Z (y a |z a )=EB_GAMP{p Y|Z (y a |z a ); y,z} represents the ath ya and z a The generalized approximate message passing algorithm based on empirical Bayesian estimation is used to estimate the transition probability model. is the m-dimensional observation vector, z=Hx, where is the observation matrix, For the n-dimensional target signal, the unknown transition probability is estimated by empirical Bayesian; Indicates the ath The probability distribution of is taken as the mean, Indicates the ath The probability distribution of is taken as the variance, is the mean of the intermediate parameters of the ath signal at the tth iteration, is the variance of the intermediate parameter of the ath signal at the tth iteration; is the i-th signal x i Based on the empirical Bayesian estimation of the generalized approximate message passing algorithm prior probability model, is the mean of the intermediate parameters of the ith signal at the tth iteration, is the variance of the intermediate parameter of the ith signal at the tth iteration, and the prior of the unknown signal is estimated by empirical Bayesian.

[0082] S5: The final estimated performance is as follows Figure 4 As shown in the figure, performance comparisons were designed for four scenarios, which were consistent with theoretical expectations. The figure shows the performance of GAMP-eb when both the signal prior and transition probabilities are unknown based on empirical Bayesian estimation. The figure shows the performance of GAMP-eb-n when the transition probabilities are unknown but the prior is known based on empirical Bayesian estimation. The figure shows the performance of GAMP when both the signal prior and transition probabilities are known. LS shows the performance when the least squares method is used. The results show that while the least squares method does not require priors and transition probabilities, it fails in this case. However, the generalized approximate message passing algorithm EB-GAMP, based on empirical Bayesian estimation, not only reconstructs the signal but also demonstrates excellent performance.

[0083] Implementation Case 2

[0084] This embodiment provides a signal prior and transition probability for generalized expected consistent signal reconstruction (GEC-SR) based on empirical Bayesian estimation.

[0085] S6: Get the channel matrix H with m rows and n columns and the observation signal y, and set the intermediate variable z = Hx

[0086] S7: Based on the model result of step S6, establish a transition probability model y=z+n based on empirical Bayesian estimation, and estimate the transition probability p(y|z) through empirical Bayesian to obtain the estimated transition probability result as follows Figure 5As shown, the true results are very close to the actual results.

[0087] S8: Obtain the transition probability p(y|z) in step S7, and further establish a signal prior model based on empirical Bayesian estimation according to the state update equation SE. The prior results are estimated based on empirical Bayesian estimation as follows: Figure 6 shown.

[0088] S9: Combining the Empirical Bayesian approach with the GEC-SR algorithm allows for effective signal recovery in the absence of prior signal knowledge and transition probability information. The core idea of ​​this approach is to use Empirical Bayesian to estimate the signal's prior distribution and state transition probabilities based on S8 and S9, and then use these estimates as input to the GEC-SR algorithm to iteratively update the signal estimate.

[0089] For the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation (EB-GEC-SR), its iterative formula is as follows

[0090] (1) Initialization:

[0091] (2) Definition:

[0092]

[0093] (3) Iteration (t=1:T):

[0094] p Y|z (y|z)=EB_GEC{p Y|Z (y|z); y,z}

[0095]

[0096]

[0097] (4) Output estimation signal:

[0098] The above variables are all expressed in vector form, 1 represents a vector whose column vectors are all 1, and 0 represents a vector whose column vector values ​​are all 0; where p Y|Z (y|z)=EB_GAMP{p Y|Z (y|z); y,z} is the transition probability model of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, is the m-dimensional observation vector, z=Hx, where is the observation matrix, For the n-dimensional target signal, the unknown transition probability is estimated by empirical Bayesian. The unknown transition probability of the generalized expected consistent signal reconstruction algorithm is estimated by empirical Bayesian. Represents the probability distribution as Take the mean, Represents the probability distribution as Take the variance; diag(·) is to find the diagonal of the matrix to form a column vector; Diag(·) is to use the elements in the vector as diagonal elements to form a matrix, and the off-diagonal elements are 0; represents vector dot division, ⊙ represents vector dot product; It means taking the mean of the probability distribution p(x|y). The variance of the probability distribution p(x|y) is taken; (·) H represents taking the conjugate transpose;

[0099] In order to estimate the prior probability model of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, the prior of the unknown signal is estimated through empirical Bayesian.

[0100] S10: Figure 7 Performance diagrams for the GEC-SR algorithm were designed for four different scenarios, and the results were consistent with theoretical expectations. The figure shows the performance of GEC-eb when both the signal prior and transition probabilities are unknown, based on empirical Bayesian estimation. The figure shows the performance of GEC-eb-n when the transition probabilities are unknown but the priors are known, based on empirical Bayesian estimation. The figure shows the performance of GEC when both the signal prior and transition probabilities are known. LS shows the performance when using the least squares method. The generalized expectation-consistent signal reconstruction algorithm EB-GEC-SR, based on empirical Bayesian estimation, also performs well when the prior and transition probabilities are completely unknown, while the least squares method LS completely fails.

Claims

1. A signal reconstruction method based on the empirical Bayesian method for processing unknown information distribution, characterized in that: The method comprises the following steps: For the problem of unknown transition probability in the generalized linear model y=Hx+w, let z=Hx, where is the m-dimensional observation vector, is the observation matrix, H is the independent and identically distributed channel matrix obeys independent and synchronous distribution For the n-dimensional target signal to be estimated, the probability distribution of w mainly describes the statistical characteristics of the intermediate variable z and the observation vector y. Y|Z (y a |z a ), Z represents the variables of the vector z as a whole, Y represents the variables of the observation vector y as a whole, based on the empirical Bayesian estimation transition probability model EB_GAMP{p Y|Z (y a |z a ); y,z} are as follows: y=z+w Where w is the transition probability to be estimated, y a represents the ath value of the observation vector y, z a Represents the ath value of the observation vector z. By processing the signal at the sending end so that its mean is 0 and its variance is 1, the distribution of z is a known Gaussian distribution. Based on the above model combined with empirical Bayesian estimation, the transition probability can be estimated; In the generalized approximate message passing or generalized expected consistent signal reconstruction algorithm, a probability distribution similar to the following will appear where p X (x) is the prior distribution of the signal, is a Gaussian distribution, whose expectation x - and variance is v - The information obtained in the previous stage of the algorithm; The mean square error (MSE) can be expressed as where the expectation is the probability p(x|x - ), the minimum mean square error (MMSE) estimated signal is expressed as In large-scale systems, replacing some variables with equivalent scalar models yields Where X~p X (x), we get a scalar model is a probability model based on empirical Bayesian estimation priors N has a mean of 0 and a variance of v - Gaussian distribution and The information obtained from the algorithm iteration in the previous stage is used to obtain the prior distribution of the unknown signal; Based on the empirical Bayesian estimation transition probability model EB_GAMP{p Y|Z (y a |z a ); y,z} and a probability model based on empirical Bayesian estimation priors It is added to the generalized approximate message passing algorithm and the generalized expected consistent signal reconstruction algorithm to perform signal recovery. The algorithm continuously updates the signal estimation value through iterative calculation until the convergence condition is reached, thus achieving accurate recovery of the original signal. The expression for the generalized approximate message passing algorithm based on empirical Bayesian estimation is Where a = 1...m, i = 1...n, when t = 1 initialization: where p(x i ) represents the i-th x of signal x i The probability distribution of p Y|Z (y a |z a )=EB_GAMP{p Y|Z (y a |z a ); y,z} represents the ath y a and z a The generalized approximate message passing algorithm based on empirical Bayesian estimation is used to estimate the transition probability model. is the m-dimensional observation vector, z=Hx, where is the observation matrix, is the n-dimensional target signal, y a represents the ath value of the observation vector y, z a represents the ath value of the observation vector z, Y represents the overall variable of the observation vector y, and H ai Represents the value of the ath row and the ith column of the matrix H; the unknown transition probability is estimated by empirical Bayes; Indicates the ath The probability distribution of is taken as the mean, Indicates the ath The probability distribution of is taken as the variance, is the mean of the intermediate parameters of the ath signal at the tth iteration, is the variance of the intermediate parameter of the ath signal at the tth iteration; is the i-th signal x i Based on the empirical Bayesian estimation of the generalized approximate message passing algorithm prior probability model, is the mean of the intermediate parameters of the ith signal at the tth iteration, is the variance of the intermediate parameter of the ith signal at the tth iteration, and the prior of the unknown signal is estimated by empirical Bayesian. The generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation is expressed as p Y|Z (y|z)=EB_GAMP{p Y|Z (y|z)(y,z} The above variables are all expressed in vector form and initialized when t=1: where p Y|Z (y|z)=EB_GAMP{p Y|Z (y|z); y,z} is the transition probability model of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, is the m-dimensional observation vector, z=Hx, where is the observation matrix, is the n-dimensional target signal, H H Denotes the conjugate transpose of the matrix H; Empirical Bayesian estimates of the unknown transition probabilities of the generalized expected consistent signal reconstruction algorithm are used to estimate the unknown transition probabilities; Represents the probability distribution as Take the mean, Represents the probability distribution as Take the variance; diag(·) is to find the diagonal of the matrix to form a column vector; Diag(·) is to use the elements in the vector as diagonal elements to form a matrix, and the off-diagonal elements are 0; represents vector dot division, ⊙ represents vector dot product; It means taking the mean of the probability distribution p(x|y). The variance of the probability distribution p(x|y) is taken; (·) H represents taking the conjugate transpose; In order to estimate the prior probability model of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, the prior of the unknown signal is estimated through empirical Bayesian.

2. The signal reconstruction method for processing unknown information distribution based on the empirical Bayesian method according to claim 1, characterized in that: Establish an empirical Bayesian estimation model. The empirical Bayesian model form is as follows: X i =aΘ i +Oh i ,i=1,…,K Where α is the unknown channel coefficient, which combines the effects of channel fading and residual phase offset. Since the role of α is to rotate and scale the signal constellation, it is assumed that α = 1 without loss of generality; {X i } is a sequence composed of observation values, {Θ i } is a sequence of signals that are independent and identically distributed, {Ω i } is an independent and identically distributed sample sequence.

3. The signal reconstruction method for processing unknown information distribution based on the empirical Bayesian method according to claim 2, characterized in that: Establishing a sample interval model of the estimated variable based on empirical Bayesian estimation includes the following steps: In the empirical Bayes model, for the unknown probability density g(θ), it has generated independent and unobservable random samples. Each Θ i From p i (X i |Θ i ) generates independent observations X i , Edge density f i (X i ) is the previous g and p i The Bayesian deconvolution problem is to recover the probability density of g from these data. f(x)=∫ T p(x∣θ)g(θ)dθ The prior density g(θ) is a mixture of discrete distribution and continuous distribution. For the convenience of explanation and calculation, it is assumed that the sample space T of parameter θ is finite and discrete. T=(θ (1) ,i (2) ,…,θ (t) ) where Θ i The sample set is T, and g(θ) is estimated from the observation value X. The process of selecting T is determined by some numerical experiments. Specifically, by changing the range of T or adjusting the degree of refinement between samples, we observe whether these changes significantly affect the estimated prior distribution g(θ); t represents the number of samples; Same X i The observation sample space φ is also assumed to be finite and discrete, φ=(x (1) ,x (2) ,…,x (α) ) Among them, for X i The discretization of the observation sample space does not pose a restriction, because φ can be regarded as X i All order statistics for the values.

4. The signal reconstruction method for processing unknown information distribution based on the empirical Bayesian method according to claim 3, characterized in that: Establishing an empirical Bayesian estimation model to estimate probability distribution includes the following steps: The priori g(θ) is represented by a vector g=(g1,g2,…,g t ) T , similarly, the marginal probability f(x) vector is represented as f=(f1,f2,…,f α ) T , in vector g and vector f, the sum of their elements is 1, Let p kj =P{X=x (k) |Θ=θ (j) }, where p kj represents the elements in row k and column j, which usually represent the transition from discrete state θ (j) to the observed value x (k) The transition probability, the convolution relationship between g(θ) and f(x) can be expressed as Define a matrix P of dimension α×t = (p kj ), according to the above convolution relationship, it can be transformed into matrix multiplication f=Pg Empirical Bayesian modeling is divided into g-model and f-model. This time, the g-model is applied. It is assumed that the prior distribution g belongs to a p-parameter exponential family on T, which can be expressed in discrete form as: Where g(θ) is the prior distribution to be estimated, which represents the probability distribution of the parameter θ, and η=(η1,η2,…,η p ) is the natural parameter vector of the model, which contains p parameters. These parameters determine the specific form of the exponential family, h j (θ) is a t-dimensional column vector of sufficient statistics defined on θ, A(η) is a normalization constant, and the above formula can be further written in the following vector form, g(θ)=e Qη / c(n) Q is a t×p matrix, where t is the number of discrete points in the sample space T and p is the number of parameters of the model. This matrix is ​​usually used to describe the relationship between the sample space T and the parameter η, and c(η) is a normalization constant.

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