Signal reconstruction for processing unknown information distribution based on empirical Bayesian method

Abstract

The invention discloses a method for processing signal reconstruction with unknown transition probability or unknown signal prior and transition probability based on an empirical Bayesian method. For the problem that the transition probability in the generalized linear model is unknown, a transition probability model based on empirical Bayesian estimation is established, so that the transition probability of the generalized linear model is obtained. According to the estimated transition probability and the state updating equation, a scalar model is obtained, and a probability model based on empirical Bayesian estimation prior can be established, so that prior distribution of unknown signals is obtained. And adding a transition probability model based on empirical Bayesian estimation and a probability model based on empirical Bayesian estimation prior into an algorithm, and carrying out signal recovery. The algorithm continuously updates a signal estimation value through iterative calculation until a convergence condition is reached, and accurate recovery of an original signal is realized. According to the method, the signal can be reconstructed under the condition that the transition probability is unknown or the signal prior and the transition probability are unknown through empirical Bayesian estimation.

Description

Technical Field

[0001] The 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] In wireless communication systems, signal processing and information transmission are core technical fields. However, due to the environmental complexity in practical applications, such as multipath effects, interference and noise, the received signals are often subject to various interferences and distortions. These factors greatly affect the performance of the communication system, especially in environments with weak signals or strong noise.

[0003] In the generalized linear model y=Hx+n, the traditional signal estimation methods include the more common generalized approximate message passing (GAMP) and generalized expectation consistent signal recovery (GEC-SR). The GAMP algorithm and the GEC-SR algorithm are two advanced algorithms widely used for signal recovery in large-scale linear systems. These algorithms have significant advantages in solving problems such as sparse signal recovery and compressed sensing. However, the performance of the GAMP and GEC algorithms in practical applications greatly depends on the characteristics of the prior information and transition probabilities of the signal. The transition probability characterizes the statistical characteristics of the intermediate variable Hx and the observation vector y. If the prior information and transition probabilities are unknown or inaccurate, the recovery effect of the algorithm will be greatly reduced. These algorithms usually rely on obtaining the signal model and the distribution of transition probabilities in advance. However, in practical applications, the distribution of signals and transition probabilities may not necessarily obtain the characteristics of the real signals and transition probabilities, which will lead to a decrease in estimation accuracy or even failure to estimate. Therefore, how to accurately estimate signals under unknown prior information and transition probabilities has become an important research topic.

[0004] Based on the empirical Bayesian method, an adaptive signal estimation method is provided by using observed data to estimate the prior distribution and transition probability distribution of the signal. Compared with the traditional Bayesian method, the empirical Bayesian method does not need to know the prior distribution of the signal in advance, but infers the prior distribution and transition probability through the data itself, so that it can better adapt to the complex environment in practical applications.

[0005] Based on the empirical Bayesian method, the present invention adaptively estimates the prior distribution and transition probability of the signal and applies it to the GAMP and GEC-SR algorithms. In this way, the accuracy of signal recovery and the robustness of the system can be significantly improved without relying on preset prior information and transition probabilities. This provides an efficient solution for signal processing in complex wireless communication environments. Summary of the invention

[0006] The purpose of the present invention is to solve the problem that the GAMP and GEC-SR algorithms cannot recover signals under the condition of unknown signal prior distribution and transition probability in wireless communication systems. By introducing an adaptive estimation method based on empirical Bayesian, the present invention solves the problem that the signal can be recovered under the condition of unknown signal prior distribution and transition probability, thus improving the signal estimation efficiency.

[0007] The accuracy of the measurement and the robustness of the system.

[0008] The present invention proposes a signal prior distribution and transition probability estimation method based on the empirical Bayes method, and applies it to the GAMP and GEC algorithms. The present invention provides the following technical solutions:

[0009] 1. For the problem of unknown transition probability 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.

[0010] 2. Based on the estimated transition probability 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.

[0011] 3. Add the probability model based on empirical Bayesian estimation of transition probability and the probability model based on empirical Bayesian estimation of priors to the GAMP or GEC algorithm for signal recovery. The algorithm continuously updates the signal estimation value through iterative calculation until the convergence condition is reached, thus realizing accurate recovery of the original signal.

[0012] Furthermore, the step 1 is specifically as follows:

[0013] 1.1 Empirical Bayes is an estimation method that combines Bayesian statistics and frequentist statistical methods. In Bayesian statistics, parameters are usually regarded as random variables with a certain prior distribution. The core idea of ​​the empirical Bayes method is to infer or estimate the prior distribution through observed data without fully knowing the prior distribution.

[0014] The empirical Bayes model is as follows:

[0015] X i =αΘi +Ω i , i=1,...,m

[0016] Where α is an 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.

[0017] In the empirical Bayes model, for the unknown probability density g(θ) (which may be a discrete distribution), it has generated independent realizations of unobservable random samples.

[0018]

[0019] Each Θ i From p i (X i |Θ i ) generates independent observations X i ,

[0020]

[0021] 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.

[0022] f(x)=∫ T p(x|θ)g(θ)dθ

[0023] 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 (m) )

[0024] where Θ iThe sample set is T, and g(θ) is estimated from the observations X. The process of choosing T can be determined by some numerical experiments. Specifically, by changing the range of T or adjusting the degree of refinement between samples, we can observe whether these changes significantly affect the estimated prior distribution g(θ). If it is found that a wider range or finer samples can significantly improve the accuracy of the prior estimate, then it makes sense to adjust these parameters. For the number of samples in T, m, it is found that even if m is increased, it will not bring much benefit. Therefore, when choosing the number of support points m, it is usually not necessary to pursue extremely high refinement.

[0025] Same X i The observation sample space φ is also assumed to be finite and discrete,

[0026] φ=(x (1) , x (2) , ..., x (n) )

[0027] For X i The discretization of the observation sample space does not constitute a restriction, because φ can be regarded as X i All order statistics of the values. Whether in the case of discrete data or continuous data, the sample space φ and the selection 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.

[0028] The priori g(θ) is represented by the vector g = (g 1 , g 2 , ..., g m ) T , similarly, the marginal probability f(x) vector is expressed as f=(f 1 , f 2 , ..., f n ) T In vector g and vector f, the sum of their elements is 1.

[0029] Let p kj =P{X=x (k) |Θ=θ (j)}, where p kj represents the kth row and jth column element, which usually represents the transition from the discrete state θ (j) to the observed value x (k) The convolutional relationship between g(θ) and f(x) can be expressed as

[0030]

[0031] Define a matrix P of dimension n×m = (p kj), according to the above convolution relationship, it can be transformed into matrix multiplication

[0032] f=P9

[0033] Empirical Bayesian modeling is divided into g-model and f-model. This article mainly talks about 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:

[0034]

[0035] Where g(θ) is the prior distribution to be estimated, which represents the probability distribution of parameter θ. 1 , η 2 , ..., η 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 write the above equation into the following vector form,

[0036] g(θ)=e Qθ / c(θ)

[0037] Q is an m×p matrix, where m 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 parameters θ. c(θ) is a normalization constant.

[0038] 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

[0039] y=Hx+n

[0040] Let z = Hx, where is the m-dimensional observation vector, is the observation matrix, H is the independent and identically distributed channel matrix, and is the target signal to be estimated in n dimensions. The probability distribution of n 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:

[0041] y=z+n

[0042] Where n is the transition probability to be estimated. By processing the signal at the transmitting end so that its mean is 0 and its variance is 1, the distribution of z becomes a known Gaussian distribution. Based on the above model combined with empirical Bayesian estimation, the transition probability can be estimated.

[0043] Furthermore, the step 2 is specifically as follows:

[0044] 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, probability distributions similar to the following will appear

[0045]

[0046] where p X (x) is the prior distribution of the signal, is a Gaussian distribution, whose expectation x - and the variance is v - This is the information obtained in the previous stage of the algorithm.

[0047] The mean square error (MSE) of the above estimate can be expressed as

[0048]

[0049] where the expectation is the probability p(x|x - ). The minimum mean square error (MMSE) estimated signal is expressed as

[0050]

[0051] In large-scale systems, replacing some variables with equivalent scalar models yields

[0052]

[0053] Where X~p X (x).

[0054] According to the scalar model obtained from the above state update equation, a probability model based on empirical Bayesian estimation prior is established in combination with empirical Bayesian estimation to obtain the prior distribution of the unknown signal. The established empirical Bayesian estimation transfer probability model and the probability model based on empirical Bayesian estimation prior are added to the GAMP or GEC-SR algorithm for signal recovery.

[0055] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0056] Traditional GAMP and GEC-SR algorithms usually 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 effects. The present invention adaptively estimates the prior distribution and transition probability of the signal through the empirical Bayesian method, and the prior information can be dynamically acquired. In the absence of a preset model, more realistic prior information can be dynamically acquired, greatly improving the accuracy of signal recovery.

[0057] 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 improves the signal recovery accuracy and system robustness of the GAMP and GEC-SR algorithms in complex environments, reduces dependence on prior models, optimizes computational efficiency, and provides effective technical support for performance improvement of wireless communication systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 Schematic diagram of the signal reconstruction system model based on empirical Baye estimation of unknown signal priors and transition probabilities.

[0059] Figure 2 To estimate the unknown transition probability of the generalized approximate message passing algorithm based on empirical Bayesian estimation, the actual transition probability distribution is a mixed Gaussian distribution. This experiment uses two Gaussian mixtures. The weight of the first Gaussian distribution is l 1 =0.5, mean 1 =-2, signal-to-noise ratio SNR 1 =13db, the weight of the second Gaussian distribution is l 2 =0.5, mean 2 =2, signal-to-noise ratio SNR 2 =13db. The figure shows the real mixed Gaussian distribution probability density and the estimated probability density.

[0060] Figure 3 In order to estimate the prior distribution of unknown signals of the generalized approximate message passing algorithm based on empirical Bayesian estimation, real numbers are used in the experiment. The same is true in the field of complex numbers. The prior is four real numbers, which is equivalent to the prior for 16QAM distribution.

[0061] Figure 4The following is a schematic diagram of the performance of the generalized approximate message passing algorithm in four different situations. The performance of GAMP-eb in the figure is based on the empirical Bayesian estimation of the signal prior and the transition probability when both are unknown. The performance of GAMP-eb-n in the figure is based on the empirical Bayesian estimation of the unknown transition probability but known prior. The performance of GAMP in the figure is when both the signal prior and the transition probability are known. LS is the performance when the least squares method is used. The multi-input multi-output model of this experiment has m=1500 and n=500.

[0062] Figure 5 In order to estimate the unknown transition probability of the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation, the actual transition probability distribution is a mixed Gaussian distribution. This experiment uses two Gaussian mixtures. The weight value of the first Gaussian distribution is l 1 =0.5, mean 1 =-2, signal-to-noise ratio SNR 1 =12db, the weight value of the second Gaussian distribution is l 2 =0.5, mean 2 =2, signal-to-noise ratio SNR 2 =12db. The figure shows the real mixed Gaussian distribution probability density and the estimated probability density.

[0063] Figure 6 In 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 16QAM distribution.

[0064] Figure 7 The following is a schematic diagram of the performance of the generalized expected consistent signal reconstruction algorithm in four different situations. In the figure, GEC-eb is the performance based on the empirical Bayesian estimation when both the signal prior and the transition probability are unknown. In the figure, GEC-eb-n is the performance based on the empirical Bayesian estimation when the transition probability is unknown but the prior is known. In the figure, GEC is the performance when both the signal prior and the transition probability are known. LS is the performance when the least squares method is used. The multi-input multi-output model of this experiment has m=1500 and n=500. DETAILED DESCRIPTION

[0065] In order to facilitate the understanding of the present invention, the present invention will be described more fully below with reference to the relevant drawings. Preferred embodiments of the present invention are provided in the drawings. However, the present invention can be implemented in many different forms and is not limited to the embodiments described herein. On the contrary, the purpose of providing these embodiments is to make the understanding of the disclosure of the present invention more thorough and comprehensive.

[0066] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those generally understood by those skilled in the art of the present invention. The terms used in the specification of the present invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more related listed items. The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0067] It is hereby declared 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, and 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. The focus of this patent is to recover the signal when two important information are missing, and this patent also further improves the application scope of the recovered signal.

[0068] The overall process of this program refers to Figure 1 .

[0069] Implementation Case 1

[0070] This embodiment provides a signal prior and transition probability based on empirical Bayesian estimation of generalized approximate message passing (GAMP).

[0071] S1: Get the channel matrix H with m rows and n columns and the observed signal y, and set the intermediate variable z = Hx

[0072] S2: Based on the model results of step S1, establish an empirical Bayesian estimated 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.

[0073] S3: In step S2, the transition probability p(y|z) is obtained, and a signal prior model based on the empirical Bayesian estimation is further established according to the state update equation SE. The prior results are estimated according to the empirical Bayesian estimation as follows Figure 3 shown.

[0074] S4: By combining the obtained empirical Bayesian estimated transition probability model and the probability model based on empirical Bayesian estimated prior with the GAMP algorithm, the signal can be restored in the absence of signal prior and transition probability. The algorithm continuously updates the signal estimation value through iterative calculation until the convergence condition is reached, thus realizing accurate recovery of the original signal.

[0075] For the generalized approximate message passing algorithm based on empirical Bayesian estimation (EB-GAMP), the iteration formula is as follows:

[0076] (1) Initialization:

[0077] (2) Definition:

[0078]

[0079] (3) Iteration (t=1:T):

[0080]

[0081] (4) Output estimation signal:

[0082] 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 y a and z a The generalized approximate message passing algorithm transition probability model 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 Bayes; Indicates that for the ath The probability distribution of is taken as the mean, Indicates that for the ath The probability distribution of takes the variance, is the mean value of the intermediate parameter 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 probability model of the generalized approximate message passing algorithm prior, is the mean value of the intermediate parameter 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 Bayes;

[0083] S5: The final estimated performance is as follows Figure 4 As shown in the figure, the performance comparison of four cases is designed, which is in line with theoretical expectations. Among them, GAMP-eb in the figure is the performance of the case where the signal prior and transition probability are unknown based on empirical Bayesian estimation. The performance of GAMP-eb-n in the figure is based on the performance of the case where the transition probability is unknown but the prior is known based on empirical Bayesian estimation. The performance of GAMP in the figure is the performance when both the signal prior and the transition probability are known. LS is the performance when the least squares method is used. The results show that although the least squares method does not require priors and transition probabilities, the algorithm fails in this case, while the generalized approximate message passing algorithm EB-GAMP based on empirical Bayesian estimation can not only reconstruct the signal, but also show good performance.

[0084] Implementation Case 2

[0085] This embodiment provides a signal prior and transition probability for reconstructing GEC-SR based on empirical Bayesian estimation of generalized expected consistent signals.

[0086] S6: Get the channel matrix H with m rows and n columns and the observed signal y, and set the intermediate variable z = Hx

[0087] S7: According to the model result of step S6, establish a transition probability model y=z+m 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 5 As shown, the true results are very close to the actual results.

[0088] S8: In step S7, the transition probability p(y|z) is obtained, and a signal prior model based on the empirical Bayesian estimation is further established according to the state update equation SE. The prior result is estimated according to the empirical Bayesian estimation. Figure 6 shown.

[0089] S9: Combining the empirical Bayesian method with the GEC-SR algorithm can effectively recover the signal in the absence of prior knowledge of the signal and transition probability information. The core idea of ​​this method is to use empirical Bayesian to estimate the prior distribution and state transition probability of the signal according to S8 and S9, and then use these estimated values ​​as the input of the GEC-SR algorithm to iteratively update the estimated value of the signal.

[0090] For the generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation (EB-GEC-SR), its iteration is as follows

[0091] (1) Initialization:

[0092] (2) Definition:

[0093]

[0094] (3) Iteration (t=1:T):

[0095] p Y|Z (y|z)=EB_GEC{p Y|Z (y|z); y,z}

[0096]

[0097]

[0098] (4) Output estimation signal:

[0099] 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 Get 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 non-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 It means taking the conjugate transpose;

[0100] 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.

[0101] S10: Figure 7Four performance diagrams for the GEC-SR algorithm were designed, and the results were consistent with theoretical expectations. The GEC-eb in the figure is based on the performance of the empirical Bayesian estimation when both the signal prior and the transition probability are unknown. The GEC-eb-n in the figure is based on the empirical Bayesian estimation when the transition probability is unknown but the prior is known. The GEC in the figure is the performance when both the signal prior and the transition probability are known. LS is the performance when the least squares method is used. The generalized expected consistent signal reconstruction algorithm EB-GEC-SR based on the empirical Bayesian estimation can also show good performance when the prior and the transition probability are completely unknown, while the least squares method LS is completely ineffective.

Claims

1. Signal reconstruction based on the empirical Bayesian method to process unknown information distribution, characterized by: The method comprises the following steps: For the problem of unknown transition probability 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 state update equation, a scalar model is obtained, and a probability model based on empirical Bayesian estimation prior can be established to obtain the prior distribution of unknown signals; The transfer probability model based on empirical Bayesian estimation and the probability model based on empirical Bayesian estimation prior 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 achieving accurate recovery of the original signal.

2. The signal reconstruction based on the empirical Bayesian method for processing unknown information distribution according to claim 1, characterized in that: Establish an empirical Bayesian estimation model. The model form of empirical Bayesian is as follows: X i =aΘ i +Oh i ,i=1,…,m 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 based on the empirical Bayesian method for processing unknown information distribution according to claim 2, characterized in that: The sample interval model of the estimated variable is established based on the empirical Bayesian estimation, including the following steps: In the empirical Bayes model, for the unknown probability density g(θ) (which may be a discrete distribution), it has generated independent realizations of 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) may usually be a mixture of discrete distribution and continuous distribution. For ease 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 (m) ) where Θ i The sample set is T, and g(θ) is estimated from the observed value X. The process of selecting T can be determined by some numerical experiments. Specifically, by changing the range of T or adjusting the precision between samples, we can observe whether these changes significantly affect the estimated prior distribution g(θ); if it is found that a wider range or finer samples can significantly improve the accuracy of the prior estimate, then it makes sense to adjust these parameters; for the number of samples m in T, it is found that even if m is increased, it will not bring much benefit. Therefore, when selecting the number of support points m, it is usually not necessary to pursue extremely high precision; Same X i The observation sample space φ is also assumed to be finite and discrete, φ=(x (1) ,x (2) ,…,x (n) ) For X i The discretization of the observation sample space does not constitute a restriction, because φ can be regarded as X i All order statistics of values, whether in the case of discrete data or continuous data, can be defined by appropriate methods for the sample space φ and the selection range T without being troubled by practical restrictions. This flexibility allows the model settings to be adjusted as needed in practical applications.

4. The signal reconstruction based on the empirical Bayesian method for processing unknown information distribution 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 m ) T , similarly, the marginal probability f(x) vector is represented as f=(f1,f2,…,f n ) 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 kth row and jth column element, which usually represents the transition from the 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 n×m = (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 parameter θ, 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. j (θ) is a sufficient statistic 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 Qθ / c(θ) Q is an m×p matrix, where m 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 θ, and c(θ) is a normalization constant.

5. The signal reconstruction based on the empirical Bayesian method for processing unknown information distribution according to claim 4, characterized in that: Establishing a transition probability model based on empirical Bayesian estimation to obtain the transition probability of the generalized linear model also includes the following steps: The generalized linear model can be expressed as y=Hx+n 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, is the target signal to be estimated in n dimensions. The probability distribution of n 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: y=z+n Where n is the transition probability to be estimated. By processing the signal at the transmitting end to make its mean 0 and variance 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.

6. The method for signal reconstruction based on the empirical Bayesian method for processing unknown information distribution according to claim 5, characterized in that: According to the state update equation, a scalar model is obtained, and a probability model based on the empirical Bayesian estimation prior can be established to obtain the prior distribution of the unknown signal, which also includes the following steps: 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 the variance is v - The information obtained in the previous stage of the algorithm; The mean square error (MSE) of the above estimate 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).

7. The signal reconstruction based on the empirical Bayesian method for processing unknown information distribution according to claims 5 and 6, characterized in that: The empirical Bayesian estimation-based transition probability model and the empirical Bayesian estimation-based prior probability model are added to the algorithm to perform signal recovery. The expression of the generalized approximate message passing algorithm based on empirical Bayesian estimation is: 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 y a and z a The generalized approximate message passing algorithm transition probability model 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 Bayes; Indicates that for the ath The probability distribution of is taken as the mean, Indicates that for the ath The probability distribution of takes the variance, is the mean value of the intermediate parameter 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 probability model of the generalized approximate message passing algorithm prior, is the mean value of the intermediate parameter 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 Bayes; The generalized expected consistent signal reconstruction algorithm based on empirical Bayesian estimation is expressed as p Y|Z (y|z)=EB_GEC{p Y|Z (y|z);y,z} The above variables are all expressed in vector form, 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 Get 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 non-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 It means 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.

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