Low-complexity EM-MAMP combined prior parameter and signal recovery method

Abstract

The invention discloses a low-complexity EM-MAMP combined prior parameter and signal recovery method, which is suitable for a noisy linear system with a right unitary invariant matrix, and learns prior parameters while realizing signal recovery. The EM-MAMP is composed of a memory linear estimator and a non-linear estimator, approximate posterior probabilities of signals are obtained through a memory matching filter and minimum mean square error demodulation, and EM updating rules of the memory linear estimator and the non-linear estimator are deduced based on the posterior probabilities. A covariance dynamic correction strategy and optimal sampling are provided, the difference between an estimated covariance and a real covariance caused by EM updating is solved, and the robustness of EM-MAMP is improved. According to the method, for large-scale and small-scale systems, the same or better performance as MAMP can be realized, and the convergence speed is higher.

Description

Technical Field

[0001] The present invention relates to the field of signal transmission, and in particular to a low-complexity EM-MAMP joint prior parameter and signal recovery method. Background Art

[0002] Consider a noisy linear system y = Ax + n, where A is a known M×N matrix, y is the known observation, x is the unknown signal to be estimated, and n is additive white Gaussian noise (AWGN). The prior distribution of the signal x is x ∼ p(x|θ1), and the noise distribution is where θ1 and θ2 are the parameter vectors of the signal prior distribution and the parameter of the noise distribution respectively (note that θ2 > 0). The goal of this problem is to estimate the signal x from y and A and estimate the parameter θ ≡ {θ1, θ2} in the prior distribution. The signal recovery of this model is a classical problem and has wide applications in fields such as compressive sensing, signal processing, and wireless communication.

[0003] When the prior distribution parameters θ ≡ {θ1, θ2} are known, this problem degenerates into a classical linear noisy estimation problem. Approximate Message Passing (AMP) is a class of efficient algorithms for linear noisy estimation problems. For the sake of rigor, the present invention defaults to large-scale systems (M, N → ∞, the ratio M / N is fixed) and the unique fixed-point assumption.

[0004] When the elements of the matrix A are independent and identically distributed (IID), the proposed AMP algorithm is Bayes-optimal and has a complexity as low as O(MN). However, when the elements in the matrix A are non-IID, the performance of AMP is poor.

[0005] The Orthogonal / Vector AMP (OAMP / VAMP) algorithm is applicable to right unitary invariant matrices (including IID Gaussian matrices, specific correlated and ill-conditioned matrices), and OAMP / VAMP is Bayes-optimal. However, due to the need for matrix inversion operations, the computational complexity of OAMP / VAMP is as high as O(M 2 N). To reduce the complexity, the Convolutional AMP (CAMP) algorithm is proposed. However, the convergence of CAMP cannot be guaranteed. When the matrix is ill-conditioned (the condition number is large), CAMP may diverge. In the Memory AMP (MAMP) algorithm framework, AMP, OAMP / VAMP, and CAMP can be unified into the MAMP framework. And when the matrix A is right unitary invariant, it is Bayes-optimal. More importantly, the analytically optimal vector damping guarantees its convergence and significantly improves the convergence speed of the algorithm.

[0006] When the prior distribution parameter θ≡{θ1,θ2} is unknown, the AMP - like algorithms face new challenges. Traditional AMP - like algorithms require an accurate known prior distribution. Therefore, when the prior distribution contains unknown parameters, the Expectation Maximization (EM) method can be used to perform parameter estimation during the iteration process. The EM - AMP method incorporates the EM parameter estimation into the AMP iteration, enabling simultaneous parameter estimation and signal detection. The EM - OAMP / VAMP method, which combines the advantages of OAMP / VAMP applicable to right unitary invariant matrices, further generalizes the EM - AMP method to be applicable to right unitary invariant matrices, but at the cost of complexity.

[0007] In the prior art, EM - AMP is only applicable to independent and identically - distributed Gaussian matrices. When the condition number of the observation matrix A is large, the performance of EM - AMP deteriorates sharply. EM - OAMP / VAMP involves matrix inversion operations in each iteration, with a very high complexity of the order of N cubed, while the complexity of EM - AMP is of the order of N squared. Summary of the Invention

[0008] The object of the present invention is to propose a low - complexity EM - MAMP joint prior parameter and signal recovery method in view of the deficiencies of the prior art.

[0009] The object of the present invention is achieved through the following technical solutions: A low - complexity EM - MAMP joint prior parameter and signal recovery method, comprising:

[0010] S1. Receive known observations and construct a noisy linear system;

[0011] S2. Perform parameter initialization of MAMP, using the initialized prior parameter to be estimated and the signal to be recovered as the initial input, where the prior parameter to be estimated includes the prior distribution parameter of the signal to be estimated and the prior distribution parameter of the noise in the system;

[0012] S3. Use a memory linear estimator and a non - linear estimator that include expectation maximization to perform alternating iterations;

[0013] The memory linear estimator includes: The input contains the signals to be recovered obtained in all previous iterations. Use the MLE function to estimate the signals to be recovered and orthogonalize them. Calculate the confidence estimate of the prior distribution of the signal to be estimated in the current iteration according to the estimate output before orthogonalization. Perform EM update on the prior parameters of the noise in the noisy linear system according to the confidence estimate and optimize the parameters;

[0014] The non - linear estimator includes: the output of the input memory linear estimator, estimating the signal to be recovered using the NLE function and performing orthogonality and damping, and performing EM updates on the Bernoulli - Gaussian sparsity rate and activation variance in the current iteration based on the estimated output before orthogonality and optimizing the parameters;

[0015] S4. After meeting the termination iteration condition, output the prior parameters to be estimated and the signal to be recovered, and state evolution can be performed.

[0016] Further, the process of receiving the known observation and constructing the noisy linear system specifically includes: y = Ax + n, where A is a known M×N matrix, y is the known observation, x is the unknown signal to be estimated, n is additive white Gaussian noise, the prior distribution of the signal x is x ∼ p(x|θ1), and the noise distribution is where θ1 and θ2 are the parameter vectors of the signal prior distribution and the noise distribution respectively.

[0017] Further, in the noisy linear system, perform singular value decomposition on the known matrix, so that A = U∑V T , where ∑ is a diagonal singular value matrix, U and V are left and right singular vector matrices respectively. Let U = I, where I is the identity matrix, and replace the large - scale random unitary matrix.

[0018] Further, the replacement of the large - scale random unitary matrix is specifically: approximate the large - scale random unitary matrix V with a scrambled FFT matrix T = ΠF, where Π is a random permutation matrix and F is a normalized FFT matrix.

[0019] Further, the calculation process of the memory linear estimator includes: Input the estimated value X t = [x1, x2, …, x t which contains memory terms, ψ t , ξ t are weight coefficients, and the transformation matrix in the estimator

[0020] MLE: r t = γ t (X t , θ 2,t ),

[0021]

[0022] where is the non - orthogonal MLE function, γ t (·) is the orthogonal MLE function, r t is the estimated output of γ t (·) at the t - th time, is the estimated output for the t-th , is the confidence estimate of p(x|y,θ t ). The parameter represents the average of the maximum and minimum eigenvalues of matrix AA H , that is p t,i are the normalization coefficient and the orthogonal coefficient respectively. θ≡{θ1,θ2}, where θ1 and θ2 are the parameter vectors of the signal prior distribution and the noise distribution respectively, and the subscript t represents the t-th iteration.

[0023] Furthermore, the calculation process of the non-linear estimator includes:

[0024]

[0025] where is the non-linear estimator function that has been orthogonalized and not damped, and φ t (·) is the NLE function that has been damped,

[0026] Furthermore, ξ t The calculation method includes analytical weights or empirical weights;

[0027] The analytical weights are:

[0028]

[0029] where

[0030]

[0031] where

[0032] The empirical weights are:

[0033]

[0034] Furthermore, the EM update of the prior parameters of the noise includes:

[0035] Estimate and the corresponding error expansion is

[0036]

[0037] where and f t =x t -x are the errors of and x t respectively, and

[0038]

[0039] The EM update for the MLE learning of the noise variance is given below: Let For \(i \geq 0\), \(j \geq 0\), \(t \geq (i + j)\), define

[0040]

[0041] Then the EM update for the noise variance is

[0042]

[0043] where

[0044] K t,1 = k t,1 + k t,2 - 2k t,3 + k t,4 x ,

[0045]

[0046] denotes the estimated term of the noise variance where is the confidence estimate of \(p(x|y,\theta\) t ). The residual terms and are used to correct the estimation error of the noise variance . To avoid the noise variance being negative, the absolute value is taken for the final estimate. is the polynomial coefficient of MLE, and \(w\) t represents the matrix \(W\) t = A H B t One - Nth of the trace of \(A\), that is

[0047] Furthermore, the prior distribution of the signal is set to Bernoulli - Gaussian. Therefore, the prior distribution \(p(x|\theta_1)\) of the signal \(x\) is written as

[0048]

[0049] where \(n = 1,\ldots,N\), \(\beta\in(0,1]\) and \(v\) x > 0 are the sparsity rate and the activation variance of the Bernoulli - Gaussian respectively; the EM update of the signal prior parameters \(\{\beta,v\) x \} is similar to the method in EM - AMP. However, different from the symbol - by - symbol estimation in EM - AMP, the estimation of the activation variance \(v\) x requires the entire estimated vector The prior parameters of the signal θ1 = {β, v x} are

[0050]

[0051]

[0052] where is the estimate of β corresponding to each {x n}, and are respectively the output estimate and variance of

[0053] On the other hand, the specification of the present invention also provides a low-complexity EM-MAMP joint prior parameter and signal recovery device, including a memory and one or more processors. Executable code is stored in the memory. When the processor executes the executable code, the low-complexity EM-MAMP joint prior parameter and signal recovery method described above is implemented.

[0054] Advantages of the present invention:

[0055] The EM-MAMP algorithm proposed by the present invention avoids matrix inverse operations, so the complexity is extremely low. The present invention is applicable to right unitary invariant matrices and can also converge to the same performance as EM-OAMP / VAMP, that is, maximum a posteriori optimal, when the matrix condition number κ is very large. It solves the problem that the EM-AMP algorithm does not converge under high condition numbers and the problem of high complexity of EM-OAMP / VAMP. In addition, Figure 5 the EM-MAMP framework shown is compatible with the EM-AMP algorithm and the EM-OAMP / VAMP algorithm. When only the estimated values x t , r t , of the current iteration are passed during the iteration process, and the damping vector ζ t+1 = [1, 0,..., 0] T , EM-MAMP degenerates into EM-OAMP / VAMP. Description of the drawings

[0056] Figure 1 is the flowchart of the EM-MAMP algorithm provided by the embodiment of the present invention;

[0057] Figure 2 is the comparison diagram of EM-MAMP and MAMP under different condition numbers provided by the embodiment of the present invention;

[0058] Figure 3 is the performance comparison of EM-AMP, EM-OAMP / VAMP, and EM-MAMP provided by the embodiment of the present invention;

[0059] Figure 4 The comparison chart of the running times of EM-AMP, EM-OAMP / VAMP, and EM-MAMP provided by the embodiments of the present invention;

[0060] Figure 5 The state evolution (SE) diagram of EM-MAMP provided by the embodiments of the present invention;

[0061] Figure 6 The covariance matrix diagram at the NLE end provided by the embodiments of the present invention;

[0062] Figure 7 The schematic diagram of a low-complexity EM-MAMP joint prior parameter and signal recovery device provided by the embodiments of the present invention. Detailed implementation manners

[0063] Consider the noisy linear system y = Ax + n, where A is a known M×N matrix, y is the known observation, x is the unknown signal to be estimated, and n is additive white Gaussian noise (AWGN). The prior distribution of the signal x is x ∼ p(x|θ1), and the noise distribution is where θ1 and θ2 are the parameter vectors of the signal prior distribution and the parameters of the noise distribution respectively (note that θ2 > 0). The objective of this problem is to estimate the signal x from y and A and estimate the parameters θ ≡ {θ1, θ2} in the prior distribution.

[0064] The following embodiments are the specific processes of EM-MAMP for compressive sensing signal estimation and parameter estimation of Bernoulli-Gaussian signals with unknown parameters. This embodiment first gives a computer program simulation case on Matlab software, and then gives the corresponding hardware implementation device.

[0065] Embodiment:

[0066] Consider a compressive sensing problem, given A and y, to recover the signal x and simultaneously learn the prior parameters θ ≡ {θ1, θ2}. In this embodiment, the prior distribution of the signal x is independently and identically distributed Bernoulli-Gaussian, that is, for any n,

[0067]

[0068] where θ1 = {β, v x}, δ(·) is the Dirac function, represents the probability density function of a Gaussian distribution with a mean of 0 and a variance of v x . The true sparsity rate and variance are β = 0.1 and v x = 10 respectively. Therefore, the variance of each signal symbol x n is 1, and the signal-to-noise ratio is SNR = θ2.

[0069] Let the singular value decomposition (SVD) of matrix A be A = UΣV T , where Σ is the diagonal singular value matrix, and U and V are the left and right singular vector matrices respectively. Then

[0070] y = UΣV T x + n.

[0071] To further reduce the algorithm complexity, this embodiment uses this matrix model for calculation. In particular, since U T n and n have the same distribution. Without loss of generality, let U = I, where I is the identity matrix. Additionally, to further simplify the operation, this embodiment uses the scrambled FFT matrix to approximate the large-scale random unitary matrix, that is, V T = ΠF, where Π is the random permutation matrix and F is the normalized FFT matrix. Of course, other random unitary matrices can also be used. Set the eigenvalues {d i} of Σ as: d i / d i+1 = and where the distribution is the minimum dimension and the maximum dimension of matrix A, that is the condition number of matrix A is κ ≥ 1. The prior parameters to be estimated are initialized as initial sparsity initial variance

[0072] Define as the non-orthogonal memory linear estimator (MLE) function, γ t (·) as the orthogonalized MLE function, as the orthogonalized non-damped non-linear estimator (NLE) function, φ t (·) as the damped NLE function, is the estimated output at the t-th , r t is the estimated output of γ t (·) at the t-th time, x t is the estimated output of φ t (·) at the t-th time. The algorithm flow of EM-MAMP is as follows: Consider where the input estimated value X t = [x1, x2, …, x t contains memory terms, ψ t , ξ t are the weight coefficients, and the transformation matrix in the estimator starts with t = 1, initialize the parameters to be estimated {θ 1,1 , θ 2,1},

[0073] MLE: r t = γ t (X t , θ 2,t ),

[0074]

[0075] where

[0076]

[0077] where and are the confidence estimates of p(x|y, θ t ), p t,i are the normalization coefficient and the orthogonal coefficient respectively, with values of

[0078]

[0079] where

[0080] W t ≡ A T B t A,

[0081]

[0082] The EM update steps for the MLE learning noise variance are as follows: Let For i ≥ 0, j ≥ 0, t ≥ (i + j), define

[0083]

[0084] Then the EM update of the noise variance is

[0085]

[0086] where

[0087] K t,1 = k t,1 + k t,2 - 2k t,3 + k t,4 ,

[0088]

[0089] To avoid the noise variance being negative, take the absolute value of the final estimate.

[0090] For the EM update of the NLE, the signal prior parameters θ1 = {β, v x} are

[0091]

[0092]

[0093] where is the estimate of β corresponding to each {x n}, and and are the output estimate and variance of respectively.

[0094] The flowchart of the EM-MAMP algorithm is as shown in Figure 1 This invention can adopt two weight ξ t calculation methods: analytical weight and empirical weight.

[0095] Analytical weight:

[0096]

[0097] where

[0098]

[0099] where

[0100] The parameter w t represents the matrix W t = A H B t One Nth of the trace of A, that is The parameter represents the average of the maximum eigenvalue and the minimum eigenvalue of the matrix AA H , that is The parameter represents the variance of the output estimate value of the NLE function without damping . The parameter is the polynomial coefficient of MLE. θ 2,t represents the estimated value of θ2 in the t-th iteration.

[0101] Empirical weight:

[0102]

[0103] The specific process of this invention using SE to track the performance of EM-MAMP is as shown in Figure 5 Assume that {θ1, θ2} is unknown, starting from , initialize {θ1, θ2}, then

[0104]

[0105] Iterations are performed through the SE functions of MLE and NLE respectively to obtain the SE's estimation of the posterior output MSE in each iteration. In the simulation, the variance of the SE estimation is compared with the true MSE. The matching of the two indicates that the SE tracks the performance of EM-MAMP very accurately.

[0106] The present invention can converge to the maximum a posteriori optimum whether using analytical weights or empirical weights, and its performance is comparable to that of MAMP in the case of known parameters. Figure 2 Shows the comparison of the MSE performance between EM-MAMP (the present invention) with analytical weights and empirical weights and MAMP under different condition numbers. The parameters are M = 8192, N = 16384, SNR = 35dB, κ ∈ {10, 100, 3000}, and "Ana" and "Emp" represent analytical weights and empirical weights respectively. At the same time, Figure 2 the variance of the SE estimation in [the relevant content] basically completely matches the actual MSE. This indicates that the SE tracks the performance of EM-MAMP with very high accuracy.

[0107] Figure 3 Shows that EM-MAMP can converge to a performance comparable to that of EM-OAMP / VAMP and can still converge under high condition numbers.

[0108] Compared with the EM-OAMP / VAMP algorithm, the present invention avoids large-scale matrix inversion operations, so the complexity is reduced by one order of magnitude. Using the scrambled FFT matrix to approximate the large-scale unitary matrix further reduces the complexity. The comparison of the running times of EM-AMP, EM-OAMP / VAMP, and EM-MAMP is as Figure 4 shown. It can be seen that the present invention has a running time comparable to that of EM-AMP and is far lower than the running time of EM-OAMP / VAMP.

[0109] Corresponding to the foregoing embodiment of a low-complexity EM-MAMP joint prior parameter and signal recovery method, the present invention also provides an embodiment of a low-complexity EM-MAMP joint prior parameter and signal recovery device.

[0110] See Figure 7 A low-complexity EM-MAMP joint prior parameter and signal recovery device provided by an embodiment of the present invention includes a memory and one or more processors. An executable code is stored in the memory. When the processor executes the executable code, it is used to implement a low-complexity EM-MAMP joint prior parameter and signal recovery method in the above embodiment.

[0111] An embodiment of a low-complexity EM-MAMP joint prior parameter and signal recovery device provided by the present invention can be applied to any device with data processing capabilities, and such a device with data processing capabilities can be a device or apparatus such as a computer. The device embodiment can be implemented by software, or by hardware or a combination of software and hardware. Taking software implementation as an example, as a logically meaningful device, it is formed by the processor of any device with data processing capabilities reading the corresponding computer program instructions in the non-volatile memory into the memory for operation. At the hardware level, as Figure 7 shown, it is a hardware structure diagram of any device with data processing capabilities where a low-complexity EM-MAMP joint prior parameter and signal recovery device provided by the present invention is located. In addition to Figure 7 the processor, memory, network interface, and non-volatile memory shown, generally, according to the actual functions of any device with data processing capabilities where the device in the embodiment is located, other hardware may also be included, which will not be elaborated here.

[0112] For the implementation processes of the functions and roles of each unit in the above device, please refer to the implementation processes of the corresponding steps in the above method for details, which will not be elaborated here.

[0113] For the device embodiment, since it basically corresponds to the method embodiment, the relevant parts can refer to the partial description of the method embodiment. The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place, or distributed to multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the solution of the present invention. Those of ordinary skill in the art can understand and implement it without creative efforts.

[0114] The embodiment of the present invention also provides a computer-readable storage medium, on which a program is stored. When the program is executed by a processor, it implements a low-complexity EM-MAMP joint prior parameter and signal recovery method in the above embodiment.

[0115] The computer-readable storage medium may be an internal storage unit of any device with data processing capabilities described in any of the foregoing embodiments, such as a hard disk or memory. The computer-readable storage medium may also be an external storage device of any device with data processing capabilities, such as a plug-in hard disk, a Smart Media Card (SMC), an SD card, a Flash Card, etc. equipped on the device. Further, the computer-readable storage medium may also include both an internal storage unit and an external storage device of any device with data processing capabilities. The computer-readable storage medium is used to store the computer program and other programs and data required by any device with data processing capabilities, and may also be used to temporarily store the data that has been output or will be output.

[0116] The present invention also provides a computer program product, including a computer program, which when executed by a processor, implements the described method for jointly recovering low-complexity EM-MAMP prior parameters and signals.

[0117] After considering the specification and practicing the disclosure herein, those skilled in the art will readily conceive of other embodiments of the present application. The present application is intended to cover any variations, uses, or adaptations of the present application, which follow the general principles of the present application and include well-known common general knowledge or conventional technical means in the technical field not disclosed in the present application. The specification and examples are only regarded as exemplary, and the true scope and spirit of the present application are pointed out by the claims.

[0118] It should be understood that the above general description and the following detailed description are only exemplary and explanatory, and cannot limit the present application. The present application is not limited to the exact structure already described and shown in the drawings, and various modifications and changes can be made without departing from its scope. The scope of the present application is only limited by the appended claims.

Claims

1. A low-complexity EM-MAMP combined prior parameter and signal recovery method, characterized in that: include: S1, receive known observations and construct a noisy linear system; S2, performing parameter initialization of MAMP, taking the initialized prior parameters to be estimated and the signal to be recovered as initial input, wherein the prior parameters to be estimated include the prior distribution parameters of the signal to be estimated and the prior distribution parameters of the noise in the system; S3, using alternating iterations of a memory linear estimator and a nonlinear estimator involving expectation maximization; The memory linear estimator includes: the input includes the signal to be recovered obtained in all previous iterations, the MLE function is used to estimate and orthogonalize the signal to be recovered, the confidence estimate of the prior distribution of the signal to be estimated in this iteration is calculated according to the estimated output before orthogonalization, and the prior parameters of the noise in the noisy linear system are updated by EM according to the confidence estimate and the parameters are optimized; The nonlinear estimator includes: inputting the output of the memory linear estimator, using the NLE function to estimate the signal to be restored and perform orthogonalization and damping, and performing EM update and parameter optimization on the Bernoulli Gaussian sparsity rate and activation variance in this iteration according to the estimated output before orthogonalization; S4. After the termination conditions of the iteration are met, the prior parameters to be estimated and the signals to be recovered are output, and the state evolution can be performed.

2. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 1, characterized in that: The receiving of known observations and construction of a noisy linear system specifically includes: y=Ax+n, where A is an M×N known matrix, p is a known observation, x is an unknown signal to be estimated, n is additive Gaussian white noise, the signal x prior distribution is x~p(x|θ1), and the noise distribution is Where θ1 and θ2 are the parameter vector of the signal prior distribution and the parameter of the noise distribution respectively.

3. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 2, characterized in that: In the noisy linear system, the known matrix is ​​decomposed into A=U∑V T , where ∑ is the diagonal singular value matrix, U and V are the left and right singular vector matrices respectively, let U = I, where I is the identity matrix, and replace the large-scale random unitary matrix.

4. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 3, characterized in that: The replacement of the large-scale random unitary matrix is ​​specifically: using the disrupted FFT matrix to approximate the large-scale random unitary matrix V T =ΠF, where Π is the random permutation matrix and F is the normalized FFT matrix.

5. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 1, characterized in that: The calculation process of the memory linear estimator includes: Enter the estimated value X t =[x1,x2,…,x t ] contains the memory term, ψ t , t is the weight coefficient, the transformation matrix in the estimator MLE:r t =c t (X t ,i 2,t ), in is the non-orthogonal MLE function, γ t (·) is the orthogonal MLE function, r t is the tth γ t The estimated output of (·) is For the tth time The estimated output is is p(x|y,θ t ) confidence estimate, parameter Represents the matrix AA H The average value of the maximum and minimum eigenvalues ​​of p t,i are the normalization coefficient and orthogonal coefficient respectively, θ≡{θ1,θ2}, θ1 and θ2 are the parameter vector of the signal prior distribution and the parameters of the noise distribution respectively, and the subscript t represents the tth iteration.

6. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 5, characterized in that: The calculation process of the nonlinear estimator includes: in is the orthogonal nonlinear estimator function without damping, φ t (·) is the NLE function that has been damped.

7. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 6, characterized in that: ξ t The calculation methods include analytical weights or empirical weights; The parsing weights are: in in The experience weights are:

8. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 6, characterized in that: The EM updating of the prior parameters of the noise includes: estimate and the corresponding error expansion is in and f t =x t -x are and x t The error of The MLE learning noise variance is given below EM update: Order For i ≥ 0, j ≥ 0, t ≥ (i + j), define Then the noise variance The EM is updated to in K t,1 =k t,1 +k t,2 -2k t,3 +k t,4 , represents the noise variance The estimated term of is p(x|y,θ t ) is a confidence estimate of the residual term. and To correct the noise variance To avoid the noise variance being negative, the absolute value of the final estimate is taken. is the polynomial coefficient of MLE, w t Representation Matrix W t =A H B t One Nth of the trace of A, that is, 9. A low-complexity EM-MAMP combined prior parameter and signal recovery method according to claim 6, characterized in that: The prior distribution of the signal is set to Bernoulli Gaussian, so the prior distribution p(x|θ1) of the signal x is written as where n = 1,…,N, β∈(0,1] and v x >0 are the sparsity rate and activation variance of Bernoulli Gaussian respectively; the signal prior parameters {β,v x The EM update of} is similar to the method in EM-AMP, but different from the symbol-by-symbol estimation in EM-AMP, the activation variance v x The estimation of requires the entire estimation vector Signal prior parameters θ1={β,v x }for in is each {x n The corresponding estimate of β is and They are The output estimates and variances of .

10. A low-complexity EM-MAMP combined prior parameter and signal recovery device, comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that: When the processor executes the executable code, a low-complexity EM-MAMP combined prior parameter and signal recovery method as described in any one of claims 1-9 is implemented.

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