Low complexity optimal mamp receiver and encoding method for massive mimo

Abstract

The application discloses a low-complexity optimal MAMP receiver coding method for large-scale MIMO, which adopts an MAMP algorithm to estimate a received signal y at a receiving end, combines all damping operations and orthogonal operations of the MAMP into a linear detection module MLD, obtains a low-complexity MAMP after equivalent transformation, uses a variational SE (VSE) function to represent an asymptotic MSE performance, and uses an I-MMSE lemma to calculate an optimal coding design principle and a maximum reachable rate of the low-complexity MAMP. The low-complexity optimal MAMP receiver for large-scale MIMO has the linear detection module MLD to perform orthogonal operation, damping operation and linear detection processing on the received signal y, and has the nonlinear detection module NLD to perform demodulation and decoding processing on the received signal y, and has the characteristics of low complexity and optimal constrained capacity, is suitable for an encoded GMIMO communication system, and uses a simple back-off damping ratio which is more robust than an optimal analytical damping used by a previously uncoded system.

Description

Technical Field

[0001] This invention belongs to the field of communication technology, and relates to a low-complexity optimal MAMP receiver coding method for large-scale MIMO, as well as a low-complexity optimal MAMP receiver for large-scale MIMO. Background Technology

[0002] Massive Multiple-Input Multiple-Output (MIMO) leverages the spatial multiplexing of a large number of antennas and signal processing algorithms to improve system capacity and spectral efficiency, making it one of the key physical layer technologies for current 5G and future 6G. However, most traditional MIMO technologies are only applicable to ideal communication system models, namely a finite number of antennas, no coding constraints, Gaussian channels, channel state information (CSI) available to the transceiver, and independent identically distributed (IID) channel matrices. For more practical generalized MIMO (GMIMO) system models, which include: 1) massive numbers of users and antennas; 2) practical coding constraints; 3) arbitrary input distribution; 4) right-unitary invariant channel matrix; and 5) channel matrix available only to the receiver, traditional MIMO technologies are not suitable.

[0003] Traditional iterative linear minimum mean square error (LMMSE) receivers can achieve Gaussian channel capacity for multi-user (MU) MIMO, but are limited by the high complexity of LMMSE estimation and the requirement of an ideal Gaussian channel, making them difficult to apply in practical discrete channels. Approximate message-passing (AMP) algorithms can solve this problem. J. Barbier and N. Macros et al., in "Mutual information and optimality of approximate message-passing inrandom linear estimation," proved through state evolution (SE) that AMP meets the Bayesian optimality of estimation. However, AMP is only applicable to independent identically distributed (IID) channel matrices. For more complex non-IID channel matrices, the performance of the AMP algorithm deteriorates or even diverges. J. Ma and L. Ping et al. proposed an Orthogonal Approximate Message Passing (OAMP) algorithm in their paper "Orthogonal AMP," and S. Rangan and P. Schniter et al. proposed a Vector Approximate Message Passing (VAMP) algorithm in their paper "Vector Approximate Message Passing." These algorithms are applicable to a wider range of right unitary invariant matrices. They mitigate linear interference through LMMSE and overcome correlation problems in iteration through orthogonalization. However, due to the high complexity of LMMSE, it is difficult to efficiently apply OAMP / VAMP to large-scale systems. In 2022, L. Liu, S. Huang, and BMKurkoski et al. proposed a low-complexity memory approximate message passing (MAMP) algorithm in their paper "Memory AMP," and proposed a theoretically optimized damping vector to guarantee and accelerate MAMP convergence. MAMP has also been proven to be the Bayesian optimal detection algorithm for right unitary invariant channel matrices. However, these algorithms only focus on signal estimation and detection in uncoded systems, neglecting the impact of coding constraints on detection, and therefore cannot characterize the actual error-free recovery capability of the system. Furthermore, because the MAMP algorithm employs a local memory estimator with memory, it uses a complex multidimensional state evolution (SE) transfer function when evaluating its asymptotic optimality. However, existing realizable rate analysis and optimal coding principles are based on the single-input single-output (SISO) transfer function and cannot be directly extended to MAMP.

[0004] In terms of information-theoretic optimality analysis, for coded MIMO systems with IID channel matrices and arbitrary input signals, the realizable rate analysis and information-theoretic optimality proof of AMP are based on scalar SE. Specifically, while satisfying the error-free decoding condition, the optimal coding principle is derived by tracking the scalar estimation variance between linear detectors (LD) and nonlinear detectors (NLD). For right-unitary invariant channel matrices and arbitrary input signaling, OAMP / VAMP can achieve limited capacity in point-to-point (P2P) GMIMO. Unlike AMP, the orthogonalization in the LD and NLD of OAMP / VAMP violates the minimum mean square error (MMSE) property, so the mutual information-MMSE (I-MMSE) lemma cannot be directly used when analyzing optimality. Meanwhile, some scholars have designed corresponding optimized low-density parity-check (LDPC) codes for AMP and OAMP / VAMP based on SE and VSE respectively, achieving optimal performance. However, the information-theoretic optimality of AMP and OAMP / VAMP is limited by the IID channel matrix and the high computational complexity, making it difficult to effectively apply to large-scale GMIMO. Summary of the Invention

[0005] The purpose of this invention is to provide a low-complexity optimal MAMP receiver coding method for large-scale MIMO, which features low complexity and information-theoretic optimality.

[0006] Another objective of this invention is to provide a low-complexity optimal MAMP receiver for large-scale MIMO, characterized by optimal constrained capacity.

[0007] The technical solution adopted in this invention is a low-complexity optimal MAMP receiver coding method for large-scale MIMO. The specific steps are as follows: At the receiver, the MAMP algorithm is used to estimate the received signal y. Then, all damping and orthogonal operations of the MAMP are merged into the linear detection module MLD to obtain the equivalent transformed low-complexity MAMP. The linear detection module MLD performs orthogonal operations, damping operations, and linear detection processing on the received signal y. The nonlinear detection module NLD performs demodulation and decoding processing on the received signal y. The variational SE (VSE) function is used to represent the asymptotic MSE performance of the equivalent transformed low-complexity MAMP. The optimal coding design principle and maximum achievable rate of the low-complexity MAMP in coded GMIMO are calculated using the I-MMSE lemma.

[0008] The invention is further characterized by:

[0009] The specific steps are as follows:

[0010] Step 1: Establish a signal transmission model for the coded GMIMO system. At the receiving end, use the MAMP algorithm to estimate the received signal y.

[0011] Step 2: Use the covariance matrix of the state evolution SE to represent the asymptotic MSE performance of MAMP in Step 1.

[0012] Step 3: Combine all damping and orthogonal operations of MAMP in Step 1 into the linear detection module MLD to obtain the equivalent transformed low-complexity MAMP. The linear detection module MLD performs orthogonal operations, damping operations and linear detection processing on the received signal y. The nonlinear detection module NLD performs demodulation and decoding processing on the received signal y. The state evolution SE in Step 2 is equivalently processed to obtain the one-dimensional variational state evolution SE. The asymptotic MSE performance of the equivalent transformed low-complexity MAMP is represented by the variational SE (VSE) function.

[0013] Step 4: Based on the constraint capacity optimality principle and the I-MMSE lemma, integrate the variational SE (VSE) function in Step 3 to obtain the optimal coding design principle and maximum achievable rate of low-complexity MAMP in coded GMIMO communication.

[0014] Step 1 is implemented in the following steps:

[0015] Step 1.1: Assume the GMIMO system has N transmit antennas and M receive antennas. The information sequence length on each transmit antenna is L. The transmitted signal on the channel is x, and the received signal is y. At time l, the transmitted signal on the channel is represented as... T is the matrix transpose, and the signal captured by the receiver is... Represented as: y l =Ax l +n l l=1,...,L; where A represents the channel matrix, A∈C M×N , n~CN(0,σ 2 I) is additive white Gaussian noise, where I is the identity matrix, and σ 2 Indicates the noise variance;

[0016] The received signal y can be rewritten as follows: Using the MAMP algorithm to estimate the received signal y, the received signal y can be rewritten as follows:

[0017] MLD Γ:y=Ax+n (1);

[0018] Code constraint

[0019] The transmitted signal x satisfies the encoding constraint Φ C And it conforms to any distribution;

[0020] Step 1.2: According to formula (1), the MAMP of the encoded GMIMO system is expressed as:

[0021] MLD:

[0022] NLD:

[0023] In equation (2), t is the number of iterations, and r t X is the output signal of the linear detector MLD during the t-th iteration. t =[x1,...x t ] represents the estimated value of the transmitted signal x in the previous t iterations, γ t This represents the deorthogonalization of the memory linear detector function. p is the normalized parameter of the linear detector MLD. t The orthogonalization parameters for the linear detector MLD are... This represents the memory-matched filter function. It relates to the linear constraint Γ, that is:

[0024]

[0025] Among them, A H Let H denote the conjugate transpose of the channel matrix A, and let H denote the conjugate transpose operator. The specific calculation is as follows:

[0026]

[0027] Where, at t=0, λ min and λ max They are AA H The minimum and maximum eigenvalues ​​of a matrix, θ t Let ξ be the relaxation parameter. t For the weights, the relaxation parameter θ t and weight ξ t Used to improve the convergence speed of MAMP receivers;

[0028] In equation (3), φ represents the nonlinear detector NLD function after adding a damping vector. t (·) represents the normalized and orthogonalized NLD detector function. in Represents the NLD function of the nonlinear detector The corresponding demodulation and posterior probability decoding functions in the nonlinear detection module, i.e. Let represent the nonlinear detector NLD function directly estimated by unnormalized orthogonalization, and It is Lipschitz continuous, and E{} denotes the conditional mean. and wt ζ represents the normalized parameter and the orthogonalized parameter of the nonlinear detector NLD, respectively. t+1 It is the back-off damping vector, which is calculated as follows:

[0029]

[0030] φ represents the NLD function of the nonlinear detector in the t-th iteration. t The output variance of (·), when φ t When the variance of the output value of (·) increases, the estimated output value and variance of the previous iteration are used as... The current output.

[0031] Step 2 is as follows:

[0032] The asymptotic mean square error property of the estimation error is represented using the covariance matrix:

[0033]

[0034] Where i and j represent the row and column positions in the covariance matrix. The covariance of the estimation error of the linear detector in the i-th and j-th iterations of the MLD function is expressed by the following formula: Where g i =r i -x,g j =r j -x represents the estimation error of the linear detector MLD in the i-th and j-th iterations, respectively. The covariance matrix represents the estimation error of the linear detector. The covariance matrix represents the estimation error of the nonlinear detector. The covariance of the estimation error of the nonlinear detector in the i-th and j-th iterations is expressed by the following formula: f i =x i -x,f j =x j -x represents nonlinear detection. The estimation errors at the i-th and j-th iterations,

[0035] Furthermore, based on the orthogonality and independent identically distributed properties, the asymptotic MSE performance of MAMP can be obtained through the γ of the state evolution SE. SE (·)and Function representation, that is:

[0036] MLD:

[0037] NLD:

[0038] The function transformations of both MLD and NLD are multidimensional.

[0039] Step 3 is as follows:

[0040] According to the fixed-point consistency theorem of MAMP and OAMP / VAMP, for the same demodulation and posterior probability decoding function... The state evolution SE of MAMP converges to the same SE fixed point as the SISO SE of OAMP / VAMP, which is all orthogonal and backtracking damped vectors ζ. t+1 The operation is merged into MLD, that is, the right side of the NLD in equation (3) [X t ,φ t (r t )]·ζ t+1 Damping operation and formula The orthogonal and normalization operations in the original code have been merged into the linear detection module MLD, leaving only the nonlinear detection module NLD. The equivalent transformation of MAMP is obtained, and the expression for the equivalent MAMP is:

[0041] MLD:

[0042] NLD:

[0043] Where η t (·) is a multidimensional MLD function, containing the functions in formula (2). And the damping and orthogonal operations of formula (3), Denotes the demodulation and posterior probability decoder function in a nonlinear detector (NLD). The output posterior estimate;

[0044] The transfer function in the NLD of the equivalent MAMP is a single-input single-output mode, containing a memory function η. t The MLD transfer function of (·) is a multidimensional function, which is not conducive to the optimality analysis of MAMP.

[0045] Using the fixed-point consistency theorem of MAMP and OAMP / VAMP, and with the help of the VSE analysis of OAMP, the optimal coding principle and achievable rate of the VSE of MAMP after the equivalent transformation are obtained. Therefore, the asymptotic MSE performance of the low-complexity MAMP after the equivalent transformation is expressed by variational SE (VSE):

[0046] MLD:

[0047] NLD:

[0048] in, This represents the ratio of the input signal to the interference noise in the NLD. As in equation (11) Input, For the case where i = t and j = t in formula (7), the calculation formula is: g t =r t -x represents the estimation error of the transformed linear detector MLD. Represents the transformed nonlinear detector The output mean square error The calculation formula is f t Represents the nonlinear detection after transformation The estimation error, f t =x t -x, and 1≤t'≤t, The function is The inverse of the matrix, The function is the MSE function of the linear minimum mean square error estimate (LMMSE), satisfying the following conditions: z is an additive white Gaussian noise vector with zero mean and variance of an identity matrix. This represents the MMSE function of the posterior probability decoder.

[0049] Step 4 is as follows:

[0050] Post-gain demodulation transfer function It is a decoding transfer function The upper limit, that is:

[0051]

[0052] in,

[0053] Assuming the iteration converges and There is a unique fixed point between them. According to the principle of optimality, if and only if the following condition is met:

[0054]

[0055] At this time, MAMP can achieve error-free decoding;

[0056] According to formula (15) and the I-MMSE lemma, a fixed decoding transfer function is obtained. The speed that MAMP can achieve is:

[0057]

[0058] Where d represents the limit of integration of calculus. satisfy The ratio of input signal to interference noise when the nonlinear detector NLD is correctly decoded in expression (13);

[0059] Based on the constraint capacity optimality proof of OAMP / VAMP in equation (15) and the rate expression for MAMP realization in equation (16), the maximum achievable rate of MAMP is calculated as follows:

[0060]

[0061] The optimal coding design principle of MAMP is:

[0062]

[0063] When satisfied At that time, among them It can achieve the maximum error-free transmission rate of MAMP.

[0064] Another technical solution adopted in this invention is a low-complexity optimal MAMP receiver for large-scale MIMO, including a linear detection module MLD and a nonlinear detection module NLD. The linear detection module MLD corresponds to the linear constraint and performs orthogonal operation, damping operation and linear detection processing on the received signal y. The nonlinear detection module NLD consists of a demodulator and a posterior probability decoder and performs demodulation and decoding processing on the received signal y.

[0065] The beneficial effects of this invention are:

[0066] (1) The low-complexity optimal MAMP receiver for large-scale MIMO of the present invention has the optimal coding principle, can achieve the maximum rate, and constrains the optimal capacity.

[0067] (2) The low-complexity optimal MAMP receiver coding method for large-scale MIMO of the present invention constructs the variational SE (VSE) of OAMP / VAMP by merging all orthogonal operations into the linear detector module. Based on this, the I-MMSE lemma is used to calculate the achievable rate and optimal coding principle of the MAMP algorithm for coding GMIMO system.

[0068] (3) Based on the SE fixed-point consistency lemma of MAMP and OAMP / VAMP, the simplified SISO variational SE of MAMP is calculated with the help of the SE of OAMP / VAMP, and the realizable rate of MAMP is obtained. The optimal MAMP coding scheme with low complexity is calculated with the maximum achievable rate as the target. The maximum achievable rate is equal to the constraint capacity of GMIMO, which proves the optimal constraint capacity of the MAMP receiver in the coded GMIMO communication system.

[0069] (4) Regarding the damping vector in the MAMP algorithm of GMIMO, the simple back-off damping used in this invention is more robust than the optimized analytical damping used in the previous uncoded system. Attached Figure Description

[0070] Figure 1 It is the MAMP and its state evolution transfer function before the equivalent transformation in the method of this invention;

[0071] Figure 2 This is a block diagram of the low-complexity optimal MAMP receiver and the equivalent variational SE (VSE) transfer function in the method of this invention;

[0072] Figure 3 This is a performance comparison chart of MAMP and OAMP using analytical damping and back-off damping respectively in the method of this invention;

[0073] Figure 4 This is a graph showing the reachable rate of the variational SE in the method of this invention.

[0074] Figure 5 This is a comparison of the bit error rate curves of the MAMP and OAMP algorithms of the present invention.

[0075] Figure 6 This is a comparison chart of the running times of the MAMP and OAMP algorithms of the present invention. Detailed Implementation

[0076] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0077] This invention addresses low-complexity optimal MAMP receivers for massive MIMO, aiming to recover the transmitted signal x from the received signal y to the greatest extent possible, such as... Figure 2 As shown, it includes a linear detection module MLD and a nonlinear detection module NLD. The linear detection module MLD corresponds to the linear constraint and performs orthogonal operation, damping operation and linear detection processing on the received signal y. The nonlinear detection module NLD consists of a demodulator and a posterior probability decoder and performs demodulation and decoding processing on the received signal y.

[0078] This invention presents a low-complexity optimal MAMP receiver coding method for large-scale MIMO. At the receiver, the MAMP algorithm is used to estimate the received signal y. Then, all damping and orthogonal operations of the MAMP are merged into the linear detection module (MLD), resulting in an equivalent transformed low-complexity MAMP, such as... Figure 2As shown, the linear detection module MLD performs orthogonal operation, damping operation and linear detection processing on the received signal y, and the nonlinear detection module NLD performs demodulation and decoding processing on the received signal y. The variational SE (VSE) function is used to represent the asymptotic MSE performance of the low-complexity MAMP after the equivalent transformation. The optimal coding design principle and maximum achievable rate of the low-complexity MAMP in coded GMIMO are calculated using the I-MMSE lemma.

[0079] The specific steps are as follows:

[0080] Step 1: Establish a signal transmission model for the coded GMIMO system. At the receiving end, use the MAMP algorithm to estimate the received signal y.

[0081] Suppose a GMIMO system has N transmit antennas and M receive antennas, with each transmit antenna having an information sequence of length L. The transmitted signal is x, and the received signal is y. At time l, the transmitted signal on the channel is represented as... If T is the matrix transpose, then the signal captured by the receiver... Represented as: y l =Ax l +n l l=1,...,L; where A represents the channel matrix, A∈C M×N , n~CN(0,σ 2 I) is additive white Gaussian noise, where I is the identity matrix, and σ 2 Indicates the noise variance;

[0082] like Figure 1 As shown, the MAMP receiver before the equivalent transformation consists of a linear detector MLD and a nonlinear detector NLD. The MLD corresponds to the linear constraint and performs orthogonal operations and linear detection processing on the received signal y. The NLD consists of a demodulator and a posterior probability decoder, and performs damping, orthogonalization, demodulation, and decoding processing on the received signal y. The received signal y is estimated using the MAMP algorithm and rewritten as follows:

[0083] MLD Γ:y=Ax+n (1);

[0084] Code constraint

[0085] The transmitted signal x satisfies the encoding constraint Φ C It can conform to any distribution, such as QPSK, QAM, Gaussian distribution, Bernoulli Gaussian distribution, etc.

[0086] According to formula (1), the MAMP of the coded GMIMO system can be represented as:

[0087] MLD:

[0088] NLD:

[0089] In equation (2), t is the number of iterations, and r t X is the output signal of the linear detector MLD during the t-th iteration. t =[x1,...x t ] represents the estimated value of the transmitted signal x in the previous t iterations, γ t This represents the deorthogonalization of the memory linear detector function. p is the normalized parameter of the linear detector MLD. t The orthogonalization parameters for the linear detector MLD are... and p t To ensure the orthogonality of the MAMP algorithm, This represents the memory-matched filter function. It relates to the linear constraint Γ, that is:

[0090]

[0091]

[0092] Among them, A H Let H denote the conjugate transpose of the channel matrix A, and let H denote the conjugate transpose operator. At t=0, λ min and λ max They are AA H The minimum and maximum eigenvalues ​​of a matrix, θ t Let ξ be the relaxation parameter. t For the weights, the relaxation parameter θ t and weight ξ t Used to improve the convergence speed of MAMP receivers.

[0093] In equation (3), φ represents the nonlinear detector NLD function after adding a damping vector. t (·) represents the normalized and orthogonalized NLD detector function. in Represents the NLD function of the nonlinear detector The corresponding demodulation and posterior probability decoding functions in the nonlinear detection module, i.e. Let represent the nonlinear detector NLD function directly estimated by unnormalized orthogonalization, and It is Lipschitz continuous, and E{} denotes the conditional mean. and w tζ represents the normalized parameter and the orthogonalized parameter of the nonlinear detector NLD, respectively. t+1 It is the back-off damping vector, which is calculated as follows:

[0094]

[0095] φ represents the NLD function of the nonlinear detector in the t-th iteration. t The output variance of (·) indicates that when φ t When the variance of the output value of (·) increases, the estimated output value and variance of the previous iteration are used as... The current output is used to prevent MAMP from diverging during iteration. From Figure 3 (a) Performance comparison chart of MAMP and OAMP using analytical damping and back-off damping when the ill-conditioned matrix condition number k is 10. Figure 3 (b) Performance comparison of MAMP and OAMP using analytical damping and back-damping when the ill-conditioned matrix condition number k is 50. It can be seen that when the ill-conditioned matrix condition number k is large, the performance of back-damping is more stable than that of analytical damping.

[0096] Step 2: Use the covariance matrix of the state evolution SE to represent the asymptotic MSE performance of MAMP in Step 1.

[0097] Since long-memory matched filtering is used in MLD, its asymptotic performance is evaluated using the covariance matrix of the state evolution SE, that is, the asymptotic mean square error of the estimation error is represented by the covariance matrix:

[0098]

[0099] Where i and j represent the row and column positions in the covariance matrix. The covariance of the estimation error of the linear detector in the i-th and j-th iterations of the MLD function is expressed by the following formula: Where g i =r i -x,g j =r j -x represents the estimation error of the linear detector MLD in the i-th and j-th iterations, respectively. The covariance matrix represents the estimation error of the linear detector. The covariance matrix represents the estimation error of the nonlinear detector. The covariance of the estimation error of the nonlinear detector in the i-th and j-th iterations is expressed by the following formula: f i =x i -x,f j =x j-x represents nonlinear detection. The estimation error at the i-th and j-th iterations;

[0100] Furthermore, based on the orthogonality and independent identically distributed properties, the asymptotic MSE performance of MAMP can be obtained through the γ of the state evolution SE. SE (·)and Function representation, that is:

[0101] MLD:

[0102] NLD:

[0103] Formulas (8) and (9) are as follows Figure 1 As shown, since the function transformations of MLD and NLD are multidimensional, the analysis is relatively complex. It is necessary to transform them to make the transfer function equivalent to a single-input single-output form similar to OAMP / VAMP, thereby simplifying the realizability analysis and optimization code design of MAMP.

[0104] Step 3: Combine all damping and orthogonal operations of MAMP in Step 1 into the linear detection module MLD to obtain the equivalent transformed low-complexity MAMP. The linear detection module MLD performs orthogonal operations, damping operations and linear detection processing on the received signal y. The nonlinear detection module NLD performs demodulation and decoding processing on the received signal y. The state evolution SE in Step 2 is equivalently processed to obtain the one-dimensional variational state evolution SE. The asymptotic MSE performance of the equivalent transformed low-complexity MAMP is represented by the variational SE (VSE) function.

[0105] To overcome the complexity of multidimensional SE analysis while facilitating the analysis of MAMP's optimal coding criterion and achievable rate, all orthogonal and backtracking damping vectors ζ are... t+1 The operation is merged into MLD, that is, the right side of the equation (3) NLD [X t ,φ t (r t )]·ζ t+1 Damping operation, and formula The orthogonal and normalization operations in the linear detection module MLD are merged into the nonlinear detection module NLD, leaving only the nonlinear detection module NLD. An equivalent transformation of MAMP can be obtained, where the expression for the equivalent MAMP is:

[0106] MLD:

[0107] NLD:

[0108] Where η t(·) is a multidimensional MLD function, containing the functions in formula (2). And the damping and orthogonal operations of formula (3), Denotes the demodulation and posterior probability decoder function in a nonlinear detector (NLD). The output posterior estimate, Figure 2 (a) is a graphical representation of the MAMP receiver after the equivalent transformation. In this case, the transfer function in the NLD of the equivalent MAMP is already in single-input single-output mode. However, it contains the memory function η. t The MLD transfer function of (·) is still multidimensional, which is not conducive to the optimality theoretical analysis of MAMP. In order to further simplify the theoretical analysis process of MAMP, the following variational SE for the maximum reachability analysis of MAMP is proposed.

[0109] For decoding functions with the same demodulation and posterior probability The MAMP and OAMP / VAMP algorithms are discussed. MAMP's multidimensional SE converges to the same SE fixed point as OAMP / VAMP's SISO SE. Using the fixed-point consistency theorem of MAMP and OAMP / VAMP, and leveraging OAMP's VSE, the optimal coding principle and achievable rate of MAMP's VSE after equivalent transformation are analyzed.

[0110] like Figure 2 As shown in (b), the asymptotic MSE performance of the low-complexity MAMP after the equivalent transformation is expressed by variational SE (VSE):

[0111] MLD:

[0112] NLD:

[0113] in, This represents the ratio of the input signal to the interference noise in the NLD. As in equation (11) Input, For the case where i = t and j = t in formula (7), the calculation formula is: g t =r t -x represents the estimation error of the transformed linear detector MLD. Represents the transformed nonlinear detector The output mean square error The calculation formula is f t Represents the nonlinear detection after transformation The estimation error, f t =x t -x, and 1≤t'≤t, The function is The inverse of the matrix, The function is the MSE function of the linear minimum mean square error estimate (LMMSE), satisfying the following conditions: z is an additive white Gaussian noise vector with zero mean and variance of an identity matrix. This represents the MMSE function of the posterior probability decoder.

[0114] Although VSE cannot be used to characterize the MSE performance of MAMP in each iteration, it can be used to accurately analyze the achievable rate and coding principles.

[0115] Step 4: Based on the constraint capacity optimality principle and the I-MMSE lemma, integrate the variational SE (VSE) function from Step 3 to calculate the optimal coding design principle and maximum achievable rate of low-complexity MAMP in GMIMO communication.

[0116] Since encoding inevitably introduces gain, therefore... Figure 4 In the demodulation transfer function It should be the decode transfer function. The upper limit, calculation formula That is

[0117]

[0118] Suppose that the iteration converges at... and There is a unique fixed point between them. Design FEC code: To ensure and There are available decoding channels between them, and according to the principle of optimality, this is true if and only if the following conditions are met:

[0119]

[0120] At this time, MAMP can achieve error-free decoding.

[0121] According to formula (15) and the I-MMSE lemma, a fixed decoding transfer function is obtained. The speed that MAMP can achieve is:

[0122]

[0123] In the formula, d represents the limit of integration of calculus. satisfy The ratio of input signal to interference noise when the nonlinear detector NLD is correctly decoded in expression (13);

[0124] Based on the proof of the constrained capacity optimality of OAMP / VAMP in equation (15) and the expression for the realizable rate of MAMP in equation (16), the maximum achievable rate of MAMP can be derived as shown in equation (17). That is, MAMP can achieve the same maximum achievable rate as OAMP / VAMP, which shows that MAMP is constrained capacity optimal in coded GMIMO communication systems.

[0125]

[0126] The optimal coding design principle of MAMP is:

[0127]

[0128] When satisfied At that time, among them It can achieve the maximum error-free transmission rate of MAMP.

[0129] The performance of the MAMP algorithm in the optimally encoded GMIMO system designed according to the above rules approximates the performance of the OAMP algorithm using the same encoding structure, as shown in the bit error rate curves for example. Figure 5 The MAMP algorithm constructs orthogonalization operations between local linear memory estimators and nonlinear memory estimators, ensuring the asymptotically independent and identically distributed Gaussian nature of the estimation errors, thereby simplifying computation and reducing the complexity of the estimation algorithm. It achieves lower computational complexity while maintaining performance. Figure 6 This is a comparison of the running time of the MAMP algorithm and the OAMP algorithm when the number of antennas N is 500 and 5000.

[0130] Depend on Figure 6 It can be observed that when N=500, MAMP and OAMP / VAMP are at 10 -5 The difference between the BER curves at each location is within 0.1 dB. When the number of antennas is increased to N = 5000, MAMP can achieve the same BER performance as OAMP / VAMP, but with much lower complexity. Furthermore, to verify the advantages of the coding scheme in this application, Figure 5 The BER performance of MAMPs with point-to-point (P2P) regular LDPC codes (Re-LDPC) and irregular LDPC codes (IRe-LDPC) is also compared. The results show that the MAMP with the optimal code design has a gain of 1.6–5.0 dB compared to the MAMP with P2P LDPC encoding. This indicates that in coded GMIMO systems, the Bayesian optimal MAMP with P2P LDPC decoding is no longer the optimal design.

[0131] To better and more intuitively demonstrate the low complexity advantage of MAMP, Figure 6A comparison of runtime between MAMP and OAMP / VAMP is presented. The target BER is set to 2 × 10⁻⁶. -4 When N = 500, MAMP's runtime is only 30% of OAMP / VAMP's. When N increases to 5000, OAMP / VAMP's runtime increases dramatically by 16,000%, from 75.56s to 12420.79s. Under the same conditions, MAMP requires only 0.4% of the time to achieve the same performance as OAMP / VAMP. Therefore, compared to OAMP / VAMP, MAMP can achieve the information theory limit of GMIMO with significantly lower complexity.

[0132] Example 1: When the channel matrix A is an ill-conditioned matrix, the condition number k of the ill-conditioned matrix is ​​set to 10, the number of transmit antennas N is set to 500, and the decoding performance reaches 10. -5 At that time, the difference between the signal-to-noise ratio required by the MAMP receiver designed with the above optimal coding principle and the signal-to-noise ratio required to achieve the constrained capacity is 1.20d.

[0133] Example 2: When the channel matrix A is an ill-conditioned matrix, the condition number k of the ill-conditioned matrix is ​​set to 50, the number of transmit antennas N is set to 500, and the decoding performance reaches 10. -5 At that time, the difference between the signal-to-noise ratio required by the MAMP receiver designed with the above optimal coding principle and the signal-to-noise ratio required to achieve the constrained capacity is 2.75 dB.

[0134] Example 3: When the channel matrix A is an ill-conditioned matrix, the condition number k of the ill-conditioned matrix is ​​set to 10, the number of transmit antennas N is set to 5000, and the decoding performance reaches 10. -5 At that time, the difference between the signal-to-noise ratio required by the MAMP receiver designed with the above optimal coding principle and the signal-to-noise ratio required to achieve the constrained capacity is 2.85 dB.

[0135] Example 4: When the channel matrix A is an ill-conditioned matrix, the condition number k of the ill-conditioned matrix is ​​set to 10, the number of transmit antennas N is set to 5000, and the decoding performance reaches 10. -5 At that time, the difference between the signal-to-noise ratio required by the MAMP receiver designed with the above optimal coding principle and the signal-to-noise ratio required to achieve the constrained capacity is 6.45 dB.

Claims

1. A low-complexity optimal MAMP receiver coding method for large-scale MIMO, characterized in that, The specific steps are as follows: Step 1: Establish a signal transmission model for the coded GMIMO system. At the receiving end, use the MAMP algorithm to estimate the received signal y. Step 2: Use the covariance matrix of the state evolution SE to represent the asymptotic MSE performance of MAMP in Step 1. Step 3: Based on the fixed-point consistency theorem of MAMP and OAMP / VAMP, all damping and orthogonal operations of MAMP in Step 1 are merged into the linear detection module MLD to obtain the equivalent transformation low-complexity MAMP. The linear detection module MLD performs orthogonal operations, damping operations and linear detection processing on the received signal y. The nonlinear detection module NLD performs demodulation and decoding processing on the received signal y. The state evolution SE in Step 2 is equivalently processed to obtain the one-dimensional variational state evolution SE. The asymptotic MSE performance of the equivalent transformation low-complexity MAMP is represented by the variational SE (VSE) function. Step 4: Based on the constraint capacity optimality principle and the I-MMSE lemma, integrate the variational SE (VSE) function in Step 3 to obtain the optimal coding design principle and maximum achievable rate of low-complexity MAMP in coded GMIMO communication.

2. The low-complexity optimal MAMP receiver coding method for large-scale MIMO according to claim 1, characterized in that, Step 1 is implemented in the following steps: Step 1.1: Set up a GMIMO system with N transmit antennas and M receive antennas. The information sequence length on each transmit antenna is L, and the signal transmitted on the channel is... x， The receiver receives signal y, at the th... At what time, the signal transmitted on the channel is represented as T is the matrix transpose, and the signal captured by the receiver. Represented as: ;in, Represents the channel matrix. , It is additive white Gaussian noise, where I is the identity matrix. Indicates the noise variance; The received signal y can be rewritten as follows: Using the MAMP algorithm to estimate the received signal y, the received signal y can be rewritten as follows: MLD (1); Code constraint and ; Send signal x Satisfy coding constraints And it conforms to any distribution; Step 1.2: According to formula (1), the MAMP of the encoded GMIMO system is expressed as: MLD: (2); NLD: (3); In equation (2), t is the number of iterations. It is the output signal of the linear detector MLD at the t-th iteration. This represents the sent signal in the previous t iterations. x The estimated value, This represents the deorthogonalization of the memory linear detector function. For the normalized parameters of the linear detector MLD, The orthogonalization parameters for the linear detector MLD are... This represents the memory-matched filter function. With linear constraints Relevant, namely: （4）； in, Representing the channel matrix The conjugate transpose of , where H denotes the conjugate transpose operator. The specific calculation is as follows: （5）； Where, at t=0, , , , and They are The minimum and maximum eigenvalues ​​of a matrix. For relaxation parameters, For weights, relaxation parameters and weight Used to improve the convergence speed of MAMP receivers; In equation (3), This represents the NLD function of the nonlinear detector after adding a damping vector. This represents the normalized and orthogonalized NLD function of the nonlinear detector. ,in Represents the NLD function of the nonlinear detector The corresponding demodulation and posterior probability decoding functions in the nonlinear detection module, i.e. Let represent the nonlinear detector NLD function directly estimated by unnormalized orthogonalization, and It is Lipschitz continuous. Indicates the conditional mean. and These are the normalized parameters and orthogonalization parameters of the nonlinear detector NLD, respectively. It is the back-off damping vector, which is calculated as follows: （6）； v φ Denotes the NLD function of the nonlinear detector in the t-th iteration. The output variance when When the variance of the output value increases, the estimated output value and variance of the previous iteration are used as... The current output.

3. The low-complexity optimal MAMP receiver coding method for large-scale MIMO according to claim 2, characterized in that, Step 2 is described in detail below: The asymptotic mean square error of the estimation error is represented using the SE covariance matrix of the state evolution: ， （7）； Where i and j represent the row and column positions in the covariance matrix. The covariance of the estimation error of the linear detector in the i-th and j-th iterations of the MLD function is expressed by the following formula: ,in , These represent the estimation errors of the linear detector MLD in the i-th and j-th iterations, respectively. The covariance matrix represents the estimation error of the linear detector. The covariance matrix represents the estimation error of the nonlinear detector. The covariance of the estimation error of the nonlinear detector in the i-th and j-th iterations is expressed by the following formula: , , , respectively represent nonlinear detection NLD The estimation errors at the i-th and j-th iterations, Furthermore, based on the orthogonality and independent identically distributed properties, the asymptotic MSE performance of MAMP can be obtained through state evolution SE. and Function representation, that is: MLD: (8); NLD: (9); The function transformations of both MLD and NLD are multidimensional.

4. The low-complexity optimal MAMP receiver coding method for large-scale MIMO according to claim 3, characterized in that, Step 3 is as follows: For the same demodulation and posterior probability decoding function The state evolution SE of MAMP converges to the same SE fixed point as the SISO SE of OAMP / VAMP, i.e., all orthogonal and backtracking damped vectors. The operation is merged into MLD, that is, the right side of the NLD in equation (3) is... Damping operation and formula The orthogonal and normalization operations in the original code have been merged into the linear detection module MLD, leaving only the nonlinear detection module NLD. We obtain the equivalent transformation of MAMP, and the expression for the equivalent MAMP is: MLD: (10); NLD: (11); in It is a multidimensional MLD function, containing the formula (2) And the damping and orthogonal operations of formula (3), Denotes the demodulation and posterior probability decoder function in a nonlinear detector (NLD). The output posterior estimate; The transfer function in the NLD of the equivalent MAMP is a single-input single-output mode and includes a memory function. The MLD transfer function is a multidimensional function, which is not conducive to the optimality analysis of MAMP. Using the fixed-point consistency theorem of MAMP and OAMP / VAMP, and with the help of the VSE analysis of OAMP, the optimal coding principle and achievable rate of the VSE of MAMP after the equivalent transformation are obtained. Therefore, the asymptotic MSE performance of the low-complexity MAMP after the equivalent transformation is expressed by variational SE (VSE): MLD: (12); NLD: (13); in, This represents the ratio of the input signal to the interference noise in the NLD. , As in equation (11) Input, For the case where i=t and j=t in formula (7), the calculation formula is: , This represents the estimation error of the transformed linear detector MLD. This represents the transformed nonlinear detector NLD. The output mean square error The calculation formula is , f t Represents the transformed nonlinear detection NLD The estimation error, ,and , The function is The inverse of the matrix, The function is the MSE function of the linear minimum mean square error estimate (LMMSE), satisfying the following conditions: z is an additive white Gaussian noise vector with zero mean and variance of an identity matrix. This represents the MMSE function of the posterior probability decoder.

5. The low-complexity optimal MAMP receiver coding method for large-scale MIMO according to claim 4, characterized in that, Step 4 is as follows: Post-gain demodulation transfer function It is a decoding transfer function The upper limit, that is: for (14), in, ; Assuming the iteration converges and There is a unique fixed point between them. According to the principle of optimality, if and only if the following condition is met: for (15); At this time, MAMP can achieve error-free decoding; According to formula (15) and the I-MMSE lemma, a fixed decoding transfer function is obtained. The speed that MAMP can achieve is: （16）； in, This indicates the limits of integration in calculus. satisfy The ratio of the input signal to the interference noise when the nonlinear detector NLD is correctly decoded in equation (13); Based on the proof of the constraint capacity optimality of OAMP / VAMP in equation (15) and the realizable rate expression of MAMP in equation (16), the maximum achievable rate of MAMP is calculated as follows: （17）； The optimal coding design principle of MAMP is: （18）； When satisfied At that time, among them It can achieve the maximum error-free transmission rate of MAMP.

6. A low-complexity optimal MAMP receiver for massive MIMO, characterized in that, Encoding is performed using the low-complexity optimal MAMP receiver coding method for large-scale MIMO as described in any one of claims 1-5, including a linear detection module MLD and a nonlinear detection module NLD. The linear detection module MLD corresponds to the linear constraint and performs orthogonal operation, damping operation and linear detection processing on the received signal y. The nonlinear detection module (NLD) consists of a demodulator and a posterior probability decoder. The NLD demodulates and decodes the received signal y.

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