A Channel Estimation Method Based on Delay-Doppler Domain

Abstract

This application relates to a channel estimation method based on the delay-Doppler domain. The method includes: establishing a channel model, constructing a sparse signal recovery task model based on the structural sparsity of the channel; establishing a factor graph model based on a joint probability model and Bayesian methods; calculating the messages passed between function nodes and variable nodes, and between function nodes and observation nodes, according to message passing rules; updating the confidence scores of each node based on the messages passed between function nodes and variable nodes, and between function nodes and observation nodes; adjusting the auxiliary probability density function based on the confidence scores of each node, calculating the variance and mean of the observation nodes, and iteratively updating the variance and mean of the observation nodes to obtain the channel in the sparse signal recovery task model. This application can improve the performance and efficiency of the algorithm.

Description

Technical Field

[0001] This application relates to the field of wireless communication technology, and in particular to a channel estimation method based on the delay-Doppler domain. Background Technology

[0002] Both 4G and the emerging 5G primarily employ Orthogonal Frequency Division Multiplexing (OFDM) as their modulation technique. However, on high-speed trains traveling at speeds up to 500 km / h, the Doppler shift caused by the highly mobile environment disrupts the orthogonality of subcarriers, leading to a degradation in communication system performance. Due to the increased demands for data rates, flexibility, and reliability in 5G cellular systems, it is necessary to consider new modulation formats. Unlike OFDM, Orthogonal Time Frequency Space (OTFS) modulation modulates information in the delay-Doppler (DD) domain, rather than the classical time-frequency (TF) domain.

[0003] In related technologies, research on orthogonal time-frequency spatial modulation multiple-input multiple-output (MIMO-OTFS) systems has increased significantly. The introduced novel OTFS framework differs from the (SP) method, which superimposes a single pilot onto the data; it employs multiple superimposed pilots (SPs). This architectural modification aims to mitigate interference between data pilots and improve channel estimation accuracy. However, this also presents a challenge: data symbols must be treated as interference during channel estimation, increasing the complexity of signal detection. Embedded pilot-assisted channel estimation schemes require adding a guard zone between pilot symbols. Traditional channel estimation methods incur significant pilot overhead, making them difficult to scale to large-scale MIMO systems. Furthermore, embedded pilot channel estimation suffers from high pilot overhead, a high peak-to-average power ratio, and poor performance.

[0004] Therefore, it is necessary to provide a new technical solution to improve one or more of the problems existing in the above solutions.

[0005] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this application, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0006] The purpose of this application is to provide a channel estimation method based on the delay-Doppler domain, thereby overcoming, at least to some extent, one or more problems caused by the limitations and defects of related technologies.

[0007] A channel estimation method based on the delay-Doppler domain, according to an embodiment of this application, includes:

[0008] Establish a channel model, and construct the channel model into a sparse signal recovery task model based on the structural sparsity of the channel;

[0009] A factor graph model is established based on the joint probability model and Bayesian methods; wherein, the nodes in the factor graph model include: function nodes, variable nodes, and observation nodes;

[0010] Calculate the messages passed between the function node and the variable node, and between the function node and the observation node in the factor graph model, according to the message passing rules;

[0011] The confidence level of each node is updated based on the messages passed between the function node and the variable node, and the messages passed between the function node and the observation node.

[0012] The variance and mean of the observation nodes are calculated based on the confidence level of each node, and the variance and mean of the observation nodes are iteratively updated to obtain the channel in the sparse signal recovery task model.

[0013] In the embodiments of this application, the step of establishing a channel model, which involves constructing the channel model into a sparse signal recovery task model based on the structural sparsity of the channel, includes:

[0014] In an orthogonal time-frequency spatial modulation multiple-input multiple-output system, the time-varying channel consisting of Q dominant propagation paths, with the time-varying channel model of the f-th transmit antenna and the r-th receive antenna, is as follows:

[0015]

[0016] In the formula, h q，r v q，r τ q，r ψ q，r Let f represent channel gain, Doppler shift, propagation delay, and phase difference, respectively, where f = 0, 1, ..., N. t -1, r = 0, 1, ..., N r -1; l represents the time delay index, j represents the complex part of the exponential function e, k represents the Doppler index, q = 0, 1, ..., Q-1, P rc (.) represents the position in T. q，r The band-limited pulse shaping filter response is calculated at point Ts, where Ts represents the system sampling interval.

[0017] In the embodiments of this application, the step of establishing a channel model, which involves constructing the channel model into a sparse signal recovery task model based on the structural sparsity of the channel, includes:

[0018] By employing a non-orthogonal pilot mode, pilot signals and data are simultaneously transmitted within orthogonal time-frequency space (OTFS) symbols, transforming an M×N symbol sequence into a two-dimensional data block. Where M and N represent the number of subcarriers in the frequency domain and the number in the time domain, respectively, and the number transmitted by the f-th transmit antenna is... The symbol, after inverse Fourier transform, is represented as:

[0019]

[0020] In the formula, F M Let F represent the M-point Fourier transform matrix. N H represents the normalized N-point Fast Fourier Transform matrix, H represents the conjugate transpose, and ISFFT represents the inverse Fourier transform.

[0021] The time-domain signal transmitted in OTFS is first applied to the inverse Fourier transform. Then, a Heisenberg transform is performed to obtain the transmission time-domain signal S in OTFS. f The expression is:

[0022]

[0023] G tx =diag[g tx (0),..,g tx (t),.,g tx ((M-1)T s / M)] (4);

[0024] In the formula, G tx The diagonal matrix g is obtained by sampling the transmitted pulse. tx (t) represents the t-th transmitted pulse;

[0025] Processing the transmission time-domain signal in OTFS, the end-to-end delay-Doppler domain input-output model from the f-th transmit antenna to the r-th receive antenna is obtained as follows:

[0026]

[0027] In the formula, R fr Indicates the received signal;

[0028] Vectorizing the input-output model of MIMO-OTFS yields the vectorized input-output relationship, expressed as follows:

[0029]

[0030] In the formula, This represents the channel in the time-delay-Doppler domain. It is additive Gaussian random white noise, where ll′ represents the delay index exponent at the f-th transmit antenna, and kk′ represents the Doppler index exponent at the f-th transmit antenna.

[0031] After performing spatial discrete Fourier transform on equations (1) and (6) along the antenna index f, the sparse signal recovery task model is obtained, and its expression is:

[0032] y = Φh + w(7);

[0033] y = {y1,...,y i ,...,y P} T y i Let h represent the measured values, which can be represented by the dimension MgNgNt×1, and w be the noise vector, which can also be represented by the dimension MgNg×1. For ease of representation, let θ = (1,...,i,...,P), C = (1,…,c,…,C), and Φ represent the dictionary matrix dimension P×C, where P = M P N P C=M g N g N t ;

[0034] In the embodiments of this application, the step of establishing a factor graph model based on the joint probability model and Bayesian methods includes:

[0035] The joint probability density function in formula (7) is decomposed into factors and further expressed as a hierarchical prior model;

[0036] p(y,h,λ,γ,η,α)=p(λ)Π i p(y i |h,λ)p(α i )Π c p(h c |γ c )p(γ c |η c )p(η c (8);

[0037] In the formula, i and c belong to the set θ, C;

[0038] The set of latent variables in the constructed factor graph is Δ = (λ, γ) c η c α i h c In the set λ, γ c η c α i h cThey also represent variable nodes in the probability factor graph, α i To reduce computational complexity, a scalar factor is introduced, p(y i |h,λ) denotes an approximate Gaussian distribution function, p(λ) denotes a gamma distribution, and p(h) c |γ c p(η) represents the approximate Gaussian distribution function. c ) represents the gamma distribution, p(α) i ) represents a Gaussian distribution.

[0039] In the embodiments of this application, the set of function nodes in the factor graph model is defined as a set. Will Divided into two parts: and in, Represents a set of functions. Represents a set of functions. Represents a function node. Represent any i, Partially propagated using BP messages, in Partially utilizes MF variational average field message aggregation.

[0040] In embodiments of this application, calculating the messages passed between the function node and the variable node in the factor graph model and the messages passed between the function node and the observation node according to the message passing rules includes:

[0041] From function node The expression for the message to variable node λ is:

[0042]

[0043] In the formula, b(α) i ) represents variable node α i The confidence level, exp(·) represents the exponential function;

[0044] Using the BP message propagation rules, assuming that the variable node h c To function node News Given that it follows a Gaussian distribution, then from the function node to variable node h c News The expression is as follows:

[0045]

[0046] In the formula, Indicates from function node to variable node h c The mean of the message Indicates from function node to variable node h c The variance corresponding to the message.

[0047] In embodiments of this application, calculating the messages passed between the function node and the variable node in the factor graph model and the messages passed between the function node and the observation node according to the message passing rules includes:

[0048] From function node to variable node h c The mean of the message and variance They are represented as follows:

[0049]

[0050] In the formula, Φ ic Represents the perception matrix, From observation node h c To function node The message being conveyed This represents the mean of an approximate Gaussian distribution. This represents the variance of an approximately Gaussian distribution. Represents variable node h c To function node The news Indicates the initial estimate;

[0051] Assume starting from variable node α i Transmit to function node Prior information Given that the confidence level b(γ) of the random variable node γ is known and all follow a Gaussian distribution, and according to the Central Limit Theorem, we can obtain that from the variable node h... c To function node News The expression is:

[0052]

[0053] In the formula, For Φ ic The adjoint matrix, |Φ ic | 2 Denotes the Frobenius norm of a matrix. h represents the variable node c The mean of an approximate Gaussian distribution, V i q The variable node h represents the variable.c The variance of the approximate Gaussian distribution, from the function nodes to variable node h c The expression for the message is:

[0054]

[0055] In the formula, b(γ) c ) represents the variable node γ c The confidence level.

[0056] In embodiments of this application, updating the confidence level of each node based on the messages transmitted between the function node and the variable node and the messages transmitted between the function node and the observation node includes:

[0057] By updating and The transmitted message is obtained from the observation node h. c confidence level b(h) c To update the product of the messages passed between the two nodes:

[0058]

[0059] Among them, observation node h c The observation node h follows a Gaussian distribution. c variance and observation node h c mean They are as follows:

[0060]

[0061] In the formula, Represents variable node γ c The mean;

[0062] From observation node h c To function node The message is:

[0063]

[0064] The transmitted message can be approximated by a Gaussian distribution (20) with variance. and mean Each as

[0065]

[0066] Utilizing the BP message propagation rules again, calculate the function node. To variable node α i News And calculate the conditional probability distribution among variables in the probability factor graph model.

[0067] According to previous iterations and Multiplication yields the variable node α i confidence level b(α) i ), b(α) i The expression for ) is as follows:

[0068]

[0069] In the formula, V i α Representing variable nodes α respectively i The mean and variance of.

[0070] In embodiments of this application, the step of calculating the variance and mean of the observation nodes based on the confidence levels of each node, obtaining an approximate posterior distribution of the latent variables, iteratively updating the variance and mean of the observation nodes, and obtaining the channel in the received signal includes:

[0071] Construct an auxiliary function Q(Δ) and minimize its divergence KL(q(Δ)||p(Δ|y)) with (8) and the posterior probability. Then, in subsequent updates, the auxiliary function Q(Δ) is obtained through message updates of the factor graph:

[0072] Q(Δ)=q(γ)q(λ)q(η)q(α)q(h) (29);

[0073] Where q(λ), q(γ), q(η), q(α), and q(h) are the approximate posterior distributions of the latent variables. In the following derivation, let <··> q(·) To represent the expectation under q(·), the approximate posterior distribution of q(h) requires... and Multiplying these two messages yields an approximate posterior Gaussian probability density function, whose variance matrix and mean are given below:

[0074]

[0075] In the embodiments of this application, the step of calculating the variance and mean of the observation nodes based on the confidence level of each node, and iteratively updating the variance and mean of the observation nodes to obtain the channel in the sparse signal recovery task model includes:

[0076] Updates and messages and The product is proportional, q(η) is an approximate posterior distribution, and its calculation method is given by the following formula, ε, a c b c For constants:

[0077]

[0078] The approximate posterior distribution function of q(λ) corresponds to the generalized inverse Gaussian and generalized anti-Gaussian distributions, which can be given in closed form and obtained through formulas (19) and (34). Expectation under q(γ):

[0079]

[0080] K p (·) denotes the p-th order second-kind modified Bessel function

[11] , and the approximate posterior distribution function of q(λ) is represented by the shape parameter J and the rate parameter J. gamma function, The approximate posterior distribution is given by (26) to (28), and the expectation of λ under q(λ) is obtained:

[0081]

[0082] To reduce complexity, equations (30) and (31) are processed in blocks. In subsequent updates, it is assumed that the vector h is decomposed into ρ blocks, and the mean of each block is... and variance The update can be approximated by an auxiliary probability function:

[0083]

[0084] In the embodiments of this application, the step of calculating the variance and mean of the approximate posterior distribution function of the observation node based on the confidence level of each node and minimizing its divergence, and iteratively updating the variance and mean of the observation node to obtain the channel in the sparse signal recovery task model, includes:

[0085] Mean of variable node h and variance as follows:

[0086]

[0087] The technical solutions provided by the embodiments of this application may include the following beneficial effects:

[0088] In one embodiment of this application, the equivalent channel in the time-delay-Doppler domain exhibits good structural sparsity, distinguishability, compactness, and stability through the above method, allowing the channel estimation problem to be treated as a sparse signal recovery problem. Next, a Bayesian prior model is introduced to assist in channel estimation. Finally, a hybrid message-passing technique is used to process the factor graph model constructed based on the prior model. This method not only overcomes the problem of large pilot overhead in traditional channel estimation methods but also finds local optima through iterative optimization, improving algorithm performance and efficiency.

[0089] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and do not limit this application. Attached Figure Description

[0090] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0091] Figure 1 The flowchart illustrates the steps of a channel estimation method based on the delay-Doppler domain in an exemplary embodiment of this application.

[0092] Figure 2 This illustration schematically shows a MIMO-OTFS system in an exemplary embodiment of this application;

[0093] Figure 3 This illustration schematically shows a factor graph model in an exemplary embodiment of this application;

[0094] Figure 4 This schematic diagram illustrates a performance comparison of normalized mean square error and signal-to-noise ratio for different channel estimation methods in an exemplary embodiment of this application.

[0095] Figure 5 This schematic diagram illustrates a comparison of the normalized mean square error and user speed of different channel estimation methods in an exemplary embodiment of this application.

[0096] Figure 6 This schematic diagram illustrates a performance comparison of the normalized mean square error of different channel estimation methods with the number of antennas at the base station in an exemplary embodiment of this application.

[0097] Figure 7 This schematic diagram illustrates a performance comparison of normalized mean square error versus pilot overhead ratio for different channel estimation methods in an exemplary embodiment of this application.

[0098] Figure 8 This schematic diagram illustrates a performance comparison of bit error rate and signal-to-noise ratio for different channel estimation methods in an exemplary embodiment of this application.

[0099] Figure 9 This illustration shows the running time of different channel estimation methods at the same signal-to-noise ratio in an exemplary embodiment of this application.

[0100] Figure 10 This illustration demonstrates the convergence of the channel estimation method based on the delay-Doppler domain in the exemplary embodiments of this application on the normalized mean square error. Detailed Implementation

[0101] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided to make this application more comprehensive and complete, and to fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0102] Furthermore, the accompanying drawings are merely illustrative of this application and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0103] This example implementation first provides a channel estimation method based on the delay-Doppler domain. (Reference) Figure 1 As shown, the method may include:

[0104] Step S101: Establish a channel model. Based on the structural sparsity of the channel, construct the channel model into a sparse signal recovery task model.

[0105] Step S102: Establish a factor graph model based on the joint probability model and Bayesian model; wherein, the nodes in the factor graph model include: function nodes, variable nodes and observation nodes.

[0106] Step S103: Calculate the messages passed between function nodes and variable nodes, and between function nodes and observation nodes in the factor graph model, according to the message passing rules.

[0107] Step S104: Update the confidence of each node based on the messages passed between the function node and the variable node, and between the function node and the observation node.

[0108] Step S105: Calculate the variance and mean of the observation nodes based on the confidence level of each node, and iteratively update the variance and mean of the observation nodes to obtain the channel in the sparse signal recovery task model.

[0109] In one embodiment of this application, the equivalent channel in the time-delay-Doppler domain exhibits good structural sparsity, distinguishability, compactness, and stability through the above method, allowing the channel estimation problem to be treated as a sparse signal recovery problem. Next, a Bayesian prior model is introduced to assist in channel estimation. Finally, a hybrid message-passing technique is used to process the factor graph model constructed based on the prior model. This method not only overcomes the problem of large pilot overhead in traditional channel estimation methods but also finds local optima through iterative optimization, improving algorithm performance and efficiency.

[0110] Below, we will refer to Figures 2 to 3 The steps of the method described above in this example embodiment will be explained in more detail.

[0111] In step S101, as Figure 2 As shown, consider a downlink MIMO-OTFS system, which includes a base station and N r users (i.e.) Figure 2 (mobile users in the region), where the base station is equipped with N t Each user is equipped with one receiving antenna and one transmitting antenna. The base station transmits an M×N (M and N are the number of resource units in the time delay dimension and Doppler dimension) orthogonal time-frequency spatial modulation (OTFS) signal. OTFS decomposes the signal into orthogonal basis functions in the time and frequency dimensions and maps the signal onto a time-frequency grid. The information symbol is mapped to X in the Doppler domain. f [k,L] represents the OTFS signal transmitted by the f-th transmit antenna, where 0≤k≤N-1, 0≤L≤M-1, and k and L represent the grid indexes of Doppler and time delay, respectively.

[0112] In step S1011, a channel model is established. Based on the structural sparsity of the channel, the channel model is constructed into a sparse signal recovery task model, including:

[0113] In an orthogonal time-frequency spatial modulation multiple-input multiple-output system, the time-varying channel consisting of Q dominant propagation paths, with the time-varying channel model of the f-th transmit antenna and the r-th receive antenna, is as follows:

[0114]

[0115] In the formula, hq,r v q,r τ q,r ψ q,r Let f represent channel gain, Doppler shift, propagation delay, and phase difference, respectively, where f = 0, 1, ..., N. t -1, r = 0, 1, ..., N r -1; l represents the time delay index, j represents the complex part of the exponential function e, k represents the Doppler index, q = 0, 1, ..., Q-1, P rc (.) represents the value at τ q,r The band-limited pulse shaping filter response is calculated at point R, where Ts represents the system sampling interval. The departure angle (Angle of Departure, AoD) corresponding to the q-th path of the receiving antenna r is denoted as Θ. qr , and Θ qr The relevant spatial angle can be expressed as λ is the antenna spacing, where λ is the wavelength of the carrier frequency.

[0116] Due to the large number of base station (BS) antennas, the pilot overhead is substantial. To reduce the overall pilot overhead of the OTFS massive MIMO system, a non-orthogonal pilot mode is employed. This mode arranges the pilots and data in orthogonal time-frequency space OTFS symbols for simultaneous transmission, rearranging a sequence of M×N symbols into a two-dimensional data block. Where M and N represent the number of subcarriers in the frequency domain and the number in the time domain, respectively, and the pilot lengths along the Doppler dimension and the time delay dimension are N, respectively. p and M p Interference between pilot and data symbols is mitigated by using guard intervals, with guard intervals of Ng and 2Mg respectively. The f-th transmitting antenna transmits... The symbol, after inverse Fourier transform, is represented as:

[0117]

[0118] In the formula, F M Let F represent the M-point Fourier transform matrix. N H represents the normalized N-point Fast Fourier Transform matrix, H represents the conjugate transpose, and ISFFT represents the inverse Fourier transform.

[0119] The time-domain signal transmitted in OTFS is first applied to the inverse Fourier transform. Then, a Heisenberg transform is performed to obtain the transmission time-domain signal S in OTFS. f The expression is:

[0120]

[0121] G tx =diag[gtx (0),..,g tx (t),.,g tx ((M-1)T s / M)] (4);

[0122] In the formula, G tx The diagonal matrix g is obtained by sampling the transmitted pulse. tx (t) represents the t-th transmitted pulse;

[0123] Processing the transmission time-domain signal in OTFS, the end-to-end delay-Doppler domain input-output model from the f-th transmit antenna to the r-th receive antenna is obtained as follows:

[0124]

[0125] In the formula, R fr Indicates the received signal;

[0126] Vectorizing the input-output model of MIMO-OTFS yields the vectorized input-output relationship, expressed as follows:

[0127]

[0128] In the formula, This represents the channel in the time-delay-Doppler domain. It is additive Gaussian random white noise, where ll′ represents the delay index exponent at the f-th transmit antenna, and kk′ represents the Doppler index exponent at the f-th transmit antenna.

[0129] After performing spatial discrete Fourier transform on equations (1) and (6) along the antenna index f, the sparse signal recovery task model is obtained, and its expression is:

[0130] y = Φh + w(7);

[0131] y = {y1,...,y i ,...,y P} T y i Let h represent the channel, whose dimension can be represented by MgNgNt×1, and w be the noise vector, whose dimension can be represented by MgNg×1. For ease of representation, let θ=(1,…,i,…,P), C=(1,...,c,...,C), and Φ represent the dictionary matrix dimension P×C, where P=M P N P C=M g N g N t .

[0132] It should be noted that the time-domain signals transmitted in OTFS are processed, including adding a cyclic prefix (CP) to the beginning of each user-generated time-domain signal to avoid interference. Signal S f If the signal reaches the receiver via the channel, then the f-th transmitting antenna and the... The signal r received between the receiving antennas is folded into a matrix. After removing the CP, the Wigner transform and Fast Fourier Transform (SFFT) are performed to obtain the end-to-end time delay-Doppler domain input-output model from the f-th transmit antenna to the r-th receive antenna.

[0133] Based on the sparsity of the channel, Bayesian inference combined with a factor graph model is introduced to reconstruct the channel h in the sparse signal recovery task model.

[0134] In step S102, a factor graph model is established based on the joint probability model and Bayesian model; wherein, the nodes in the factor graph model include: function nodes, variable nodes and observation nodes.

[0135] In step S1021, a factor graph model is established based on the joint probability model and Bayesian model, including:

[0136] According to Bayesian theory, the joint probability density function expressed in formula (7) is decomposed into a conditional probability product form, as shown in formula (8). Since Gaussian multiplication is involved in parameter updates, taking the logarithm may lead to excessively large coefficients. To solve this problem, an additional variable node α is introduced to reduce complexity. Figure 3 The figure shows the factor plot of the improved scalar form of the 4-L hierarchical model, where i = (1,..,P) and c = (1,...,C). Since the specific expressions for each unknown parameter (γ,λ,η) cannot be obtained, the improved variational message passing (VMP) algorithm is chosen to approximate the maximum a posteriori estimate and express it as a 4-L hierarchical model. This hierarchical prior model can be expressed as Equation (8) according to Equation (7).

[0137] First, the joint probability density function in formula (7) is decomposed into factors, and further expressed as a hierarchical prior model (8): where i, c belong to the set θ, C:

[0138] p(y,h,λ,γ,η,α)=p(λ)Π i p(y i |h,λ)p(α i )Π c p(h c |γ c )p(γ c |η c )p(η c (8);

[0139] The set of latent variables in the constructed factor graph is Δ = (λ, γ) c η c α i h c In the set λ, γ c η c α i h c They also represent variable nodes in the probability factor graph, α i To reduce computational complexity, a scalar factor is introduced, p(y i |h,λ) denotes an approximate Gaussian distribution function, p(λ) denotes a gamma distribution, and p(h) c |γ c p(η) represents the approximate Gaussian distribution function. c ) represents the gamma distribution, p(α) i ) represents a Gaussian distribution.

[0140] In step S1022, hybrid message passing is employed in the factor graph model. This method provides greater flexibility in designing message passing algorithms, making them more suitable for solving complex variational inference problems. Specifically, HMP (Hybrid Message Passing) rules can implement two message computation rules from the same factor node to the connected variable node through different edges. For example... Figure 3 As shown, the blue circles, green squares, and shaded gray circles represent variable nodes, function nodes, and observations, respectively. This application will calculate the normalized message for each node in the graph according to different message passing rules.

[0141] Therefore, the set of function nodes in the factor graph model is defined as a set. Will Divided into two parts: and in, Represents a set of functions. Represents a set of functions. Represents a function node. Represent any i, Partially propagated using BP messages, in Partially utilizes MF variational average field message aggregation.

[0142] In step S103, the messages passed between function nodes and variable nodes, as well as the messages passed between function nodes and observation nodes in the factor graph model, are calculated according to the message passing rules.

[0143] In step S1031, the messages passed between function nodes and variable nodes, and between function nodes and observation nodes in the factor graph model, are calculated according to the message passing rules, including:

[0144] like Figure 3 As shown, a scalar α is introduced into the factor graph. i As a variable node that transmits messages, assuming the confidence of variable node λ is called b(λ), a variational mean field method is used to handle complex probabilistic models. By transforming the primal problem into an optimization problem, an approximate distribution is sought to represent the primal distribution, in order to efficiently handle high-dimensional data and complex models. From function nodes... The expression for the message to variable node λ is:

[0145]

[0146] In the formula, y i Represents the measured value, b(α) i ) represents variable node α i The confidence level, exp(·) represents the exponential function;

[0147] Using the BP message propagation rules, assuming that the variable node h c To function node News Given that it follows a Gaussian distribution, then from the function node to variable node h c News The expression is as follows:

[0148]

[0149] In the formula, Indicates from function node to variable node h c The mean of the message Indicates from function node to variable node h c The variance corresponding to the message.

[0150] From function node to variable node h c The mean of the message and variance They are represented as follows:

[0151]

[0152] In the formula, Φ ic Represents the perception matrix, From observation node h c To function node The message being conveyed This represents the mean of an approximate Gaussian distribution. This represents the variance of an approximately Gaussian distribution. Represents variable node h c To function node The news Indicates the initial estimate;

[0153] Assume starting from variable node α i Transmit to function node Prior information Given that the confidence level b(γ) of the random variable node γ is known and all follow a Gaussian distribution, and according to the Central Limit Theorem, we can obtain that from the variable node h... c To function node News The expression is:

[0154]

[0155] In the formula, For Φ ic The adjoint matrix, |Φ ic | 2 Denotes the Frobenius norm of a matrix. h represents the variable node c The mean of an approximate Gaussian distribution, V i q The variable node h represents the variable. c The variance of the approximate Gaussian distribution, from the function nodes to variable node h c The expression for the message is:

[0156]

[0157] In the formula, b(γ) c ) represents the variable node γ c The confidence level.

[0158] By updating and The transmitted message is obtained from the observation node h. c confidence level b(h) c To update the product of the messages passed between the two nodes:

[0159]

[0160] Among them, observation node h c The observation node h follows a Gaussian distribution. c variance and observation node h c mean They are as follows:

[0161]

[0162] In the formula, Represents variable node γ c The mean;

[0163] From observation node h c To function node The message is:

[0164]

[0165] Among them, observation node h c variance and mean They are as follows:

[0166]

[0167]

[0168] Utilizing the BP message propagation rules again, calculate the function node. To variable node α i News And calculate the conditional probability distribution among variables in the probability factor graph model.

[0169] According to previous iterations and Multiplication yields the variable node α i confidence level b(α) i ), b(α) i The expression for ) is as follows:

[0170]

[0171] In the formula, V i α Representing variable nodes α respectively i Mean and variance:

[0172] Construct an auxiliary function Q(Δ) and minimize its (Kullback-Leibler, KL) divergence KL(q(Δ)||p(Δ|y)) with (8) and the posterior probability. Then, in subsequent updates, the auxiliary function Q(Δ) is obtained through message updates of the factor graph:

[0173] Q(Δ)=q(γ)q(λ)q(η)q(α)q(h) (29);

[0174] Where q(λ), q(γ), q(η), q(α), and q(h) are the approximate posterior distributions of the latent variables. In the following derivation, let <··> q(·) Let represent the expected value under q(·). The approximate posterior distribution of q(h) requires ... and Multiplying these two messages yields an approximate posterior Gaussian probability density function, whose variance matrix and mean are given by the following Φ. H Here is the conjugate transpose matrix:

[0175]

[0176] In step S105, the variance and mean of the observation nodes are calculated based on the confidence level of each node, and the variance and mean of the observation nodes are iteratively updated to obtain the channel in the sparse signal recovery task model.

[0177] In step S1051, the variance and mean of the observed nodes are calculated based on the confidence level of each node, and the variance and mean of the observed nodes are iteratively updated to obtain the channel in the sparse signal recovery task model, including:

[0178] In the first iteration, the mean value of the observed node h needs to be calculated. With variance Then, update each parameter in subsequent iterations.

[0179] Updates and messages and The product is proportional, q(η) is an approximate posterior distribution, and its calculation method is given by the following formula, ε, a c b c For constants:

[0180]

[0181] The approximate posterior distribution function of q(λ) corresponds to the generalized inverse Gaussian (GIG) distribution, which can be given in closed form and obtained through formulas (20) and (34). Expectation under q(γ):

[0182]

[0183] K p (·) denotes the p-th order second-kind modified Bessel function

[11] , and the approximate posterior distribution function of q(λ) is represented by the shape parameter J and the rate parameter J. gamma function, The approximate posterior distribution is given by equations (26) to (28), yielding the expectation of λ under q(λ):

[0184]

[0185] Because the computational complexity increases when the factor graph dimension is large, equations (30) and (31) are divided into blocks to reduce complexity. In subsequent updates, it is assumed that the channel h is decomposed into ρ blocks, and the mean of each block is... and variance The update can be approximated by an auxiliary probability function. Here is the conjugate transpose matrix:

[0186]

[0187] In step S1052, the variance and mean of the approximate posterior distribution function of the observed nodes are calculated based on the confidence level of each node, and its divergence is minimized. The variance and mean of the observed nodes are then iteratively updated to obtain the channel in the sparse signal recovery task model, including:

[0188] Mean of variable node h and variance as follows:

[0189]

[0190] As shown in the probability factor diagram, the connections between nodes are closely intertwined, leading to many different message passing rules. This application summarizes the corresponding message calculation process in the channel estimation method. As shown in Table 1, It is initialized before the iteration process and outputs a stable channel estimation result after repeated iterations, i.e., output formula (41). After T iterations, the complete pseudocode is shown in Table 1.

[0191] Table 1 MIMO-OTFS Channel Estimation Methods

[0192]

[0193] The complexity of this application is analyzed below.

[0194] The complexity in this application mainly comes from the message passing part, specifically the update operations involving the calculation of the mean and variance in the Gaussian approximation. Since the channel estimation method based on the delay-Doppler domain proposed in this application does not involve matrix inversion, its complexity is primarily concentrated in matrix multiplication operations. The complexity of calculating the Frobenius norm of the sensing matrix is... The complexity of each iteration is attributed to the multiplication calculations in rows 6 and 7 of Table 1, with a complexity of approximately [missing information]. The overall complexity is then...

[0195] The following simulation will further illustrate this application.

[0196] In the OTFS system, we introduce Normalized Mean Square Error (NMSE) and Bit Error Rate (BER) as metrics to evaluate the proposed method. The bit error rate is also called the bit error rate. The pilot overhead ratio η, which is the ratio of the OTFS symbol dimension occupied by the pilot to M×N, is expressed by the following formula:

[0197]

[0198] The pilot overhead ratio η is defined as the ratio between the number of resource units in pilot transmission and the total number of resource units in the delay-Doppler domain.

[0199] Table 2 Simulation Parameters

[0200] parameter value carrier frequency 4GHz Number of subcarriers 30 OTFS symbol number 600 Subcarrier spacing 15KHz Channel Model LTE channel model User rate 10-160m / s

[0201] In the simulation process of this application, simulation results for channel estimation of the OTFS system are presented and compared with traditional channel estimation methods, including Orthogonal Matching Pursuit (OMP), 3D-SOMP (3D-Simplified Orthogonal Matching Pursuit), and pulse-based channel estimation methods. A standardized spatial channel model was simulated in the simulation. Detailed system parameters are summarized in Table 2, mainly focusing on the normalized mean square error performance, the number of base station (BS) antennas, signal-to-noise ratio (SNR), number of iterations, algorithm runtime, and user speed.

[0202] Figure 4 The image shows a performance comparison between NMSE and signal-to-noise ratio. The antenna count was set to 64, pilot overhead ratio to 50%, and user speed to 150 m / s. Figure 4 As can be seen, the channel estimation method based on the delay-Doppler domain proposed in this application outperforms the 3D-SOMP channel estimation method for recovering sparse channels. However, this method has high pilot overhead and computational complexity. Traditional pulse-based channel estimation methods suffer from high NMSE due to interference between multiple antennas.

[0203] exist Figure 5The paper presents a performance comparison between NMSE and user speed ν. The pilot overhead ratio is set to (η = 50%), and the signal-to-noise ratio is 5 dB. It can be observed that the channel estimation NMSE gradually decreases as the user speed increases. This is mainly due to two factors: the influence of user speed on the time-varying characteristics of the channel and sampling errors, leading to a significant decrease in NMSE performance. When the user speed reaches 100 m / s, the channel estimation NMSE performance tends to stabilize.

[0204] exist Figure 6 The paper presents a comparison of the NMSE performance of the methods under different numbers of antennas at the base station, with a user speed of 150 m / s. It can be seen that in traditional channel estimation methods, as the number of antennas increases, the NMSE gradually exceeds 10. -1 This is because when the number of BS antennas is large and the pilot overhead ratio is constant, the interval between two adjacent pulses is insufficient. The figure shows that, under the condition of the same number of base station antennas, the proposed channel estimation method based on the delay-Doppler domain has a lower NMSE value and is applicable to a large number of BS antennas.

[0205] exist Figure 7 The paper presents a performance comparison of NMSE and pilot overhead ratio η, with 16 BS antennas and a user speed of 100 m / s. Traditional pulse-based methods suffer from insufficient pilot overhead when the pilot overhead ratio is low, making them susceptible to interference from adjacent pulses and thus reducing the NMSE of the pulse-based channel estimation method. In contrast, the delay-Doppler domain-based channel estimation method in this application uses non-orthogonal pilots, resulting in a significantly lower NMSE than the 3D-SOMP channel estimation method at the same pilot overhead ratio.

[0206] exist Figure 8 The paper presents a comparison between bit error rate (BER) and signal-to-noise ratio (SNR), clearly demonstrating the performance of the channel estimation method under different SNR conditions and the reliability of the system at various SNR levels. Assuming a user's movement speed of 100 meters per second, under this high mobility condition, the proposed method also achieves relatively good BER performance, very similar to its performance under perfect channel state conditions in the OTFS system.

[0207] exist Figure 9 The paper presents a comparison of the calculated NMSE and runtime under different signal-to-noise ratios (SNRs). It can be observed that when the SNR is greater than 10 dB, the proposed delay-Doppler domain-based channel estimation method has a shorter runtime and lower computational complexity compared to the 3D-SOMP channel estimation method. However, the shorter runtime may also affect the estimation accuracy. Therefore, a balance between method complexity and runtime needs to be considered comprehensively.

[0208] exist Figure 10 The figure presents the convergence of the Bayesian-based channel estimation method under different signal-to-noise ratio (SNR) settings. It can be observed from the figure that the NMSE converges after 15-20 iterations, and the required number of iterations is slightly larger at higher SNRs. This is because, in cases with good signal strength, the NMSE achievable by the delay-Doppler domain-based channel estimation method is smaller, requiring more iterations to meet the iteration stopping condition.

[0209] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein.

Claims

1. A channel estimation method based on the delay-Doppler domain, characterized in that, The method includes: Establish a channel model, and construct the channel model into a sparse signal recovery task model based on the structural sparsity of the channel; A factor graph model is established based on the joint probability model and Bayesian methods; wherein, the nodes in the factor graph model include: function nodes, variable nodes, and observation nodes; Calculate the messages passed between the function node and the variable node, and between the function node and the observation node in the factor graph model, according to the message passing rules; The confidence level of each node is updated based on the messages passed between the function node and the variable node, and the messages passed between the function node and the observation node. The variance and mean of the observation nodes are calculated based on the confidence level of each node, and the variance and mean of the observation nodes are iteratively updated to obtain the channel in the sparse signal recovery task model.

2. The channel estimation method based on the delay-Doppler domain according to claim 1, characterized in that, The establishment of the channel model, based on the structural sparsity of the channel, constructs the channel model into a sparse signal recovery task model, including: In an orthogonal time-frequency spatial modulation multiple-input multiple-output system, the time-varying channel consisting of Q dominant propagation paths, with the time-varying channel model of the f-th transmit antenna and the r-th receive antenna, is as follows: （1）； In the formula, , , , Let f represent channel gain, Doppler shift, propagation delay, and phase difference, respectively, where f = 0, 1, ..., N. t 1, r=0,1,...,N r 1; Indicates a delay index. This represents the complex part of the exponential function e. Represents the Doppler index, q=1, ...,Q, In order to be in The band-limited pulse shaping filter response is calculated at point Ts, where Ts represents the system sampling interval.

3. The channel estimation method based on the delay-Doppler domain according to claim 2, characterized in that, The establishment of the channel model, based on the structural sparsity of the channel, constructs the channel model into a sparse signal recovery task model, including: By employing a non-orthogonal pilot mode, pilot signals and data are simultaneously transmitted within orthogonal time-frequency space (OTFS) symbols, transforming a sequence of symbols of length M×N into a two-dimensional data block X. DD X DD ∈ M×N Where M and N represent the number of subcarriers in the frequency domain and the number in the time domain, respectively, and the number of subcarriers transmitted by the f-th transmit antenna is... The symbol, after inverse Fourier transform, is represented as: （2）； In the formula, Denotes the M-point Fourier transform matrix. Let H denote the normalized N-point Fast Fourier Transform matrix, and let H denote the conjugate transpose. Indicates the inverse Fourier transform; The time-domain signal transmitted in OTFS is first applied to the inverse Fourier transform. Then, a Heisenberg transform is performed to obtain the transmission time-domain signal in OTFS. The expression is: （3）； （4）； In the formula, It is a diagonal matrix obtained by sampling the transmitted pulse. This represents the t-th transmitted pulse; Processing the transmission time-domain signal in OTFS, the end-to-end delay-Doppler domain input-output model from the f-th transmit antenna to the r-th receive antenna is obtained as follows: （5）； In the formula, Indicates the received signal; Vectorizing the input-output model of MIMO-OTFS yields the vectorized input-output relationship, expressed as follows: （6）； In the formula, This represents the channel in the time-delay-Doppler domain. It is additive Gaussian random white noise. This represents the delay index exponent at the f-th transmit antenna. This represents the Doppler index at the f-th transmitting antenna; After performing spatial discrete Fourier transform on equations (1) and (6) along the antenna index f, the sparse signal recovery task model is obtained, and its expression is: （7）； y i Here, h represents the measured value, and its dimension is MgNgNt×1. The noise vector, whose dimension can be represented by MgNg×1, is shown below for ease of representation. , This represents a dictionary matrix with dimensions P×C, where... , .

4. The channel estimation method based on the delay-Doppler domain according to claim 3, characterized in that, The establishment of the factor graph model based on the joint probability model and Bayesian methods includes: The joint probability density function in formula (8) is decomposed into factors and further expressed as a hierarchical prior model; （8）； In the formula, i and c belong to the set ; The set of latent variables in the construct factor graph is In the set These also represent variable nodes in the probability factor graph. i To reduce computational complexity, a scalar factor is introduced. This represents an approximate Gaussian distribution function. Represents the gamma distribution. This represents an approximate Gaussian distribution function. Represents the gamma distribution. This indicates a Gaussian distribution.

5. The channel estimation method based on the delay-Doppler domain according to claim 4, characterized in that, The set of function nodes in the factor graph model is defined as a set. ,Will Divided into two parts: and ;in, Represents a set of functions. Represents a set of functions. Represents a function node. Represent any i, , , Partially propagated using BP messages, in Partially utilizes MF variational average field message aggregation.

6. The channel estimation method based on the delay-Doppler domain according to claim 5, characterized in that, The step of calculating the messages passed between the function node and the variable node in the factor graph model, and the messages passed between the function node and the observation node, according to the message passing rules, includes: From function node To the variable node The expression for the message is: （9）； In the formula, Represents variable nodes Confidence level, Represents an exponential function; Using the BP message propagation rules, assuming the variable node h c To function node News Given that it follows a Gaussian distribution, then from the function node to variable node h c News The expression is as follows: （10）； In the formula, Indicates from function node to variable node h c The mean of the message Indicates from function node to variable node h c The variance corresponding to the message.

7. The channel estimation method based on the delay-Doppler domain according to claim 6, characterized in that, The step of calculating the messages passed between the function node and the variable node in the factor graph model, and the messages passed between the function node and the observation node, according to the message passing rules, includes: From function node to variable node h c The mean of the message and variance They are represented as follows: （11）； （12）； In the formula, Represents the perception matrix, The message being conveyed This represents the mean of an approximate Gaussian distribution. This represents the variance of an approximately Gaussian distribution. Represents variable node h c To function node The news Indicates the initial estimate; Assuming from variable node Transmit to function node Prior information Given that the confidence level b(γ) of the random variable node γ is known and all follow a Gaussian distribution, and according to the Central Limit Theorem, we can obtain that from the variable node h... c To function node News The expression is: （13）； （14）； （15）； In the formula, for The adjoint matrix, Denotes the Frobenius norm of a matrix. h represents the variable node c The mean of an approximate Gaussian distribution. The variable node h represents the variable. c The variance of the approximate Gaussian distribution, from the function nodes to variable node h c The expression for the message is: , （16）； In the formula, Represents variable nodes The confidence level.

8. The channel estimation method based on the delay-Doppler domain according to claim 7, characterized in that, The step of updating the confidence level of each node based on the messages passed between the function node and the variable node, and the messages passed between the function node and the observation node, includes: By updating The transmitted message is received by the observation node. confidence level To update the product of messages passed between two nodes: （17）； Among them, observation nodes The observation nodes satisfy a Gaussian distribution. variance and observation nodes mean They are as follows: （18）； （19）； In the formula, Represents variable nodes The mean; The message being conveyed is: （20）； The transmitted message can be approximated by a Gaussian distribution (20) with variance. and mean They are as follows: （21）； （22）； Utilizing the BP message propagation rules again, calculate the function node. To the variable node News And calculate the conditional probability distribution among variables in the probability factor graph model. ; (twenty three); （24）； （25）； According to previous iterations and Multiplication yields variable nodes , The expression is as follows: （26）； （27）； （28）； In the formula, Representing variable nodes respectively The mean and variance of.

9. The channel estimation method based on the delay-Doppler domain according to claim 8, characterized in that, The step of calculating the variance and mean of the observed nodes based on the confidence level of each node, obtaining the approximate posterior distribution of the latent variables, iteratively updating the variance and mean of the observed nodes, and obtaining the channel in the received signal includes: Constructing auxiliary functions And make its divergence with formula (8) and posterior probability. Minimize this, and then obtain the auxiliary function through message updates of the factor graph in subsequent updates. : （29）； in, It is the approximate posterior distribution of the latent variables. In the following derivation, let Indicates in The approximate posterior distribution of q(h) under the expected value needs to be... Multiplying these two messages yields a Gaussian probability density function that approximates the posterior distribution, whose variance matrix and mean are given below: （30）； （31）； （32）； Updates and messages and The product is proportional. As an approximate posterior distribution, its calculation method is given by the following formula: , For constants: （33）； （34）； The approximate posterior distribution function corresponding to the generalized inverse Gaussian distribution can be given in closed form and obtained through formulas (19) and (34). exist The following expectations: （35）； express The second-order modified Bessel function of the second kind The approximate posterior distribution function is represented by having shape parameter J and velocity parameter J. gamma function, The approximate posterior distribution is given by (26) to (28). exist The following expectations: （36）； （37）； （38）； Formulas (30) and (31) are processed in blocks to reduce complexity. In subsequent updates, it is assumed that h is decomposed into Blocks, average value of each block and variance The update can be approximated by an auxiliary probability function: （39）； （40）； In the formula, This represents the conjugate transpose matrix.

10. The channel estimation method based on the delay-Doppler domain according to claim 9, characterized in that, The step of calculating the variance and mean of the approximate posterior distribution function of the observed nodes based on the confidence level of each node and minimizing its divergence, and iteratively updating the variance and mean of the observed nodes to obtain the channel in the received signal, includes: Mean of variable node h and variance as follows: （41）； （42）。

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