A Channel Estimation Method for Broadband Millimeter-Wave MIMO-OFDM Systems Based on Beam Tilt Effect

Abstract

This invention relates to the field of millimeter-wave broadband wireless communication and is a channel estimation method for broadband millimeter-wave MIMO-OFDM systems based on beam tilt effect. It includes the following steps: (1) constructing a model of the MIMO-OFDM system; (2) modeling the channel estimation cost function, reformulating the system model for each subcarrier as a frequency-dependent sparse signal recovery problem; (3) updating the channel gain and noise variance; (4) updating the signal covariance matrix; (5) updating the angle estimate; and (6) reconstructing the channel for each subcarrier using a low-complexity algorithm to obtain high-precision channel estimation results. Compared to traditional schemes, this method achieves more accurate channel estimation results while maintaining acceptable computational complexity.

Description

Technical Field

[0001] This invention relates to the field of millimeter-wave broadband wireless communication and is a channel estimation method for broadband millimeter-wave MIMO-OFDM systems based on beam tilt effect. Background Technology

[0002] Millimeter-wave (mmWave) communication is considered one of the most promising candidate technologies for future wireless communication networks. Specifically, the millimeter-wave band can achieve greater communication bandwidth at higher carrier frequencies, thus providing more than ten times the data transmission rate compared to traditional wireless communication. However, with the dramatic increase in frequency bands and the number of antenna elements, adopting a fully digital architecture design, i.e., allocating a dedicated radio frequency (RF) link to each antenna element, will become prohibitively expensive in terms of hardware cost and power consumption.

[0003] To address this issue, a feasible solution is to employ a hybrid analog and digital architecture. To date, numerous channel estimation schemes have been developed for millimeter-wave massive multiple-input multiple-output (MIMO) systems with hybrid architectures. Specifically, codebook-based channel estimation methods can utilize adaptive compressed sensing techniques. However, the actual angle of arrival may not necessarily lie at discrete grid points; this is commonly referred to as grid mismatch error.

[0004] To address this issue, an iterative weighted channel estimation method was proposed, which utilizes gradient descent to make the estimated grid points approximate the true angles. Furthermore, by leveraging the shared sparse channel structure information among different subcarriers, a method based on Spatial Alternating Generalized Expectation-Maximization (SAGE) was also proposed.

[0005] A significant drawback of the aforementioned channel estimation methods is their failure to account for beam tilt effects, stemming from the frequency dependence of millimeter-wave broadband channels. To address this issue, a broadband channel estimation method based on Continuous Support Detection (SSD) has been proposed. By leveraging a unique frequency-correlated sparse structure, spatial orientation can be estimated by approximately capturing the total power contributed by each subcarrier. However, these methods still suffer from grid mismatch issues, which can lead to severe performance degradation. Summary of the Invention

[0006] This invention simultaneously considers the grid mismatch problem and beam tilt effect, proposing a novel frequency-dependent dictionary channel estimation method. Specifically, by utilizing the angular sparse structure of the broadband channel, a frequency-dependent overcomplete dictionary is designed for each subcarrier. Then, a certain sparse prior distribution is used to model the joint spatial sparsity of the entire broadband channel, resulting in a joint sparse signal recovery problem. To solve this non-convex optimization problem, a block alternation optimization method is used to recombine the original problem into a series of parameter estimation subproblems, each of which can be efficiently solved. Notably, the positions of grid points are treated as adjustable variables to reduce discrete estimation errors, allowing for iterative optimization of the corresponding dictionary. However, these iterative updates can lead to high computational complexity. To further reduce the computational burden, a threshold method is used to remove grid points whose contribution to channel reconstruction is negligible.

[0007] Technical solution of the present invention:

[0008] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized by the following steps:

[0009] Step 1: Construct a model of the orthogonal frequency division multiplexing (MIMO-OFDM) system;

[0010] Step 2: Model the channel estimation cost function, and reformulate the system model of each subcarrier as a frequency-dependent sparse signal recovery problem;

[0011] Step 3: Update the channel gain and noise variance;

[0012] Step 4: Update the signal covariance matrix;

[0013] Step 5: Update the angle estimate.

[0014] Step 6: Reconstruct the channel of each subcarrier using a low-complexity algorithm to obtain high-precision channel estimation results.

[0015] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that step 1 includes the following steps:

[0016] Step 1.1, consider an uplink time-division duplex (TDD) millimeter-wave multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) system, where the base station (BS) uses N antennas and N... RFTo simultaneously serve K single-antenna users using multiple radio frequency links, this invention considers the Saleh-Valenzuela channel model, which is widely used in the frequency domain. Specifically, the channel for a user on subcarrier m, m = 1, 2, ..., M, can be represented as:

[0017]

[0018] Where L is the number of distinguishable paths, α l and τ l These are the path gain and time delay of the l-th path, respectively; Let f represent the frequency of subcarrier m, where f c and f s These are the center carrier frequency and the system bandwidth (sampling rate), respectively, a(φ) l,m ) represents the array response; for a typical uniform linear array (ULA) with N elements, a(φ) l,m ) can be represented as here, It is the spatial direction of the incoming signal of the l-th path on subcarrier m, where c represents the speed of light, and θ l It represents the physical direction of the l-th path, where d represents the antenna spacing, designed as half the wavelength of the center carrier frequency, i.e. It is important to note that, unlike the traditional array response vector in the narrowband case, the broadband array response vector a(φ) l,m It is frequency-dependent;

[0019] Step 1.2: In a typical TDD system, users need to send pilot sequences to the base station (BS) for channel estimation. Although the high-frequency millimeter-wave band shortens the channel coherence time, the channel can still be considered approximately constant over multiple consecutive sampling periods due to the large bandwidth. In this invention, it is assumed that K users send orthogonal pilot signals, which allows channel estimation for each user to be performed independently; to simplify the problem, this invention discusses the channel estimation problem under specific user conditions; specifically, using s m,q This represents the known pilot signal transmitted on subcarrier m in training frame q, which, after being combined at the receiver, results in N at the base station. RF ×1 Received sample y m,q It can be written as:

[0020]

[0021] Where, n m,q and Let n represent additive noise and signal pre-encoder, respectively; assuming noise n m,q To be related to signal s m,qIndependent and conforming to a Gaussian distribution η is the covariance matrix parameter of the noise; due to limitations in the analog hardware architecture, a frequency-flat hybrid combiner W is used. q (Fixed on different subcarriers) for processing;

[0022] Step 1.3, by transmitting y for Q consecutive pilot signals m,q The values ​​q = 1, 2, ..., Q are stacked into a vector to obtain the overall observation results. As shown below:

[0023]

[0024] For the sake of simplicity, let's assume s m,q =1,q =1,2,…,Q, and define and These are used as augmented preencoder and noise vector, respectively.

[0025] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that step 2 includes the following steps:

[0026] Step 2.1, by utilizing the inherent structure of the millimeter-wave channel, the channel vector hm in step 1.1 is re-expressed as...

[0027]

[0028] Where A m (θ)=[a(φ 1,m ),a(φ 2,m ),…,a(φ L,m )],θ=[θ1,θ2,…,θ L ] T Specifically, θ l L = 1, 2, ..., L represents the physical direction of the l-th path, embedded in φ l,m In, that is also, It is a vector of channel gain.

[0029] Step 2.2, in order to obtain the observed result y m (See step (1.3) for recovering θ and This problem is addressed using compressed sensing technology. Specifically, by leveraging the angular sparsity of the channel vector, h is approximated. m for:

[0030]

[0031] in, It is through a complete dictionary. is the sparse channel gain vector, where G >> L.

[0032] Step 2.3, Define Where ψ = [ψ1, ψ2, ..., ψ G ] T It is a set of sampled grids that covers the potential angular domain. Therefore, the system model in step 1.3 is re-represented as:

[0033]

[0034] in, It is an equivalent perception matrix.

[0035] Step 2.4, assuming that the noise contained in the q-th frame of subcarrier m follows a Gaussian distribution. Therefore, the augmented noise vector is obtained. The distribution is in Therefore, {y m The likelihood function of} can be expressed as:

[0036]

[0037] in,

[0038] Step 2.5, The maximum likelihood estimate is the minimum weighted l2 norm solution of the formula in step 1.4. However, such a solution is non-sparse because it neglects H. v The row sparse structure property. To address this issue, a sparse prior distribution is used to describe H. v Its special structure. Therefore, assume... The following forms are obtained:

[0039]

[0040] Among them, Λ G =diag{λ}, λ=[λ1,λ2,…,λ G ] T By sharing the channel gain covariance matrix Λ G For subcarriers of different frequencies A joint sparsity constraint was applied.

[0041] Step 2.6, combining steps 2.4 and 2.5, expresses the channel estimation problem as:

[0042]

[0043] Step 2.7, in order to alleviate the burden on the overcomplete dictionary {B} m The grid mismatch error caused by the fixed sampling grid point constraint is addressed by assuming the sampling grid set ψ is not predefined and is kept updated during iterative parameter optimization. Taking the negative logarithm of the objective function in step 2.6 and discarding irrelevant constant terms yields the following optimization:

[0044]

[0045] Step 2.8: The above optimization structure can be solved by alternating minimization. That is, the variables of one block are updated each time, while the variables of other blocks remain unchanged.

[0046] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that step 3 includes the following steps:

[0047] Step 3.1, the objective of this invention is to obtain the measurement result {y} m The physical direction θ and the corresponding path gain information are estimated from the m = 1, 2, ..., M. Therefore, this invention updates... The problem begins. Under the condition of fixed variables η, λ, and ψ, optimize... The subproblem simplifies to:

[0048]

[0049] in

[0050] Step 3.2 shows that under different subcarriers... The optimization of this is an independent weighted least squares problem, which can be updated sequentially. Therefore, the update... The objective has been restated as:

[0051]

[0052] Step 3.3, given η, λ, and ψ, yields a unique minimized value:

[0053]

[0054] Step 3.4, next, consider the subproblem concerning the noise variance η, by fixing λ and ψ are used for optimization, that is:

[0055]

[0056] Step 3.5, minimizing the objective function from step 3.4 with respect to η yields:

[0057]

[0058] The constraint η≥0 is automatically satisfied here because of the weighted least squares terms. It is not negative.

[0059] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that step 4 includes the following steps:

[0060] Step 4.1, consider other blocks (i.e. Given that η and ψ are fixed, the optimization problem of λ can be rewritten as follows: The optimization in step 2.7 can be rewritten as...

[0061]

[0062] Where λ=[λ1,λ2,…,λ G ] T The condition λ≥0 must hold true element by element.

[0063] Step 4.2, each element λ g For g = 1, 2, ..., G, update in the following order:

[0064]

[0065] Step 4.3, this subproblem has a closed-form solution, namely:

[0066]

[0067] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that step 5 includes the following steps:

[0068] Step 5.1: In this step, we consider updating ψ. It can be observed from the formula in Step 2.7 that the cost function is highly nonlinear and nonconvex with respect to ψ because the objective ψ of this invention resides in the exponential term of the pointing vector in the formula in Step 2.2. Therefore, unlike other variable blocks, a closed-form solution for updating ψ cannot be found directly.

[0069] Step 5.2: To address this problem, although this invention cannot directly obtain an analytical solution, it can find a more accurate channel estimate through gradient optimization to ensure that the cost function maintains its non-increasing characteristic during iteration. This goal is easily achieved through gradient optimization. Specifically, first, the formula in step 3.3 is substituted into the formula in step 2.7 to accelerate convergence. Then, for fixed η and λ, the optimization problem of ψ can be equivalently expressed as:

[0070]

[0071] Where ψ is embedded in the overcomplete dictionary {B} m}middle.

[0072] Step 5.3, because and Define the cost function in the formula of step 5.2 as f(ψ), and further simplify it to:

[0073]

[0074] in

[0075] In the following content, for ease of explanation, we will use... to indicate

[0076] Step 5.4, iterative optimization of ψ for each subcarrier leads to very high computational complexity, especially when the number of subcarriers is large. Therefore, this invention further divides all M subcarriers into P groups, each containing... With one subcarrier, only P dictionaries are needed. To represent the entire overcomplete dictionary

[0077] Step 5.5, to achieve this, each dictionary Designed to approximate the dictionary related to different frequencies in the p-th group, i.e., the wavelength λ of the p-th dictionary. p The average wavelength of these subcarriers selected as the p-th group, i.e., the subcarrier index. in In other words, the group index of the m-th subcarrier is determined by... Given that m = 1, 2, ..., M.

[0078] Step 5.6, considering the above approximation, the objective function in step 5.3 is re-expressed as:

[0079]

[0080] Step 5.7: The above equation can be efficiently minimized using existing methods (e.g., gradient descent). The goal is to find a new ψ. (r+1) Ensure f(ψ) (r+1) )≤f(ψ (r) ), where r represents the iterative index.

[0081] Step 5.8: Since f(ψ) is differentiable with respect to ψ, it is recommended to use gradient descent to update the parameter estimation results. Specifically, the update of ψ can be expressed as:

[0082]

[0083] Where μ (r) It's the step length.

[0084] Step 5.9, in order to calculate The gradient is first defined. in Then, using the chain rule, we can obtain the gradient of f(ψ) with respect to ψ:

[0085]

[0086] in,

[0087]

[0088]

[0089] as well as

[0090]

[0091] in,

[0092]

[0093] A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that step 6 includes the following steps:

[0094] Step 6.1, the sampling grid set is initialized using the following uniform sampling method, namely... The general guideline for choosing G is to let G >> K, which results in a finer initial mesh set, making it less likely that the search for the true physical direction θ will get stuck in an undesirable local minimum.

[0095] Step 6.2, however, on the one hand, a large value of G may lead to excessive computational burden, because the main computational task of the proposed method is caused by the inverse of the following G×G matrix: On the other hand, a denser sampling grid increases the interrelationships between the overcomplete dictionary columns, which may lead to a significant performance degradation of the proposed scheme, rendering it ineffective. To eliminate these negative effects, it is noted that the update process does not need to update the entire sampling grid point set λ, since λ is a sparse vector. Therefore, only the elements with non-zero corresponding values ​​in λ need to be updated iteratively.

[0096] Step 6.3, specifically, involves pruning the active grid point set by thresholding the elements in λ, that is, removing the smaller components in λ and their corresponding grid points in ψ. This reduces computational complexity as the number of active grid points decreases. Mathematically, as the iteration process progresses, the sampled grid point set can be reduced through ψ. Ω =ψ({g∣γ g ≥α th}) to prune, where Ω represents the set of indices corresponding to the pruning grid point set, α th It is a dictionary used to remove those elements whose contribution to the signal required for recovery is negligible. The threshold of the components in.

[0097] Step 6.4, with the pruning of potential grid point candidates, only λ and λ need to be updated. The relatively large elements in the equation. As iterative optimization proceeds, the estimated parameters ψ and It can be derived from a coarse initialization to the actual physical direction θ and path gain. Further refinement is needed.

[0098] Step 6.5: Based on these estimates, reconstruct the channel for each subcarrier using the formula from Step 2.1.

[0099] Beneficial effects:

[0100] This invention first remodels the system model for each subcarrier as a unique frequency-dependent sparse signal recovery problem. Then, by utilizing the joint sparse structure of the subcarriers, this invention proposes an iterative minimization method to estimate the channel parameters. The proposed method reduces the objective function using a block-alternating optimization approach, achieving more accurate channel estimation results compared to traditional schemes, while maintaining acceptable computational complexity. Attached Figure Description

[0101] Figure 1 This is a comparison chart of NMSE performance of the method of the present invention under different signal-to-noise ratios.

[0102] Figure 2 This is a comparison chart of NMSE performance of the method of the present invention under different snapshot numbers Q.

[0103] Figure 3 The graph showing the relationship between computational complexity and Q in the method of this invention. Detailed Implementation

[0104] This invention presents a channel estimation method for broadband millimeter-wave MIMO-OFDM systems based on beam tilt effects. It is a channel estimation scheme that estimates channel parameters with high accuracy through iterative minimization. Specifically, this invention first remodels the system model for each subcarrier as a unique frequency-dependent sparse signal recovery problem. Then, by utilizing the joint sparse structure of the subcarriers, this invention proposes an iterative minimization method to estimate the channel parameters. The proposed method reduces the objective function using a block-alternating optimization approach, achieving more accurate channel estimation results compared to traditional schemes, while maintaining acceptable computational complexity.

[0105] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0106] First, establish the system model:

[0107] Step 1. Establish a system model

[0108] Step (1.1) considers an uplink time-division duplex (TDD) millimeter-wave multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) system, where the base station (BS) uses N antennas and N... RF This invention utilizes multiple radio frequency links to simultaneously serve K single-antenna users. It considers the Saleh-Valenzuela channel model, widely used in the frequency domain. Specifically, the channel for a user on subcarriers m, m = 1, 2, ..., M, can be represented as:

[0109]

[0110] Where L is the number of distinguishable paths, α l and τ l These are the path gain and time delay of the l-th path, respectively. Let f represent the frequency of subcarrier m, where f c and f s These are the center carrier frequency and the system bandwidth (sampling rate), respectively, a(φ) l,m ) represents the array response. For example, for a typical uniform linear array (ULA) with N elements, a(φ) l,m ) can be represented as

[0111] here, It is the spatial direction of the incoming signal of the l-th path on subcarrier m, where c represents the speed of light, and θ l It represents the physical direction of the l-th path, and d represents the antenna spacing, which is usually designed to be half the wavelength of the center carrier frequency, i.e. It is important to note that, unlike the traditional array response vector in the narrowband case, the broadband array response vector a(φ)l,m It is frequency-dependent.

[0112] Step (1.2): In a typical TDD system, users need to send pilot sequences to the base station (BS) for channel estimation. Although the high-frequency millimeter-wave band shortens the channel coherence time, the channel can still be considered approximately constant over multiple consecutive sampling periods due to the large bandwidth. For example, this invention assumes that a user with a center carrier frequency of 60 GHz, a bandwidth of 2 GHz, and a maximum speed of 40 m / s will generate a coherence time of 0.125 ms. However, the sampling duration is on the order of 0.5 ns, which ensures that sufficient pilot samples are available for channel estimation. In this invention, it is assumed that K users send orthogonal pilot signals, which allows channel estimation for each user to be performed independently. To simplify the problem, this invention discusses the channel estimation problem under specific user conditions. Specifically, using s m,q This represents the known pilot signal transmitted on subcarrier m in training frame q, which, after being combined at the receiver, results in N at the base station. RF ×1 Received sample y m,q It can be written as:

[0113]

[0114] Where, n m,q and Let n represent additive noise and the signal pre-encoder, respectively. m,q To be related to signal s m,q Independent and conforming to a Gaussian distribution η is the covariance matrix parameter of the noise. Due to limitations in the analog hardware architecture, a frequency-flat hybrid combiner W is used. q (Fixed on different subcarriers) for processing.

[0115] Step (1.3) involves transmitting y through Q consecutive pilot signals. m,q The values ​​q = 1, 2, ..., Q are stacked into a vector to obtain the overall observation results. As shown below:

[0116]

[0117] For the sake of simplicity, let's assume s m,q =1,q =1,2,…,Q, and define and These are used as augmented preencoder and noise vector, respectively.

[0118] The following is the broadband channel estimation algorithm proposed in this invention:

[0119] Step 2. Problem Statement

[0120] Step (2.1), by utilizing the inherent structure of the millimeter-wave channel, the channel vector h in step (1.1) m It can be re-represented as

[0121]

[0122] Where A m (θ)=[a(φ 1,m ),a(φ 2,m ),…,a(φ L,m )],θ=[θ1,θ2,…,θ L ] T Specifically, θ l L = 1, 2, ..., L represents the physical direction of the l-th path, embedded in φ l,m In, that is also, It is a vector of channel gain.

[0123] Step (2.2), in order to obtain the observed result y m (See step (1.3) for recovering θ and This problem can be addressed using compressed sensing technology. Specifically, by leveraging the angular sparsity of the channel vector, h can be approximated. m for:

[0124]

[0125] in, It is through a complete dictionary. is the sparse channel gain vector, where G >> L.

[0126] Step (2.3), define Where ψ = [ψ1, ψ2, ..., ψ G ] T It is a set of sampled grids that covers the potential angular domain. Therefore, the system model in step (1.3) can be reformulated as:

[0127]

[0128] in, It is an equivalent perception matrix.

[0129] Step (2.4) assumes that the noise contained in the q-th frame of subcarrier m follows a Gaussian distribution. Therefore, the augmented noise vector can be obtained. The distribution is in Therefore, {ym The likelihood function of} can be expressed as:

[0130]

[0131] in,

[0132] Step (2.5) is easy to see. The maximum likelihood estimate is the minimum weighted l2 norm solution of the formula in step (1.4). However, such a solution is non-sparse because it ignores H. v The row sparse structure property of H. To address this issue, a sparse prior distribution can be used to describe H. v Its special structure. Therefore, assume... The following forms are obtained:

[0133]

[0134] Among them, Λ G =diag{λ}, λ=[λ1,λ2,…,λ G ] T By sharing the channel gain covariance matrix Λ G For subcarriers of different frequencies A joint sparsity constraint was applied.

[0135] Step (2.6), combined with steps (2.4) and (2.5), allows the channel estimation problem to be expressed as:

[0136]

[0137] Step (2.7), in order to alleviate the overcomplete dictionary {B m The grid mismatch error caused by the fixed sampling grid point constraint in} is assumed to be updated during the iterative parameter optimization process, assuming the sampling grid set ψ is not predefined. Taking the negative logarithm of the objective function in step (2.6) and discarding irrelevant constant terms, the following optimization can be obtained:

[0138]

[0139] Step (2.8), intuitively speaking, the well-structured optimization described above allows for a solution through alternating minimization. That is, the variables of one block can be updated at a time while keeping the other blocks unchanged.

[0140] Step 3. Update scheme for channel gain and noise variance

[0141] Step (3.1), the objective of this invention is to obtain the measurement result {y} mThe physical direction θ and the corresponding path gain information are estimated from the m = 1, 2, ..., M. Therefore, this invention updates... The problem begins. Under the condition of fixed variables η, λ, and ψ, optimize... The subproblem simplifies to:

[0142]

[0143] in

[0144] Step (3.2) shows that under different subcarriers The optimization of this is an independent weighted least squares problem, which can be updated sequentially. Therefore, the update... The objective has been restated as:

[0145]

[0146] Step (3.3), given η, λ, and ψ, yields a unique minimized value:

[0147]

[0148] Step (3.4), next, consider the subproblem concerning the noise variance η, by fixing λ and ψ are used for optimization, that is:

[0149]

[0150] Step (3.5) minimizes the objective function (in step (3.4)) with respect to η to obtain:

[0151]

[0152] The constraint η≥0 is automatically satisfied here because of the weighted least squares terms. It is not negative.

[0153] Step 4. λ update scheme

[0154] Step (4.1), now we discuss other blocks (i.e. Given that η and ψ are fixed, the optimization problem of λ can be specifically rewritten as follows:

[0155]

[0156] Where λ=[λ1,λ2,…,λ G ] T The condition λ≥0 must hold true element by element.

[0157] In step (4.2), we can see that each element λ g For g = 1, 2, ..., G, updates can be performed in the following order:

[0158]

[0159] In step (4.3), this subproblem has a closed-form solution, namely:

[0160]

[0161] Step 5. Update scheme for ψ

[0162] Step (5.1) involves considering the update of ψ. It can be observed from the formula in step (2.7) that the cost function is highly nonlinear and nonconvex with respect to ψ because the objective ψ of this invention resides in the exponential term of the pointing vector in the formula of step (2.2). Therefore, unlike other variable blocks, a closed-form solution for updating ψ cannot be found directly.

[0163] Step (5.2): To address this problem, although the present invention cannot directly obtain an analytical solution, it can find a more accurate channel estimate through gradient optimization to ensure that the cost function maintains its non-increasing characteristic during iteration. This goal is easily achieved through gradient optimization. Specifically, firstly, the formula in step (3.3) is substituted into the formula in step (2.7) to accelerate the convergence speed. Then, for fixed η and λ, the optimization problem of ψ can be equivalently expressed as:

[0164]

[0165] Where ψ is embedded in the overcomplete dictionary {B} m}middle.

[0166] Step (5.3), because and Defining the cost function in the formula of step (5.2) as f(ψ), it can be further simplified to:

[0167]

[0168] in

[0169] In the following content, for ease of explanation, we will use... to indicate

[0170] Step (5.4) shows that iterative optimization of ψ for each subcarrier leads to very high computational complexity, especially when the number of subcarriers is large, for example, hundreds of subcarriers. Fortunately, it is noted that the frequency-dependent array response vectors can be very similar over a small range of consecutive subcarriers across a large bandwidth. Inspired by this, the present invention further divides all M subcarriers into P groups, each group containing With one subcarrier, only P dictionaries are needed. To represent the entire overcomplete dictionary

[0171] Step (5.5), to achieve this, each dictionary It should be cleverly designed to ensure that it can approximately represent the dictionaries related to different frequencies in the p-th group. Therefore, the wavelength λ of the p-th dictionary... p The average wavelength of these subcarriers selected as the p-th group, i.e., the subcarrier index. in In other words, the group index of the m-th subcarrier is determined by... Given that m = 1, 2, ..., M.

[0172] In step (5.6), considering the above approximation, the objective function in step (5.3) can be reformulated as:

[0173]

[0174] Step (5.7) shows that the above equation can be efficiently minimized using existing methods (e.g., gradient descent). The goal is to find a new ψ. (r+1) Ensure f(ψ) (r+1) )≤f(ψ (r) ), where r represents the iterative index.

[0175] In step (5.8), since f(ψ) is differentiable with respect to ψ, it is recommended to use gradient descent to update the parameter estimation results. Specifically, the update of ψ can be expressed as:

[0176]

[0177] Where μ (r) It's the step length.

[0178] Step (5.9), in order to calculate The gradient is first defined. in Then, using the chain rule, we can obtain the gradient of f(ψ) with respect to ψ:

[0179]

[0180] in,

[0181]

[0182]

[0183] as well as

[0184]

[0185] in,

[0186]

[0187] Step 6. Implementation of a low-complexity algorithm

[0188] In step (6.1), the sampling grid set can be initialized using the following uniform sampling method, namely... The general guideline for choosing G is to let G >> K, which results in a finer initial mesh set, making it less likely that the search for the true physical direction θ will get stuck in an undesirable local minimum.

[0189] Step (6.2), however, on the one hand, a large value of G may lead to excessive computational burden, because the main computational task of the proposed method is caused by the inverse of the following G×G matrix: On the other hand, a denser sampling grid increases the interrelationships between the overcomplete dictionary columns, which may lead to a significant performance degradation of the proposed scheme, rendering it ineffective. To eliminate these negative effects, it is noted that the update process does not need to update the entire sampling grid point set λ, since λ is a sparse vector. Therefore, only the elements with non-zero corresponding values ​​in λ need to be updated iteratively.

[0190] Step (6.3) specifically involves pruning the active grid point set by thresholding the elements in λ, i.e., deleting the smaller components in λ and their corresponding grid points in ψ. This reduces computational complexity as the number of active grid points decreases. Mathematically, as the iteration process progresses, the sampled grid point set can be reduced through ψ. Ω =ψ({g∣γ g ≥α th}) to prune, where Ω represents the set of indices corresponding to the pruning grid point set, α th It is a dictionary used to remove those elements whose contribution to the signal required for recovery is negligible. The threshold of the components in.

[0191] In step (6.4), with the pruning of potential grid point candidates, only λ and λ need to be updated. The relatively large elements in the equation. As iterative optimization proceeds, the estimated parameters ψ and It can be derived from a coarse initialization to the actual physical direction θ and path gain. Further refinement is needed.

[0192] In step (6.5), based on these estimates, the channel for each subcarrier can be reconstructed using the formula in step (2.1).

[0193] Figure 1 The NMSE performance comparison is shown under different signal-to-noise ratios (SNRs), with pilot transmission using Q=8 time points. It can be seen that SOMP and the SAGE-based method perform poorly across the entire SNR range. This is because these two methods are based on the assumption of joint sparsity of the subcarrier channel, which is not strictly valid due to beam tilt effects. On the other hand, the SSD and SS-SW-OMP algorithms are much more accurate than SOMP and the SAGE-based method because they fully utilize the sparse structure of the wideband channel. Finally, by employing a grid-point refinement strategy, it can be found that the proposed method achieves the minimum NMSE.

[0194] Figure 2 The NMSE performance is shown by comparing different time-number Qs with a fixed SNR of 5dB. Figure 2 You can see and Figure 1 Similar to the trend observed in other methods, the proposed method significantly outperforms the others. Furthermore, it is noted that even when Q is large, SOMP and SAGE-based methods fail to work effectively due to severe beam tilt issues.

[0195] exist Figure 3 In this paper, the computational complexity of the tested algorithms was investigated as Q increased. The SNR was set to 5 dB, and Q was increased from 8 to 18. It was observed that the proposed method produced the highest computational complexity among the tested methods, yet it remained within an acceptable range. This is because the proposed method requires more iterations to satisfy the stopping condition.

[0196] Furthermore, as Figure 1 and Figure 2 As shown, the estimation accuracy of the four comparison methods is not as good as that of the method of the present invention.

Claims

1. A channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect, characterized in that, Includes the following steps: Step 1: Construct a model of the orthogonal frequency division multiplexing (MIMO-OFDM) system; Step 2: Model the channel estimation cost function, and reformulate the system model of each subcarrier as a frequency-dependent sparse signal recovery problem; Step 3: Update the channel gain and noise variance; Step 4: Update the signal covariance matrix; Step 5: Update the angle estimate. Step 6: Reconstruct the channel of each subcarrier using a low-complexity algorithm to obtain high-precision channel estimation results; Step 1 includes the following steps: Step 1.1, consider an uplink time-division duplex millimeter-wave multiple-input multiple-output orthogonal frequency division multiplexing system, where the base station uses N antennas and To simultaneously serve K single-antenna users using multiple radio frequency links, consider the widely used Saleh-Valenzuela channel model in the frequency domain. Specifically, for a given user on a subcarrier... The channel on can be represented as: in It is the number of distinguishable paths. and They are the first The path gain and time delay of each path; Indicates subcarrier The frequency of, among which and These are the center carrier frequency and the system bandwidth (sampling rate), respectively. Represents the array response; for those with A typical uniform linear array ULA with 1 element, It can be represented as ; here, It is the first The path in the subcarrier The spatial direction of the incoming wave signal, where Represents the speed of light. It is the first The physical direction of the path, This indicates the antenna spacing, designed to be half the wavelength of the center carrier frequency, i.e. It is important to note that, unlike the traditional array response vector in the narrowband case, the broadband array response vector... It is frequency-dependent; Step 1.2: In a typical TDD system, the user needs to send pilot sequences to the base station for channel estimation, letting... Each user sends orthogonal pilot signals, allowing channel estimation for each user to be performed independently; Represents training frames In subcarrier The previously known pilot signals transmitted are combined at the receiving end, and the result is obtained at the base station. Received samples Written as: in, and These represent additive noise and the signal pre-encoder, respectively; assuming noise To be with the signal Independent and conforming to a Gaussian distribution , These are the parameters of the noise covariance matrix; due to limitations in the analog hardware architecture, a frequency-flat hybrid combiner is used. Process it; Step 1.3, by continuously Pilot transmission Stacking them into a vector form yields the overall observation results. As shown below: and define and These are used as augmented preencoder and noise vector, respectively; Step 2 includes the following steps: Step 2.1, by utilizing the inherent structure of the millimeter-wave channel, the channel vector in Step 1.1... Represented as in , Specifically, Indicates the first The physical direction of each path is embedded in In, that is ;also, It is a vector of channel gain; Step 2.2, in order to obtain the observation results China Resumption and This problem can be addressed using compressed sensing technology; specifically, by leveraging the angular sparsity of channel vectors, it can be approximated. for: in, It is through a complete dictionary. It is a sparse channel gain vector, where ; Step 2.3, Define ,in It is a set of sampled grids that covers the potential angular domain; therefore, the system model in step 1.3 is re-represented as: in, It is an equivalent perception matrix; Step 2.4, on the subcarrier The The noise contained in the frame follows a Gaussian distribution. Therefore, the augmented noise vector is obtained. The distribution is ;therefore, The likelihood function is expressed as: in, ; Step 2.5, The maximum likelihood estimate is the minimum weighted average of the formula in step 1.

4. Normative solutions; however, such solutions are non-sparse because they neglect norms. The row sparse structure property; to address this issue, a sparse prior distribution is used to describe it. The special structure; This yields the following form: in, , By sharing the channel gain covariance matrix For subcarriers of different frequencies A joint sparsity constraint was applied; Step 2.6, combining steps 2.4 and 2.5, expresses the channel estimation problem as: Step 2.7, to mitigate the impact of overcomplete dictionaries Mesh mismatch error caused by fixed sampling grid point limitation, sampling grid set Not predefined, it is kept updated during the iterative parameter optimization process; taking the negative logarithm of the objective function in step 2.6 and discarding irrelevant constant terms, the following optimization is obtained: Step 2.8: The above optimization structure is solved by alternating minimization, that is, the variables of one block are updated each time, while the variables of other blocks remain unchanged; Step 3 includes the following steps: Step 3.1, from the measurement results Estimating physical direction And the corresponding path gain information, from the update The problem begins with fixed variables. Under the condition of optimization The subproblem simplifies to: in ; Step 3.2 shows that under different subcarriers... The optimization of this is an independent weighted least squares problem, which can be updated sequentially; therefore, the update... The objective has been restated as: Step 3.3, in the given and Under these circumstances, a unique minimized value can be obtained: Step 3.4, next, consider the noise variance. The subproblems, through fixed and To optimize, namely: Step 3.5, relative to the objective function in step 3.4 Minimize to get: The constraints here It is automatically satisfied because of the weighted least squares term. It is not negative; Step 4 includes the following steps: Step 4.1, consider its Under fixed conditions, The optimization problem is to rewrite the optimization in step 2.7 as follows: in ,Require Element-wise, it holds true. Step 4.2, each element Update in the following order: Step 4.3, this subproblem has a closed-form solution, namely: Step 5 includes the following steps: Step 5.1, in this step, consider the following: The update, as can be observed from the formula in step 2.7, is that the cost function relative to... It is highly nonlinear and nonconvex, the target It exists in the exponential term pointing to the vector in the formula of step 2.2; therefore, unlike other variable blocks, a closed-form solution for updating ψ cannot be found directly. Step 5.2: First, substitute the formula from step 3.3 into the formula from step 2.7 to accelerate the convergence speed; then, for a fixed... and , The optimization problem is equivalently represented as: in Embedded in an overcomplete dictionary middle; Step 5.3, because and blkdiag The cost function in the formula of step 5.2 is defined as... And further simplified to: in , ; For ease of explanation in the following content, we will use to indicate ; Step 5.4, put all of them Each subcarrier is divided into Groups, each group contains Only one subcarrier is needed, so that only one subcarrier is required. A dictionary To represent the entire overcomplete dictionary ; Step 5.5, for each dictionary Designed to approximate the first The dictionary related to different frequencies in the group, i.e., the first wavelength of a dictionary Selected as the The average wavelength of these subcarriers in the group, i.e., the subcarrier index. ,in In other words, the group index of the m-th subcarrier is determined by... ; The objective function in steps 5.6 and 5.3 is re-expressed as follows: Step 5.7: The above equation is minimized using gradient descent; the goal is to find a new... ,make sure ,in Indicates an iterative index; Step 5.8, because )about Differentiable, the parameter estimation results are updated using gradient descent; specifically, The update is represented as: in It is the step size; Step 5.9, in order to calculate The gradient is first defined. ,in Then, using the chain rule, we can obtain... about gradient: in, as well as in, 。 2. The channel estimation method for a broadband millimeter-wave MIMO-OFDM system based on beam tilt effect as described in claim 1, characterized in that, Step 1 includes the following steps: Step 6.1, the sampling grid set is initialized using the following uniform sampling method, namely... ; Step 6.2, for those in Update and iterate on the elements with non-zero values; Step 6.3, specifically, select by... Thresholding is applied to the elements in the data to prune the active grid point set, that is, to delete those... Components smaller than a set threshold and their values ​​in The relevant grid points in the sample; the sampled grid point set is obtained through... To prune, among which This represents the set of indices corresponding to the trimmed grid point set. It is a dictionary used to remove those elements whose contribution to the signal required for recovery is negligible. The threshold of the components in; Step 6.4, with the pruning of potential grid point candidates, only updates are needed. and Elements in the set threshold that are greater than the specified threshold; as iterative optimization proceeds, the estimated parameters... and It can move from a rough initialization to the actual physical direction. and path gain To refine; Step 6.5: Based on these estimates, reconstruct the channel for each subcarrier using the formula from Step 2.

1. .

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