A sparse variational DOA estimation method for phase missing
By introducing phase latent variables and sparse variational Bayesian inference methods, the computational complexity problem caused by phase loss in incoherent array systems is solved, and efficient DOA estimation is achieved in incoherent array systems.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- UNIV OF ELECTRONICS SCI & TECH OF CHINA
- Filing Date
- 2026-03-26
- Publication Date
- 2026-06-12
AI Technical Summary
In incoherent array systems, due to the lack of phase consistency between receiving channels, existing sparse Bayesian inference methods struggle to maintain analytical form under conditions where only amplitude observations are available, resulting in high computational complexity and insufficient algorithm convergence and stability.
By introducing a phase latent variable and employing a sparse variational Bayesian inference method, the phase, sparse signal matrix, and parameters of the received signal are updated iteratively. Through the likelihood function and prior constraints, DOA estimation is achieved.
It improves the stability and resolution of estimation in multi-target, low signal-to-noise ratio scenarios, simplifies computational complexity, and achieves efficient DOA estimation.
Smart Images
Figure CN122194048A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of array signal processing, specifically relating to a sparse variational DOA estimation method for phase loss. Background Technology
[0002] Array signal processing technology is widely used in radar, sonar, wireless communication, and passive sensing, among which DOA estimation is a core problem. Traditional DOA estimation methods are usually based on coherent array models, which can use the phase difference relationship between array elements to estimate parameters. However, in practical engineering systems, due to limitations such as hardware cost, system structure, or operating environment, the array receiving channels cannot always maintain phase consistency. In this case, there is no longer a stable and usable phase reference between the array output signals.
[0003] The array systems described above, lacking inter-channel phase consistency, are typically referred to as incoherent arrays. In incoherent arrays, the amplitude information received by the array elements can still be stably measured; therefore, signal processing problems in incoherent arrays often manifest as parameter estimation problems primarily based on amplitude observations. In recent years, some methods have attempted to introduce sparse representations or Bayesian learning into incoherent DOA estimation. However, under conditions where only amplitude observations are available, the statistical model corresponding to the received signal usually exhibits Ricean distribution characteristics, making it difficult for existing sparse Bayesian inference methods to maintain analytical form. This often necessitates reliance on numerical optimization, approximate inference, or heuristic algorithms. This not only increases computational complexity but may also lead to insufficient convergence and stability of the algorithms. Summary of the Invention
[0004] To address the aforementioned problems, this invention proposes a DOA estimation method for phase loss in incoherent arrays. When constructing the statistical model of the received signal of the incoherent array, the unobservable phase component in the received signal of the array elements is treated as a random uncertainty and introduced into the model, thus simplifying the likelihood function of the received signal. Subsequently, sparse variational Bayesian inference is used to update and iterate the phase of the received signal, the sparse signal matrix, the signal accuracy, and the noise variance until convergence. Finally, the DOA estimation result is obtained from the sparse signal matrix. This invention, based on the introduction of the received signal phase as a latent variable and the inclusion of appropriate prior constraints, improves the stability and resolution of the estimation in complex scenarios such as multiple targets and low signal-to-noise ratios.
[0005] The technical solution adopted in this invention is:
[0006] A sparse variational DOA estimation method for phase-deficient signals is proposed, which uses two uniform linear arrays to receive the target signal, with the two arrays sharing the first array element. The first array has... The first array element, the second array has If there are a number of array elements, then the two arrays have a total of array elements. There are 1 array, and the angle between the two arrays is 1. The DOA estimation method includes:
[0007] S1. Divide the spatial domain into a grid with a grid step size of [value missing]. The total number of grid points is If the target position is exactly on the grid, the amplitude of the received signal is:
[0008] ,
[0009] in, The received signal for the first array. For the signals received by other elements in the second array that do not contain the first element, yes and The guide vector matrices stacked together, For a sparse signal matrix, The number of sampling points. For all grid points, the guide vector matrix for the first array, For all grid points, the guiding vector matrix for the second array elements that do not contain the first element. It is an independent additive Gaussian noise matrix, i.e. ,in express The Line number The element of the column, i.e., the first The physical array element in the first The noise received in a quick snapshot The variance is the variance of independent additive Gaussian noise;
[0010] S2, Introducing latent phase variables ,definition ,get:
[0011] ,
[0012] in, For a complete received signal matrix with phase, express The Line number The element of the column, i.e., the first The physical array element in the first The measured values and phase in a quick snapshot Combined equivalent complex observation signals, express The Line number The element of the column, i.e., the first The physical array element in the first A snapshot of the actual measured pure amplitude signal; definition , For the phase of the ideal noise-free received signal, we obtain The likelihood function is:
[0013] ;
[0014] Transformation from Cartesian coordinates to polar coordinates Jacobian determinant for:
[0015] ,
[0016] Jacobi determinant Substitute into From the likelihood function, we obtain the complete data likelihood function:
[0017] ;
[0018] S3, Phase latent variable Variational posterior Modeled as a von Mises distribution and initialized as A uniform distribution on the surface is represented as ;
[0019] Establish a sparse signal matrix The probability density function is:
[0020] ,
[0021] in, For all grid points A collection assembled in sequence. For the first The precision parameters of the signal corresponding to each grid point. For sparse signal matrix The Middle Line number The element of the column, i.e., the first The spatial grid point at the th A signal captured in a quick shot;
[0022] For all Assign a prior distribution to Gamma:
[0023] ,
[0024] in, and The preset hyperparameters are the shape and scale parameters;
[0025] for Set an inverse Gamma prior distribution:
[0026] ,
[0027] in, and These are preset constant hyperparameters;
[0028] Before solving for the parameters using variational Bayesian inference, it is first necessary to construct a joint probability distribution model of the entire direction-finding system. This joint probability distribution includes the array's received signal matrix. The introduced latent phase matrix (For phase latent variables) The matrix composed of the sparse signal matrix, and all unknown prior parameters to be inferred (sparse signal matrix). Accuracy hyperparameters Noise variance Based on the assumption of conditional independence, the joint distribution of observed data and all latent variables and unknown parameters can be represented by the product of the complete data likelihood function and the prior distributions with independently set parameters. Therefore, within the variational Bayesian framework, this joint probability distribution is defined and decomposed as follows:
[0029] ,
[0030] The posterior distribution factor is transformed into:
[0031] ;
[0032] S4. Calculate latent variables The log-posterior distribution:
[0033] ,
[0034] in, It is a constant. The variational posterior of the phase latent variable follows a von Mises distribution:
[0035] ;
[0036] Calculate the expectation of the complex exponential phase term:
[0037] ,
[0038] in, It is a nonlinear activation function composed of Bessel functions;
[0039] S5. Calculate the sparse signal matrix The log-posterior distribution:
[0040] ;
[0041] Each column of the sparse signal matrix The variational posteriors of all are independent and follow a complex Gaussian distribution:
[0042] ;
[0043] S6, Calculation accuracy hyperparameter The log-variable posterior distribution:
[0044] ;
[0045] Each The variational posterior distribution follows a Gamma distribution:
[0046] ,
[0047] in, , It is the mean vector The One element, It is the covariance matrix The One diagonal element;
[0048] S7, Calculation The log-variable posterior distribution:
[0049] ,
[0050] in, Defined as the expected sum of reconstruction errors:
[0051] ;
[0052] The variational posterior follows an inverse Gamma distribution:
[0053] ;
[0054] S8. Approximate the true joint posterior distribution by alternately updating the posterior distributions of each latent variable and parameter. In each iteration, execute steps S4 to S7 sequentially to update the phase latent variables. von Mises distribution parameters, update the sparse signal matrix The complex Gaussian distribution parameters, update the accuracy hyperparameters. The Gamma distribution parameters are updated to determine the noise variance. The inverse Gamma distribution parameters;
[0055] After completing a single update, convergence is determined by calculating the relative change in the posterior mean of the sparse signal matrix between two adjacent iterations.
[0056] ,
[0057] in, Denotes the Frobenius norm of a matrix. For the first The complete matrix is formed by the mean of the signal matrices obtained from the next iteration. This is a preset minimum tolerance threshold. For the number of iterations, A preset maximum number of iterations is used to prevent infinite loops; if the convergence criterion is met, the posterior mean of the finally converged sparse signal matrix is used. or accuracy hyperparameter expectation To reconstruct the spatial pseudospectrum, we obtain the various spatial grid points. The average energy of the corresponding signal:
[0058] ,
[0059] Calculation yields all After analyzing the energy at each grid point, a spatial power spectrum is formed. By performing a one-dimensional spectral peak search across the entire discrete spatial grid, maximum points above a preset energy threshold are extracted. The spatial physical angles corresponding to the grid points where these extreme peaks are located are the final output target DOA estimation results.
[0060] If the convergence condition is not met, then let Return to S4 and continue iterating.
[0061] The beneficial effects of this invention are as follows: In incoherent array scenarios, the receiving channels lack stable phase consistency, and only amplitude observations are available. Therefore, estimation methods need to address the challenge of parameter inference under complex Ricean distributions. This invention introduces the missing true phase as a latent variable into the statistical model of the received signal, and uses sparse variational Bayesian inference to alternately solve for the phase, sparse signal matrix, and parameters, thereby achieving accurate estimation of the target DOA. This invention effectively decouples the nonlinear relationship between amplitude and phase, avoids complex numerical integration, has high computational efficiency, and provides good direction-finding performance. Attached Figure Description
[0062] Figure 1This is a comparison between the spatial spectrum estimated by this method and the actual target orientation. Detailed Implementation
[0063] The present invention will now be described in detail with reference to the accompanying drawings and simulation examples.
[0064] This invention sets up an estimation scenario with two uniform linear arrays (ULAs), both arrays sharing the first array element. The two arrays only receive amplitude values. By introducing a phase latent variable and then performing sparse variational Bayesian inference, DOA estimation is achieved. DOA estimation includes the following steps:
[0065] S1, the first array has The first array element, the second array has If there are a number of array elements, then the two arrays have a total of array elements. There are 1 array, and the included angle between the two arrays is 1. The spatial domain is divided into grids with a grid step size of [value missing]. The total number of grid points is Assume the target location is exactly on the grid.
[0066] The angle of incidence of each grid point with respect to the first array is... The angle of incidence for the second array is ,but:
[0067] (1)
[0068] Then the guide vectors for each grid point with respect to the two arrays are as follows:
[0069]
[0070] (2)
[0071] For the second array, the steering vector does not consider the first element. Therefore, the steering vector matrices for all grid points in both arrays are as follows:
[0072] (3)
[0073] signal matrix It is a sparse matrix, where only the rows corresponding to the grid points where the target is located are non-zero rows. Let be the number of sampling points. Then the received signal is:
[0074] (4)
[0075] S2, Order ,but
[0076] (5)
[0077] make ,but The likelihood function is:
[0078] (6)
[0079] Transformation from Cartesian coordinates to polar coordinates The Jacobian determinant is:
[0080] (7)
[0081] Therefore, substituting the Jacobian determinant into... From the likelihood function, we can obtain the complete data likelihood function:
[0082] (8)
[0083] S3. This invention will address the missing phase latent variables. Variational posterior The model is a von Mises distribution. Since there is no prior phase information in the initial state, it is initialized as follows: A uniform distribution on the surface. Mathematically, this is equivalent to initializing the concentration parameter of the von Mises distribution to zero, i.e. .
[0084] Sparse signal matrix The settings are controlled by hyperparameters and follow a zero-mean complex Gaussian prior distribution. It is assumed that each snapshot and each grid point is independent, and its probability density function is...
[0085] (9)
[0086] in, For the first Each grid corresponds to a precision parameter of the signal. When the precision parameter of a certain grid is inferred... When it approaches infinity, the corresponding signal The value will approach zero, thus achieving a sparse representation of spatial dimensions.
[0087] To make the accuracy parameters The inference can be achieved through data adaptation while maintaining the conjugate property with the complex Gaussian distribution, for all... Assign a prior distribution to Gamma:
[0088] (10)
[0089] in, and These are the preset hyperparameters for shape and scale.
[0090] For the variance of complex Gaussian noise To ensure conjugate with the likelihood function, an inverse Gamma prior distribution is set for it:
[0091] (11)
[0092] in, and These are also preset constant hyperparameters.
[0093] Within the variational Bayesian framework, the joint probability distribution can be decomposed into: Furthermore, it is assumed that the posterior distribution can be factorized as follows:
[0094] (12)
[0095] S4. Calculate the latent variables. The log-posterior distribution of the latent variable The log-posterior distribution should be equal to the joint probability distribution with respect to the remaining latent variables (signal matrix). and noise variance ) expectations:
[0096] (13)
[0097] in, It is a constant, and all subsequent ones are constants. They are not necessarily the same; they are just used to represent the concept of a constant. Substitute the complete data likelihood function constructed earlier into the above formula, take the logarithm and calculate the expectation, and then include all variables not included in the current variable. All terms can be merged into the constant term, and then the terms related to the constant term can be extracted. The related cross terms yield:
[0098] (14)
[0099] in, This is used to characterize the concentration (or precision) of the term. The above equation perfectly matches the logarithmic form of the von Mises distribution (circular normal distribution). Therefore, it can be determined that the variational posterior of the phase latent variable follows a von Mises distribution:
[0100] (15)
[0101] In order to update the signal matrix in subsequent steps It is necessary to utilize the resources available. To calculate the expectation of the complex exponential phase term:
[0102] (16)
[0103] in, It is a nonlinear activation function composed of Bessel functions.
[0104] S5. Following the line of thought above. This can be expressed as the sum of the expectations of two terms (with all other irrelevant terms classified as constants):
[0105] (17)
[0106] Define an ideal signal ,in It is the guiding vector matrix The OK, yes The Column. Define hyperparameters Expected diagonal matrix Define auxiliary vectors. (length is) ), its first Each element combines the expected value of the observed amplitude and the latent phase variable, i.e. Using the above definition, we can combine the first and second terms and write them as a quadratic form of the matrix conjugate transpose:
[0107] (18)
[0108] It is possible to discover each column of the sparse signal matrix The variational posteriors of all are independently distributed by a complex Gaussian distribution, i.e.:
[0109] (19)
[0110] S6. Based on the variational Bayesian framework, the accuracy hyperparameter... The log-variable posterior distribution is equal to the expectation of the joint probability distribution with respect to the remaining latent variables. That is:
[0111] (20)
[0112] Since the above derivation results perfectly conform to the logarithmic form, and each They are mutually independent, therefore it can be rigorously proven that each The variational posterior distribution still follows the Gamma distribution:
[0113] (twenty one)
[0114] in, , It is the mean vector The One element, It is the covariance matrix The One diagonal element.
[0115] S7, Noise Variance The logarithmic variational posterior distribution is equal to the joint probability distribution with respect to the sparse signal matrix. and phase latent variables The expectation. That is.
[0116] (twenty two)
[0117] in, Defined as the expected sum of reconstruction errors:
[0118] (twenty three)
[0119] By comparing the coefficients, it can be seen that the updated variational posterior still strictly follows the inverse Gamma distribution:
[0120] (twenty four)
[0121] S8. The true joint posterior distribution is approximated by alternately updating the posterior distributions of the latent variables and parameters. In each iteration (let the current iteration number be ), In the process, steps S4 to S7 are executed sequentially to update the phase latent variables. von Mises distribution parameters; update the sparse signal matrix. Complex Gaussian distribution parameters; update accuracy hyperparameters Gamma distribution parameters; update noise variance The inverse Gamma distribution parameter.
[0122] After completing a single full update, the algorithm performs a convergence check. It calculates the relative change (or change in the precision hyperparameter) of the posterior mean of the sparse signal matrix between two consecutive iterations. If the following convergence condition is met, the iteration terminates:
[0123] (25)
[0124] in, Denotes the Frobenius norm of a matrix. For the first The complete matrix is formed by the mean of the signal matrices obtained from the next iteration. This is a preset minimum tolerance threshold. Set a preset maximum number of iterations to prevent infinite loops. If the above conditions are not met, then let... Return to S4 and continue iterating.
[0125] After the algorithm terminates its iterations when the convergence condition is met, the system will utilize the posterior mean of the finally converged sparse signal matrix. or accuracy hyperparameter expectation To reconstruct the spatial pseudospectrum, we obtain the various spatial grid points. The average energy of the corresponding signal:
[0126] (26)
[0127] Calculation yields all After analyzing the energy at each grid point, a spatial power spectrum is formed. By performing one-dimensional peak searching across the entire discrete spatial grid, maxima points above a preset energy threshold are extracted. The spatial physical angles corresponding to these grid points containing these extreme peaks are the final output DOA (Direction of Arrival) estimation results.
[0128] Simulation example:
[0129] A non-coherent array direction finding scenario was simulated using MATLAB. It was assumed that the first and second subarrays of the non-coherent dual-linear array had 20 elements each, with the first elements of both subarrays overlapping at a 20° angle. Three real targets existed in space, with DOA angles of -40°, 0°, and 20°, respectively. The signal-to-noise ratio of the simulation environment was set to 5 dB, and the number of snapshots for received data was 100. The spatial grid range was -70° to 70°, with a grid step size of 1°. The "SBL method" refers to the sparse variational DOA estimation algorithm for phase-deficient data proposed in this invention. This algorithm was used to process the generated amplitude-only observation data, setting the maximum number of iterations to 300 and the tolerance threshold to 1e-5. The phase latent variable, sparse signal matrix, and related hyperparameters were alternately updated, and iteration stopped when the relative change in the evidence lower bound (ELBO) satisfied the convergence condition. Finally, the average energy generation spatial spectrum of the converged signal was calculated, as shown below. Figure 1 As shown, the red dashed lines represent the actual target positions, namely -40°, 0°, and 20°. Figure 1 The spatial spectrum estimated by the algorithm of this invention in this scenario is shown, and the spectral peaks in the figure accurately correspond to the orientation of the real target.
Claims
1. A sparse variational DOA estimation method for phase-deficient data, characterized in that, The target signal is received by two uniform linear arrays, with the two arrays sharing the first array element. The first array has... The first array element, the second array has If there are a number of array elements, then the two arrays have a total of array elements. There are 1 array, and the angle between the two arrays is 1. ; The DOA estimation method includes: S1. Divide the spatial domain into a grid with a grid step size of [value missing]. The total number of grid points is If the target position is exactly on the grid, the received signal is: ， in, The received signal for the first array. For the signals received by other elements in the second array that do not contain the first element, yes and The guide vector matrices stacked together, For a sparse signal matrix, The number of sampling points. For all grid points, the guide vector matrix for the first array, For all grid points, the guiding vector matrix is given for all other elements in the second array that do not contain the first element. It is an independent additive Gaussian noise matrix, i.e. ,in express The Line number The element of the column, i.e., the first The physical array element in the first The noise received in a quick snapshot The variance is the variance of independent additive Gaussian noise; S2, Introducing latent phase variables ,definition ,get: ， in, For a complete received signal matrix with phase, express The Line number The element of the column, i.e., the first The physical array element in the first The measured values and phase in a quick snapshot Combined equivalent complex observation signals, express The Line number The element of the column, i.e., the first The physical array element in the first A quick snapshot captures the actual measured pure amplitude signal; definition , For the phase of the ideal noise-free received signal, we obtain The likelihood function is: ； Transformation from Cartesian coordinates to polar coordinates Jacobian determinant for: ， Jacobi determinant Substitute into From the likelihood function, we obtain the complete data likelihood function: ； S3, Phase latent variable Variational posterior Modeled as a von Mises distribution and initialized as A uniform distribution on the surface is represented as ; Establish a sparse signal matrix The probability density function is: ， in, For all grid points A collection assembled in sequence. For the first The precision parameters of the signal corresponding to each grid point. For sparse signal matrix The Middle Line number The element of the column, i.e., the first The spatial grid point at the th A signal captured in a quick shot; For all Assign a prior distribution to Gamma: ， in, and The preset hyperparameters are the shape and scale parameters; for Set an inverse Gamma prior distribution: ， in, and These are preset constant hyperparameters; A joint probability distribution model of all variables is constructed, including the array's received signal matrix. The introduced phase latent variable Forming the latent phase matrix And all unknown prior parameters to be inferred, including the sparse signal matrix. Accuracy hyperparameters Noise variance Based on the assumption of conditional independence, the joint distribution of observed data and all latent variables and unknown parameters is represented by the product of the complete data likelihood function and the prior distributions of each parameter set independently. Within the variational Bayesian framework, the joint probability distribution is defined and decomposed as follows: ， The posterior distribution factor is transformed into: ； S4. Calculate latent variables The log-posterior distribution: ， in, It is a constant. The variational posterior of the phase latent variable follows a von Mises distribution: ； Calculate the expectation of the complex exponential phase term: ， in, It is a nonlinear activation function composed of Bessel functions; S5. Calculate the sparse signal matrix The log-posterior distribution: ； Each column of the sparse signal matrix The variational posteriors of all are independent and follow a complex Gaussian distribution: ； S6, Calculation accuracy hyperparameter The log-variable posterior distribution: ； Each The variational posterior distribution follows a Gamma distribution: ， in, , It is the mean vector The One element, It is the covariance matrix The One diagonal element; S7, Calculation The log-variable posterior distribution: ， in, Defined as the expected sum of reconstruction errors: ； The variational posterior follows an inverse Gamma distribution: ； S8. Approximate the true joint posterior distribution by alternately updating the posterior distributions of each latent variable and parameter. In each iteration, execute steps S4 to S7 sequentially to update the phase latent variables. von Mises distribution parameters, update the sparse signal matrix The complex Gaussian distribution parameters, update the accuracy hyperparameters. The Gamma distribution parameters are updated to determine the noise variance. The inverse Gamma distribution parameters; After completing a single update, convergence is determined by calculating the relative change in the posterior mean of the sparse signal matrix between two adjacent iterations. ， in, Denotes the Frobenius norm of a matrix. For the first The complete matrix is formed by the mean of the signal matrices obtained from the next iteration. This is a preset minimum tolerance threshold. For the number of iterations, A preset maximum number of iterations is used to prevent infinite loops; if the convergence criterion is met, the posterior mean of the finally converged sparse signal matrix is used. or accuracy hyperparameter expectation To reconstruct the spatial pseudospectrum, we obtain the various spatial grid points. The average energy of the corresponding signal: ， Calculation yields all After analyzing the energy at each grid point, a spatial power spectrum is formed. By performing a one-dimensional spectral peak search across the entire discrete spatial grid, maximum points above a preset energy threshold are extracted. The spatial physical angles corresponding to the grid points where these extreme peaks are located are the final output target DOA estimation results. If the convergence condition is not met, then let Return to S4 and continue iterating.