A DOA estimation method of step length control gradient descent type subspace tracking

Abstract

The present application relates to array signal processing and direction of arrival (DOA) estimation technical field, specifically step control gradient descent type subspace tracking DOA estimation method, under the maximum correlation entropy-based robust criterion, first obtain array observation snapshot vector, then set initial subspace base matrix and subspace dimension, calculate projection and error vector. Robust cost function based on maximum correlation entropy is used and gradient descent method is used to solve, so that the algorithm can maintain good tracking performance under impulse noise. According to priori estimation error and posteriori estimation error, step control mechanism is constructed, and the adaptive step size at the current time is obtained by minimizing the posteriori error energy. At the same time, in order to prevent subspace mutation, the present application designs a kind of anti-mutation mechanism, which resets the step size immediately when mutation is detected, so that the algorithm quickly re-converges. Finally, the subspace base matrix obtained by EPSRIT algorithm is converted into angle information, and DOA estimation is realized.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing and direction of arrival (DOA) estimation technology, specifically relating to a DOA estimation method for step-size controlled gradient descent subspace tracking. Background Technology

[0002] As Zheng et al. described in *Electronics*, vol. 14, no. 23, p. 4562, in array signal processing applications such as radar, sonar, array communication, and microphone array positioning, it is often necessary to estimate the direction of arrival (DOA) of incoming spatial waves to achieve functions such as target localization, interference suppression, and beamforming. DOA estimation typically relies on extracting the signal subspace or noise subspace from array observation snapshots and further converting the subspace information into angular parameters. In dynamic scenarios (such as target motion), traditional methods based on batch covariance estimation and eigenvalue decomposition have high computational overhead and slow updates; therefore, online subspace tracking algorithms have become one of the important technical approaches for dynamic DOA estimation.

[0003] In the field of subspace tracking, several classic algorithms have been widely applied. In 1995, Yang proposed the Projection Approximation Subspace Tracking (PAST) algorithm based on the Recursive Least Squares (RLS) idea and the Gradient Descent Subspace Tracking (GDST) algorithm based on the Gradient Descent idea in IEEE Transactions on Signal Processing, vol. 43, no. 1, pp. 95-107. However, from the perspective of optimization criteria, both GDST and PAST algorithms are subspace tracking algorithms based on Mean Square Error (MSE). They are sensitive to anomalous large-value samples in impulse noise or heavy-tailed non-Gaussian environments. A small number of outliers can significantly interfere with the cost function optimization and recursive update process, leading to increased steady-state error, increased fluctuations, and decreased tracking stability, thus limiting their angle estimation performance under complex noise conditions. Meanwhile, as Clavier et al. stated in the EURASIP Journal on Wireless Communications and Networking, vol. 2021, no. 1, p. 13, impulse noise, whether originating from human interference or the natural environment, is widely present in practical engineering applications. Examples include electromagnetic interference noise in mobile communications, ice breakage noise in underwater acoustic communications, and thunderstorm noise in atmospheric propagation environments. This type of noise typically exhibits significant non-Gaussian characteristics such as strong suddenness and heavy-tailed amplitude distribution, thus placing higher demands on the robustness of traditional subspace tracking methods.

[0004] To improve robustness in non-Gaussian noise, especially impulse noise environments, Zhang and Qiu proposed the MCC-PAST algorithm in DigitalSignal Processing, vol. 62, pp. 168-175, which introduces the Maximum Correntropy Criterion (MCC) based on the PAST framework. This algorithm improves subspace estimation stability and angle estimation performance under impulse noise conditions by using a robust cost function based on MCC to suppress the impact of abnormally large samples on the update. However, because it still inherits the recursive structure of the PAST algorithm, its computational complexity and implementation overhead remain relatively high. Furthermore, such algorithms typically use a forgetting factor parameter to exponentially weight historical information, and its value needs to be balanced between tracking capability and steady-state accuracy. Specifically, a smaller forgetting factor can enhance the response speed to time-varying subspaces, but it amplifies noise disturbances, leading to increased estimation jitter and steady-state error. A larger forgetting factor is beneficial for reducing steady-state error and obtaining smoother subspace estimation, but it is prone to tracking lag when the source angle changes abruptly or the environment changes rapidly. Since the statistical characteristics and dynamic changes of noise often vary over time, a fixed forgetting factor is difficult to adaptively balance performance across all time periods, thus limiting the overall performance of the algorithm in complex non-stationary scenarios.

[0005] In contrast, the gradient update form of the GDST algorithm typically consists of a simple vector outer product and a scalar step size, resulting in lower computational complexity and ease of online implementation. Therefore, it has engineering advantages in real-time dynamic processing scenarios. Furthermore, since it does not require a fixed forgetting factor to weight historical information, it avoids the parameter setting problem of repeatedly compromising between tracking capability and steady-state accuracy when determining the forgetting factor value. However, the GDST algorithm also has the following shortcomings: Firstly, the GDST algorithm typically uses a fixed step size parameter, which leads to an incompatible trade-off between convergence speed and steady-state accuracy. Specifically, when the step size is large, although the convergence speed is fast, the steady-state performance is poor; while when the step size is small, the steady-state error is low, but the convergence speed is slow, making it difficult to meet the dual requirements of fast tracking and high steady-state accuracy in dynamic scenarios. Secondly, the GDST algorithm is based on the MSE criterion, making it sensitive to impulse noise and anomalous samples, and prone to problems such as subspace estimation shift, large angle estimation errors, and even instability. To address the aforementioned issues, this invention proposes a step-size controlled gradient descent subspace tracking DOA estimation method: First, it employs a robust criterion based on maximum correlation entropy to improve the algorithm's robustness in impulse noise environments. Second, it designs a step-size control mechanism that allows the step size to adaptively change according to the iteration state, thus balancing the algorithm's convergence speed and steady-state accuracy. Third, it designs an anti-mutation mechanism that can respond quickly when mutations occur in the signal subspace, avoiding the slow reconvergence problem caused by the fixed update method in traditional methods, thereby improving the algorithm's adaptability in dynamic scenarios. Fourth, this invention avoids the additional overhead of complex matrix operations while ensuring robustness and tracking performance, resulting in lower computational complexity.

[0006] In summary, this invention maintains the advantage of low complexity while taking into account robustness, fast convergence, high steady-state accuracy, and rapid response capability under abrupt change scenarios, thus making it more suitable for dynamic DOA estimation in impulse noise environments and having good practical application value. Summary of the Invention

[0007] The purpose of this invention is to provide a step-size controlled gradient descent subspace tracking DOA estimation method. It adopts a robust criterion based on the entropy function and designs a step-size control mechanism based on the prior estimation error and the posterior estimation error. This effectively solves the contradiction between convergence speed and steady-state accuracy, and improves the algorithm's resistance to impulse noise and DOA estimation accuracy.

[0008] To achieve the above-mentioned technical objectives and effects, the present invention is implemented through the following technical solution: A step-size controlled gradient descent subspace tracking DOA estimation method is disclosed. The method employs a robust criterion based on an entropy function, constructs a step-size control mechanism using prior and posterior estimation errors, and obtains the adaptive step size at the current time by minimizing the posterior error energy. The method includes the following steps: Obtain the array observation snapshot vector at the current time. ,set up A narrowband far-field signal source is incident on by A uniform linear array composed of elements, at the current time Sampling is performed on each array element to obtain the array observation snapshot vector: in, For the first Each element at any moment The output, This indicates the transpose operation.

[0009] Based on the current time array observation snapshot vector Subspace basis matrix at the previous time step Calculate the subspace projection vector at the current time. and error vector ,in, Indicates conjugate transpose; According to the error vector A robust cost function is constructed and solved using the gradient descent method. The robust cost function is constructed using the maximum correlation entropy criterion to suppress the impact of impulse noise on DOA estimation. A step size control mechanism is constructed based on the prior and posterior estimation errors. The step size value at the current time is obtained by minimizing the posterior error energy. The expectation term in the step size control mechanism is then smoothly estimated to obtain an engineering-implementable step size update value. ; Based on the residual energy ratio, an anti-mutation mechanism is constructed to detect subspace mutations and trigger fast reconvergence processing when a subspace mutation is detected, thereby improving the tracking capability under subspace mutation conditions. Update the subspace basis matrix at the current time step according to the robust cost function and the step size control mechanism. Its update formula is: in, Pick , Indicates the front of the main diagonal One element is 1, and the rest are 0. matrix, For Gaussian kernel function, Describing the F-norm, For kernel width.

[0010] Based on the updated subspace basis matrix The ESPRIT algorithm is used to calculate the angle of arrival (OA) of each source to obtain the DOA estimation result; the above steps are iteratively executed until the snapshot ends.

[0011] Furthermore, the robust cost function is constructed using the maximum correlation entropy criterion, based on the error vector. Calculate the entropy function weighting factor ; Furthermore, regarding the fixed step size parameter Design a step size control mechanism. The step size control mechanism By analyzing the posterior estimation error The expectation of the square of the second norm with respect to Take the derivative and set it to 0 to obtain its expression. , in, , , Indicates taking the real part, Expressing expectations, , They represent The prior and posterior estimation errors, express The noise-free component.

[0012] Furthermore, due to the noise-free components of the observed signal in practice... It cannot be directly obtained, and there is an error in the prior estimation. An approximation is performed using a smoothing estimation method, wherein the smoothing estimation satisfies: in, As a smoothing factor, As the initial value, express The zero vector.

[0013] Furthermore, the expected term in the step size control mechanism is smoothly estimated to obtain the step size control mechanism for engineering implementation; calculation and The instantaneous values ​​are as follows: And estimated using exponential smoothing: Therefore, the step size control mechanism during engineering implementation is as follows: ; in, As a smoothing factor, Under normal circumstances Take 0.1.

[0014] Furthermore, the anti-mutation mechanism is implemented as follows: the instantaneous residual energy ratio at the current moment is calculated as follows. ; The residual energy ratio is exponentially smoothed to obtain the average residual energy ratio used for decision-making. ,in, It is a smoothing factor; Furthermore, the historical reference lower limit of the average residual energy ratio is recorded. And update as follows: A mutation decision threshold is constructed based on the aforementioned historical reference lower limit. ;in, This is the threshold coefficient; when it satisfies and When a subspace mutation occurs, a reconvergence process is triggered. , These two items need to be reset, that is, let And update the historical reference lower limit to the current average residual energy ratio level, i.e. ,in, The set number of preheating steps.

[0015] Furthermore, based on the updated subspace basis matrix The ESPRIT algorithm is used to calculate the angle of arrival (DOA) for each source, and the DOA estimation results are obtained.

[0016] The beneficial effects of this invention are: 1. Unlike traditional algorithms based on the MSE criterion, this invention employs a robust criterion of maximum correlation entropy. During the update process, an entropy function weighting factor is introduced, assigning lower weights to samples with abnormally large values. This effectively suppresses the disruption of gradient updates and DOA estimation processes caused by impulse noise and outliers. Therefore, this invention can maintain the numerical stability of subspace tracking even in impulse noise environments, continuously outputting accurate signal subspace estimation results, thereby improving the stability and accuracy of dynamic DOA estimation.

[0017] 2. This invention further constructs a step size control mechanism, enabling the update rate to automatically adjust according to changes in system state and error. In the initial stage of the algorithm or during periods of significant error increase due to sudden environmental changes, the step size mechanism maintains a large step size to accelerate convergence and tracking. As the algorithm gradually enters a steady state and the error tends to stabilize, the step size automatically decreases to reduce steady-state jitter, mitigate mismatch, and suppress oscillations. Therefore, this invention alleviates the inherent contradiction between convergence speed and steady-state performance when using a fixed step size.

[0018] 3. In view of the sudden changes in the signal subspace that may occur in practical applications, which may cause estimation mismatch or tracking performance degradation, this invention introduces an anti-mutation mechanism. When a mutation is detected in the system, it can quickly identify and take measures to enable the system to quickly reconverge to a stable state.

[0019] 4. The computational complexity of the PAST algorithm and the MCC-PAST algorithm is typically O(n log n). This invention employs an online update framework based on gradient descent to track the signal subspace, eliminating the need for matrix inversion and complex recursive matrix updates, with a computational complexity of only [missing information]. Therefore, the computational burden on each snapshot is lower and the implementation structure is simpler. This complexity advantage makes the present invention more suitable for real-time online snapshot-by-snap processing scenarios, improving engineering feasibility while balancing robustness and tracking performance, thereby better meeting the application requirements of dynamic DOA estimation for low latency, low complexity, and stable operation.

[0020] Of course, different embodiments of the present invention do not necessarily need to achieve all the advantages described above at the same time. Attached Figure Description

[0021] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0022] Figure 1 It is a curve comparing the actual DOA and the estimated DOA trajectory in a dynamic environment.

[0023] Figure 2 It is the root mean square error (RMSE) curve of the algorithm. Detailed Implementation

[0024] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. Example 1

[0025] The DOA estimation method for step-size controlled gradient descent subspace tracking described in this embodiment includes the following steps: A. Obtain the array observation snapshot vector at the current time. ,consider A narrowband far-field signal source is incident on by An array composed of elements, at the current moment Sample each array element to obtain the result. The snapshot vector observed by each element in, For the first Each element at the current moment The observation, This indicates the transpose operation.

[0026] B. Observe the snapshot vector based on the array. Subspace basis matrix at the previous time step Calculate the subspace projection vector at the current time. and error vector ,in, Indicates conjugate transpose; C. Constructing the entropy function weighting factors According to the error vector Constructing the entropy function weighting factor ; in, For Gaussian kernel function, This is the kernel width parameter.

[0027] D. Design a step size control mechanism D1. Construct a step size control mechanism For the fixed step size parameter Design a step size control mechanism. The step size control mechanism satisfy in, , , Indicates taking the real part, Expressing expectations, To account for the prior estimation error of the noise-free components of the observed signal, A constant used to prevent the denominator from being too small, which would cause the value to diverge.

[0028] D2. Smoothing the prior estimation error. Since the noise-free components of the observed signal cannot be directly obtained in practice, the prior estimation error is affected. An approximation is performed using a smoothing estimation method, wherein the smoothing estimation satisfies: in, As a smoothing factor, As the initial value, express The zero vector.

[0029] D3. Smoothly estimate the expected terms to ensure project feasibility. The expected term in the step size control mechanism is smoothly estimated to obtain the step size control mechanism for engineering implementation; calculation and The instantaneous values ​​are as follows: And estimated using exponential smoothing: in, As a smoothing factor, Under normal circumstances Take 0.1; Therefore, the step size control mechanism during engineering implementation is as follows: .

[0030] D4. Using the adaptive step size value obtained in step D3 Update the subspace E. Design anti-mutation mechanisms E1. Calculate the residual energy ratio and perform smoothing. Calculate the instantaneous residual energy ratio at the current moment: right Perform exponential smoothing to obtain the average residual energy ratio used for decision-making: in, This is a smoothing factor.

[0031] E2. Record the reference lower limit and construct an adaptive decision threshold. To determine the average residual energy ratio To determine whether an abnormal increase has occurred, this invention records its historical smallest value as a reference lower limit, denoted as... The lower reference limit is updated as follows: Construct a decision threshold based on this reference lower bound: in, This is the threshold coefficient, used to adjust how well the threshold follows the reference lower limit.

[0032] Simultaneously set the number of preheating steps. ,exist No mutation decision is made in the initial stage to avoid accidental triggering before the algorithm is stable.

[0033] E3, Mutation Decision and Reconvergence Triggering When satisfied and When a subspace mutation occurs, reconvergence is triggered. After triggering, the smoothing statistics of the numerator and denominator in the variable step-size formula are adjusted. and Reset to initial values ​​respectively and Meanwhile, to avoid repeated triggering during the reconvergence phase due to the residuals not yet being recovered, the reference lower bound will be updated to the current level: .

[0034] F. Use the ESPRIT algorithm to convert the obtained subspace into angle information. G. Let t = t + 1, and repeat steps A, B, C, D, E, and F until the fast-paced video ends.

[0035] Simulation experiment: To verify the effectiveness of the present invention, simulation experiments were conducted, and to create a fair comparison environment, the present invention will be compared with GDST, MCC-GDST, PAST and MCC-PAST algorithms.

[0036] Simulation conditions This experiment does not limit the array type; the following simulation uses a uniform linear array as an example, where the array includes... Each array element, the spacing between array elements The wavelength is half. Considering a three-source hybrid dynamic scenario, this is used to verify the tracking performance of the algorithm in a multi-source time-varying environment, where the first signal is defined as... The angle of the second signal decreases linearly from 20° to 5° throughout the entire observation interval, forming a continuous, uniformly changing trajectory; the angle of the third signal remains fixed at -10°. These settings encompass slow drift, rapid change, uniform time-varying, and constant angles of the signal sources, thus constituting a hybrid dynamic multi-source scenario to comprehensively evaluate the subspace tracking and DOA estimation performance of this invention under complex time-varying conditions.

[0037] Array observation snapshot vector Through formula Calculated, where, For the guiding matrix, For the first The steering vector of a signal, Is with the first The frequency associated with the signal, The source signal vector, For the first A narrowband signal. Noise vector. It follows a symmetric alpha-stable distribution, and its characteristic function is: ,in Controlling the impulse characteristics and heavy tail of noise, To control the propagation or dispersion of noise. In this simulation experiment, the parameters for alpha noise are selected as follows: , To comprehensively evaluate the performance of the DOA estimation algorithm, this invention combines qualitative and quantitative evaluation metrics. Qualitative evaluation assesses the algorithm's merits by comparing the estimated DOA with the actual DOA (e.g., whether the DOA estimation curve is smooth, closely follows the actual trajectory, and has spikes or spikes). Quantitative evaluation, on the other hand, assesses the overall accuracy of the algorithm from a numerical perspective by statistically estimating the error level between the estimated and actual DOA over the entire snapshot sequence. Specifically, it calculates the RMSE curve to quantify the deviation between the estimated and actual values. The specific calculation formula is as follows: in, For the number of Monto Carlo experiments, the number taken in this experiment is... , For time t, the first From the perspective of the second estimate, This represents the actual angle at that moment. For fair comparison, the parameter values ​​for these algorithms are shown in Table 1.

[0038] Table 1 Parameter settings for each algorithm Figure 1 This is a performance comparison chart for DOA estimation, showing the comparison between the algorithm's actual and estimated DOA trajectories. From Figure 1It can be seen that under the influence of impulse noise, the GDST algorithm exhibits severe and significant fluctuations, making estimation impossible. The PAST algorithm's tracking trajectory shows obvious spikes, while the DOA estimation curves of the MCC-PAST algorithm and the present invention do not show spikes or significant fluctuations. This indicates that the present invention and the MCC-PAST algorithm have good robustness to impulse noise. However, during the rapid angle change phase, the PAST and MCC-PAST algorithms exhibit significant tracking lag due to the limitation of their tracking performance by a fixed forgetting factor. Although the MCC-GDST algorithm also employs a robustness criterion and does not show obvious spikes, it exhibits significant estimation instability and significant estimation bias in the initial stage due to the use of a fixed step size parameter.

[0039] To further quantitatively analyze the DOA estimation performance of different algorithms, Figure 2 The comparison algorithms are given. The estimated RMSE curve, and the results are consistent with... Figure 1 The effect shown is consistent, from Figure 2 It can be further seen that, with low computational complexity, the present invention can achieve an estimation accuracy comparable to the MCC-PAST algorithm, and its impact is significantly smaller during the rapid angle change phase.

[0040] The above results demonstrate that the present invention achieves better stability, higher accuracy, and lower computational complexity in dynamic DOA estimation, and has superior performance in practical applications.

[0041] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A DOA estimation method for step-size controlled gradient descent subspace tracking, characterized in that, Includes the following steps: Obtain the array observation snapshot vector at the current time. ,consider A narrowband far-field signal source is incident on by An array composed of individual array elements, in the current Each array element is sampled at any time to obtain the result. The snapshot vectors observed by each array element: in, For the first The observation of each array element at the current moment, Indicates the transpose operation; According to the current Time array observation snapshot vector Compared to the previous moment subspace basis matrix Calculate the subspace projection vector at the current time. and error vector ,in, Indicates conjugate transpose; According to the error vector A robust cost function is constructed and solved using the gradient descent method. The robust cost function is constructed using the maximum correlation entropy criterion to suppress the impact of impulse noise on DOA estimation. A step size control mechanism is constructed based on the prior and posterior estimation errors. The step size value at the current time is obtained by minimizing the posterior error energy. The expectation term in the step size control mechanism is then smoothly estimated to obtain an engineering-implementable step size update value. ; A mutation prevention mechanism is constructed based on the residual energy ratio to detect subspace mutations and trigger fast reconvergence processing when a subspace mutation is detected, thereby improving the tracking capability under subspace mutation conditions. Update the subspace basis matrix at the current time step according to the robust cost function and the step size control mechanism. Its update formula is: Wherein, the initial value of the subspace basis matrix Pick , Indicates the front of the main diagonal One element is 1, and the rest are 0. matrix, For Gaussian kernel function, Describing the F-norm, For kernel width; Based on the updated subspace basis matrix The ESPRIT algorithm is used to calculate the angle of arrival (OA) of each source to obtain the DOA estimation result; the above steps are iteratively executed until the snapshot ends.

2. The method as described in claim 1, characterized in that, The robust cost function is constructed using the maximum correlation entropy criterion, based on the error vector. Calculate the entropy function weighting factor .

3. The method as described in claim 1, characterized in that, For the fixed step size parameter Design a step size control mechanism. The step size control mechanism By analyzing the posterior estimation error The expectation of the square of the second norm is about Take the derivative and set it to 0 to obtain its expression. , in, , , Indicates taking the real part, Expressing expectations, , They represent The prior and posterior estimation errors, express The noise-free component.

4. The method as described in claim 3, characterized in that, Because in practice, the observed signal has no noise component. It cannot be directly obtained, and there is an error in the prior estimation. An approximation is performed using a smoothing estimation method, wherein the smoothing estimation satisfies: in, As a smoothing factor, As the initial value, express The zero vector.

5. The method as described in claim 3, characterized in that, The expected term in the step size control mechanism is smoothly estimated to obtain the step size control mechanism for engineering implementation; calculation and The instantaneous values ​​are as follows: And estimated using exponential smoothing: in, As a smoothing factor, Under normal circumstances The step size is set to 0.1; therefore, the step size control mechanism during engineering implementation is as follows: .

6. The method as described in claim 1, characterized in that, The anti-mutation mechanism is implemented as follows: The instantaneous residual energy ratio at the current moment is calculated as follows: ; The residual energy ratio is exponentially smoothed to obtain the average residual energy ratio used for decision-making. ,in, This is a smoothing factor.

7. The method as described in claim 6, characterized in that, Record the historical reference lower limit of the average residual energy ratio. And update as follows: A mutation decision threshold is constructed based on the aforementioned historical reference lower limit. ;in, This is the threshold coefficient; when it satisfies and When a subspace mutation occurs, a reconvergence process is triggered. , These two items need to be reset, that is, let And update the historical reference lower limit to the current average residual energy ratio level, i.e. ,in, The set number of preheating steps.

8. The method as described in claim 1, characterized in that, Based on the updated subspace basis matrix The ESPRIT algorithm is used to calculate the angle of arrival (DOA) for each source, and the DOA estimation results are obtained.

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