A hybrid off-grid blade tip timing signal parameter estimation method based on maximum likelihood estimation

Abstract

The application discloses a hybrid off-grid blade tip timing signal parameter estimation method based on maximum likelihood estimation and belongs to the technical field of array signal processing. The maximum likelihood estimation method is used to optimize the rough estimation value of each parameter of an array signal into a precise estimation value, improve the precision of the estimation value of the parameter, and thus the weak frequency and amplitude change in the blade tip timing signal can be effectively extracted. The method has strong robustness, can effectively estimate the noise intensity, and reduces the interference of noise on parameter estimation. The hybrid off-grid blade tip timing signal parameter estimation method based on maximum likelihood estimation is flexible and can adapt to different types of signals and complex signal structures.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, and specifically relates to a method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation. Background Technology

[0002] Rotating blades constitute fundamental components of rotating machinery, such as aircraft engines and gas turbines. However, challenging and harsh operating conditions (high temperature, high speed, heavy load, etc.) pose a serious threat to the safe and stable operation of blades. To effectively monitor their operating status, blade tip timing has emerged as a non-invasive measurement method among various blade monitoring methods. However, due to the limited number of probes and restricted installation locations, previously obtained blade vibration displacements are difficult to accurately and effectively indicate the operating status of the blades.

[0003] Generally, the recovered vibration frequency and amplitude are considered crucial health indicators. Typically, blade cracks lead to reduced stiffness, which in turn causes a decrease in natural frequency. Therefore, using the frequency shift of blade vibration to indicate structural damage is a feasible and reliable method. To obtain parameters such as the frequency of blade vibration, directly applying a non-uniform discrete Fourier transform can effectively recover the spectrum; however, the "non-uniformity" introduces aliasing, affecting high-frequency results. Furthermore, other classic spectrum estimation methods, such as the Capon method and linear fitting techniques, are employed in blade tip timing signal processing. Subspace methods based on the Carathéodory-Toeplitz theorem are more suitable for blade tip timing signals because of their high resolution and applicability to various array geometries. These methods improve the rank deficiency of the source covariance matrix through spatial smoothing, thereby reducing the array aperture.

[0004] Assuming the excitation and natural frequencies in the vibration signal are precisely aligned with a predefined frequency grid, compressed sensing transforms the observation model with missing data into a system of linear equations for sparse signal reconstruction and parameter estimation. Compressed sensing requires only a single snapshot and offers higher resolution than subspace-based methods. However, the misalignment between the actual frequencies and the predefined frequency grid can lead to estimation errors. Summary of the Invention

[0005] The purpose of this invention is to provide a method for estimating parameters of hybrid off-grid leaf tip timing signals based on maximum likelihood estimation, in order to address the aforementioned problems. This method aims to improve the difficulty in extracting the weak frequency and amplitude variations in leaf tip timing signals.

[0006] The technical solution adopted in this invention is as follows: a method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation, the method comprising the following steps: Model establishment: Based on the basic principles of leaf tip timing and array signal processing theory, a sparse layout sampling model for leaf tip timing is established; The augmented covariance matrix is ​​obtained from the model: The augmented covariance matrix under coprime arrays is obtained from the sparse layout sampling model; Obtaining coarse estimates of parameters: A sparse iterative covariance estimation method based on the covariance fitting criterion is presented to obtain coarse estimates of each parameter of the array signal; Obtaining precise estimates of parameters: The coarse estimates are optimized by the asymptotically optimal maximum likelihood estimation method to obtain precise estimates of each parameter of the array signal.

[0007] Furthermore, array signal processing includes the following steps: Set a common Circles and each circle Composed of individual sensors An array of probes, the array receiving signals from the tip of a rotating blade. Narrowband frequency The vibration signal has a power corresponding to its frequency. The frequencies that constitute the vibration signal are represented as vectors. : ; in, This is the transpose of the matrix.

[0008] Furthermore, based on the fundamental principle of blade tip timing, the sensor is placed on the same plane as the vibration, and the installation angle of the blade tip timing sensor is... for: ; in, Derived from the sensor location layout vector, Indicates the first The distance between each probe and any origin point The unit is rad; The corresponding array steering vector is: , ; in, , The imaginary unit, Let be the angular velocity of the bladed disk, and The value is constant.

[0009] Furthermore, a sparse layout sampling model for leaf tip timing is established: Data is collected based on the leaf tip timing principle. The second width is The snapshot matrix, in which the first The received signal corresponding to this quick snapshot is: , ; ; ; in, Indicates a single snapshot. The unknown signal waveform represents the signal power. It is an array of manifold rectangles. It is a noise vector, and noise vector It follows a mean of 0 and a variance of . Additive white Gaussian noise; Here, "snapshot" is a direct translation of the English word "snapshot," referring to a data sample collected at a specific moment.

[0010] The sampling covariance matrix corresponding to the matrix is: ; in, for The conjugate transpose of a matrix.

[0011] Furthermore, when the sensor layout satisfies the sparse linear array requirement, the layout angle of the blade-end timing sensor is adjusted as follows: , ; in, The minimum included angle between all sensors. To indicate the first An integer representing the location of a sensor; Set up a virtual array element set consisting of a differential array: ; Among them, the virtual array element set contains A continuous set of virtual array elements, forming a uniform linear array that replaces the original sparse linear array. Construct the difference array selection matrix The first of the matrix Line 1 The elements of the column are: ; in, To round up, For weighting functions; Depend on Each virtual array element receives the output signal. : ; in, The Kronecker product of the matrices For variance, This refers to the vectorization operation of matrices. The unknown signal waveform represents the signal power. Based on the output signal of the virtual array element Obtain the augmented covariance matrix : ; in, yes The Subarrays, for The conjugate transpose of a matrix.

[0012] Furthermore, we construct an optimization problem based on the covariance fitting criterion: ; ; ; ; ; ; ; in, This represents the trace used to extract the trajectory. This indicates taking the inverse of the matrix. To augment the covariance matrix, for The conjugate transpose of a matrix. and To optimize the optimization variables of the problem, The covariance matrix under sparse representation, A diagonal matrix composed of signal power. For frequency grid The upper frequency limit of the manifold matrix constructed from the virtual uniform linear array is . , For the first The weight of each frequency corresponds to the signal power. This is the normalization factor for noise power. The covariance matrix formed by virtual matrix linear arrays Dimensions The number of frequency grids; The above optimization problem was solved using the SDPT3 convex optimization toolkit, through a frequency grid. Obtain signal power and noise variance This allows for a rough estimate of the array signal parameters.

[0013] Furthermore, preserve signal power. middle The largest value and its corresponding grid frequency We obtain a rough estimate of the signal covariance matrix: ; in, This is a preliminary estimate of the noise variance.

[0014] Furthermore, the log-likelihood function for the unconditional maximum likelihood estimation problem of a signal is established as follows: ; in, The number of data acquisitions is based on the leaf tip timing principle. For matrix The determinant, For the sampling covariance matrix, For frequency, A diagonal matrix composed of signal power; Calculate the frequency, signal power, and variance versus the likelihood function respectively. Partial derivatives: ; ; ; ; The update step size for each parameter is: ; ; ; in, It is the size of the frequency grid. and It is a scale parameter. yes initial value, It is the growth rate; By optimizing the coarse estimate using the likelihood function, the fine estimate of each parameter of the array signal is obtained.

[0015] Furthermore, set the maximum number of iterations. Repeated iteration operation: ; ; ; Among them, superscript This represents the number of iterations. When the number of iterations exceeds the maximum number of iterations Or the iteration stopping condition is met. Stop iterating when the time is right.

[0016] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are: This invention optimizes the estimated values ​​of each parameter through the maximum likelihood estimation method, thereby improving the accuracy of the parameter estimation and effectively extracting the weak frequency and amplitude changes in the blade tip timing signal. Furthermore, the method proposed in this invention has strong robustness, can effectively estimate noise intensity, and reduce the interference of noise on parameter estimation. The hybrid off-grid blade tip timing signal parameter estimation method based on maximum likelihood estimation is highly flexible and can adapt to different types of signals and complex signal structures.

[0017] This invention can not only effectively deal with irregular sensor arrangements, but also completely eliminate the influence of gridding by relying on the maximum likelihood estimation of "off-grid", thereby achieving high-precision identification of blade tip timing signal parameters. Attached Figure Description

[0018] Figure 1 This is a flowchart of the method of the present invention; Figure 2 The spectrum of the synthesized signal obtained by the method of this invention and three other methods; Figure 3 A piecewise linear comparison of the normalized frequency estimation error and amplitude estimation error obtained by the method of the present invention and other methods under different signal-to-noise ratios; Figure 4 This is a time-domain waveform of blade tip vibration obtained from the blade tip timing test bench of the present invention. Figure 5 This is a comparison of the frequency and amplitude of the leaf tip timing experimental signal obtained by the method of the present invention with those obtained by other methods over time. Figure 6 This is a comparison of frequency and amplitude estimation results obtained from the timing data of the blade tips of faulty and normal blades in this invention. Detailed Implementation

[0019] The present invention will now be described in detail with reference to the accompanying drawings.

[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0021] like Figure 1 As shown, a method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation is presented. This method includes the following steps: Step S100: Establish a model. Based on the basic principles of leaf tip timing and array signal processing theory, establish a sparse layout sampling model for leaf tip timing. Step S200: Obtain the augmented covariance matrix based on the model, and obtain the augmented covariance matrix under coprime array based on the sparse layout sampling model; Step S300: Obtain coarse estimates of the parameters, and provide a sparse iterative covariance estimation method based on the covariance fitting criterion to obtain coarse estimates of each parameter of the array signal. Step S400: Obtain the fine estimate of the parameters, optimize the coarse estimate by the asymptotically optimal maximum likelihood estimation method, and obtain the fine estimate of each parameter of the array signal.

[0022] The maximum likelihood estimation method is used to optimize the estimated values ​​of each parameter, thereby improving the accuracy of the parameter estimation and effectively extracting the weak frequency and amplitude changes in the leaf tip timing signal.

[0023] Example 1 In one embodiment of the present invention, step S100 specifically includes the following steps: Step S101: Set up a system by Circles and each circle Composed of individual sensors An array of probes, the array receiving signals from the tip of a rotating blade. Narrowband normalized frequency The vibration signal has power corresponding to each frequency as follows: The sensor location layout vector is The rotational speed was normalized to The corresponding array steering vector is , ; Step S102: Collect data using the sparse layout sampling model with leaf tip timing from step S101. The second width is The snapshot matrix, in which the first The received signal corresponding to this quick snapshot is: , ,in, Indicates a single snapshot; It is an unknown signal waveform that includes signal power. It is an array manifold matrix, noise vector It follows the variance Additive white Gaussian noise, The sampling covariance matrix corresponding to the matrix is: .

[0024] Example 2 Another embodiment of the present invention is that step S200 specifically includes the following steps: Step S201: When the sensor layout satisfies the sparse linear array requirement, adjust the layout angle of the leaf tip timing sensor as follows: , ,in, The minimum included angle between all sensors can be represented by a virtual array element set consisting of a difference array: ; And this virtual array element set contains A continuous virtual array element ,in, This virtual array of elements forms a uniform linear array, replacing the original sparse linear array. Step S202: Construct the difference array selection matrix The first of the matrix Line 1 The elements of the column are: ; The output signal is obtained from 57 virtual array elements: ; Step S203: Augmented covariance matrix Signals can be output from virtual array elements Get as ,in, yes The Subarrays.

[0025] Example 3 Another embodiment of the present invention is that step S300, obtaining a coarse estimate of the signal parameters through sparse iterative covariance estimation, includes the following steps: Construct an optimization problem based on the covariance fitting criterion: ; ; ; ; in, It is a diagonal matrix composed of signal power. It is composed of frequency grid and virtual uniform linear array The constructed manifold matrix, normalized upper frequency limit .

[0026] Solving the above optimization problem using the SDPT3 convex optimization toolkit yields a coarse estimate of the signal parameters, specifically along the frequency grid. The obtained signal power and noise variance .

[0027] Example 4 Another embodiment of the present invention is that updating the result of step S300 by the maximum likelihood estimation method in step S400 includes the following steps: Step S401: Preserve signal power middle The largest value and its corresponding grid frequency This yields a rough estimate of the signal covariance matrix. .

[0028] The log-likelihood function for the unconditional maximum likelihood estimation problem of a signal is established as follows: ; in, The number of data acquisitions is based on the leaf tip timing principle. For matrix The determinant, For the sampling covariance matrix, For frequency, A diagonal matrix composed of signal power; Step S402: Calculate the likelihood function for frequency, power, and noise variance respectively. Partial derivatives: ; ; ; in, ; The update step size for each parameter is: ; ; ; in, It is the size of the frequency grid. and It is a scale parameter. yes initial value, It is the growth rate.

[0029] Step S403: Set the maximum number of iterations The iterative operation is repeated as follows: ; ; ; The iteration count reaches the iteration stopping condition. When the iteration stops, the signal frequency is obtained. The corresponding signal power is and noise variance .

[0030] Example 5 like Figure 2 As shown, another embodiment of the present invention compares the noise robustness of the method, with the simulation parameters set as follows: a special difference comatrix is ​​considered for comparison, and coprime pairs are selected. and ,in .

[0031] The virtual ULA corresponding to this array is obtained as Its sampling interval Normalized to 1. The frequency of the synthesized signal. The power corresponding to each frequency is .

[0032] The signal snapshot matrix is ​​composed of Uniform white Gaussian noise was added to the signal to construct a signal-to-noise ratio of 10 dB. ).

[0033] Figure 2 Four methods are presented, including: a hybrid off-grid leaf-end timing signal parameter estimation method based on maximum likelihood estimation (HYBRID) provided in this embodiment; an orthogonal matched pursuit (OMP) method; a multiple signal classification (MUSIC) method; and a minimum variance distortionless response (MVDR) method, to recover the spectrum of the synthesized signal. For the main parameters of these methods, the number of sources is assumed to be... To allow for flexibility, the number of iterations in step S300 is set to 20, and the maximum number of iterations for maximum likelihood estimation is set to 120.

[0034] The frequency grid of the spectrum in the above method is set to The grid size is The results obtained by the four methods are as follows: Figure 2As shown, the red boxes represent baseline values. In Figure 2(a), the estimation results obtained by this invention are very close to the expected optimal values. Figure 2 only marks the amplitudes corresponding to the correct frequency components and retains the original spectra of MUSIC and MVDR. Without further correction, the original spectrum from OMP performs best, while the amplitude results from SPICE tend to underestimate. The original spectra obtained from MUSIC and MVDR are very weak around 0.606 and 0.726, especially MVDR, where some spurious peaks appear. After further correction, all four methods can effectively estimate the frequency of the signal, but are limited by the grid size. Except for the proposed method, OMP is the most noteworthy, with two additional frequencies appearing in the grid adjacent to the correct frequency. In the estimation of signal amplitude, the hybrid method performs best, followed by OMP, while the estimates from MUSIC and MVDR are biased towards higher values. In fact, the amplitude estimation error comes not only from frequency mismatch but also from noise interference. The proposed method simultaneously estimates the noise variance. This further improves the accuracy of signal amplitude. In a comprehensive comparison, the estimation method of this invention is superior to the other three methods.

[0035] Example 6 like Figure 3 As shown, another embodiment of the present invention further compares the noise robustness of the method proposed in this embodiment. The simulation parameters are set as follows: the signal parameters remain unchanged and are the same as in Embodiment 5. 5000 Monte Carlo simulations were performed at signal-to-noise ratios of -10dB to 15dB at intervals of 2.5dB, thereby obtaining more statistically significant results.

[0036] Figure 3(a) presents the average RMSE results of frequency estimations obtained by six methods, including the Hybrid Estimation Method (HYBRID), Orthogonal Matching Pursuit (OMP), Multiple Signal Classification (MUSIC), Minimum Variance Distortionless Response (MVDR), Sparse Iterative Covariance Estimation (SPICE), and Craméro (CRB) provided in this embodiment. CRB is used to indicate the boundary of parameter estimation under the current conditions. The effect of gridding is obvious. As the signal-to-noise ratio increases, the frequency RMSE of SPICE, OMP, and MUSIC decreases. The error tends to stabilize near the reference value, and this error limit is exactly equal to the RMSE between the frequency grid and the reference value. SPICE and OMP, both based on sparse methods, have similar frequency estimation accuracy, reaching their limits near -5dB. MUSIC, on the other hand, lags behind the former two, starting to be limited by the grid size near 5dB. Meanwhile, MVDR's frequency estimation results still lag behind. This is due to sacrificing some robustness to noise in the construction of the optimal weight vector. Unlike these methods, the hybrid method significantly overcomes the grid error through MLE. In the low SNR region, the error of the hybrid method is close to that of SPICE. However, as the SNR increases, especially when the SNR is greater than 0dB, the hybrid method almost reaches the performance of CRB. This indicates that the hybrid method is asymptotically efficient.

[0037] In Figure 3(b), the effect of gridding is also present in the amplitude estimation results, although the limits differ for various methods. SPICE exhibits the largest RMSE of amplitude, attributed to the inherent estimation error introduced by the method itself, which remains almost constant with increasing signal-to-noise ratio. The result of MUSIC is not only attributed to the grid effect; it also stems from the inherent bias introduced by directly using least squares, leading to an overestimation of amplitude, which can also be observed in Figure 2(c).

[0038] For the method provided in this embodiment, the amplitude result is similar to that of the frequency; when the signal-to-noise ratio is greater than 0 dB, the hybrid method provided in this embodiment is close to CRB. In summary, the hybrid method of this embodiment not only significantly improves the accuracy of the initial values ​​obtained from SPICE, but also shows superior performance in terms of estimation accuracy and noise robustness.

[0039] Example 7 like Figure 4 As shown, another embodiment of the present invention generates real blade tip timing experimental data using a blade tip timing test bench. A crack is pre-installed on one blade in the experimental setup to simulate a fault. Four blade tip timing probes and one OPR probe are installed, wherein the position angle of the blade tip timing probes is [missing information]. The OPR signal is used to measure rotational speed, ignoring speed fluctuations within the same rotation. To minimize the impact of noise on the blade tip timing signal measurement, air excitation is used to enhance the blade's response, thereby increasing the vibration amplitude.

[0040] Figure 4 shows a segment of the collected experimental data. The disk rotation speed was increased from 8000 rpm to 9000 rpm, held constant for 20 seconds, and then reduced back to 8000 rpm. In subsequent comparisons, the data was divided into segments of length 1040, each segment forming a... The snapshot matrix is ​​used. Based on this, parameter selection is performed for the four methods. Specifically, the frequency grid is still set to 1Hz. The sparsity level of OMP is set to 5 to account for measurement uncertainty and noise, while the maximum number of iterations for SPICE and MLE is set to 20 and 150, respectively.

[0041] Figure 5 illustrates the frequency and amplitude variations over time of the leaf tip timing experimental signal obtained using five methods: the Hybrid estimation method (HYBRID), Orthogonal Matching Pursuit (OMP), Multiple Signal Classification (MUSIC), Minimum Variance Distortionless Response (MVDR), and Sparse Iterative Covariance Estimation (SPICE). The Campbell plot, based on finite element analysis, shows the natural frequency variation with velocity represented by a red line for reference. In terms of frequency estimation, SPICE and OMP exhibit similar behavior, with frequencies still dispersed within a single frequency band. However, after improvement, the frequencies converge significantly towards the reference line of the natural frequencies, and the trend of velocity variation becomes more pronounced. Compared to OMP, the mesh has less impact on MUSIC and OMP, and their results are closer to the true values. Regarding amplitude, the results from Hybrid, MUSIC, and MVDR are very similar; the main difference lies in the smaller variance and narrower prediction interval of the Hybrid method. The accurate amplitude obtained through OMP is limited by the frequency, and frequency estimation errors lead to greater amplitude dispersion. In summary, the maximum likelihood estimation method used in this embodiment to update parameters of the blade tip timing experimental data significantly reduces estimation errors and produces better results than the other methods. Furthermore, blade tip timing data for cracked and normal blades were analyzed. Here, only the results of the hybrid method in this embodiment are compared to demonstrate its effectiveness in blade condition monitoring.

[0042] First, in Figure 6(a), the results for normal and cracked blades are compared side-by-side, with blue representing normal and orange representing cracked. It can be observed that the differences in natural frequencies between the two cases are difficult to discern and overlap, leading to misjudgments of blade condition. Figure 6(b) shows the precise frequencies, where the differences between normal and faulty blades become very pronounced, with the natural frequency of the cracked blade being 3-4 Hz lower than that of the normal blade.

[0043] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation, characterized in that, The method includes the following steps: Model establishment: Based on the basic principles of leaf tip timing and array signal processing theory, a sparse layout sampling model for leaf tip timing is established; The augmented covariance matrix is ​​obtained from the model: The augmented covariance matrix under coprime arrays is obtained from the sparse layout sampling model; Obtaining coarse estimates of parameters: A sparse iterative covariance estimation method based on the covariance fitting criterion is presented to obtain coarse estimates of each parameter of the array signal; Obtaining precise estimates of parameters: The coarse estimates are optimized by the asymptotically optimal maximum likelihood estimation method to obtain precise estimates of each parameter of the array signal.

2. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 1, characterized in that, Array signal processing includes the following steps: Set a common Circles and each circle Composed of individual sensors An array of probes, the array receiving signals from the tip of a rotating blade. Narrowband frequency The vibration signal has a frequency corresponding to a power of The frequencies that constitute the vibration signal are represented as vectors. : ； in, This is the transpose of the matrix.

3. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 2, characterized in that, Based on the fundamental principle of blade tip timing, the sensor is placed on the same plane as the vibration, and the installation angle of the blade tip timing sensor is... for: ； in, Derived from the sensor location layout vector, Indicates the first The distance between each probe and any origin point The unit is rad. For probe array; The corresponding array steering vector is: ， ； in, , The imaginary unit, Let be the angular velocity of the bladed disk, and The value is constant.

4. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 3, characterized in that, Establish a sparse layout sampling model for leaf tip timing: Data is collected based on the leaf tip timing principle. The second width is The snapshot matrix, in which the first The received signal corresponding to this quick snapshot is: ， ； ； in, Indicates a single snapshot. The unknown signal waveform represents the signal power. It is an array of manifold rectangles. Let be the noise vector, and Noise vector It follows a mean of 0 and a variance of . Additive white Gaussian noise; The sampling covariance matrix corresponding to the matrix is: ； in, for The conjugate transpose of a matrix.

5. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 3, characterized in that, When the sensor layout satisfies the sparse linear array requirement, the layout angle of the blade tip timing sensor is adjusted as follows: ， ； in, The minimum included angle between all sensors. To indicate the first An integer representing the location of a sensor; Set up a virtual array element set consisting of a differential array: ； Among them, the virtual array element set contains A continuous set of virtual array elements, forming a uniform linear array that replaces the original sparse linear array. Construct the difference array selection matrix The first of the matrix Line 1 The elements of the column are: ； in, To round up, For weighting functions; Depend on Each virtual array element receives the output signal. : ； in, The Kronecker product of the matrices. For variance, For vectorization operations on matrices, The unknown signal waveform represents the signal power. Based on the output signal of the virtual array element Obtain the augmented covariance matrix : ； in, yes The Subarrays, for The conjugate transpose of a matrix.

6. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 1, characterized in that, Construct an optimization problem based on the covariance fitting criterion: ； ； ； ； ； ； ； in, This represents the trace used to extract the trajectory. This indicates taking the inverse of the matrix. To augment the covariance matrix, for The conjugate transpose of a matrix. and To optimize the optimization variables of the problem, The covariance matrix under sparse representation, A diagonal matrix composed of signal power. For frequency grid The upper frequency limit of the manifold matrix constructed from the virtual uniform linear array is . , Let g be the weight of the signal power corresponding to the g-th frequency. This is the normalization factor for noise power. The covariance matrix formed by virtual matrix linear arrays Dimensions The number of frequency grids; The above optimization problem was solved using the SDPT3 convex optimization toolkit, through a frequency grid. Obtain signal power and noise variance This allows for a rough estimate of the array signal parameters.

7. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 6, characterized in that, Preserve signal power middle The largest value and its corresponding grid frequency We obtain a rough estimate of the signal covariance matrix: ； in, This is a preliminary estimate of the noise variance.

8. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 7, characterized in that, The log-likelihood function for the unconditional maximum likelihood estimation problem of a signal is established as follows: ； in, The number of data acquisitions is based on the leaf tip timing principle. For matrix The determinant, For the sampling covariance matrix, For frequency, A diagonal matrix composed of signal power; Calculate the frequency, signal power, and variance versus the likelihood function respectively. Partial derivatives: ； ； ； ; The update step size for each parameter is: ； ； ； in, It is the size of the frequency grid. and It is a scale parameter. yes initial value, It is the growth rate; By optimizing the coarse estimate using the likelihood function, the fine estimate of each parameter of the array signal is obtained.

9. The method for estimating parameters of hybrid off-grid leaf-end timing signals based on maximum likelihood estimation according to claim 8, characterized in that, Set the maximum number of iterations. Repeated iteration operation: ； ； ； Among them, superscript This represents the number of iterations. When the number of iterations exceeds the maximum number of iterations Or the iteration stopping condition is met. Stop iterating when the time is right.

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