A method and system for joint estimation of DOA and carrier frequency based on tensor modeling

Abstract

This invention provides a method and system for joint estimation of DOA and carrier frequency based on tensor modeling, relating to the field of array signal processing technology. By acquiring and rearranging spatiotemporally non-uniformly sampled data, a third-order observation tensor is constructed. A low-rank PARAFAC decomposition model is established, and after PARAFAC decomposition, spatial phase difference vectors and temporal delay phase difference vectors are constructed. A reference difference set is selected based on an unambiguous difference screening criterion to obtain initial estimates of target frequency parameters and spatial phase parameters. Integer ambiguity numbers are calculated by combining principal value phase observations to complete the unwrapping and compensation of spatial and temporal delay phases, obtaining unambiguous phase difference estimates for each target. Joint estimation of DOA and carrier frequency is achieved through least-squares fitting. This solves the phase entanglement problem under non-uniform sampling, reduces computational complexity, improves the resolution of closely spaced incoherent targets, and effectively improves the stability and accuracy of parameter estimation.

Description

Technical Field

[0001] This invention relates to the field of visibility meters, and in particular to a method and system for joint estimation of DOA and carrier frequency based on tensor modeling. Background Technology

[0002] In array signal processing fields such as radar, communications, and electronic reconnaissance, accurate estimation of the Direction of Arrival (DOA) and carrier frequency is a key technology for target localization, spectrum management, and interference suppression. Traditional parameter estimation methods typically assume that the signal carrier frequency is known or that all signals have the same carrier frequency. However, in real-world complex electromagnetic environments, multiple signals are often distributed across different frequency bands and their frequencies are unknown. Furthermore, due to limitations in system cost, hardware complexity, and sampling conditions, array spatial and temporal dimension sampling often exhibits non-uniform characteristics.

[0003] Existing research has proposed several joint estimation methods, but these have the following limitations: 1. Joint estimation of angle and frequency based on the ESPRIT framework relies on a uniform array structure and requires additional parameter pairing, making the algorithm relatively cumbersome. 2. Combining covariance reconstruction with ESPRIT improves the parameter recovery process and achieves automatic pairing, but it still mainly targets regular linear array structures and has limited adaptability to spatiotemporally non-uniform sampling scenarios. Another approach considers sparse sampling in both spatial and temporal dimensions, but its parameter recovery relies on two-dimensional pseudospectral search, resulting in high computational complexity, which further increases with higher resolution requirements. 3. Joint estimation of angle and frequency is achieved using trilinear decomposition, but the model is mainly based on an oversampled output structure and is not specifically designed for spatiotemporally non-uniform sampling scenarios. Summary of the Invention

[0004] The main objective of this invention is to provide a joint estimation method for DOA and carrier frequency based on tensor modeling, which solves the problems of strong dependence on regular sampling structure, cumbersome parameter recovery process, complex joint estimation implementation, limited resolution performance in close-range target scenarios, and low accuracy of parameter estimation in the prior art.

[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a joint estimation method of DOA and carrier frequency based on tensor modeling, comprising the following steps: S1: Acquire spatiotemporal non-uniform sampling data, rearrange them according to spatial dimension, temporal delay dimension and snapshot dimension, construct a third-order observation tensor, and establish the corresponding low-rank PARAFAC decomposition model; S2: Perform PARAFAC decomposition on the third-order observation tensor to obtain the spatial factor matrix, time delay factor matrix and snapshot factor matrix, and eliminate scale ambiguity by column normalization; S3: Extract the phase difference between adjacent array element pairs and adjacent time delay pairs using the normalized spatial factor matrix and time delay factor matrix respectively, and construct the spatial phase difference vector and the time delay phase difference vector; select the reference difference set according to the unambiguous difference screening criterion, and obtain the initial estimate of the target frequency parameter and spatial phase parameter through least squares fitting; S4: Use the initial estimation results to predict the fully differential phase, combine the principal value phase observation to calculate the integer ambiguity number, complete the unwrapping and compensation of the spatial and temporal delay phases, and obtain the unambiguous phase difference estimate of each target. S5: Estimate the carrier frequency based on the unambiguous time delay phase difference, combine the unambiguous spatial phase difference with the estimated frequency, and solve the signal DOA by least squares fitting to achieve joint estimation of DOA and carrier frequency.

[0006] In the preferred embodiment, in step S1, the spatiotemporal non-uniform sampling is as follows: the array adopts a nonlinear non-uniform linear array, and multiple non-uniform time delays are connected after each array element. The spatial sampling position and the time delay sampling interval are both non-uniformly distributed. Assuming there is An uncorrelated signal source is incident from the far field onto a nonlinear array, which contains There are 1 array elements, with the first element as the reference element. The array coordinates can be denoted as: Each array element is connected to The path has a non-uniform time delay, let the first one be... Road delay is ,in, The unit delay; the first The DOA and carrier frequency of each signal are: Under the narrowband and plane wave assumptions, the array receiving vector can be expressed as: (1); Among them, the baseband signal is Subscript Indicates transpose. For noise, the spatial steering matrix is ,make , If the speed is the speed of light, then each column of this matrix is ​​a spatial steering vector. ; For the One signal source, in Time is determined by narrowband conditions ,remember Then the first The observations with time delays are as follows: (2); in, The time delay steering matrix is ​​defined as follows: The time delay steering vector is By stacking all the delayed outputs, the overall observation matrix is ​​obtained: (3); in, For Khatri-Rao product, For noise; definition For spacetime guiding matrix, collect The sampling matrix is ​​obtained from the sampling of each snapshot. (4); in, The source signal matrix, This is the noise matrix.

[0007] In the preferred embodiment, in step S1, the third-order observation tensor is constructed as follows: sampling matrix To rearrange columns, the expression is: (5); in, For spatial orientation matrix, For delay steering matrix, This is the transpose of the source signal matrix. For the noise matrix, For complex fields, These are the indices for the spatial dimension, the time delay dimension, and the snapshot dimension, respectively; M is the total number of array elements, N is the total number of time delay channels, and L is the total number of snapshots.

[0008] In the preferred scheme, in step S2, the PARAFAC decomposition satisfies the Kruskal condition: (6); in, , and Each is a matrix , and The Kruskal rank; K is the total number of signal sources; The factor matrix obtained from PARAFAC decomposition is as follows: (7); in, Let be the permutation matrix. , , It is a diagonal scaling matrix. , , This is the corresponding estimation error matrix; For spatial orientation matrix, For delay steering matrix, This is the transpose of the source signal matrix; This is an estimate of the spatial factor matrix. This is the estimated value of the time delay factor matrix. This is the estimated value of the snap factor matrix; Pick and The The columns are denoted as follows: , Normalize its first element: (8); Obtain the The normalized space and time delay factor column vectors corresponding to each information source and Its specific form is and .

[0009] In the preferred embodiment, step S3, constructing the spatial phase difference vector and the time delay phase difference vector, includes: Obtain the normalized space factor matrix and delay factor matrix , for the For each information source, the phase difference estimates corresponding to adjacent array element pairs and adjacent time delay pairs are as follows: (9); in, For phase extraction operations, the principal value falls on This range; For the m-th adjacent array element, For the nth adjacent delay, define and The phase differences between adjacent array element pairs and adjacent time delay pairs are respectively denoted as... and : (10); in, , These are the phase change rates in the spatial dimension and the frequency dimension, respectively. Furthermore, define and For a fixed target Stack the phase principal values ​​corresponding to all adjacent time delay pairs and adjacent array element pairs to form a phase difference vector: (11); in, The element spacing matrix, This is the time delay interval matrix; Theoretically, phase factor and It can be obtained through the following least squares estimation. (12); in, and This represents the actual observed principal phase value vector; where, Indicates a false reversal; When the array element spacing or time delay difference If it is too large, the phase change will exceed... The relationship between the estimated value and the true value is as follows: (13); in, and It is an unknown integer vector.

[0010] In the preferred scheme, in step S3, the ambiguity-free differential screening criterion is: the element spacing and time delay difference corresponding to the ambiguity-free reference unit must satisfy the Nyquist constraint condition, expressed as: (14); in, To detect the highest carrier frequency within the frequency band; For the spacing between array elements, For time delay difference, The unit delay value, At the speed of light, This is the sine function of DOA.

[0011] In the preferred embodiment, step S3 involves obtaining initial estimates of the target frequency parameters and spatial phase parameters, specifically as follows: Vectors composed of unambiguous phase observations and Construct a linear regression model: (15); in, and These characterize the linear rate of change of the phase difference with respect to the interval in the frequency dimension and the spatial dimension, respectively. and This is the actual observed principal phase vector; The pseudo-inverse of the element spacing matrix. This is the pseudo-inverse of the time delay interval matrix; , These represent the phase change rates corresponding to the spatial dimension and the frequency dimension, respectively. To satisfy the Nyquist constraint on the element spacing matrix, The time delay interval matrix to satisfy the Nyquist constraint; Predict all phase differences, including the ambiguous segments. and The expression is: (16).

[0012] In the preferred embodiment, in step S4, the integer ambiguity number is calculated as follows: the full differential phase is predicted using the initial estimation result, and the integer ambiguity number is calculated by combining the principal value phase observation, thereby completing the unwrapping and compensation of the spatial and temporal delay phases and obtaining the unambiguous phase difference estimate of each target. Obtain all phase differences, including the blurred segments. and By comparing the predicted values ​​with the observed principal values, the number of integer fuzzy numbers can be estimated. and : (17); in, This indicates rounding to the nearest integer. and The predicted values ​​are those with fuzzy time delays and spatial phase differences; and The principal values ​​for time delay and spatial phase difference observations; The observed phase is compensated to obtain the deblurred phase estimate. and : (18).

[0013] In the preferred embodiment, in step S5, the carrier frequency and azimuth angle of the target are calculated using the compensated phase: (19); in, The frequency phase change rate corresponding to the k-th target. Let be the spatial phase change rate corresponding to the k-th target. The speed of light; This is the delay value. , The pseudo-inverses of the element spacing matrix and the time delay spacing matrix are given. , This is for spatial and temporal phase estimation after deblurring.

[0014] Secondly, the present invention provides a joint estimation system for DOA and carrier frequency based on tensor modeling, applicable to the aforementioned joint estimation method for DOA and carrier frequency based on tensor modeling, comprising: The tensor construction module is used to acquire the received data of the spatiotemporal non-uniform array, rearrange it according to the spatial dimension, time delay dimension and snapshot dimension, construct the third-order observation tensor, and establish its low-rank PARAFAC decomposition model. The PARAFAC decomposition module is used to perform PARAFAC decomposition on the third-order observation tensor to obtain the spatial factor matrix, the time delay factor matrix and the snapshot factor matrix, and to eliminate scale ambiguity by column normalization. The initial parameter calculation module is used to extract the phase difference between adjacent array element pairs and adjacent time delay pairs using the normalized spatial factor matrix and time delay factor matrix respectively, and construct the spatial phase difference vector and the time delay phase difference vector; select the reference difference set according to the unambiguous difference screening criterion, and obtain the initial estimate of the target frequency parameter and spatial phase parameter through least squares fitting; The phase unwrapping module is used to predict the fully differential phase using the initial estimation results, calculate the integer ambiguity number by combining the principal value phase observation, complete the unwrapping and compensation of the spatial and temporal delay phases, and obtain the unambiguous phase difference estimate of each target. The parameter calculation module is used to estimate the target frequency based on the compensated time delay phase difference. Then, by combining the compensated spatial phase difference and the estimated frequency, the target orientation angle is estimated through least squares fitting, thus realizing the joint estimation of angle and frequency parameters.

[0015] This invention provides a joint estimation method for DOA and carrier frequency based on tensor modeling. The method comprises the following steps: S1: Acquire and rearrange spatiotemporally non-uniformly sampled data to construct a third-order observation tensor and establish a low-rank PARAFAC decomposition model; S2: Perform PARAFAC decomposition on the third-order observation tensor; S3: Construct spatial phase difference vectors and temporal delay phase difference vectors; Select a reference difference set based on unambiguous differential selection criteria to obtain initial estimates of target frequency parameters and spatial phase parameters; S4: Calculate integer ambiguity numbers by combining principal phase observations to complete the unwrapping and compensation of spatial and temporal delay phases, obtaining unambiguous phase difference estimates for each target; S5: Combine the unambiguous spatial phase difference with the estimated frequency, solve the signal DOA through least-squares fitting, achieving joint estimation of DOA and carrier frequency while maintaining the structural integrity of the multi-dimensional observation data; Through unambiguous differential selection and phase unwrapping mechanisms, the phase entanglement problem under non-uniform sampling is effectively solved; It eliminates the need for two-dimensional pseudospectral search, reducing computational complexity, improving the resolution of closely spaced incoherent targets, and effectively improving the stability and accuracy of parameter estimation. Attached Figure Description

[0016] The present invention will be further described below with reference to the accompanying drawings and embodiments: Figure 1 This is a flowchart of the joint estimation method of the present invention; Figure 2 This is a schematic diagram of the spatiotemporal non-uniform sampling system to which this invention is applicable; Figure 3 This is a scatter plot of the simulation results of the joint estimation of this invention. Detailed Implementation

[0017] Example 1 like Figure 1-3 As shown, a joint estimation method for DOA and carrier frequency based on tensor modeling includes the following steps: S1: Acquire spatiotemporally non-uniform sampling data, rearrange them according to spatial dimension, temporal delay dimension and snapshot dimension, construct a third-order observation tensor, and establish the corresponding low-rank PARAFAC decomposition model.

[0018] S2: Perform PARAFAC decomposition on the third-order observation tensor to obtain the spatial factor matrix, time delay factor matrix and snapshot factor matrix, and eliminate scale ambiguity by column normalization.

[0019] S3: Extract the phase difference between adjacent array element pairs and adjacent time delay pairs using the normalized spatial factor matrix and time delay factor matrix respectively, and construct the spatial phase difference vector and the time delay phase difference vector; select the reference difference set according to the unambiguous difference screening criterion, and obtain the initial estimate of the target frequency parameter and spatial phase parameter through least squares fitting.

[0020] S4: Use the initial estimation results to predict the fully differential phase, combine the principal value phase observation to calculate the integer ambiguity number, complete the unwrapping and compensation of the spatial and temporal delay phases, and obtain the unambiguous phase difference estimate of each target.

[0021] S5: Estimate the carrier frequency based on the unambiguous time delay phase difference, combine the unambiguous spatial phase difference with the estimated frequency, and solve the signal DOA by least squares fitting to achieve joint estimation of DOA and carrier frequency.

[0022] This embodiment is applicable to spatiotemporally non-uniform sampling scenarios. By constructing a third-order observation tensor and establishing a low-rank PARAFAC decomposition model, it achieves joint recovery of angle and frequency parameters while maintaining the structural integrity of multidimensional observation data. Through unambiguous difference filtering and phase unwrapping mechanisms, it effectively solves the phase entanglement problem under non-uniform sampling. It eliminates the need for two-dimensional pseudospectral search, reduces computational complexity, improves the ability to distinguish closely spaced incoherent targets, and effectively improves the stability and accuracy of parameter estimation.

[0023] In the preferred embodiment, in step S1, the spatiotemporal non-uniform sampling is as follows: the array adopts a nonlinear non-uniform linear array, and multiple non-uniform time delays are connected after each array element. The spatial sampling position and the time delay sampling interval are both non-uniformly distributed. This embodiment considers the spatial aspect from a modeling perspective. An uncorrelated signal source is incident from the far field onto a nonlinear array, which contains There are 1 array elements, with the first element as the reference element. The array coordinates can be denoted as: Each array element is connected to The path has a non-uniform time delay, let the first one be... Road delay is ,in, The unit delay; the first The DOA and carrier frequency of each signal are: Under the narrowband and plane wave assumptions, the array receiving vector can be expressed as: (1); Among them, the baseband signal is Subscript Indicates transpose. For noise, the spatial steering matrix is ,make , If the speed is the speed of light, then each column of this matrix is ​​a spatial steering vector. .

[0024] For the One signal source, in Time is determined by narrowband conditions ,remember Then the first The observation with time delay is (2); in, The time delay steering matrix is ​​defined as follows: The time delay steering vector is .

[0025] Stacking all the delayed outputs yields the overall observation matrix: (3); in, For Khatri-Rao product, It is noise.

[0026] definition For spacetime guiding matrix, collect The sampling matrix is ​​obtained from the samples taken in each snapshot: (4); in, The source signal matrix, This is the noise matrix.

[0027] This embodiment rearranges the sampling matrix column-wise into a third-order tensor consisting of spatial, temporal, and snapshot dimensions, thus fully preserving the multidimensional structural information of the spatiotemporally non-uniform sampling data and improving the integrity of tensor decomposition and parameter estimation data.

[0028] In the preferred scheme, in step S1, the construction method of the third-order observation tensor is to construct the sampling matrix. To rearrange columns, the expression is: (5); in, For spatial orientation matrix, For delay steering matrix, This is the transpose of the source signal matrix. For the noise matrix, For complex fields, These are the indices for the spatial dimension, the time delay dimension, and the snapshot dimension, respectively; M is the total number of array elements, N is the total number of time delay channels, and L is the total number of snapshots.

[0029] This embodiment ensures the essential uniqueness of the decomposition result by using PARAFAC decomposition that satisfies Kruskal's condition; the spatial factor matrix and the time delay factor matrix obtained by decomposition share the same permutation matrix, realizing automatic pairing of angle parameters and frequency parameters, simplifying the parameter recovery process, and avoiding the cumbersome parameter pairing steps in traditional methods.

[0030] In the preferred scheme, in step S2, the PARAFAC decomposition is essentially unique when the following Kruskal condition is satisfied: (6); in, , and Each is a matrix , and The Kruskal rank of a given matrix, i.e., for any given matrix, is determined by any Kruskal rank. The largest integer that can be reached when all columns are linearly independent K represents the total number of signal sources. After PARAFAC decomposition, estimates of these matrices can be obtained. , and .

[0031] Since the PARAFAC decomposition involves column permutation fuzziness and column scaling fuzziness, the resulting factor matrix is: (7); in, Let be the permutation matrix. , , It is a diagonal scaling matrix. , , This is the corresponding estimation error matrix; For spatial orientation matrix, For delay steering matrix, This is the transpose of the source signal matrix; This is an estimate of the spatial factor matrix. This is the estimated value of the time delay factor matrix. This is the estimated value of the snap factor matrix.

[0032] After completing the PARAFAC decomposition, take and The The columns are denoted as follows: , To eliminate column scaling blur and common phase blur, the first element is normalized: (8); Thus obtain the first The normalized space and time delay factor column vectors corresponding to each information source and Its specific form is and .

[0033] In the preferred embodiment, step S3, constructing the spatial phase difference vector and the time delay phase difference vector, includes: Obtain the normalized space factor matrix and delay factor matrix , for the For each information source, the phase difference estimates corresponding to adjacent array element pairs and adjacent time delay pairs are as follows: (9); in, For phase extraction operations, the principal value falls on This range; For the m-th adjacent array element, Let n be the delay of the nth adjacent time. (Definition) and The phase differences between adjacent array element pairs and adjacent time delay pairs are respectively denoted as... and : (10); in, , These are the phase change rates in the spatial dimension and the frequency dimension, respectively.

[0034] Furthermore, define and For a fixed target Stack the phase principal values ​​corresponding to all adjacent time delay pairs and adjacent array element pairs to form a phase difference vector: (11); in, The element spacing matrix, This is the time delay interval matrix.

[0035] Theoretically, phase factor and It can be obtained through the following least squares estimation: (12); in, and Let be the principal value vector of the phase as actually observed; where, It indicates a false reversal.

[0036] When the array element spacing or time delay difference If it is too large, the phase change will exceed... The relationship between the estimated value and the true value is as follows: (13); in, and is an unknown integer vector representing the integer ambiguity number corresponding to each phase difference.

[0037] In this embodiment, a linear regression model is constructed using an unambiguous reference difference set. Initial estimates of the target frequency parameters and spatial phase parameters are obtained through least squares fitting. The calculation is simple and the stability is high, which improves the accuracy of subsequent phase unwrapping.

[0038] In the preferred scheme, in step S3, the ambiguity-free differential screening criterion is: the element spacing and time delay difference corresponding to the ambiguity-free reference unit must satisfy the Nyquist constraint condition, expressed as: (14); in, To detect the highest carrier frequency within the frequency band; For the spacing between array elements, For time delay difference, The unit delay value, At the speed of light, This is the sine function of DOA.

[0039] All those that meet the conditions and Construct vectors respectively and The corresponding unambiguous phase observations constitute a vector. and .

[0040] This embodiment improves the reliability of linear regression sample data and effectively solves the phase ambiguity problem caused by excessive array element spacing or time delay difference by setting an unambiguous difference screening criterion based on Nyquist constraints.

[0041] In the preferred embodiment, step S3 involves obtaining initial estimates of the target frequency parameters and spatial phase parameters, specifically as follows: As shown in equation (11), there is a linear relationship between the phase difference and the interval. A vector is formed using unambiguous phase observations. and Construct a linear regression model: (15); in, and These characterize the linear rate of change of the phase difference with respect to the interval in the frequency dimension and the spatial dimension, respectively. and This is the actual observed principal phase vector; The pseudo-inverse of the element spacing matrix. This is the pseudo-inverse of the time delay interval matrix; , These represent the phase change rates corresponding to the spatial dimension and the frequency dimension, respectively. To satisfy the Nyquist constraint on the element spacing matrix, The time delay interval matrix to satisfy the Nyquist constraint.

[0042] Predict all phase differences, including the ambiguous segments. and The expression is: (16).

[0043] In the preferred scheme, in step S4, the integer ambiguity number is calculated as follows: the initial estimation result is used to predict the fully differential phase, and the integer ambiguity number is calculated by combining the principal value phase observation, thereby completing the unwrapping and compensation of the spatial and temporal delay phases and obtaining the unambiguous phase difference estimate of each target.

[0044] Obtain all phase differences, including the blurred segments. and By comparing the predicted values ​​with the observed principal values, the number of integer fuzzy numbers can be estimated. and : (17); in, This indicates rounding to the nearest integer. and The predicted values ​​are those with fuzzy time delays and spatial phase differences; and These are the principal values ​​for time delay and spatial phase difference observations.

[0045] The observed phase is compensated to obtain the deblurred phase estimate. and : (18).

[0046] This embodiment predicts the full phase difference through initial estimation, accurately calculates the integer ambiguity number by combining principal phase observation, and completes phase unwrapping and compensation. This mechanism can effectively handle the phase entanglement problem under non-uniform structures and significantly improve the accuracy and robustness of phase estimation.

[0047] In the preferred embodiment, in step S5, the carrier frequency and direction angle of the target signal are calculated using the compensated phase: (19); in, The frequency phase change rate corresponding to the k-th target. Let be the spatial phase change rate corresponding to the k-th target. The speed of light; This is the delay value. , The pseudo-inverses of the element spacing matrix and the time delay spacing matrix are given. , This is for spatial and temporal phase estimation after deblurring.

[0048] This embodiment directly estimates the target carrier frequency by using the compensated time delay phase difference. The calculation formula is simple and does not require iterative search, thus improving computational efficiency. By using the compensated spatial phase difference and the estimated frequency, the target orientation angle is estimated through least squares fitting, realizing the joint recovery of angle and frequency parameters, reducing computational complexity and improving the real-time performance of data processing.

[0049] This embodiment is applicable to scenarios where the spatial array element positions are non-uniformly distributed and the temporal delay sampling intervals are non-uniformly distributed. It reduces the dependence on regular arrays and regular time delay structures, expands the scope of application of the method, and meets the non-uniform sampling requirements caused by cost and hardware limitations in actual systems.

[0050] The effectiveness of this embodiment is illustrated with a specific example. Assume there are K=3 far-field random source signals with DOAs of -0.6°, 0.1°, and 0.8°, and carrier frequencies of 5.5MHz, 8.5MHz, and 9.5MHz, respectively. Assume the number of array elements M=12, the number of delay channels N=10, and that some array elements and delays satisfy the Nyquist constraint condition of formula (14).

[0051] like Figure 1 The diagram shown is a schematic of a spacetime non-uniform sampling system, illustrating the non-uniform distribution of spatial array element positions. and the non-uniform distribution of time delay dimension sampling intervals This visually demonstrates the spatiotemporal non-uniform sampling scenario targeted by the present invention.

[0052] like Figure 2 As shown, under the conditions of L=400 snapshots and a signal-to-noise ratio of 5dB, the method of this embodiment is used to jointly estimate DOA and carrier frequency. The scatter plot results of 100 Monte Carlo simulations show the distribution of the estimation results for three incoherent target signals. It can be seen that the scheme of this embodiment can achieve joint estimation of DOA and carrier frequency in spatiotemporal non-uniform sampling scenarios, and has a strong ability to distinguish incoherent targets with small angular intervals, thus improving the stability and accuracy of parameter prediction.

[0053] This embodiment constructs a tensor observation model and performs low-rank decomposition to achieve joint recovery of angle and frequency parameters while maintaining the integrity of the multidimensional data structure. Addressing the challenges of phase entanglement and parameter recovery under non-uniform structures, a post-processing recovery mechanism is designed based on initial slope estimation using adjacent array element pairs, progressive phase unwinding, and residual screening with weighted estimation, effectively improving the stability and accuracy of parameter estimation. This method enables joint estimation of DOA and carrier frequency under non-uniform sampling conditions.

[0054] Example 2 Further illustrating with reference to Example 1, a joint estimation system for DOA and carrier frequency based on tensor modeling, applicable to the joint estimation method for DOA and carrier frequency based on tensor modeling in Example 1, includes: The tensor construction module is used to acquire the received data of the spatiotemporal nonuniform array, rearrange it according to the spatial dimension, time delay dimension and snapshot dimension, construct the third-order observation tensor, and establish its low-rank PARAFAC decomposition model.

[0055] The PARAFAC decomposition module is used to perform PARAFAC decomposition on the third-order observation tensor to obtain the spatial factor matrix, the time delay factor matrix, and the snapshot factor matrix, and eliminates scale ambiguity through column normalization.

[0056] The initial parameter calculation module is used to extract the phase difference between adjacent array element pairs and adjacent time delay pairs using the normalized spatial factor matrix and time delay factor matrix, respectively, and construct the spatial phase difference vector and the time delay phase difference vector; select the reference difference set according to the unambiguous difference screening criterion, and obtain the initial estimate of the target frequency parameter and spatial phase parameter through least squares fitting.

[0057] The phase unwrapping module is used to predict the fully differential phase using the initial estimation results, calculate the integer ambiguity number by combining the principal value phase observation, complete the unwrapping and compensation of the spatial and temporal delay phases, and obtain the unambiguous phase difference estimate of each target. The parameter calculation module is used to estimate the target frequency based on the compensated time delay phase difference. Then, by combining the compensated spatial phase difference and the estimated frequency, the target direction angle is estimated through least squares fitting, thus realizing the joint estimation of DOA and carrier frequency.

[0058] This embodiment provides the working process, working details and technical effects of a joint estimation method for DOA and carrier frequency based on tensor modeling, which can be referred to in Embodiment 1 and will not be repeated here.

[0059] The above embodiments are merely preferred technical solutions of the present invention and should not be considered as limitations on the present invention. The scope of protection of the present invention should be limited to the technical solutions described in the claims, including equivalent substitutions of the technical features described in the claims. That is, equivalent substitutions and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. A method for joint estimation of DOA and carrier frequency based on tensor modeling, characterized in that, Includes the following steps: S1: Acquire spatiotemporal non-uniform sampling data, rearrange them according to spatial dimension, temporal delay dimension and snapshot dimension, construct a third-order observation tensor, and establish the corresponding low-rank PARAFAC decomposition model; S2: Perform PARAFAC decomposition on the third-order observation tensor to obtain the spatial factor matrix, time delay factor matrix and snapshot factor matrix, and eliminate scale ambiguity by column normalization; S3: Extract the phase difference between adjacent array element pairs and adjacent time delay pairs using the normalized spatial factor matrix and time delay factor matrix respectively, and construct the spatial phase difference vector and the time delay phase difference vector; select the reference difference set according to the unambiguous difference screening criterion, and obtain the initial estimate of the target frequency parameter and spatial phase parameter through least squares fitting; S4: Use the initial estimation results to predict the fully differential phase, combine the principal value phase observation to calculate the integer ambiguity number, complete the unwrapping and compensation of the spatial and temporal delay phases, and obtain the unambiguous phase difference estimate of each target. S5: Estimate the carrier frequency based on the unambiguous time delay phase difference, combine the unambiguous spatial phase difference with the estimated frequency, and solve the signal DOA by least squares fitting to achieve joint estimation of DOA and carrier frequency.

2. The method for joint estimation of DOA and carrier frequency based on tensor modeling according to claim 1, characterized in that, In step S1, the spatiotemporal non-uniform sampling is as follows: the array adopts a nonlinear non-uniform linear array, and multiple non-uniform time delays are connected after each array element. The spatial sampling position and the time delay sampling interval are both non-uniformly distributed. Assuming there is An uncorrelated signal source is incident from the far field onto a nonlinear array, which contains There are several array elements, with the first element as the reference element. The array coordinates can be denoted as: Each array element is connected to The path has a non-uniform time delay, let the first one be... Road delay is ,in, The unit delay; the first The DOA and carrier frequency of each signal are: Under the narrowband and plane wave assumptions, the array receiving vector can be expressed as: (1)； Among them, the baseband signal is Subscript Indicates transpose. For noise, the spatial steering matrix is ,make , If the speed is the speed of light, then each column of this matrix is ​​a spatial steering vector. ; For the first signal source, at time instant the narrowband condition has , and , then the observation of the first time delay is: (2)； in, The time delay steering matrix is ​​defined as follows: The time delay steering vector is By stacking all the delayed outputs, the overall observation matrix is ​​obtained: (3)； wherein is the Khatri-Rao product, is noise; Definitions For the spatial steering matrix, collect samples over shots to get a sample matrix: (4)； wherein is a source signal matrix, is a noise matrix.

3. The method of claim 1, wherein, In step S1, the third-order observation tensor is constructed as follows: The sampling matrix reordering by column, the expression is: (5)； in, For spatial orientation matrix, For delay steering matrix, This is the transpose of the source signal matrix. For the noise matrix, For complex fields, These are the indices for the spatial dimension, the time delay dimension, and the snapshot dimension, respectively; M is the total number of array elements, N is the total number of time delay channels, and L is the total number of snapshots.

4. The method of claim 1, wherein, In step S2, the PARAFAC decomposition satisfies the Kruskal condition: (6)； in, , and Each is a matrix , and The Kruskal rank; K is the total number of signal sources; The factor matrix obtained from PARAFAC decomposition is as follows: (7)； in, Let be the permutation matrix. , , It is a diagonal scaling matrix. , , This is the corresponding estimation error matrix; For spatial orientation matrix, For delay steering matrix, This is the transpose of the source signal matrix; This is an estimate of the spatial factor matrix. This is the estimated value of the time delay factor matrix. This is the estimated value of the snap factor matrix; Take With the first column, respectively, denoted , , the first element normalization is performed: (8)； Obtain the normalized space corresponding to the first source and the delay factor column vector and , the specific form is and .

5. The method of claim 1, wherein, In step S3, constructing the spatial phase difference vector and the time delay phase difference vector includes: obtaining a normalized spatial factor matrix and a delay factor matrix for the first source, the phase difference estimates corresponding to adjacent element pairs and adjacent delay pairs are respectively (9)； in, For phase extraction operations, the principal value falls on This range; For the m-th adjacent array element, For the nth adjacent time delay, then define and The phase differences between adjacent array element pairs and adjacent time delay pairs are respectively denoted as... and : (10)； wherein , respectively the phase variation rate of the spatial dimension, the frequency dimension. Further, define and for the fixed target Stack all the adjacent time delay pairs and the corresponding phase principal value of the adjacent element pairs respectively to form the corresponding phase difference vector: (11)； wherein, is a matrix of inter-element spacings, is a matrix of time delays; Theoretical phase factor With This can be obtained by least squares estimation as follows: (12)； wherein with is the actual observed phase principal eigenvector; wherein denotes the pseudo-inverse; When the array element spacing or time delay difference If it is too large, the phase change will exceed... The relationship between the estimated value and the true value is as follows: (13)； wherein with is an unknown integer vector.

6. The method for joint estimation of DOA and carrier frequency based on tensor modeling according to claim 1, characterized in that, In step S3, the unambiguous difference screening criterion is: the element spacing and time delay difference corresponding to the unambiguous reference element must satisfy the Nyquist constraint condition, expressed as: (14)； in, To detect the highest carrier frequency within the frequency band; For the spacing between array elements, For time delay difference, The unit delay value, At the speed of light, This is the sine function of DOA.

7. The method of claim 1, wherein, In step S3, initial estimates of the target frequency parameters and spatial phase parameters are obtained, specifically as follows: Utilizing unambiguous phase observation vectors with , a linear regression model is constructed: (15)； in, and These characterize the linear rate of change of the phase difference with respect to the interval in the frequency dimension and the spatial dimension, respectively. and This is the actual observed principal phase vector; The pseudo-inverse of the element spacing matrix. This is the pseudo-inverse of the time delay interval matrix; , These represent the phase change rates corresponding to the spatial dimension and the frequency dimension, respectively. To satisfy the Nyquist constraint on the element spacing matrix, The time delay interval matrix to satisfy the Nyquist constraint; predicting the entire phase difference including the ambiguous section with , the expression is (16)。 8. The method of claim 1, wherein, In step S4, the integer ambiguity number is calculated as follows: the full differential phase is predicted using the initial estimation result, and the integer ambiguity number is calculated by combining the principal value phase observation, thereby completing the unwrapping and compensation of the spatial and temporal delay phases and obtaining the unambiguous phase difference estimate of each target. Obtain all phase differences, including the blurred segments. and By comparing the predicted values ​​with the observed principal values, the number of integer fuzzy numbers can be estimated. and : (17)； wherein, denotes rounding off to the nearest integer; and are the fuzzy delay and spatial phase difference predictors; and are the delay and spatial phase difference observation primary values; Compensating the observed phases to obtain deblurred phase estimates with : (18)。 9. The method of claim 1, wherein, In step S5, the carrier frequency and direction angle of the target are calculated using the compensated phase: (19)； in, The frequency phase change rate corresponding to the k-th target. Let be the spatial phase change rate corresponding to the k-th target. The speed of light; This is the delay value. , The pseudo-inverses of the element spacing matrix and the time delay spacing matrix are given. , This is for spatial and temporal phase estimation after deblurring.

10. A joint estimation system for DOA and carrier frequency based on tensor modeling, applicable to the joint estimation method for DOA and carrier frequency based on tensor modeling as described in any one of claims 1-9, characterized in that, include: The tensor construction module is used to acquire the received data of the spatiotemporal non-uniform array, rearrange it according to the spatial dimension, time delay dimension and snapshot dimension, construct the third-order observation tensor, and establish its low-rank PARAFAC decomposition model. The PARAFAC decomposition module is used to perform PARAFAC decomposition on the third-order observation tensor to obtain the spatial factor matrix, the time delay factor matrix and the snapshot factor matrix, and to eliminate scale ambiguity by column normalization. The initial parameter calculation module is used to extract the phase difference between adjacent array element pairs and adjacent time delay pairs using the normalized spatial factor matrix and time delay factor matrix, respectively, and to construct the spatial phase difference vector and the time delay phase difference vector. A reference difference set is selected based on the unambiguous difference screening criterion, and the initial estimates of the target frequency parameters and spatial phase parameters are obtained through least squares fitting. The phase unwrapping module is used to predict the fully differential phase using the initial estimation results, calculate the integer ambiguity number by combining the principal value phase observation, complete the unwrapping and compensation of the spatial and temporal delay phases, and obtain the unambiguous phase difference estimate of each target. The parameter calculation module is used to estimate the target frequency based on the compensated time delay phase difference. Then, by combining the compensated spatial phase difference and the estimated frequency, the target orientation angle is estimated through least squares fitting, thus realizing the joint estimation of angle and frequency parameters.

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