CN117970235A - Unitary ESPRIT two-dimensional DOA estimation method based on virtual array interpolation - Google Patents
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Abstract
The invention discloses a unitary ESPRIT two-dimensional DOA estimation method based on virtual array interpolation. The patent method introduces the thinking of virtual array interpolation to construct a uniform linear virtual array, utilizes convex optimization to reconstruct a covariance matrix of an equivalent received signal, optimizes the first element of a corresponding vector of the covariance matrix, and finally carries out DOA estimation on the reconstructed covariance matrix by applying unitary rotation invariant technology and an overall least squares method. Compared with DOA estimation by using the largest continuous uniform array of the extended virtual array, the method fully utilizes the information contained in the virtual array, improves the degree of freedom and resolution of estimation, simultaneously realizes automatic matching of azimuth angle and pitch angle, and verifies the effectiveness of the method by using simulation results.
Description
Technical Field
The invention belongs to the field of array signal processing, and particularly relates to a unitary ESPRIT two-dimensional DOA estimation method based on virtual array interpolation.
Background
In array signal processing, a DOA estimation algorithm is widely applied to the fields of radar, sonar, wireless communication and the like, a DOA estimation technology based on a uniform linear array is mature, an array structure is simple, and the DOA estimation method is the most commonly used array DOA estimation method. In recent years, compared with the traditional array, the sparse array has been paid attention to widely because the degree of freedom of the array can be effectively improved, the sparse array can fully reduce the number of array elements on the premise of ensuring the performance, has larger array aperture and lower side lobe level under the condition that the number of the array elements is the same, and improves the precision, resolution and degree of freedom of a direction finding algorithm by resolving the positions and weights of the array elements. The mutual mass array is expanded into a virtual array in a certain mode, the number of the array elements of the virtual array is more than that of the actual array elements, and the degree of freedom can be increased to a certain extent. The virtual array obtained by mutual mass array expansion is not a linear uniform array due to the lack of array elements, and the accurate direction of arrival cannot be obtained by using the traditional DOA estimation method. Therefore, the existing parallel mutual array DOA estimation algorithm only utilizes the maximum continuous array element part, and the problems of low array element utilization rate, insufficient algorithm precision, unmatched azimuth angle and pitch angle and the like caused by hole deletion still exist.
Aiming at the problems, the patent aims at a virtual array obtained by expanding a parallel mutual array, carries out array interpolation on the virtual array, constructs a uniform linear virtual array, applies the interpolated mutual array to the parallel array, thereby increasing the degree of freedom, carries out DOA Estimation on a covariance matrix reconstructed by interpolation by using a unitary rotation invariant technique (ESPRIT) and a total least square (Total Least Squares, TLS) method, converts two-dimensional DOA Estimation into two one-dimensional DOA Estimation, fully utilizes information contained in the virtual array, fully utilizes array element information, and improves the Estimation degree of freedom and resolution.
Disclosure of Invention
Aiming at the defects of the existing two-dimensional DOA estimation method, the patent provides a unitary ESPRIT two-dimensional DOA estimation method based on virtual array interpolation, which is suitable for a parallel mutual-mass sensor array, wherein the parallel mutual-mass sensor array comprises a subarray 1 and a subarray 2, the subarray 1 is unfolded along a Y axis, the number of array elements of the subarray 2 is the same as that of the subarray 1, the subarray 2 and the subarray 1 are placed in parallel, and are placed on an XOY plane, the distance between the subarray 2 and the subarray 1 is d, wherein d is half wavelength of a signal, namely d=lambda/2; the subarray 1 is formed by nesting a pair of uniform linear arrays, the first uniform linear array is composed of M sensors with intervals Nd, the M sensors are respectively positioned on Y-axis {0, nd,2Nd, (M-1) Nd }, the second uniform linear array is composed of N sensors with intervals Md, and the N sensors are respectively positioned on Y-axis {0, md,2Md, (N-1) Md }, wherein M and N are prime integers, the two linear arrays form a prime linear array, and the prime linear array is composed of M+N-1 sensors; the coordinates of the array elements of the subarray 2 in the X-axis direction are d, and the coordinates of the array elements in the Y-axis direction are the same as the coordinates of the corresponding array elements of the subarray 11; by means ofAnd θ k represents the azimuth and pitch angles of the kth (k=1, …, K) incident source, respectively, and α k and β k represent the direction angles of the incident signal and the X-axis and Y-axis, respectively, the relationship between the direction angles and the azimuth and pitch angles being: /(I)AndThe method is characterized in that: the DOA estimation includes the steps of:
Step one: the array received data of subarrays 1 and 2 are X (t) =as (t) +n x (t) and Y (t) =aΩ s (t) +n y (t), respectively, where s (t) =s 1(t),s2(t),…,sK(t)]T represents the incident signal vector, n x (t) and n y (t) are additive white gaussian noise data vectors received by subarrays 1 and 2, respectively, and the noise is independent of the incident signal, a= [ a (a 1),…,a(αK) ] is the manifold matrix of subarray 1, where K represents the number of incident signals, S= { z i, i=1, 2, …, m+n-1} represents the set of array element positions in subarray 1 and subarray 2,/>In the formula, [. Cndot. ] T represents the transpose of the matrix, and diag {. Cndot. } represents the diagonal matrix;
Step two: calculation of the auto-covariance matrix of subarray 1 and subarray 1 And cross covariance matrix of subarray 1 and subarray 2The calculation method comprises the following steps: /(I) L is the number of snapshots, [. Cndot. ] H represents the conjugate transpose of the matrix;
Step three: vectorization V 1 and V 2,/>, were obtainedWherein vec (& gt) is to stretch the matrix into long vectors by columns, de-duplicate the elements from v 1 and sort to obtain a virtual array/>
Step four: interpolation is carried out on the virtual array v 1 to obtain signals of the virtual arrayWherein the method comprises the steps of [] i Denotes the virtual signal of the virtual sensor at position i;
Step five: signals to virtual arrays Performing toeplitz matrix reconstruction to obtain an initial covariance matrixWherein/>To/>A top-ranked toeplitz matrix;
Step six: covariance matrix solved in the fifth step Substituting the problem of convex optimization into the model,
Solving the problem by using a convex optimization tool box to obtain a covariance matrix T (z) after interpolation, wherein P is a AND matrixProjection matrix of the same dimension, matrix/>The elements in P correspond to vector elements of the interpolation uniform array, but the element value of the matrix P corresponds to the interpolation array element is 0, the rest is 1, T (z) is an Hermite Toeplitz matrix taking the vector z as the first column, tau represents regularization parameters,/>Representing the Hadamard product, |·| F represents the Frobenius norm, tr (·) represents the trace of the matrix, and rank (·) is the rank of the matrix;
step seven: unitary transformation of the resulting interpolated reconstruction matrix T Wherein the method comprises the steps of
For/>An anti-diagonal identity matrix having an element on the anti-diagonal of 1, and the other elements being equal to 0,Representing a dimension of/>Zero matrix of/>Representing dimensions asIs, (. Cndot.) * represents the conjugate of the matrix;
Step eight: performing feature decomposition on R f to obtain a signal subspace E s to obtain ψ r＝(K1Es)+K2Es, wherein (. Cndot.) + represents the inverse of the matrix, re (. Cndot.) represents the real part of the matrix, and Im (. Cndot.) represents the imaginary part of the matrix;
step nine: performing feature decomposition on ψ r to obtain feature values Then the alpha angle can be estimated as
Step ten: and step nine, obtainingCarry-inObtained estimate/>Wherein/>Representing the Cronecker product operation, structure/>
Step eleven: singular value decomposition is carried out on W, the right singular vector v s,vs1 corresponding to the minimum singular value is the front K term of v s, v s2 is the last term of v s, and calculation is carried outCalculating the elements to obtain the estimation of the beta angle
Step twelve: obtained by step nineAnd step eleven/>Estimate of azimuth and pitch angles can be obtained
Compared with the prior art, the technical scheme provided by the invention has the following technical effects:
(1) The virtual array interpolation method is applied to two-dimensional DOA estimation of the parallel mutual mass array, a continuous uniform linear array is constructed, and the degree of freedom is increased;
(2) The traditional intersubstance array is subjected to virtual interpolation, so that the aperture of the array is improved, and the estimation accuracy of two-dimensional DOA estimation is improved;
(3) According to the invention, spectrum peak searching is not needed, so that the calculation complexity is greatly reduced;
(4) The method of the invention realizes the automatic matching of the azimuth angle and the pitch angle.
Description of the drawings:
FIG. 1 is a diagram of an actual scenario application of the present system;
FIG. 2 is a two-dimensional DOA estimation result of the signal processing method of the present patent;
FIG. 3 is a graph of the root mean square error of azimuth and pitch angles versus signal to noise ratio before and after virtual interpolation in the signal processing method of the present patent;
FIG. 4 is a graph showing the relation between the root mean square error and the signal to noise ratio in the signal processing method of the present patent;
FIG. 5 is a plot of root mean square error versus data length for the signal processing method of the present patent;
The specific embodiment is as follows:
The invention will now be further described with reference to examples, figures:
First embodiment: figure 1 shows a parallel reciprocal array model used in the present invention. The parallel mutual mass sensor array comprises a subarray 1 and a subarray 2, wherein the subarray 1 is unfolded along a Y axis, the number of array elements of the subarray 2 is the same as that of the array structure of the subarray 1, the subarray 2 and the subarray 1 are placed in parallel and are placed on an XOY plane, the distance between the subarray 2 and the subarray 1 is d=0.03 m, and d=lambda/2; the subarray 1 is formed by nesting a pair of uniform linear arrays, the first uniform linear array consists of 3 sensors with a spacing of 5d, the 3 sensors are respectively positioned on positions of Y-axis {0,5d,10d }, the second uniform linear array consists of 5 sensors with a spacing of 3d, and are respectively positioned on positions of Y-axis {0,3d,6d,9d,12d }, the two linear arrays form a mutual linear array, and the mutual linear array consists of 7 sensors; the coordinates of the array elements of the subarray 2 in the X-axis direction are all 0.03m, and the coordinates of the array elements in the Y-axis direction are the same as the coordinates of the corresponding array elements of the subarray 1; let 3 far field, narrowband and uncorrelated signals be incident on a parallel mutual mass array, And θ k represent azimuth and pitch angles of the kth (k=1, 2, 3) source, respectively, the three incident signals (azimuth, pitch angle) are (25 °,40 °), (35 °,50 °), (45 °,60 °), the signal-to-noise ratio is 10dB, and the snapshot number is l=200.
The method adopts the conditions to carry out the direction estimation of the arrival, and comprises the following specific implementation processes:
Step one: the array received data of subarrays 1 and 2 are X (t) = As (t) +n x (t) and Y (t) = aΩ s (t) +n y (t), respectively, where s (t) = [ s 1(t),s2(t),s3(t)]T ] represents an incident signal vector, n x (t) and n y (t) are additive white gaussian noise data vectors received by subarrays 1 and 2, respectively, and noise is independent of the incident signal, a= [ a (a 1),a(α2),a(α3) ] is a manifold matrix of subarray 1, where the number of incident signals is 3, S= { z i, i=1, 2, …,7} represents the set of array element positions in subarray 1 and subarray 2,/>In the formula, [. Cndot. ] T represents the transpose of the matrix, and diag {. Cndot. } represents the diagonal matrix;
Step two: calculation of the auto-covariance matrix of subarray 1 and subarray 1 And cross covariance matrix of subarray 1 and subarray 2The calculation method comprises the following steps: /(I) [ ] H Represents the conjugate transpose of the matrix;
Step three: vectorization V 1 and V 2,/>, were obtainedWherein vec (& gt) is to stretch the matrix into long vectors by columns, de-duplicate the elements from v 1 and sort to obtain a virtual array/>
Step four: for virtual arraysInterpolation is carried out to obtain a virtual array signal/>Wherein the method comprises the steps of [] i Denotes the virtual signal of the virtual sensor at position i;
Step five: signals to virtual arrays Performing toeplitz matrix reconstruction to obtain an initial covariance matrixWherein/>To/>A top-ranked toeplitz matrix;
Step six: covariance matrix solved in the fifth step Substituting the problem of convex optimization into the model,
Solving the problem by using a convex optimization tool box to obtain a covariance matrix T (z) after interpolation, wherein P is a AND matrixProjection matrix of the same dimension, matrix/>The elements in P correspond to vector elements of the interpolation uniform array, but the element value of the matrix P corresponds to the interpolation array element is 0, the rest is 1, T (z) is an Hermite Toeplitz matrix taking the vector z as the first column, tau represents regularization parameters,/>Representing the Hadamard product, |·| F represents the Frobenius norm, tr (·) represents the trace of the matrix, and rank (·) is the rank of the matrix
Step seven: unitary transformation of the resulting interpolated reconstruction matrix TWherein the method comprises the steps ofII 12 is a 12×12 anti-diagonal identity matrix, the elements on the anti-diagonal are 1, and the other elements are all equal to 0, 0 12×1 represents a zero matrix with 12×1 dimensions, I 12 represents an identity matrix with 12×12 dimensions, (. Cndot.) * represents the conjugate of the matrix;
Step eight: performing feature decomposition on R f to obtain a signal subspace E s to obtain ψ r＝(K1Es)+K2Es, wherein J2＝[024×1,I24×24]，/>(. Cndot.) + represents the inverse of the matrix, re (. Cndot.) represents the real part of the matrix, and Im (. Cndot.) represents the imaginary part of the matrix;
step nine: performing feature decomposition on ψ r to obtain feature values Then the alpha angle can be estimated as
Step ten: : and step nine, obtainingCarry-in/>Obtained estimate/>Wherein/>Representing the Cronecker product operation, structure/>
Step eleven: singular value decomposition is carried out on W, the right singular vector v s,vs1 corresponding to the minimum singular value is the front K term of v s, v s2 is the last term of v s, and calculation is carried outCalculating the elements to obtain the estimation of the beta angle
Step twelve: obtained by step nineAnd step eleven/>Estimate of azimuth and pitch angles can be obtainedThe simulation of the present patent method based on the above conditions is performed by MATLAB simulation software, and the angle estimation result of the present patent method is shown in fig. 2, and when the incident signals (azimuth angle, pitch angle) are respectively (25 °,40 °), (35 °,50 °), (45 °,60 °), the obtained estimation angles are (25.07 °,40.05 °), (34.61 °,50.34 °), (45.7 °,59.78 °), and the experimental result indicates that the present patent method can accurately locate all signal sources.
Example 2: the signal processing method intercepts the relation curve of the root mean square error and the signal to noise ratio of the azimuth angle and the pitch angle of the continuous array element before virtual interpolation and after virtual interpolation, displays the accurate DOA estimation result, and the obtained effect diagram is shown in figure 3. The application conditions of the method of the invention are as follows:
The method is based on a parallel mutual mass array model as shown in fig. 1, wherein a parallel mutual mass linear array consists of subarrays 1 and subarrays 2, each subarray is formed by nesting two sparse uniform linear arrays, M=3, N=5, the number of array elements of each array is 7, and the distance between adjacent elements in the array and the two parallel linear arrays is 0.03M. The number of signals is k=2, the two incident signals (azimuth angle, pitch angle) are (30 °,25 °), (45 °,60 °), and the snapshot number is l=200. The signal to noise ratio was varied from-5 dB, and each step was increased from 4dB to 15dB, 500 independent monte-cary experiments were performed, and simulation analysis was performed using MATLAB simulation software, and the resulting performance analysis versus curve was shown in fig. 3.
As can be seen from fig. 4, the azimuth and pitch angles of the method herein and the root mean square error of the azimuth and pitch angles of the method of intercepting successive array elements decrease with increasing signal to noise ratio. The method of the invention increases the degree of freedom and improves the utilization rate of the array elements due to the virtual interpolation, so the performance of the method of the invention is superior to that of a method for intercepting continuous array elements.
Example 3: the relation curve of the root mean square error and the signal to noise ratio of the signal processing method is displayed, the DOA estimation accurate result is displayed, the obtained effect diagram is shown in figure 4, and meanwhile, a comparison diagram of performance analysis of other methods (the DOA-Matrix method of Dai Xiangrui, the improved PM algorithm of Li Jianfeng and the CCM method) is also provided. The application conditions of the method of the invention are as follows:
The method is based on a parallel mutual mass array model as shown in fig. 1, wherein a parallel mutual mass linear array consists of subarrays 1 and subarrays 2, each subarray is formed by nesting two sparse uniform linear arrays, M=3, N=5, the number of array elements of each array is 7, and the distance between adjacent elements in the array and the two parallel linear arrays is 0.03M. The number of the signals is k=3, the three incident signals (azimuth angle, pitch angle) are (25 °), respectively (35 °,50 °), respectively (45 °,55 °), and the snapshot number is l=200. The signal to noise ratio was varied from 0dB, and 500 independent monte-cary experiments were performed with 5dB to 20dB steps each, and simulation analysis was performed using MATLAB simulation software, and the resulting performance analysis versus curve is shown in fig. 4.
As can be seen from fig. 4, the root mean square error of both methods decreases with increasing signal-to-noise ratio. The method has the minimum root mean square error, and has better performance than other three methods and stronger estimation performance under low signal-to-noise ratio as seen from the integral effect.
Example 4: the relation curve of root mean square error and snapshot number of the signal processing method is displayed, the DOA estimation accurate result is displayed, the obtained effect diagram is shown in figure 4, and meanwhile, a comparison diagram of performance analysis of other methods ((Dai Xiangrui DOA-Matrix method, li Jianfeng improved PM algorithm and CCM method) is also provided:
the method is based on a parallel mutual mass array model as shown in fig. 1, wherein a parallel mutual mass linear array consists of subarrays 1 and subarrays 2, each subarray is formed by nesting two sparse uniform linear arrays, M=3, N=5, the number of array elements of each array is 7, and the distance between adjacent elements in the array and the two parallel linear arrays is 0.03M. The number of the signals is K=3, the three incident signals (azimuth angle and pitch angle) are (25 degrees, 40 degrees, 35 degrees, 50 degrees, 45 degrees and 55 degrees respectively), and the snapshot number is L=200. The signal to noise ratio was varied from 0dB, and 500 independent monte-cary experiments were performed with 0dB to 20dB step size each, and simulation analysis was performed using MATLAB simulation software, and the resulting performance analysis versus curve is shown in fig. 5.
As can be seen from fig. 5, the root mean square error of both methods decreases with increasing snapshot count. The method has the least square error of azimuth angles, and has better azimuth angle performance than other three methods and stronger estimation performance under low snapshot.
The specific examples described herein are offered by way of illustration only. Those skilled in the art may make various modifications, additions or substitutions to the described embodiments without departing from the invention or the scope thereof as defined in the accompanying claims.
Claims (1)
1. The unitary ESPRIT two-dimensional DOA estimation method based on virtual array interpolation is suitable for a parallel mutual quality sensor array, wherein the parallel mutual quality sensor array comprises a subarray 1 and a subarray 2, the subarray 1 is unfolded along a Y axis, the number of array elements of the subarray 2 is the same as that of the subarray 1, the subarray 2 and the subarray 1 are placed in parallel and are placed on an XOY plane, the distance between the subarray 2 and the subarray 1 is d, and d is half wavelength of a signal, namely d=lambda/2; the subarray 1 is formed by nesting a pair of uniform linear arrays, the first uniform linear array consists of M sensors with intervals Nd, the M sensors are respectively positioned on Y-axis {0, nd,2Nd, …, (M-1) Nd } positions, the second uniform linear array consists of N sensors with intervals Md, and the N sensors are respectively positioned on Y-axis {0, md,2Md, … (N-1) Md } positions, wherein M and N are prime integers, the two linear arrays form a prime linear array, and the prime linear array consists of M+N-1 sensors; the coordinates of the array elements of the subarray 2 in the X-axis direction are d, and the coordinates of the array elements in the Y-axis direction are the same as the coordinates of the corresponding array elements of the subarray 1; by means ofAnd θ k represents the azimuth and pitch angles of the kth (k=1, …, K) incident source, respectively, and α k and β k represent the direction angles of the incident signal and the X-axis and Y-axis, respectively, the relationship between the direction angles and the azimuth and pitch angles being: /(I)And/>The method is characterized in that: the DOA estimation includes the steps of:
Step one: the array received data of subarrays 1 and 2 are X (t) = As (t) +n x (t) and Y (t) = aΩ s (t) +n y (t), respectively, where s (t) = [ s 1(t),s2(t),…,sK(t)]T ] represents an incident signal vector, n x (t) and n y (t) are additive white gaussian noise data vectors received by subarrays 1 and 2, respectively, and noise is independent of the incident signal, a= [ a (a 1),…,a(αK) ] is a manifold matrix of subarray 1, where K represents the number of incident signals, S= { z i, i=1, 2, …, m+n-1} represents the set of array element positions in subarray 1 and subarray 2,/>In the formula, [. Cndot. ] T represents the transpose of the matrix, and diag {. Cndot. } represents the diagonal matrix;
Step two: calculation of the auto-covariance matrix of subarray 1 and subarray 1 And cross covariance matrix/>, of subarrays 1 and 2The calculation method comprises the following steps: /(I) L is the number of snapshots, [. Cndot. ] H represents the conjugate transpose of the matrix;
Step three: vectorization V 1 and V 2,/>, were obtainedWherein vec (& gt) is to stretch the matrix into long vectors by columns, de-duplicate the elements from v 1 and sort to obtain a virtual array/>
Step four: for virtual arraysInterpolation is carried out to obtain the signal/>, of the virtual arrayWherein the method comprises the steps of [ · ] i Represents the virtual signal of the virtual sensor at location i;
Step five: signals to virtual arrays Performing toeplitz matrix reconstruction to obtain an initial covariance matrixWherein/>To/>A top-ranked toeplitz matrix;
Step six: covariance matrix solved in the fifth step Substituting the problem of convex optimization into the model,
Solving the problem by using a convex optimization tool box to obtain a covariance matrix T (z) after interpolation, wherein P is a AND matrixProjection matrix of the same dimension, matrix/>The elements in P correspond to vector elements of the interpolation uniform array, but the element value of the matrix P corresponds to the interpolation array element is 0, the rest is 1, T (z) is an Hermite Toeplitz matrix taking the vector z as the first column, tau represents regularization parameters,/>Representing the Hadamard product, |·| F represents the Frobenius norm, tr (·) represents the trace of the matrix, and rank (·) is the rank of the matrix;
step seven: unitary transformation of the resulting interpolated reconstruction matrix T Wherein the method comprises the steps of
For/>An anti-diagonal identity matrix having an element on the anti-diagonal of 1, and the other elements being equal to 0, Representing a dimension of/>Zero matrix of/>Representing dimensions asIs, (. Cndot.) * represents the conjugate of the matrix;
Step eight: performing feature decomposition on R f to obtain a signal subspace E s to obtain ψ r＝(K1Es)+K2Es, wherein (. Cndot.) + represents the inverse of the matrix, re (. Cndot.) represents the real part of the matrix, and Im (. Cndot.) represents the imaginary part of the matrix;
step nine: performing feature decomposition on ψ r to obtain feature values Then the alpha angle can be estimated as
Step ten: and step nine, obtainingCarry-inObtained estimate/>Wherein/>Representing the Cronecker product operation, structure/>
Step eleven: singular value decomposition is carried out on W, the right singular vector v s,vs1 corresponding to the minimum singular value is the front K term of v s, v s2 is the last term of v s, and calculation is carried outPair/>Is calculated to obtain an estimate of the beta angle
Step twelve: obtained by step nineAnd step eleven/>Estimate of azimuth and pitch angles can be obtained
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