UUV broadside parallel co-prime array DOA estimation method

Abstract

The invention belongs to the crossing field of information and ocean science and technology, and discloses a DOA estimation method for a UUV broadside parallel co-prime array. According to the method, firstly, a virtual extension array is generated by utilizing an array auto-covariance and cross-covariance matrix, then, Toeplitz completion is carried out on'holes' in the virtual array through trace norm regularization and nuclear norm regularization constraints, a complete covariance structure is recovered, and finally, an extension matrix is constructed, and rotation invariance of a signal subspace of the extension matrix is utilized, so that the covariance structure of the virtual array is obtained. The pitch angle and the azimuth angle of the target are jointly solved, automatic angle pairing is achieved, and the effectiveness and the reliability of the method are verified through simulation results. According to the method, aiming at the application challenges that the UUV broadside array space is limited and the underwater environment is complex, all virtual array elements of the broadside parallel co-prime array are fully utilized, the array freedom degree and estimation precision are remarkably improved, the target detection capacity of the UUV is effectively enhanced, and the method has high practical engineering application value.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of next-generation information technology and marine science and technology, specifically relating to a high-precision, high-degree-of-freedom target orientation estimation method for flank arrays of unmanned underwater vehicles (UUVs). Based on parallel coprime array virtual interpolation and extended matrix technology, this invention achieves two-dimensional direction-of-arrival (DOA) estimation of underwater acoustic targets, and can be widely applied in fields such as autonomous navigation of underwater UUVs, underwater target detection and identification, marine environmental monitoring, and underwater communication. Background Technology

[0002] As a crucial vehicle for ocean exploration and development, the target detection and location estimation capabilities of the flank sonar arrays of unmanned underwater vehicles (UUVs) directly impact their situational awareness and autonomous operation performance. Limited by the size and structure of UUV platforms, flank arrays typically face challenges such as a limited number of array elements, small array aperture, and low space utilization, resulting in bottlenecks in target resolution and estimation accuracy for traditional uniform arrays. Furthermore, the complex underwater acoustic environment presents challenges such as multipath interference, low signal-to-noise ratio, and limited snapshots, further restricting the performance of existing DOA estimation methods.

[0003] Traditional UUV side arrays often employ uniform linear arrays, which, limited by half-wavelength spacing, cannot accommodate enough array elements within a confined space, resulting in low degrees of freedom and insufficient angular resolution. While sparse arrays (such as coprime arrays and nested arrays) can expand the virtual aperture with the same number of elements, the virtual arrays generated by their differential co-arrays are often non-uniform, exhibiting "holes." Existing underwater target estimation methods typically only estimate the largest continuous portion of the virtual array, discarding the information carried by discontinuous elements, leading to underutilization of array resources. Furthermore, two-dimensional DOA estimation often relies on computationally intensive two-dimensional spectral peak searches, which is insufficient to meet the real-time processing requirements of UUVs. To address these issues, the Virtual Interpolation Unitary Transform (VIU-ESPRIT) method proposed by Sun Gaoli et al. only reconstructs the autocovariance matrix, failing to fully utilize the cross-covariance information between arrays, thus limiting estimation performance in underwater environments with low signal-to-noise ratios and limited snapshots.

[0004] To address this issue, this invention proposes a method for estimating the DOA (Direction of Area) of a UUV (Universal Vehicle) side-mounted parallel coprime array. This method constructs a parallel coprime array and utilizes its coprime property to generate a large-aperture virtual array. It then reconstructs the complete covariance structure by interpolating and regularizing the "holes" in the virtual array. Finally, it constructs an extended matrix and utilizes rotation invariance to achieve automatic matching and estimation of two-dimensional angles. This invention aims to solve practical problems such as improving target detection capabilities, enhancing robustness in low signal-to-noise ratio environments, and reducing computational complexity for UUV side-mounted arrays under space-constrained conditions. Summary of the Invention

[0005] This invention addresses the challenges of limited array aperture and strong underwater environmental interference in target orientation estimation using UUV side-mounted arrays. It proposes a method for DOA estimation using a parallel coprime array on the UUV side. This method is applicable to parallel coprime sensor arrays, which consist of subarray 1 and subarray 2. Subarray 2 has the same number of elements as subarray 1, and the spacing between subarray 1 and subarray 2 is d. , The wavelength of the signal is represented; subarray 1 is a coprime array composed of a pair of nested uniform linear arrays. The first uniform linear array consists of M sensors spaced Nd apart, and the sum of its element position sets is... The second uniform linear array consists of N sensors spaced Md apart, and the set of its element positions is as follows: Where M and N are coprime integers, the two uniform linear arrays share the first sensor, and the coprime array consists of M+N-1 sensors; subarray 2 is parallel to subarray 1 and above subarray 1, with a distance d between the two subarrays. and They represent the first The azimuth and elevation angles of the incident signal source are used. This represents the angle between the incident signal and subarray 1;

[0006] A method for estimating the DOA of a parallel coprime matrix on the side of a UUV includes the following steps:

[0007] Step 1: The received data for subarray 1 and subarray 2 are respectively and Calculate the autocovariance matrix of subarray 1. and the cross-covariance matrix of subarray 1 and subarray 2 Where: L represents the number of snapshots, Represents the conjugate transpose of a matrix;

[0008] Step 2: Vectorization get ,in This means stretching the matrix column-wise into a long vector, from... Remove duplicate elements and sort to obtain a virtual array Vectorization get ,from Remove duplicate elements and sort to obtain a virtual array ;

[0009] Step 3: Calculate the set of non-uniform virtual array positions based on the array parameters. ,according to Define a set of uniform virtual array positions ;

[0010] Step 4: Based on the calculation results of Step 2 and Step 3, construct the extended vector. and , Indicates position Virtual signals from virtual sensors at the location;

[0011] Step 5: For vectors Perform Toeplitz reconstruction on the elements in the array to obtain In the formula express The One element; , Represents a set The number of elements;

[0012] Step Six: Utilize the information from Step Five Constructing an autocovariance matrix with Hermitian positive semi-definite and Toeplitz structures, the reconstruction problem can be formulated as follows:

[0013] ,

[0014] Solving using the CVX Toolbox ,in Is with matrix Projection matrices of the same dimension, matrices The corresponding interpolation element has a value of 0, and the rest have a value of 1. Represents the regularization parameter. This represents the Hadamard product. Represents the trace of a matrix. Denotes the square of the Frobenius norm. ;

[0015] Step 7: For vectors Perform Toeplitz reconstruction on the elements in the array to obtain In the formula express The One element;

[0016] Step 8: Obtain the result from the calculation in Step 7. To construct a non-Hermitian cross-covariance matrix, the cross-covariance matrix is ​​reconstructed using the nuclear norm regularization method. ,

[0017] ,

[0018] in, This represents a Toeplitz matrix with c as the first column and r as the first row, where Is with matrix Projection matrices of the same dimension, matrices The corresponding interpolation element has a value of 0, and the rest have a value of 1. , To assist the positive semidefinite matrix, Represents the regularization parameter;

[0019] Step 9: Based on the interpolated covariance matrix obtained in Step 6 And obtained in step eight Construct the extended matrix ;

[0020] Step 10: [Regarding...] Perform singular value decomposition to extract the signal subspace. ,right Divide into blocks , and for The matrix, using , Constructing a matrix ;

[0021] Step Eleven: [Regarding...] Eigenvalue decomposition, It can be seen from The corresponding eigenvalues ​​are obtained from the data. , This indicates the number of incident signal sources; the elevation angle can be obtained through the eigenvalues. In the formula This represents the phase angle when taken as a complex number;

[0022] Step Twelve: Utilize eigenvector matrix after eigenvalue decomposition Reconstruct the direction matrix Based on the phase relationship between adjacent elements, we can obtain , Indicates the number of incident sources, where , , Represents the square of the 2-norm;

[0023] Step Thirteen: Obtained through Step Twelve , can be obtained Angle estimates , Indicates the number of incident sources;

[0024] Step Fourteen: Based on the results obtained in Step Thirteen And obtained in step eleven An estimated value of the azimuth can be obtained. , This indicates the number of incident sources.

[0025] Furthermore, the process of removing duplicate elements and sorting in step two specifically includes:

[0026] A1: Define the set of uniform virtual array positions ,in ;

[0027] A2: For each virtual array element position traverse all actual array element pairs ,in, and They represent the first and second digits in the actual array, respectively. The and the first Index of each array element , Given the actual total number of array elements, find the one that satisfies... The array element pairs, among which This refers to the actual positions of the array elements;

[0028] A3: For each pair of array elements that satisfy the conditions Collect the corresponding covariance matrix element values. or ,in Represents the autocovariance matrix of submatrix 1 The element in the m-th row and n-th column, Representing the cross-covariance matrix The element in the m-th row and n-th column is used to average the collected values ​​to obtain the virtual sensor signal value at that location: , ,in, This indicates that all conditions satisfying the positional difference are... The set of array element pairs;

[0029] A4: If a certain position There is no corresponding actual array element pair, that is If the signal value at that location is missing, then that location signal value is recorded as missing. ;

[0030] A5: Arrange all positions in order to obtain the virtual array signal vector. and .

[0031] Compared with the prior art, the technical solution of the present invention has the following technical effects:

[0032] (1) By interpolating and optimizing the "holes" of the virtual array, the present invention utilizes all virtual array elements and achieves up to 2M(N-1)+1 degrees of freedom, which is much higher than the actual number of array elements and the method of truncating continuous array elements, and can estimate more signal sources.

[0033] (2) The method of the present invention effectively recovers the missing cross-covariance information by optimizing and reconstructing the nuclear norm. Combined with all aperture information, it significantly improves the angle estimation accuracy, especially under the conditions of low signal-to-noise ratio and few snapshots.

[0034] (3) The method of the present invention transforms the two-dimensional estimation problem into two one-dimensional problems and realizes them through vectorized operations, which completely avoids the computationally intensive two-dimensional spectral peak search. The algorithm is highly efficient, and the estimated azimuth and elevation angles are automatically and correctly paired without the need for additional pairing operations. Attached Figure Description

[0035] Figure 1 This is an array model diagram of the signal processing method of the present invention;

[0036] Figure 2 This is a flowchart of the signal processing method of the present invention;

[0037] Figure 3 This is the two-dimensional DOA estimation result of the signal processing method of this invention;

[0038] Figure 4 This is the curve showing the relationship between the root mean square error of the azimuth angle and the signal-to-noise ratio in the signal processing method of this invention.

[0039] Figure 5 This is the curve showing the relationship between the root mean square error of the elevation angle and the signal-to-noise ratio in the signal processing method of this invention.

[0040] Figure 6 This is the curve showing the relationship between the root mean square error of the azimuth angle and the number of snapshots in the signal processing method of this invention.

[0041] Figure 7 This is the curve showing the relationship between the root mean square error of the pitch angle and the number of snapshots in the signal processing method of this invention. Detailed Implementation

[0042] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0043] First embodiment: Figure 1 The parallel coprime array model used in this invention is presented. The parallel coprime array consists of subarray 1 and subarray 2. The number of elements in subarray 2 is the same as the array structure of subarray 1, and the spacing between subarray 1 and subarray 2 is d = 0.03 m. , The wavelength of the signal is represented by subarray 1, which is a coprime array composed of a pair of nested uniform linear arrays. The first uniform linear array consists of three sensors spaced 5d apart, and the sum of its element position sets is... The second uniform linear array consists of five sensors spaced 3d apart, and the set of its element positions is as follows: Two uniform linear arrays share the first sensor. This coprime linear array consists of 7 sensors. Subarray 2 is parallel to subarray 1 and located above subarray 1. The distance between the two subarrays is d = 0.03 m. Assume two far-field narrowband uncorrelated signals are incident on the parallel coprime array. and They represent the first The azimuth and elevation angles of the incident signal source are used. This represents the direction angle between the incident signal and the Y-axis. The relationship between this direction angle and the azimuth and elevation angles is as follows: Let the two incident signals (azimuth and elevation angles) be (30°, 10°) and (45°, 20°) respectively, the signal-to-noise ratio be 10dB, and the number of snapshots be L=500.

[0044] The above conditions are used to estimate the azimuth of arrival. Figure 2 A flowchart of the signal processing method of the present invention is provided; the specific implementation process of the method of the present invention is as follows:

[0045] Step 1: The received data for subarray 1 and subarray 2 are respectively and Calculate the autocovariance matrix of subarray 1. and the cross-covariance matrix of subarray 1 and subarray 2 Where: L=500 represents the number of snapshots, Represents the conjugate transpose of a matrix;

[0046] Step 2: Vectorization get ,in This means stretching the matrix column-wise into a long vector, from... Remove duplicate elements and sort to obtain a virtual array Vectorization get ,from Remove duplicate elements and sort to obtain a virtual array ;

[0047] Step 3: Calculate the set of non-uniform virtual array positions based on the array parameters. ,according to Define a set of uniform virtual array positions ;

[0048] Step 4: Based on the calculation results of Step 2 and Step 3, construct the extended vector. and , Indicates position Virtual signals from virtual sensors at the location;

[0049] Step 5: For vectors Perform Toeplitz reconstruction on the elements in the array to obtain In the formula express The One element;

[0050] Step Six: Utilize the information from Step Five Constructing an autocovariance matrix with Hermitian positive semi-definite and Toeplitz structures, the reconstruction problem can be formulated as follows:

[0051] ,

[0052] Solving using the CVX Toolbox ,in Is with matrix Projection matrices of the same dimension, matrices The corresponding interpolation element has a value of 0, and the rest have a value of 1. Represents the regularization parameter. This represents the Hadamard product. Represents the trace of a matrix. Denotes the square of the Frobenius norm;

[0053] Step 7: For vectors Perform Toeplitz reconstruction on the elements in the array to obtain In the formula express The One element;

[0054] Step 8: Obtain the result from the calculation in Step 7. To construct a non-Hermitian cross-covariance matrix, the cross-covariance matrix is ​​reconstructed using the nuclear norm regularization method. ,

[0055] ,

[0056] in, This represents a Toeplitz matrix with c as the first column and r as the first row, where Is with matrix Projection matrices of the same dimension, matrices The corresponding interpolation element has a value of 0, and the rest have a value of 1. , To assist the positive semidefinite matrix, Represents the regularization parameter;

[0057] Step 9: Based on the interpolated covariance matrix obtained in Step 6 And obtained in step eight Construct the extended matrix ;

[0058] Step 10: [Regarding...] Perform singular value decomposition to extract the signal subspace. ,right Divide into blocks , and for The matrix, using , Constructing a matrix ;

[0059] Step Eleven: [Regarding...] Eigenvalue decomposition, It can be seen from The corresponding eigenvalues ​​are obtained from the data. , This indicates the number of incident signal sources; the elevation angle can be obtained through the eigenvalues. In the formula This represents the phase angle when taken as a complex number;

[0060] Step Twelve: Utilize eigenvector matrix after eigenvalue decomposition Reconstruct the direction matrix Based on the phase relationship between adjacent elements, we can obtain , Indicates the number of incident sources, where , , Represents the square of the 2-norm;

[0061] Step Thirteen: Obtained through Step Twelve , can be obtained Angle estimates , Indicates the number of incident sources;

[0062] Step Fourteen: Based on the results obtained in Step Thirteen And obtained in step eleven An estimated value of the azimuth can be obtained. , This indicates the number of incident sources.

[0063] Step two, which involves removing duplicate elements and sorting them, specifically includes:

[0064] A1: Define the set of uniform virtual array positions ,in ;

[0065] A2: For each virtual array element position traverse all actual array element pairs ,in, and They represent the first and second digits in the actual array, respectively. The and the first Index of each array element Find the satisfying The array element pairs, among which This refers to the actual positions of the array elements;

[0066] A3: For each pair of array elements that satisfy the conditions Collect the corresponding covariance matrix element values. or ,in Represents the autocovariance matrix of submatrix 1 The element in the m-th row and n-th column, Representing the cross-covariance matrix The element in the m-th row and n-th column is used to average the collected values ​​to obtain the virtual sensor signal value at that location: , ,in, This indicates that all conditions satisfying the positional difference are... The set of array element pairs;

[0067] A4: If a certain position There is no corresponding actual array element pair, that is If the signal value at that location is missing, then that location signal value is recorded as missing. ;

[0068] A5: Arrange all positions in order to obtain the virtual array signal vector. and .

[0069] The method of the present invention based on the above conditions was simulated using MATLAB simulation software, and the angle estimation results of the method of the present invention are as follows. Figure 3 As shown, when the incident signals (azimuth and elevation angles) are (30°, 10°) and (45°, 20°) respectively, the estimated angles obtained are (30.02°, 9.97°) and (44.98°, 20.02°). Figure 3 As can be seen from the experimental results, the method of this invention can accurately locate the position of the signal source.

[0070] Second embodiment: The relationship curve between the root mean square error and the signal-to-noise ratio of the signal processing method of the present invention is shown in the figure. Figure 4 and Figure 5 As shown, a performance comparison chart between the present invention and the parallel coprime matrix virtual interpolation unitary transform method is also provided. The application conditions of the method of this invention are as follows:

[0071] The method of this invention is based on an array model of parallel coprime matrices, as follows: Figure 1 As shown, the parallel coprime linear array consists of subarray 1 and subarray 2. Each subarray is composed of two nested sparse uniform linear arrays, with M=3, N=5, and 7 array elements per subarray. The spacing between the two parallel linear arrays is 0.03m. The number of signals K=2, and the two incident signals (azimuth and elevation angles) are (25°, 5°) and (40°, 15°), respectively. A snapshot count L=200 is used, and the signal-to-noise ratio (SNR) starts from 0dB and increases in step sizes of 2dB to 10dB. 500 independent Monte Carlo experiments are performed, and simulation analysis is conducted using MATLAB software. The resulting performance analysis curves are shown below. Figure 4 and Figure 5 As shown.

[0072] Depend on Figure 4 It can be seen that the root mean square error of the azimuth angle of both methods decreases with the increase of the signal-to-noise ratio. Among them, the root mean square error of the method of the present invention is the smallest. From the overall effect, the method of the present invention performs better than the other method. It also has strong estimation performance at low signal-to-noise ratio and can accurately estimate the azimuth angle of the signal.

[0073] Depend on Figure 5 It can be seen that the root mean square error of the pitch angle of both methods decreases with the increase of the signal-to-noise ratio. Among them, the root mean square error of the method of the present invention is the smallest. From the overall effect, the method of the present invention performs better than the other method and can accurately estimate the pitch angle of the signal.

[0074] Third embodiment: The relationship curve between the root mean square error and the number of snapshots in the signal processing method of the present invention is shown below. Figure 6 and Figure 7 As shown, a performance comparison chart between the present invention and the parallel coprime matrix virtual interpolation unitary transform method is also provided. The application conditions of the method of this invention are as follows:

[0075] The method of this invention is based on an array model of parallel coprime matrices, as follows: Figure 1 As shown, the parallel coprime linear array consists of subarray 1 and subarray 2. Each subarray is composed of two nested sparse uniform linear arrays, with M=3, N=5, and 7 array elements per subarray. The distance between the two parallel linear arrays is 0.03m, and the number of signals K=2. The two incident signals (azimuth and elevation angles) are (25°, 5°) and (40°, 15°), respectively. A signal-to-noise ratio of 10dB is used, and the number of snapshots starts from 100 and increases in step size from 100 to 500. 500 independent Monte Carlo experiments are performed, and MATLAB simulation software is used for simulation analysis. The resulting performance analysis curves are shown below. Figure 6 and Figure 7 As shown.

[0076] Depend on Figure 6It can be seen that the root mean square error of the azimuth angle of both methods decreases with the increase of the number of snapshots. Among them, the root mean square error of the method of the present invention is the smallest. From the overall effect, the method of the present invention performs better than the other method. It also has strong estimation performance at low number of snapshots and can accurately estimate the azimuth angle of the signal.

[0077] Depend on Figure 7 It can be seen that the root mean square error of the pitch angle of both methods decreases with the increase of the number of snapshots. Among them, the root mean square error of the method of the present invention is the smallest. From the overall effect, the method of the present invention performs better than the other method and can accurately estimate the pitch angle of the signal.

[0078] The specific examples described herein are merely illustrative of the invention. Those skilled in the art to which this invention pertains may make various modifications, additions, or similar substitutions to the described specific examples without departing from the invention or exceeding the scope defined by the appended claims.

Claims

1. A UUV side parallel-coprime array DOA estimation method, the parallel-coprime array is composed of subarray 1 and subarray 2, the number of elements of subarray 2 is the same as the array structure and subarray 1, the spacing between subarray 1 and subarray 2 is d, , representing the wavelength of the signal; subarray 1 is a coprime array nested by a pair of uniform linear arrays, the first uniform linear array is composed of M sensors with a spacing of Nd, the element position set is , and the second uniform linear array is composed of N sensors with a spacing of Md, the element position set is , wherein M and N are coprime integers, the two uniform linear arrays share the first sensor, and the coprime array is composed of M+N-1 sensors in total; subarray 2 is parallel to subarray 1 and above subarray 1, the spacing between the two subarrays is d, and the azimuth angle and the elevation angle of the first incident source are represented by and respectively, , and the angle between the incident signal and subarray 1 is represented by A UUV side parallel interlaced array DOA estimation method includes the following steps: ​ Step one: the received data of subarray 1 and subarray 2 are respectively and , the auto-covariance matrix of subarray 1 is calculated as and the cross-covariance matrix of subarray 1 and subarray 2 is calculated as wherein L represents the number of snapshots, represents the conjugate transpose of a matrix; Step two: vectorization Obtain where represents stretching the matrix by column into a long vector, from remove duplicate elements and sort to obtain a virtual array , vectorization Obtain , from remove duplicate elements and sort to obtain a virtual array ; Step three: according to the array parameters, calculate the non-uniform virtual array position set , according to define the uniform virtual array position set ; Step four: constructing the spreading vector based on the results of step two and step three and , representing a virtual signal of a virtual sensor at a position ; Step 5: For vectors Perform Toeplitz reconstruction on the elements in the array to obtain In the formula express The one element, , Represents a set The number of elements; Step six: Use the result of step five to construct the matrix Constructing a self-covariance matrix with Hermitian positive semi-definite and Toeplitz structure, the reconstruction problem can be formulated as ， Solved by CVX toolbox where is a projection matrix of the same dimension as matrix is a matrix corresponding to the element values of the interpolation elements are 0, the rest is 1, denotes the regularization parameter, denotes the Hadamard product, denotes the trace of the matrix, denotes the square of the Frobenius norm, ; Step seven: Toeplitz reconstruction of the elements in the vector results in where denotes the th element of . Step eight: Construct the non-Hermitian cross-covariance matrix by the result of step seven , and reconstruct the cross-covariance matrix by the kernel norm regularization method , ， wherein represents a Toeplitz matrix with c as the first column and r as the first row, wherein is a projection matrix of the same dimension as the matrix is a projection matrix of the same dimension as the matrix corresponding to the element value of the interpolation element is 0, and the rest is 1, , is an auxiliary positive semi-definite matrix, represents a regularization parameter; Step nine: Construct the extended matrix from the interpolated covariance matrix obtained in step six and the matrix obtained in step eight Step ten: Construct the extended matrix from the interpolated covariance matrix obtained in step six ; Step 10: [Regarding...] Perform singular value decomposition to extract the signal subspace. ,right Divide into blocks , and for The matrix, using , Constructing a matrix ; Step eleven: obtaining the corresponding eigenvalues from the eigenvalue decomposition, where N represents the number of incident sources, and the elevation angle is obtained from the eigenvalues where φ represents the phase angle of the complex number.​​​​​ Step twelve: using the eigenvector matrix after eigen decomposition , reconstructing the direction matrix , the phase relationship between adjacent elements can be obtained , represents the number of incident sources, where , , represents the square of the 2-norm; Step thirteen: obtaining the estimate of the angle by the result of step twelve , obtainable estimate of the angle , denotes the number of incident sources; Step fourteen: obtaining an estimate of the azimuth angle from the result of step thirteen and the result of step eleven an estimate of the azimuth angle , denotes the number of incident sources.

2. The UUV flank parallel interdigitated array DOA estimation method of claim 1, wherein: The process of removing duplicate elements and sorting in step two is as follows: A1 : Define a set of uniform virtual array positions wherein ; A2: For each virtual array element position , go through all actual array element pairs , where and are the indices of the th and th array element in the actual array , , find the array element pair that satisfies , where is the actual array element position; A3: for each element pair satisfying the condition , collect the corresponding covariance matrix element value or , wherein represents the mth row and nth column element of the subarray 1 autocovariance matrix represents the mth row and nth column element of the cross-covariance matrix , and take the average of the collected values to obtain the virtual sensor signal value at this position: , , wherein represents the set of all element pairs satisfying the position difference ;​ A4: If a position has no corresponding actual element pair, i.e. then the position signal value is noted as missing ; A5: Arrange all positions in order to get a virtual array signal vector and .

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