Virtual array extension-based Toeplitz matrix reconstruction method and system

Abstract

The invention provides a Toeplitz matrix reconstruction method and a Toeplitz matrix reconstruction system based on virtual array extension. The method comprises the following steps: constructing an original receiving array; the method comprises the following steps: receiving a space signal in real time through an original receiving array, and constructing a virtual extension array covariance matrix through vectorization; performing DOA estimation and bidirectional smoothing processing on the coherent signal to obtain a smoothed covariance matrix; according to the method, the direction of arrival of coherent signals can be effectively estimated, the effective aperture of the array can be expanded, compared with an existing matrix reconstruction method, the direction of arrival of more signals can be effectively estimated, and higher DOA estimation precision is achieved when the direction of arrival of the same number of signals is estimated.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of array signal processing, and particularly relates to a Toeplitz matrix reconstruction method and system based on virtual array expansion. BACKGROUND

[0002] The ionosphere is an important part of the near space environment and a natural medium for short-wave communication. Various disturbances near the ground can be uploaded to the ionosphere and interact with the ionized components. The ionosphere has an important influence on wireless communication, satellite navigation, measurement and human activities. In the ionosphere detection system, the Direction of Arrival (DOA) estimation technology of the signal is the core link to realize accurate ionosphere characteristic analysis and space target positioning. In the traditional method, the characteristic space decomposition algorithm based on array signal processing, such as the MUSIC algorithm, is widely used in DOA estimation, but these technologies are not adaptive to the high dynamic ionosphere environment, especially when facing the multipath effect. Due to the existence of a large number of coherent signals, the rank of the coherent signal covariance matrix is lost, and the estimation accuracy is easily sharply reduced or even invalid.

[0003] The core idea of the coherent signal DOA estimation is to solve the rank loss of the matrix, reduce the correlation of the signal through the algorithm, restore the dimension of the signal subspace after processing or transformation to the same as the number of signals, and thus improve the estimation performance of the target in the coherent signal environment. A typical decorrelation algorithm is the spatial smoothing algorithm, but this algorithm is usually only applicable to equidistant uniform linear arrays, and the algorithm needs to divide the array into several sub-arrays, then performs spatial smoothing to restore the rank of the signal covariance matrix. Not only is the implementation complex, but also the dimension of the corrected matrix is less than that of the original matrix, and the decorrelation performance of the algorithm has certain limitations under low signal-to-noise ratio conditions. Another simple and efficient algorithm is the matrix reconstruction algorithm, which usually rearranges the characteristic vectors or covariance matrix elements of the received signals to reconstruct one or more Toeplitz matrices. The rank of the Toeplitz matrix is only related to the DOA of the signal and is not affected by the correlation of the signal, which can restore the rank of the signal covariance matrix and realize decorrelation at the same time. However, the matrix reconstruction algorithm also performs dimension reduction processing on the matrix. When the number of arrays is M, the dimension of the reduced matrix is (M+1) / 2. If the number of signals is greater than the dimension of the reduced matrix, the algorithm cannot effectively estimate the DOA of the coherent signal. At this time, it is necessary to increase the number of array elements so that the dimension of the reduced matrix is greater than the number of signals to effectively estimate the DOA of the coherent signal. However, increasing the number of array elements will increase the cost of the receiving array. It is of great research significance and practical value to solve the problem of effectively estimating more DOAs of coherent signals while using fewer array elements in the ionosphere detection system.

[0004] To solve this problem, the application provides a Toeplitz matrix reconstruction method and system based on virtual array expansion. After the virtual expansion based on the fourth-order cumulant, the effective aperture of the array is expanded. At this time, combined with the matrix reconstruction method, not only the direction of arrival of the coherent signal can be effectively estimated, but also more signals can be effectively estimated compared with the existing matrix reconstruction method, and the DOA estimation accuracy is higher when estimating the same number of signals. SUMMARY

[0005] In order to effectively improve the DOA estimation effect of Wuhan multi-channel ionospheric sounding system (Wuhan Multi-channel Ionospheric Sounding System, WMISS) on coherent signals, the application provides a Toeplitz matrix reconstruction method based on virtual array expansion. When the multi-channel receiver of the WMISS system processes the received signal, first, the fourth-order cumulant matrix of the received signal is constructed, and the covariance matrix of the virtual array after expansion is obtained by removing the redundant items in the fourth-order cumulant matrix ; secondly, the elements of the 0th row to row of are selected to construct a plurality of Toeplitz matrices , and then is multiplied by and summed to obtain a matrix containing complete covariance matrix information; finally, the two-way smoothing processing is performed on to obtain , and the DOA estimation of the coherent signal is realized by combining with the ESPRIT algorithm. The algorithm can not only effectively estimate the direction of arrival of the coherent signal, but also expand the effective aperture of the receiving antenna array, and can effectively estimate more signals compared with the existing matrix reconstruction method, and has higher DOA estimation accuracy when estimating the same number of signals.

[0006] To solve the above technical problems, the application provides a Toeplitz matrix reconstruction method and system based on virtual array expansion.

[0007] The technical scheme of the method of the application is a Toeplitz matrix reconstruction method based on virtual array expansion, which specifically comprises the following steps: Step 1: constructing an original receiving array; Step 2: receiving spatial signals in real time through the original receiving array, and constructing a virtual expansion array covariance matrix through vectorization; Step 3: DOA estimation of coherent signals, and two-way smoothing processing to obtain a smoothed covariance matrix; As a preference, the original receiving array in step 1 is a uniform linear array consisting of array elements with an array element spacing of The first array element position of the original receiving array is defined as the reference point as the coordinate origin. The position of the th array element is ; As a preference, the spatial signal in step 2 is defined as follows: The expression at the th moment is:

[0008] Where, is the slowly varying amplitude at the th moment, is the slowly varying phase at the th moment, is the signal carrier frequency, , is the propagation speed of light in vacuum, is the signal wavelength; Suppose there are signals incident on the ULA of array elements, these signals can be coherent signals, incoherent signals, or mixed signals of coherent and incoherent signals; As a preference, the virtual extended array covariance matrix is constructed by vectorization in step 2 as follows: Then the vector form of the receiving array can be expressed as:

[0009] Where, is the output vector of the array at the th moment, is the output of the th array element at the th moment, is the steering vector matrix of the dimensional array, is the th column steering vector of the steering vector matrix of the dimensional array, is the array element spacing, is the angle of arrival of the th signal, is the signal wavelength, is the dimensional signal source vector, is the th spatial signal at the th moment, is the th spatial noise at time , is the th spatial noise at time The fourth-order cumulant matrix of is constructed as follows:

[0010] wherein denotes the cumulant operation, is the th element of the fourth-order cumulant of the output vector of the array at time is the output vector of the array at time denotes the expectation, denotes the Kronecker product, denotes the conjugate, denotes the conjugate transpose; The virtual extended array direction vector obtained by constructing the fourth-order cumulant matrix is: wherein

[0011] is the reference array element position vector, is the signal wavelength, and it can be seen from the above formula that the array element corresponding to the term of the direction vector is a virtual array element, the array element corresponding to the term of the direction vector is a real array element, and both the virtual array element and the real array element have repeated redundant terms. The receiving array before virtual extension is a uniform linear array composed of array elements, and the center array element position of the array is at the coordinate origin, and the direction vector thereof can be expressed as ; The receiving array after virtual extension is a uniform linear array composed of array elements, and the center array element position of the array is at the coordinate origin, and the direction vector thereof can be expressed as:

[0012]

[0013] The redundant terms in the fourth-order cumulant matrix are removed, the row and column elements corresponding to the direction vector of the extended array are retained, and the virtual extended array covariance matrix corresponding to the direction vector of the extended array is obtained: ​​​​​​​​​​​​​ in, is the fourth-order cumulant matrix Middle Row, No. Column element. ; .

[0014] Preferably, step 3 performs DOA estimation on the coherent signal as follows: The dimension of the Toeplitz matrix reconstructed by any row element of the array covariance matrix before expansion is , can be up to The coherent signal is used to estimate the DOA; For the virtual extended array covariance matrix , the dimension of the reconstructed matrix is , can be up to The coherent signal is used to estimate the DOA; Select Any line, such as The Toeplitz matrix is ​​constructed by row:

[0015] in, is the virtual extended array covariance matrix Middle Row, No. Column element. , .

[0016] because No. to The information of row reconstruction has the conjugate symmetry property. and Contains the same statistics. Optionally include a virtual extended array covariance matrix Complete information from line 0 to line Construct multiple Toeplitz matrices by row , and then and Multiplying and summing gives:

[0017] in, Indicates summation, is a matrix No. Toeplitz matrix constructed by rows; Preferably, the smoothed covariance matrix obtained by the bidirectional smoothing process in step 3 is as follows: In order to further improve the algorithm performance, the following formula is used The bidirectional smoothing processing can obtain:

[0018] Wherein, The matrix is represented by the anti-diagonal element 1 and the rest of the elements 0.

[0019] The equivalent data covariance matrix obtained is The ESPRIT algorithm is combined to realize the DOA estimation of the coherent signal.

[0020] The technical scheme of the system of the application is a Toeplitz matrix reconstruction system based on virtual array expansion, which is specifically as follows: The array receiving module is used to construct the original receiving array. The virtual extended array covariance matrix construction module receives the spatial signal in real time through the original receiving array, and constructs the virtual extended array covariance matrix through vectorization. The smoothed covariance matrix construction module estimates the DOA of the coherent signal, and obtains the smoothed covariance matrix through bidirectional smoothing processing. The algorithm disclosed by the application can not only effectively estimate the direction of arrival of the coherent signal, but also can expand the effective aperture of the array, and compared with the existing matrix reconstruction method, can effectively estimate the direction of arrival of more signals, and has higher DOA estimation accuracy when estimating the direction of arrival of the same number of signals. BRIEF DESCRIPTION OF DRAWINGS

[0021] Figure 1 The method flowchart of the embodiment of the application; Figure 2 The transmission schematic diagram of the signal of the embodiment of the application in the ionosphere; Figure 3 The block diagram of the Wuhan multi-channel ionosphere detection system of the embodiment of the application; Figure 4 The schematic diagram of the receiving array before virtual expansion of the embodiment of the application through the DOA estimation of the Toeplitz algorithm; Figure 5 The schematic diagram of the receiving array after virtual expansion of the embodiment of the application through the DOA estimation of the Toeplitz algorithm; Figure 6 The DOA estimation result diagram of the original receiving array element number of the embodiment of the application is 5, and the signal number is 4; Figure 7 The result diagram of the DOA estimation root mean square error and the successful resolution of the original receiving array element number of the embodiment of the application is 5, and the signal number is 2, and the change relationship result diagram of the signal-to-noise ratio; Figure 8: Result diagram of the relationship between the DOA estimation root mean square error and the success resolution as a function of the number of snapshots when the number of original receiving array elements is 5 and the number of signals is 2 according to an embodiment of the present invention; Figure 9 : Result diagram of the relationship between the DOA estimation root mean square error and success resolution as a function of angular interval when the number of original receiving array elements is 5 and the number of signals is 2 according to an embodiment of the present invention; DETAILED DESCRIPTION The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0022] In specific implementation, the method proposed in the technical solution of the present invention can be automatically run by those skilled in the art using computer software technology. System devices that implement the method, such as computer-readable storage media that store the corresponding computer program of the technical solution of the present invention and computer equipment that runs the corresponding computer program, should also be within the scope of protection of the present invention.

[0023] like Figure 2 As shown in the figure, during signal transmission, the transmitting antenna transmits the signal toward the ionosphere. After reaching the ionosphere, the signal is reflected by the ionosphere and transmitted to the receiving antenna array for reception. However, in actual signal transmission, co-channel interference and multipath effects cause the presence of a large number of coherent signals. In this case, decorrelation processing is required at the receiving end to effectively estimate the signal's direction of arrival.

[0024] like Figure 3 This is the overall block diagram of the Wuhan Multi-channel Ionospheric Sounding System (WMISS), which mainly includes a transmitting channel, a multi-channel receiver, a time-frequency synchronization module, and a communication module. The transmitting channel is mainly responsible for transmitting the signal. The main function of the multi-channel receiver is to extract the signal submerged in background noise and interference. The function of the time-frequency synchronization module is to ensure that the transmitting channel and the receiving device are synchronized in time and frequency. The role of the communication module is to complete the upper computer's issuance of the underlying parameters and instructions to the detection equipment, and to ensure that the underlying data of the device is transmitted to the upper computer. The present invention is an algorithm used in the multi-channel receiver of the WMISS system, which can perform decorrelation processing on the received coherent signals and realize DOA estimation. This patent conducts the following simulation experiments on the processing of coherent signals at the receiving end of the Wuhan Multi-channel Ionospheric Sounding System: A specific embodiment of the method of the present application is a Toeplitz matrix reconstruction method based on virtual array expansion, which is specifically as follows: Step 1: constructing an original receiving array; The original receiving array in step 1 is composed of a uniform linear array with array elements, and the array element spacing is The first array element position of the original receiving array is defined as the coordinate origin, as a reference point. The position of the th array element is ; Step 2: receiving spatial signals in real time through the original receiving array, and constructing a virtual extended array covariance matrix through vectorization; The expression at the th moment is:

[0025] Wherein, is the slowly varying amplitude at the th moment, is the slowly varying phase at the th moment, is the signal carrier frequency, , is the propagation speed of light in vacuum, is the signal wavelength.

[0026] Suppose that signals are incident on the ULA of 5 array elements in space, and the signals can be coherent signals, incoherent signals, or mixed signals of coherent and incoherent signals; The vector form of the receiving array can be expressed as:

[0027] Wherein, is the output vector of the array at the th moment, is the output of the th array element at the th moment, is the steering vector matrix of the dimensional array, is the steering vector of the th column of the steering vector matrix of the dimensional array, is the array element spacing, is the angle of arrival of the th signal, is the signal wavelength, is the dimensional signal source vector, For the The moment A spatial signal, For the Moment dimensional channel noise, , For the The moment It is assumed that the noise in each channel is independent of each other, obeys Gaussian distribution, and is independent of the signal source; right The construction of the fourth-order cumulant matrix yields:

[0028] in, Indicates the operation of finding the cumulative amount, Is the array The output vector of the fourth-order cumulant elements, For the array The output vector at time , Expressing hope, represents the Kronecker product, Indicates the conjugate, It means to find the conjugate transpose; The virtual extended array direction vector obtained by constructing the fourth-order cumulant matrix is:

[0029] in, is the reference element position vector, is the signal wavelength. From the above formula, we can see that the direction vector is The array element corresponding to the item is a virtual array element, and the direction vector is The array elements corresponding to the items are real array elements, and both the virtual array elements and the real array elements have repeated redundant items.

[0030] The receiving array before virtual expansion is a uniform linear array composed of 5 array elements. Taking the origin as the reference point, the center element of the array is located at the origin of the coordinate system, and its direction vector can be expressed as ; The virtually extended receiving array is a uniform linear array consisting of 9 elements. Taking the origin as the reference point, the center element of the array is located at the origin of the coordinate system. Its direction vector can be expressed as:

[0031] Remove the fourth-order cumulant matrix The redundant items in retain the row and column elements corresponding to the direction vector of the expanded array, and obtain the virtual expanded array covariance matrix corresponding to the direction vector of the expanded array:

[0032] in, is the fourth-order cumulant matrix Middle Row, No. Column element. ; .

[0033] Step 3: Perform DOA estimation on the coherent signal and perform bidirectional smoothing to obtain the smoothed covariance matrix; The dimension of the Toeplitz matrix reconstructed by any row element of the array covariance matrix before expansion is , DOA estimation can be performed on up to 2 coherent signals; For the virtual extended array covariance matrix , the dimension of the reconstructed matrix is , DOA estimation can be performed on up to 4 coherent signals; Select Any line, such as The Toeplitz matrix is ​​constructed by row:

[0034] in, is the virtual extended array covariance matrix Middle Row, No. Column element. , .

[0035] because No. to The information of row reconstruction has the conjugate symmetry property. and Contains the same statistics. Optionally include a virtual extended array covariance matrix The 0th to 4th rows of the complete information construct multiple Toeplitz matrices , and then and Multiplying and summing gives:

[0036] in, Indicates summation, is a matrix No. Toeplitz matrix constructed by rows.

[0037] In order to further improve the performance of the algorithm, Performing bidirectional smoothing gives:

[0038] in, Represents a matrix where the anti-diagonal elements are 1 and the rest of the elements are 0.

[0039] The equivalent data covariance matrix obtained is Combined with ESPRIT algorithm, DOA estimation of coherent signals is realized.

[0040] like Figure 1 As shown in the figure, the present invention is a Toeplitz matrix reconstruction method based on virtual array expansion. First, the fourth-order cumulant matrix of the coherent received signal is constructed, and the covariance matrix of the expanded virtual array is obtained by removing the redundant items in the fourth-order cumulant matrix. ; Secondly, choose Row 0 to Construct multiple Toeplitz matrices using the elements of the rows , and then and Multiply and sum to get a matrix containing the complete covariance matrix information ; Finally Perform bidirectional smoothing to obtain , and Combined with ESPRIT algorithm, DOA estimation of coherent signals is realized.

[0041] like Figure 2 As shown in FIG, during the signal transmission process, the transmitting antenna transmits the signal to the ionosphere. After the signal reaches the ionosphere, it is reflected by the ionosphere and transmitted to the receiving antenna array for reception.

[0042] like Figure 3 The figure shows the overall block diagram of the Wuhan Multi-channel Ionospheric Sounding System (WMISS). It primarily consists of a transmitter channel, a multi-channel receiver, a time-frequency synchronization module, and a communication module. The transmitter channel is responsible for signal transmission. The multi-channel receiver's primary function is to extract signals buried in background noise and interference. The time-frequency synchronization module ensures time and frequency synchronization between the transmitter channel and the receiver. The communication module allows the host computer to issue low-level parameters and commands to the detection equipment and ensures the transmission of low-level data from the equipment to the host computer.

[0043] like Figure 4As shown, the receiving array before virtual expansion is composed of The uniform linear array consists of array elements, and the array element spacing is , taking the origin as the reference point, the center element of the array is located at the origin of the coordinate system, and its direction vector can be expressed as . Receive data matrix for array before virtual expansion The covariance matrix obtained is reconstructed by Toeplitz matrix, and the dimension of the reconstructed matrix is , can be up to The DOA estimation is performed based on the coherent signals.

[0044] like Figure 5 As shown, the receiving array after virtual expansion based on the fourth-order cumulant is composed of The uniform linear array consists of array elements, and the array element spacing is , taking the origin as the reference point, the center element of the array is located at the origin of the coordinate system, and its direction vector can be expressed as . Remove the fourth-order cumulant matrix The redundant items in , retain the row and column elements corresponding to the expanded array direction vector, and obtain the virtual expanded array covariance matrix corresponding to the expanded array direction vector. The virtual expanded array covariance matrix is ​​reconstructed by Toeplitz matrix, and the reconstructed matrix dimension is , can be up to The DOA estimation is performed based on the coherent signals.

[0045] The root mean square error (RMSE) of DOA estimation is defined as:

[0046] in, is the number of Monte Carlo experiments, is the number of incident signals, For the In the Monte Carlo experiment The DOA estimate of the signal.

[0047] Simulation experiment 1: Assume that the original receiving array is a uniform linear array with the number of array elements , array element spacing , signal-to-noise ratio , snapshot number After the virtual expansion based on the fourth-order cumulant, the array becomes a uniform linear array with 9 elements, and the azimuth angle is selected as 、 The two coherent signals and 、 two independent signals, a total of four signals, 50 times of Monte Carlo experiments are carried out, and simulation results are shown in Figure 6 As shown in Figure 6 , the algorithm can effectively estimate the directions of arrival of the four signals, and the number of directions of arrival that can be estimated is twice that of the existing matrix reconstruction method.

[0048] Simulation experiment 2: assuming that the original receiving array is a uniform linear array, the number of array elements , the array element spacing , the number of snapshots . Two coherent signals with azimuth angles of , are selected, the signal-to-noise ratio changes in the range of , 100 times of Monte Carlo experiments are carried out, and simulation results are shown in Figure 7 . As shown in Figure 7 , with the continuous increase of the signal-to-noise ratio, the DOA estimation accuracy of different methods gradually improves, and the algorithm always maintains the best DOA estimation accuracy under different signal-to-noise ratios.

[0049] Simulation experiment 3: assuming that the original receiving array is a uniform linear array, the number of array elements , the array element spacing , the signal-to-noise ratio . Two coherent signals with azimuth angles of , are selected, the number of snapshots changes in the range of , 100 times of Monte Carlo experiments are carried out, and simulation results are shown in Figure 8 . As shown in Figure 8 , with the continuous increase of the number of snapshots, the DOA estimation accuracy of different methods gradually improves, and the algorithm always maintains the best DOA estimation accuracy under different numbers of snapshots.

[0050] Simulation experiment 4: assuming that the original receiving array is a uniform linear array, the number of array elements , the array element spacing , the signal-to-noise ratio , the number of snapshots . Two coherent signals with azimuth angles of , are selected, the angle interval changes in the range of , 100 times of Monte Carlo experiments are carried out, and simulation results are shown in Figure 9 . As shown in Figure 9 , with the continuous increase of the angle interval, the DOA estimation accuracy of different methods gradually improves, and the algorithm always maintains the best DOA estimation accuracy under different angle intervals.

[0051] It should be understood that parts of the specification not specifically described in detail are part of the prior art.

[0052] It should be understood that the above description of the embodiments is more detailed and is not therefore considered as limiting the scope of patent protection of the present application. Any person skilled in the art, under the guidance of the present application, can make substitutions or modifications without departing from the scope of protection of the present application, and all such substitutions or modifications fall within the scope of protection of the present application. The scope of protection of the present application should be subject to the appended claims.

Claims

1. A Toeplitz matrix reconstruction method based on virtual array expansion, characterized in that: The following steps are involved: Step 1: Construct the original receiving array; Step 2: Receive the spatial signal in real time through the original receiving array and construct the virtual extended array covariance matrix through vectorization; Step 3: Perform DOA estimation on the coherent signal and perform bidirectional smoothing to obtain the smoothed covariance matrix.

2. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 1, characterized in that: The original receiving array described in step 1 is composed of The uniform linear array consists of array elements, and the array element spacing is , the first element position of the original receiving array is taken as the coordinate origin and defined as the reference point; No. The position of the array element is .

3. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 2, characterized in that: The spatial signal in step 2 is defined as follows: In the The expression of time is: in, For the The slow change of time, For the The slowly changing phase of time, is the signal carrier frequency, , is the speed of light in vacuum, is the signal wavelength; Assume that the space has A signal is incident on On the ULA of the array element, this The signal can be a coherent signal, an incoherent signal, or a mixed signal of coherent and incoherent signals.

4. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 3, characterized in that: The virtual extended array covariance matrix constructed by vectorization in step 2 is as follows: The vector form of the receiving array can be expressed as: in, For the array The output vector at time , For the The array element is in the Output at any moment, for dimensional array of steering vector matrices, for The first dimension of the steering vector matrix of the Column-oriented vector, is the array element spacing, For the The angle of arrival of the signal, is the signal wavelength, for Weixin Source Vector, For the The moment A spatial signal, For the Moment dimensional channel noise, , For the The moment It is assumed that the noises in each channel are independent of each other, obey Gaussian distribution, and are independent of the signal source.

5. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 4, characterized in that: right The construction of the fourth-order cumulant matrix yields: in, Indicates the operation of finding the cumulative amount, Is the array The output vector of the fourth-order cumulant elements, For the array The output vector at time , Expressing hope, represents the Kronecker product, Indicates the conjugate, It means to find the conjugate transpose; The virtual extended array direction vector obtained by constructing the fourth-order cumulant matrix is: in, is the reference element position vector, is the signal wavelength. From the above formula, we can see that the direction vector is The array element corresponding to the item is a virtual array element, and the direction vector is The array elements corresponding to the items are real array elements, and both the virtual array elements and the real array elements have repeated redundant items.

6. The Toeplitz matrix reconstruction method based on virtual array extension according to claim 5, characterized in that: The receiving array before virtual expansion is composed of The uniform linear array is composed of array elements. With the origin as the reference point, the center element of the array is located at the origin of the coordinate system, and its direction vector can be expressed as ； The virtual extended receiving array is composed of The uniform linear array is composed of array elements. With the origin as the reference point, the center element of the array is located at the origin of the coordinate system. Its direction vector can be expressed as: 。 7. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 6, characterized in that: Remove the fourth-order cumulant matrix The redundant items in retain the row and column elements corresponding to the direction vector of the expanded array, and obtain the virtual expanded array covariance matrix corresponding to the direction vector of the expanded array: in, is the fourth-order cumulant matrix Middle Row, No. Column Elements 。 8. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 7, characterized in that: Step 3 performs DOA estimation on the coherent signal as follows: The dimension of the Toeplitz matrix reconstructed by any row element of the array covariance matrix before expansion is , can be up to The coherent signal is used to estimate the DOA; For the virtual extended array covariance matrix , the dimension of the reconstructed matrix is , can be up to The coherent signal is used to estimate the DOA; Select Any line, such as The Toeplitz matrix can be constructed by row: in, is the virtual extended array covariance matrix Middle Row, No. Column elements; , ; because No. to The information of row reconstruction has the conjugate symmetry property. and Contains the same statistics; optionally includes a virtual extended array covariance matrix Complete information from line 0 to line Construct multiple Toeplitz matrices by row , and then and Multiplying and summing gives: in, Indicates summation, is a matrix No. Toeplitz matrix constructed by rows.

9. The Toeplitz matrix reconstruction method based on virtual array expansion according to claim 8, characterized in that: The smoothed covariance matrix obtained by the bidirectional smoothing process in step 3 is as follows: In order to further improve the performance of the algorithm, Performing bidirectional smoothing gives: in, Represents a matrix where the elements on the anti-diagonal are 1 and the rest are 0; The equivalent data covariance matrix obtained is Combined with ESPRIT algorithm, DOA estimation of coherent signals is realized.

10. A Toeplitz matrix reconstruction system based on virtual array expansion, characterized by: Array receiving module, used to build the original receiving array; A virtual extended array covariance matrix construction module receives spatial signals in real time through the original receiving array and constructs the virtual extended array covariance matrix through vectorization; The smoothed covariance matrix construction module performs DOA estimation on the coherent signal and obtains the smoothed covariance matrix through bidirectional smoothing.

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