Cyclostationary signal direction of arrival estimation method based on co-prime array

Abstract

The invention relates to a cyclostationary signal direction-of-arrival estimation method based on a co-prime array, and belongs to the technical field of direction-of-arrival estimation in array signal processing. The method comprises the following steps: establishing a co-prime array based on two sparse uniform arrays, establishing an array receiving signal according to the co-prime array established in the step 1, constructing a cyclic autocorrelation matrix output by the co-prime array, constructing a virtual array model through vectorization processing, expanding an array aperture, improving the degree of freedom, and finally solving the direction of arrival by using a TLS-ESPRIT method. The method has the advantages that the direction of arrival estimation of the cyclostationary signal can be realized, the calculation complexity is low, and the estimation degree of freedom and the estimation precision are superior.

Description

Technical Field

[0001] The invention belongs to the technical field of direction of arrival estimation in array signal processing, and in particular relates to a method for estimating the direction of arrival of a cyclostationary signal based on a coprime array. Background Art

[0002] In modern communications, radar, sonar and other related fields, direction of arrival estimation DOA (Direction of Arrival) is a key technology in signal processing. DOA estimation methods are widely used in array signal processing, especially inferring the direction of the arriving signal through multiple receiving units of the array antenna. This technology plays an important role in many fields such as wireless communication, intelligent transportation, unmanned driving, positioning systems and military monitoring. Cyclostationary signals are a type of signal. Unlike traditional stationary signals, the statistical characteristics of cyclostationary signals (such as mean, covariance matrix, etc.) will change within a certain period. Compared with stationary signals, cyclostationary signals have richer structural characteristics. They not only have stable statistical characteristics in the average sense, but also their signal statistical characteristics will change periodically within a periodic time window. Cyclostationary signals not only show stable statistical characteristics within a long time window of time, but also their statistical characteristics within a short period will change periodically over time. This makes cyclostationary signals have a more complex and richer structure than stationary signals, which can provide more information in signal processing and bring higher accuracy and robustness to signal analysis and estimation.

[0003] Coprime arrays are a method to improve the array signal processing performance by selecting antenna array configurations with coprime spacing. Traditional array structures usually require array elements to be arranged at uniform intervals, while coprime arrays construct arrays by using different array element spacings with a coprime relationship, thereby avoiding the spatial redundancy problem that exists in conventional uniform arrays. Coprime arrays can solve the problem of underdetermined signal source estimation through their special design, that is, they can estimate more signal sources by providing more spatial sampling information when the number of array elements is insufficient. In the past, DOA estimation algorithms for cyclostationary signals often focused on further improving computational complexity and estimation accuracy, ignoring some other important issues. These methods still rely on the number of useful signals being less than the number of physical array elements of the array antenna. This makes these algorithms unable to handle situations where the number of signal sources is greater than the number of array antennas, and thus unable to achieve underdetermined estimation of useful signals, that is, unable to accurately estimate the directions of multiple signal sources, especially when there are too many signal sources. Summary of the invention

[0004] The present invention provides a method for estimating the direction of arrival of cyclostationary signals based on coprime arrays to solve the problem that the number of useful signals must be less than the number of physical array elements of the array antenna in the past methods for processing cyclostationary signals. Coprime arrays are used to solve the problem that the number of signal sources is greater than the number of array antennas, and the TLS-ESPRIT method is used to further improve the estimation accuracy and reduce the computational complexity.

[0005] The technical solution adopted by the present invention comprises the following steps:

[0006] Step 1: Construct a coprime array by combining two sparse uniform linear arrays;

[0007] Step 2: Based on the coprime array constructed in step 1, establish an array receiving signal model;

[0008] Step 3: Obtain the corresponding circulant correlation matrix from the received signal model;

[0009] Step 4: Using column vectorization technology, the circulant correlation matrix of the array is converted into a virtual array model;

[0010] Step 5: Select the largest part of continuous virtual array elements in the virtual array to form a virtual uniform linear array, and process it using a spatial smoothing method to obtain a forward and backward smoothing matrix;

[0011] Step 6: Perform eigenvalue decomposition on the forward and backward smoothing matrices, and use the TLS-ESPRIT algorithm to estimate the direction of arrival and give the direction angle.

[0012] The coprime array used in step 1 of the present invention is a non-uniform linear array, which is composed of two sparse uniform linear arrays A and B. Subarrays A and B share a reference array element. The first physical array element position of subarrays A and B is the same. Take two coprime positive integers M and N. d=λ / 2 represents the unit array spacing. The wavelength of the signal is λ. The array element spacing of subarray A is set to Nd, and the number of array elements is 2M; the array element spacing of subarray B is set to Md, and the number of array elements is N. Therefore, the total number of array elements of the coprime array is L=2M+N-1.

[0013] In step 2 of the present invention, the signal is set to be a far-field narrowband independent cyclostationary signal, with a total of K incident directions (θ 1 ,θ 2 ,…,θ K ) k (t) incident on the array and are parallel and coplanar, θ k is the angle between the incident angle of the kth signal and the array normal, and the array receiving signal model is established as:

[0014]

[0015] Where t = 1, 2, ..., T, T is the number of snapshots, N(t) is additive Gaussian white noise with a mean of 0 and a variance of σ 2 ,S(t)=[s 1 (t),s 2 (t),…,s K (t)] T , [·] T represents the transfer rank, a(θ K ) is the steering vector of the array, which can be expressed as:

[0016]

[0017] Where: d l is the distance between the lth (l=1,2,…,L)th array element and the first array element; d l ∈P, P is the set of mutually prime array element positions:

[0018] P={0,-Nd,-2Nd,…,-(2M-1)Nd}∪{0,Md,2Md,…,(N-1)Md} (3)

[0019] Then the direction matrix A of the array can be expressed as:

[0020] A=[a(θ 1 ),a(θ 2 ),…,a(θ K )] (4)

[0021] The circular correlation matrix of the array signal is obtained by collecting the received signal obtained by snapshot.

[0022] In step 3 of the present invention, according to the cyclostationary theory, assuming that the noise and the signal are independent of each other, at the cyclic frequency α, K α The signal has cyclostationary characteristics, K α ≤K, the cyclic correlation function output by the pth array element and the qth array element (p,q=1,2,…,L) is:

[0023]

[0024] In the formula, P lp , P lq ∈P, [·] * represents conjugate calculation;

[0025] It can be further simplified as:

[0026]

[0027] The useful signals are not cyclically correlated with each other and do not have cyclic correlation, and the noise and other influencing factors do not have cyclostationary characteristics at the cyclic frequency α;

[0028] In the formula, is the circular correlation function of the received signal

[0029] Then the cyclic correlation matrix of the received signal model is obtained as:

[0030]

[0031] In step 4 of the present invention, according to the cyclic correlation function and cyclic correlation matrix of step 3, the matrix expression of the array output can be derived as follows:

[0032]

[0033] In the formula, is the autocorrelation matrix of the incident signal, let the kth (k=1,2,…,K α ) The power of the incident signal is but:

[0034]

[0035] Column-vectorize the circular correlation matrix:

[0036]

[0037] In the formula,

[0038]

[0039] We can get a virtual array, let B(θ 1 ,θ 2 ,…,θ Kα )=(A * αA) is the steering vector of the virtual array. For any element in B The formula or To represent, where m,m 1 ,m 2 ∈[0,2M-1],n,n 1 ,n 2 ∈[0,N-1], B is the set of horizontal distance differences between any two physical array elements in the coprime array due to different positions;

[0040] The self-difference position set and mutual difference position set of two sub-matrices of a coprime matrix are defined as follows:

[0041]

[0042] Then the position difference set of coprime arrays is D p =D AA ∪D BB ∪D AB , after removing the repeated redundant parts, the virtual array model is:

[0043]

[0044] Where Fc is the selection matrix, when it first appears (d a -d b )=mL c When F c =1, otherwise F c =0,d a d b Respectively represent the horizontal distances of the ath and bth physical array elements in subarrays A and B relative to the first reference array element, m = 1, ..., 2L c -1;L c =MN+M-1; a=b=1,2,…,L, the array response of the virtual array generated after the selection is expressed as:

[0045]

[0046] In step 5 of the present invention, the new non-uniform linear array obtained according to step 4, -(MN+M-1) to MN+M-1 is the longest continuous uniform linear array part, the length is 2(MN+M)-1, and the front and back smoothing process is adopted, MN+M is set as the length of a single smooth segment, and L=MN+M is the number of smooth segments:

[0047]

[0048] in: represents the array output of the forward smoothed ith smoothed sub-matrix,

[0049] The cyclic autocorrelation matrices of all forward smoothed subsegments are averaged to obtain:

[0050]

[0051] In the formula, [·] H Indicates the conjugate rank calculation. Similarly, the average of the backward smooth correlation matrix, that is, the inverse matrix of the forward smooth matrix, can be obtained as follows:

[0052]

[0053] Therefore, the forward and backward smoothed correlation matrix is:

[0054]

[0055] The forward and backward smoothed correlation matrix obtained in step 5 in step 6 of the present invention is Perform eigenvalue decomposition on it:

[0056]

[0057] Where U s is the signal subspace, U n is the noise subspace, Λ s and Λ n are the characteristic values ​​of signal and noise respectively, and U s The first to the second to last columns constitute the subspace U sx , take U s The second to last columns constitute the subspace U sy , merge the two subspaces to form a new matrix, and calculate the matrix P, that is:

[0058]

[0059] Perform eigenvalue decomposition on P to obtain the eigenvalue and eigenvector E, and divide E into a combination of 4 matrices, namely:

[0060]

[0061] Through the matrix E 12 and E 22 Calculate the matrix F, that is:

[0062] F=-E 12 E 22 -1 (twenty four)

[0063] Perform eigenvalue decomposition on the matrix F to obtain the eigenvalue λ. The angle between its real and imaginary parts is the direction angle:

[0064]

[0065] In the formula, λ i The i-th eigenvalue, is the real part of the eigenvalue, |λ i | is the modulus of the eigenvalue.

[0066] The present invention has the following advantages:

[0067] The use of mutually prime arrays can achieve higher angular resolution at a smaller array scale, and can more accurately estimate the direction of arrival (DOA) of multiple signals in an environment with dense signal sources. The use of mutually prime array element spacing avoids the spatial redundancy problem of elements in the array. Due to its sparsity and flexible array design, it can better adapt to complex signal environments, especially under conditions such as multipath effects, signal interference, and noise. The present invention can provide higher estimation accuracy and robustness in complex wireless communications, radar, and sonar applications;

[0068] Compared with the previous methods for processing the direction of arrival of cyclostationary signals, this method can break through the limitation of the number of physical array elements and estimate the direction of arrival with physical array elements that are less than the number of signal sources. By adopting mutually prime array element spacing, the present invention avoids the problem of spatial redundancy of elements in the array. Traditional uniform arrays often have too small array element spacing, resulting in spatial sampling redundancy of the array, thereby reducing the effectiveness of the array. The present invention reduces unnecessary redundancy and improves the spatial sampling efficiency of the array by optimizing the array element spacing design;

[0069] This method can accurately identify cyclostationary signals and other interference signals, and can effectively distinguish signals in complex environments by using cyclostationary characteristics;

[0070] Compared with other algorithms for direction of arrival parameter estimation, the present invention has superiority in algorithm complexity while ensuring algorithm accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 is a schematic diagram of a coprime array structure used in an embodiment of the present invention;

[0072] Figure 2 is a flow chart of the present invention;

[0073] Figure 3 is a comparison diagram of the MUSIC degrees of freedom of the present invention and a one-dimensional uniform linear array provided in Experimental Example 1;

[0074] Figure 4 This is a comparison diagram of the screening of useful cyclically correlated signals by the present invention and one-dimensional uniform linear array MUSIC provided in Experimental Example 2;

[0075] Figure 5 This is a comparison diagram of the root mean square error between the present invention and the one-dimensional uniform linear array MUSIC under different signal-to-noise ratios provided in Experimental Example 3. DETAILED DESCRIPTION

[0076] The following steps are included, see Figure 2 :

[0077] Step 1: Construct a coprime array by combining two sparse uniform linear arrays;

[0078] The coprime array used is a non-uniform linear array, such as Figure 1 As shown, it is composed of two sparse uniform linear arrays A and B. Subarrays A and B share a reference array element. The first physical array element position of subarrays A and B is the same. Take two mutually prime positive integers M and N. d = λ / 2 represents the unit array spacing. The wavelength of the signal is λ. The array element spacing of subarray A is set to Nd, and the number of array elements is 2M; the array element spacing of subarray B is Md, and the number of array elements is N. Therefore, the total number of array elements of the mutually prime array is L = 2M + N-1;

[0079] Step 2: Based on the coprime array constructed in step 1, establish an array receiving signal model;

[0080] The signal is assumed to be a far-field narrowband independent cyclostationary signal, with a total of K incident directions (θ 1 ,θ 2 ,…,θ K ) k (t) incident on the array and are parallel and coplanar, θ k is the angle between the incident angle of the kth signal and the array normal, and the array receiving signal model is established as:

[0081]

[0082] Where t = 1, 2, ..., T, T is the number of snapshots, N(t) is additive Gaussian white noise with a mean of 0 and a variance of σ 2 ,S(t)=[s 1 (t),s 2 (t),…,s K (t)] T , [·] T represents the transfer rank, a(θ K ) is the steering vector of the array, which can be expressed as:

[0083]

[0084] Where: d l is the distance between the lth (l=1,2,…,L)th array element and the first array element; d l ∈P, P is the set of mutually prime array element positions:

[0085] P={0,-Nd,-2Nd,…,-(2M-1)Nd}∪{0,Md,2Md,…,(N-1)Md} (3)

[0086] Then the direction matrix A of the array can be expressed as:

[0087] A=[a(θ 1 ),a(θ2 ),…,a(θ K )] (4)

[0088] The circular correlation matrix of the array signal is obtained by collecting the received signal obtained by snapshot.

[0089] Step 3: Obtain the corresponding circulant correlation matrix from the received signal model;

[0090] According to the cyclostationary theory, assuming that the noise and signal are independent of each other, at the cyclic frequency α, K α The signal has cyclostationary characteristics (K α ≤K), the cyclic correlation function output by the p-th array element and the q-th array element (p,q=1,2,…,L) is:

[0091]

[0092] In the formula, P lp , P lq ∈P, [·] * represents conjugate calculation;

[0093] It can be further simplified as:

[0094]

[0095] The useful signals are not cyclically correlated with each other and do not have cyclic correlation, and the noise and other influencing factors do not have cyclostationary characteristics at the cyclic frequency α;

[0096] In the formula, is the circular correlation function of the received signal Then the cyclic correlation matrix of the received signal model can be obtained as:

[0097]

[0098] Step 4: Using column vectorization technology, the circulant correlation matrix of the array is converted into a virtual array model;

[0099] According to the cyclic correlation function and cyclic correlation matrix, the matrix expression of the array output can be derived as follows:

[0100]

[0101] In the formula, is the autocorrelation matrix of the incident signal, let the kth (k=1,2,…,K α ) The power of the incident signal is but:

[0102]

[0103] Column-vectorize the circular correlation matrix:

[0104]

[0105] In the formula,

[0106]

[0107] We can get a virtual array, let is the steering vector of the virtual array. For any element in B The formula or To represent, where m,m 1 ,m 2 ∈[0,2M-1],n,n 1 ,n 2 ∈[0,N-1], B is the set of horizontal distance differences between any two physical array elements in the coprime array due to different positions;

[0108] The self-difference position set and mutual difference position set of two sub-matrices of a coprime matrix are defined as follows:

[0109]

[0110] Then the position difference set of coprime arrays is D p =D AA ∪D BB ∪D AB The virtual array model obtained after removing the repeated redundant parts is:

[0111]

[0112] Among them, F c To select the matrix, when it first appears (d a -d b )=mL c When F c =1, otherwise F c =0.d a d b Respectively represent the horizontal distances of the ath and bth physical array elements in subarrays A and B relative to the first reference array element. m=1,…,2L c -1;L c =MN+M-1; a=b=1,2,…,L. The array response of the virtual array generated after the selection can be expressed as:

[0113]

[0114] Step 5: Select the largest part of continuous virtual array elements in the virtual array to form a virtual uniform linear array, and process it using a spatial smoothing method to obtain a forward and backward smoothing matrix;

[0115] According to the virtual array obtained in step 4, -(MN+M-1) to MN+M-1 is the longest continuous uniform linear array part, with a length of 2(MN+M)-1. The front and back smoothing process is adopted, and MN+M is set as the length of a single smooth segment, and L=MN+M is the number of smooth segments:

[0116]

[0117] in: represents the array output of the forward smoothed ith smoothed sub-matrix,

[0118] The cyclic autocorrelation matrices of all forward smoothed subsegments are averaged to obtain:

[0119]

[0120] In the formula, [·] H Indicates the conjugate rank calculation. Similarly, the average of the backward smooth correlation matrix, that is, the reverse matrix of the forward smooth matrix, can be obtained as follows:

[0121]

[0122] Therefore, the forward and backward smoothed correlation matrix is:

[0123]

[0124] Step 6: Perform eigenvalue decomposition on the forward and backward smoothing matrices, and use the TLS-ESPRIT algorithm to estimate the direction of arrival and give the direction angle.

[0125] According to the forward and backward smoothed correlation matrix obtained in step 5 Perform eigenvalue decomposition on it:

[0126]

[0127] Where U s is the signal subspace, U n is the noise subspace, Λ s and Λ n are the characteristic values ​​of signal and noise respectively. Take U s The first to the second to last columns constitute the subspace U sx , take U s The second to last columns constitute the subspace U sy, merge the two subspaces to form a new matrix, and calculate the matrix P, that is:

[0128]

[0129] Perform eigenvalue decomposition on P to obtain the eigenvalue and eigenvector E, and divide E into a combination of 4 matrices, namely:

[0130]

[0131] Through the matrix E 12 and E 22 Calculate the matrix F, that is:

[0132] F=-E 12 E 22 -1 (twenty four)

[0133] Perform eigenvalue decomposition on the matrix F to obtain the eigenvalue λ. The angle between its real and imaginary parts is the direction angle:

[0134]

[0135] In the formula, λ i The i-th eigenvalue, is the real part of the eigenvalue, |λ i | is the modulus of the eigenvalue.

[0136] The effects of the present invention are further illustrated by experimental examples below.

[0137] Experimental Example 1:

[0138] In order to verify that the proposed algorithm has obvious advantages in algorithm freedom, a comparative experiment is designed according to the above steps with the MUSIC algorithm of a one-dimensional uniform linear array. Assume that there are 9 far-field BPSK signals with directions uniformly distributed between -60° and 60° incident on the array, and the carrier frequency f c =1.6Ghz, sampling frequency f s =1.6Ghz, signal code rate is 0.5MHz, cycle frequency α=2f c1 , number of snapshots T = 500, signal-to-noise ratio SNR = 5dB.

[0139] from Figure 3 It can be observed that the conventional MUSIC algorithm has failed and is completely unable to perform effective DOA estimation, and the spectrum peaks are disordered. The proposed algorithm uses the advantages of coprime arrays to break through the limitation of the number of physical array elements, and can accurately estimate the direction of the signal, with clear and sharp spectrum peaks.

[0140] Experimental Example 2:

[0141] In order to verify that the proposed algorithm can effectively screen useful signals, it is assumed that there are three far-field BPSK signals with an angle of 1°, 1° and 30°, and the carrier frequency f c1 =f c2 =1.6Ghz and f c3 =1Ghz, sampling frequency f s =1.6Ghz, signal code rate is 0.5MHz, cycle frequency α=2f c1 , snapshot number T = 500, signal-to-noise ratio SNR = 5dB, and observe its comparison with the MUSIC algorithm.

[0142] from Figure 4 It can be observed that the MUSIC algorithm of the one-dimensional uniform linear array has a peak at the 30° angle of the interference signal, and fails to accurately estimate the incoming wave directions of the cyclostationary signal at -1° and 1°, indicating that the cyclostationary signal and the interference signal cannot be distinguished. The algorithm proposed in the present invention uses the cyclostationary characteristics of the signal to accurately identify the cyclostationary signal and the interference signal, accurately estimates the incoming wave directions of -1° and 1°, and has a sharp spectrum peak, showing that the present invention can effectively screen useful signals.

[0143] Experimental Example 3:

[0144] In order to verify that the proposed algorithm has advantages in estimation performance, the root mean square error RMSE is selected as the performance indicator. Assuming that there are three far-field BPSK signals incident on the array, the carrier frequency f c =1.6Ghz, sampling frequency f s =1.6Ghz, signal code rate is 0.5MHz, cycle frequency α=2f c1 , the number of snapshots T = 500, the signal-to-noise ratio SNR increases from -20 to 30 with a step size of 5, and observe its comparison with the MUSIC algorithm.

[0145] from Figure 5 It can be observed that the root mean square error RMSE decreases with the increase of the signal-to-noise ratio, and the conventional MUSIC algorithm decreases significantly less than the algorithm proposed in the present invention, which proves that the present algorithm is superior in estimation performance.

[0146] The above content is a further detailed description of the present invention in combination with specific implementation methods, but it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, several deductions or substitutions can be made without departing from the idea of ​​the present invention, which should be regarded as falling within the scope of protection of the present invention.

Claims

1. A method for estimating the direction of arrival of cyclostationary signals based on coprime arrays, characterized in that: The following steps are involved: Step 1: Construct a coprime array by combining two sparse uniform linear arrays; Step 2: Based on the coprime array constructed in step 1, establish an array receiving signal model; Step 3: Obtain the corresponding circulant correlation matrix from the received signal model; Step 4: Using column vectorization technology, the circulant correlation matrix of the array is converted into a virtual array model; Step 5: Select the largest part of continuous virtual array elements in the virtual array to form a virtual uniform linear array, and process it using a spatial smoothing method to obtain a forward and backward smoothing matrix; Step 6: Perform eigenvalue decomposition on the forward and backward smoothing matrices, and use the TLS-ESPRIT algorithm to estimate the direction of arrival and give the direction angle.

2. The method for estimating the direction of arrival of a cyclostationary signal based on a coprime array according to claim 1, characterized in that: The mutually prime array used in the step 1 is a non-uniform linear array, which is composed of two sparse uniform linear arrays A and B. Subarrays A and B share a reference array element. The first physical array element positions of subarrays A and B are the same. Two mutually prime positive integers M and N are taken. d=λ / 2 represents the unit array spacing. The wavelength of the signal is λ. The array element spacing of subarray A is set to Nd, and the number of array elements is 2M; the array element spacing of subarray B is set to Md, and the number of array elements is N. Therefore, the total number of array elements of the mutually prime array is L=2M+N-1.

3. The method for estimating the direction of arrival of cyclostationary signals based on coprime arrays according to claim 1, characterized in that: In step 2, the signal is set to be a far-field narrowband independent cyclostationary signal, with a total of K incident directions (θ1, θ2, …, θ K ) k (t) incident on the array and are parallel and coplanar, θ k is the angle between the incident angle of the kth signal and the array normal, and the array receiving signal model is established as: Where t = 1, 2, ..., T, T is the number of snapshots, N(t) is additive Gaussian white noise with a mean of 0 and a variance of σ 2 ,S(t)=[s1(t),s2(t),…,s K (t)] T , [·] T represents the transfer rank, a(θ K ) is the steering vector of the array, which can be expressed as: Where: d l is the distance between the lth (l=1,2,…,L)th array element and the first array element; d l ∈P, P is the set of mutually prime array element positions: P={0,-Nd,-2Nd,…,-(2M-1)Nd}∪{0,Md,2Md,…,(N-1)Md} (3) Then the direction matrix A of the array can be expressed as: A=[a(θ1),a(θ2),…,a(θ K )] (4) The circular correlation matrix of the array signal is obtained by collecting the received signal obtained by snapshot.

4. The method for estimating the direction of arrival of a cyclostationary signal based on a coprime array according to claim 1, characterized in that: In step 3, according to the cyclostationary theory, assuming that the noise and the signal are independent of each other, at the cyclic frequency α, K α The signal has cyclostationary characteristics, K α ≤K, the cyclic correlation function output by the pth array element and the qth array element (p,q=1,2,…,L) is: In the formula, P lp , P lq ∈P, [·] * represents conjugate calculation; Further simplified to: The useful signals are not cyclically correlated with each other and do not have cyclic correlation, and the noise and other influencing factors do not have cyclostationary characteristics at the cyclic frequency α; In the formula, is the circular correlation function of the received signal Then the cyclic correlation matrix of the received signal model is obtained as:

5. The method for estimating the direction of arrival of a cyclostationary signal based on a coprime array according to claim 1, characterized in that: In step 4, according to the cyclic correlation function and cyclic correlation matrix of step 3, the matrix expression of the array output can be derived as follows: In the formula, is the autocorrelation matrix of the incident signal, let the kth (k=1,2,…,K α ) The power of the incident signal is but: Column-vectorize the circular correlation matrix: In the formula, We can get a virtual array, let is the steering vector of the virtual array. For any element in B The formula or To represent, where m,m1,m2∈[0,2M-1], n,n1,n2∈[0,N-1], B is the set of horizontal distance differences between any two physical array elements in the coprime array due to different positions; The self-difference position set and mutual difference position set of two sub-matrices of a coprime matrix are defined as follows: Then the position difference set of coprime arrays is D p =D AA ∪D BB ∪D AB , after removing the repeated redundant parts, the virtual array model is: Where Fc is the selection matrix, when it first appears (d a -d b )=mL c When F c =1, otherwise F c =0,d a ,d b Respectively represent the horizontal distances of the ath and bth physical array elements in subarrays A and B relative to the first reference array element, m = 1, ..., 2L c -1;L c =MN+M-1; a=b=1,2,…,L, the array response of the virtual array generated after the selection is expressed as:

6. The method for estimating the direction of arrival of cyclostationary signals based on coprime arrays according to claim 1, characterized in that: In step 5, according to the virtual array obtained in step 4, -(MN+M-1) to MN+M-1 are the longest continuous uniform linear array parts, with a length of 2(MN+M)-1. The front and back smoothing process is adopted, and MN+M is set as the length of a single smooth segment, and L=MN+M is the number of smooth segments: in: represents the array output of the forward smoothed ith smoothed sub-matrix, The cyclic autocorrelation matrices of all forward smoothed subsegments are averaged to obtain: In the formula, [·] H Indicates the conjugate rank calculation. Similarly, the average of the backward smooth correlation matrix, that is, the inverse matrix of the forward smooth matrix, can be obtained as follows: Therefore, the forward and backward smoothed correlation matrix is:

7. The method for estimating the direction of arrival of cyclostationary signals based on coprime arrays according to claim 1, characterized in that: In step 6, the forward and backward smoothed correlation matrix obtained in step 5 is Perform eigenvalue decomposition on it: Where U s is the signal subspace, U n is the noise subspace, Λ s and Λ n are the characteristic values ​​of signal and noise respectively, and U s The first to the second to last columns constitute the subspace U sx , take U s The second to last columns of the subspace U sy , merge the two subspaces to form a new matrix, and calculate the matrix P, that is: Perform eigenvalue decomposition on P to obtain the eigenvalue and eigenvector E, and divide E into a combination of 4 matrices, namely: Through the matrix E 12 and E 22 Calculate the matrix F, that is: F=-E 12 AND 22 -1 (24) Perform eigenvalue decomposition on the matrix F to obtain the eigenvalue λ. The angle between its real and imaginary parts is the direction angle: In the formula, λ i The i-th eigenvalue, is the real part of the eigenvalue, |λ i | is the modulus of the eigenvalue.

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