Unambiguous search-free co-prime array DoA estimation method

Abstract

The invention discloses an unambiguous search-free co-prime array DoA estimation method, and belongs to the field of array signal processing. The implementation method comprises the following steps of: firstly, dividing a co-prime array into two independent sub-arrays, and presetting and calculating a plurality of graphic parameters for reflecting a sub-array relationship according to antenna structure parameters; then a Root-MUSIC algorithm is completed in two sub-arrays of the co-prime array, and estimation results of the two sub-arrays are corrected by means of the previously calculated imaging parameters and a threshold judgment method; and finally, completing joint optimal estimation of the two sub-arrays according to a self-adaptive weighting formula. Under the same number of array elements, the co-prime array can obtain a larger antenna aperture and higher estimation precision, and the array design difficulty and the mutual coupling effect of the antenna can be reduced. According to the method, the phase ambiguity problem is eliminated through graphic analysis and a geometric method, joint optimal estimation of the co-prime array is realized by means of an adaptive weighting scheme, and the estimation precision can be improved while the ambiguity problem is avoided.

Description

Technical Field

[0001] The invention relates to a target positioning and direction finding technology in the field of wireless communications, and in particular to a non-ambiguous and search-free coprime array DoA estimation method, belonging to the field of array signal processing. Background Art

[0002] As one of the core research directions in the field of communications, target positioning has been widely used in many important application scenarios such as millimeter wave radar, autonomous driving technology, and 5G space division multiple access communication. These fields place extremely high demands on accurate target positioning technology, and Direction of Arrival (DoA) estimation is a key component of target positioning. This technology can accurately determine the direction of arrival of the target signal, assisting the communication system in achieving environmental perception and maintaining stable communication.

[0003] With the continuous improvement of communication rates and frequency bands, DoA estimation technology faces increasingly higher requirements for accuracy and real-time performance. In this context, coprime arrays have gradually become a popular research direction for solving high-precision DoA estimation problems due to their unique performance advantages. Coprime arrays consist of two uniform linear arrays that satisfy the coprime relationship. Their unique structural design does not need to follow the half-wavelength limitation of traditional antenna arrays. This feature not only significantly expands the array aperture, but also effectively reduces the complexity of antenna design, while reducing the impact of mutual coupling effects on array performance. These advantages make coprime arrays show great application potential in the field of communications.

[0004] However, coprime arrays also face major challenges in their application. When the traditional DoA estimation algorithm is directly applied to coprime arrays, a serious phase ambiguity problem will occur due to the increase in the spacing between the uniform linear arrays that make up the coprime array. The phase ambiguity problem makes the results of DoA estimation completely invalid, seriously affecting the reliability and accuracy of target positioning. Researchers have conducted a lot of exploration on the phase ambiguity problem of coprime arrays and proposed a variety of improved algorithms. However, existing solutions generally have the problems of high computational complexity and insufficient real-time performance. Therefore, how to fully utilize the high degree of freedom and high resolution performance of coprime arrays while combating the phase ambiguity problem to achieve fast and reliable DoA estimation has become an important research topic in the field of communications. Summary of the invention

[0005] In order to solve the problem of unreliable positioning results caused by phase ambiguity when coprime arrays are performing target positioning, the purpose of the present invention is to provide an ambiguity-free and search-free coprime array DoA estimation method, which visualizes the phase ambiguity problem by a graphical method, eliminates the phase ambiguity problem by graphical analysis and geometric methods, and realizes the joint optimal estimation of coprime arrays with the help of an adaptive weighting scheme, thereby improving the estimation accuracy while avoiding the ambiguity problem.

[0006] The present invention discloses a non-ambiguous and search-free coprime array DoA estimation method, comprising the following steps:

[0007] Step 1: Construct a coprime array consisting of two uniform linear arrays. The two uniform linear arrays have M1 and M2 sensors, respectively, corresponding to different array element spacings d1 = M2λ / 2 and d2 = M1λ / 2. Where λ is the signal wavelength, and M1 and M2 are prime numbers. The two uniform linear arrays are arranged in the same dimension and share the first sensor array element. The total number of sensors N = M1+M2-1.

[0008] First, we analyze the mapping relationship between the fuzzy estimated solution and the true value under a given array structure. Let the relationship parameter A = M1 / M2 of the two uniform linear arrays; define p i,x Denotes the blur coefficient of the i-th uniform linear array in the x-th angle domain, and defines D x Indicates the preset distance of the xth angle domain.

[0009] When x = 0, let p i,x =0(i∈1,2), D x = 0. According to formula (1), the positive critical angle θ of the angle domain corresponding to x = 0 and the uniform linear array number i corresponding to the critical angle are calculated.

[0010] {θ,i}=min{θ i =|sin -1 ((p i,x +1) / M i )||i=1,2} (1)

[0011] If the critical angle not calculated does not satisfy θ≥90°, then:

[0012] x=x+1 (2)

[0013] p i,x =p i,x-1 +1 (3)

[0014]

[0015] in At this time, we get two blur coefficients p that are both related to the linear array when x=1 i,x and preset distance D x . The critical angle θ at this time is calculated again according to formula (1). If the requirement of θ ≥ 90° is not met, the calculation steps of formula (2)-(5) are repeated again. Until the critical angle θ corresponding to x meets the requirement of θ ≥ 90°, at this time x indicates the number of angle domains divided when θ > 0°, and the number of angle domains is further recorded as X = x.

[0016] For the case of θ<0°, directly let

[0017] p i,-x =p i,x (6)

[0018] D -x =D x (7)

[0019] where x = 1,…,X. We obtain a set of parameters {(A,p i,x ,D x )|x∈{-X,…,0,…,X}}.

[0020] Step 2: Use the coprime array constructed in step 1 to receive K uncorrelated far-field narrowband signals. The incident direction of each signal is θ k The angle set θ = {θ1, θ2, ... θ k ,…,θ K |θ k ∈(-90°,90°)}. At this time, the sampling data X of the i-th uniform linear array in the coprime array at time t i (t) is expressed as:

[0021]

[0022] where i∈{1,2} corresponds to two uniform linear arrays, s(t)=[s1(t),s2(t),…,s k (t),…,s K (t)] T represents the independent source signal vector, s k (t) represents the sampling waveform of the kth signal, N i (t) is the Gaussian white noise vector on the i-th uniform linear array, A i =[a i (θ1),a i (θ2),…,a i (θ k ),…,a i (θ K )] is the direction matrix of the i-th uniform linear array of the coprime array, a i (θ k ) indicates that the k-th signal corresponds to the direction vector on the ith uniform linear array.

[0023] Step 3: Based on the sampling data model established in step 2, calculate the covariance matrix R of the two uniform linear arrays under L sampling snapshots i , and perform eigendecomposition on the covariance matrix:

[0024]

[0025] Sort the results of eigendecomposition in descending order of eigenvalues. and They correspond to the diagonal matrix composed of the first K eigenvalues ​​and the last M eigenvalues ​​in the sorted eigenvalues. i -K eigenvalues ​​form a diagonal matrix, and and They are and The corresponding feature vector.

[0026] Define the i-th uniform linear array with respect to U i,n Polynomial function of The polynomial coefficients Intermediate parameters Calculate f i (z i )=0(|z i |≤1), we get M i -1 characteristic root, take out the K complex roots closest to the unit circle Then, the blurred spatial phase difference of the kth incident signal estimated by the i-th uniform linear array is calculated according to formula (10):

[0027]

[0028] Where arg(·) represents the phase angle, and mod(a,b) represents the remainder of a / b. According to equation (10), the estimated fuzzy spatial phase difference set of the two uniform linear arrays can be calculated: and

[0029] Step 4: Calculate the Cartesian product of the two fuzzy spatial phase difference sets obtained in step 3:

[0030]

[0031] According to formula (12), the distance from each Cartesian product in set F corresponding to the point on the two-dimensional plane to the straight line passing through the origin and with a slope equal to the relationship parameter A is calculated.

[0032]

[0033] Compare the calculated result with the preset distance D calculated in step 1 x Matching is performed and the threshold method is used to extract the legal value. This step is expressed as an optimization problem, as shown in formula (13):

[0034]

[0035] The threshold ε is set as:

[0036]

[0037] The solution set of the optimization problem in equation (13) corresponds to the legal matching results of the two uniform linear arrays. "Legal" means that these matches correspond to the same incident signal. Based on this solution set, the unambiguous estimated phase difference is calculated. The specific calculation process is:

[0038]

[0039] in represents the unambiguous estimate of the kth signal after correction on the i-th uniform line array, p i.z is the ambiguity coefficient of the coprime array obtained in step 1. Then we get an unambiguous phase difference matching set:

[0040]

[0041] Step 5: Based on the matching set obtained in step 4 and the relationship between the signal phase difference and the incident angle given by equation (10), an unambiguous angle matching set is obtained:

[0042]

[0043] According to equation (18), the final joint estimate of the phase difference of the coprime array to the input signal is calculated:

[0044]

[0045] Then we get the angle set where ρ k To realize the adaptive weighting coefficient of joint estimation, the calculation formula is expressed as:

[0046]

[0047] gather is the polynomial function f calculated in step 3 i (z i )=0(|z i |≤1) to obtain M i -1 characteristic root. is an intermediate parameter used to simplify the formula expression, and its calculation formula is given by (20):

[0048]

[0049] in(·) * represents conjugation, It means taking the real part.

[0050] Step 6: Set the angle calculated in step 5 As the final estimation result output, the final estimation result is the obtained unambiguous DoA estimation result, that is, the unambiguous search-free coprime array DoA estimation is achieved.

[0051] Beneficial effects:

[0052] 1. Compared with the traditional uniform linear array, the present invention discloses a non-ambiguous and search-free coprime array DoA estimation method, which is implemented based on the unfolded coprime array. With the same number of array elements, the coprime array can obtain a larger antenna aperture and higher estimation accuracy, and can reduce the difficulty of array design and the mutual coupling effect of the antenna.

[0053] 2. The present invention discloses a non-ambiguous and search-free DoA estimation method for coprime arrays, which uses the relationship parameters of the two uniform linear arrays that constitute the coprime array to achieve rapid elimination of the phase ambiguity problem unique to the coprime array. Compared with the existing ambiguity elimination algorithm, the present invention visualizes the phase ambiguity problem by using a problem graphical method, eliminates the phase ambiguity problem by using graphical analysis and geometric methods, and achieves rapid elimination of the phase ambiguity problem of the coprime array without the need for a search operation, with lower computational complexity, thereby improving the DoA estimation efficiency of the coprime array.

[0054] 3. The present invention discloses a non-ambiguous and search-free coprime array DoA estimation method, which uses adaptive weighting to achieve optimal estimation of two uniform linear arrays of a joint coprime array, and the obtained estimation result is optimal in the sense of minimum mean square error. Compared with the mean weighted scheme in the traditional scheme, the estimation performance of the present invention is always better than or at least equal to the mean weighted scheme. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0056] Figure 1 The present invention discloses a flow chart of a non-ambiguous and search-free coprime array DoA estimation method.

[0057] Figure 2 Schematic diagram of the unfolded coprime array structure used in the present invention.

[0058] Figure 3 The present invention proposes a graphical schematic diagram of the relationship between the uniform linear arrays of the coprime array based on two uniform linear arrays of M1=7 and M2=5.

[0059] Figure 4 In order to bring the estimated results into Figure 3 The matching result diagram of the graphical schematic diagram shown.

[0060] Figure 5 This is a graph showing the relationship between the estimated root mean square error of the method of the present invention and other methods and the change in input signal-to-noise ratio.

[0061] Figure 6 This is a graph showing the relationship between the estimated root mean square error of the method of the present invention and other methods and the number of input snapshots.

[0062] Figure 7 This is a graph showing the relationship between the number of complex multiplications involved in a single mean square calculation and the size of the matrix as estimated by the method of the present invention and other methods. DETAILED DESCRIPTION

[0063] The present invention is further described and described in detail below with reference to the accompanying drawings and embodiments.

[0064] like Figure 1 As shown, the present embodiment discloses a non-ambiguous and search-free coprime array DoA estimation method, and the specific implementation steps are as follows:

[0065] Step 1: Refer to Figure 2 The unfolded coprime array structure shown. Two uniform linear arrays share the same array element at the origin, and the number of array elements and the array element spacing of the two uniform linear arrays need to satisfy certain coprime constraints. For example, take two uniform linear arrays with array element numbers M1=7 and M2=5 respectively, the two uniform linear arrays are arranged in opposite directions in the same dimension, and their array element spacing should satisfy d1=M2λ / 2=5λ / 2 and d2=M1λ / 2=7λ / 2. Where λ is the signal wavelength. The total number of array elements N=M1+M2-1=11.

[0066] The relationship between the estimated spatial phase difference of the two uniform linear arrays of the coprime array and the actual incident angle is converted into the x-axis and y-axis of a two-dimensional coordinate system:

[0067]

[0068] in represents the spatial phase difference estimated by the i-th uniform linear array, u=sin(θ) represents the sine of the incident angle, and mod(a,b) represents the remainder of calculating a / b. The relationship curve of the estimation results of two uniform linear arrays of coprime arrays can be drawn, as shown in Figure 2 The figure has straight lines, and the number of coprime array angle domains in the invention is exactly This corresponds to the angle domain when θ>0°. These straight lines have the same slope, corresponding to the relationship parameter A=M1 / M2 between the two uniform linear arrays in the invention content. The x-axis and y-axis of the two-dimensional coordinate are the spatial phase difference obtained by estimating the two uniform linear arrays. This corresponds to all the possible estimates of the two uniform linear arrays, and the drawn straight line corresponds to the legal result of the combined estimation results of the two uniform linear arrays. Only when the estimation result is on the drawn "legal straight line" is the corresponding estimation result real and meaningful. Further, the fuzzy coefficient p of the i-th array in the x-th angle domain calculated in the invention content is 1,x and p 2,x The corresponding straight line offset on the x-axis and y-axis. The preset distance D calculated in the invention content x This corresponds to the distance between each straight line and the straight line passing through the origin.

[0069] When x = 0, let p i,x =0(i∈1,2), D x = 0. According to formula (22), the positive critical angle θ of the angular domain corresponding to x = 0 and the uniform linear array number i corresponding to the critical angle are calculated.

[0070] {θ,i}=min{θ i =|sin -1 ((p i,x +1) / M i )||i=1,2} (22)

[0071] If the critical angle not calculated does not satisfy θ≥90°, then:

[0072] x=x+1 (23)

[0073] p i,x =p i,x-1 +1 (24)

[0074]

[0075] in Then, the critical angle θ at this time is calculated according to formula (22). If the requirement of θ ≥ 90° is not met, the calculation steps of formulas (23) to (26) are repeated again until the critical angle θ corresponding to x meets the requirement of θ ≥ 90°.

[0076] For the case of θ<0°, directly let

[0077] p i,-x =p i,x (27)

[0078] D -x =D x (28)

[0079] Where x = 1, ..., X. Then we need to describe the relationship between the two uniform linear arrays of the coprime array by the graphical parameters {(A, p i,x ,Dx )|x∈{-X,…,0,…,X}}. This is also the first step of the "imaging method" in the present invention. According to this operation, the imaging parameter calculation can be completed quickly offline.

[0080] Step 2: Consider the actual scenario where millimeter wave radar or 5G base station uses coprime arrays for multi-target positioning. At this time, it is necessary to first complete the reliable wave direction estimation of multiple targets. Suppose there are K (K<min{M1,M2}) signals with different incident directions in space, and the incident direction of each signal is θ k After frequency conversion, analog-to-digital conversion and other operations, the sampling data X of the i-th linear array of the coprime array at time t can be obtained: i (t):

[0081]

[0082] where i∈{1,2} corresponds to two uniform linear arrays, s(t)=[s1(t),s2(t),…,s k (t),…,s K (t)] T represents the independent source signal vector, s k (t) represents the sampling waveform of the kth signal, N i (t) is the Gaussian white noise vector on the i-th uniform linear array, A i =[a i (θ1),a i (θ2),…,a i (θ k ),…,a i (θ K )] is the direction matrix of the i-th uniform linear array of the coprime array, and the two uniform linear arrays have different forms of direction vectors:

[0083]

[0084] Step 3: According to the sampling data model, calculate the sampling covariance matrix of the two uniform linear arrays under L sampling snapshots:

[0085]

[0086] Where (·) represents the estimation of the true value. Perform eigendecomposition on the covariance matrix of the two uniform linear arrays and divide the signal space and the noise space according to the number of incident signals, and we get:

[0087]

[0088] Sort the results of eigendecomposition in descending order of eigenvalues. and They correspond to the diagonal matrix composed of the first K eigenvalues ​​and the last M eigenvalues ​​in the sorted eigenvalues. i -A diagonal matrix consisting of K eigenvalues. and They are and The corresponding feature vector.

[0089] Then define about U i,n Polynomial function of :

[0090]

[0091] The polynomial coefficients Intermediate parameters Calculate f i (z i )=0(|z i |≤1), we get M i -1 characteristic root, take out the K complex roots closest to the unit circle Then, the spatial phase difference corresponding to the kth incident signal obtained by the i-th uniform linear array using the above root-finding algorithm (this operation is essentially the Root-MUSIC algorithm) is calculated according to formula (35):

[0092]

[0093] Step 4: Use the Root-MUSIC algorithm to obtain a set of ambiguous spatial phase differences between the two uniform linear arrays. and Pair any elements of to get their Cartesian product:

[0094]

[0095] Draw the elements of set F to Figure 3 In the two-dimensional coordinate system depicting two uniform line arrays, we get Figure 4 The matching diagram of the estimation results shown in FIG. As mentioned before, the two-dimensional coordinate system shows all possible estimation results of the coprime array, and the points on the multiple straight lines correspond to the legal estimation results. By excluding the paired points outside the straight line, the illegal results caused by phase ambiguity are excluded. Considering the existence of noise in the actual environment, the threshold method is used here to compare the calculated results with the preset distance D calculated by formula (26). x Match and extract legal values. This process can be expressed as an optimization problem:

[0096]

[0097] The threshold ε is set as:

[0098]

[0099] The solution set of the optimization problem in formula (37) corresponds to Figure 4 The legal pairings are marked in , and the remaining pairs are all incorrect estimates due to phase ambiguity. The legal pairings correspond to the estimated phase difference without ambiguity, which is specifically expressed as:

[0100]

[0101] in represents the unambiguous estimate of the kth incident signal after correction on the ith uniform linear array, p i.z is the fuzzy coefficient describing the uniform linear array constituting the coprime array calculated by equations (24) and (25).

[0102] Step 5: Convert the calculated phase difference into an angle and express it in the form of a set:

[0103]

[0104] Substitute this set into the construction formula of the joint optimal estimate:

[0105]

[0106] Get the set of unambiguous optimal estimated angles where ρ k To realize the adaptive weighting coefficient of joint estimation, the calculation formula is expressed as:

[0107]

[0108] gather is the M obtained by calculating the polynomial equation when the polynomial function (34) is equal to 0. i -1 characteristic root. is an intermediate parameter used to simplify the formula expression. Its specific calculation formula is given by (43):

[0109]

[0110] in(·) * represents conjugation, It means taking the real part.

[0111] Step 6: Assembly The final estimated result obtained by this solution. Output set The ambiguity-free and search-free DoA estimation proposed by the present invention is completed.

[0112] In order to illustrate the implementation effect of the embodiments of the present invention, the present invention has established several comparative experiments. As described in the above example, two uniform linear arrays with array elements M1=7 and M2=5 respectively are used to form a coprime array, and three unrelated signal sources incident from three angles [-47.31°, -35.87°, 18.01°] are received. The number of repetitions J=1000. The proposed scheme, the traditional DECOM scheme, and the half-wavelength uniform linear array with the same number of array elements using the Root-MUSIC scheme and the estimated root mean square error (RMSE) of the four cases of CRLB of the array are compared. Figure 5 The above RMSE curve is shown as it changes with the input signal-to-noise ratio SNR (the number of snapshots of the covariance matrix calculation is L=200). Figure 6 The above RMSE curve is shown as the number of snapshots L of the calculated covariance matrix changes (input signal-to-noise ratio SNR = 10 dB).

[0113] Similarly, in order to demonstrate the low computational complexity of the embodiments of the present invention. Figure 7 The graph shows how the complex multiplication calculation amount curves involved in the solution of the present invention and the traditional DECOM algorithm change with the number of overall array elements N of the coprime array.

[0114] According to the above simulation, it can be seen that the present invention, because it is implemented based on a coprime array, obtains an estimation performance that is significantly better than that of a traditional ULA array. In addition, the estimation accuracy and computational complexity of the present invention are significantly improved compared to traditional solutions.

[0115] The specific examples described in the present invention are merely illustrative of the present invention and are not intended to limit the present application. A person skilled in the art of the present invention may make various modifications, supplements or substitute the specific examples described in a similar manner, but this will not deviate from the present invention or exceed the scope defined by the appended claims.

Claims

1. A non-ambiguous and search-free coprime array DoA estimation method, characterized by: The following steps are included: Step 1: Construct a coprime array consisting of two uniform linear arrays; the two uniform linear arrays have M1 and M2 sensors respectively, corresponding to different array element spacings d1 = M2λ / 2, d2 = M1λ / 2; where λ is the signal wavelength, and M1 and M2 are prime numbers to each other; Let the relationship parameter of the two uniform linear arrays be A = M1 / M2; define p i,x Denotes the blur coefficient of the i-th uniform linear array in the x-th angle domain, and defines D x represents the preset distance of the xth angle domain; when x = 0, let p i,x =0(i∈1,2), D x =0; according to formula (1), calculate the positive critical angle θ of the angle domain corresponding to x=0 and the uniform linear array number i corresponding to the critical angle; {θ,i}=min{θ i =|sin -1 ((p i,x +1) / M i )||i=1,2} (1) If the critical angle not calculated does not satisfy θ≥90°, then let: x=x+1 (2) p i,x =p i,x-1 +1 (3) in Then, the critical angle θ at this time is calculated again according to formula (1). If the requirement of θ ≥ 90° is not satisfied, the calculation steps of formulas (2) to (5) are repeated again until the critical angle θ corresponding to x satisfies the requirement of θ ≥ 90°, and the number of angle domains divided when θ > 0° is recorded as X = x. For the case of θ<0°, directly set: p i,-x =p i,x (6) D -x =D x (7) where x = 1,…,X; thus we get a set of parameters {(A,p i,x ,D x )|x∈{-X,…,0,…,X}}; Step 2: Use the coprime array constructed in step 1 to receive K uncorrelated far-field narrowband signals; the incident direction of each signal is θ k The angle set θ = {θ1, θ2, ... θ k ,…,θ K |θ k ∈(-90°,90°)}; At this time, the sampling data X of the i-th uniform linear array in the coprime array at time t i (t) is expressed as: where i∈{1,2} corresponds to two uniform linear arrays, s(t)=[s1(t),s2(t),…,s k (t),…,s K (t)] T represents the independent source signal vector, s k (t) represents the sampling waveform of the kth signal, N i (t) is the Gaussian white noise vector on the i-th uniform linear array, A i =[a i (θ1),a i (θ2),…,a i (θ k ),…,a i (θ K )] is the direction matrix of the i-th uniform linear array of the coprime array, a i (θ k ) represents the direction vector of the k-th signal corresponding to the i-th uniform linear array; Step 3: Based on the sampling data model established in step 2, calculate the covariance matrix R of the two uniform linear arrays under L sampling snapshots i , and perform eigendecomposition on the covariance matrix: Sort the results of eigendecomposition in descending order of eigenvalues. and They correspond to the diagonal matrix composed of the first K eigenvalues ​​and the last M eigenvalues ​​in the sorted eigenvalues. i -K eigenvalues ​​form a diagonal matrix, and and They are and The corresponding eigenvector; Define the i-th uniform linear array with respect to U i,n Polynomial function of The polynomial coefficients Intermediate parameters Calculate f i (z i )=0(|z i |≤1), the K complex roots closest to the unit circle Then, the blurred spatial phase difference of the kth incident signal estimated by the i-th uniform linear array is calculated according to formula (10): Where arg(·) represents the phase angle, and mod(a,b) represents the remainder of a / b. Then the set of fuzzy spatial phase differences between two uniform linear arrays can be calculated: and Step 4: Calculate the Cartesian product of the two fuzzy spatial phase difference sets obtained in step 3: According to formula (12), the distance of each Cartesian product in the set F to the straight line on the two-dimensional plane that passes through the origin and has a slope equal to the relationship parameter A is calculated; Use the optimization problem shown in formula (13) to eliminate illegal values: The threshold ε is set as: According to the solution set of equation (13), the estimated phase difference with respect to the unambiguous phase is calculated: in represents the unambiguous estimate of the kth signal after correction on the i-th uniform linear array; then the unambiguous phase difference matching set is obtained: Step 5: Based on the matching set obtained in step 4 and the relationship between the signal phase difference and the incident angle given by equation (10), an unambiguous angle matching set is obtained: According to equation (18), the final joint estimate of the phase difference of the coprime array to the input signal is calculated: Then we get the angle set where ρ k To realize the adaptive weighting coefficient of joint estimation, the calculation formula is expressed as: is an intermediate parameter used to simplify the formula expression, and its calculation formula is given by (20): in(·) * represents conjugation, represents taking the real part; Step 6: Set the angle calculated in step 5 The final estimation result is output as the final estimation result, which is the obtained unambiguous DoA estimation result.

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