CN111965591B - Direction-finding estimation method based on fourth-order cumulant vectorization DFT - Google Patents
Direction-finding estimation method based on fourth-order cumulant vectorization DFT Download PDFInfo
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Abstract
The invention discloses a direction finding estimation method based on fourth-order cumulant vectorization DFT, which comprises the following steps: obtaining a fourth-order cumulant matrix from the received signals of the nested array and carrying out vectorization processing on the fourth-order cumulant matrix to obtain a vector z; ordering z, deleting redundancy to obtain received signal z of continuous virtual array element 1 (ii) a Constructing DFT matrix F, calculating vector y = Fz 1 And obtaining the position of the maximum K peak values
And constructing a phase rotation matrix, and obtaining an offset phase through spectrum peak search in a small range to obtain a parameter estimation result. The method can fully balance the complexity and the angle estimation performance, and breaks through the limitation that the traditional sparse array angle estimation method has good angle estimation performance but higher complexity or has lower calculation complexity but angle estimation performance.
Description
Technical Field
The invention relates to the technical field of array signal processing, in particular to a direction-finding estimation method based on fourth-order cumulant vectorization DFT.
Background
Direction-of-arrival estimation is one of the hot problems in array signal processing research, and conventional subspace methods which are widely applied in the fields of radar, sonar, wireless communication and the like only use a second-order statistic array autocorrelation matrix, and assume that a signal source is a random variable or a random process with Gaussian distribution in a signal model. For the case where the signal is non-gaussian, the information is not fully included in the second order statistics, and a large amount of useful information is also included in the higher order cumulants. A more accurate correlation function matrix can be obtained using the high order cumulants. In addition, since the high-order cumulant is insensitive to the gaussian process, the additive noise, whether it is white gaussian noise or color noise, can theoretically suppress the noise completely. High-order statistics as a powerful signal processing tool has been widely applied in the fields of communication, radar, sonar, geophysical and biomedical science. Early DOA estimation is based on uniform linear arrays, and in order to avoid the problem of angle ambiguity, the spacing between array elements of a conventional array is usually required to be less than or equal to half the wavelength of a received signal, however, strong mutual coupling influence is brought by too close distance between array elements, and thus estimation accuracy is reduced. Scholars have proposed the concept of sparse arrays. Common sparse arrays mainly include co-prime arrays, nested arrays, minimum redundant arrays, and the like. These sparse arrays have the advantages of expanding the array aperture, increasing the degree of freedom (DOF) and reducing the mutual coupling effect between array elements.
As the fourth-order cumulant has good angle estimation performance in the DOA estimation process of a non-Gaussian source, and the appearance of the sparse array further improves the DOA estimation performance, the combination of the fourth-order cumulant method and the sparse array is proposed by scholars. The existing methods are mainly applied to sparse arrays of different array manifolds by combining space smooth subspace algorithms such as SS-MUSIC or SS-ESPRIT after four-order cumulant Vectorization (VFOC), but the methods either have the problems that all virtual array elements cannot be fully utilized, the degree of freedom is reduced, or the complexity is higher due to large-range angle search.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a direction finding estimation method based on four-order cumulant vectorization DFT, which utilizes a DFT algorithm to complete DOA estimation after vectorization, can effectively utilize all array elements of a two-stage virtual array, obtains fine estimation after obtaining initial estimation and searches in a small range, effectively reduces complexity and ensures angle estimation performance.
In order to realize the purpose, the invention adopts the following technical scheme:
a direction-finding estimation method based on four-order cumulant vectorization DFT comprises the following steps:
s1, establishing a mathematical model of a nested array receiving signal; the nested array is composed of two uniform linear arrays, and the array element number of the first-stage linear sub-array is M 1 The 0 distance between array elements is d = lambda/2; the number of array elements of the second-stage linear sub-array is M 2 With a spacing d between array elements 2 ＝(M 1 +1)dThe distance between the two subarrays is d = lambda/2; the total number of array elements is 2M;
s2, obtaining a fourth-order cumulant matrix R from the received signals X of the nested arrays 4 Vectorizing the vector to obtain a vector z, and vectorized R 4 The corresponding direction vector matrix is set as Λ (theta);
s3, sequencing z according to the position of the secondary virtual array element corresponding to the lambda (theta), and deleting redundancy to obtain a received signal z of the continuous virtual array element 1 ；
S4, performing initial estimation: constructing DFT matrix F, calculating vector y = Fz 1 And obtaining the maximum K peak positions
k＝1，2，…，K；
S5, carrying out fine estimation: constructing a phase rotation matrix phi (eta), and obtaining an offset phase eta by searching a spectrum peak in a small range of (-pi/T, pi/T) k Obtaining a parameter estimation result; wherein the offset phase eta k E (-pi/T, pi/T) is such that the rotated steering vector will have and only one non-zero element.
The technical problem to be solved by the invention is as follows: how to provide a four-order cumulant vectorization DFT direction finding estimation method in signal angle parameter estimation (the signal source is a non-Gaussian random process with the mean value of 0) under the incidence condition of far-field narrow-band uncorrelated multiple signal sources in a nested array. The invention avoids the high complexity caused by the need of spectral peak search when the SS-MUSIC algorithm carries out one-dimensional angle estimation, and simultaneously ensures the performance of signal angle parameter estimation. Simulation results show that the algorithm is superior to an FOC-MUSIC algorithm in angle estimation performance, is slightly superior to a VFOC-SSMUSIC algorithm, and is far lower in complexity than a large-range spectral peak search algorithm.
In order to optimize the technical scheme, the specific measures adopted further comprise:
further, in step S1, the establishing of the mathematical model of the nested array received signal means,
for the entire nested array, the received signal is represented as:
X＝A(θ)S+N
wherein: a (θ) = [ a (θ) = 1 )，a(θ 2 )，...，a(θ K )]∈C 2M×K ；
In the formula, A (theta) represents a direction matrix, S represents a non-Gaussian signal source with the average value of 0, and L represents a fast beat number;
expressed as mean 0 and variance σ 2 White additive gaussian noise.
Further, in step S2, a fourth-order cumulant matrix R is obtained from the received signals X of the nested matrix 4 The vectorization processing of the vector z to obtain the vector z comprises the following steps:
s21, solving a fourth-order cumulant matrix of the received signals:
wherein:
b (theta) is an array manifold expanded by using the fourth-order cumulant array;
s22, adopting a virtualization method to carry out on the fourth-order cumulant matrix R 4 And (3) processing:
z＝vec(R 4 )＝vec[B(θ)C s B H (θ)]＝Λ(θ)p
wherein:
and is
Denotes the power of the kth source, K =1,2, \ 8230;, K;
s23, expressing the position of the first-level continuous virtual array element corresponding to B (theta) as follows:
L 1 ＝{-M a d，-(M a -1)d，...，0，...，(M a -1)d，M a d}，M a ＝M(M+1)-1
and expressing the closed-form solution of the position of the second-level virtual array element corresponding to the lambda (theta) as:
L 2 ＝{-M b d，-(M b -1)d，...，0，...，(M b -1)d，M b d}，M b ＝2M a ＝2(M(M+1)-1)。
further, R after vectorization 4 The corresponding directional vector matrix Λ (θ) is extended from the two differential arrays a (θ).
Further, in step S3, sorting and deleting the redundancy processing for z according to the position of the secondary virtual array element corresponding to Λ (θ) to obtain a received signal z of the continuous virtual array element 1 Comprises the following steps:
s31, according to L 2 Sequencing z by the positions of the middle-level and second-level virtual array elements and removing redundant array element construction z 1 ：
Wherein the content of the first and second substances,
is a matrix that is Λ (θ) redundancy-elimination and sorting, based on>
Represents except for M b (2M) wherein element +1 is 1 and all the other elements are 0 b + 1) x 1-dimensional column vectors; directional matrix of differential array A 1 is:
further, in step S4, the performing initial estimation: constructing DFT matrix F, calculating vector y = Fz 1 And obtaining the maximum K peak positions
K =1, 2.., K, the process comprises the steps of:
s41, defining a normalized DFT matrix
Comprises the following steps:
wherein the (p, q) th element of the matrix F is
T＝2M b +1=4 (M + 1) -1) +1 represents that the DFT transform correspondence distribution is [ -M b d，M b d]Total number of long uniform linear array elements in the range; />
S42, setting the direction vector of the kth signal of the secondary virtual array as a v (q k ) K =1,2, \ 8230;, K, direction vector after DFT processing is:
the qth element is:
wherein q is k ＝T sinθ k When the ratio of the carbon atoms to the carbon atoms is an integer,
has and only has the q th k Each element is not zero; q. q.s k Not an integer, in combination with a non-integer number>
Only with q k Adjacent elements are not zero, and the rest elements are zero;
s43, assume the received signal vector after DFT transform as y ini ＝Fz 1 The positions of the corresponding K maximum peaks are recorded as
The initial DOA estimate is then:
further, in step S5, the performing fine estimation: constructing a phase rotation matrix phi (eta), and obtaining an offset phase eta by searching a spectrum peak in a small range of (-pi/T, pi/T) k The process of obtaining the parameter estimation result comprises the following steps:
s51, defining a phase rotation matrix Φ (η) as:
wherein, eta belongs to (-pi/T, pi/T) is offset phase;
s52, expressing the rotation guide vector as:
then its qth element is:
s53, obtaining eta by searching in a small range of (-pi/T, pi/T) k ：
Wherein, the first and the second end of the pipe are connected with each other,
is the ^ th of the matrix F>
A row;
the result of the accurate parameter estimation is then:
the beneficial effects of the invention are:
(1) The four-order cumulant matrix vectorization method corresponds to the two-stage virtual continuous array of NA, the DOF is fully improved, the loss of the degree of freedom in the traditional space smoothing algorithm is avoided, and the method can be applied to more information source estimation.
(2) The characteristics of the DFT initial estimation can be used for estimating the number of the information sources. According to the method, only the DFT part needs to be subjected to low-frequency angle search in a local range for fine estimation, the complexity is low, and the angle search ensures the angle estimation performance.
(3) The invention can realize one-dimensional DOA estimation with higher resolution, and the calculation complexity is lower than that of FOC-MUSIC algorithm, VFOC-SSMUSIC algorithm and VFOC-SSESPRIT algorithm.
(4) The method can realize the one-dimensional DOA estimation with higher resolution, and the angle estimation performance of the method is higher than that of an FOC-MUSIC algorithm and slightly better than that of a VFOC-SSMUSIC algorithm.
Drawings
Fig. 1 is a schematic diagram of a nested array.
FIG. 2 is a diagram of the position and weight of a primary virtual array element.
FIG. 3 is a diagram of the positions and weights of two-level virtual array elements.
Fig. 4 is a comparison of several different array element positions.
FIG. 5 is a multiple parameter estimation dot diagram of the present invention.
Fig. 6 is a graph comparing performance of the DOA estimation algorithm under different fast beat numbers according to the present invention.
FIG. 7 is a graph comparing performance of the DOA estimation algorithm under different array element numbers.
FIG. 8 is a comparison graph of the initial estimation and the fine estimation of the present invention and the angle estimation performance of the VFOC-SSMUSIC algorithm and the FOC-NA algorithm under the same array structure and the same fast beat number conditions with different signal-to-noise ratios.
Fig. 9 is a flow chart of the direction-finding estimation method based on the fourth-order cumulant vectorized DFT of the present invention.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
It should be noted that the terms "upper", "lower", "left", "right", "front", "back", etc. used in the present invention are for clarity of description only, and are not intended to limit the scope of the present invention, and the relative relationship between the terms and the terms may be changed or adjusted without substantial technical change.
The symbols represent: in the invention (.) T ，(·) H (·) -1 And (·) * Respectively expressed as transpose, conjugate transpose, inversion and conjugate operation. Bold upper case letters denote matrices, bold lower case letters denote vectors,
indicating a Kronecker product,. Indicates a Khatri-Rao product,. Vec (-) indicates matrix vectorization,. Angle (-) indicates a phase angle of the solved complex number.
With reference to fig. 9, the present invention provides a direction finding estimation method based on fourth-order cumulant vectorized DFT, and the direction finding estimation method includes the following steps:
s1, establishing a mathematical model of the nested array received signal. The nested array is composed of two uniform linear arrays, and the array element number of the first-stage linear sub-array is M 1 The 0 distance between array elements is d = lambda/2; the number of array elements of the second-stage linear sub-array is M 2 With spacing between array elements of d 2 ＝(M 1 + 1) d, the spacing between the two sub-arrays being d = λ/2; the total number of array elements is 2M.
S2, obtaining a fourth-order cumulant matrix R from the received signals X of the nested arrays 4 Vectorizing the vector to obtain a vector z, and vectorized R 4 The corresponding directional vector matrix is set to Λ (θ).
S3, sequencing z according to the position of the secondary virtual array element corresponding to the lambda (theta), and deleting redundancy to obtain a received signal z of the continuous virtual array element 1 。
S4, performing initial estimation: constructing DFT matrix F, calculating vector y = Fz 1 And obtaining the maximum K peak positions
k＝1，2，…，K。
S5, carrying out fine estimation: constructing a phase rotation matrix phi (eta), and obtaining an offset phase eta by searching a spectrum peak in a small range of (-pi/T, pi/T) k Obtaining a parameter estimation result; wherein the offset phase eta k E (-pi/T, pi/T) is such that the rotated steering vector will have and only one non-zero element.
Assuming that K narrow-band far-field incoherent sources are incident on the nested array in the space, the direction of arrival of one dimension of the source is theta k (k =1,2,l,k). The nested array can be expressed as two uniform linear arrays which are connected in series, and the array element number of the first-stage linear sub-array is M 1 The 0 distance between array elements is d = lambda/2,the number of array elements of the second-stage linear sub-array is M 2 With spacing between array elements of d 2 ＝(M 1 + 1) d, the spacing between the two sub-arrays being d = λ/2, where λ represents the wavelength. The nested array involved in the invention is shown in FIG. 1, and the array elements of two sub-arrays of the two-level nested array are assumed to be M without loss of generality 1 ＝M 2 And (c) = M. Firstly, obtaining an cumulant matrix according to an array signal mathematical model, vectorizing, sequencing and removing redundancy on the cumulant matrix, then constructing a DFT matrix and carrying out initial estimation, and finally constructing a phase rotation matrix and carrying out fine estimation and obtaining an angle parameter estimation value of an information source signal. In this example, a nested array DFT direction finding estimation method based on fourth-order cumulant vectorization is specifically implemented as follows:
step 1: establishing a mathematical model of a nested array received signal
For the entire nested array, the received signal can be expressed as:
X＝A(θ)S+N (1)
wherein: a (θ) = [ a (θ) = 1 )，a(θ 2 )，...，a(θ K )]∈C 2M×K ；
In the formula, A (theta) represents a direction matrix, s represents a non-Gaussian signal source with the mean value of 0, and L represents a fast beat number;
expressed as mean 0 and variance σ 2 White additive gaussian noise.
Step 2: calculating a fourth-order cumulant matrix R 4 Vectorizing the vector to obtain a vector z:
the fourth-order cumulant method utilizes the fourth-order cumulant characteristics of the received data to construct a high-order virtual array model, expands the aperture of the virtual array and realizes DOA estimation with high DOF. We first solve the fourth order cumulant matrix of the received signal
Wherein
B (theta) is the array manifold expanded by the fourth-order cumulant array
In order to further improve DOA estimation precision, a virtualization method is adopted for a fourth-order cumulant matrix R 4 Is treated->
z＝vec(R 4 )＝vec[B(θ)C s B H (θ)]＝Λ(θ)p (5)
Wherein
And is
Denotes the power of the kth source, K =1,2, \ 8230;, K.
Vectorized R 4 The corresponding directional vector matrix Λ (theta) can be expanded by considering a two-stage differential array A (theta)
The position of the first-level continuous virtual array element corresponding to B (θ) is shown in fig. 2, and can be represented as:
L 1 ＝{-M a d，-(M a -1)d，...，0，...，(M a -1)d，M a d}，M a ＝M(M+1)-1 (7)
the position of the second-level virtual array element corresponding to the lambda (theta) is shown in FIG. 3, and the closed-form solution can be expressed as
L 2 ＝{-M b d，-(M b -1)d，...，0，...，(M b -1)d，M b d}，M b ＝2M a ＝2(M(M+1)-1) (8)
Obviously, the positions of the secondary virtual array elements are also completely continuous. At the same time, we present a map of the position of several different array elements, as shown in fig. 4.
According to L 2 Sequencing z by the positions of the middle-level and second-level virtual array elements and removing redundant array element construction z 1 ，
Wherein the content of the first and second substances,
is a matrix of Λ (θ) redundancy elimination and sorting, @>
Represents except for M b (2M) wherein element +1 is 1 and all the other elements are 0 6 + 1) x l dimensional column vectors; directional matrix a of differential array 1 Comprises the following steps:
and 3, step 3: performing initial estimation, constructing DFT matrix F, and finding vector y = Fz 1 Position of maximum K peaks
The DFT algorithm requires a continuous uniform array, and equation (8) is shownA fully continuous second-level virtual matrix corresponding to the direction matrix Λ (theta) of the clear z and having a range of [ -M [ ] b d，M b d]The array element number is T =2M through a long uniform linear array with the array element spacing of d b +1=4 (M + 1) -1) +1. Obviously, z is obtained by sorting z according to the positions of the second-level virtual array elements and removing redundant array elements 1 The extent and continuity of the virtual array elements is not changed.
Defining a normalized DFT matrix
Is composed of
Wherein the (p, q) th element of the matrix F is
Let the direction vector of the kth (K =1,2, \8230;, K) signal of the second-level virtual array be a v (θ k ) The structure can be made according to the formula (8). The direction vector after DFT processing is
The q-th element thereof is
As can be seen from the structure of formula (13), q k ＝T sinθ k When the ratio of the alkyl group to the alkyl group is an integer,
has and only has the q th k (eg.T＝45，θ k = arcsin (2/45)) elements are not zero; q. q of k (eg.T＝45，θ k =30 °) is not an integer>
Only with q k Several adjacent elements are not zero, and the rest elements are zero. So that it can be determined by looking for>
Approximate position q of a non-zero element k To theta k An initial estimation is performed.
In practical application, the received signal z can be obtained by 1 A DFT transform is performed to obtain an angle estimate. Assume that the received signal vector after DFT conversion is y ini ＝Fz 1 Noting that the K maximum peak positions are
The initial DOA is estimated to be
And 4, step 4: carrying out fine estimation, constructing a phase rotation matrix phi (eta), and obtaining a parameter estimation result;
from the above analysis, when
When the angle is not an integer, the accuracy of the angle estimation cannot be improved. In order to further improve the estimation accuracy of the algorithm, phase rotation is introduced to compensate for errors.
Defining a phase rotation matrix phi (eta) of
Wherein eta ∈ (-pi/T, pi/T) is the offset phase;
s52, expressing the rotation guide vector as:
then its qth element is
Obviously, there must be an offset phase η k E (-pi/T, pi/T) such that the equation
Is established when>
Will have and only one non-zero element>
Therefore, the introduction of the phase rotation effectively solves the problem that non-zero elements in the initial estimation are not unique, so that more accurate angle estimation can be obtained. Eta k Can be found by searching within a small range of (-pi/T, pi/T), i.e.
Wherein the content of the first and second substances,
is the ^ th of the matrix F>
And (6) a row. The result of the accurate parameter estimation is
The method of the invention has the following operation complexity analysis:
the operation complexity of the algorithm is analyzed, and the method specifically comprises the following steps: wherein, the sizes of the nested array subarrays 1 and the subarrays 2 are both M.The source number is K, and the fast beat number is L. The complex multiplication times are taken as the basis, and the main complexity of the algorithm in the section comprises the following steps: the fourth order cumulant matrix needs to be calculated by O { (2M) 4 L, calculating the DFT transformed received signal vector needs O { T } 2 And the fine search process needs O { nT }, wherein n represents the phase search times in the fine search process, T represents the total number of secondary virtual array elements, and T =4 (M (M + 1) -1) +1, n =100, and the total complexity is O { (2M) 4 L+T 2 + nT }. The complexity of the DOA estimation algorithm mainly comes from spectral peak search, and the algorithm only needs to search the information source angle in a small range, so that the complexity of the algorithm is lower under the same array structure.
FIG. 5 is a lattice diagram of the angle estimation of the algorithm of the present invention. Consider that K =13 uncorrelated narrowband signals are incident on M 1 ＝M 2 In a two-level nested linear array with = M =6, the azimuth angles of signals are uniformly distributed between 0 ° and 65 °, L =1000, and SNR =0dB. Fig. 5 shows the estimation result of the proposed algorithm. As can be seen from the figure, the algorithm can effectively estimate the source angle, and the estimated source number is larger than the actual array element number.
FIG. 6 is a graph of the angular estimation performance of the algorithm of the present invention at different snapshots. The number of fast beats increases, i.e., the sampled data increases. It can be derived from the graph that the angular estimation performance of the algorithm becomes better as the number of snapshots increases. Wherein the angle parameter (theta) of the incident signal 1 ，θ 2 ) = (5 °,40 °), nested array size M 1 ＝M 2 = M =5, the range of the fine estimation angle search is (-pi/T, pi/T), and the number of angle searches is 100.
FIG. 7 is a graph of angle estimation performance of the algorithm of the present invention under different array elements. The number of elements of the array increases, i.e. the diversity gain achieved by the receiving antenna increases. It can be derived from the graph that the angle estimation performance of the algorithm becomes better as the number of array elements increases. Wherein the angle parameter (theta) of the incident signal 1 ，θ 2 ) =5 °,40 °, number of snapshots L =1000, range of the fine estimation angle search is (-pi/T, pi/T), and number of times of the fine estimation angle search is 100.
FIG. 8 is the initial estimate, the fine estimate and the VFOC-SSMU of the proposed algorithmAnd (5) simulation comparison results of the SIC algorithm and the FOC-NA algorithm in different signal-to-noise ratios. As shown in FIG. 8, the angle estimation performance of the proposed algorithm is higher than that of the FOC-NA algorithm and lower than that of the VFOC-SSMUSIC algorithm. Wherein the angle parameter (theta) of the incident signal 1 ，θ 2 ) = (5 °,40 °), nested array size M 1 ＝M 2 = M =5. The precision estimation angle search range of the algorithm is (-pi/T, pi/T), and the angle search times are 100.
The above are only preferred embodiments of the present invention, and the scope of the present invention is not limited to the above examples, and all technical solutions that fall under the spirit of the present invention belong to the scope of the present invention. It should be noted that modifications and adaptations to those skilled in the art without departing from the principles of the present invention may be apparent to those skilled in the relevant art and are intended to be within the scope of the present invention.
Claims (5)
1. A direction-finding estimation method based on fourth-order cumulant vectorization DFT is characterized by comprising the following steps:
s1, establishing a mathematical model of a nested array receiving signal; the nested array is composed of two uniform linear arrays, and the number of array elements of a first-stage linear sub-array is M 1 The distance between the array elements is d = lambda/2; the number of array elements of the second-stage linear sub-array is M 2 With spacing between array elements of d 2 ＝(M 1 + 1) d, the spacing between the two sub-arrays being d = λ/2; the total number of array elements is 2M;
s2, obtaining a fourth-order cumulant matrix R from the received signals X of the nested array 4 Vectorizing the vector to obtain a vector z, and vectorized R 4 The corresponding direction vector matrix is set as Λ (theta);
s3, sequencing z according to the position of the secondary virtual array element corresponding to the lambda (theta), and deleting redundancy to obtain a received signal z of the continuous virtual array element 1 ；
S4, performing initial estimation: constructing DFT matrix F, calculating vector y = Fz 1 And obtaining the position of the maximum K peak values
S5, carrying out fine estimation: constructing a phase rotation matrix phi (eta), and obtaining an offset phase eta by searching a spectrum peak in a small range of (-pi/T, pi/T) k Obtaining a parameter estimation result; wherein the offset phase eta k The element belongs to (-pi/T, pi/T) so that the rotary steering vector has one and only one nonzero element, and T is the array element number of the continuous virtual linear array;
in step S1, the establishing of the mathematical model of the nested array received signal means that,
for the entire nested array, the received signal is represented as:
X＝A(θ)S+N
wherein: a (θ) = [ a (θ) = 1 )，a(θ 2 )，...，a(θ K )]∈C 2M×K ；
In the formula, A (theta) represents a direction matrix, S represents a non-Gaussian signal source with the mean value of 0, and L represents a fast beat number;
expressed as mean 0 and variance σ 2 Additive white gaussian noise of (1);
in step S2, a fourth-order cumulant matrix R is obtained from the received signal X of the nested array 4 The vectorization processing is carried out to obtain the vector z, and the vectorization processing comprises the following steps:
s21, solving a fourth-order cumulant matrix of the received signals:
wherein:
wherein B (theta) is an array manifold expanded by a fourth-order cumulant array;
s22, adopting a virtualization method to carry out on the fourth-order cumulant matrix R 4 And (3) processing:
z＝vec(R 4 )＝vec[B(θ)C s B H (θ)]＝Λ(θ)p
wherein:
and is
Denotes the power of the kth source, K =1,2, \ 8230;, K;
and S23, expressing the position of the first-level continuous virtual array element corresponding to the B (theta) as follows:
L 1 ＝{-M a d，-(M a -1)d，...，0，...，(M a -1)d，M a d}，M a ＝M(M+1)-1
and expressing a closed-form solution of the position of the secondary virtual array element corresponding to the lambda (theta) as follows:
L 2 ＝{-M b d，-(M b -1)d，...，0，...，(M b -1)d，M b d}，M b ＝2M a ＝2(M(M+1)-1)。
2. the direction-finding estimation method based on fourth-order cumulant vectorization DFT as claimed in claim 1, wherein R after vectorization 4 The corresponding direction vector matrix lambda (theta) is expanded by the two-stage differential array A (theta).
3. The direction-finding estimation method based on fourth-order cumulant vectorization DFT as claimed in claim 1, wherein in step S3, the step of ordering and removing redundancy processing on z according to the position of the second-order virtual array element corresponding to Λ (θ) to obtain the received signal z of continuous virtual array element 1 Comprises the following steps:
s31 according to L 2 Sequencing z by the positions of the middle-level and second-level virtual array elements and removing redundant array element construction z 1 ：
Wherein the content of the first and second substances,
is a matrix that is Λ (θ) redundancy-elimination and sorting, based on>
Represents except for M b +1 element is 1 and all other elements are 0 (2M) b + 1) x 1-dimensional column vectors; directional matrix Λ of differential array 1 Comprises the following steps:
4. the direction-finding estimation method based on the fourth-order cumulant vectorized DFT according to claim 3, wherein in step S4, the initial estimation is performed: constructing DFT matrix F, calculating vector y = Fz 1 And obtaining the position of the maximum K peak values
Comprises the following steps:
s41, defining a normalized DFT matrix
Comprises the following steps:
wherein the (p, q) th element of the matrix F is
Indicating that the DFT transform is correspondingly distributed in [ -M ] b d，M b d]Total number of long uniform linear array elements in the range;
s42, setting the direction vector of the kth signal of the secondary virtual array as a v (θ k ) K =1,2, \ 8230;, K, direction vector after DFT processing is:
the qth element is:
wherein q is k ＝Tsinθ k When the ratio of the carbon atoms to the carbon atoms is an integer,
has and only has the q th k Each element is not zero; q. q of k When not an integer, is greater than or equal to>
Only with q k Several adjacent elements are notZero, and the rest elements are zero;
s43, assume the received signal vector after DFT transform as y ini ＝Fz 1 The positions of the corresponding K maximum peaks are recorded as
Then the initial DOA estimate is:
5. the direction-finding estimation method based on fourth-order cumulant vectorized DFT according to claim 4, wherein in step S5, the fine estimation is performed: constructing a phase rotation matrix phi (eta), and obtaining an offset phase eta by searching a spectrum peak in a small range of (-pi/T, pi/T) k The process of obtaining the parameter estimation result comprises the following steps:
s51, defining a phase rotation matrix Φ (η) as:
wherein, eta belongs to (-pi/T, pi/T) is offset phase;
s52, expressing the rotation guide vector as:
then its qth element is:
s53, obtaining eta by searching in a small range of (-pi/T, pi/T) k ：
Wherein the content of the first and second substances,
is the th of matrix F>
A row;
the result of the accurate parameter estimation is then:
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