CN111965591A - 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 PDF

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CN111965591A
CN111965591A CN202010703136.4A CN202010703136A CN111965591A CN 111965591 A CN111965591 A CN 111965591A CN 202010703136 A CN202010703136 A CN 202010703136A CN 111965591 A CN111965591 A CN 111965591A
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张小飞
叶长波
李宝宝
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Nanjing University of Aeronautics and Astronautics
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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 element1(ii) a Constructing DFT matrix F, calculating vector y ═ Fz1And obtaining the position of the maximum K peak values
Figure DDA0002593650660000011
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

Direction-finding estimation method based on fourth-order cumulant vectorization DFT
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 a non-gaussian signal, the information is not completely included in the second order statistics, and a large amount of useful information is 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 achieve the purpose, the invention adopts the following technical scheme:
a direction-finding estimation method based on fourth-order cumulant vectorization DFT comprises 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 number of array elements of a first-stage linear sub-array is M1The 0 distance between array elements is d ═ lambda/2; the number of array elements of the second-stage linear sub-array is M2With spacing between array elements of d2＝(M1+1) d, the distance between two subarrays is 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 matrix4Vectorizing the vector to obtain a vector z, and vectorized R4Setting a corresponding direction vector matrix as lambda (theta);
s3, sorting 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 continuous virtual array elements1；
S4, performing initial estimation: constructing DFT matrix F, calculating vector y ═ Fz1And obtaining the position of the maximum K peak values
Figure RE-GDA0002662804650000021
And 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)kObtaining a parameter estimation result; wherein the phase η is shiftedkE (-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-based 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 building 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:
Figure RE-GDA0002662804650000022
Figure RE-GDA0002662804650000023
Figure RE-GDA0002662804650000024
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;
Figure RE-GDA0002662804650000025
expressed as mean 0 and variance σ2White additive gaussian noise.
Further, in step S2, the fourth-order cumulant matrix R is obtained from the received signal X of the nested matrix4The 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:
Figure RE-GDA0002662804650000031
wherein:
Figure RE-GDA0002662804650000032
Figure RE-GDA0002662804650000033
b (theta) is an array manifold expanded by using the fourth-order cumulant array;
s22, adopting virtualization method to perform fourth-order cumulant matrix R4And (3) processing:
z＝vec(R4)＝vec[B(θ)CsBH(θ)]＝Λ(θ)p
wherein:
Figure RE-GDA0002662804650000034
and is
Figure RE-GDA0002662804650000035
Denotes the power of the kth source, K ═ 1,2, …, K;
s23, the position of the first-order continuous virtual array element corresponding to B (θ) is expressed as:
L1＝{-Mad,-(Ma-1)d,...,0,...,(Ma-1)d,Mad},Ma＝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:
L2＝{-Mbd,-(Mb-1)d,...,0,...,(Mb-1)d,Mbd},Mb＝2Ma＝2(M(M+1)-1)。
further, vectorized R4The corresponding direction vector matrix lambda (theta) is expanded by the two-stage differential array A (theta).
Further, in step S3, the step of sorting and deleting the redundancy processing for z according to the position of the secondary virtual array element corresponding to Λ (θ) to obtain the receiving information of the continuous virtual array elementNumber z1Comprises the following steps:
s31, according to L2Ordering z by the positions of the middle-level and second-level virtual array elements and removing redundant array element construction z1：
Figure RE-GDA0002662804650000036
Wherein the content of the first and second substances,
Figure RE-GDA0002662804650000037
is a matrix with redundancy removed and ordered by a (theta),
Figure RE-GDA0002662804650000038
represents except for Mb+1 element is 1 and all other elements are 0 (2M)b+1) x 1-dimensional column vectors; directional matrix Λ of differential array1Comprises the following steps:
Figure RE-GDA0002662804650000041
further, in step S4, the performing initial estimation: constructing DFT matrix F, calculating vector y ═ Fz1And obtaining the position of the maximum K peak values
Figure RE-GDA0002662804650000042
Comprises the following steps:
s41, defining a normalized DFT matrix
Figure RE-GDA0002662804650000043
Comprises the following steps:
Figure RE-GDA0002662804650000044
wherein the (p, q) th element of the matrix F is
Figure RE-GDA0002662804650000045
T＝2Mb+1 ═ 4(M +1) -1) +1 denotes the distribution of DFT transform correspondence between [ -Mbd,Mbd]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 av(θk) K is 1,2, …, K, and the direction vector after DFT processing is:
Figure RE-GDA0002662804650000046
the qth element is:
Figure RE-GDA0002662804650000047
wherein q isk＝TsinθkWhen the ratio of the alkyl group to the alkyl group is an integer,
Figure RE-GDA0002662804650000048
has and only has the q thkEach element is not zero; q. q.skWhen the number of the carbon atoms is not an integer,
Figure RE-GDA0002662804650000049
only with qkAdjacent elements are not zero, and the rest elements are zero;
s43, assume that the received signal vector after DFT transform is yini＝Fz1The positions of the corresponding K maximum peaks are recorded as
Figure RE-GDA00026628046500000410
The initial DOA estimate is then:
Figure RE-GDA00026628046500000411
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)kThe process of obtaining the parameter estimation result includes the following steps:
s51, defining a phase rotation matrix Φ (η) as:
Figure RE-GDA0002662804650000051
wherein, eta belongs to (-pi/N, pi/N) is offset phase;
s52, expressing the rotation guide vector as:
Figure RE-GDA0002662804650000052
then its qth element is:
Figure RE-GDA0002662804650000053
s53, finding eta by searching in a small range of (-pi/T, pi/T)k：
Figure RE-GDA0002662804650000054
Wherein the content of the first and second substances,
Figure RE-GDA0002662804650000055
is the first of matrix F
Figure RE-GDA0002662804650000056
A row;
the result of the accurate parameter estimation is then:
Figure RE-GDA0002662804650000057
the invention has the beneficial effects that:
(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 perform 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 and VFOC-SSMUSIC algorithm and is close to that of 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.
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 is not limited by the technical contents of the essential changes.
The symbols represent: in the invention (.)T,(·)H(·)-1And (·)*Respectively expressed as transpose, conjugate transpose, inversion and conjugate operation. Bold upper case letters denote matrices, bold lower case letters denote vectors,
Figure RE-GDA0002662804650000061
indicates Kronecker product,. indicates Khatri-Rao product,. vec (. cndot.) indicates matrix vectorization, and angle (. cndot.) indicates the phase angle of the 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:
and S1, establishing a mathematical model of the nested array received 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 M1The 0 distance between array elements is d ═ lambda/2; the number of array elements of the second-stage linear sub-array is M2With spacing between array elements of d2＝(M1+1) d, the distance between two subarrays is 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 matrix4Vectorizing the vector to obtain a vector z, and vectorized R4The corresponding directional vector matrix is set to Λ (θ).
S3, sorting 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 continuous virtual array elements1。
S4, performing initial estimation: constructing DFT matrix F, calculating vector y ═ Fz1And obtaining the position of the maximum K peak values
Figure RE-GDA0002662804650000071
And S5, carrying out fine estimation: constructing a phase rotation matrix phi (eta), and obtaining the offset phase by searching a spectrum peak in a small range of (-pi/T, pi/T)ηkObtaining a parameter estimation result; wherein the phase η is shiftedkE (-pi/T, pi/T) is such that the rotated steering vector will have and only one non-zero element.
The space is assumed to have K narrow-band far-field incoherent sources incident on the nested array, and the one-dimensional direction of arrival of the sources is thetak(K ═ 1,2, …, 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 M1The 0 distance between array elements is d ═ lambda/2, and the number of array elements of second-stage linear sub-array is M2With spacing between array elements of d2＝(M1+1) d, the spacing between the two sub-arrays being d ═ λ/2, where λ denotes 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 generality1＝M2M. 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. The nested array vector DFT direction finding estimation method based on fourth-order cumulant in the embodiment is specifically realized as follows:
step 1: establishing mathematical model of nested array received signal
For the entire nested array, the received signal can be expressed as:
X＝A(θ)S+N (1)
wherein the content of the first and second substances,
Figure RE-GDA0002662804650000072
represents a direction matrix, and
Figure RE-GDA0002662804650000073
Figure RE-GDA0002662804650000074
represents a non-Gaussian signal source with a mean value of 0, and
Figure RE-GDA0002662804650000075
l denotes fastThe number of beats.
Figure RE-GDA0002662804650000076
Expressed as mean 0 and variance σ2White additive gaussian noise.
Step 2: calculating a fourth-order cumulant matrix R4Vectorizing 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
Figure RE-GDA0002662804650000077
Wherein
Figure RE-GDA0002662804650000081
Figure RE-GDA0002662804650000082
B (theta) is the array manifold expanded by the fourth-order cumulant array
Figure RE-GDA0002662804650000083
In order to further improve DOA estimation precision, a virtualization method is adopted for a fourth-order cumulant matrix R4To perform treatment
z＝vec(R4)＝vec[B(θ)CsBH(θ)]＝Λ(θ)p (5)
Wherein
Figure RE-GDA0002662804650000084
And is
Figure RE-GDA0002662804650000085
Representing the power of the kth source.
Vectorized R4The corresponding directional vector matrix Λ (theta) can be expanded by considering a two-stage differential array A (theta)
Figure RE-GDA0002662804650000086
The position of the first-level continuous virtual array element corresponding to B (θ) is shown in fig. 2, and can be represented as:
L1＝{-Mad,-(Ma-1)d,...,0,...,(Ma-1)d,Mad},Ma＝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
L2＝{-Mbd,-(Mb-1)d,...,0,...,(Mb-1)d,Mbd},Mb＝2Ma＝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 L2Ordering z by the positions of the middle-level and second-level virtual array elements and removing redundant array element construction z1，
Figure RE-GDA0002662804650000087
Wherein the content of the first and second substances,
Figure RE-GDA0002662804650000088
is a matrix with redundancy removed and ordered by a (theta),
Figure RE-GDA0002662804650000089
represents except for Mb+1 element is 1 and all other elements are 0 (2M)b+1) × 1-dimensional column vector. From the above, the direction matrix Λ of the differential array1Is composed of
Figure RE-GDA0002662804650000091
And step 3: performing initial estimation, constructing DFT matrix F, and finding vector y ═ Fz1Position of maximum K peaks
The DFT algorithm requires a continuous uniform array, and equation (8) has indicated a fully continuous two-level virtual array corresponding to the directional matrix Λ (θ) of z, and is in the range of [ -M [ ]bd,Mbd]By long uniform linear array with array element spacing of d, the number of array elements is T ═ 2Mb+1 ═ 4(M +1) -1) + 1. Obviously, z is obtained by ordering z according to the positions of the secondary virtual array elements and removing redundant array elements1The extent and continuity of the virtual array elements is not changed.
Defining a normalized DFT matrix
Figure RE-GDA0002662804650000092
Is composed of
Figure RE-GDA0002662804650000093
Wherein the (p, q) th element of the matrix F is
Figure RE-GDA0002662804650000094
Let the direction vector of the kth (K is 1,2, …, K) signal of the second-level virtual array be av(θk) The structure can be made according to the formula (8). The direction vector after DFT processing is
Figure RE-GDA0002662804650000095
The q-th element thereof is
Figure RE-GDA0002662804650000096
As can be seen from the structure of formula (13), qk＝TsinθkWhen the ratio of the alkyl group to the alkyl group is an integer,
Figure RE-GDA0002662804650000097
has and only has the q thk(eg.T＝45,θkArcsin (2/45)) elements are not zero; q. q.sk(eg.T＝45,θk30 deg.) is not an integer,
Figure RE-GDA0002662804650000098
only with qkSeveral adjacent elements are not zero, and the rest elements are zero. So it can be found
Figure RE-GDA0002662804650000099
Approximate position q of a non-zero elementkTo thetakAn initial estimation is performed.
In practical application, the received signal z can be obtained by1A DFT transform is performed to obtain an angle estimate. Assume that the received signal vector after DFT conversion is yini＝Fz1Noting that the K maximum peak positions are
Figure RE-GDA0002662804650000101
The initial DOA is estimated to be
Figure RE-GDA0002662804650000102
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
Figure RE-GDA0002662804650000103
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
Figure RE-GDA0002662804650000104
Where η ∈ (- π/N, π/N) is the offset phase.
The steering vector of rotation is expressed as
Figure RE-GDA0002662804650000105
Then its qth element is
Figure RE-GDA0002662804650000106
Obviously, there must be an offset phase ηkE (-pi/T, pi/T) such that the equation
Figure RE-GDA0002662804650000107
Is established when
Figure RE-GDA0002662804650000108
Will have one and only one non-zero element
Figure RE-GDA0002662804650000109
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. EtakCan be found by searching within a small range of (-pi/T, pi/T), i.e.
Figure RE-GDA0002662804650000111
Wherein the content of the first and second substances,
Figure RE-GDA0002662804650000112
is the first of matrix F
Figure RE-GDA0002662804650000113
And (6) rows. The result of the accurate parameter estimation is
Figure RE-GDA0002662804650000114
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)4L, calculating the DFT transformed received signal vector needs O { T }2And O { nT } is needed in the fine search process, wherein n represents the phase search times in the fine search process, T represents the total number of secondary virtual array elements, T is 4(M (M +1) -1) +1, n is 100, and the total complexity is O { (2M)4L+T2+ 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 the incidence of K-13 uncorrelated narrow-band signals into M1＝M2In the two-level nested linear array with M being 6, the azimuth angles of signals are uniformly distributed between 0 DEG and 65 DEG, L being 1000, and SNR being 0 dB. 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 greater 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 signal1,θ2) The size of the nested array is M (5 degrees, 40 degrees)1＝M2The 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 the 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 signal1,θ2) The fast beat number L is 1000, the range of the fine estimation angle search is (-pi/T, pi/T), and the number of the fine estimation angle searches is 100.
FIG. 8 shows the simulation comparison results of the initial estimation, the fine estimation, the VFOC-SSMUSIC algorithm and the FOC-NA algorithm of the proposed algorithm at different signal-to-noise ratios. As shown in FIG. 8, the angle estimation performance of the proposed algorithm is higher than that of FOC-NA algorithm and lower than that of VFOC-SSMUSIC algorithm. Wherein the angle parameter (theta) of the incident signal1,θ2) The size of the nested array is M (5 degrees, 40 degrees)1＝M2M5. The precision estimation angle search range of the algorithm is (-pi/T, pi/T), and the angle search times are 100.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (7)

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 the nested array received 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 M1The 0 distance between array elements is d ═ lambda/2; the number of array elements of the second-stage linear sub-array is M2With spacing between array elements of d2＝(M1+1) d, the distance between two subarrays is 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 matrix4Vectorizing the vector to obtain a vector z, and vectorized R4Setting a corresponding direction vector matrix as lambda (theta);
s3, sorting and deleting the redundancy of z according to the position of the secondary virtual array element corresponding to the lambda (theta) to obtain continuous virtual array elementsReceived signal z of array element1；
S4, performing initial estimation: constructing DFT matrix F, calculating vector y ═ Fz1And obtaining the position of the maximum K peak values
Figure FDA0002593650630000011
And 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)kObtaining a parameter estimation result; wherein the phase η is shiftedkE (-pi/T, pi/T) is such that the rotated steering vector will have and only one non-zero element.
2. The direction-finding estimation method based on fourth-order cumulant vectorized DFT as claimed in claim 1, wherein in step S1, the building of the mathematical model of the nested array received signals means,
for the entire nested array, the received signal is represented as:
X＝A(θ)S+N
wherein:
Figure FDA0002593650630000012
Figure FDA0002593650630000013
Figure FDA0002593650630000014
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;
Figure FDA0002593650630000015
expressed as mean 0 and variance σ2White additive gaussian noise.
3. The direction-finding estimation method based on fourth-order cumulant vectorized DFT as claimed in claim 2, wherein in step S2, the fourth-order cumulant matrix R is obtained from the received signals X of the nested array4The 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:
Figure FDA0002593650630000021
wherein:
Figure FDA0002593650630000022
Figure FDA0002593650630000023
in the formula, B (theta) is an array manifold expanded by a fourth-order cumulant array;
s22, adopting virtualization method to perform fourth-order cumulant matrix R4And (3) processing:
z＝vec(R4)＝vec[B(θ)CsBH(θ)]＝Λ(θ)p
wherein:
Figure FDA0002593650630000024
and is
Figure FDA0002593650630000025
Figure FDA0002593650630000026
Denotes the power of the kth source, K ═ 1,2, …, K;
s23, the position of the first-order continuous virtual array element corresponding to B (θ) is expressed as:
L1＝{-Mad,-(Ma-1)d,...,0,...,(Ma-1)d,Mad},Ma＝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:
L2＝{-Mbd,-(Mb-1)d,...,0,...,(Mb-1)d,Mbd},Mb＝2Ma＝2(M(M+1)-1)。
4. the direction-finding estimation method based on fourth-order cumulant vectorized DFT as claimed in claim 3, wherein R after vectorization4The corresponding direction vector matrix lambda (theta) is expanded by the two-stage differential array A (theta).
5. The direction-finding estimation method based on the fourth-order cumulant vectorized DFT as claimed in claim 3, wherein in step S3, the step S3 is performed to sequence z according to the position of the second-order virtual array element corresponding to Λ (θ) and remove redundancy to obtain the received signal z of the continuous virtual array element1Comprises the following steps:
s31, according to L2Ordering z by the positions of the middle-level and second-level virtual array elements and removing redundant array element construction z1：
Figure FDA0002593650630000027
Wherein the content of the first and second substances,
Figure FDA0002593650630000031
is a matrix with redundancy removed and ordered by a (theta),
Figure FDA0002593650630000032
represents except for Mb+1 element is 1 and all other elements are 0 (2M)b+1) x 1-dimensional column vectors; directional matrix Λ of differential array1Comprises the following steps:
Figure FDA0002593650630000033
6. the direction-finding estimation method based on fourth-order cumulant vectorized DFT as claimed in claim 5, wherein in step S4, the initial estimation is performed: constructing DFT matrix F, calculating vector y ═ Fz1And obtaining the position of the maximum K peak values
Figure FDA0002593650630000034
Comprises the following steps:
s41, defining a normalized DFT matrix
Figure FDA0002593650630000035
Comprises the following steps:
Figure FDA0002593650630000036
wherein the (p, q) th element of the matrix F is
Figure FDA0002593650630000037
T＝2Mb+1 ═ 4(M +1) -1) +1 denotes the distribution of DFT transform correspondence between [ -Mbd,Mbd]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 av(θk) K is 1,2, …, K, and the direction vector after DFT processing is:
Figure FDA0002593650630000038
the qth element is:
Figure FDA0002593650630000039
wherein q isk＝TsinθkWhen the ratio of the alkyl group to the alkyl group is an integer,
Figure FDA00025936506300000310
has and only has the q thkEach element is not zero; q. q.skWhen the number of the carbon atoms is not an integer,
Figure FDA00025936506300000311
only with qkAdjacent elements are not zero, and the rest elements are zero;
s43, assume that the received signal vector after DFT transform is yini＝Fz1The positions of the corresponding K maximum peaks are recorded as
Figure FDA0002593650630000041
The initial DOA estimate is then:
Figure FDA0002593650630000042
7. the direction-finding estimation method based on fourth-order cumulant vectorized DFT as claimed in claim 6, wherein in step S5, the performing the 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)kThe process of obtaining the parameter estimation result includes the following steps:
s51, defining a phase rotation matrix Φ (η) as:
Figure FDA0002593650630000043
wherein, eta belongs to (-pi/N, pi/N) is offset phase;
s52, expressing the rotation guide vector as:
Figure FDA0002593650630000044
then its qth element is:
Figure FDA0002593650630000045
s53, finding eta by searching in a small range of (-pi/T, pi/T)k：
Figure FDA0002593650630000046
Wherein the content of the first and second substances,
Figure FDA0002593650630000047
is the first of matrix F
Figure FDA0002593650630000048
A row;
the result of the accurate parameter estimation is then:
Figure FDA0002593650630000049
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