CN114325568A - Nested array non-circular signal DOA estimation method based on BNC in impulse noise environment - Google Patents

Nested array non-circular signal DOA estimation method based on BNC in impulse noise environment Download PDF

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CN114325568A
CN114325568A CN202111598949.2A CN202111598949A CN114325568A CN 114325568 A CN114325568 A CN 114325568A CN 202111598949 A CN202111598949 A CN 202111598949A CN 114325568 A CN114325568 A CN 114325568A
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array
matrix
bnc
signal
virtual
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董续东
张小飞
孙萌
赵君
钱洋
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Nanjing University of Aeronautics and Astronautics
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a BNC-based nested array non-circular signal DOA estimation method in an impulse noise environment, which receives signals through an array antenna of a nested array structure to obtain received signal information x (t); constructing expanded signal information X (t) according to the non-circular characteristics of the signal and calculating the BNC matrix R thereofBNC(ii) a R is to beBNCVectorizing and rearranging to obtain a virtual array receiving signal z; removing redundancy of the virtual array received signal z to obtain a new virtual uniform linear array received signal
Figure DDA0003431147510000011
To pair
Figure DDA0003431147510000012
A medium difference matrix, andrespectively performing space smoothing on continuous parts of the array with the array element spacing of half wavelength, splicing to obtain virtual array information Y, and constructing a covariance matrix R for the YYAnd then obtaining the accurate estimation of DOA by using a dimension reduction MUSIC method. The invention uses the bounded nonlinear function to restrain the abnormal value in the received signal, constructs a BNC matrix, carries out vectorization processing, rearranges and removes redundancy to obtain the virtual uniform linear array received signal information with the array element spacing of half wavelength, and adopts the dimension reduction MUSIC method to reduce the calculation complexity.

Description

Nested array non-circular signal DOA estimation method based on BNC in impulse noise environment
Technical Field
The invention belongs to the field of direction of arrival (DOA) estimation, and particularly relates to a nested array non-circular signal DOA estimation method based on Bounded Nonlinear Covariance (BNC) in an impulse noise environment.
Background
In recent years, a new type of sparse array has gained much attention, namely a nested array, which is formed by combining two uniform linear arrays. N is a radical of1+N2The nested array of array elements can obtain 2N2(N1+1) -1, while Uniform Linear Array (ULA) with the same number of array elements can only obtain N1+N2A DOF of 1. Therefore, the nested array structure greatly improves the number of detectable information sources and can obtain the improvement of the angle estimation performance.
Furthermore, most DOA estimation methods in sparse arrays assume that the environmental noise is gaussian distributed. However, in practice, the noise often shows a non-gaussian characteristic and may show a certain impulse noise characteristic. Under high impulse noise environment, the second moment of the received signal no longer exists, and the performance of these DOA estimation methods is significantly degraded. Therefore, a new technical solution is needed to solve the above problems.
Disclosure of Invention
The purpose of the invention is as follows: in order to solve the problems in the prior art, the invention provides a nested array non-circular signal DOA estimation method based on BNC in an impulse noise environment.
The technical scheme is as follows: the invention provides a BNC-based nested array non-circular signal DOA estimation method in an impulse noise environment, which comprises the following steps of:
(1) receiving signals through an array antenna of a nested array structure to obtain received signal information x (t);
(2) constructing the expanded signal information X (t) and calculating its BNC matrix RBNC；
(3) R is to beBNCVectorizing to obtain a vectorized virtual array receiving signal z;
(4) removing redundancy of the virtual array received signal z to obtain a new virtual uniform linear array received signal
Figure BDA0003431147490000011
(5) To pair
Figure BDA0003431147490000012
Respectively performing spatial smoothing on continuous parts of the medium difference array and the sum array with the array element spacing of half wavelength, splicing to obtain virtual array information Y, and constructing a covariance matrix R for YYAnd then obtaining the accurate estimation of DOA by using a dimension reduction MUSIC method.
Further, the array antenna of the nested array structure in the step (1) is composed of two array elements, the number of which is N respectively1And N2Is composed of N sub-arrays1The uniform linear array element interval is d0The number of array elements is N2Is (N)1+1)d0Wherein d is0λ/2, λ is the carrier wavelength.
Further, the signal receiving information x (t) in step (1) is:
x(t)＝AΦs0(t)+n(t)
wherein a ═ a (θ)1),…,a(θk),…a(θK)]Is a matrix of directions, and the direction matrix,
Figure BDA0003431147490000021
is a vector of the direction of the light,
Figure BDA0003431147490000022
is a non-circular phase matrix of which,
Figure BDA0003431147490000023
representing the k-th signalNon-circular phase, j is an imaginary unit, n (t) is an impulse noise term which obeys the stable distribution of the symmetrical characteristic index alpha, and alpha is more than 0 and less than or equal to 2; s0(t)＝[s01(t),…,s0k(t),…s0K(t)]TIs a vector of signals, where s0k(t) represents the kth signal.
Further, the step (2) comprises the steps of:
(21) by using the non-circular characteristics of the signal, the expanded array receives the signal as follows:
Figure BDA0003431147490000024
wherein
Figure BDA0003431147490000025
In order to extend the direction matrix,
Figure BDA0003431147490000026
to amplify the impulse noise term; []*For conjugation, the received signal information X is composed of X (t), which is the tth time domain snapshot in X;
(22) calculating BNC matrix R according to array received signal X (t)BNC：
Figure BDA0003431147490000027
g(X(t))＝score(real(X(t)))+j·score(imag(X(t)))
Figure BDA0003431147490000028
Wherein score (x) is a cauchy kernel function, wherein λ1And λ2Is an adjustable parameter for adjusting the almost linear region of the signal, j is an imaginary unit, real (·),
Figure BDA0003431147490000031
and 2]HThe real part operation, the imaginary part operation, the expected operation and the conjugate transpose are respectively expressed.
Further, the virtual uniform line array receiving signal z in the step (3) is:
z＝Jvec(RBNC)＝J(B*⊙B)sBNC+JΓ
Figure BDA0003431147490000032
Figure BDA0003431147490000033
where vec (-) represents a vectorization operation,
Figure BDA0003431147490000034
in order to extend the direction matrix,
Figure BDA0003431147490000035
is the spreading direction vector, sBNCRepresenting signal energy, gamma representing a stretched vector of an impulse noise term, gamma representing a vector of the impulse noise term;
Figure BDA0003431147490000036
indicates a Kronecker product, which indicates a K-R product;
Figure BDA0003431147490000037
is a row switching matrix, and
Figure BDA0003431147490000038
IPan identity matrix of P × P, 0PA zero matrix representing P x P is shown,
Figure BDA0003431147490000039
represents 2P2×2P2Zero matrix of (1), P ═ N1+N2。
Further, the new virtual uniform linear array receiving signal of step (4)
Figure BDA00034311474900000310
Comprises the following steps:
Figure BDA00034311474900000311
Figure BDA00034311474900000312
Figure BDA00034311474900000313
Figure BDA00034311474900000314
Figure BDA00034311474900000315
wherein
Figure BDA00034311474900000316
And
Figure BDA00034311474900000317
a direction vector matrix, Γ, of a difference matrix, respectively of a sum matrix1，Γ4And Γ2，Γ3Are respectively
Figure BDA00034311474900000318
The corresponding difference matrix and the noise vector of the continuous array element part of the matrix.
Further, the virtual array information Y in step (5) is:
Figure BDA0003431147490000041
wherein M is1＝N1N2+N2-1，M2＝N1N2+N1+N2-1；
Figure BDA0003431147490000042
And
Figure BDA0003431147490000043
is to
Figure BDA0003431147490000044
Figure BDA0003431147490000045
And
Figure BDA0003431147490000046
carrying out spatial smoothing to obtain a smooth matrix; constructing a covariance matrix R of virtual array information YY：
RY＝YYH/(M1+1)。
Has the advantages that: compared with the prior art, the invention has the beneficial effects that: the non-circular characteristic of the signal is utilized, so that the DOA estimation precision and the number of the estimable information sources are improved; the invention uses Bounded Nonlinear Function (BNF) to restrain abnormal value in the received signal, constructs BNC matrix, carries out vectorization processing, rearranges and removes redundancy to obtain virtual uniform linear array received signal information with half-wavelength array element spacing, and adopts dimension-reducing MUSIC method to reduce calculation complexity.
Drawings
FIG. 1 is a schematic diagram of a nested array configuration of the present invention;
fig. 2 is a schematic diagram of a structure of a nested linear array virtual array according to the present invention;
FIG. 3 is a diagram of 13 sources incident on a nested array, where N is1＝4,N2When the characteristic index alpha is 1.5 under the impulse noise environment, the method adopts a single MC experiment DOA estimation spectrum peak searching schematic diagram;
FIG. 4 is a diagram of when 3 sources are incident on a nested array, where N is1＝4,N2Run 500 MC experiments as 5The method and other algorithms are adopted to obtain the RMSE performance schematic diagram under the conditions of different generalized signal-to-noise ratios when alpha is 1.5;
FIG. 5 is a diagram of when 3 sources are incident on a nested array, where N is1＝4,N2Running 500 MC experiments with the method and other algorithms of the invention under different snapshot conditions when alpha is 1.5, wherein the RMSE performance is schematically shown;
FIG. 6 is a graph of 3 sources incident on a nested array, where N is1＝4,N2Run 500 MC experiments with RMSE performance schematic of the method of the invention and other algorithms under different characteristic index α conditions, 5.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
Impulse noise does not appear to be common in the field of array signal processing, and most DOA estimation methods assume that the noise is a gaussian noise model. However, in practical cases, the noise is composed of irregular pulses or spikes with short duration and large amplitude, and the conventional second-order statistics are no longer applicable. This impulse noise can be modeled by a stationary distribution of α, which is very well applicable, and its characteristic function φ (u) can be expressed as:
Figure BDA0003431147490000051
Figure BDA0003431147490000052
Figure BDA0003431147490000053
wherein u is a variable of the characteristic function, alpha is more than 0 and less than or equal to 2 is a characteristic index, j is an imaginary unit, gamma is a dispersion parameter, and the meaning of the dispersion parameter is consistent with the variance of Gaussian distribution; β is a skewness parameter, δ is a position parameter, and a distribution when β ═ δ ═ 0 is a symmetric α stable (S α S) distribution.
The invention provides a BNC-based nested array non-circular signal DOA estimation method in an impulse noise environment, which specifically comprises the following steps:
step 1: and receiving the signal through the array antenna in a nested array structure to obtain received signal information x (t).
The array antenna structure shown in fig. 1 is composed of two array elements, N1And N2Is composed of N sub-arrays1The uniform linear array element interval is d0The number of array elements is N2Is (N)1+1)d0Wherein d is0And lambda/2 is half wavelength, the position set of the array element is as follows:
Figure BDA0003431147490000054
order to
Figure BDA0003431147490000055
sort (. cndot.) is an array spacing sort operation from small to large with the first array element as the reference frame, liIs the ith array element position after the array element positions are sorted from small to large, and l 10. Suppose K DOAs are each θkK is 1,2, …, K0(t) is incident on the nested linear arrays as shown in fig. 1, the array receive signal can be expressed as:
x(t)＝AΦs0(t)+n(t)
wherein a ═ a (θ)1),…,a(θk),…a(θK)]Is a matrix of directions, and the direction matrix,
Figure BDA0003431147490000061
is a vector of the direction of the light,
Figure BDA0003431147490000062
is a non-circular phase matrix of which,
Figure BDA0003431147490000063
expressing the non-circular phase of the kth signal, j being an imaginary unit, n (t) being the impulse noise which follows a stable distribution of the symmetric characteristic index alphaThe sound term is more than 0 and less than or equal to 2; s0(t)＝[s01(t),…,s0k(t),…s0K(t)]TIs a vector of signals, where s0k(t) represents the kth signal.
Step 2: constructing expanded signal information X (t) according to the non-circular characteristics of the signal and calculating the BNC matrix R thereofBNC。
By using the non-circular characteristics of the signal, the expanded array receives the signal as follows:
Figure BDA0003431147490000064
wherein
Figure BDA0003431147490000065
In order to augment the directional matrix,
Figure BDA0003431147490000066
to amplify the impulse noise term; []*For conjugate operation, the received signal information X is composed of X (t), which is the t-th time domain snapshot in X.
Calculating the BNC matrix RBNC：
Calculating BNC matrix R according to array received signal X (t)BNC：
Figure BDA0003431147490000067
g(X(t))＝score(real(X(t)))+j·score(imag(X(t)))
Figure BDA0003431147490000068
Wherein score (x) is a cauchy kernel function, wherein λ1And λ2Is an adjustable parameter for adjusting the almost linear region of the signal. j is an imaginary unit, real (),
Figure BDA0003431147490000069
and 2]HThe real part operation, the imaginary part operation, the expected operation and the conjugate transpose are respectively expressed.
And step 3: r is to beBNCVectorization and rearrangement processing are carried out to obtain a virtual array receiving signal z.
The covariance matrix R obtained in the step 2BNCVectorization processing and rearrangement are carried out to obtain a virtual uniform linear array receiving signal z:
Figure BDA0003431147490000071
Figure BDA0003431147490000072
Figure BDA0003431147490000073
Figure BDA0003431147490000074
Figure BDA0003431147490000075
where vec (-) represents a vectorization operation,
Figure BDA0003431147490000076
in order to extend the direction matrix,
Figure BDA0003431147490000077
is the spreading direction vector, sBNCRepresents the signal energy, Γ represents the stretched vector of the impulse noise term, and Γ' ═ J Γ represents the stretched vector of the impulse noise term after realignment.
Figure BDA0003431147490000078
Which represents the product of the Kronecker reaction,l represents a K-R product;
Figure BDA0003431147490000079
is a row switching matrix, and
Figure BDA00034311474900000710
IPan identity matrix of P × P, 0PA zero matrix representing P x P is shown,
Figure BDA00034311474900000711
represents 2P2×2P2Zero matrix of (1), P ═ N1+N2；
Figure BDA00034311474900000712
And
Figure BDA00034311474900000713
representing the kth signal information of the differential array virtual array,
Figure BDA00034311474900000714
and
Figure BDA00034311474900000715
and information of the k-th signal of the virtual array of the negative sum array and the positive sum array respectively. z is a radical ofD，
Figure BDA00034311474900000716
And
Figure BDA00034311474900000717
the virtual array information respectively represents a difference array, a negative sum array and a positive sum array, and can be specifically represented as follows:
Figure BDA00034311474900000718
Figure BDA00034311474900000719
Figure BDA00034311474900000720
wherein gamma isD，
Figure BDA0003431147490000081
And
Figure BDA0003431147490000082
the noise vectors of the difference matrix, the negative sum matrix and the positive sum matrix are respectively represented.
Since the virtual array of the nested array is composed of a sum array and a difference array, the following definitions are provided:
difference matrix:
Figure BDA0003431147490000083
positive sum matrix:
Figure BDA0003431147490000084
negative sum matrix:
Figure BDA0003431147490000085
it can be shown that the difference matrix zDThe range of the continuous uniform linear array is [ -M [)1,M1]d0,M1＝N1N2+N2-1, positive sum matrix
Figure BDA0003431147490000086
Has a range of [0, M ] of continuous uniform linear arrays2]d0Negative sum matrix
Figure BDA0003431147490000087
Range of the continuous uniform linear array is [ -M2,0]d0Wherein M is2＝N1N2+N1+N2-1，R32MN + M-1, N as shown in fig. 21＝3,N2A virtual array when 3.
And 4, step 4: rearranging and de-redundancy a virtual array received signal zTo obtain a new virtual uniform linear array receiving signal
Figure BDA0003431147490000088
For the virtual array received signal z, based on the difference array D, and the sum array S+S _ virtual array element position sorting, intercepting z continuous array element part virtual array information to obtain virtual uniform linear array receiving signal with half-wavelength array element spacing
Figure BDA0003431147490000089
Figure BDA00034311474900000810
Figure BDA00034311474900000811
Figure BDA00034311474900000812
Figure BDA00034311474900000813
Figure BDA00034311474900000814
Figure BDA00034311474900000815
Figure BDA00034311474900000816
Figure BDA00034311474900000817
Figure BDA00034311474900000818
Figure BDA0003431147490000091
Figure BDA0003431147490000092
Figure BDA0003431147490000093
Figure BDA0003431147490000094
Wherein gamma is1，Γ4And Γ2，Γ3Are respectively
Figure BDA0003431147490000095
The corresponding difference matrix and the noise vector of the continuous array element part of the matrix. In fact, it is possible to use,
Figure BDA0003431147490000096
shown in FIG. 2 as M1＝4,M2A virtual array when 4.
And 5: to pair
Figure BDA0003431147490000097
The elements of the sum-difference matrix are spaced by a continuous half-wavelength (i.e. the sum-difference matrix is a continuous part)
Figure BDA0003431147490000098
And
Figure BDA0003431147490000099
) Respectively performing spatial smoothing and splicingObtaining virtual array information Y and constructing covariance matrix R for YYAnd then obtaining the accurate estimation of DOA by using a dimension reduction MUSIC method.
To pair
Figure BDA00034311474900000910
And
Figure BDA00034311474900000911
performing a spatial smoothing algorithm:
Figure BDA00034311474900000912
Figure BDA00034311474900000913
Figure BDA00034311474900000914
Figure BDA00034311474900000915
Figure BDA00034311474900000916
Figure BDA00034311474900000917
wherein M is1+1＝N1N2+N2The number of times of smoothing of the sum matrix and the number of times of smoothing of the difference matrix,
Figure BDA00034311474900000918
representation difference matrix
Figure BDA00034311474900000919
M in1+2-M to 2M1+2-m elements;
Figure BDA00034311474900000920
representing a sum matrix
Figure BDA00034311474900000921
M in1+2-M to Mth2+2-m elements;
Figure BDA00034311474900000922
representing a sum matrix
Figure BDA00034311474900000923
M in1+2-M to Mth2+2-m elements; integrating new difference array and sum array to construct larger virtual array information
Figure BDA00034311474900000924
The corresponding covariance matrix is:
RY＝YYH/(M1+1)
the above formula can therefore be regarded as one consisting of M1The covariance matrix of the Uniform Linear Array (ULA) with +1 elements can be directly used in the dimension-reduced MUSIC estimation algorithm, and 2M can be estimated2-M1Compared with the traditional DOA estimation algorithm of the circular signals, the +3 information sources improve certain degree of freedom and estimation precision.
Obtain the smoothed covariance matrix RYObtaining a noise subspace U after the characteristic value is decomposedNAn accurate estimate f of the DOA of the signal is obtained by the following spectral peak search functionRD-MUSIC：
Figure BDA0003431147490000101
Figure BDA0003431147490000102
Wherein e ═ 010]TBikdiag {. denotes a matrix block diagonalization operation. T is matrix transposition, and theta is DOA grid value of spectrum peak search.
The dimension of P (theta) determines the maximum number of sources that can be estimated, and
Figure BDA0003431147490000103
therefore, the spatial degree of freedom obtained by the method is 2M2-M1+3。
The complex multiplication times are taken as the evaluation standard of the calculation complexity, and the complexity of the method mainly comprises the following steps: the complexity of the BNC matrix is O { (4P)2+8P) L }, the complexity of performing spatial smoothing is:
Figure BDA0003431147490000104
wherein M is0＝2M2-M1+3, covariance matrix RYComplexity of feature decomposition is
Figure BDA0003431147490000105
The complexity required for the dimension-reducing MUSIC method is
Figure BDA0003431147490000106
n is the number of searches, so the overall complexity of the method of the invention is:
Figure BDA0003431147490000107
in order to verify the effect of the above method, multiple simulation experiments are performed in this embodiment, and the experimental performance is analyzed. In an impulse noise environment, the generalized signal-to-noise ratio is defined as:
Figure BDA0003431147490000108
the performance estimation criterion is a joint mean square error (RMSE) defined as:
Figure BDA0003431147490000111
wherein the content of the first and second substances,
Figure BDA0003431147490000112
the method is an accurate estimation value of the kth information source DOA in the jth Monte Carlo process, wherein K represents the number of information sources, and MC represents the number of Monte Carlo tests.
The invention (NA-NC-BNC-RDMUSIC) compares with the existing method which comprises the following steps: the method comprises a traditional uniform linear array phase fraction low-order moment MUSIC (PFLOM-MUSIC) method, a MUSIC (CRCO-MUSIC) method based on correlation entropy and a traditional uniform linear array non-circular signal bounded nonlinear covariance dimension reduction MUSIC (NC-BNC-RDMUSIC) method.
Fig. 3 is a graph of the spectral peak search obtained by the present invention when K-13 sources are incident on the nested array, DOA is-30 ° +5 ° (K-1), K-1, …,13, and this example only runs one MC experiment. At the moment, the element numbers of the sub-arrays of the nested array are respectively N1＝4,N 25, the snapshot number L is 500, and GSNR is 10 dB. The impulse noise characteristic index α is 1.5, and it can be seen that the present invention can obtain an accurate DOA estimate.
Fig. 4 is a comparison of algorithm performance under different GSNRs with α being 1.5 and snapshot L being 500, and the azimuth angles of 3 sources are [0 °,10 °,20 ° ] for 500 MC experiments. It can be seen that the invention has better DOA estimation performance under the same generalized signal-to-noise ratio condition.
Fig. 5 is a comparison of algorithm performance at different snapshot numbers with GSNR 5dB and α 1.5, and 3 sources with azimuth angles of [0 °,10 °,20 ° ] running 500 MC experiments. It can be seen that the performance of the method is improved along with the increase of the number of snapshots, and the estimation performance of the method is superior to that of other estimation methods under the same snapshot condition.
Fig. 6 is a comparison of algorithm performance under different characteristic indexes, with GSNR 5dB and snapshot L500, and the azimuth angles of 3 sources are [0 °,10 °,20 ° ] for 500 MC experiments. It can be seen that the performance of the method is improved along with the increase of the characteristic index alpha, and the method has better estimation performance under the same alpha condition.
In summary, from the analysis of the simulation effect diagram, it can be known that the nested array non-circular signal DOA estimation method based on the BNC in the impulse noise environment provided by the present invention realizes the DOA accurate estimation in the nested array impulse noise environment. The degree of freedom is improved, and the estimation performance is superior to that of the traditional uniform linear array DOA estimation method.
The embodiments of the present invention have been described in detail with reference to the drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the gist of the present invention.

Claims (7)

1. A BNC-based nested array non-circular signal DOA estimation method in an impulse noise environment is characterized by comprising the following steps:
(1) receiving signals through an array antenna of a nested array structure to obtain received signal information x (t);
(2) constructing the expanded signal information X (t) and calculating its BNC matrix RBNC；
(3) R is to beBNCVectorizing to obtain a vectorized virtual array receiving signal z;
(4) removing redundancy of the virtual array received signal z to obtain a new virtual uniform linear array received signal
Figure FDA0003431147480000011
(5) To pair
Figure FDA0003431147480000012
Respectively performing spatial smoothing on continuous parts of the medium difference array and the sum array with the array element spacing of half wavelength, splicing to obtain virtual array information Y, and constructing a covariance matrix R for YYAnd then obtaining the accurate estimation of DOA by using a dimension reduction MUSIC method.
2. The BNC-based nested array in an impulsive noise environment of claim 1The method for estimating the DOA of the column non-circular signals is characterized in that the array antenna of the nested array structure in the step (1) consists of two array elements which are N respectively1And N2Is composed of N sub-arrays1The uniform linear array element interval is d0The number of array elements is N2Is (N)1+1)d0Wherein d is0λ/2, λ is the carrier wavelength.
3. The BNC-based nested array non-circular signal DOA estimation method according to claim 1, wherein said signal receiving information x (t) in step (1) is:
x(t)＝AΦs0(t)+n(t)
wherein a ═ a (θ)1),…,a(θk),…a(θK)]Is a matrix of directions, and the direction matrix,
Figure FDA0003431147480000013
is a vector of the direction of the light,
Figure FDA0003431147480000014
is a non-circular phase matrix of which,
Figure FDA0003431147480000015
expressing the non-circular phase of the kth signal, j is an imaginary number unit, n (t) is a pulse noise term which is subjected to stable distribution of a symmetric characteristic index alpha, and alpha is more than 0 and less than or equal to 2; s0(t)＝[s01(t),…,s0k(t),…s0K(t)]TIs a vector of signals, where s0k(t) represents the kth signal.
4. The BNC-based nested array non-circular signal DOA estimation method according to claim 1, wherein said step (2) comprises the steps of:
(21) by using the non-circular characteristics of the signal, the expanded array receives the signal as follows:
Figure FDA0003431147480000021
wherein
Figure FDA0003431147480000022
In order to extend the direction matrix,
Figure FDA0003431147480000023
to amplify the impulse noise term; []*For conjugation, the received signal information X is composed of X (t), which is the tth time domain snapshot in X;
(22) calculating BNC matrix R according to array received signal X (t)BNC：
Figure FDA0003431147480000024
g(X(t))＝score(real(X(t)))+j·score(imag(X(t)))
Figure FDA0003431147480000025
λ1＝0.4,λ1＝(1+λ1 2)/2＝0.58
Wherein score (x) is a cauchy kernel function, wherein λ1And λ2Is an adjustable parameter for adjusting the almost linear region of the signal, j is an imaginary unit, real (·),
Figure FDA0003431147480000026
and 2]HThe real part operation, the imaginary part operation, the expected operation and the conjugate transpose are respectively expressed.
5. The BNC-based nested array non-circular signal DOA estimation method according to claim 1, wherein said virtual uniform line array received signal z in step (3) is:
z＝Jvec(RBNC)＝J(B*⊙B)sBNC+JΓ
Figure FDA0003431147480000027
Figure FDA0003431147480000028
where vec (-) represents a vectorization operation,
Figure FDA0003431147480000029
in order to extend the direction matrix,
Figure FDA00034311474800000210
is the spreading direction vector, sBNCRepresenting signal energy, gamma representing a stretched vector of an impulse noise term, gamma representing a vector of the impulse noise term;
Figure FDA00034311474800000211
indicates a Kronecker product, which indicates a K-R product;
Figure FDA0003431147480000031
is a row switching matrix, and
Figure FDA0003431147480000032
IPan identity matrix of P × P, 0PA zero matrix representing P x P is shown,
Figure FDA0003431147480000033
represents 2P2×2P2Zero matrix of (1), P ═ N1+N2。
6. The BNC-based nested array non-circular signal DOA estimation method in an impulse noise environment according to claim 1, wherein the step (4) is performed on the new virtual uniform linear array received signal
Figure FDA00034311474800000318
Comprises the following steps:
Figure FDA0003431147480000034
Figure FDA0003431147480000035
Figure FDA0003431147480000036
Figure FDA0003431147480000037
Figure FDA0003431147480000038
wherein
Figure FDA0003431147480000039
And
Figure FDA00034311474800000310
a direction vector matrix, Γ, of a difference matrix, respectively of a sum matrix1，Γ4And Γ2，Γ3Are respectively
Figure FDA00034311474800000311
The corresponding difference matrix and the noise vector of the continuous array element part of the matrix.
7. The BNC-based nested array non-circular signal DOA estimation method according to claim 1, wherein said virtual array information Y of step (5) is:
Figure FDA00034311474800000312
wherein M is1＝N1N2+N2-1，M2＝N1N2+N1+N2-1；
Figure FDA00034311474800000313
And
Figure FDA00034311474800000314
is to
Figure FDA00034311474800000315
Figure FDA00034311474800000316
And
Figure FDA00034311474800000317
carrying out spatial smoothing to obtain a smooth matrix; constructing a covariance matrix R of virtual array information YY：
RY＝YYH/(M1+1)。
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* Cited by examiner, † Cited by third party
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CN115825863B (en) * 2022-12-16 2023-12-29 南京航空航天大学 Method for rapidly and directly positioning non-circular signal under impact noise

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