CN111175690A - Joint diagonalization L-type MIMO radar circle and non-circle mixed direction finding method - Google Patents

Joint diagonalization L-type MIMO radar circle and non-circle mixed direction finding method Download PDF

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CN111175690A
CN111175690A CN201911035936.7A CN201911035936A CN111175690A CN 111175690 A CN111175690 A CN 111175690A CN 201911035936 A CN201911035936 A CN 201911035936A CN 111175690 A CN111175690 A CN 111175690A
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circle
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陈华
方嘉雄
刘永红
谢璐璠
潘锐
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Ningbo University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
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    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae

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Abstract

The invention belongs to the field of radar signal processing, aims to provide a two-dimensional ESPRIT algorithm for estimating 2D-DOD and 2D-DOA, can realize automatic matching of angles, can solve the problem of angle doubling, and has better estimation performance. The technical scheme includes that a new data vector is constructed by using data vectors received by a multi-input multi-output MIMO radar and conjugate thereof, and then a method of estimating ESPRIT by using high-resolution parameters is adopted, and two-dimensional departure direction 2D-DOD and two-dimensional arrival direction 2D-DOA are estimated by performing joint diagonalization on a direction matrix containing non-circular information. The invention is mainly applied to radar signal processing occasions.

Description

Joint diagonalization L-type MIMO radar circle and non-circle mixed direction finding method
Technical Field
The invention belongs to the field of radar signal processing, and particularly relates to an L-shaped MIMO-based radar.
Background
The designed MIMO radar is a novel radar system using Multiple-Input Multiple-Output (MIMO) technology, and has many potential advantages, and in recent years, the combined estimation technology of direction of departure (DOD) and direction of arrival (DOA) has received extensive attention and research in MIMO radar. For the one-dimensional (1D) DOD and DOA (direction of arrival) estimation problem, the 2D-DOA estimation method in array signal processing is mostly directly utilized. The two-dimensional (2D) 2D-DOD and 2D-DOA estimation problems are discussed less. On the other hand, due to the non-circular characteristic of the non-circular signal, the angle estimation precision can be improved, more signals can be detected, a series of algorithms are provided by combining the non-circular characteristic with the bistatic MIMO radar, but signals received by the algorithms cannot be estimated under the condition that the circular signal and the non-circular signal coexist on the premise of pure non-circular. To solve this problem, there is a literature that proposes an ESPRIT algorithm for DOD and DOA estimation when circles and non-circles coexist in a bistatic MIMO radar. The above work only investigated the 1D-DOD and 1D-DOA estimation problems and required parameter pairing. At present, the 2D-DOD and 2D-DOA estimation problems are still rarely and hardly researched by utilizing the non-circular characteristic.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a two-dimensional ESPRIT algorithm for estimating 2D-DOD and 2D-DOA, can realize automatic matching of angles, can solve the problem of angle doubling and has better estimation performance. The technical scheme includes that a new data vector is constructed by using data vectors received by a multi-input multi-output MIMO radar and conjugate thereof, and then a method of estimating ESPRIT by using high-resolution parameters is adopted, and two-dimensional departure direction 2D-DOD and two-dimensional arrival direction 2D-DOA are estimated by performing joint diagonalization on a direction matrix containing non-circular information.
The method comprises the following specific steps:
step 1 for extended data vectors
Figure BDA0002251491140000011
Performing singular value decomposition to obtain
Figure BDA0002251491140000012
Step 2, constructing a direction matrix set according to the formula (12)
Figure BDA0002251491140000013
Gl＝(Kl2US)+Kl1US＝EΘlEH,l＝1,2,3,4 (12)
Wherein, Kl1For strictly non-circular mixing of signals with respect to angle thetak1Of the selection matrix, Kl2Mixing signals for circles about an angle thetak1E is thetaklThe feature vector of (2);
step 3 is to
Figure BDA0002251491140000014
Joint diagonalization is performed to obtain a unitary matrix
Figure BDA0002251491140000015
Step 4 of calculating the eigenvalues according to equation (14)
Figure BDA0002251491140000016
Then estimated by equation (15)
Figure BDA0002251491140000017
Figure BDA0002251491140000018
Figure BDA0002251491140000021
Each circular signal is considered to be two virtually exact non-circular signals of equal angle, so θklAll have K' angles, but in practice, only K incident signals exist, circle and non-circle signals are distinguished according to the number of repeated impact occurring in the solved angle, and theta is obtained according to the averaging mean square methodkl,c；
Step 5 estimates the 2D-DOD and 2D-DOA of the circle signal by equation (16).
Figure BDA0002251491140000022
The concrete steps are detailed as follows:
(a) signal receiving model
The MIMO radar is composed of an L-shaped transmitting array and an L-shaped receiving array, and the number of array elements of the transmitting array is M ═ M1+M2-1, wherein there is M1And M2The array elements are respectively positioned on X and Y axes, and the number of the array elements of the receiving array is N ═ N1+N2-1, wherein there is N1And N2Each array element is respectively positioned on an X 'axis and a Y' axis, four subarrays are uniform linear arrays, the distances of the array elements are equal and are set as d, d is taken to be lambda/2, lambda is the wavelength of a signal, and K signals s are assumed to existk(t), K is 1,2, …, K is incident on the array, K is Kn+KcThe direction of the kth signal is denoted as (θ)k1,θk2,θk3,θk4) Wherein (theta)k1,θk2) Is the 2D-DOD of the kth signal, (theta)k3,θk4) For 2D-DOA of the kth signal, the incident signal includes KnA non-circular signal sn,k(t),k＝1,2,…,KnAnd KcIndividual circle signal sc,k(t),k＝1,2,…,KcThe output vector x (t) of the array at sample t is expressed as:
x(t)＝C(θi1,θi2,θi3,θi4)s(t)+n(t) (1)
wherein x (t) ═ x1(t),…,xMN(t)]TA matrix of received data representing an array; c ═ C1,c2,…,cK]TA popular matrix representing an extended virtual array,
Figure BDA0002251491140000023
is an extended virtual array manifold vector, akAnd bkRespectively, M × 1 dimensional transmit and N × 1 dimensional receive array popularity vectors, respectively, denoted as
Figure BDA0002251491140000024
Figure BDA0002251491140000025
n(t)＝[n1(t),…,nMN(t)]TRepresenting an additive white Gaussian noise matrix with the mean value of zero and the variance of sigma2；s(t)＝[s1(t),…,sK(t)]TRepresenting the vector of the mixed incident source signal, re-representing s (t) as
Figure BDA0002251491140000026
Wherein
Figure BDA0002251491140000027
For non-circular phase of non-circular signal, phi is K × K' dimensional matrix, K ═ Kn+2KcThe K' x 1-dimensional vector includes KnA strictly non-circular signal sn(t) and a circle signal scK of (t)cReal part of
Figure BDA0002251491140000031
And KcImaginary part
Figure BDA0002251491140000032
Rewriting manifold matrix C to
C＝[C1(θk1,n,θk2,n,θk3,n,θk4,n) C2(θk1,c,θk2,c,θk3,c,θk4,c)]＝[C1C2](3)
Wherein, C1Is MN × KnSteering matrix for strictly non-circular signals, C1Is MN × KcA steering matrix of the circle signals;
by substituting formula (2) and formula (3) into formula (1)
Figure BDA0002251491140000033
For simplicity of illustration, some of the following will omit the angle pair (θ)k1,θk2,θk3,θk4) And a time t;
(b) four-dimensional parameter estimation
To utilize the strictly non-circular nature of strictly non-circular signals and the virtually strictly non-circular nature of circular signals, a data vector x and its conjugate correspondingly coupled data vector are combined into a new data vector y
Figure BDA0002251491140000034
Wherein, γMNIs an MN x MN dimensional switching matrix,
Figure BDA0002251491140000035
extending the guide vector for 2MN multiplied by K' with the expression of
Figure BDA0002251491140000036
Figure BDA0002251491140000037
Is a 2MN x 1 noise vector,
Figure BDA0002251491140000038
and carrying out singular value decomposition on the Y to obtain:
Figure BDA0002251491140000039
wherein, 2MN is multiplied by K' dimension USAnd 2MN x (2MN-K') dimension UNThe left singular signal subspace and the noise subspace, L x K' dimension V, respectivelySAnd L × (2MN-K') dimension VNRespectively a right singular signal subspace and a noise subspace, L being the snap-shot number, sigmaS＝diag(λ1,λ2,…,λK') Sum-sigmaN＝diag(λK'+1,λK'+2,…,λ2MN) Respectively representing diagonal arrays consisting of K 'and 2NM-K' characteristic values;
two selection matrices are defined as follows
Figure BDA00022514911400000310
Figure BDA00022514911400000311
Strictly non-circular and circular mixed signals with respect to the angle thetak1Is selected as
Figure BDA00022514911400000312
Figure BDA0002251491140000041
Therein is provided with
Figure BDA0002251491140000042
Similarly resulting in an angle thetaklIs selected (K)l1,Kl2)；
Based on a selection matrix Kl1And Kl2Defining the angle theta based on the principles of the non-circular ESPRIT algorithmklIs given by a direction matrix Gl；
Gl＝(Kl2US)+Kl1US＝EΘlEH(12)
Wherein E is a K 'multiplied by K' unitary matrix and comprises a diagonal matrix theta of angle informationlIs expressed as
Figure BDA0002251491140000043
G in the formula (12)lThe joint diagonalization condition is satisfied. Define a set G ═ { G ═ G1,G2,G3,G4Get a unitary matrix E ═ E based on joint diagonalization method1,e2,…,eK']E is θklCharacteristic vector of (a), thetaklFeature values of (2) are preserved in a joint diagonalization processMaintain a one-to-one correspondence without angle pairing, θklIs calculated by the following formula
Figure BDA0002251491140000044
Then, it is easily obtained from the formula (14)
Figure BDA0002251491140000045
Each circular signal is considered to be two virtually exact non-circular signals of equal angle, so θklThere are K' angles. However, in practice, only K incident signals exist, so that the circle and non-circle signals are distinguished according to the number of repeated impact occurring in the solved angle, and the final non-circle signal angle theta is obtained according to the mean square method of taking the mean square because the two estimated angles of the circle signals are reliablekl,c(l＝1,2,3,4)，
Figure BDA0002251491140000046
Wherein
Figure BDA0002251491140000047
And
Figure BDA0002251491140000048
first and second angles estimated for the non-circular signal, respectively.
The invention has the characteristics and beneficial effects that:
the method is based on the L-type MIMO radar, and realizes the joint estimation and automatic pairing of the 2D DOD and the 2D DOA by fully utilizing the non-circular information of the signals when the radar is considered to simultaneously send circular and non-circular signals, and can distinguish the mixed signals, and the conclusion is verified by a simulation experiment.
Description of the drawings:
FIG. 12A scatterplot of D DOD and 2D DOA estimates.
FIG. 2 is a flow chart of a MIMO radar circle and non-circle hybrid direction finding method according to the present invention.
Detailed Description
The method belongs to the field of radar signal processing, and particularly relates to a joint diagonalization-ESPRIT direction finding algorithm which is based on an L-type MIMO radar, and is used for joint estimation and automatic pairing of a two-dimensional (2D) departure angle (DOD) and a 2D arrival angle (DOA) under the condition that circular and non-circular signals coexist.
Under the condition of coexistence of circular and non-circular signals in the bistatic MIMO radar, the invention provides a two-dimensional ESPRIT algorithm for estimating 2D-DOD and 2D-DOA based on a joint diagonalization technology. The invention can realize automatic matching of the angles, can solve the problem of angle combination and has better estimation performance.
The technical scheme adopted by the invention is as follows: an extended virtual array is obtained on a receiving array by fully utilizing an L-type MIMO array structure, a new data vector is constructed by utilizing the received data vector and a conjugate thereof, and then 2D-DOD and 2D-DOA are estimated by adopting an ESPRIT method and carrying out joint diagonalization on a direction matrix containing non-circular information.
Description of the symbols: (.)*，(·)T，(·)+And (·)HRespectively, conjugate, transpose, pseudo-inverse, and conjugate transpose. diag (·) denotes a diagonalization operation; blkdiag (·) denotes the generation of a block diagonal matrix;
Figure BDA0002251491140000051
representing kron product operation; i iskRepresenting a k-dimensional identity matrix; arg (·) denotes calculating the phase angle of the complex number;
Figure BDA0002251491140000052
an estimated value of the symbol θ is indicated.
The specific technical scheme is as follows:
(a) signal receiving model
Consider a MIMO radar consisting of an L-shaped transmit array and an L-shaped receive array, the transmit array having M-M array elements1+M2-1, wherein there is M1And M2The array elements are respectively positioned on X and Y axes, and the number of the array elements of the receiving array is N ═ N1+N2-1, wherein there is N1And N2Array of unitsThe elements lie on the X 'and Y' axes, respectively. All four sub-arrays are uniform linear arrays, the array element intervals are equal and are set as d, d is equal to lambda/2, and lambda is the wavelength of the signal. Suppose that K (K ═ K) is providedn+Kc) A signal sk(t), K is 1,2, …, K is incident on the array, and the direction of the kth signal is denoted as (θ)k1,θk2,θk3,θk4) Wherein (theta)k1,θk2) Is the 2D-DOD of the kth signal, (theta)k3,θk4) For 2D-DOA of the kth signal, the incident signal includes KnA non-circular signal sn,k(t),k＝1,2,…,KnAnd KcIndividual circle signal sc,k(t),k＝1,2,…,KcThe output vector x (t) of the array at sample t, is represented as
x(t)＝C(θi1,θi2,θi3,θi4)s(t)+n(t) (1)
Wherein x (t) ═ x1(t),…,xMN(t)]TA matrix of received data representing an array; c ═ C1,c2,…,cK]TA popular matrix representing an extended virtual array,
Figure BDA0002251491140000053
is an extended virtual array manifold vector, akAnd bkRespectively, mx 1-dimensional transmit and nx1-dimensional receive array popularity vectors, respectively, may be represented as
Figure BDA0002251491140000054
Figure BDA0002251491140000055
n(t)＝[n1(t),…,nMN(t)]TRepresenting an additive white Gaussian noise matrix with the mean value of zero and the variance of sigma2；s(t)＝[s1(t),…,sK(t)]TRepresenting the vector of the mixed incident source signal, s (t) can be re-represented as
Figure BDA0002251491140000061
Wherein
Figure BDA0002251491140000062
For non-circular phase of non-circular signal, phi is K × K' dimensional matrix, K ═ Kn+2KcThe K' x 1-dimensional vector includes KnA strictly non-circular signal sn(t) and a circle signal scK of (t)cReal part of
Figure BDA0002251491140000063
And KcImaginary part
Figure BDA0002251491140000064
Rewriting manifold matrix C to
C＝[C1(θk1,n,θk2,n,θk3,n,θk4,n) C2(θk1,c,θk2,c,θk3,c,θk4,c)]＝[C1C2](3)
Wherein, C1Is MN × KnSteering matrix for strictly non-circular signals, C1Is MN × KcA steering matrix of circular signals.
By substituting formula (2) and formula (3) into formula (1)
Figure BDA0002251491140000065
For simplicity of illustration, some of the following will omit the angle pair (θ)k1,θk2,θk3,θk4) And a time t.
(b) Four-dimensional parameter estimation
To utilize the strictly non-circular nature of strictly non-circular signals and the virtually strictly non-circular nature of circular signals, a data vector x and its conjugate correspondingly coupled data vector are combined into a new data vector y
Figure BDA0002251491140000066
Wherein, γMNIs an MN x MN dimensional switching matrix,
Figure BDA0002251491140000067
extending the guide vector for 2MN multiplied by K' with the expression of
Figure BDA0002251491140000068
Figure BDA0002251491140000069
Is a 2MN x 1 noise vector,
Figure BDA00022514911400000610
the singular value decomposition of Y can be obtained
Figure BDA00022514911400000611
Wherein, 2MN is multiplied by K' dimension USAnd 2MN x (2MN-K') dimension UNThe left singular signal subspace and the noise subspace, L x K' dimension V, respectivelySAnd L × (2MN-K') dimension VNThe signal subspace and the noise subspace are right singular signals and noise subspaces respectively, and L is a snapshot number. SigmaS＝diag(λ1,λ2,…,λK') Sum-sigmaN＝diag(λK'+1,λK'+2,…,λ2MN) Respectively, representing diagonal arrays of K 'and 2NM-K' eigenvalues.
Two selection matrices are defined as follows
Figure BDA00022514911400000612
Figure BDA0002251491140000071
Strictly non-circular and circular mixed signals with respect to the angle thetak1Is selected as
Figure BDA0002251491140000072
Figure BDA0002251491140000073
Therein is provided with
Figure BDA0002251491140000074
Similarly, the angle θ can be obtainedkl( l 2,3,4) selection matrix (K)l1,Kl2),l＝2,3,4。
Based on a selection matrix Kl1And Kl2Defining the angle theta based on the principles of the non-circular ESPRIT algorithmklDirection matrix G of (1, 2,3,4)l
Gl＝(Kl2US)+Kl1US＝EΘlEH,l＝1,2,3,4 (12)
Wherein E is a K 'multiplied by K' unitary matrix and comprises a diagonal matrix theta of angle informationlIs expressed as
Figure BDA0002251491140000075
It is apparent that G in the formula (12)lThe joint diagonalization condition is satisfied. Define a set G ═ { G ═ G1,G2,G3,G4Get a unitary matrix E ═ E based on joint diagonalization method1,e2,…,eK']E is θkl(1, 2,3,4), θklThe characteristic values of the method keep a one-to-one corresponding relation in the joint diagonalization process, and angle pairing is not needed. ThetaklThe characteristic value of (A) can be calculated by the following formula
Figure BDA0002251491140000076
Then, it is easily obtained from the formula (14)
Figure BDA0002251491140000077
Each circular signal is considered to be two virtually exact non-circular signals of equal angle, so θklThere are K' angles. However, in practice, there are only K incident signals, so that it is possible to distinguish between circular and non-circular signals according to the number of repetitions occurring at the solution angle. Because two estimated angles of the circular signal are reliable, the final non-circular signal angle theta can be obtained according to the mean square methodkl,c(l＝1,2,3,4)，
Figure BDA0002251491140000078
Wherein
Figure BDA0002251491140000079
And
Figure BDA00022514911400000710
first and second angles estimated for the non-circular signal, respectively.
In order to verify the resolution performance of the invention, the array element number of the L-shaped MIMO array is set to be M1＝M2＝N1＝N2And 5, setting the array element spacing d as a half wavelength lambda/2. There are 2 strictly non-circular (BPSK) and 2 circular (QPSK) signals. The angle parameters of BPSK signals are (85 °,70 °,100 °,105 °) and (90 °,95 °,115 °,100 °), and the angle parameters of QPSK signals are (105 °,85 °,80 °,80 °) and (75 °,90 °,95 °,75 °). SNR is 10dB, snapshot number is 500, Mc200. As can be seen from the 2D-DOD and 2D-DOA distributions of the mixed signal in FIG. 1, the present invention can successfully estimate the two-dimensional angle of the circular and non-circular mixed signal and can effectively distinguish BPSK from QPSK signals.
Step 1 for extended data vectors
Figure BDA0002251491140000081
Performing singular value decomposition to obtain
Figure BDA0002251491140000082
Step 2, constructing a direction matrix set according to the formula (12)
Figure BDA0002251491140000083
Step 3 is to
Figure BDA0002251491140000084
Joint diagonalization is performed to obtain a unitary matrix
Figure BDA0002251491140000085
Step 4 of calculating the eigenvalues according to equation (14)
Figure BDA0002251491140000086
Then estimated by equation (15)
Figure BDA0002251491140000087
Step 5 estimates the 2D-DOD and 2D-DOA of the circle signal by equation (16).

Claims (3)

1. A joint diagonalization L-type MIMO radar circle and non-circle mixed direction finding method is characterized in that a new data vector is constructed by using data vectors received by a multi-input multi-output MIMO radar and a conjugate of the data vectors, and then a high-resolution parameter estimation ESPRIT method is adopted, and a two-dimensional wave departure direction 2D-DOD and a two-dimensional wave arrival direction 2D-DOA are estimated on the basis of joint diagonalization of a direction matrix containing non-circle information.
2. The joint diagonalization L-type MIMO radar circle and non-circle hybrid direction finding method according to claim 1, which is characterized by comprising the following steps:
step 1 for extended data vectors
Figure FDA0002251491130000011
Performing singular value decomposition to obtain
Figure FDA0002251491130000012
Step 2, constructing a direction matrix set according to the formula (12)
Figure FDA0002251491130000013
Gl＝(Kl2US)+Kl1US＝EΘlEH,l＝1,2,3,4 (12)
Wherein, Kl1For strictly non-circular mixing of signals with respect to angle thetak1Of the selection matrix, Kl2Mixing signals for circles about an angle thetak1E is thetaklThe feature vector of (2);
step 3 is to
Figure FDA0002251491130000014
Joint diagonalization is performed to obtain a unitary matrix
Figure FDA0002251491130000015
Step 4 of calculating the eigenvalues according to equation (14)
Figure FDA0002251491130000016
Then estimated by equation (15)
Figure FDA0002251491130000017
Figure FDA0002251491130000018
Figure FDA0002251491130000019
Each circular signal is considered to be two virtually exact non-circular signals of equal angle, so θklAll have K' angles, but in practice, only K incident signals exist, circle and non-circle signals are distinguished according to the number of repeated impact occurring in the solved angle, and theta is obtained according to the averaging mean square methodkl,c；
Step 5 estimating 2D-DOD and 2D-DOA of the circle signal by equation (16)
Figure FDA00022514911300000110
3. The joint diagonalization L-type MIMO radar circle and non-circle hybrid direction finding method as claimed in claim 1, characterized by comprising the following concrete steps:
(a) signal receiving model
The MIMO radar is composed of an L-shaped transmitting array and an L-shaped receiving array, and the number of array elements of the transmitting array is M ═ M1+M2-1, wherein there is M1And M2The array elements are respectively positioned on X and Y axes, and the number of the array elements of the receiving array is N ═ N1+N2-1, wherein there is N1And N2Each array element is respectively positioned on an X 'axis and a Y' axis, four subarrays are uniform linear arrays, the distances of the array elements are equal and are set as d, d is taken to be lambda/2, lambda is the wavelength of a signal, and K signals s are assumed to existk(t), K is 1,2, K is incident on the array, K is Kn+KcThe direction of the kth signal is denoted as (θ)k1,θk2,θk3,θk4) Wherein (theta)k1,θk2) Is the 2D-DOD of the kth signal, (theta)k3,θk4) For 2D-DOA of the kth signal, the incident signal includes KnA non-circular signal sn,k(t),k＝1,2,,KnAnd KcIndividual circle signal sc,k(t),k＝1,2,,KcThe output vector x (t) of the array at sample t is expressed as:
x(t)＝C(θi1,θi2,θi3,θi4)s(t)+n(t) (1)
wherein x (t) ═ x1(t),,xMN(t)]TA matrix of received data representing an array; c ═ C1,c2,,cK]TA popular matrix representing an extended virtual array,
Figure FDA0002251491130000021
is an extended virtual array manifold vector, akAnd bkRespectively, M × 1 dimensional transmit and N × 1 dimensional receive array popularity vectors, respectively, denoted as
Figure FDA0002251491130000022
Figure FDA0002251491130000023
n(t)＝[n1(t),,nMN(t)]TRepresenting an additive white Gaussian noise matrix with the mean value of zero and the variance of sigma2；s(t)＝[s1(t),,sK(t)]TRepresenting the vector of the mixed incident source signal, re-representing s (t) as
Figure FDA0002251491130000024
Wherein
Figure FDA0002251491130000025
Figure FDA0002251491130000026
For non-circular phase of non-circular signal, phi is K × K' dimensional matrix, K ═ Kn+2KcThe K' x 1-dimensional vector includes KnA strictly non-circular signal sn(t) and a circle signal scK of (t)cReal part of
Figure FDA0002251491130000027
And KcImaginary part
Figure FDA0002251491130000028
Rewriting manifold matrix C to
C＝[C1(θk1,n,θk2,n,θk3,n,θk4,n) C2(θk1,c,θk2,c,θk3,c,θk4,c)]＝[C1C2](3)
Wherein, C1Is MN × KnSteering matrix for strictly non-circular signals, C1Is MN × KcA steering matrix of the circle signals;
by substituting formula (2) and formula (3) into formula (1)
Figure FDA0002251491130000029
For simplicity of illustration, some of the following will omit the angle pair (θ)k1,θk2,θk3,θk4) And a time t;
(b) four-dimensional parameter estimation
To utilize the strictly non-circular nature of strictly non-circular signals and the virtually strictly non-circular nature of circular signals, a data vector x and its conjugate correspondingly coupled data vector are combined into a new data vector y
Figure FDA00022514911300000210
Wherein, γMNIs an MN x MN dimensional switching matrix,
Figure FDA00022514911300000211
extending the guide vector for 2MN multiplied by K' with the expression of
Figure FDA0002251491130000031
Figure FDA0002251491130000032
Is a 2MN x 1 noise vector,
Figure FDA0002251491130000033
and carrying out singular value decomposition on the Y to obtain:
Figure FDA0002251491130000034
wherein, 2MN is multiplied by K' dimension USAnd 2MN x (2MN-K') dimension UNThe left singular signal subspace and the noise subspace, L x K' dimension V, respectivelySAnd L × (2MN-K') dimension VNRespectively a right singular signal subspace and a noise subspace, L being the snap-shot number, sigmaS＝diag(λ1,λ2,,λK') Sum-sigmaN＝diag(λK'+1,λK'+2,,λ2MN) Respectively representing diagonal arrays consisting of K 'and 2NM-K' characteristic values;
two selection matrices are defined as follows
Figure FDA0002251491130000035
Figure FDA0002251491130000036
Strictly non-circular and circular mixed signals with respect to the angle thetak1Is selected as
Figure FDA0002251491130000039
Figure FDA00022514911300000310
Therein is provided with
Figure FDA0002251491130000037
Similarly resulting in an angle thetaklIs selected (K)l1,Kl2)；
Based on a selection matrix Kl1And Kl2Defining the angle theta based on the principles of the non-circular ESPRIT algorithmklIs given by a direction matrix Gl；
Gl＝(Kl2US)+Kl1US＝EΘlEH(12)
Wherein E is a K 'multiplied by K' unitary matrix and comprises a diagonal matrix theta of angle informationlIs expressed as
Figure FDA0002251491130000038
G in the formula (12)lThe joint diagonalization condition is satisfied. Define a set G ═ { G ═ G1,G2,G3,G4Get a unitary matrix E ═ E based on joint diagonalization method1,e2,,eK']E is θklCharacteristic vector of (a), thetaklThe characteristic values of the two-dimensional image are kept in one-to-one correspondence in the joint diagonalization process, angle pairing is not needed, and theta isklIs calculated by the following formula
Figure FDA0002251491130000041
Then, it is easily obtained from the formula (14)
Figure FDA0002251491130000042
Each circular signal is considered to be two virtually exact non-circular signals of equal angle, so θklThere are K' angles. However, in practice, only K incident signals exist, so that the circle and non-circle signals are distinguished according to the number of repeated impact occurring in the solved angle, and the final non-circle signal angle theta is obtained according to the mean square method of taking the mean square because the two estimated angles of the circle signals are reliablekl,c(l＝1,2,3,4)，
Figure FDA0002251491130000043
Wherein
Figure FDA0002251491130000044
And
Figure FDA0002251491130000045
first and second angles estimated for the non-circular signal, respectively.
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* Cited by examiner, † Cited by third party
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CN113341371A (en) * 2021-05-31 2021-09-03 电子科技大学 DOA estimation method based on L array and two-dimensional ESPRIT algorithm
CN113341371B (en) * 2021-05-31 2022-03-08 电子科技大学 DOA estimation method based on L array and two-dimensional ESPRIT algorithm

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