CN115169378A - Mobile sparse receiving array angle estimation method - Google Patents

Mobile sparse receiving array angle estimation method Download PDF

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CN115169378A
CN115169378A CN202210509255.5A CN202210509255A CN115169378A CN 115169378 A CN115169378 A CN 115169378A CN 202210509255 A CN202210509255 A CN 202210509255A CN 115169378 A CN115169378 A CN 115169378A
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汪跃先
雒帅
李建瀛
王伶
张兆林
杨欣
韩闯
谢坚
陶明亮
粟嘉
邢自健
宫延云
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Abstract

The invention provides a mobile sparse receiving array angle estimation method, which comprises the steps of firstly expanding the array element spacing of various traditional array structures, transmitting a periodic signal related to the moving speed of a receiving array at the transmitting end of an MIMO radar, further deducing the space-time invariance of the receiving signal, then expanding the array aperture of the receiving end by utilizing a passive aperture synthesis technology, and finally processing the received data by utilizing the methods of straightening, extracting, tensor smoothing, parallel factor decomposition and the like, thereby realizing the estimation of a target angle. The method greatly improves the array freedom degree of the system, further utilizes inherent multidimensional structural information in data by using a tensor smoothing method, fully utilizes the array aperture of the MIMO radar by using a Khatri-Rao product, finally realizes the resolving and automatic pairing of the target angle by an ESPRIT algorithm, and greatly improves the parameter estimation precision of the target.

Description

Mobile sparse receiving array angle estimation method
Technical Field
The invention relates to the field of MIMO radars, in particular to an angle estimation method of an MIMO radar, which is suitable for application scenes of a receiving array working on a mobile platform.
Background
In both the civil and military fields, the realization of accurate positioning of the target has important practical significance. Due to the adoption of the bistatic MIMO radar, the departure angle and the arrival angle of the target can be obtained at the receiving end at the same time, and then the positioning calculation of the target can be realized by utilizing the cross positioning technology without additional distance information. The bistatic MIMO radar also has the advantages of high resolution, strong anti-interference capability, more detectable targets, far distance between the transmitting array and the receiving array, strong self-protection capability and the like, and is widely researched by scholars at home and abroad in the past decade.
Since the nested array and the co-prime array were proposed, the parameter estimation problem of the sparse array presents a booming situation. The sparse array has the advantages of higher array degree of freedom, lower array element mutual coupling and the like under the condition of the same physical array elements, so the sparse array is also applied to bistatic MIMO radar for parameter estimation of targets, and the degree of freedom is improved.
In practical applications, many application platforms are mobile, and in recent years, researchers propose to combine sparse arrays and passive aperture synthesis techniques to further improve the degree of freedom of the arrays and the parameter estimation accuracy of targets. Therefore, the sparse array movement is applied to the bistatic MIMO radar to improve the parameter estimation performance, and the method has profound research significance.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides an angle estimation method of a mobile sparse receiving array. The invention provides an MIMO radar angle estimation method using a sparse mobile receiving array, aiming at further improving the array degree of freedom of a bistatic MIMO radar. The method is used for firstly expanding the array element spacing of various traditional array structures (generally about 1-3 times), transmitting a periodic signal related to the moving speed of a receiving array at the transmitting end of the MIMO radar, further deducing the space-time invariance of the receiving signal, and then expanding the array aperture of the receiving end by utilizing the passive aperture synthesis technology. And finally, processing the received data by using methods such as straightening, extraction, tensor smoothing, parallel factor decomposition and the like, thereby realizing the estimation of the target angle.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: establishing a signal receiving model of the bistatic MIMO radar in a sparse receiving array moving scene;
and 2, step: signals transmitted among array elements of the MIMO radar transmitting array are mutually orthogonal, and when the mutual coupling influence of the array elements is not considered, the signals received in a pulse are subjected to matched filtering, and a data model of the received signals is obtained as follows:
Figure BDA0003637276490000021
wherein B = (B) t ⊙B r ) Instead, it is a Khatri-Rao product,
Figure BDA0003637276490000022
and
Figure BDA0003637276490000023
the array manifold matrix is respectively a receiving array and a transmitting array, and the guiding vectors of the transmitting array and the receiving array to different directions are respectively:
Figure BDA0003637276490000024
Figure BDA0003637276490000025
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003637276490000026
and theta k Respectively a wave departure angle and a wave arrival angle of the target to be detected;
considering the doppler shift due to the movement of the receiving array, the received signal vector is expressed as:
Figure BDA0003637276490000027
wherein(s) 1 (t)，s 2 (t)，...，s K (t)) is obtained after matched filtering of the target transmission signal, and L is a fast sampling beat number, i.e., the number of pulses in a CPI coherence interval, t =1,2,. Gtoreq, L;
and 3, step 3: since the receiving array is shifted only half a wavelength and the target is located in the far field, the angle of the target is a constant, and at t + τ, the received signal is represented as:
Figure BDA0003637276490000028
wherein τ is the time required for the receiving array to move half a wavelength, satisfying v τ = d;
and 4, step 4: the received signal vector is:
Figure BDA0003637276490000029
since the transmitted signal is a signal with a period of time τ, it has s k (t+τ)＝s k (t), the received signal vector is thus represented as:
Figure BDA0003637276490000031
wherein
Figure BDA0003637276490000032
Is a Hadamard product and v τ = d, so x (t + τ) is expressed as:
Figure BDA0003637276490000033
wherein
Figure BDA0003637276490000034
And 5: then, splicing the received signals x (t) and x (t + tau), thereby obtaining a new array received signal by using a passive aperture synthesis technology:
y(t)＝(B t ⊙B rc )s(t)+w(t)＝B c s(t)+w(t)
wherein B is rc ＝[b rc (θ 1 )，...，b rc (θ K )]And the new array receive steering vector is:
Figure BDA0003637276490000035
then, covariance statistics is carried out on array received data y (t) to obtain:
Figure BDA0003637276490000036
wherein
Figure BDA0003637276490000037
As a signal covariance matrix
Figure BDA0003637276490000038
For the noise covariance matrix, using
Figure BDA0003637276490000039
Estimating R y ；
Step 6: and straightening the receiving covariance data of the array to obtain:
Figure BDA00036372764900000310
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00036372764900000311
Figure BDA00036372764900000312
and I = vec (I) 2MN )；
Eliminate the vector r y The redundant items are rearranged to obtain a virtual receiving vector r of the signal 0 ：
Figure BDA00036372764900000313
Wherein i 0 Is a column vector with zero elements except the middle element of 1, and the virtual transmitting and receiving array flow pattern is
Figure BDA0003637276490000041
Seed B r0 ＝[b r0 (θ 1 )，...，b r0 (θ K )]And virtually receive and transmit steering vectorsComprises the following steps:
Figure BDA0003637276490000042
Figure BDA0003637276490000043
wherein M is 0 ＝M 2 (M 1 + 1) and N 0 ＝3N 1 (N 2 + 1) +2, thus showing that the degrees of freedom are expanded nearly three times by the array movement;
and 7: two selection matrices are then defined, which are used to select the virtual received vector r 0 Carrying out the following operations:
Figure BDA0003637276490000044
Figure BDA0003637276490000045
wherein
Figure BDA0003637276490000046
And is provided with
Figure BDA0003637276490000047
Through two selection matrices, get the first
Figure BDA0003637276490000048
An individual vector
Figure BDA0003637276490000049
Expressed as:
Figure BDA00036372764900000410
(symbol)
Figure BDA00036372764900000411
is the product of Kronecker, and B z ＝(B tz ⊙B rz )，
Figure BDA00036372764900000412
B rz ＝[b rz (θ 1 )，...，b rz (θ K )]Wherein:
Figure BDA00036372764900000413
Figure BDA00036372764900000414
Figure BDA00036372764900000415
Figure BDA00036372764900000416
according to the matrix smoothing method, a new virtual covariance matrix is reconstructed:
Figure BDA00036372764900000417
and step 8: the method adopts a tensor smoothing method, and firstly defines the virtual single snapshot receiving tensor as follows:
Figure BDA00036372764900000418
wherein
Figure BDA00036372764900000419
And is provided with
Figure BDA00036372764900000420
A unit diagonal tensor of kxkxk;
defining a fourth order tensor:
Figure BDA0003637276490000051
wherein
Figure BDA0003637276490000052
Represents the outer product, and then uses the following formula to obtain a new virtual covariance matrix R 0 ：
Figure BDA0003637276490000053
To obtain R 0 ＝B z C T +N；
Arranging the three-order tensor to obtain a third-order tensor
Figure BDA0003637276490000054
Then parallel factorization is used to obtain
Figure BDA0003637276490000055
And step 9: in order to fully utilize the transmitting and receiving array apertures of the MIMO radar, the obtained
Figure BDA0003637276490000056
And
Figure BDA0003637276490000057
performing a Khatri-Rao product operation, i.e.
Figure BDA0003637276490000058
To solve for the angle using the ESPRIT algorithm,
Figure BDA0003637276490000059
and
Figure BDA00036372764900000510
(
Figure BDA00036372764900000511
and
Figure BDA00036372764900000512
) Four selection matrices are defined, and a matrix is selected respectively
Figure BDA00036372764900000513
Front M of 0 -1(N 0 -1) lines and postX M 0 -1(N 0 -1) rows;
step 10: using the rotational invariance of the array, the following is obtained:
Figure BDA00036372764900000514
due to the fact that
Figure BDA00036372764900000515
Is formed in which
Figure BDA00036372764900000516
Is composed of
Figure BDA00036372764900000517
The signal subspace obtained after singular value decomposition, T, is a full-rank square matrix, and therefore contains the matrix psi of the target angle information t And Ψ r Obtained by the following formula:
Figure BDA00036372764900000518
wherein the symbols
Figure BDA00036372764900000519
Represents the Moore-Penrose pseudoinverse, hence according to Ψ t ＝TD t T -1 And Ψ r ＝TD r T -1 Obtaining the sum of the information of the target wave departure angle through characteristic value decompositionTwo diagonal arrays D of angle of arrival information t And D r And the angle values of the targets can be automatically paired; finally according to
Figure BDA0003637276490000061
Obtaining angle estimation values capable of automatic pairing, wherein
Figure BDA0003637276490000062
Seed of a species of rice
Figure BDA0003637276490000063
Are respectively a matrix D t And D r Kth diagonal elements.
In the step 1, a transmitting array of the bistatic MIMO radar has M array elements, a receiving array has N array elements, and K far-field irrelevant targets to be detected are arranged in a space; the transmitting array is a classic nested array type, and the sub-array 1 thereof has M 1 Each array element, subarray 2 has M 2 An array element; the receiving array is a sparse array obtained by expanding the array element spacing of the classical nested array, and a sub-array 1 of the sparse array has N 1 Each array element, subarray 2 having N 2 The array elements and the receiving array move at a constant speed v along a straight line; the schematic diagram of the array structure is shown in fig. 2 in detail, and the array element positions of the transmitting array and the receiving array are respectively:
Figure BDA0003637276490000064
Figure BDA0003637276490000065
where d = λ/2, λ being the carrier wavelength of the received signal.
The beneficial effects of the invention are: according to the invention, the mobile sparse array is introduced into the receiving end of the bistatic MIMO radar, the array degree of freedom of the system is greatly improved, the tensor smoothing method is used for further utilizing inherent multi-dimensional structural information in data, the array aperture of the MIMO radar is fully utilized by using the Khatri-Rao product, finally, the resolving and automatic pairing of the target angle are realized by the ESPRIT algorithm, and the parameter estimation precision of the target is greatly improved.
Drawings
Fig. 1 is a flow chart of a mobile sparse-based receive array angle estimation method.
Fig. 2 is a schematic diagram of direction finding of a bistatic MIMO radar system, wherein fig. 2 (a) is a schematic diagram of a bistatic MIMO radar transmitting end, and fig. 2 (b) is a schematic diagram of a bistatic MIMO radar mobile receiving end.
Fig. 3 is a schematic diagram of the maximum recognizable target number in different methods, fig. 3 (a) is a result of angle estimation performed on 12 targets when the receiving end uses a classical nested array, and fig. 3 (b) is a result of angle estimation performed on 40 targets when the receiving end uses a moving sparse array.
Fig. 4 shows the root mean square error as a function of the signal to noise ratio for angle estimation of 3 targets by different methods.
Fig. 5 shows the root mean square error of the angle estimates for 3 targets as a function of the number of fast beats.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
The invention applies the moving sparse array to bistatic MIMO to improve the degree of freedom of the system, and further improves the estimation precision of the parameters by a tensor smoothing method.
As shown in fig. 1, the present invention provides a mobile sparse receiving array angle estimation method, which includes the steps of:
step 1: setting initialization parameters;
in the scheme, the transmitting array of the MIMO radar is set as a classical nested array, and a sub-array 1 of the array has M 1 Each array element, subarray 2 has M 2 And the transmitting array element transmits completely orthogonal signals. The receiving array is a sparse array obtained by expanding the array element spacing of the classical nested arrayColumn with subarrays 1 having N 1 Each array element, subarray 2 has N 2 The array elements are arranged, and the receiving array moves at a constant speed v along a straight line.
Step 2: establishing tensor signal model of received data
Transmitting periodic signals with the period of half wavelength required by the movement of the receiving array at a transmitting end, obtaining the received signals by utilizing a passive aperture synthesis technology, and carrying out covariance statistics to obtain
Figure BDA0003637276490000071
Then, a new tensor receiving model is obtained through operations of straightening, extracting, tensor smoothing, rearranging and the like
Figure BDA0003637276490000072
Then parallel factorization is used to obtain
Figure BDA0003637276490000073
And step 3: obtained by operation of the Khatri-Rao product
Figure BDA0003637276490000074
For is to
Figure BDA0003637276490000075
Singular value decomposition is carried out to obtain a signal subspace
Figure BDA0003637276490000076
And 4, step 4: then, two diagonal arrays D containing target departure angle information and arrival angle information are obtained by using an ESPRIT algorithm t And D r 。
And 5: final estimation by target direction
Figure BDA0003637276490000077
And obtaining the target angle estimated value of automatic pairing.
Fig. 3 is at signal-to-noise ratio SNR =5dB; sampling fast beat number L =100, and Radar Cross Section (RCS) amplitude is selected as alpha = [1,1 =] T The Doppler frequency shift of the target is uniformly distributed between f = (200, 950)/2000, the transmitting arrays are all selected as uniform linear arrays with the array element number M =3, and the receiving arrays are respectively selected as N 1 =2 and N 2 Classical nested and moving sparse arrays of = 2. For two different receiving arrays, schematic diagrams of estimation results of 12 targets and 40 targets are respectively selected. Fig. 3 shows that the transmitting arrays of the MIMO radar are all set to be uniform linear arrays, while the receiving arrays use classical nested arrays and the other sparse moving array tested by the invention, and it can be seen that the method can perform parameter estimation of more targets compared with the uncorrected method.
Fig. 4 selects the target number as 3 and the doppler frequency as f = [200,400,850 =] T /2000, the directions of the targets are respectively
Figure BDA0003637276490000081
When the sampling snapshot number is L =200, the emitting arrays are all selected as array element numbers M 1 ＝2，M 2 A classical nested array of =3, with the receive array selected to be N 1 ＝2，N 2 The method comprises the following steps of =3 classical nested arrays, an ESPRIT algorithm (ESPRIT) for matrix reconstruction, a parallel factor algorithm (MS-PARAFAC) for matrix smoothing, and a root mean square error variation graph (RMS-PARAFAC) for parallel factor algorithms (TS-PARAFAC) for tensor smoothing under different signal-to-noise ratios (SNRs).
Fig. 4 is a direction estimation error curve under different signal-to-noise ratios when the sampling fast beat number is selected and L =200 is selected in the experiment of the present invention, and it can be seen that compared with a matrix model algorithm and a parallel factorization algorithm based on matrix smoothing, the method of the present invention has higher estimation accuracy.
FIG. 5 selects the target number of 3 and the Doppler frequency of f = [200,400,850 = [200,400 ]] T /2000, target directions are respectively
Figure BDA0003637276490000082
And selecting SNR =5dB by the signal-to-noise ratio, and using an ESPRIT algorithm (ESPRIT) of matrix reconstruction, a parallel factor algorithm (MS-PARAFAC) of matrix smoothing, and a root mean square error change diagram of a parallel factor algorithm (TS-PARAFAC) of tensor smoothing under different sampling fast-afraid numbers L.
Fig. 5 is a direction estimation error curve under different sampling fast beat numbers when the signal-to-noise ratio SNR =5dB is selected in the experiment of the present invention, and it can be seen that the method of the present invention has higher accuracy compared with a matrix model algorithm and a parallel factorization algorithm based on matrix smoothing.
In conclusion, the application of the mobile sparse array to the receiving end of the bistatic MIMO radar can greatly improve the array freedom of the system, increase the maximum recognizable target number of the system, and further improve the estimation accuracy of the target by using a tensor smoothing algorithm.

Claims (2)

1. A method for estimating the angle of a mobile sparse receiving array is characterized by comprising the following steps:
step 1: establishing a receiving signal model of the bistatic MIMO radar in a sparse receiving array mobile scene;
step 2: signals transmitted among array elements of the MIMO radar transmitting array are mutually orthogonal, and when the mutual coupling influence of the array elements is not considered, the signals received in one pulse are subjected to matched filtering, and a data model of the received signals is obtained as follows:
Figure FDA0003637276480000011
wherein B = (B) t ⊙B r ) Instead, it is a Khatri-Rao product,
Figure FDA0003637276480000012
and
Figure FDA0003637276480000013
the array manifold matrix is respectively a receiving array and a transmitting array, and the guiding vectors of the transmitting array and the receiving array to different directions are respectively:
Figure FDA0003637276480000014
Figure FDA0003637276480000015
wherein the content of the first and second substances,
Figure FDA0003637276480000016
and theta k Respectively is the departure angle and the arrival angle of the target to be measured;
considering the doppler shift due to the movement of the receiving array, the received signal vector is expressed as:
Figure FDA0003637276480000017
wherein(s) 1 (t),s 2 (t),…,s K (t)) is obtained after matched filtering of the target transmission signal, and L is a sampling fast beat number, i.e. the number of pulses in a CPI coherence interval, t =1,2, \ 8230;
and step 3: since the receiving array is moved only a half wavelength distance and the target is located in the far field, the angle of the target is a constant, and at t + τ, the received signal is expressed as:
Figure FDA0003637276480000018
where τ is the time required for the receive array to move half a wavelength, satisfying v τ = d;
and 4, step 4: the received signal vector is:
Figure FDA0003637276480000021
Figure FDA0003637276480000022
Figure FDA0003637276480000023
since the transmitted signal is a signal with a period of time τ, there is s k (t+τ)＝s k (t), so the received signal vector is represented as:
Figure FDA0003637276480000024
wherein
Figure FDA0003637276480000025
Is a Hadamard product and v τ = d, so x (t + τ) is expressed as:
Figure FDA0003637276480000026
wherein
Figure FDA0003637276480000027
And 5: then, splicing the received signals x (t) and x (t + tau), thereby obtaining a new array received signal by using a passive aperture synthesis technology:
y(t)＝(B t ⊙B rc )s(t)+w(t)＝B c s(t)+w(t)
wherein B is rc ＝[b rc (θ 1 )，...，b rc (θ K )]And the new array receive steering vector is:
Figure FDA0003637276480000028
then, covariance statistics is carried out on array received data y (t) to obtain:
Figure FDA0003637276480000029
wherein
Figure FDA00036372764800000210
As a signal covariance matrix
Figure FDA00036372764800000211
For the noise covariance matrix, using
Figure FDA00036372764800000212
Estimating R y ；
Step 6: and straightening the receiving covariance data of the array to obtain:
Figure FDA00036372764800000213
wherein the content of the first and second substances,
Figure FDA0003637276480000031
and I = vec (I) 2MN )；
Removing the vector r y The redundant items are rearranged to obtain a virtual signal receiving vector r 0 ：
Figure FDA0003637276480000032
Wherein i 0 Is a column vector with zero elements except the middle element of 1, and a virtual transmit and receive array streamIs formed by
Figure FDA0003637276480000033
And B r0 ＝[b r0 (θ 1 ),…,b r0 (θ K )]And the virtual receive and transmit steering vectors are:
Figure FDA0003637276480000034
Figure FDA0003637276480000035
wherein M is 0 ＝M 2 (M 1 + 1) and N 0 ＝3N 1 (N 2 + 1) +2, thus it is seen that the degrees of freedom are expanded nearly three times by the array movement;
and 7: defining two selection matrices for the virtual received vector r 0 Carrying out the following operations:
Figure FDA0003637276480000036
Figure FDA0003637276480000037
wherein
Figure FDA0003637276480000038
M 0 And is
Figure FDA0003637276480000039
Through two selection matrices, get
Figure FDA00036372764800000310
An individual vector
Figure FDA00036372764800000311
Expressed as:
Figure FDA00036372764800000312
(symbol)
Figure FDA00036372764800000313
is a product of Kronecker, and B z ＝(B tz ⊙B rz )，
Figure FDA00036372764800000314
B rz ＝[b rz (θ 1 ),…,b rz (θ K )]Wherein:
Figure FDA00036372764800000315
Figure FDA00036372764800000316
Figure FDA00036372764800000317
Figure FDA00036372764800000318
according to the matrix smoothing method, a new virtual covariance matrix is reconstructed:
Figure FDA0003637276480000041
and 8: the method comprises the following steps of firstly defining a virtual single snapshot receiving tensor as follows by adopting a tensor smoothing method:
Figure FDA0003637276480000042
wherein
Figure FDA0003637276480000043
And is
Figure FDA0003637276480000044
Is a unit diagonal tensor of kxkxk;
defining a fourth order tensor:
Figure FDA0003637276480000045
wherein
Figure FDA0003637276480000046
Representing the outer product, and then using the following formula, obtaining a new virtual covariance matrix R 0 ：
Figure FDA0003637276480000047
To obtain R 0 ＝B z C T +N；
Arranging them to obtain a third order tensor
Figure FDA0003637276480000048
Then parallel factorization is used to obtain
Figure FDA0003637276480000049
And step 9: in order to fully utilize the transmitting and receiving array apertures of the MIMO radar, the obtained
Figure FDA00036372764800000410
And
Figure FDA00036372764800000411
performing a Khatri-Rao product operation, i.e.
Figure FDA00036372764800000412
To solve the angle using the ESPRIT algorithm,
Figure FDA00036372764800000413
and
Figure FDA00036372764800000414
(
Figure FDA00036372764800000415
and
Figure FDA00036372764800000416
) Four selection matrices are defined, and a matrix is selected respectively
Figure FDA00036372764800000417
Front M of 0 -1(N 0 -1) lines and postM 0 -1(N 0 -1) rows;
step 10: using the rotational invariance of the array, the following is obtained:
Figure FDA00036372764800000418
due to the fact that
Figure FDA0003637276480000051
Is formed in which
Figure FDA0003637276480000052
Is composed of
Figure FDA0003637276480000053
The signal subspace obtained after singular value decomposition, T, is a full-rank square matrix, and therefore contains the matrix psi of the target angle information t And Ψ r Obtained by the following formula:
Figure FDA0003637276480000054
wherein the symbols
Figure FDA0003637276480000055
Represents the Moore-Penrose pseudoinverse, hence according to Ψ t ＝TD t T -1 And Ψ r ＝TD r T -1 Obtaining two diagonal arrays D containing target departure angle information and arrival angle information through eigenvalue decomposition t And D r And the angle values of the targets can be automatically paired; finally according to
Figure FDA0003637276480000056
Obtaining angle estimation values capable of automatic pairing, wherein
Figure FDA0003637276480000057
And
Figure FDA0003637276480000058
are respectively a matrix D t And D r Kth diagonal elements.
2. The method of claim 1, wherein the angle estimation method comprises:
in the step 1, a transmitting array of the bistatic MIMO radar has M array elements, a receiving array has N array elements, and K far-field irrelevant targets to be detected are arranged in a space; the transmitting array is a classic nested array type, and the sub-array 1 thereof has M 1 Each array element, subarray 2 hasM 2 An array element; the receiving array is a sparse array obtained by expanding the array element spacing of the classical nested array, and a sub-array 1 of the sparse array has N 1 Each array element, subarray 2 has N 2 The array elements and the receiving array move at a constant speed v along a straight line; the array element positions of the transmitting array and the receiving array are respectively as follows:
Figure FDA0003637276480000059
Figure FDA00036372764800000510
where d = λ/2, λ being the carrier wavelength of the received signal.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112269172A (en) * 2020-10-14 2021-01-26 中国人民解放军空军工程大学 Nested MIMO radar angle estimation method and device based on tensor structure
CN112269172B (en) * 2020-10-14 2024-03-08 中国人民解放军空军工程大学 Nested MIMO radar angle estimation method and device based on tensor structure

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