CN112698263A - Orthogonal propagation operator-based single-basis co-prime MIMO array DOA estimation algorithm - Google Patents

Orthogonal propagation operator-based single-basis co-prime MIMO array DOA estimation algorithm Download PDF

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CN112698263A
CN112698263A CN202011245206.2A CN202011245206A CN112698263A CN 112698263 A CN112698263 A CN 112698263A CN 202011245206 A CN202011245206 A CN 202011245206A CN 112698263 A CN112698263 A CN 112698263A
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周围
冉靖萱
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Chongqing University of Post and Telecommunications
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Abstract

The invention relates to the technical field of radar direction finding, in particular to a single-base co-prime MIMO array DOA estimation algorithm based on an orthogonal propagation operator, which comprises the following steps: a pair of single-row co-prime arrays are used as a transmitting-receiving array to form a single-base MIMO array; establishing a received signal model after Doppler compensation and matched filtering, and carrying out covariance processing on the received signal to obtain a covariance matrix; solving a propagation operator by the covariance matrix, constructing a noise subspace by the propagation operator, carrying out orthogonalization treatment on the noise subspace, and replacing the noise subspace in the MUSIC algorithm with the orthogonalized noise subspace to carry out spectrum peak search of the orthogonal propagation operator. The algorithm arranges the transmitting and receiving arrays of the MIMO radar according to the rule of the co-prime array, fully utilizes the advantages of the co-prime and the MIMO array, greatly improves the DOF of the algorithm by utilizing the virtual array flow pattern matrix of the MIMO array, and avoids the characteristic value decomposition in the MUSIC algorithm by the OPM algorithm.

Description

Orthogonal propagation operator-based single-basis co-prime MIMO array DOA estimation algorithm
Technical Field
The invention relates to the technical field of radar direction finding, in particular to a single-base co-prime MIMO array DOA estimation algorithm based on an orthogonal propagation operator.
Background
A Multiple Input Multiple Output (MIMO) radar is a new type of radar widely used in the communication field in recent years, and it is first possible to trace back to a proposal of using a plurality of antennas on the transmitting and receiving sides of a communication system proposed by bell laboratories and the like in the middle of 1990 s, and until 2003, the concept of the MIMO radar has been proposed by researchers. MIMO radar uses multiple elements to transmit and receive signals synchronously to perform centralized signal processing, which has significant advantages for applications such as increasing channel capacity, improving signal detection and target parameter estimation performance.
Direction Of Arrival (DOA) estimation is widely used in the fields Of communications, medical imaging, astronomical imaging, seismology, sonar, etc., and is an important Direction in the research Of MIMO radar. The recently proposed MIMO radar DOA Estimation is mostly based on uniform line array combined with Multiple Signal Classification (MUSIC) and rotation Invariance (ESPRIT). However, the aperture Of the uniform line array is limited because the nyquist sampling theorem must be satisfied when using the uniform line array, the spacing between the array elements is not more than half the wavelength Of the signal, and the Degree Of Freedom (DOF) Of the uniform line array is also limited by the number Of the array elements. In this context, the literature (VAIDYANATHAN P P and PAL P.Sparse sensing with co-prime samples and arrays [ J ]. IEEE Transactions on Signal Processing,2011,59(2): 573-586. doi:10.1109/TSP.2010.2089682.) proposes the concept of an coprime array, which lays the foundation for estimation of DOA of coprime arrays.
Compared with a uniform array, the co-prime array can perform undersampling on incident signals, so that the spacing limit between array elements is broken through, and the array can obtain the DOF far exceeding the number of the array elements. The DECOM algorithm proposed in the literature (ZHOU Chengwei, SHI Zhiguo, GU Yujie, et al, DECOM: DOA estimation with combined MUSIC for coprime array [ C ].2013 International Conference on Wireless communication and Signal Processing, Handzhou, China,2013: 1-5.) decomposes the coprime array into two coprime sparse uniform linear sub-arrays, estimates the DOA on each sub-array using the MUSIC algorithm, and finds the peak where the two spatial spectra coincide using the prime property as the true DOA. The method can avoid phase ambiguity caused by a sparse array, but only can distinguish DOA of a single information source, and subarray decomposition inevitably causes information loss so as to reduce estimation precision and performance, although the method in the literature (SHI J, HU G, ZHANG X, et al. generalized co-prime MIMO radar for DOA estimation with enhanced details of free [ J ]. IEEE Sensors Journal,2018,18(3): 1203-1212) overcomes the problem of the single information source, the problem of information loss is still not solved. In the literature (jiaa Y, Chen C, Zhong X, et al.doa estimation of coherent and coherent targets based on homogeneous MIMO array [ J ]. Digital Signal Processing,2019,94.), the co-prime array is split into N sparse transmitting antennas and 2M-1 sparse receiving antennas for DOA estimation, so that joint estimation of coherent signals and incoherent signals is realized, and although the MIMO array is adopted, the limitation of the number of physical antennas on the degree of freedom is broken through, and the degree of freedom is improved to some extent, but the improvement of the degree of freedom is still not obvious. The document (PAL P and VAIDYANATHAN P P. copy sampling and the MUSIC algorithm [ C ]. Proceedings of 2011 Digital Signal Processing and Signal Processing reduction Meeting, Sedona, USA,2011:289 and 294.) proposes a virtual domain Nyquist matching method, and utilizes an expanded virtual uniform array manifold structure to combine with a spatial smoothing technique and a MUSIC algorithm to carry out DOA estimation.
In summary, the existing radar direction finding technology has the following problems:
1. the resolution and the DOF of the radar are respectively determined by the aperture of the radar and the number of array elements of the radar, so that the array elements are used as little as possible while the aperture and the DOF of the array are improved;
2. the MUSIC algorithm of the uniform linear array is extremely high in complexity, and how to reduce the calculation complexity of the algorithm;
3. the accuracy of the uniform line array MUSIC algorithm is low, namely the Root Mean Square Error (RMSE) is large.
Disclosure of Invention
In view of this, the present invention provides a single-radix co-prime MIMO array DOA estimation algorithm based on an orthogonal propagation operator, which arranges the transmitting and receiving arrays of the MIMO radar according to the rules of the co-prime array, fully utilizes the advantages of the co-prime and MIMO arrays, greatly improves the DOF of the algorithm by using the virtual array flow pattern matrix of the MIMO array, and avoids the eigenvalue decomposition in the MUSIC algorithm by using the OPM algorithm.
The invention solves the technical problems by the following technical means:
a single-basis co-prime MIMO array DOA estimation algorithm based on an orthogonal propagation operator comprises the following steps:
establishing a radar model, and forming a monostatic MIMO array by using a pair of single-row co-prime arrays as a receiving and transmitting array;
establishing a signal model, establishing a received signal model after Doppler compensation and matched filtering, and carrying out covariance processing on the received signal to obtain a covariance matrix;
and establishing a DOA estimation algorithm, solving a propagation operator by using the covariance matrix, constructing a noise subspace by using the propagation operator, performing orthogonalization treatment on the noise subspace, and performing spectrum peak search on the orthogonal propagation operator by using the orthogonalized noise subspace to replace the noise subspace in the MUSIC algorithm.
Further, the transceiving arrays are identical, the number of the array elements is M + N-1, and the number set form of each array element position of the transmitting array and the receiving array in the transceiving arrays is represented as:
Figure BDA0002769771780000031
wherein M is the number of array elements of a sub-array 1 in a single-row co-prime array, N is the number of array elements of a sub-array 2 in a single-row co-prime array, Md is the array element interval of the sub-array 1, Nd is the array element interval of the sub-array 2,
Figure BDA0002769771780000032
λ is the signal wavelength.
Further, the first array elements of the sub-array 1 and the sub-array 2 are coincident, the number of the array elements is more than N, and M and N are relatively prime.
Further, the received signal model is established as follows:
suppose there are K uncorrelated far-field targets in space with respective azimuth angles θ1,θ2,...,θKThen the received signal at the receiving array at time t is represented as follows:
Figure BDA0002769771780000033
wherein s (t) ═ s1(t),s2(t),...,sk(t)]TBeing a waveform vector of the signal, alphak、fkRespectively the radar reflection coefficient and the Doppler frequency shift of the Kth target signal, n (t) is the mean value of 0 and the variance is sigma2The vector of white gaussian noise of (a),
Figure BDA0002769771780000034
lambda is the wavelength of the transmitted and received signals,
Figure BDA0002769771780000035
i＝1,2,...,M+N-1；
the received signal model after Doppler compensation and matched filtering is established as follows:
x(t)＝Aα(t)+n(t)
wherein the content of the first and second substances,
Figure BDA0002769771780000036
array manifold of steering matrix for the monostatic MIMO array, ar(theta) and at(theta) is the steering vector of the array transmitting end and receiving end respectively, and alpha (t) ([ alpha ]1(t),α2(t),...,αk(t)]Is a vector of signal waveforms containing RCS.
Further, the covariance matrix is as follows:
Rxx＝E[x(t)xH(t)]＝ARssAH+σ2I
wherein the content of the first and second substances,
Figure BDA0002769771780000037
the time average of the number of snapshot samples L is taken to approximate the ideal covariance matrix as follows:
Figure BDA0002769771780000041
further, the solution method of the propagation operator is as follows:
setting the number of transmitting arrays or receiving arrays of the single-ground MIMO array as NeAssuming an array manifold of steering matrices for a monostatic MIMO array, M + N-1
Figure BDA0002769771780000042
Column full rank, then there are K rows in a that are linearly independent, and the other rows are linearly represented by the K rows, and a is partitioned as follows:
Figure BDA0002769771780000043
wherein the content of the first and second substances,
Figure BDA0002769771780000044
let A1Full rank, being a non-singular matrix, i.e. A1Rows are independent of each other, A2Can be prepared from A1Is obtained by linear transformation, thus having
A2＝PHA1
Where P is the propagation operator and the definition matrix Q is
Figure BDA0002769771780000045
Then there is
Figure BDA0002769771780000046
Therefore, the propagation operator is solved by the covariance matrix, for RxxThe blocking is performed as follows:
Figure BDA0002769771780000047
where H is GP, P is obtained by least squares
Figure BDA0002769771780000048
Further, the noise subspace is represented as follows:
Figure BDA0002769771780000049
further, the orthogonalization of the noise subspace is as follows:
Figure BDA00027697717800000410
further, the orthogonal propagation operator spectral peak search is as follows:
Figure BDA00027697717800000411
the invention provides a single-base co-prime MIMO array DOA estimation algorithm based on an orthogonal propagation operator, which combines a co-prime array with an MIMO radar for the first time, and provides the co-prime MIMO array DOA estimation algorithm based on the orthogonal propagation operator, wherein the algorithm arranges the transmitting and receiving arrays of the MIMO radar according to the rule of the co-prime array, fully utilizes the advantages of the co-prime array and the MIMO array, greatly improves the DOF of the algorithm by utilizing the virtual array flow pattern matrix of the MIMO array, avoids the characteristic value decomposition in the MUSIC algorithm by the OPM algorithm, and has the technical advantages of low complexity, high resolution, high DOF, high precision and the like.
Drawings
FIG. 1 is a diagram of the geometry of a relatively prime monostatic MIMO array;
FIG. 2 is a DOA estimation space spectrum of different algorithms when SNR is 0 dB;
FIG. 3 is a DOA estimated space spectrum of different algorithms at SNR-15 dB;
FIG. 4 is a small angle DOA estimated space spectrogram of different algorithms when SNR is-15 dB;
FIG. 5 is a diagram of a multi-target space spectrum with SNR of 5 dB;
figure 6 is a plot of root mean square error as a function of SNR.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The Orthogonal propagation operator-Based single-base co-prime MIMO Array DOA Estimation Algorithm of this embodiment combines a co-prime Array with a MIMO radar for the first time, and proposes a co prime MIMO Array DOA Estimation Algorithm (CP-MIMO-OPM) Based on an Orthogonal propagation operator (Orthogonal propagation Method, OPM). The algorithm arranges the transmitting and receiving arrays of the MIMO radar according to the rule of the co-prime array, fully utilizes the advantages of the co-prime array and the MIMO array, greatly improves the DOF of the algorithm by utilizing the virtual array flow pattern matrix of the MIMO array, and avoids the characteristic value decomposition in the MUSIC algorithm by the OPM algorithm. The algorithm comprises the following steps:
establishing a radar model, and forming a monostatic MIMO array by using a pair of single-row co-prime arrays as a receiving and transmitting array;
establishing a signal model, establishing a received signal model after Doppler compensation and matched filtering, and carrying out covariance processing on the received signal to obtain a covariance matrix;
and establishing a DOA estimation algorithm, solving a propagation operator by using the covariance matrix, constructing a noise subspace by using the propagation operator, performing orthogonalization treatment on the noise subspace, and performing spectrum peak search on the orthogonal propagation operator by using the orthogonalized noise subspace to replace the noise subspace in the MUSIC algorithm.
Specifically, the estimation algorithm of the single-basis co-prime MIMO array DOA based on the orthogonal propagation operator in this embodiment is as follows:
s1, establishing a radar model
A pair of single-row co-prime arrays are used as a transmitting-receiving array to form a single-base MIMO array, and a single-base MIMO array model is shown in FIG. 1, wherein the single-row co-prime array has the following characteristics:
(1) m and N are the number of array elements of the subarray 1 and the subarray 2 respectively;
(2) m > N and M and N are coprime;
(3) the first array elements of the two sub-arrays are overlapped;
(4) the receiving and transmitting arrays are completely the same, and the number of array elements is M + N-1;
(5) the array element interval of the sub-array 1 is Md, the array element interval of the sub-array 2 is Nd, wherein
Figure BDA0002769771780000061
λ is the signal wavelength.
Figure BDA0002769771780000062
S2, establishing a signal model
Suppose there are K uncorrelated far-field targets in space with respective azimuth angles θ1,θ2,...,θKThen the received signal at the receiving array at time t is represented as follows:
Figure BDA0002769771780000063
wherein s (t) ═ s1(t),s2(t),...,sk(t)]TBeing a waveform vector of the signal, alphak、fkRespectively the radar reflection coefficient and the Doppler frequency shift of the Kth target signal, n (t) is the mean value of 0 and the variance is sigma2The vector of white gaussian noise of (a),
Figure BDA0002769771780000064
lambda is the wavelength of the transmitted and received signals,
Figure BDA0002769771780000065
i＝1,2,...,M+N-1。
the received signal model after Doppler compensation and matched filtering is established as follows:
x(t)＝Aα(t)+n(t) (3)
wherein the content of the first and second substances,
Figure BDA0002769771780000066
array manifold of steering matrix for the monostatic MIMO array, ar(theta) and at(theta) is the steering vector of the array transmitting end and receiving end respectively, and alpha (t) ([ alpha ]1(t),α2(t),...,αk(t)]Is a vector of signal waveforms containing RCS.
The covariance processing of the received signal is as follows:
Rxx＝E[x(t)xH(t)]＝ARssAH+σ2I (4)
wherein the content of the first and second substances,
Figure BDA0002769771780000071
the time average of the number of snapshot samples L is taken to approximate the ideal covariance matrix as follows:
Figure BDA0002769771780000072
s3, establishing a DOA estimation algorithm
The PM algorithm uses matrix linear operation with lower complexity to replace eigenvalue decomposition in the MUSIC algorithm to obtain a noise subspace of the signal, so that the complexity of the operation is reduced. For convenience of calculation, the number of transmitting arrays or receiving arrays of the monostatic MIMO array is set as NeAssuming an array manifold of steering matrices for a monostatic MIMO array, M + N-1
Figure BDA0002769771780000073
Column full rank, then there are K rows in a that are linearly independent, and the other rows are linearly represented by the K rows, and a is partitioned as follows:
Figure BDA0002769771780000074
wherein the content of the first and second substances,
Figure BDA0002769771780000075
let A1Full rank, being a non-singular matrix, i.e. A1Rows are independent of each other, A2Can be prepared from A1Is obtained by linear transformation, thus having
A2＝PHA1 (7)
Where P is the propagation operator and the definition matrix Q is
Figure BDA0002769771780000076
Then there is
Figure BDA0002769771780000077
Therefore, the propagation operator is solved by the covariance matrix, for RxxThe blocking is performed as follows:
Figure BDA0002769771780000078
where H is GP, P is obtained by least squares
Figure BDA0002769771780000079
The noise subspace constructed by the propagation operator can be expressed as follows:
Figure BDA00027697717800000710
the noise subspace derived by the propagation operator method being non-orthogonalThe performance of the method is different from that of the MUSIC algorithm, and the propagation operator after orthogonal processing has performance closer to that of the MUSIC algorithm, so that the method can be used for solving the problem of low performance of the method
Figure BDA00027697717800000711
The orthogonalization process is performed as follows:
Figure BDA0002769771780000081
finally, the noise subspace in the formula (12) is used
Figure BDA0002769771780000082
The orthogonal propagation operator spectral peak search is performed instead of the noise subspace in the MUSIC algorithm as follows:
Figure BDA0002769771780000083
the orthogonal propagation operator-based single-basis co-prime MIMO array DOA estimation algorithm of the embodiment has the technical advantages of low complexity, high resolution, high DOF, high precision and the like, and specifically comprises the following steps:
1. low complexity
The amount of one complex multiplication can be regarded as four real multiplication operations. The complexity comparison between the proposed CoPrime MIMO Array DOA Estimation Algorithm (Coprime MIMO Array DOA Estimation Algorithm) Based on Orthogonal propagation Operator (OPM) and the CoPrime MIMO Array DOA Estimation Algorithm (Coprime MIMO Array DOA Estimation Method Based on MUSIC) Based on MUSIC Algorithm is measured by analyzing the number of real multiplication operations of the Algorithm.
The operation amount of the CP-MIMO-OPM algorithm mainly comes from three parts of solving of a covariance matrix, calculation of a propagation operator and final spectral peak search, and if the search step is 0.01, the complexity of solving of the covariance matrix is
Figure BDA0002769771780000089
The complexity of the calculation of the propagation operator P is o (3P)2Ne+K3) The complexity of the spectral peak search is
Figure BDA0002769771780000084
The total complexity is thus expressed as
Figure BDA0002769771780000085
Because the complexity of the spectral peak search is only related to the array dimension and the search step, the complexity of the spectral peak search of the two algorithms is uniformly expressed by o (C) for convenience of representation on the premise of the same array and search step, and the complexity of the CP-MIMO-PM algorithm can be simplified into that of the CP-MIMO-PM algorithm
Figure BDA0002769771780000086
In addition, the complexity of eigenvalue decomposition in the MUSIC algorithm can be expressed as
Figure BDA0002769771780000087
Table 1 shows the complexity contrast of the proposed algorithm with the CP-MIMO-MUSIC algorithm.
Figure BDA0002769771780000088
TABLE 1
It can be seen that the complexity of the CP-MIMO-OPM algorithm increases in the fourth power as the number of sub-arrays increases, while the complexity of the CP-MIMO-MUSIC algorithm increases in the sixth power as eigenvalue decomposition is required. The complexity of the CP-MIMO-OPM algorithm is much less than that of the CP-MIMO-MUSIC algorithm.
2. High resolution and DOF
The performance of the proposed algorithm is compared by comparing the spatial spectra of various algorithms through simulation. Assuming that the number of sub-arrays 1 and 2 in the co-prime MIMO array is M-7 and N-5, respectively, the target number is θ1＝-50，θ2＝30.3°，θ3＝15.4°，θ4＝30°，θ560.75 degrees, fast beat number L1000, interval of spectral peak search (step)Is 0.01 deg. Fig. 2 and fig. 3 are spatial spectrum peak position simulation diagrams of the proposed CP-MIMO-OPM algorithm, CP-MIMO-MUSIC algorithm, uniform line array MUSIC algorithm (ULA-MUSIC), and uniform MIMO line array MUSIC algorithm (ULA-MIMO-MUSIC), respectively, when the signal-to-noise ratio is 0dB and-15 dB.
Fig. 2 shows that when the SNR is 0dB, four algorithms can effectively identify the DOA, and the CP-MIMO-OPM algorithm almost completely coincides with the spatial spectrum of the CP-MIMO-MUSIC, so that it can be concluded that the performance of the algorithm proposed by the present invention is very close to that of the CP-MIMO-MUSIC algorithm when the SNR is 0dB or more. Fig. 3 shows that the ULA-MUSIC algorithm has been unable to accurately identify DOAs when the signal-to-noise ratio SNR is-15 dB, and the remaining three algorithms can still accurately identify DOAs. Although the ULA-MIMO-MUSIC algorithm can also accurately identify the DOA at a very low signal-to-noise ratio, as shown in fig. 4, when the remaining conditions are unchanged and the same signal-to-noise ratio SNR is-15 dB and the incident angles are 0 ° and 3 °, respectively, the ULA-MIMO algorithm and the ULA-MIMO-MUSIC algorithm cannot distinguish the DOA of the two signals, and the small-angle resolution capability of the CP-MIMO-OPM is far better than that of the ULA-MIMO-MUSIC algorithm. Fig. 5 is a simulation diagram of DOA estimation spatial spectrum of the algorithm when multiple targets simultaneously appear, and it can be seen that when the SNR is 10dB, the CP-MIMO-OPM algorithm can simultaneously and well distinguish the DOAs of 41 signal sources, the DOF of the order of magnitude cannot be reached by the conventional linear and co-prime arrays, and the multi-target spectral peaks are very sharp, so that the DOF of the algorithm is greatly improved compared with the conventional uniform linear array.
3. High precision
The performance of the CP-MIMO-OPM algorithm and the ULA-MIMO-MUSIC algorithm is measured by comparing the Root Mean Square Error (RMSE) of the two algorithms with the change of the signal-to-noise ratio. The root mean square error formula is as follows
Figure BDA0002769771780000091
Wherein K is the number of signals, H is the number of Monte Carlo experiments,
Figure BDA0002769771780000092
for the h experiment of the k signalDOA estimate, θkIs the actual value of the signal DOA.
The monte carlo frequency H is set to 100, and the signal-to-noise ratio SNR is taken at equal intervals from-11 to 6 to obtain simulation graphs of the two algorithms, as shown in fig. 6. It can be seen that the error of both algorithms decreases as the signal-to-noise ratio increases. The CP-MIMO-OPM algorithm provided by the invention combines a co-prime array, the size of the real aperture of the array can be expressed as max (M (N-1) d, N (M-1) d) and is far larger than the real aperture (N + M-1) d of the ULA-MIMO-MUSIC algorithm, so that the algorithm of the invention has smaller root mean square error and higher precision under the same condition.
Although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the spirit and scope of the invention as defined in the appended claims. The techniques, shapes, and configurations not described in detail in the present invention are all known techniques.

Claims (9)

1. A single-basis co-prime MIMO array DOA estimation algorithm based on an orthogonal propagation operator is characterized by comprising the following steps:
establishing a radar model, and forming a monostatic MIMO array by using a pair of single-row co-prime arrays as a receiving and transmitting array;
establishing a signal model, establishing a received signal model after Doppler compensation and matched filtering, and carrying out covariance processing on the received signal to obtain a covariance matrix, and approximately estimating an ideal covariance matrix;
and establishing a DOA estimation algorithm, solving a propagation operator by using the covariance matrix, constructing a noise subspace by using the propagation operator, performing orthogonalization treatment on the noise subspace, and performing spectrum peak search on the orthogonal propagation operator by using the orthogonalized noise subspace to replace the noise subspace in the MUSIC algorithm.
2. The orthogonal propagation operator-based single-radix co-prime MIMO array DOA estimation algorithm according to claim 1, wherein the transceiving arrays are identical, the number of the array elements is M + N-1, and the number set form of each array element position of the transmitting array and the receiving array in the transceiving arrays is represented as:
Figure FDA0002769771770000011
wherein M is the number of array elements of a sub-array 1 in a single-row co-prime array, N is the number of array elements of a sub-array 2 in a single-row co-prime array, Md is the array element interval of the sub-array 1, Nd is the array element interval of the sub-array 2,
Figure FDA0002769771770000012
λ is the signal wavelength.
3. A single basis co-prime MIMO array DOA estimation algorithm based on orthogonal propagation operators according to claim 2, characterized in that the first array elements of sub-array 1 and sub-array 2 coincide, the number of array elements M > N and M and N are co-prime.
4. The orthogonal propagation operator-based single-radix co-prime MIMO array DOA estimation algorithm of claim 3, wherein the received signal model is established as follows:
suppose there are K uncorrelated far-field targets in space with respective azimuth angles θ1,θ2,...,θKThen the received signal at the receiving array at time t is represented as follows:
Figure FDA0002769771770000013
wherein s (t) ═ s1(t),s2(t),...,sk(t)]TBeing a waveform vector of the signal, alphak、fkRespectively the radar reflection coefficient and the Doppler frequency shift of the Kth target signal, n (t) is the mean value of 0 and the variance is sigma2The vector of white gaussian noise of (a),
Figure FDA0002769771770000014
lambda is the wavelength of the transmitted and received signals,
Figure FDA0002769771770000015
i＝1,2,...,M+N-1；
the received signal model after Doppler compensation and matched filtering is established as follows:
x(t)＝Aα(t)+n(t)
wherein the content of the first and second substances,
Figure FDA0002769771770000021
array manifold of steering matrix for the monostatic MIMO array, ar(theta) and at(theta) is the steering vector of the array transmitting end and receiving end respectively, and alpha (t) ([ alpha ]1(t),α2(t),...,αk(t)]Is a vector of signal waveforms containing RCS.
5. The orthogonal propagation operator-based single-radix co-prime MIMO array DOA estimation algorithm of claim 4, wherein the covariance matrix is as follows:
Rxx＝E[x(t)xH(t)]＝ARssAH+σ2I
wherein the content of the first and second substances,
Figure FDA0002769771770000022
the time average of the number of snapshot samples L is taken to approximate the ideal covariance matrix as follows:
Figure FDA0002769771770000023
6. the orthogonal propagation operator-based single-radix co-prime MIMO array DOA estimation algorithm according to claim 5, wherein the propagation operator solution method is as follows:
setting the number of transmitting arrays or receiving arrays of the single-ground MIMO array as NeAssuming an array manifold of steering matrices for a monostatic MIMO array, M + N-1
Figure FDA0002769771770000024
Column full rank, then there are K rows in a that are linearly independent, and the other rows are linearly represented by the K rows, and a is partitioned as follows:
Figure FDA0002769771770000025
wherein the content of the first and second substances,
Figure FDA0002769771770000026
let A1Full rank, being a non-singular matrix, i.e. A1Rows are independent of each other, A2Can be prepared from A1Is obtained by linear transformation, thus having
A2＝PHA1
Where P is the propagation operator and the definition matrix Q is
Figure FDA0002769771770000027
Then there is
Figure FDA0002769771770000028
Therefore, the propagation operator is solved by the covariance matrix, for RxxThe blocking is performed as follows:
Figure FDA0002769771770000031
where H is GP, P is obtained by least squares
Figure FDA0002769771770000032
7. The orthogonal propagation operator-based single-basis co-prime MIMO array DOA estimation algorithm according to claim 6, wherein the noise subspace is expressed as follows:
Figure FDA0002769771770000033
8. the orthogonal propagation operator-based single-radix co-prime MIMO array DOA estimation algorithm according to claim 7, wherein the orthogonalization of the noise subspace is as follows:
Figure FDA0002769771770000034
9. the orthogonal propagation operator based single-radix co-prime MIMO array DOA estimation algorithm of claim 8, wherein the orthogonal propagation operator spectral peak search is as follows:
Figure FDA0002769771770000035
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