CN108828551B - Flexible MIMO radar mixed target DOA estimation method based on compressed sensing - Google Patents

Flexible MIMO radar mixed target DOA estimation method based on compressed sensing Download PDF

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CN108828551B
CN108828551B CN201810989016.8A CN201810989016A CN108828551B CN 108828551 B CN108828551 B CN 108828551B CN 201810989016 A CN201810989016 A CN 201810989016A CN 108828551 B CN108828551 B CN 108828551B
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mimo radar
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CN108828551A (en
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师俊朋
胡国平
周豪
张秦
冯子昂
刘梦波
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Air Force Engineering University of PLA
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    • 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
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Abstract

The invention discloses a flexible multiple-Input and multiple-Output (MIMO) radar mixed target Direction of Arrival (DOA) estimation method based on Compressed Sensing (CS), which relates to the technical field of array signal processing and aims at the structural design of a sparse array MIMO radar and the DOA estimation of a mixed target to mainly solve the following two problems: (1) a Flexible MIMO radar structure is designed and defined as a Flexible Array with Flexible Inter-element Spacing (SA-FIS). (2) A reduced complexity two-step CS algorithm is proposed to fully utilize the total virtual array elements. By modifying and removing off-diagonal elements in the target covariance matrix, the improved CS algorithm can identify only diagonal elements therein. Since the conventional CS algorithm needs to estimate all non-zero elements, the present invention can improve estimation performance and reduce complexity by estimating fewer number of elements.

Description

Flexible MIMO radar mixed target DOA estimation method based on compressed sensing
Technical Field
The invention relates to the technical field of array signal processing, in particular to a flexible MIMO radar mixed target DOA estimation method based on compressed sensing.
Background
In order to improve the upper limit of the degree of freedom under the condition of a given physical array element number, the research on the extension of the virtual array element of the sparse array MIMO radar from the perspective of a 'joint array' gradually draws attention from the academic world and makes great progress. The nested MIMO radar can obtain O (M) by utilizing O (M) array elements2) Or O (M)3) The degree of freedom of (a) is,but the mutual coupling rate is relatively large due to the dense arrangement of the sub-arrays. The larger array element spacing of the co-prime MIMO radar further reduces the mutual coupling rate, and O (M + N) array elements can be used for obtaining O (MN) freedom degree. But the structural design research of the sparse array MIMO radar of the system is not carried out.
In recent years, research on mixed-target (including incoherent target and coherent target) DOA estimation algorithms has attracted much attention. In general, due to the fact that the source covariance matrix is not full of rank, the estimation performance of the traditional subspace algorithm is sharply reduced. Many decoherence algorithms are proposed one after the other and make significant progress, such as spatial smoothing algorithm, Toeplitz matrix reconstruction method, maximum likelihood algorithm, CS algorithm, etc. In order to improve the target detection quantity, two-step angle estimation is respectively carried out on an incoherent target and a coherent target by establishing a target information separation matrix, for example, a space difference method firstly adopts a traditional subspace algorithm to estimate an incoherent information source, and then adopts a space difference technology to subtract an incoherent part to obtain coherent information source information; the oblique projection technique separates incoherent and coherent sources by establishing a projection matrix, and then performs angle estimation and the like by adopting a traditional subspace-like algorithm.
The research aiming at the DOA estimation algorithm of the sparse array MIMO radar virtual echo signal is greatly improved, and the method mainly comprises a space smoothing algorithm and a CS algorithm. Compared with a spatial smoothing algorithm, the CS can overcome the aperture loss caused by subarray smoothing, and can utilize discrete parts in a virtual array element structure. However, the conventional method is mainly suitable for incoherent target DOA estimation, and relatively few researches are carried out on sparse array MIMO radar mixed target DOA estimation algorithm. The traditional algorithm adopts the CS algorithm to solve the DOA estimation problem under the structure of the sum and difference combined array, but the traditional CS algorithm has larger operation amount because the non-diagonal elements in the target covariance matrix after the vectorization need to be estimated. For example, when K targets are fully coherent, the array element to be estimated is K2This causes a sharp drop in estimation performance and an increase in the amount of computation.
The prior art has the following defects: 1) the structural design of the sparse array MIMO radar based on the degree of freedom and mutual coupling joint optimization is not carried out; 2) the traditional CS algorithm has higher operation amount when carrying out sparse array MIMO radar mixed target DOA estimation.
Disclosure of Invention
The embodiment of the invention provides a flexible MIMO radar mixed target DOA estimation method based on compressed sensing, which is used for solving the problems in the prior art.
A flexible MIMO radar mixed target DOA estimation method based on compressed sensing comprises the following steps:
step one, establishing a flexible MIMO radar echo signal model: firstly, performing matched filtering on a transmitting array and a receiving array of a flexible MIMO radar (SA-FIS) to obtain an array echo signal vector model, and then obtaining a covariance matrix R and a vectorization covariance matrix R of the array echo signal according to the array echo signal vector model x (t);
step two, carrying out structure optimization on the flexible MIMO radar echo signal model: because the array flow pattern matrix B in the vectorization covariance matrix r conforms to the characteristics of a sum-difference combined array, a proper expansion factor is selected by analyzing the combined array structure of the array flow pattern matrix B, so that more virtual array elements of the flexible MIMO radar are obtained;
and thirdly, aiming at the flexible MIMO radar echo signal model after structure optimization, performing mixed target DOA estimation by adopting a complexity-reduction two-step CS algorithm: and establishing a complexity-reducing two-step CS algorithm model by estimating and correcting redundant items, and removing repeated rows in the vectorization covariance matrix r by combining the total virtual array element arrangement sequence to obtain a new signal model.
Preferably, the specific steps of the first step are as follows:
the transmitting array and the receiving array of the flexible MIMO radar (namely SA-FIS) are composed of sparse uniform linear arrays, so that the total physical array element number is T-M + N;
the transmitting array is provided with M array elements, and the distance between the array elements is alpha d; the receiving array is provided with N array elements, and the spacing between the array elements is beta d; alpha and beta are coprime expansion factors, d is unit array element distance and is set as lambda/2, and lambda is signal wavelength;
array element position set of transmitting array and receiving array
Figure GDA0003352433910000033
And
Figure GDA0003352433910000034
comprises the following steps:
Figure GDA0003352433910000031
wherein m and n are integers;
k far-field incoherent and coherent mixed targets are arranged, and the information source direction set is theta ═ thetakK is 1,2, …, K, where the number of incoherent objects and the number of coherent objects are K, respectivelyuAnd KcI.e. K ═ Ku+Kc(ii) a Suppose KcEach target satisfies a full coherence condition; then the vector model of the array echo signal after matched filtering is:
Figure GDA0003352433910000032
wherein:
Figure GDA0003352433910000041
wherein,
Figure GDA0003352433910000042
is the reflection coefficient of the kth target when the fast beat number is t;
Figure GDA0003352433910000043
is the k-th0Attenuation coefficient (i.e., coherence coefficient) of an object, for convenience of description, it is assumed
Figure GDA0003352433910000044
[·]TFor matrix transpose operations, diag (-) is a diagonal operation,
Figure GDA0003352433910000049
and
Figure GDA0003352433910000045
respectively representing Khatri-Rao product and Kronecker product;
n (t) is an independent and equally distributed additive white Gaussian noise vector satisfying CN (0, sigma)2)；
And is
Figure GDA00033524339100000410
At＝[at(θ1),at(θ2),…,at(θK)] (5)
Ar＝[ar(θ1),at(θ2),…,ar(θK)] (6)
Figure GDA0003352433910000046
at(θk) And ar(θk) Direction vectors of kth targets of the transmitting array and the receiving array respectively are as follows:
Figure GDA0003352433910000047
Figure GDA0003352433910000048
the covariance matrix of the array echo signal obtained from the echo signal model of equation (2) is:
R＝E[x(t)xH(t)]＝ARsAH+σ2IMN (9)
wherein:
Rs＝E[s(t)sH(t)] (10)
is a target covariance matrix, AHFor the complex conjugate transpose operation of matrix A, IMNThe matrix is MN multiplied by MN dimension unit matrix;
when the fast beat number is L (t 1, …, L), its sample covariance matrix is approximated as:
Figure GDA0003352433910000051
vectorized covariance matrix R yields:
r＝vec(R)＝Bvec(Rs)+σ2vec(IMN) (12)
wherein,
Figure GDA0003352433910000052
A*representing a matrix complex conjugate operation, vec (-) representing a matrix vectorization operation.
Preferably, the specific steps of the second step are as follows: array element position collection using transmit and receive arrays of formula (1)
Figure GDA0003352433910000053
And
Figure GDA0003352433910000054
obtaining:
the sum and difference joint array set of the flexible MIMO radar SA-FIS is as follows:
Figure GDA0003352433910000055
wherein m is0,n0Is an integer which is the number of the whole,
Figure GDA0003352433910000056
Figure GDA0003352433910000057
Figure GDA0003352433910000058
for further analysis
Figure GDA0003352433910000059
The degree of freedom of the method can obtain the following virtual array element distribution condition of the flexible MIMO radar SA-FIS: defining a set of 'sum and difference joint arrays' for flexible MIMO radar SA-FIS by vectorizing a covariance matrix R
Figure GDA00033524339100000510
The flexible MIMO radar SA-FIS has the following characteristics:
(a) the coprime expansion factor satisfies: alpha is more than or equal to 1 and less than or equal to 2N-1, beta is more than or equal to 1 and less than or equal to 2M-1;
(b) collection
Figure GDA00033524339100000511
Has a continuous virtual array element range of [ -c, c]Wherein, c ═ α M + β N- α β -1;
(c)
Figure GDA0003352433910000061
wherein the total number of virtual array elements is 2g +1, g ═ alpha (M-1) + beta (N-1) - (alpha-1) (beta-1)/2;
the spatial smoothing technology can only utilize continuous virtual array elements, and the degree of freedom is c + 1; the CS algorithm can utilize all virtual array elements, and the degree of freedom is 2g + 1; to select the appropriate spreading factor to obtain more virtual array elements, the total virtual array element number and the continuous virtual array element number are optimized as follows:
(3) total number of virtual array elements g
Determining the optimal distribution structures of the expansion factors alpha and beta and the total physical array element number at the transmitting end and the receiving end by optimizing the parameter g, wherein the optimization objective function is as follows:
Figure GDA0003352433910000062
(4) number of consecutive virtual array elements c
Similarly, an objective function related to the number c of the optimized array elements is established as follows:
Figure GDA0003352433910000063
preferably, the third step specifically comprises: according to equation (9), the covariance matrix of the array echo signals can be re-expressed as:
Figure GDA0003352433910000064
wherein,
Figure GDA0003352433910000065
vectorizing the matrix R to obtain:
Figure GDA0003352433910000066
wherein,
Figure GDA0003352433910000071
Figure GDA0003352433910000072
signal energy for the kth target;
the first term is shown by the formula (17)
Figure GDA0003352433910000073
From the target covariance matrix RsThe diagonal element composition of (2), the second term
Figure GDA00033524339100000710
From RsFor a mixed target consisting of coherent and incoherent targets, the non-zero off-diagonal elements consist of coherent target coherence coefficients; as can be seen from equation (17), the diagonal elements of the target covariance matrix can be directly estimated by using the first term to determine the target angle, and the second term can be directly treated as the redundant term;
and (3) establishing a complexity-reducing two-step CS algorithm model by estimating and correcting redundant items by combining the model analysis, wherein the specific contents are as follows:
(1) for the echo signal vector model in the formula (2), the estimated value of the target signal s (t) obtained by adopting the least mean square technique is as follows:
Figure GDA0003352433910000075
wherein, assuming matrix A row full rank, A+＝AH(AAH)-1Then the target covariance matrix can be expressed as:
Figure GDA0003352433910000076
according to equation (19), the second term in equation (17) is represented as:
Figure GDA0003352433910000077
using the estimated value of equation (19), equation (17) is re-expressed as:
Figure GDA0003352433910000078
wherein the parameter lambda1∈[0,1]；
The new signal model obtained by removing the repeated rows in the data vector r in combination with the total virtual array element arrangement order is:
Figure GDA0003352433910000079
wherein, the vector e is a row vector of 2g +1, the element of the g +1 row is 1, and the other rows are 0;
Figure GDA0003352433910000087
is related to a virtual array element(2g +1) xK dimensional matrix, B, corresponding in position0Is the corresponding (2g + 1). times.K2A dimension matrix;
setting a search vector to θ ═ θjAnd j is 1,2, …, P, then the optimization objective function can be established according to equation (22) as:
Figure GDA0003352433910000081
wherein:
Figure GDA0003352433910000082
Figure GDA0003352433910000083
the noise statistic is known, and eta is a regularization parameter;
the optimization model in equation (23) is solved by using LASSO algorithm, and for the convenience of explanation of step two, the solution is defined as:
Figure GDA0003352433910000084
(2) the second term in the formula (23) is calculated according to the search vector theta, and because theta has a large angle error, the redundant term error is relatively large; for this purpose, the estimation result θ in equation (26) can be used(1)Revising the second term to further improve the estimation performance;
according to the estimation result theta(1)The new target covariance matrix that can be obtained is:
Figure GDA0003352433910000085
wherein:
Figure GDA0003352433910000086
according to formula (27):
Figure GDA0003352433910000091
Figure GDA0003352433910000092
a new objective optimization function can be constructed according to equation (30) as:
Figure GDA0003352433910000093
wherein the parameter lambda2∈[0,1]；
As can be seen by comparing the expressions (23) and (31), the second term after correction
Figure GDA0003352433910000094
More accurate, hence λ1≤λ2Therefore, the algorithm has lower operation amount and higher estimation performance; meanwhile, the angle can be estimated again from equation (31):
Figure GDA0003352433910000095
it is to be noted that the parameter λ1,λ2Are constants whose values are influenced by the echo signal model, so that for a given signal model the optimum value can be determined by an exhaustive method.
By estimating and correcting the target covariance matrix, the algorithm provided by the invention has lower operation amount and higher estimation performance than the traditional CS algorithm, is also suitable for the estimation of the incoherent target DOA, can improve the estimation performance by inhibiting the target correlation coefficient error, but needs to carry out twice optimization solution. The invention has the beneficial effects that: obtaining larger degree of freedom through virtual array element optimization;
array element mutual coupling is reduced by increasing the array element spacing of the transmitting array and the receiving array;
the estimation precision is improved and the algorithm operation amount is reduced by estimating and suppressing the off-diagonal elements of the signal covariance matrix.
Drawings
Fig. 1 is a schematic diagram of a flexible MIMO radar based on a compressed sensing flexible MIMO radar mixed target DOA estimation method according to an embodiment of the present invention;
fig. 2 is a schematic diagram of a flexible MIMO radar based on compressed sensing when different β values of a flexible MIMO radar hybrid target DOA estimation method are provided in an embodiment of the present invention;
fig. 3 is a schematic diagram of distribution of virtual array elements of a MIMO radar according to a flexible MIMO radar mixed target DOA estimation method based on compressive sensing according to an embodiment of the present invention;
fig. 4 is a schematic diagram of operation time of a conventional CS and two-step CS algorithm of a flexible MIMO radar mixed target DOA estimation method based on compressed sensing according to an embodiment of the present invention;
FIG. 5 is a graph showing the variation of different algorithms RMSE with SNR according to the flexible MIMO radar mixed target DOA estimation method based on compressive sensing provided by the embodiment of the present invention;
FIG. 6 is a graph showing the relationship between RMSE (RMSE) and snapshot number changes in different algorithms of a DOA (direction of arrival) estimation method for a flexible MIMO (multiple input multiple output) radar mixed target based on compressive sensing according to an embodiment of the present invention;
FIG. 7 is a diagram of the variation of different array RMSE with SNR according to the flexible MIMO radar mixed target DOA estimation method based on compressive sensing provided by the embodiment of the present invention;
FIG. 8 is a graph showing the relationship between the variation of different array RMSE with snapshot numbers in a flexible MIMO radar mixed target DOA estimation method based on compressive sensing according to an embodiment of the present invention;
fig. 9 is a normalized spatial spectrum of different algorithms of a flexible MIMO radar mixed target DOA estimation method based on compressed sensing according to an embodiment of the present invention;
fig. 10 is a normalized spatial spectrum of different sparse array structures of a flexible MIMO radar mixed target DOA estimation method based on compressed sensing according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention are clearly and completely described below with reference to the drawings in the embodiments of the present invention, but it should be understood that the scope of the present invention is not limited by the specific embodiments.
The invention has the following aims: firstly, a more flexible sparse array MIMO radar structure is established to inhibit mutual coupling and increase the degree of freedom, wherein the current sparse array MIMO structure is in a special form; and secondly, a complexity reduction CS algorithm is provided for mixed target DOA estimation.
Referring to fig. 1, a schematic diagram of a flexible MIMO radar based on a compressed sensing flexible MIMO radar mixed target DOA estimation method according to an embodiment of the present invention,
step one, establishing a flexible MIMO radar echo signal model
The transmitting array and the receiving array of the flexible MIMO radar (namely SA-FIS) are composed of sparse uniform linear arrays; the transmitting array is provided with M array elements, and the distance between the array elements is alpha d; the receiving array is provided with N array elements, and the spacing between the array elements is beta d; alpha and beta are coprime expansion factors, d is unit array element distance and is set as lambda/2, and lambda is signal wavelength;
thus, the total number of physical array elements is T ═ M + N;
array element position set of transmitting array and receiving array
Figure GDA0003352433910000111
And
Figure GDA0003352433910000112
comprises the following steps:
Figure GDA0003352433910000113
wherein m and n are integers;
k far-field incoherent and coherent mixed targets are arranged, and the information source direction set is theta ═ thetak,k＝1,2…, K }, wherein the number of incoherent objects and the number of coherent objects are respectively KuAnd KcI.e. K ═ Ku+Kc(ii) a Suppose KcEach target satisfies a full coherence condition; then the vector model of the array echo signal after matched filtering is:
Figure GDA0003352433910000114
wherein:
Figure GDA0003352433910000121
wherein,
Figure GDA0003352433910000122
is the reflection coefficient of the kth target when the fast beat number is t;
Figure GDA0003352433910000123
is the k-th0Attenuation coefficient (i.e., coherence coefficient) of an object, for convenience of description, it is assumed
Figure GDA0003352433910000124
[·]TFor matrix transpose operations, diag (-) is a diagonal operation,
Figure GDA0003352433910000125
and
Figure GDA0003352433910000126
respectively representing Khatri-Rao product and Kronecker product;
n (t) is an independent and equally distributed additive white Gaussian noise vector satisfying CN (0, sigma)2)；
And is
Figure GDA0003352433910000127
At＝[at(θ1),at(θ2),…,at(θK)] (5)
Ar＝[ar(θ1),at(θ2),…,ar(θK)] (6)
Figure GDA0003352433910000128
at(θk) And ar(θk) Direction vectors of kth targets of the transmitting array and the receiving array respectively are as follows:
Figure GDA0003352433910000129
Figure GDA00033524339100001210
the covariance matrix of the array echo signal obtained from the echo signal model of equation (2) is:
R＝E[x(t)xH(t)]＝ARsAH+σ2IMN (9)
wherein:
Rs＝E[s(t)sH(t)] (10)
is a target covariance matrix, AHFor the complex conjugate transpose operation of matrix A, IMNThe matrix is MN multiplied by MN dimension unit matrix;
in fact, when the fast beat number is L (t ═ 1, …, L), its sample covariance matrix is approximated as:
Figure GDA0003352433910000131
the vectorized covariance matrix R may be obtained:
r＝vec(R)＝Bvec(Rs)+σ2vec(IMN) (12)
wherein,
Figure GDA0003352433910000132
A*representing a matrix complex conjugate operation, vec (-) representing a matrix vectorization operation;
step two, carrying out structure optimization design on flexible MIMO radar echo signal model
According to the formula (12), the array flow pattern matrix B of the vector r conforms to the characteristics of a sum-difference joint array, so that the virtual array element expansion condition of the flexible MIMO radar is explored by analyzing the joint array structure of the matrix B; then the array element position set of the transmitting array and the receiving array is used by the formula (1)
Figure GDA0003352433910000133
And
Figure GDA0003352433910000134
the following can be obtained:
the sum and difference joint array set of the flexible MIMO radar SA-FIS is as follows:
Figure GDA0003352433910000135
wherein m is0,n0Is an integer which is the number of the whole,
Figure GDA0003352433910000136
Figure GDA0003352433910000137
Figure GDA0003352433910000138
for further analysis
Figure GDA0003352433910000139
The degree of freedom of the method can obtain the following virtual array element distribution condition of the flexible MIMO radar SA-FIS: defining a set of 'sum-difference joint arrays' for flexible MIMO radar SA-FIS by vectorizing a covariance matrix RCombination of Chinese herbs
Figure GDA00033524339100001310
The flexible MIMO radar SA-FIS has the following characteristics:
(a) the coprime expansion factor satisfies: alpha is more than or equal to 1 and less than or equal to 2N-1, beta is more than or equal to 1 and less than or equal to 2M-1;
(b) collection
Figure GDA0003352433910000141
Has a continuous virtual array element range of [ -c, c]Wherein, c ═ α M + β N- α β -1;
(c)
Figure GDA0003352433910000142
wherein the total number of virtual array elements is 2g +1, g ═ alpha (M-1) + beta (N-1) - (alpha-1) (beta-1)/2;
the spatial smoothing technology can only utilize continuous virtual array elements, and the degree of freedom is c + 1; the CS algorithm can utilize all virtual array elements, and the degree of freedom is 2g + 1;
to select the appropriate spreading factor to obtain more virtual array elements, the total virtual array element number and the continuous virtual array element number are optimized as follows:
(1) total number of virtual array elements g
Determining the optimal distribution structures of the expansion factors alpha and beta and the total physical array element number at the transmitting end and the receiving end by optimizing the parameter g, wherein the optimization objective function is as follows:
Figure GDA0003352433910000143
the results of equation (14) are then shown in Table 1 using the AM-GM inequality:
TABLE 1 results of solving equation (14)
Figure GDA0003352433910000144
(2) Number of consecutive virtual array elements c
Similarly, an objective function related to the number c of the optimized array elements is established as follows:
Figure GDA0003352433910000145
the result of the solution of formula (15) is α ═ 2N-1, β ═ 1, or α ═ 1, β ═ 2M-1;
in this case, the variable c may take the maximum value of cmax2MN-M-N, wherein M and N are the same as in table 1, respectively;
the following conclusions can be drawn according to the optimization results:
(1) when the expansion factor alpha (beta) takes the maximum value, the total number of the virtual array elements is kept unchanged, namely equal to 2 MN-M-N;
(2) combining the optimization results of equations (14) and (15), it can be seen that when the expansion factor α (β) takes the maximum value, the variable g increases with the increase of β (α)maxRemains unchanged and c decreases as it increases.
Referring to fig. 2, a schematic diagram of a flexible MIMO radar with different β values for a flexible MIMO radar hybrid target DOA estimation method based on compressed sensing according to an embodiment of the present invention shows continuous virtual array elements and total virtual array element numbers with different β values, where M is N is 3, and α is 5; as can be seen from fig. 2, the total number of virtual array elements remains constant at all times, with a value equal to 25, while the number of successive array elements decreases with increasing β, with values of 25, 21, 13, respectively.
(3) As can be seen from fig. 2, compared to the relatively prime MIMO radar structure, the SA-FIS does not need the precondition of "relatively prime number of transmit and receive array elements", and thus, the SA-FIS can meet the requirement of any physical array element number. In addition, when cmax＝gmaxIn this case, the transmitting or receiving array is a dense array structure (α is 1 or β is 1), and then a large mutual coupling rate exists, which further affects the angle estimation performance. However, when using the total number of virtual array elements for DOA estimation, the mutual coupling can be suppressed by increasing the array element spacing. Therefore, the CS algorithm is more suitable for SA-FIS from the viewpoint of suppressing mutual coupling. Finally, varying array element spacing (i.e., α ≠ 1 and β ≠ 1) also makes SA-FIS more practical to apply.
(4) To illustrate the virtual aperture extension advantages of SA-FIS over traditional nested and joint array MIMO radar, table 2 gives the number of virtual array elements for the relevant array structure, where M and N are assumed to be relatively prime integers. As can be seen from Table 2, the SA-FIS has higher degree of freedom than other array structures, and can further inhibit the mutual coupling effect of the array elements by increasing the spacing of the array elements.
Table 2 virtual array element number comparison of related array structures
Step three, aiming at the optimized mixed target DOA estimation algorithm of the flexible MIMO radar echo signal model
Performing mixed target DOA estimation by using a complexity-reducing two-step CS algorithm aiming at the MIMO structure optimized in the second step;
according to equation (9), the covariance matrix of the array echo signals can be re-expressed as:
Figure GDA0003352433910000161
wherein,
Figure GDA0003352433910000162
vectorizing the matrix R yields:
Figure GDA0003352433910000163
wherein,
Figure GDA0003352433910000164
Figure GDA0003352433910000165
signal energy for the kth target;
the first term is shown by the formula (17)
Figure GDA0003352433910000166
From the target covariance matrix RsThe diagonal element composition of (2), the second term
Figure GDA0003352433910000167
From RsFor a mixed target consisting of coherent and incoherent targets, the non-zero off-diagonal elements consist of coherent target coherence coefficients; as can be seen from equation (17), the diagonal elements of the target covariance matrix can be directly estimated by using the first term to determine the target angle, and the second term can be directly treated as the redundant term;
and (3) establishing a complexity-reducing two-step CS algorithm model by estimating and correcting redundant items by combining the model analysis, wherein the specific contents are as follows:
(1) for the echo signal vector model in the formula (2), the estimated value of the target signal s (t) obtained by using the least mean square technique is:
Figure GDA0003352433910000171
wherein, assuming matrix A row full rank, A+＝AH(AAH)-1Then the target covariance matrix can be expressed as:
Figure GDA0003352433910000172
according to equation (19), the second term in equation (17) can be expressed as:
Figure GDA0003352433910000173
using the estimate of equation (19), equation (17) can be re-expressed as:
Figure GDA0003352433910000174
wherein the parameter lambda1∈[0,1]；
The new signal model obtained by removing the repeated rows in the data vector r in combination with the total virtual array element arrangement order is:
Figure GDA0003352433910000175
wherein, the vector e is a row vector of 2g +1, the element of the g +1 row is 1, and the other rows are 0;
Figure GDA0003352433910000176
is a (2g +1) xK dimensional matrix corresponding to the position of the virtual array element, B0Is the corresponding (2g + 1). times.K2A dimension matrix;
setting a search vector to θ ═ θjAnd j is 1,2, …, P, then the optimization objective function can be established according to equation (22) as:
Figure GDA0003352433910000177
wherein:
Figure GDA0003352433910000178
Figure GDA0003352433910000179
the noise statistic is known, and eta is a regularization parameter;
the optimization model in equation (23) is solved by using LASSO algorithm, and for the convenience of explanation of step two, the solution is defined as:
Figure GDA0003352433910000181
(2) the second term in equation (23) is calculated from the search vector θ, and since θ has a large angle error, the redundant term error is relatively large; for this purpose, the estimation result θ in equation (26) can be used(1)Revising the second term to further improve the estimation performance;
according to the estimation result theta(1)The new target covariance matrix that can be obtained is:
Figure GDA0003352433910000182
wherein:
Figure GDA0003352433910000183
according to formula (27):
Figure GDA0003352433910000184
Figure GDA0003352433910000185
a new objective optimization function can be constructed according to equation (30) as:
Figure GDA0003352433910000186
wherein the parameter lambda2∈[0,1]；
As can be seen by comparing the expressions (23) and (31), the second term after correction
Figure GDA0003352433910000187
More accurate, hence λ1≤λ2(ii) a The re-estimated angle from equation (31) is:
Figure GDA0003352433910000191
it is to be noted that the parameter λ1,λ2Are constants whose values are influenced by the echo signal model, so that for a given signal model an optimum value, such as λ, can be determined by an exhaustive method1,λ2＝0.1,0.2,…,1。
Example 1: establishing flexible MIMO radar echo signal model
Referring to fig. 3, a schematic diagram of a distribution of virtual array elements of a MIMO radar related to a flexible MIMO radar hybrid target DOA estimation method based on compressed sensing according to an embodiment of the present invention is provided, where it is assumed that the array elements of an SA-FIS transmit array and a receive array are M-4 and N-3. Then, according to theorem 1, alpha is more than or equal to 1 and less than or equal to 5, and beta is more than or equal to 1 and less than or equal to 7. In combination with the related sparse array MIMO radar structures in table 2, fig. 3 gives the virtual array element distribution of each array structure, wherein the flexible co-prime MIMO radar satisfies
Figure GDA0003352433910000192
p is 2, SA-FIS satisfies α is 5, β is 3. As can be seen from FIG. 3, the transmitting or receiving array of the nested subarray MIMO radar is a dense array, and other array structures are all formed by sparse arrays, so that the mutual coupling rate of the nested subarray MIMO radar is the highest. In particular, the SA-FIS is capable of acquiring 35 virtual array elements, of which [ -13,13]The internal virtual array elements are continuous; a traditional co-prime MIMO radar has 29 virtual array elements, and a flexible co-prime MIMO radar has 25 virtual array elements, wherein the two virtual array elements are arranged in [ -11,11]The virtual array elements in the range are continuous; the nested subarray MIMO radar has only 23 continuous virtual array elements. Therefore, more freedom and lower array element mutual coupling can be obtained by optimizing the parameters alpha, beta, SA-FIS, so that the method has better estimation performance.
Example 2 improvement of CS Algorithm operating time
Referring to fig. 4, a schematic diagram of operation time of a conventional CS and two-step CS algorithm of a flexible MIMO radar mixed target DOA estimation method based on compressed sensing provided by an embodiment of the present invention, where an SNR is 10dB, a fast beat number is 200, a search range is [0 °,40 °]The search step is 1 deg., N is 3, and M varies in the range of [2,8 ]]. Let α be 5, β be 3, and the three target directions be θ1＝10°、θ2＝20°、θ3At 30 deg., where the latter two objects are coherent, the corresponding coherence coefficients are 0.9exp (j1.1 pi) and 0.8exp (j0.75 pi). Therefore, as can be seen from fig. 4, the two-step CS algorithm has a lower operation amount than the conventional CS algorithm.
Example 3 mean square error of improved CS Algorithm
Reference is made to fig. 5 and6, first, compare the conventional CS algorithm, l1-estimated performance of SVD and two-step CS algorithms, and CRB provides a lower bound on estimated performance. Assuming that the target position and the coherence coefficient are the same as those in embodiment 2, M is 2, N is 3, α is 5, β is 3, and the search step is 0.05 °. FIG. 5 shows the variation of RMSE with SNR for a fast beat count of 200. FIG. 6 shows the RMSE as a function of fast beat number with a SNR of 0 dB. As can be seen from FIGS. 5-6, the CS algorithm, l, is improved1Both SVD and conventional CS algorithms increase with increasing SNR, number of snapshots. The second step of improving the CS algorithm can improve the estimation performance by correcting the target covariance matrix, so the estimation precision is superior to that of the traditional CS algorithm; however, the estimation performance of the target covariance matrix estimated in the step one is weaker than that of the traditional CS algorithm, and meanwhile, the estimation performance of the improved CS algorithm is better than that of the traditional CS algorithm; l1SVD is the worst performance estimation since it has a low degree of freedom because only the "sum-joint array" of echo signals can be utilized.
Subsequently, the estimated performance between different array structures is compared by using the improved CS algorithm, wherein the CRB of the SA-FIS is used as the lower estimated boundary, and the three target directions are theta1＝5°、θ2＝10°、θ 315 deg., the last two objects are coherent, corresponding coherence coefficients of 0.8exp (j0.9 pi) and 0.65exp (j0.85 pi), and a search step size of 0.05 deg..
Referring to fig. 7 and 8, fig. 7 depicts RMSE as a function of SNR with a fast beat count of 200. FIG. 8 depicts the variation of RMSE with fast beat number, with SNR of 0 dB. As can be seen from fig. 7-8, by using discrete virtual array elements, the flexible co-prime MIMO radar estimates better performance than the nested sub-array MIMO radar; similarly, conventional co-prime MIMO is superior to flexible co-prime architectures due to the presence of more discrete virtual array elements. By optimizing the expansion factors alpha, beta, SA-FIS, more virtual array elements can be obtained, and therefore the estimation performance is better.
Example 4: improved CS algorithm angle resolution
Referring to fig. 9 and 10, first, different algorithms (including the conventional CS algorithm, l) are compared1SVD and two-step CS algorithms) assuming the presence of three neighboring targets, the position is θ1＝20°、θ2＝23°、θ2The last two objects are coherent with corresponding coherence coefficients of 0.9exp (j1.1 pi) and 0.8exp (j0.75 pi), respectively, with the dashed lines representing the true angular direction. Fig. 9 shows the CS space spectrum of two algorithms in SA-FIS configuration with SNR of 20dB, fast beat number of 200, M4, N3, α 5, β 6, search range of [0 °,90 °]The search step is 0.5 °. As can be seen from FIG. 9, the two-step CS algorithm can resolve three nearby targets, but the two-step CS algorithm and l1The SVD method cannot identify intermediate targets.
Subsequently, the angular resolution between different array structures is compared using a modified CS algorithm, where the three adjacent target positions are θ1＝20°、θ2＝24°、θ3The latter two objects are coherent, the coherence factor is the same as in fig. 9, and the dashed line indicates the true angular direction. Fig. 10 shows the CS spatial spectrum under different MIMO configurations, where SNR is 10dB, fast beat number is 200, M is 4, N is 3, α is 5, β is 3, and search range is [0 °,90 °]The search step is 0.5 °. As can be seen from fig. 10, SA-FIS can resolve the three neighboring targets, whereas conventional co-prime MIMO, flexible co-prime MIMO and nested subarray MIMO cannot resolve the second target. Meanwhile, compared with the traditional co-prime MIMO radar, the SA-FIS has higher estimation precision, especially for the first target.
In summary, the present invention mainly solves the following two problems for sparse array MIMO radar structure design and mixed target DOA estimation:
(1) a flexible MIMO radar structure is designed and defined as a flexible array element spacing sparse array, namely SA-FIS. In particular, the SA-FIS can increase the array element spacing of the transmitting and receiving arrays by utilizing two co-prime spreading factors, and theoretical analysis shows that the traditional nested and co-prime MIMO radar is in a special structure. According to the concept of 'sum and difference joint array', the system deduces closed-form solutions of a co-prime expansion factor, continuous virtual array elements and total virtual array elements. The optimized structure proves that the SA-FIS can obtain more virtual array elements under the condition of inhibiting mutual coupling.
(2) A reduced complexity two-step CS algorithm is proposed to fully utilize the total virtual array elements. By modifying and removing off-diagonal elements in the target covariance matrix, the improved CS algorithm can identify only diagonal elements therein. Since the conventional CS algorithm needs to estimate all non-zero elements, the improved algorithm of the present invention can improve estimation performance and reduce complexity by estimating fewer number of elements.
The above disclosure is only one specific embodiment of the present invention, however, the present invention is not limited thereto, and any variations that can be made by those skilled in the art are intended to fall within the scope of the present invention.

Claims (4)

1. A flexible MIMO radar mixed target DOA estimation method based on compressed sensing is characterized by comprising the following steps:
step one, establishing a flexible MIMO radar echo signal model: firstly, performing matched filtering on a transmitting array and a receiving array of the flexible MIMO radar to obtain an array echo signal vector model, and then obtaining a covariance matrix R and a vectorization covariance matrix R of the array echo signal according to the array echo signal vector model x (t);
step two, carrying out structure optimization on the flexible MIMO radar echo signal model: because the array flow pattern matrix B in the vectorization covariance matrix r conforms to the characteristics of a sum-difference combined array, a proper expansion factor is selected by analyzing the combined array structure of the array flow pattern matrix B, so that more virtual array elements of the flexible MIMO radar are obtained;
and thirdly, aiming at the flexible MIMO radar echo signal model after structure optimization, performing mixed target DOA estimation by adopting a complexity-reduction two-step CS algorithm: and establishing a complexity-reducing two-step CS algorithm model by estimating and correcting redundant items, and removing repeated rows in the vectorization covariance matrix r by combining the total virtual array element arrangement sequence to obtain a new signal model.
2. The method of claim 1, wherein the step one comprises the following specific steps:
the transmitting array and the receiving array of the flexible MIMO radar are composed of sparse uniform linear arrays, so that the total physical array element number is T which is M + N;
the transmitting array is provided with M array elements, and the distance between the array elements is alpha d; the receiving array is provided with N array elements, and the spacing between the array elements is beta d; alpha and beta are coprime expansion factors, d is unit array element distance and is set as lambda/2, and lambda is signal wavelength;
array element position set of transmitting array and receiving array
Figure FDA0003352433900000011
And
Figure FDA0003352433900000012
comprises the following steps:
Figure FDA0003352433900000021
wherein m and n are integers;
k far-field incoherent and coherent mixed targets are arranged, and the information source direction set is theta ═ thetakK is 1,2, …, K, where the number of incoherent objects and the number of coherent objects are K, respectivelyuAnd KcI.e. K ═ Ku+Kc(ii) a Suppose KcEach target satisfies a full coherence condition; then the vector model of the array echo signal after matched filtering is:
Figure FDA0003352433900000022
wherein:
Figure FDA0003352433900000023
wherein,
Figure FDA0003352433900000028
is the reflection coefficient of the kth target when the fast beat number is t;
Figure FDA0003352433900000029
is the k-th0Attenuation coefficient of an object, for convenience of description, it is assumed
Figure FDA0003352433900000027
[·]TFor matrix transpose operations, diag (-) is a diagonal operation,
Figure FDA0003352433900000024
and
Figure FDA0003352433900000025
respectively representing Khatri-Rao product and Kronecker product;
n (t) is an independent and equally distributed additive white Gaussian noise vector satisfying CN (0, sigma)2)；
And is
Figure FDA0003352433900000026
At＝[at(θ1),at(θ2),…,at(θK)] (5)
Ar＝[ar(θ1),at(θ2),…,ar(θK)] (6)
Figure FDA0003352433900000031
at(θk) And ar(θk) Direction vectors of kth targets of the transmitting array and the receiving array respectively are as follows:
Figure FDA0003352433900000032
Figure FDA0003352433900000033
the covariance matrix of the array echo signal obtained from the echo signal model of equation (2) is:
R＝E[x(t)xH(t)]＝ARsAH+σ2IMN (9)
wherein:
Rs＝E[s(t)sH(t)] (10)
is a target covariance matrix, AHFor the complex conjugate transpose operation of matrix A, IMNThe matrix is MN multiplied by MN dimension unit matrix;
when the fast beat number is L, t is 1, …, L, its sample covariance matrix is approximated as:
Figure FDA0003352433900000034
vectorized covariance matrix R yields:
r＝vec(R)＝Bvec(Rs)+σ2vec(IMN) (12)
wherein,
Figure FDA0003352433900000035
A*representing a matrix complex conjugate operation, vec (-) representing a matrix vectorization operation.
3. The method as claimed in claim 2, wherein the specific steps of the second step are as follows: array element position collection using transmit and receive arrays of formula (1)
Figure FDA0003352433900000036
And
Figure FDA0003352433900000037
obtaining:
the sum and difference joint array set of the flexible MIMO radar SA-FIS is as follows:
Figure FDA0003352433900000038
wherein m is0,n0Is an integer which is the number of the whole,
Figure FDA0003352433900000039
Figure FDA0003352433900000041
Figure FDA0003352433900000042
for further analysis
Figure FDA0003352433900000043
The degree of freedom of the method can obtain the following virtual array element distribution condition of the flexible MIMO radar SA-FIS: defining a set of 'sum and difference joint arrays' for flexible MIMO radar SA-FIS by vectorizing a covariance matrix R
Figure FDA0003352433900000044
The flexible MIMO radar SA-FIS has the following characteristics:
(a) the coprime expansion factor satisfies: alpha is more than or equal to 1 and less than or equal to 2N-1, beta is more than or equal to 1 and less than or equal to 2M-1;
(b) collection
Figure FDA0003352433900000047
Has a continuous virtual array element range of [ -c, c]Wherein, c ═ α M + β N- α β -1;
(c)
Figure FDA0003352433900000048
wherein the total number of virtual array elements is 2g +1, g ═ alpha (M-1) + beta (N-1) - (alpha-1) (beta-1)/2;
the spatial smoothing technology can only utilize continuous virtual array elements, and the degree of freedom is c + 1; the CS algorithm can utilize all virtual array elements, and the degree of freedom is 2g + 1; to select the appropriate spreading factor to obtain more virtual array elements, the total virtual array element number and the continuous virtual array element number are optimized as follows:
(1) total number of virtual array elements g
Determining the optimal distribution structures of the expansion factors alpha and beta and the total physical array element number at the transmitting end and the receiving end by optimizing the parameter g, wherein the optimization objective function is as follows:
Figure FDA0003352433900000045
(2) number of consecutive virtual array elements c
Similarly, an objective function related to the number c of the optimized array elements is established as follows:
Figure FDA0003352433900000046
4. the method according to claim 3, wherein the specific steps of the third step are as follows: according to equation (9), the covariance matrix of the array echo signals can be re-expressed as:
Figure FDA0003352433900000051
wherein,
Figure FDA0003352433900000052
vectorizing the matrix R to obtain:
Figure FDA0003352433900000053
wherein,
Figure FDA0003352433900000054
Figure FDA0003352433900000055
signal energy for the kth target;
the first term is shown by the formula (17)
Figure FDA0003352433900000056
From the target covariance matrix RsThe diagonal element composition of (2), the second term
Figure FDA0003352433900000057
From RsFor a mixed target consisting of coherent and incoherent targets, the non-zero off-diagonal elements consist of coherent target coherence coefficients; as can be seen from equation (17), the diagonal elements of the target covariance matrix can be directly estimated by using the first term to determine the target angle, and the second term can be directly treated as the redundant term;
and (3) establishing a complexity-reducing two-step CS algorithm model by estimating and correcting redundant items by combining the model analysis, wherein the specific contents are as follows:
(1) for the echo signal vector model in the formula (2), the estimated value of the target signal s (t) obtained by adopting the least mean square technique is as follows:
Figure FDA0003352433900000058
wherein, assuming matrix A row full rank, A+＝AH(AAH)-1Then the target covariance matrix can be expressed as:
Figure FDA0003352433900000059
according to equation (19), the second term in equation (17) is represented as:
Figure FDA0003352433900000061
using the estimated value of equation (19), equation (17) is re-expressed as:
Figure FDA0003352433900000062
wherein the parameter lambda1∈[0,1]；
The new signal model obtained by removing the repeated rows in the data vector r in combination with the total virtual array element arrangement order is:
Figure FDA0003352433900000063
wherein, the vector e is a row vector of 2g +1, the element of the g +1 row is 1, and the other rows are 0;
Figure FDA0003352433900000064
is a (2g +1) xK dimensional matrix corresponding to the position of the virtual array element, B0Is the corresponding (2g + 1). times.K2A dimension matrix;
setting a search vector to θ ═ θjAnd j is 1,2, …, P, then the optimization objective function can be established according to equation (22) as:
Figure FDA0003352433900000065
wherein:
Figure FDA0003352433900000066
Figure FDA0003352433900000067
the noise statistic is known, and eta is a regularization parameter;
the optimization model in equation (23) is solved by using LASSO algorithm, and for the convenience of explanation of step two, the solution is defined as:
Figure FDA0003352433900000068
(2) the second term in the formula (23) is calculated according to the search vector theta, and because theta has a large angle error, the redundant term error is relatively large; for this purpose, the estimation result θ in equation (26) can be used(1)Revising the second term to further improve the estimation performance;
according to the estimation result theta(1)The new target covariance matrix that can be obtained is:
Figure FDA0003352433900000071
wherein:
Figure FDA0003352433900000072
according to formula (27):
Figure FDA0003352433900000073
Figure FDA0003352433900000074
a new objective optimization function can be constructed according to equation (30) as:
Figure FDA0003352433900000075
wherein the parameter lambda2∈[0,1]；
As can be seen by comparing the expressions (23) and (31), the second term after correction
Figure FDA0003352433900000076
More accurate, hence λ1≤λ2Therefore, the algorithm has lower operation amount and higher estimation performance; meanwhile, the angle can be estimated again from equation (31):
Figure FDA0003352433900000077
it is to be noted that the parameter λ1,λ2Are constants whose values are influenced by the echo signal model, so that for a given signal model the optimum value can be determined by an exhaustive method.
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