CN111505613B - MIMO radar transmitting antenna arrangement method based on virtual antenna Kuhn-Munkres algorithm - Google Patents

MIMO radar transmitting antenna arrangement method based on virtual antenna Kuhn-Munkres algorithm Download PDF

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CN111505613B
CN111505613B CN202010297463.4A CN202010297463A CN111505613B CN 111505613 B CN111505613 B CN 111505613B CN 202010297463 A CN202010297463 A CN 202010297463A CN 111505613 B CN111505613 B CN 111505613B
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孙丽
何茜
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a MIMO radar transmitting antenna arrangement method based on a virtual antenna Kuhn-Munkres algorithm, and belongs to the field of signal processing. The MIMO radar transmitting antenna arrangement method takes the output signal-to-noise ratio of the detector as the detection performance evaluation index under the Neyman-Pearson criterion. In this method, in conjunction with the matching theory in the bipartite graph, we consider the transmit antennas and the optional grid points as vertices in the bipartite graph, and consider the performance impact produced by the antenna placement as edges with weights in the graph. Since the number of transmit antennas and grid points is different, the original antenna placement problem is translated into an MWM problem in an asymmetric bipartite graph. Next, a symmetric bipartite graph is constructed by adding virtual antennas, so that the Kuhn-Munkres algorithm can be applied to select the optimal position of the transmitting antenna. We call this virtual antenna based KM algorithm the VKM method. With our proposed VKM method, the optimal position for placing the transmit antennas can be chosen within the allowed region and the complexity is low.

Description

MIMO radar transmitting antenna arrangement method based on virtual antenna Kuhn-Munkres algorithm
Technical Field
The invention belongs to the field of signal processing, relates to the problem of transmitting antenna arrangement in the technical field of MIMO radar target detection, and is suitable for MIMO radar transmitting antenna layout design.
Background
The concept of MIMO (Multiple Input Multiple output) radar is originally derived from the Multiple Input Multiple output technology in communication, and is divided into two main categories, i.e., co-located MIMO radar and split MIMO radar. For the MIMO radar system with the split antennas, targets can be simultaneously detected from various angles, so that the gain of space diversity is realized, and the MIMO radar system has higher target detection performance. The position selection of the antenna has an important influence on the performance of the radar system, and under the condition that the electromagnetic environment is increasingly complex, how to perform antenna layout to better improve the performance of the radar system becomes a content which is widely concerned by scholars at home and abroad.
According to the radar signal processing theory, the more the number of antennas, the better the performance of the system. For the receiving end, the more the number of antennas at the transmitting end is, the more matched filtered signals the receiving end needs to process, and the higher the computational complexity is. Therefore, in practical applications, how to perform layout design using limited transmit antenna resources becomes an important research issue in view of saving system software and hardware resources and ensuring system performance. At present, there are many researches on antenna arrangement of a MIMO radar system, and for such optimization problems, an exhaustive search is generally adopted to obtain an optimal antenna position design scheme. However, the exhaustive search method is computationally expensive and is not suitable for systems that require computational cost. Therefore, some suboptimal methods with fast computation, such as greedy algorithm, are designed, which is a commonly used solution in the optimization problem. Although the greedy algorithm performs well in terms of computation speed, it does not guarantee a globally optimal solution.
In consideration of the intuitive and simple modeling by using Graph Theory (Graph Theory), and the related algorithm has the advantages of low computational complexity, capability of obtaining the optimal solution of the problem and the like, the method is commonly used in the communication field to solve the problem of resource allocation. In the problem of communication resource allocation, a Matching (Matching) algorithm in a Weighted bipartite graph (bipartite graph) is commonly used, and the algorithm mainly includes Maximum-Weighted Matching (MWM) and smooth Matching. For MWM problems, the classical algorithm is Kuhn-Munkres (KM) algorithm (see J. Munkres, "Algorithms for the alignment and transfer schemes", Journal of the Society of Industrial and Applied Mathesics, vol.5, No.1, pp.32-38, March 1957), which can obtain a maximum weighted match in polynomial time, i.e., the optimal solution to the problem sought. In a radar system, a transmitting antenna is also a system resource, and how to allocate different antennas to different positions to achieve the optimal system performance is also a problem of resource allocation. Thus. For the transmit antenna placement problem, we can model it as a MWM problem and then solve the problem in conjunction with the KM algorithm.
Disclosure of Invention
The invention provides a method for arranging MIMO radar transmitting antennas by taking the output signal-to-noise ratio of a detector as a detection performance evaluation index under the Neyman-Pearson criterion in combination with the knowledge of graph theory. In this method, in conjunction with the matching theory in the bipartite graph, we consider the transmit antennas and the optional grid points as vertices in the bipartite graph, and consider the performance impact produced by the antenna placement as edges with weights in the graph. Because the number of transmit antennas and grid points is different, the original antenna placement problem is translated into the MWM problem in an asymmetric bipartite graph. Next, a symmetric bipartite graph is constructed by adding virtual antennas, and the Kuhn-munkres (km) algorithm can be applied to select the optimal position of the transmitting antenna. The virtual-antenna-based KM algorithm (virtual-antenna based KM) is called VKM method. With our proposed VKM method, the optimal position for placing the transmit antennas can be chosen within the allowed area and the complexity is low.
The technical scheme of the invention is that the MIMO radar transmitting antenna arrangement method based on the virtual antenna Kuhn-Munkres algorithm comprises the following steps:
step 1: setting N receiving antennas to be fixed in position, setting the number of transmitting antennas to be M, and dividing an area for placing the transmitting antennas into Q (Q is more than M) grid points which can be uniformly divided or randomly divided;
step 2: establishing a MIMO radar receiving signal model, and defining a binary variable amqE {0,1}, M1, M, Q1, Q, if the mth transmitting antenna is placed at the qth grid point, then amq1, otherwise amq0; let the time domain sampling number be K and the sampling interval be TsWriting all observed values of the N receiving antennas into a vector r;
r＝[r1[1],...,r1[K],...,rN[K]]T
wherein the content of the first and second substances,
Figure BDA0002452725430000021
w hereinn[k]Denotes the nth receiving antenna is at kTsThe noise at time is assumed to be white, zero mean complex Gaussian random noise in time and space, and the variance is
Figure BDA0002452725430000022
Conform to
Figure BDA0002452725430000023
(ii) a gaussian distribution of; zetanqAnd τnqRespectively representing the reflection coefficient and the time delay between the nth receiving antenna and the qth selectable grid point, the reflection coefficient obeying
Figure BDA0002452725430000024
A Gaussian distribution of (1), wherein
Figure BDA0002452725430000025
Represents a correspondence ζnqThe variance of (a); sm(t) is a transmission signal waveform of the mth transmission antenna, EmRepresenting the emission energy, dR,nAnd dT,qRespectively representing the distance between the nth receiving antenna and the qth grid point to the target; for subsequent analysis, let
Figure BDA0002452725430000031
Satisfy the requirement of
Figure BDA0002452725430000032
The distribution of the gaussian component of (a) is,
Figure BDA0002452725430000033
the representation corresponds to hnmA variance of (a), and
Figure BDA0002452725430000034
and step 3: a binary hypothesis testing problem is constructed,
Figure BDA0002452725430000035
indicating the presence of the object(s),
Figure BDA0002452725430000036
indicating that the target is not present; calculating a log-likelihood ratio to obtain an optimal detector;
Figure BDA0002452725430000037
Figure BDA0002452725430000038
covariance matrix, operator, representing noise
Figure BDA0002452725430000039
Representing a mathematical expectation. Wherein the noise vector is defined as w ═ w1[1],...,w1[K],...,wN[K]]TElement w thereofn[k]Is corresponding to an observation vector r ═ r1[1],...,r1[K],...,rN[K]]TObserved value r in (1)n[k]The noise of (2);
Figure BDA00024527254300000310
representing a covariance matrix of the received signal in the presence of the target; the final expression of T is calculated as
Figure BDA00024527254300000311
Wherein the content of the first and second substances,
Figure BDA00024527254300000312
denotes the matched filtered output signal at the nth receiving antenna, corresponding to the mth transmitting antenna placed at grid point q, the symbol (·)*Represents a conjugate operation;
and 4, step 4: calculating the output signal-to-noise ratio eta of the detector
Figure BDA00024527254300000313
Here, to simplify the expression, let
Figure BDA00024527254300000314
And 5: establishing an optimization problem based on the output signal-to-noise ratio of the detector
Figure BDA00024527254300000315
Figure BDA00024527254300000316
Figure BDA00024527254300000317
amq∈{0,1}
The optimization problem is a binary integer programming problem;
step 6: according to the knowledge of graph theory, a bipartite graph G ═ V is constructed1,V2E) to represent the relationship of the M transmit antennas to the Q selectable grid points; wherein, the vertex corresponding to the transmitting antenna uses AmM is 1, M denotes a group V1＝{A1,...,AMG is used as the vertex corresponding to the optional gridqQ is 12＝{G1,...,GQ}; e represents a connection V1And V2The edge set of the middle vertex defines the weight of the edge between the mth transmitting antenna and the qth grid point as
Figure BDA0002452725430000041
Constructing a weight matrix W by using the weights, wherein W is in dimension of M multiplied by Q;
and 7: by adding Q-M virtual antennas, set V1Extension to V1'＝{A1,...,AM,AM+1,...,AQIn which { A }M+1,...,AQDenotes the added virtual antenna vertex; for V1The edge connected with the virtual vertex in the middle is set to have a weight value of 0; then it is expanded into dimensions by adding Q-M all 0 rows in WA matrix W' with degree Q × Q;
and 8: applying a KM algorithm to the weight matrix W' obtained in the step 7 to obtain a group a with the maximum weight summqAnd (4) taking values, and neglecting the added virtual antenna vertexes to obtain corresponding results between the M transmitting antennas and the Q grid points.
The method provided by the invention can make an optimal transmitting antenna arrangement scheme under the condition of saving system resources, and ensure the target detection performance of the system. Compared with the conventional exhaustive search method, the computational complexity is greatly reduced and is O (Q)3) The algorithm runs fast. Therefore, this method is an effective and fast method of solving the antenna arrangement problem.
Drawings
Figure 1 shows the process of constructing a weighted bipartite graph from a research problem.
Fig. 2 is a schematic diagram of the position distribution of candidate grid points in this experiment.
Fig. 3 is an ROC curve diagram under the condition that the transmission energy of each transmission antenna is different but the target reflection coefficient variance of each path is the same, and includes an optimal antenna position scheme obtained by exhaustive search, an antenna position scheme obtained by the VKM method provided by the present invention, a scheme obtained based on a greedy algorithm, a random scheme, and an ROC curve of a worst scheme.
Fig. 4 is an ROC curve under the condition that the transmitting energy of each transmitting antenna is different and the variance of the target reflection coefficient of each path is different, including an optimal antenna position scheme selected by exhaustive search, an antenna position scheme selected by the VKM method provided by the present invention, a scheme selected based on a greedy algorithm, a random scheme, and a worst scheme.
Table 1 lists the actual computation execution times of the VKM method and the exhaustive search method proposed by the present invention, and the ratio of their actual times to the theoretical complexity.
Detailed Description
For convenience of description, the following definitions are first made:
bold lowercase letters represent vectors, bold uppercase letters represent momentsBattle array (.)*For conjugation, (. cndot)TIs a transposition ofHFor conjugate transfer, Diag {. denotes a block diagonal matrix, Diag {. denotes a diagonal matrix, IKFor a unit matrix of order K, det (-) represents the determinant of the matrix.
Figure BDA0002452725430000051
Representing a mathematical expectation, Var {. is } representing a variance.
Consider a MIMO radar system with N receive antennas, the location of which is known as (x)R,n,yR,n) N1., N, M transmit antennas have Q selectable grid points, located at (x)T,q,yT,q) Q1.., Q, in which the receiving antenna and the transmitting antenna are each separated by a large interval. The m-th transmitting antenna transmits signals of
Figure BDA0002452725430000052
Normalized waveform of
Figure BDA0002452725430000053
Representing the emission energy, TsDenotes the sampling interval, with the sampling instants K, K being 1. Assuming a possible point target is located at (x, y), the time delay from the qth selectable grid point location to the nth receive antenna via the target reflection is expressed as
Figure BDA0002452725430000054
Wherein d isT,qAnd dR,nRespectively representing the location of the q-th grid point and the distance from the nth receiving antenna to the target, and c represents the speed of light. For the convenience of explanation, a binary selection variable a is introducedmq
Figure BDA0002452725430000055
The binary variable satisfies
Figure BDA0002452725430000056
Because one grid point is allocated to at most one transmitting antenna and satisfies
Figure BDA0002452725430000057
Representing a transmit antenna, a grid must be selected as its location.
The nth receiving antenna is at kTsThe received signal of the moment is
Figure BDA0002452725430000058
Wherein
Figure BDA0002452725430000059
ζnqRepresenting the reflection coefficient between the receiving antenna n and the transmitting antenna placed at the q-th grid point, which is assumed to be a zero-mean complex Gaussian random variable satisfying
Figure BDA0002452725430000061
Accordingly, the method can be used for solving the problems that,
Figure BDA0002452725430000062
and is
Figure BDA0002452725430000063
In the formula (5), wn[k]The assumption is that a time and space white, zero mean complex gaussian random noise,
Figure BDA0002452725430000064
defining an observation vector at the nth receive antenna as rn＝[rn[1],rn[2],...,rn[K]]TThe observation vectors of all N receiving antennas can be expressed as
r＝[r1 T,r2 T,...,rN T]T＝Sh+w (7)
Wherein S ═ Diag { S ═ S1,S2,...,SNIs a matrix that collects all the transmitted signals,
Sn＝[sn[1],sn[2],...,sn[K]]T (8)
Figure BDA0002452725430000065
in equation (7), vector
Figure BDA0002452725430000066
hn＝[hn1,hn2,...,hnM]TOf a noise vector
Figure BDA0002452725430000067
wn＝[wn[1],wn[2],...,wn[K]]T. From the foregoing assumptions, we can know that h is a zero-mean complex Gaussian vector with a covariance matrix of
Figure BDA0002452725430000068
The covariance of the noise vector w is
Figure BDA0002452725430000069
To simplify the following analysis, we assume that the signals originating from the different transmit antennas m and m' are approximately orthogonal and for different time delays τ of interestm、τm'Maintaining near orthogonality
Figure BDA00024527254300000610
According to equation (7), target detection can be given by the following binary hypothesis testing problem
Figure BDA00024527254300000611
The optimum detector is
Figure BDA00024527254300000612
Threshold gamma false alarm probability PFAIf so, the log likelihood ratio expression is as follows
Figure BDA00024527254300000613
R in the above formula is represented by
Figure BDA00024527254300000614
Under the assumption, the covariance matrix of the received signal
Figure BDA00024527254300000615
Limiting the false alarm probability P according to the Neyman-Pearson criterionFAUnder the condition of ≦ alpha, if L (r) is greater than the threshold value gamma, then
Figure BDA00024527254300000616
And if so, otherwise,
Figure BDA00024527254300000617
this is true. Next, the compound represented by the formula (13) can be obtained
Figure BDA0002452725430000071
Where T represents the detection statistic, when the threshold becomes γ' ═ γ -ln [ det (C)w)]+ln[det(R)]. Next, a specific expression of the detection statistic T is calculated. According to the matrix inversion theorem, there are
Figure BDA0002452725430000072
And due to CwIs a diagonal matrix so that the detection statistic can be rewritten as
Figure BDA0002452725430000073
Based on the assumption of orthogonal signals in (8) and (10), there are
Figure BDA0002452725430000074
Substituting (18) into (17) to obtain
Figure BDA0002452725430000075
Wherein
Figure BDA0002452725430000076
Representing the matched filtered output signal at the receive antenna n corresponding to the mth transmit antenna placed at the qth grid point. Our goal is to determine the selection variable amqTo maximize the detection probability PD. To calculate PDIt is necessary to calculate T separately
Figure BDA0002452725430000077
And
Figure BDA0002452725430000078
probability density function (pdf) below. Analysis (19) shows that T obeys the generalized chi-square distribution with the degree of freedom of 2NM, the pdf calculation is more complex, and P isDIs also subject to a false alarm probability PFAAnd the influence of the threshold value gamma, and directly solving for PDIt is cumbersome and therefore an angle consideration solution is needed.
The representation η of the signal-to-noise ratio of the detector output is defined below
Figure BDA0002452725430000079
It can approximately characterize the detection performance. Calculate η, first calculate ynmqIs expected to
Figure BDA00024527254300000710
And ynmqVariance of (2)
Figure BDA0002452725430000081
Figure BDA0002452725430000082
Then substituting (20) the expectation and variance in (21) and (22) to calculate
Figure BDA0002452725430000083
Here, substitution is made for the sake of simplifying the expression of (23)
Figure BDA0002452725430000084
Order to
Figure BDA0002452725430000085
After the above operations, we get the optimization problem
Figure BDA0002452725430000086
Note that P1 is a binary integer programming problem, belonging to the NP-hard problem,typically with an exponential computational complexity. To obtain an optimal solution to the problem P1, Q (Q-1) × … × (Q-M +1) antenna combinations with selectable grid points must be searched exhaustively with a complexity of O (Q!/(Q-M)!). When M and Q become large, the computational complexity can be large. From our current knowledge, there are many classically valid algorithms in graph theory that can be used to solve similar allocation problems. Therefore, the method for converting P1 into the Maximum Weighted Matching (MWM) problem in a two-part graph is considered, and the Kuhn-Munkres (KM) algorithm is combined to obtain the optimal solution within polynomial time, wherein the complexity is O (Q)3)。
According to the bipartite graph definition and the problem we are to solve, construct a complete bipartite graph G (V)1,V2E) to represent the relationship between the transmit antennas and the optional grid point locations, as shown in fig. 1 (a). Wherein, set V1＝{A1,...,AMPoints in denotes the transmit antenna, base | V1M, set V2＝{G1,...,GQPoints in denotes optional grid points, base | V2|＝Q；
Set E { (a)m,Gq) 1,. Q,. M, Q, contains a connection V1And V2The edges of the middle vertex. For the edge between the m-th transmitting antenna and the q-th grid point position, the weight is set as
Figure BDA0002452725430000091
The weight matrix formed by the weights is
Figure BDA0002452725430000092
Figure BDA0002452725430000093
Finally, the MWM problem was found to be as follows
Figure BDA0002452725430000094
We can find that the problem P2 is equivalent to P1, which can be optimally solved using the Kuhn-munkres (km) algorithm. Since the classical KM algorithm is applied to a symmetric bipartite graph, i.e. the cardinality of two vertex subsets is equal, we propose a virtual-antenna-based KM (VKM) method below.
By adding Q-M virtual antennas, the antenna vertexes are collected into a V1Extension to V1'＝{A1,...,AM,AM+1,...,AQIs made | V1|＝|V2And setting the edge weight value corresponding to the virtual vertex as 0. As shown in fig. 1(b), the virtual vertices and virtual edges are indicated by dashed lines. The weights of all edges including the virtual top points are put together to construct a new weight matrix
Figure BDA0002452725430000095
Figure BDA0002452725430000096
The KM algorithm can then be applied to W', and the optimal solution to problem P3 can be obtained.
The specific steps for the VKM method are as follows:
initialization: (1) calculating the weight w according to (24) and (26)mqM1,., M, Q1,., Q, constructing a weight matrix in (27); (2) adding Q-M virtual antennas, and adding V1Extension to V1', then add Q-M all-zero rows to W, constructing a new weight matrix W', as shown in (29).
Step 1: for each row in W', subtracting the largest element of the row from each element thereof; for each column in W', the largest element of that column is subtracted from each of its elements.
Step 2: considering a certain 0 element in the matrix, if the row and column where the 0 is located do not have an asterisked 0 (i.e. 0 ″), then the 0 element is asterisked as 0 ″; the above process of adding an asterisk is repeated for each 0 element in the matrix.
And step 3: covering each column with 0, if all columns of the matrix are covered, then the element in the original matrix W' corresponding to each 0 position is the selected matching result, and the algorithm is ended; otherwise, go to step 4.
And 4, step 4: selecting a 0 not covered by the scribe line, and annotating the 0' with the 0', if the 0' is not located in the row, turning to the step 5; if there is 0, the row with 0 is overwritten and the column with 0 is overwritten. The above process is repeated until all 0's are covered, and go to step 6.
And 5: a sequence of alternating 0 x and 0' was constructed as follows: by Z0Denotes 0' uncovered with Z1Represents Z00 of the column (if any), by Z2Represents Z10' of the row. This process is repeated until the column in which a certain 0 'is located contains no 0, the alternating sequence ending at that 0'; in the resulting alternating sequence, 0's asterisk was removed and 0's asterisk was changed to 0. All 0's in the matrix are erased and all row overlays go to step 3.
Step 6: the largest uncovered element in the matrix is denoted by h, which is added to each covered row and then subtracted from each covered column. Go to step 4.
And after the algorithm is finished, outputting the result, namely the optimal matching. And neglecting the virtual antenna vertex added in the matching result to obtain the optimal position of each transmitting antenna.
Two simulation examples and comparative analysis of algorithm complexity are given for the transmit antenna placement problem based on the virtual antenna Kuhn-munkres (vkm) method.
The simulation parameters are set as follows: assume a transmit waveform of
Figure BDA0002452725430000101
0＜kTs＜T，T＝KTsRepresents the total observation time, Ts0.25 mus, K4000 and
Figure BDA0002452725430000102
indicating the frequency gain between the m and m +1 th transmit antennas. In a two-dimensional cartesian coordinate system, assuming that a target position to be detected is an origin (x, y) ═ 0,0 km, two receiver positions are (x)R,1,yR,1) (-1,0) km and (x)R,2,yR,2)＝(1,0)km。
In simulation 1, assuming that the transmission energy of each transmission antenna is different, the reflection coefficient variance is the same for different n and q, and is set as
Figure BDA0002452725430000103
Assuming that there are 6 transmit antennas to be located, the transmit energy is set to [ E [ [ E ]1,E2,E3,E4,E5,E6]＝ 1012×[1,2,5,8,6,4]There are 12 optional grid points, randomly distributed in the allowed area, and the position distribution diagram is shown in fig. 2. Fig. 3 shows ROC curves corresponding to antenna positions obtained under different methods, where a diamond identifier represents the optimal selection obtained by exhaustive search, a square identifier is the selection of the VKM method proposed in this patent, a circle identifier is the selection based on the greedy algorithm, a triangle identifier is the random selection, and a plus sign identifier represents the worst selection obtained by exhaustive search. It can be seen that the optimal performance same as that of exhaustive search can be obtained by applying the VKM method, and the method is low in computation complexity and high in computation speed. The performance obtained by selection based on the greedy algorithm is suboptimal, the performance of random selection is lower than that of the greedy algorithm, and the performance of worst selection is worst.
In simulation 2, it is assumed that the transmission energy of each transmission antenna is different and the reflection coefficient variance is also different for different n and q, and the reflection coefficient variance is randomly generated in a gaussian distribution. Assuming that there are 6 transmitting antennas to be located, the transmitting energy is [ E ]1,E2,E3,E4,E5,E6]＝1012×[2,4,6,10,9,5]The position distribution of the 12 optional grid points is also shown in fig. 2. Fig. 4 shows ROC curves corresponding to antenna positions obtained by different methods. It can be seen that the optimum performance as the exhaustive search can still be obtained using the VKM method, and the complex is calculatedLow complexity and high operation speed. The performance obtained by selection based on the greedy algorithm is suboptimal, the performance of random selection is lower than that of the greedy algorithm, and the performance of worst selection is worst.
Regarding the analysis of algorithm complexity, the complexity ratio of exhaustive search to VKM method is defined as ε ═ Q! L ((Q-M)! Q3) Epsilon is used to illustrate the theoretical complexity difference between the two methods. In addition, the ratio of the actual execution time of the exhaustive search and the VKM method at the local computer is defined as
Figure BDA0002452725430000113
The actual computation times and their ratios for the exhaustive search and VKM methods are listed in Table 1, respectively
Figure BDA0002452725430000112
Compared with the theoretical complexity ratio epsilon, the change trend of the theoretical complexity and the actual execution time is basically consistent, and the complexity of the VKM method is far lower than that of exhaustive search along with the increase of Q and M. By contrast, the VKM method proposed in the present invention for transmit antenna placement is a better performing algorithm.
TABLE 1
Figure BDA0002452725430000111

Claims (1)

1. A MIMO radar transmitting antenna arrangement method based on a virtual antenna Kuhn-Munkres algorithm comprises the following steps:
step 1: setting N receiving antennas to be fixed in position, setting the number of transmitting antennas to be M, and dividing an area for placing the transmitting antennas into Q grid points, wherein Q is more than M, and the Q grid points can be uniformly divided or randomly divided;
step 2: establishing a MIMO radar receiving signal model, and defining a binary variable amqE {0,1}, M1, M, Q1, Q, if the mth transmitting antenna is placed at the qth grid point, then amq1, otherwise amq0; setting the time domain sampling number as K, samplingSample interval of TsWriting all observed values of the N receiving antennas into a vector r;
r＝[r1[1],...,r1[K],...,rN[K]]T
wherein the content of the first and second substances,
Figure FDA0003503807840000011
w hereinn[k]Denotes the nth receiving antenna is at kTsThe noise at a moment is assumed to be white, zero mean complex Gaussian random noise in time and space, and the variance is
Figure FDA0003503807840000012
Meet with
Figure FDA0003503807840000013
(ii) a gaussian distribution of; zetanqAnd τnqRespectively representing the reflection coefficient and the time delay between the nth receiving antenna and the qth selectable grid point, the reflection coefficient obeying
Figure FDA0003503807840000014
A Gaussian distribution of wherein
Figure FDA0003503807840000015
Represents a correspondence ζnqThe variance of (a); sm(t) is a transmission signal waveform of the mth transmission antenna, EmRepresenting the emission energy, dR,nAnd dT,qRespectively representing the distance between the nth receiving antenna and the qth grid point to the target; for subsequent analysis, let
Figure FDA0003503807840000016
Satisfy the requirement of
Figure FDA0003503807840000017
The distribution of the gaussian component of (a) is,
Figure FDA0003503807840000018
the representation corresponds to hnmA variance of (a), and
Figure FDA0003503807840000019
and step 3: a binary hypothesis testing problem is constructed,
Figure FDA00035038078400000110
indicating the presence of the object(s),
Figure FDA00035038078400000111
indicating that the target is not present; calculating a log-likelihood ratio to obtain an optimal detector;
Figure FDA00035038078400000112
Figure FDA00035038078400000113
covariance matrix, operator, representing noise
Figure FDA00035038078400000114
Expressing the mathematical expectation; wherein the noise vector is defined as w ═ w1[1],...,w1[K],...,wN[K]]TElement w thereofn[k]Is corresponding to an observation vector r ═ r1[1],...,r1[K],...,rN[K]]TObserved value r in (1)n[k]The noise of (2);
Figure FDA00035038078400000115
representing a covariance matrix of the received signal in the presence of the target; the final expression of T is calculated as
Figure FDA0003503807840000021
Wherein the content of the first and second substances,
Figure FDA0003503807840000022
denotes the matched filtered output signal at the nth receiving antenna, corresponding to the mth transmitting antenna placed at grid point q, the symbol (·)*Represents a conjugate operation;
and 4, step 4: calculating the output signal-to-noise ratio eta of the detector
Figure FDA0003503807840000023
To simplify the expression here, let
Figure FDA0003503807840000024
And 5: establishing an optimization problem based on the output signal-to-noise ratio of the detector
Figure FDA0003503807840000025
Figure FDA0003503807840000026
Figure FDA0003503807840000027
amq∈{0,1}
The optimization problem is a binary integer programming problem;
step 6: according to the knowledge of graph theory, a bipartite graph G ═ V is constructed1,V2E) to represent the relationship of the M transmit antennas to the Q selectable grid points; wherein, the vertex corresponding to the transmitting antenna uses AmM is 1, M denotes a group V1＝{A1,...,AMG is used as the vertex corresponding to the optional gridqQ is 12＝{G1,...,GQ}; e represents a connection V1And V2The edge set of the middle vertex defines the weight of the edge between the mth transmitting antenna and the qth grid point as
Figure FDA0003503807840000028
Constructing a weight matrix W by using the weight, wherein W is in dimension of M multiplied by Q;
and 7: by adding Q-M virtual antennas, set V1Extension to V1'＝{A1,...,AM,AM+1,...,AQTherein { A })M+1,...,AQDenotes the added virtual antenna vertex; for V1The edge connected with the virtual vertex in the middle is set to have a weight value of 0; then, adding Q-M all-0 rows in W, and expanding the W into a matrix W' with the dimension of Q multiplied by Q;
and 8: applying a KM algorithm to the weight matrix W' obtained in the step 7 to obtain a group a with the maximum weight summqAnd (4) taking values, and neglecting the added virtual antenna vertexes to obtain corresponding results between the M transmitting antennas and the Q grid points.
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