CN114879137A - Estimation method for direction of arrival of co-prime matrix based on deep learning reconstruction covariance matrix - Google Patents

Estimation method for direction of arrival of co-prime matrix based on deep learning reconstruction covariance matrix Download PDF

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CN114879137A
CN114879137A CN202210661582.2A CN202210661582A CN114879137A CN 114879137 A CN114879137 A CN 114879137A CN 202210661582 A CN202210661582 A CN 202210661582A CN 114879137 A CN114879137 A CN 114879137A
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matrix
covariance matrix
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潘玉剑
胡星宇
徐欣
杨阿锋
刘晴
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Hangzhou Dianzi University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/14—Systems for determining direction or deviation from predetermined direction
    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
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    • G06N3/00—Computing arrangements based on biological models
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    • G06N3/04—Architecture, e.g. interconnection topology
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/08—Learning methods
    • G06N3/084—Backpropagation, e.g. using gradient descent
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Abstract

The invention discloses a method for estimating the direction of arrival of a co-prime matrix based on a deep learning reconstruction covariance matrix. The method comprises the following steps: arranging a coprime array; outputting and modeling a relatively prime array; modeling a covariance matrix of the deep learning label; generating a deep learning network training data set; designing a deep learning network architecture and constructing a semi-positive definite covariance matrix; setting a loss function and training a deep learning network; reconstructing a semi-positive definite covariance matrix by using a deep learning network, and estimating the number of signal sources; and estimating the arrival direction of the signal by using the reconstructed semi-positive definite covariance matrix. The method fully utilizes the strong fitting capability of deep learning, directly reconstructs the covariance matrix of the uniform linear array through the output of the deep learning network model, and equivalently realizes the interpolation problem of the virtual array holes. The corresponding estimation performance of the direction of arrival is improved compared with that of the traditional signal processing method, and the estimated root mean square error is closer to the lower limit of the Clarmero theory.

Description

Estimation method for direction of arrival of co-prime matrix based on deep learning reconstruction covariance matrix
Technical Field
The invention belongs to the field of array signal processing, particularly relates to a method for estimating the direction of arrival of a signal based on a co-prime array by using a deep learning method, and particularly relates to a method for estimating the direction of arrival of a co-prime array based on a deep learning reconstruction covariance matrix.
Background
The direction of arrival estimation technology utilizes an array to estimate the incoming wave direction of a signal, and has important application in the military and civil technical fields of radar, communication and the like. The uniform linear array is a common array type, but has the following disadvantages. First, there is an upper limit to the array element spacing in order to ensure unambiguous estimation of the direction of arrival. Therefore, the aperture can be increased only by increasing the number of array elements, resulting in increased system cost; secondly, the small array element spacing can cause strong array element mutual coupling, which leads to the reduction of estimation performance; and finally, the degree of freedom of the uniform linear array is limited by the number of physical array elements, and the number of signal sources which can be estimated is less than the number of physical array elements.
In order to avoid the above problems, researchers have proposed sparse arrays, which are arranged in a non-uniform array element pitch. The co-prime array is a novel sparse array. Compared with the traditional minimum redundant array, the design of the co-prime array can be based on an analytic expression. The co-prime array has smaller array element cross coupling, and can generate a virtual array with increased freedom degree. Based on the virtual array, the estimation scene with the signal source number larger than the physical array element number can be dealt with. The most classical algorithm for dealing with the estimation problem of the direction of arrival of the co-prime matrix is a spatial smoothing method, see document 1: (Pal P, Vaidyanathan P. printing sampling and the MUSIC algorithm [ C ]// Digital Signal Processing Workshop and IEEE Signal Processing reduction Workshop (DSP/SPE),2011 IEEE. IEEE, 2011.). However, because the virtual array corresponding to the co-prime array has discontinuous holes, the method can only utilize the output of the continuous array elements positioned in the middle position in the virtual array. In order to increase the utilization rate of the array elements, discontinuous holes in the virtual array can be filled by an interpolation method, so that the output of all the array elements is utilized to improve the estimation performance of the direction of arrival. Document 2: (Y Pan, Luo G Q. effective Direction-of-Arrival-Estimation with Coprime Array [ J ]. Signal Processing,2021,184(3):108061.) the proposed zero-valued interpolation method realizes more accurate Estimation of Direction of Arrival on the co-prime matrix than other interpolation methods. However, the estimated root mean square error obtained by the estimation method has a small distance from the lower limit of the Cramer-Rao theory, and the performance of the estimation method still has a space for improvement.
Disclosure of Invention
The invention aims to provide a method for estimating the direction of arrival of a co-prime matrix based on a deep learning reconstruction covariance matrix, aiming at the defects of the prior art. The method provided by the invention comprises the following steps:
step one, coprime array arrangement; setting a positive integer pair M, N and M < N, constructing a pair of sparse uniform linear array sub-arrays, wherein the first sub-array comprises N antenna array elements with the distance Md, the positions of the array elements are 0, Md,. multidot., (N-1) Md, the second sub-array comprises 2M antenna array elements with the distance Nd, and the positions of the array elements are 0, Nd,. multidot., (2M-1) Nd;
wherein d represents a distance, half of the signal wavelength λ;
the first array elements of the two sub-arrays are overlapped and arranged in the same direction, and because M and N are relatively prime integers, all the array elements of the overlapped array except the first array elements cannot be overlapped, so that a relatively prime array with 2M + N-1 array elements can be formed, and the positions of the array elements form a vector l;
step two, outputting and modeling a co-prime array;
from [ theta ] 1 ,θ 2 ,...,θ K ] T K narrow-band signals in the direction are incident to a co-prime array with 2M + N-1 array elements;
wherein, theta k K is the incoming wave direction of the kth incident signal, and K is 1, 2. The output model of the array at time t is: x (t) ═ a (θ) s (t) + η (t); eta (t) epsilon CN (0, sigma) 2 I 2M+N-1 ) Representing the noise vector, CN (·,. cndot.) representing the complex Gaussian distribution, the first parametric location representing the mean vector, the second parametric location representing the covariance matrix, σ 2 Representing the noise power I 2M+N-1 Representing a 2M + N-1 dimensional identity matrix; s (t) ═ s 1 (t),...,s K (t)] T Vector of K signals representing time t, s k (t) represents s: (t) the kth element, (. C) T Denotes transposition, s (t) epsilon CN (0, R) s )，R s Is a signal covariance matrix; a (θ) ═ a (θ) 1 ),...,a(θ K )]A flow pattern matrix which is an array, wherein
Figure BDA0003688362480000021
As a guide vector,/ i The ith element, i, representing l, is 1, 2M + N-1,
Figure BDA0003688362480000022
step three, modeling a covariance matrix of the deep learning label; after the holes are filled and the space is smoothed in the virtual array of the co-prime array in the step one, a virtual uniform linear array with (2M-1) N array elements is obtained, and a corresponding covariance matrix is used for estimation of the direction of arrival;
using the theoretical covariance matrix of the uniform linear array as a label for network training;
generating a deep learning network training data set; the data set comprises P pieces of data, and each piece of data consists of a vector z and a corresponding label R;
designing a deep learning network architecture and constructing a semi-positive definite covariance matrix; the neural network is set as a full-connection network, and the input layer dimension is (2M + N-1) 2 Output layer dimension of 2((2M-1) N) 2 The method comprises the following steps that L hidden layers are provided, each hidden layer contains m neurons, an activation function exists between every two hidden layers, and the activation function adopts ReLU or Tanh; the neural network is denoted as y ═ f (z), the network outputs
Figure BDA0003688362480000031
Constructing a semi-positive definite covariance matrix R of (2M-1) N rows (2M-1) N columns by using y F ；
Step six, setting a loss function and training a deep learning network;
setting a loss function of a training neural network to
Figure BDA0003688362480000032
Wherein vec (·) representation matrix vectorized according to columns, | · | | non-conductive 2 Representing 2 norm, left-hand multiplication of vector -1/2 Indicating whitening decorrelation, wherein
Figure BDA0003688362480000033
Representation matrix R T Kronecker product with matrix R;
training the network through a back propagation algorithm based on the loss function to update network parameters, and obtaining the neural network after the training is completed
Figure BDA0003688362480000034
Step seven, reconstructing a semi-positive definite covariance matrix by using a deep learning network, and estimating a signal source number K'; forming new vector z' for test data and inputting into neural network
Figure BDA0003688362480000035
In (1), obtaining neural network output
Figure BDA0003688362480000036
Semi-positive definite covariance matrix R 'reconstructed from y' F The method is the same as the method in the step five; estimating the number of signal sources; step eight, estimating the direction of arrival of the signal by using the reconstructed semi-positive definite covariance matrix; to reconstructed R' F Performing characteristic decomposition, and extracting a signal subspace U formed by the characteristic vectors corresponding to the K' maximum characteristic values s (ii) a Determining a polynomial
Figure BDA0003688362480000037
K's are close to the root x on the unit circle k′ K '═ 1, 2.. K', where p (x) · x · 1 x (2M-1)N-1 ] T Then the direction of the signal is estimated as:
Figure BDA0003688362480000038
arg (·) represents the argument of the complex, arcsin [ ·]Representing an inverted sine.
Preferably, theFrom [ theta ] in step three 1 ,θ 2 ,...,θ K ] T K narrow-band signals of the direction are incident to a uniform linear array with (2M-1) N array elements. The theoretical covariance matrix of the array is:
Figure BDA0003688362480000039
I (2M-1)N denotes a (2M-1) N-dimensional identity matrix, A ULA (theta) represents an array flow pattern matrix of virtual uniform linear arrays, wherein A ULA (θ)＝[a ULA (θ 1 ),...,a ULA (θ K )]Guide vector
Figure BDA0003688362480000041
Preferably, the data generation of step four includes the following sub-steps:
substeps four to one, in the signal-to-noise ratio range (SNR) min ,SNR max ) The SNR is selected as SNR, and s (T) and eta (T) of T snapshots are randomly generated, wherein T is 1,2
SNR＝10log 10 (mean(diag(R s ))/σ 2 ) (ii) a Wherein, diag ((-)) represents that the diagonal elements of the corresponding matrix are taken to form a vector, and mean () represents that the mean value of the corresponding vector elements is taken;
angle of incidence range [ -theta [ ] max ,θ max ]Dividing the vector into O subintervals at intervals of delta theta, randomly selecting K subintervals in the O subintervals each time, and randomly generating an angle in each subinterval in the K subintervals to form a vector theta';
theta 'contains the incident angles of K signal sources and is used for generating a flow pattern matrix A (theta') of a co-prime array;
bringing s (T), η (T) and a (θ') into a relatively prime array output model to obtain an array output of x (T), where T is 1,2,. T;
step four or two, calculating a sample covariance matrix
Figure BDA0003688362480000042
(·) H Representing a conjugate transpose, take
Figure BDA0003688362480000043
The real and imaginary parts of the upper triangular element and the diagonal element form a vector z, and z is (2M + N-1) in common 2 A real number element;
and a fourth substep and a third substep, wherein a flow pattern matrix A of the uniform linear array with (2M-1) N array elements is generated by the vector theta ULA (θ'), calculating according to the third step to obtain the label R.
Preferably, in the fifth step, R F The construction of (1) comprises the following substeps:
substep five one, take the front ((2M-1) N) of the network output y 2 Value of
Figure BDA0003688362480000044
Constructing a real part matrix F of an auxiliary matrix F r The construction mode adopts row filling or column filling;
substep V two, take the post ((2M-1) N) of network output y 2 Value of
Figure BDA0003688362480000045
Constructing the imaginary matrix F of the auxiliary matrix F according to the method of the substep five to one i ；
Substeps five and three, the imaginary matrix F is processed i And a real part matrix F r Combining to construct an auxiliary matrix F ═ F r +jF i ；
And a substep of fifthly, constructing a semi-positive definite covariance matrix R by using the auxiliary matrix F F ：R F ＝FF H 。
Preferably, in the seventh step, estimating the number of signal sources includes the following steps:
to R' F Carrying out characteristic decomposition to obtain characteristic values v arranged from small to large 1 ≤v 2 ≤...≤v (2M-1)N-1 ≤v (2M-1)N (ii) a Calculating a difference of the characteristic values
Figure BDA0003688362480000051
Computing sequences
Figure BDA0003688362480000052
Variance of (2)
Figure BDA0003688362480000053
Figure BDA0003688362480000054
Finally, the number of signal sources is estimated
Figure BDA0003688362480000055
Figure BDA0003688362480000056
Means taking j, an intermediate parameter that minimizes h (j)
Figure BDA0003688362480000057
Compared with the prior art, the invention has the following beneficial effects:
1. in the method for estimating the direction of arrival based on deep learning, the covariance matrix used for estimating the number of signal sources and estimating the direction of arrival is constructed by the auxiliary matrix output by the network, and the semipositive nature of the constructed covariance matrix is ensured compared with the method for constructing the covariance matrix by directly using the network output. The semi-positive qualitative model is more consistent with a general array signal processing model.
2. The method can directly reconstruct the covariance matrix of the uniform linear array from the sample covariance matrix output by the co-prime array through the output of the deep learning network model, thereby equivalently realizing the interpolation problem of virtual array holes, and realizing the estimation of the direction of arrival without the grid based on the covariance matrix
3. The loss function used for network training is whitened, so that the correlation among elements in an error vector in the loss function is reduced, and the performance of the neural network-based algorithm can be improved.
4. Based on strong nonlinear fitting capability of a deep learning network, the direction-of-arrival estimation performance of the method is improved compared with that of a traditional signal processing method, and the estimated root mean square error is closer to the lower limit of the Clalmelo theory.
Drawings
FIG. 1 is a block diagram of the overall flow of the method of the present invention;
FIG. 2 is a schematic diagram of a sparse uniform linear subarray having an array element number of 2M;
FIG. 3 is a schematic diagram of a sparse uniform linear subarray having an array element number N;
FIG. 4 is a schematic diagram of a relatively prime array comprising 2M + N-1 array elements;
FIG. 5 shows the estimation result of the direction of arrival at different SNR according to the present invention;
FIG. 6 shows the estimation result of the direction of arrival at different snapshots.
Detailed Description
The technical solution of the present invention is further specifically described below by way of specific examples in conjunction with the accompanying drawings.
Example 1
The following describes the embodiments and effects of the present invention in further detail with reference to the accompanying drawings.
As shown in fig. 1, the present invention includes the following:
step 1, coprime array arrangement; in each sparse array layout, the advantages of the relatively prime array that the direction-finding freedom can be improved and the relatively small mutual coupling between array elements can be more consistent with practical application. Consider a set of co-prime positive integer pairs M, N, assuming that M < N, a sparse, uniform linear array sub-array is first constructed, where the first sub-array, as shown in fig. 2, contains N antenna elements with a spacing Md, whose element positions are 0, Md. The second sub-array, as shown in fig. 3, includes 2M antenna elements with a spacing Nd, and the positions of the elements are 0, Nd, (2M-1) Nd. d represents the distance, half the signal wavelength λ. The first array elements of the two sub-arrays are superposed and arranged in the same direction to obtain a co-prime array as shown in figure 4, because M and N are co-prime integers, all the array elements except the first array elements of the superposed array can not be overlapped, so that the co-prime array with 2M + N-1 array elements can be formed, and the positions of all the array elements form a vector l
Step 2, output modeling of a co-prime array; from [ theta ] 1 ,θ 2 ,...,θ K ] T K narrow-band signals in the direction are incident to a relatively prime array with 2M + N-1 array elements. Wherein, theta k K is the incoming wave direction of the kth incident signal, and K is 1, 2. The output model of the array at time t is: x (t) ═ a (θ) s (t) + η (t); eta (t) belongs to CN (0, sigma) 2 I 2M+N-1 ) Representing the noise vector, CN (·,. cndot.) representing the complex Gaussian distribution, the first parametric location representing the mean vector, the second parametric location representing the covariance matrix, σ 2 Representing the noise power I 2M+N-1 Representing a 2M + N-1 dimensional identity matrix; s (t) ═ s 1 (t),...,s K (t)] T Vector of K signals representing time t, s k (t) denotes the kth element of s (t) (. cndot.) T Denotes transposition, s (t) epsilon CN (0, R) s )，R s Is a signal covariance matrix; a (θ) ═ a (θ) 1 ),...,a(θ K )]A flow pattern matrix which is an array, wherein
Figure BDA0003688362480000061
As a guide vector,/ i The ith element, i, representing l, is 1, 2M + N-1,
Figure BDA0003688362480000062
step 3, modeling a covariance matrix for deep learning labels; and (3) filling holes in the virtual array of the co-prime array in the step (1) and performing spatial smoothing treatment to obtain a virtual uniform linear array with (2M-1) N array elements, wherein the corresponding covariance matrix can be used for estimating the direction of arrival. The theoretical covariance matrix of the uniform line array is therefore used as a label for network training. From [ theta ] 1 ,θ 2 ,...θ K ] T The directional K narrow-band signals are incident on a uniform line array with (2M-1) N array elements. The theoretical covariance matrix of the array is:
Figure BDA0003688362480000071
I (2M-1)N denotes a (2M-1) N-dimensional identity matrix, A ULA (theta) an array flow pattern matrix of virtual uniform linear arrays, A ULA (θ)＝[a ULA (θ 1 ),...,a ULA (θ K )]Guide vector
Figure BDA0003688362480000072
Step 4, generating a deep learning network training data set; the data set contains P pieces of data, each piece of data consists of a vector z and a corresponding label R, and the generation mode of each piece of data is as follows:
first, in the signal-to-noise ratio range (SNR) min ,SNR max ) And randomly selecting the signal-to-noise ratio as SNR, and randomly generating s (T) and eta (T) of T snapshots, wherein T is 1,2 10 (mean(diag(R s ))/σ 2 ) (ii) a Wherein, diag (circle) represents taking the diagonal elements of the corresponding matrix to form a vector, and mean (circle) represents taking the mean value of the corresponding vector elements. Angle of incidence range [ -theta [ ] max ,θ max ]And dividing the vector into O subintervals at intervals of delta theta, randomly selecting K subintervals in the O subintervals each time, and randomly generating an angle in each subinterval in the K subintervals to form a vector theta'. Theta 'contains the incident angles of K signal sources and is used for generating a flow pattern matrix A (theta') of a relatively prime array. And(s), (T), eta (T) and A (theta') are substituted into the relatively prime array output model to obtain an array output x (T), wherein T is 1, 2.
Then, a sample covariance matrix is calculated
Figure BDA0003688362480000073
(·) H Representing a conjugate transpose, take
Figure BDA0003688362480000074
The real and imaginary parts of the upper triangular element and the diagonal element form a vector z, and z is (2M + N-1) in common 2 A real element. Due to the fact that
Figure BDA0003688362480000075
Is a conjugate symmetric matrix to
Figure BDA0003688362480000076
Middle element r i,j Comprises the following steps: r is i,j ＝r j,i * Is established, (.) * Means taking the conjugate, hence only taking
Figure BDA0003688362480000077
The data of the diagonal elements and the real part and the imaginary part of the upper triangle can cover all information, and meanwhile, the complexity of the network can be reduced, and the training speed of the network can be accelerated.
Finally, a flow pattern matrix A of uniform linear arrays with (2M-1) N array elements is generated from the vector theta ULA (θ'), calculating according to step 3 to obtain the label R. The existing estimation algorithm of the direction of arrival of the co-prime array is to firstly obtain a covariance matrix under an equivalent uniform linear array after matrix vectorization processing through hole filling, space smoothing and the like, and then estimate by using a method of the uniform linear array. In the invention, the covariance matrix of the equivalent uniform linear array model is reconstructed by inputting the sample covariance output by the co-prime array into the network model, and the interpolation of the discontinuous holes of the co-prime array is realized by utilizing the strong fitting capability of deep learning, thereby obtaining better effect than the traditional method.
Step 5, designing a deep learning network architecture and constructing a semi-positive definite covariance matrix; the neural network is set to be a full-connection network, and the input layer dimension of the neural network is (2M + N-1) 2 Output layer dimension of 2((2M-1) N) 2 The method comprises the following steps of providing L hidden layers, wherein each hidden layer comprises m neurons, an activation function exists between every two hidden layers, and the activation function can adopt ReLU or Tanh and the like; the neural network is denoted as y ═ f (z), the network outputs
Figure BDA0003688362480000081
Obtaining a semi-positive definite covariance matrix R of (2M-1) N rows and (2M-1) N columns by using a y structure F ：
(3-1) taking the front of network output ((2M-1) N) 2 Value of
Figure BDA0003688362480000082
Constructing a real part matrix F of an auxiliary matrix F r The structure mode can adopt line filling or pressingColumn filling or other possible filling means;
(3-2) taking the network output y ((2M-1) N) 2 Value of
Figure BDA0003688362480000083
Constructing an imaginary matrix F of the auxiliary matrix F according to the method (3-1) i ；
(3-3) combining the imaginary matrix F i And a real part matrix F r Combining to construct an auxiliary matrix F ═ F r +jF i ；
(3-4) construction of a semi-positive definite covariance matrix R Using the auxiliary matrix F F ：R F ＝FF H 。
The covariance matrix is a conjugate symmetric matrix and is semi-positive definite, and the matrix obtained by directly constructing the covariance matrix through the upper triangular part of the network output matrix is a conjugate symmetric matrix but cannot be guaranteed to be a semi-positive definite matrix, so that the method of outputting an auxiliary matrix by a network and constructing the covariance matrix through the auxiliary matrix is adopted, the conjugate symmetry and the semi-positive definite of the matrix can be guaranteed simultaneously, the constructed matrix is more fit with the characteristics of a real covariance matrix, and the estimation of the direction of arrival of subsequent signals is facilitated. Step 6, setting a loss function and training a deep learning network;
setting a loss function of a training neural network to
Figure BDA0003688362480000084
Wherein vec (·) representation matrix vectorized according to columns, | · | | non-conductive 2 Representing 2 norm, left-hand multiplication of vector -1/2 Indicating whitening decorrelation, wherein
Figure BDA0003688362480000085
Representation matrix R T Kronecker product with matrix R. Training the network through a back propagation algorithm based on the loss function to update network parameters, and obtaining the neural network after the training is completed
Figure BDA0003688362480000086
Step 7, reconstructing a semi-positive definite covariance matrix by using a deep learning network, and estimating a signal source number K';
forming new vector z' for test data and inputting into neural network
Figure BDA0003688362480000087
In (1), obtaining neural network output
Figure BDA0003688362480000088
Semi-positive definite covariance matrix R 'reconstructed from y' F Method and step 3 construction of R from y F The method is the same; estimating the number of signal sources according to the following method:
to R' F Carrying out characteristic decomposition to obtain characteristic values v arranged from small to large 1 ≤v 2 ≤...≤v (2M-1)N-1 ≤v (2M-1)N (ii) a Calculating a difference of the characteristic values
Figure BDA0003688362480000091
Computing sequences
Figure BDA0003688362480000092
Variance of (2)
Figure BDA0003688362480000093
Figure BDA0003688362480000094
Finally, the number of signal sources is estimated
Figure BDA0003688362480000095
Figure BDA0003688362480000096
Means taking j, an intermediate parameter that minimizes h (j)
Figure BDA0003688362480000097
Step 8, estimating the direction of arrival of the signal by using the reconstructed semi-positive definite covariance matrix;
to reconstructed R' F Performing feature decomposition to extract K' piecesThe eigenvector corresponding to the largest eigenvalue constitutes the signal subspace U s (ii) a Determining a polynomial
Figure BDA0003688362480000098
K's are close to the root x on the unit circle k′ K '1, 2, K', wherein p (x) 1 x (2M-1)N-1 ] T Then the direction of the signal is estimated as:
Figure BDA0003688362480000099
arg (·) represents the argument of the complex, arcsin [ ·]Representing an inverted sine.
The effect of the present invention is verified by combining with the simulation example, and in order to verify the performance of the present invention, the spatial smoothing method in the background art document 1 and the nulling interpolation method in the document 2, as well as the theoretical cramer-circle, are compared with the method proposed in the present invention. All statistical results are based on monte carlo experiments.
Simulation example 1: the performance of the direction of arrival estimation of the method under different signal-to-noise ratios is verified and compared. The number M of two sub-array elements of the relatively prime array is 2, N is 5, and the number K of signal sources is 9 and the fast beat number T is 500. The signal source incidence angles were [ -55 °, -41.25 °, -27.5 °, -13.75 °,0 °,13.75 °,27.5 °,41.25 °,55 ° ] 1000 monte carlo experiments were performed, and the signal-to-noise ratio was selected from-10 dB to 20dB to generate test data. Specific network parameters of a deep learning network are set as shown in table 1, each hidden layer neuron of a full-connection network containing 5 hidden layers is 4096, an activation function selects a ReLU, an optimizer selects Adam, the maximum epoch number is set to 500, the Batch size is set to 2048, and the initial learning rate is 0.001. The angle range of the training data is [ -60 degrees and 60 degrees ], the angle is divided into 20 sub-intervals by selecting Δ θ to be 6 degrees, each subspace of 9 sub-intervals is selected at each time, an angle is randomly selected to generate training data, and 839800 training data are generated in total. The estimation and comparison result of the direction of arrival is shown in fig. 5, and under the condition of low signal-to-noise ratio, the method proposed by the invention is superior to the method of spatial smoothing and has slightly poorer effect than the method of zero-valued interpolation. The method of the invention is closer to the cramer-perot boundary as the signal-to-noise ratio increases and is clearly superior to the other two methods.
Table 1 network parameter table
Figure BDA0003688362480000101
Simulation example 2: and verifying and comparing the estimation effect of the direction of arrival of the method under different snapshot numbers. Under the parameter setting of the simulation example 1, the direction of arrival is estimated by using three methods respectively for data with the signal-to-noise ratio fixed at 0dB and the fast beat number varying from 100 to 1000, and the estimation results of the three methods are compared as shown in FIG. 6. The result shows that under the condition of less snapshot numbers, the method of the invention is obviously superior to other methods and is closer to the Clarmerico boundary. When the number of snapshots is about 1000, the method of the invention has similar effect with the method of zero interpolation.
The above description is only exemplary of the invention and should not be taken as limiting the scope of the invention, which is intended to cover any variations, equivalents, and modifications that are within the spirit and scope of the invention.

Claims (5)

1. The method for estimating the direction of arrival of the co-prime matrix based on the deep learning reconstruction covariance matrix is characterized by comprising the following steps of:
step one, coprime array arrangement; setting a positive integer pair M, N and M < N, constructing a pair of sparse uniform linear array sub-arrays, wherein the first sub-array comprises N antenna array elements with the distance Md, the positions of the array elements are 0, Md,. multidot., (N-1) Md, the second sub-array comprises 2M antenna array elements with the distance Nd, and the positions of the array elements are 0, Nd,. multidot., (2M-1) Nd;
wherein d represents a distance, half of the signal wavelength λ;
the first array elements of the two sub-arrays are overlapped and arranged in the same direction, and because M and N are relatively prime integers, all the array elements of the overlapped array except the first array elements cannot be overlapped, so that a relatively prime array with 2M + N-1 array elements can be formed, and the positions of the array elements form a vector l;
step two, outputting and modeling a co-prime array;
from [ theta ] 1 ,θ 2 ,...,θ K ] T K narrow-band signals in the direction are incident to a co-prime array with 2M + N-1 array elements;
wherein, theta k K is the incoming wave direction of the kth incident signal, and K is 1, 2. The output model of the array at time t is: x (t) ═ a (θ) s (t) + η (t); eta (t) belongs to CN (0, sigma) 2 I 2M+N-1 ) Representing the noise vector, CN (·,. cndot.) representing the complex Gaussian distribution, the first parametric location representing the mean vector, the second parametric location representing the covariance matrix, σ 2 Representing the noise power I 2M+N-1 Representing a 2M + N-1 dimensional identity matrix; s (t) ═ s 1 (t),..,s K (t)] T Vector of K signals representing time t, s k (t) denotes the kth element of s (t) (. cndot.) T Denotes transposition, s (t) epsilon CN (0, R) s )，R s Is a signal covariance matrix; a (θ) ═ a (θ) 1 ),..,a(θ K )]A flow pattern matrix which is an array, wherein
Figure FDA0003688362470000011
As a guide vector,/ i The ith element, i, representing l, is 1, 2M + N-1,
Figure FDA0003688362470000012
step three, modeling a covariance matrix of the deep learning label;
after the holes are filled and the space is smoothed in the virtual array of the co-prime array in the step one, a virtual uniform linear array with (2M-1) N array elements is obtained, and a corresponding covariance matrix is used for estimation of the direction of arrival;
using the theoretical covariance matrix of the uniform linear array as a label for network training;
generating a deep learning network training data set; the data set comprises P pieces of data, and each piece of data consists of a vector z and a corresponding label R;
designing a deep learning network architecture and constructing a semi-positive definite covariance matrix;
the neural network is set as a full-connection network, and the input layer dimension is (2M + N-1) 2 Output layer dimension of 2((2M-1) N) 2 The method comprises the following steps that L hidden layers are provided, each hidden layer contains m neurons, an activation function exists between every two hidden layers, and the activation function adopts ReLU or Tanh; the neural network is denoted as y ═ f (z), the network outputs
Figure FDA0003688362470000021
Constructing a semi-positive definite covariance matrix R of (2M-1) N rows (2M-1) N columns by using y F ；
Step six, setting a loss function and training a deep learning network;
setting a loss function of a training neural network to
Figure FDA0003688362470000022
Wherein vec (·) representation matrix vectorized according to columns, | · | | non-conductive 2 2 norm, left-times sigma, representing the solution to the vector -1/2 Indicating whitening decorrelation, wherein
Figure FDA0003688362470000023
Figure FDA0003688362470000024
Representation matrix R T Kronecker product with matrix R;
training the network through a back propagation algorithm based on the loss function to update network parameters, and obtaining the neural network after the training is completed
Figure FDA0003688362470000025
Step seven, reconstructing a semi-positive definite covariance matrix by using a deep learning network, and estimating a signal source number K'; forming new vector z' for test data and inputting into neural network
Figure FDA0003688362470000026
In (1), obtaining neural network output
Figure FDA0003688362470000027
Semi-positive definite covariance matrix R 'reconstructed from y' F The method is the same as the method in the step five; estimating the number of signal sources;
step eight, estimating the direction of arrival of the signal by using the reconstructed semi-positive definite covariance matrix;
to reconstructed R' F Performing characteristic decomposition, and extracting a signal subspace U formed by the characteristic vectors corresponding to the K' maximum characteristic values s (ii) a Determining a polynomial
Figure FDA0003688362470000028
K's are close to the root x on the unit circle k′ K '═ 1, 2.. K', where p (x) · x · 1 x (2M-1)N-1 ] T Then the direction of the signal is estimated as:
Figure FDA0003688362470000029
arg (·) represents the argument of the complex, arcsin [ ·]Representing the arcsine.
2. The method for estimating the direction of arrival of the co-prime matrix based on the deep learning reconstructed covariance matrix as claimed in claim 1, wherein the theoretical covariance matrix in the third step is calculated as:
Figure FDA00036883624700000210
I (2M-1)N represents a (2M-1) N-dimensional identity matrix,
A ULA (theta) represents an array flow pattern matrix of virtual uniform linear arrays, wherein,
A ULA (θ)＝[a ULA (θ 1 ),...,a ULA (θ K )]guide vector
Figure FDA0003688362470000031
From [ theta ] 1 ,θ 2 ,...,θ K ] T K narrow-band signals in the direction are obtained by being incident to a uniform linear array with (2M-1) N array elements.
3. The method for estimating the direction of arrival of the co-prime matrix based on the deep learning reconstructed covariance matrix as claimed in claim 1, wherein the data generation of the fourth step comprises the following sub-steps:
substeps four to one, in the signal-to-noise ratio range (SNR) min ,SNR max ) And randomly selecting the signal-to-noise ratio as SNR, and randomly generating s (T) and eta (T) of T snapshots, wherein T is 1,2, and T, and SNR is 10log 10 (mean(diag(R s ))/σ 2 ) (ii) a Wherein, diag ((-)) represents that the diagonal elements of the corresponding matrix are taken to form a vector, and mean () represents that the mean value of the corresponding vector elements is taken;
angle of incidence range [ -theta [ ] max ,θ max ]Dividing the vector into O subintervals at intervals of delta theta, randomly selecting K subintervals in the O subintervals each time, and randomly generating an angle in each subinterval in the K subintervals to form a vector theta';
theta 'contains the incident angles of K signal sources and is used for generating a flow pattern matrix A (theta') of a co-prime array;
bringing s (T), η (T) and a (θ') into a relatively prime array output model to obtain an array output of x (T), where T is 1,2,. T;
step four or two, calculating a sample covariance matrix
Figure FDA0003688362470000032
(·) H Representing a conjugate transpose, take
Figure FDA0003688362470000033
The real and imaginary parts of the upper triangular element and the diagonal element form a vector z, and z is (2M + N-1) in common 2 A real number element;
substeps four and three, from the vector θ', generating a vector having (2M-1) NFlow pattern matrix A of uniform linear array of array elements ULA (θ'), calculating according to the third step to obtain the label R.
4. The method according to claim 1, wherein in the fifth step, R is F The construction of (1) comprises the following substeps:
substep five one, take the front of network output y ((2M-1) N) 2 Value of
Figure FDA0003688362470000034
Constructing a real part matrix F of an auxiliary matrix F r The construction mode adopts row filling or column filling;
substep five two, take the back ((2M-1) N) of the network output y 2 Value of
Figure FDA0003688362470000041
Constructing the imaginary matrix F of the auxiliary matrix F according to the method of the substep five to one i ；
Substeps five and three, the imaginary matrix F is processed i And a real part matrix F r Combining to construct an auxiliary matrix F ═ F r +jF i ；
And a substep of fifthly, constructing a semi-positive definite covariance matrix R by using the auxiliary matrix F F ：R F ＝FF H 。
5. The method for estimating the direction of arrival of the co-prime matrix based on the deep learning reconstructed covariance matrix as claimed in claim 1, wherein the estimating the number of signal sources in the seventh step comprises the following steps:
to R' F Performing characteristic decomposition to obtain characteristic values arranged from small to large
v 1 ≤v 2 ≤...≤v (2M-1)N-1 ≤v (2M-1)N ；
Calculating a difference of the characteristic values
Figure FDA0003688362470000042
Computing sequences
Figure FDA0003688362470000043
Variance of (2)
Figure FDA0003688362470000044
j ═ 1., (2M-1) N-1; finally, the number of signal sources is estimated
Figure FDA0003688362470000045
Figure FDA0003688362470000046
Means taking j, an intermediate parameter that minimizes h (j)
Figure FDA0003688362470000047
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