CN113360841B

CN113360841B - Distributed MIMO radar target positioning performance calculation method based on supervised learning

Info

Publication number: CN113360841B
Application number: CN202110543520.7A
Authority: CN
Inventors: 何茜; 纪瑞明; 叶沙兵
Original assignee: University of Electronic Science and Technology of China
Current assignee: University of Electronic Science and Technology of China
Priority date: 2021-05-19
Filing date: 2021-05-19
Publication date: 2022-05-03
Anticipated expiration: 2041-05-19
Also published as: CN113360841A

Abstract

The method discloses a distributed MIMO radar target positioning performance calculation method based on supervised learning, and belongs to the technical field of signal processing. By using the method, the MSE performance boundary of the distributed MIMO radar target positioning task realized by using the fully-connected neural network can be obtained based on the prior statistical characteristics of the sample set. According to the performance bound, the optimization of the neural network topological structure can be realized, so that the performance of the neural network topological structure is close to the performance bound based on the traditional method.

Description

Distributed MIMO radar target positioning performance calculation method based on supervised learning

Technical Field

The invention belongs to the technical field of signal processing, and particularly relates to performance analysis for positioning a target by using a supervised learning method, which is suitable for the problem of target positioning of a distributed MIMO radar.

Background

The Multiple Input Multiple output (MIMO for short) radar is a new type of radar that can adapt to complex environment, and its basic idea is to arrange Multiple transmitting antennas and receiving antennas at different spatial positions, where each transmitting antenna transmits signals with different waveforms, and the receiving antennas receive echo signals and transmit them to a processing center for joint processing. The MIMO radar with the split antennas has the advantages in the aspect of waveform gain, also has space diversity gain which is not possessed by the traditional radar, has great potential in the aspects of relieving fading, improving resolution, inhibiting interference, resisting stealth and the like, and can obviously improve the target positioning performance of the radar.

For the problem of parameter estimation such as target positioning, the traditional methods such as maximum likelihood estimation are generally adopted. The Cramer-Rao Bound (CRB) defines a lower limit for the variance of the unbiased estimate based on the conventional method, i.e. the variance of the unbiased estimate can only approach the CRB without limitation, but not less than the CRB. In the field of radar parameter estimation, CRB has great practical significance, and it has proven to be one of the most effective lower bounds for evaluating radar system parameter estimation performance.

With the rapid development of deep learning, the application of the deep learning to the field of radar signal processing becomes a new research hotspot. Currently, because the computation complexity is controllable and the function fitting capability is strong, a neural network based on supervised learning is used as a typical deep learning model to solve the radar target positioning problem. In document 1(s.pak, b.k.chalise and b.himed, "Target Localization in Multi-static Radar Systems with adaptive Neural Networks," International Radar Conference (Radar), Toulon, France,2019, pp.1-5.), the author proposes a Target Localization method based on a Neural network for a Passive Radar scene consisting of multiple transmit-receive antennas and a single Target. In document 2(w.zhu and m.zhang, "a Deep Learning Architecture for Broadband DOA Estimation," IEEE 19th International Conference on Communication Technology (ICCT), Xi' an, China,2019, pp.244-247.), the authors propose a Broadband DOA Estimation method based on neural networks. Including the above documents, when constructing a sample set, noise or error often exists in the input of the network, which affects the parameter estimation performance of the neural network. The method aims to research the performance of distributed MIMO radar target positioning based on supervised learning on the premise of knowing the prior statistical characteristics (including noise or Error statistical characteristics) of a sample set, gives a Mean Square Error (MSE) boundary of the distributed MIMO radar target positioning based on supervised learning, and compares the MSE boundary with a CRB (Mean Square Error) boundary of the distributed MIMO radar target positioning based on the traditional method.

Disclosure of Invention

The invention aims at solving the technical problem of the prior art that the performance of distributed MIMO radar target positioning based on supervised learning is calculated on the premise of the prior statistical characteristics of a known sample set, the MSE boundary of the distributed MIMO radar target positioning is given, and the MSE boundary is compared with the CRB of the distributed MIMO radar target positioning based on the traditional method.

The technical scheme of the invention is a distributed MIMO radar target positioning performance calculation method based on supervised learning, which comprises the following steps:

step 1: setting a split antenna MIMO radar which is provided with M single antenna transmitters and N single antenna receivers, wherein a static target with an unknown position exists in a scene; at the receiving end, the received signals corresponding to all NM transceiving paths are:

v_nm＝[v_nm[1],v_nm[2],...,v_nm[K]]^T

＝ξ_nmu_nm+q_nm

u_nm＝[u_nm[1],...,u_nm[K]]^T

q_nm＝[q_nm[1],...,q_nm[K]]^T

the position of the mth transmitter is (x) in the two-dimensional Cartesian coordinate system_tm,y_tm) Where M is 1, 2.. multidot.m, the nth receiver is located at (x)_rn,y_rn) Where N is 1,2,.. times.n, assuming a stationary target position of ω ═ x, y]^TIs the parameter to be estimated. The transmission signals of the transmitters are assumed to be orthogonal, and are approximately orthogonal after undergoing different time delays; transmitting signal s of mth transmitter_m(t) in kT_sSampled value of time being

Where K1, 2, K denotes a sampling number, T_sRepresenting the sampling time interval, E represents the total transmitted signal energy of all transmitters, assuming that the signal energy transmitted by each transmitter is the same here, then

Representing the transmitted signal energy of a single transmitter. Assuming that the waveform of the transmitted signal is normalized

ξ_nmThe complex reflection coefficient corresponding to the nm receiving and transmitting path is constant in an observation interval and is determined to be unknown; q. q.s_nm[k]Indicating that the corresponding nm receiving and transmitting path is at kT_sNoise at the moment; tau is_nmRepresents the time delay corresponding to the nm-th transceiving path, d_tmRepresenting the distance of the target from the m-th transmitter, d_rnRepresents the distance between the target and the nth receiver, and c represents the speed of light; v. of_nmIndicating the received signal corresponding to the nm-th transceiving path, v_nRepresenting the signal received by the nth receiver; e {. is taken as an average value,

represents a pair q_nm[k]The conjugation is taken out and the reaction is carried out,

representing the variance of the noise, delta [ ·]Is a unit impulse function;

step 2: computing a joint estimation time delay τ_nmAnd reflection coefficient xi_nmCRB of (2);

let τ be_nmHas a maximum likelihood estimate of

Defining all unknown parameter vectors as

ω_non,nm＝[τ_nm,ξ_Rnm,ξ_Inm]^T

In which ξ_RnmAnd xi_InmRespectively represent xi_nmThe real and imaginary parts of (c); according to the formula

Wherein

Re {. is used for taking a real part, a superscript H is used for taking a conjugate transpose,

calculate ω_non,nmCorresponding Fisher information matrix J (omega)_non,nm) Then J^-1(ω_non,nm) The first row and column element is τ_nmCRB of (i) that

In the ideal case of the water-cooled turbine,

progressive obedience mean value of τ_nmVariance is

Is a Gaussian distribution of

For the nth receiver, the delay estimation vector

Has a probability distribution of

Wherein

Delay estimation vectors for all NM transmit and receive paths

Has a probability distribution of

τ_ML～N(τ,CRB_τ)

Wherein

And step 3: calculating an average MSE boundary ACRB for estimating coordinates of two dimensions x and y;

according to the formula

Where ω [ i ] represents the ith element of the target position ω,

and is

Calculating a target position ω ═ x, y]^TThe ith row and jth column elements of the corresponding Fisher information matrix J (ω), where i is 1,2, and J is 1, 2; then according to the formula

CRB_x＝[J^-1(ω)]₁₁

CRB_y＝[J^-1(ω)]₂₂

CRBs for x and y can be obtained respectively; final calculation formula

The average MSE boundary ACRB for estimating the x-dimension coordinates and the y-dimension coordinates by adopting a traditional method can be obtained;

and 4, step 4: constructing a network topological structure and giving out prior statistical characteristics of a sample set by adopting a full-connection neural network method;

the neural network comprises an input layer, L hidden layers and an output layer; the input layer is composed of Y₀Each node is composed of a first hidden layer of Y_lA node, wherein L is 1,2, L, and the output layer is Y_L+1And each node is formed. The output layer adopts linear output, and other layers adopt ReLU as a nonlinear activation function;

the sample set consists of a training sample set and a test sample set; suppose a training sample set of Z samples is given as

Wherein r is_0,zAnd theta_zRespectively, an input and a label corresponding to the Z-th sample, wherein Z is 1, 2. Suppose the y-th sample for the z-th sample₀An input characteristic

Subject to mean being zero variance of

Noise or error of Gaussian distribution, and different input characteristics are independent; besides, it is assumed that the samples are independent of each other, and the probability of obtaining each sample from the sample set is the same, i.e. all 1/Z; obtaining a test sample set by the same method

Wherein Z_testRepresenting the number of samples of the test sample set;

the mapping from time delay to target position is realized by using a fully-connected neural network; in order to implement the mapping relationship, when constructing the required training sample, Z position points are discretely taken in the potential region of the target, and the probability that the target exists in each position point is the same, and each position point corresponds to one sample. For the z-th sample, the label is the target location ω^(z)＝[x^(z),y^(z)]^TI.e. theta_z＝ω^(z)(ii) a Network input for all NM strip transceivingDelay estimation vector of path

The probability distribution of which has been calculated by step 2, i.e.

r_true,z＝τ^(z)，

Constructing Z samples into a training sample set

Namely, it is

Considering that a static target exists in a scene, constructing a test sample

And ω^(test)Respectively representing the time delay estimated value and the real position corresponding to the target to be tested, and obtaining the time delay estimated value and the real position in the same way

And 5: carrying out forward propagation of the neural network based on the prior statistical characteristics of the sample set;

since the input of the network is a delay estimation value which is a random variable subject to Gaussian distribution, it is assumed that for the hidden layer and the output layer, both the output of linear transformation and the output of nonlinear transformation through ReLU are subject to Gaussian distribution, and the elements in each layer are independent;

for the hidden layer, i.e. when L1, 2, L, let W_lAnd b_lRespectively representing the weight matrix and the offset vector of the transformation from the l-1 st layer output to the l-1 th layer output, a_l,zAnd r_l,zIndividual watchShowing the output of the l-th layer after linear transformation and nonlinear transformation, phi (-) represents a ReLU activation function, and the forward propagation expression is as follows:

wherein,

and

respectively represent r_l,z,a_l,zAnd b_lY of_lThe number of the elements is one,

represents W_lY of_lA column vector of row elements;

obeying a gaussian distribution, according to the formula:

respectively calculate

Mean value of

Sum variance

Wherein

Represents W_lThe y th_lLine y_l-1Elements of a column; after ReLU nonlinear transformation, according to the formula

Respectively calculate

Mean value of

Sum variance

Wherein

Phi (-) and

respectively representing a standard normal cumulative distribution function and a standard normal distribution function;

for the output layer, the forward propagation formula is obtained in the same way

According to the formula

Separately computing network outputs

Mean value of

Sum variance

Step 6: carrying out back propagation of the neural network based on the prior statistical characteristics of the sample set;

for the output layer, i.e. when L ═ L + 1:

when the value of i is equal to j,

when i ≠ j,

wherein i is 1,2_L+1，j＝1,2,...,Y_L+1；

For hidden layers, i.e. when L ═ L, L-1

Wherein

Then by the formula

Respectively obtaining loss functions loss corresponding to the z-th training sample_zFor all network weight parameters

And all bias parameters

Partial derivatives of (A) by formula

Obtaining loss function loss of training sample set to all network weight parameters

And all bias parameters

Finally, obtaining the network parameters after iterative updating based on an Adam optimization algorithm;

and 7: initializing the network weight parameter and the bias parameter according to the Hommine initialization strategy, and setting the iterative updating times as P. Using a training sample set D_trainAnd repeating the step 5 and the step 6, and iteratively updating the network weight parameter and the bias parameter for P times. Using test sample sets D_testSubstituting the network parameters after P times of iterative update into step 5, and obtaining the average value of network output through forward propagation

Sum variance

Formula for calculation

Obtaining the measured Performance index MSE_test；

And 8: repeating the step 7 for G times in total according to the formula;

obtaining average test performance index MMSE after initializing network weight parameters and bias parameters for G times_testIn which MSE_test,gThe performance index of the test at the g time is expressed; MMSE_testNamely, a full-connection neural network method is adoptedThe mean MSE boundary is estimated for the x and y two dimensional coordinates.

By utilizing the steps, the MSE performance boundary for realizing the distributed MIMO radar target positioning task by utilizing the fully-connected neural network can be obtained based on the prior statistical characteristics of the sample set. According to the performance bound, the optimization of the neural network topological structure can be realized, so that the performance of the neural network topological structure is close to the performance bound based on the traditional method.

Drawings

FIG. 1 is a schematic diagram of a fully connected neural network topology.

Fig. 2 is a schematic diagram of the position of the MIMO radar with a single antenna when M is 3 and N is 2.

Fig. 3 is a schematic diagram of the variation of NNAB and ACRB with SNR for different network topologies.

Detailed Description

For convenience of description, the following definitions are first made:

()^Tis a transposition of^HFor conjugate transposition, Re {. cndot } is a practical part, Diag {. cndot } represents a block diagonal matrix, E {. cndot } is a mean value, Var {. cndot } is a variance, and | | · | | is a norm.

Consider a split antenna MIMO radar with M widely spaced single antenna transmitters and N widely spaced single antenna receivers with a stationary target in the scene whose position is unknown. The position of the mth transmitter is (x) in the two-dimensional Cartesian coordinate system_tm,y_tm) Where M is 1, 2.. multidot.m, the nth receiver is located at (x)_rn,y_rn) Where N is 1,2, N, assuming a stationary target position of ω ═ x, y]^TIs the parameter to be estimated. It is assumed that the transmitted signals of the transmitters are orthogonal to each other and are also approximately orthogonal after being subjected to different time delays. Transmitting signal s of mth transmitter_m(t) in kT_sSampled value of time being

Where K1, 2, K denotes a sampling number, T_sRepresenting the sampling time interval, E represents the total transmitted signal energy of all transmitters, assuming that each transmitter transmits a signal energyThe amounts are the same, then

Since the transmitted signals are orthogonal to each other, each receiver can effectively separate out the components of the echo from the M transmitters. Then transmitted by the mth transmitter via the nth receiver at kT_sThe signal received at a moment can be expressed as

In which ξ_nmAnd the complex reflection coefficient of the corresponding nm transceiving path is shown. Assumption xi_nmIs constant over the observation interval and is certainly unknown. q. q.s_nm[k]Corresponding to the nm-th transceiving path at kT_sNoise at the moment. It is assumed that the noise follows a complex Gaussian distribution and is independent of each other in space and time, i.e.

τ_nmThe time delay corresponding to the nm-th transceiving path can be expressed as

Wherein

Indicating the distance of the target from the mth transmitter,

representing the distance between the target and the nth receiver and c the speed of light.

Order to

Considering K sample samples, equation (1) can be written in vector form

Wherein

q_nm＝[q_nm[1],...,q_nm[K]]^TWith a covariance matrix of

Considering NM receiving and transmitting paths to obtain all received signals as

Wherein

Representing the signal received by the nth receiver.

The invention uses a distributed MIMO radar estimator, i.e. for the nth receiver, the position of the target does not need to be estimated directly, but first locally from the received signal v_nEstimating the time delay tau experienced by signals transmitted by M transmitters_n＝[τ_n1,τ_n2,...,τ_nM]^TThen the estimated result is

Passing into a fusion center that combines the estimates from the N receivers

And then the position parameter omega of the target is estimated. When the fusion center fuses the time delay estimation value to estimate the target position parameter, the performance of positioning the target by utilizing the fully-connected neural network is analyzed, the MSE boundary of the target is given, and the MSE boundary is compared with the MSE boundary-CRB of the traditional method.

Step 1: determining a received signal v corresponding to the nm-th transceiving path by a signal model (6)_nmThe signals v of the NM paths are obtained from equations (7) and (8).

Step 2: the step calculates the joint estimation time delay tau_nmAnd reflection coefficient xi_nmCRB of (1).

Suppose that the nm-th receiving and transmitting path time delay tau_nmHas a maximum likelihood estimate of

Due to complex reflection coefficient xi_nmAlso determined to be unknown, all unknown parameter vectors need to be defined as

ω_non,nm＝[τ_nm,ξ_Rnm,ξ_Inm]^T (9)

In which ξ_RnmAnd xi_InmRespectively represent xi_nmReal and imaginary parts of (c). Since the parameter vector to be estimated is ω_non,nmWith dimension 3, so the Fisher information matrix J (ω)_non,nm) Is a 3 x 3 matrix with the formula

Wherein

Then J^-1(ω_non,nm) The first row and column element is τ_nmCRB of (i) that

In the ideal case, i.e. when the signal to noise ratio is high or the number of samples is sufficiently large,

progressive obedience mean of τ_nmVariance is

Is a Gaussian distribution of

For the nth receiver, the noises of the corresponding M transceiving paths are mutually independent in space, and the time delay estimation vector

Can be expressed as

Wherein

Considering that there are a total of N receivers and the noise between different receivers is independent of each other, the delay estimation vectors corresponding to N receivers, i.e. all NM transmit-receive paths, can be obtained

The probability distribution of which can be expressed as

τ_ML～N(τ,CRB_τ) (18)

Wherein

And step 3: the step adopts a traditional method to calculate an average MSE boundary for estimating x and y dimensional coordinates.

Target position ω ═ x, y]^TThe corresponding Fisher information matrix J (omega) is a 2 x 2 matrix according to the formula

Wherein,

and is provided with

Row i and column J elements of J (ω) can be obtained, where i is 1,2, and J is 1,2, and thus J (ω). According to the formula

CRB_x＝[J^-1(ω)]₁₁ (24)

CRB_y＝[J^-1(ω)]₂₂ (25)

CRBs for x and y can be obtained, respectively. Final calculation formula

The average MSE bound for the x and y dimensional coordinates estimated using conventional methods can be found, and we refer to it as acrb (average crb).

And 4, step 4: the method adopts a full-connection neural network method, constructs a network topological structure and gives the prior statistical characteristics of the sample set.

The neural network comprises an input layer, L hidden layers and an output layer. The input layer is composed of Y₀Each node is composed of a first hidden layer of Y_lA node, wherein L is 1,2, L, and the output layer is Y_L+1And each node is formed. The output layer uses linear output and the other layers use ReLU as a non-linear activation function, as shown in fig. 1.

The sample set is composed of a training sample set and a testing sample set. Suppose a training sample set of Z samples is given as

Wherein

And

the input and the label corresponding to the Z (Z ═ 1, 2.., Z) th sample are respectively represented. Suppose the y-th sample for the z-th sample₀An input characteristic

Subject to mean being zero variance of

Gaussian distributed noise or error, and different input characteristics are independent of each other, then the input r_0,zObey the following distribution

Wherein

And

respectively representing the mean vector and the covariance matrix corresponding to the input of the z-th sample. In addition to this, it is assumed that the individual samples are independent of each other and that the probability of taking each sample from the set of samples is the same, i.e. all 1/Z. The same way can obtain a test sample set

Wherein Z_testRepresenting the number of samples in the test sample set.

Consider the use of a fully connected neural network to implement the mapping of the time delay to the target location. When a training sample required for positioning is constructed, Z position points are discretely taken in a potential area of the target, and the probability that the target exists in each position point is the same, wherein each position point corresponds to one sample. For the z-th sample, the label is the target location ω^(z)＝[x^(z),y^(z)]^TI.e. theta_z＝ω^(z)(ii) a Network input is time delay estimation vector of all NM receiving and transmitting paths

The probability distribution is given by equation (18), i.e.

r_true,z＝τ^(z)，

Constructing Z samples into a training sample set

Namely, it is

Considering that a static target exists in a scene, constructing a test sample

And 5:

this step derives the forward propagation process of the neural network based on the prior statistical properties of the sample set.

Since the input of the network is a delay estimation value which is a random variable subject to Gaussian distribution, it is assumed that for the hidden layer and the output layer, both the output of linear transformation and the output subjected to ReLU nonlinear transformation are subject to Gaussian distribution, and the elements in each layer are independent of each other.

For the hidden layer, the output of the L-1 layer is taken as the input of the L layer, where L is 1,2

r_l,z＝φ(a_l,z)＝φ(W_lr_l-1,z+b_l) (28)

Wherein

And

respectively representing the weight matrix and the offset vector of the transformation from the l-1 st layer output to the l-1 th layer output, a_l,z＝W_lr_l-1,z+b_lRepresents the output after linear transformation, phi (-) represents the ReLU activation function. Rewriting formula (28) to

Wherein,

and

represents W_lY of_lA column vector of row elements.

Obeying a Gaussian distribution according to the formula

Respectively calculate

Mean value of

Sum variance

Wherein

Represents W_lThe y th_lLine y_l-1The elements of the column. After ReLU nonlinear transformation, according to the formula

Respectively calculate

Mean value of

Sum variance

Wherein

Φ (-) represents a standard normal cumulative distribution function,

representing a standard normal distribution function.

For the output layer, the final output of the network is represented as

r_L+1,z＝a_L+1,z＝W_L+1r_L,z+b_L+1 (34)

Wherein W_L+1And b_L+1Respectively representing the weight matrix and the offset vector of the output layer. The rewriting formula (34) is

In the same way according to the formula

Separately computing network outputs

Mean value of

Sum variance

Step 6: this step derives the back propagation process of the neural network based on the prior statistical properties of the sample set.

When L ═ L +1, i.e. for the output layer:

when the value of i is equal to j,

when i ≠ j,

wherein i is 1,2_L+1，j＝1,2,...,Y_L+1。

When L ═ L, L-1., 1, i.e., for the hidden layer, equation (38) (39) is used as the initialization equation, and the equation is iteratively calculated

Wherein

Then by the formula

And all bias parameters

Partial derivatives of (A) by formula

And all bias parameters

Finally, updating the biased first-order moment estimation and the biased second-order moment estimation respectively based on an Adam optimization algorithm, and iteratively updating network parameters after the deviations of the first-order moment and the second-order moment are corrected respectively.

And 7: initializing the network weight parameter and the bias parameter according to the Hommine initialization strategy, and setting the iterative updating times as P. Using a training sample set D_trainAnd repeating the step 5 and the step 6, and iteratively updating the network weight parameter and the bias parameter for P times. Using test sample sets D_testSubstituting the network parameters after P times of iterative update into step 5, and obtaining the average value of the network output through forward propagation

Sum variance

Formula for calculation

Obtaining the tested performance index MSE_test。

And step 8: repeating step 7 for G times according to formula

Obtaining average test performance index MMSE after initializing network weight parameters and bias parameters for G times_testIn which MSE_test,gThe g-th test performance index is shown. MMSE_testNamely, a fully connected neural Network method is adopted to estimate the Average MSE boundary of x and y dimensional coordinates, which is called NNAB (neural Network Average bound).

Working principle of the invention

1. In step 2, the likelihood function can be expressed as a function of the signal model (6)

Then the log-likelihood function is

Wherein C is₁Is and ω_non,nmAn unrelated item. The Fisher information matrix is expressed as

According to document 3(S.Kay, "Fundamentals of Statistical Signal Processing: Estimation Theory," Prentice-Hall. Englewood Cliffs, NJ,1993.), the elements in line i and column j can be expressed as

Wherein, i is 1,2,3, j is 1,2, 3. From the formula (57), the formula (10) can be obtained. Further, the formula (16) can be obtained from document 4(S.Fortunati, F.Gini, M.S.Greco, "Performance bases for Parameter Estimation units Misspecified Models: Fundamental Finding and Applications," IEEE Signal Processing Magazine, vol.34, No.6, pp.142-157, Nov.2017.).

2. In step 3,. tau._MLIs a random vector of target unknown parameters omega, and estimates the vector tau according to the time delay_MLCan obtain a likelihood function

Then the log-likelihood function is

Wherein

Is a constant term. The Fisher information matrix can be expressed as

Equation (19) can be obtained according to reference 3(S.Kay, "Fundamentals of Statistical Signal Processing: Estimation Theory," Prentice-Hall. Englewood Cliffs, NJ, 1993).

3. In step 5, in the calculation

Mean value of

Sum variance

Due to linear transformationOutput a_l,zEach element is independent of the other, so the formula

The formula (30) (31) can be obtained. In the calculation of

Mean value of

Sum variance

Due to

Wherein erf (-) represents an error function, and the standard normal cumulative distribution function and the error function have a relationship of

The formula (32) and (33) can be obtained by substituting the formula (65) for the formulae (63) and (64).

4. In step 6, the minimum MSE is taken as an optimization criterion, and a loss function corresponding to the training sample set is defined as

Wherein loss_zRepresents the corresponding loss function of the z-th training sample, and the expression is

Then, loss_zFor all network weight parameters

And all bias parameters

Respectively are

Thus, the formula (48) (49) is obtained. By the formula

The formula (40) can be obtained, and the formulae (41), (42) and (43) can be obtained in the same manner. By the formula

The formula (44) can be obtained. By the formula

Formula (45) can be obtained. By the formula

The formula (46) can be obtained. By the formula

The formula (47) can be obtained.

5. In step 7, in order to measure the output test performance of each node on average at the output end, the test performance index of the test sample set is defined as

Namely, formula (52).

6. In step 8, since the selection of the initial parameters of the network determines whether the final training can converge, the convergence speed and the generalization effect, the strategy of initializing mocam is a random initialization strategy, and in order to analyze the average test performance generated by the random characteristics of the initialization parameters, the formula (53) is defined as the performance boundary of the neural network.

Aiming at the problem of distributed MIMO radar target positioning, the performance of the traditional method and the supervised learning method are respectively analyzed by simulation, wherein simulation parameters are set as follows:

the simulation uses signals of

Wherein the signal period T is 0.1ms and the carrier frequency f_c＝1GHz，f_mFor the transmitting frequency of the mth transmitter, specific parameters will be given later, and the total transmitting signal energy E of the transmitter is 10¹⁴Defining a signal-to-noise ratio (SNR) of

Suppose a stationary target is located at (50, -40) m. Assume that each transmitter and receiver is located 500m from a reference point (0,0) m, respectively. Considering incoherent scenes, the antenna will face different directions at the target location, i.e. within different target beamwidths, and thus will produce different reflection coefficients. To fully exploit the advantages of incoherent estimation, assuming that the M transmitters are uniformly distributed over an angle [0,2 π ], the angle of the mth transmitter is then

N receivers are also uniformly distributed within [0,2 π) angle, the angle of the nth receiver is then

Considering the case of the number of antennas with M being 3 and N being 2, the antenna placement positions are as shown in fig. 2. Suppose that the transmission frequencies of the 3 transmitters are respectively f₁＝100kHz，f₂200kHz and f ₃300 kHz. In order to make the sampling frequency satisfy the Nyquist theorem, assume the sampling frequency f_s1MHz, so the sampling interval T_sThe number of samples K is 100 at 1 μ s.

For the supervised learning method, it is assumed that targets exist within the range of [ -350, -250] mx [250,350] m, [ -50,50] mx [ -50,50] m and [250,350] mx [ -350, -250] m, and the possibility of existing at each position is the same. In the stage of generating the sample set, it is assumed that a stationary object is located at a point of the two-dimensional cartesian coordinate system with a gap of 1m, that is, a total of Z101 × 101 × 3 30603 possible position sampling points. Setting SNR from 5dB to 20dB and interval to 3dB for the z-th specific position point, generating received data according to a signal model (7), and estimating time delay corresponding to all paths based on a maximum likelihood estimation method, wherein the probability distribution of time delay estimation values is given by an equation (18), and CRB of time delay estimation can be obtained by an equation (15). And taking the time delay estimation value with random characteristics corresponding to the specific position point as input data of the network, and taking the position coordinate corresponding to the specific position point as a label, so that a training sample set comprising 30603 samples can be obtained for each SNR by considering all position sampling points. As only one target exists in the scene, the test sample can be obtained in the process of constructing the training sample set. It should be noted that the samples corresponding to each SNR are processed separately.

In addition, because the input delay estimation value is extremely small, and the network is difficult to distinguish the delays corresponding to different positions, the delay estimation value multiplied by the speed of light c is considered as the input of the network, so that the average value of the network input becomes the average value of the delay estimation value multiplied by c, and the variance becomes the variance of the delay estimation value multiplied by c²This operation is adopted in the simulation analysis. The dimension of the input data is the product of the number of transmitting antennas and the number of receiving antennas, and a scenario where M is 3 and N is 2 is considered here, so the dimension of the input data is NM 6, i.e. the number of nodes of the input layer of the network is 6, and the dimension of the output layer is determined by the dimension of the coordinate system, so the dimension is 2, i.e. the number of nodes of the output end is 2. Considering the prior statistical characteristics of a sample set, training and testing the networks with topological structures of 6-5-5-2 ', 6-10-10-2 ' and 6-20-20-20-20-2 ' for multiple times respectively to obtain the MSE (mean square error) boundary NNAB of the neural network, wherein 6 in the 6-10-10-10-2 ' represents the number of nodes of an input layer, 2 represents the number of nodes of an output layer, the middle 10-10-10 ' represents three hidden layers in total, and the number of nodes of each layer is 10, and the other same principles. It should be noted that abnormal results caused by gradient disappearance or gradient explosion during training are discarded.

Figure 3 compares NNAB and ACRB for different topology scenarios. In the figure, the central circle red line, the plus sign green line and the square blue line respectively represent NNAB change curves with SNR under the conditions that the topological structures are 6-5-5-2, 6-10-10-10-2 and 6-20-20-20-20-2, and the diamond black line represents ACRB change curves with SNR. As can be seen, NNAB decreases with increasing network computational complexity and approaches ACRB gradually. For a network with a topology of 6-20-20-20-20-20-2, the NNAB is very close to the ACRB when the SNR is 5dB to 14dB, and the two trends are consistent. ACRB is obtained with the knowledge of the model information (including transmitter and receiver locations) for the time delay to target location mapping, but NNAB is obtained with this model information unknown, which illustrates that with increased computational complexity, the network can learn this model information efficiently. On the other hand, both ACRB and NNAB are obtained on the premise of knowing the statistical properties of the delay estimation values, which shows that the statistical information can be effectively utilized as long as the network computation complexity is high enough. Furthermore, it can be seen that both NNAB and ACRB decrease with increasing signal-to-noise ratio, since increasing signal-to-noise ratio results in a more accurate estimate of the network input, delay. When the signal to noise ratio is high, such as 17dB to 20dB, the effectiveness of NNAB approximating ACRB is not as good as 5dB to 14dB, indicating that the performance of the network is limited when the signal to noise ratio is high.

In addition, as can be seen from the figure, the computational complexity of the network changed from "6-5-5-2" to "6-10-10-10-2" is increased 215, and the computational complexity of the network changed from "6-10-10-10-2" to "6-20-20-20-20-20-2" is increased 1480, but the latter is far less than the former in terms of the magnitude of performance improvement, which indicates that the learning capability of the neural network with high computational complexity for the sample can reach a saturated state, and as long as a proper network topology is selected, the performance with little complexity can be achieved.

Claims

1. A distributed MIMO radar target positioning performance calculation method based on supervised learning comprises the following steps:

v_nm＝[v_nm[1],v_nm[2],...,v_nm[K]]^T

＝ξ_nmu_nm+q_nm

u_nm＝[u_nm[1],...,u_nm[K]]^T

q_nm＝[q_nm[1],...,q_nm[K]]^T

the position of the mth transmitter is (x) in the two-dimensional Cartesian coordinate system_tm,y_tm) Where M is 1, 2.. multidot.m, the nth receiver is located at (x)_rn,y_rn) Wherein n is 1N, assuming a stationary target position of ω ═ x, y]^TIs a parameter to be estimated; the transmission signals of the transmitters are assumed to be orthogonal and are approximately orthogonal after being subjected to different time delays; transmitting signal s of mth transmitter_m(t) in kT_sSampled value of time being

Representing the transmitted signal energy of a single transmitter; assuming that the waveform of the transmitted signal is normalized

ξ_nmThe complex reflection coefficient corresponding to the nm receiving and transmitting path is constant in an observation interval and is determined to be unknown; q. q.s_nm[k]Indicating that the corresponding nm receiving and transmitting path is at kT_sNoise at the moment; tau is_nmRepresenting the time delay corresponding to the nm-th transceiving path, d_tmRepresenting the distance of the target from the m-th transmitter, d_rnRepresents the distance between the target and the nth receiver, and c represents the speed of light; v. of_nmIndicating the received signal corresponding to the nm-th transceiving path, v_nRepresenting the signal received by the nth receiver; e {. is taken as an average value,

representing the variance of the noise, delta [ ·]Is a unit impulse function;

step 2: computing a joint estimate of time delay tau_nmAnd reflection coefficient xi_nmCRB of (2);

let τ be_nmHas a maximum likelihood estimate of

Defining all unknown parameter vectors as

ω_non,nm＝[τ_nm,ξ_Rnm,ξ_Inm]^T

Wherein

Re {. is used for taking a real part, the superscript H is used for taking a conjugate transpose,

calculating omega_non,nmCorresponding Fisher information matrix J (omega)_non,nm) Then J^-1(ω_non,nm) The first row and column element is τ_nmCRB of (i) that

In the ideal case of the water-cooled turbine,

progressive obedience mean of τ_nmVariance is

Is a Gaussian distribution of

For the nth receiver, the delay estimation vector

Has a probability distribution of

Wherein

Delay estimation vectors for all NM transmit and receive paths

Has a probability distribution of τ_ML～N(τ,CRB_τ)

Wherein

neural network packageComprises an input layer, L hidden layers and an output layer; the input layer is composed of Y₀Each node is composed of a first hidden layer of Y_lA node, wherein L is 1,2, L, and the output layer is Y_L+1Each node is formed; the output layer adopts linear output, and other layers adopt ReLU as a nonlinear activation function;

Wherein r is_0,zAnd theta_zRespectively, an input and a tag corresponding to a Z-th sample, wherein Z is 1, 2. Suppose the y-th sample for the z-th sample₀An input characteristic

Subject to mean being zero variance of

Wherein Z_testRepresenting the number of samples of the test sample set;

the mapping from time delay to target position is realized by using a fully-connected neural network; in order to realize the mapping relation, when a required training sample is constructed, Z position points are discretely taken in a potential region of a target, the probability that the target exists in each position point is the same, and each position point corresponds to one sample; for the z-th sample, the label is the target location ω^(z)＝[x^(z),y^(z)]^TI.e. theta_z＝ω^(z)(ii) a Network input is time delay estimation vector of all NM receiving and transmitting paths

The probability distribution of which has been calculated by step 2, i.e.

r_true,z＝τ^(z)，

Constructing Z samples into a training sample set

Namely, it is

Considering that a static target exists in a scene, constructing a test sample

and 7: initializing a network weight parameter and a bias parameter according to a Hommine initialization strategy, and setting the iteration updating times as P; using a training sample set D_trainRepeating the step 5 and the step 6, and iteratively updating the network weight parameter and the bias parameter for P times; using test sample sets D_testSubstituting the network parameters after the iteration updating for P times into the step 5Forward propagation to obtain the mean of the network output

Sum variance

Formula for calculation

Obtaining the tested performance index MSE_test；

And 8: repeating the step 7 for G times in total according to the formula;

obtaining average test performance index MMSE after initializing network weight parameters and bias parameters for G times_testIn which MSE_test,gThe performance index of the test at the g time is expressed; MMSE_testNamely, a fully connected neural network method is adopted to estimate the average MSE boundary of x and y dimensional coordinates.

2. The method for calculating the target positioning performance of the distributed MIMO radar based on supervised learning as claimed in claim 1, wherein the specific method in step 3 is as follows:

according to the formula

Where ω [ i ] represents the ith element of the target position ω,

and is provided with

CRB_x＝[J^-1(ω)]₁₁

CRB_y＝[J^-1(ω)]₂₂

CRBs for x and y can be obtained respectively; final calculation formula

The average MSE boundary ACRB estimated using conventional methods for x and y dimensional coordinates can be obtained.

3. The method for calculating the target positioning performance of the distributed MIMO radar based on supervised learning as claimed in claim 1, wherein the specific method in step 5 is as follows:

for the hidden layer, i.e. when L1, 2, L, let W_lAnd b_lRespectively representing the weight matrix and the offset vector of the transformation from the output of layer l-1 to the output of layer l, a_l,zAnd r_l,zRespectively representing the output of the l-th layer after linear transformation and nonlinear transformation, phi (·) represents a ReLU activation function, and the forward propagation expression is as follows:

wherein,

and

represents W_lY of_lA column vector of row elements;

obeying a gaussian distribution, according to the formula:

respectively calculate

Mean value of

Sum variance

Wherein

Respectively calculate

Mean value of

Sum variance

Wherein

Phi (-) and

According to the formula

Separately computing network outputs

Mean value of

Sum variance

4. The method for calculating the target positioning performance of the distributed MIMO radar based on supervised learning as claimed in claim 1, wherein the specific method in step 6 is as follows:

for the output layer, i.e. when L ═ L + 1:

when the value of i is equal to j,