CN109239649B - Novel co-prime array DOA estimation method under array error condition - Google Patents
Novel co-prime array DOA estimation method under array error condition Download PDFInfo
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Abstract
The invention performs DOA estimation on the co-prime array under the array error condition. When array errors exist, the accuracy of DOA estimation of a co-prime array is reduced as a novel array, and related research is few. Under the background, the invention mainly solves the DOA estimation problem of the co-prime array under the array error condition, and mainly performs the following work: 1) A general signal model of a co-prime array in the presence of array errors is presented, which can separate out the error parameters. 2) A sparse Bayesian model based on Bessel prior is provided for estimation of the co-prime array DOA under the array error condition. 3) Aiming at the problem of grid mismatch, a grid mismatch root-finding algorithm of a co-prime array is provided. Through a series of simulation and analysis, the algorithm provided by the invention has higher precision and accuracy under the condition that the sampling point and the signal-to-noise ratio are unchanged.
Description
Technical Field
Array errors encountered in the array antenna engineering process mainly include array element directional diagram errors, array element position errors, array element mutual coupling, channel amplitude phase errors, channel frequency band errors and the like. The directional diagram error and the frequency band error can be equivalent to a channel amplitude-phase error under a specific condition, so that the research on the array error mainly focuses on array element position error, array element mutual coupling and channel amplitude-phase error. The position error of the array element is related to the direction of an incoming wave, and is generally called direction-dependent error; the mutual coupling of array elements and the channel amplitude-phase error are independent of the incoming wave direction, and are called direction independent errors.
Background
Early array error correction was achieved by measuring, interpolating, and storing the array flow patterns, which were costly to implement and not effective. After the 90 s of the 20 th century, the research on array error correction gradually developed towards parameter estimation, and error parameters were estimated according to observation data by modeling various errors. The array error correction method of the parameter estimation type can be divided into two types, namely active correction and passive correction (self-correction). Active correction is carried out by arranging an auxiliary information source with a precisely known azimuth in space and carrying out off-line estimation on array errors according to observation data. The algorithm is based on the premise that the direction of the auxiliary information source is accurately known, the incoming wave direction of the auxiliary information source does not need to be estimated in the correction process, the calculation amount is small, and when the incoming wave direction of the auxiliary information source has deviation, the deviation of the parameter estimation result is large. The self-correction algorithm carries out joint estimation on the incoming wave direction and the error parameters of the information source according to the observation data, does not need to set an auxiliary information source, has lower system cost compared with active correction, can carry out online estimation and realizes real-time correction. However, since the self-correction algorithm relates to parameter joint estimation, the corresponding high-dimensional nonlinear optimization problem is huge in calculation amount, and the global optimal solution is difficult to obtain, so that huge challenges are brought to the research of the self-correction algorithm.
Related research on the problem of error correction of the coprime array is scarce, and few documents can be searched at present. For the problem of array element Mutual coupling, document [1] carries out exploratory research on estimation of a Mutual-prime array DOA under a Mutual coupling condition, firstly measures the Mutual coupling effect among array elements in the Mutual-prime array by using a Receiving Mutual coupling Impedance Method (RMIM), and then jointly estimates an arrival angle and a Mutual coupling Matrix by using a hybrid parameter Covariance Matrix adaptive iteration algorithm (CMA-ES); the literature [2] researches the influence of array element mutual coupling on the DOA estimation performance of three sparse arrays and provides two mutual coupling compensation methods. In the method, the mutual coupling effect is partially known, and a mutual coupling matrix and an arrival angle are updated by respectively applying sparse signal reconstruction and CMA-ES in an iterative process; the other method assumes that mutual coupling is completely unknown, and minimizes a cost function by using CMA-ES so as to realize simultaneous estimation of a mutual coupling matrix, signal source energy and a signal arrival angle. Although the two documents realize the DOA estimation problem of the co-prime array under the cross coupling condition, the CMA-ES algorithm has large computation amount and long time consumption, and is difficult to realize the real-time correction of errors.
For the channel amplitude and phase error problem, no published research result about amplitude and phase error correction of a co-prime array is available at present, but a document 3,4 researches the amplitude and phase error correction problem of a nested array and can provide reference for the co-prime array. The document [3] respectively processes the amplitude error and the phase error, estimates the amplitude error by using partial Toeplitz of a nested array received data covariance matrix, and directly estimates the source arrival angle by constructing a Sparse Total Least Square (STLS) problem instead of directly estimating an error parameter for the phase error. Document [4] analyzes the robustness of the algorithm based on document [3], and expands the correction algorithm to any non-uniform linear array.
Aiming at the problem of array element position errors, a literature [5] analyzes the influence of array element position disturbance on a co-prime array DOA estimation algorithm based on a virtual array, and provides a sparse reconstruction type iterative algorithm to realize the joint estimation of the DOA and the array element position errors. The method comprises the following steps that a document [6] estimates an array flow pattern of a virtual array by using first-order Taylor series expansion, constructs a 'double affine' model, performs dimension reduction processing on the basis, converts the problem into a linear underdetermined DOA estimation problem, and finally solves the problem by using an iteration method in a document [5], so that the problem of array element position error correction of a uniform linear array and a co-prime array is solved from the perspective of the virtual array.
By integrating the current research results, the co-prime array has the advantages of strong resolving power, high estimation precision, strong information source processing capability and the like, and meanwhile, the special array structure brings difficulty for error correction, and the main points are as follows:
(1) Array elements are sparsely arranged, so that the covariance matrix does not have Toeplitz performance any more, and a well-researched uniform linear array error correction method is difficult to be suitable for a co-prime array;
(2) The key of the DOA estimation algorithm based on the co-prime array to expand the direction-finding freedom degree is that a virtual array is constructed, and in the process, the operation of covariance enables array errors to multiply, so that the difficulty of error correction is increased.
In view of the above difficulties, the problem of error correction of co-prime arrays is not well solved, and the problems of large computation amount, poor real-time performance and the like generally exist in the existing results. Therefore, the invention researches the array error correction problem of the co-prime array.
Disclosure of Invention
1) A general signal model of the co-prime array when the array error exists is provided, the model can separate out error parameters and is convenient for subsequent processing, and the features of the co-prime array are expanded on the basis of the general model in the SBAC algorithm [7] corresponding to the content in the specification specific implementation mode (1).
2) Provides a sparse Bayesian model based on Bessel prior to carry out co-prime array DOA estimation under the array error condition, corresponding to the content in the specific implementation modes (2) a and (2) b of the specification, the Bessel prior sparse Bayesian model [13] is applied to the cross-prime array error correction DOA estimation problem.
3) Aiming at the problem of grid mismatch, a grid mismatch root-finding algorithm of a co-prime array is provided for reducing the complexity of a model, and the method can be suitable for more sparse grids and has lower operation complexity. Corresponding to the content in the section (2) c, the grid mismatch root-seeking processing method [14] is improved and upgraded according to the structural characteristics of the relatively prime array.
Drawings
FIG. 1 is a Bayesian structure model diagram.
FIG. 2 is a graph showing the effect of signal-to-noise ratio variation on algorithm performance under coupling error conditions, where graph (a) shows the RMSE variation with signal-to-noise ratio, graph (b) shows the success rate variation with signal-to-noise ratio, and graph (c) shows the operation time variation with signal-to-noise ratio.
FIG. 3 is a graph showing the effect of coupling error condition sampling point number change on algorithm performance, where sub-graph (a) shows the RMSE change with the sampling point number, sub-graph (b) shows the success rate change with the sampling point number, and sub-graph (c) shows the operation time change with the sampling point number.
Fig. 4 shows the influence of the signal-to-noise ratio variation on the algorithm performance under the amplitude-phase error condition, wherein a subgraph (a) shows the RMSE variation with the signal-to-noise ratio, a subgraph (b) shows the success rate variation with the signal-to-noise ratio, and a subgraph (c) shows the operation time variation with the signal-to-noise ratio.
FIG. 5 shows the effect of the number of samples on the performance of the algorithm under the condition of amplitude-phase error, wherein a sub-graph (a) shows the change of RMSE with the number of samples, a sub-graph (b) shows the change of success rate with the number of samples, and a sub-graph (c) shows the change of operation time with the number of samples.
FIG. 6 shows the effect of SNR variation on algorithm performance under position error conditions, where graph (a) shows RMSE variation with SNR, graph (b) shows success rate variation with SNR, and graph (c) shows operation time variation with SNR.
Fig. 7 shows the influence of the change of the sampling point number on the performance of the algorithm under the condition of the position error, wherein a subgraph (a) shows the change of the RMSE with the sampling point number, a subgraph (b) shows the change of the success rate with the sampling point number, and a subgraph (c) shows the change of the operation time with the sampling point number.
Detailed Description
1) General model of array errors in the Presence of
a) Signal model for array element position error
When array position errors exist, the output signal model of the obtained co-prime array is as follows:
y(t)＝A p (θ)x(t)+n(t) (1)
wherein Y (t) is the t-th column of the array output Y,
is an incoming wave signal
T column of (c), t e [ L]，A p And (theta) is an array fashion matrix containing array element position errors. Is provided with
Is a co-prime array
Array element position error vector of individual array element, d 0 The basic array element spacing is = lambda/2, and A can be obtained p The expression of (θ) is:
in DOA estimation, the relative position difference of the array elements is usually considered, and the theoretical solution set of the position error of the array elements can be obtained as
Coefficient of performance
Is composed of
1, selecting the first array element as a reference array element for enhancing the uniqueness of the solution, and calculating the relative position error of other array elements and the first array element, namely
Is provided with
For the relative array element position error vector to be estimated, C p ＝diag([0,c p ])，A p1 (θ)＝A(θ)diag([j2πd 0 sinθ 1 /λ 0 ,...,j2πd 0 sinθ G /λ 0 ]) And A (theta) is the array popular matrix under the condition of error, and when the position error of the array element is smaller, the pair is
Performing a first-order Taylor expansion to obtain:
A p (θ)＝A(θ)+C p A p1 (θ) (3)
in the formula:
in the formula (I), the compound is shown in the specification,
is a and c p An independent parameter matrix. To formula (5) about c p Taking the derivative, we can get:
in the formula (I), the compound is shown in the specification,
is composed of
The (c) th column of (a),
only the (i +1 ) th element is 1, and the other elements are 0,c p,i Is c p The ith element of (1). Substituting equation (4) into equation (1) can obtain a signal model when an array element position error exists as follows:
b) Signal model for existence of array element amplitude phase error
When array element amplitude phase errors exist, the output signal model of the co-prime array is as follows:
y(t)＝A g (θ)x(t)+n(t)＝C g A(θ)x(t)+n(t) (8)
in the formula (I), the compound is shown in the specification,
and
are respectively co-prime arrays
Amplitude gain error and initial phase error of individual array elements, A g (θ) is the array prevalence matrix containing the magnitude-phase error, expressed as:
amplitude-phase errors, similar to the case of array element position errorsTheoretical solution set of difference is C ge ＝b 0 C g Coefficient of
And b is 0 Not equal to 0, in order to enhance the uniqueness of the solution, the first array element is selected as a reference array element, and the relative error between other array elements and the first array element, namely alpha, is calculated 1 ＝1，
To C in formula (8) g A (theta) x (t) is developed to obtain:
in the formula (I), the compound is shown in the specification,
is provided with
With respect to c of formula (11) g Taking the derivative, we can get:
in the formula (I), the compound is shown in the specification,
is composed of
The (c) th column of (a),
only the (i +1 ) th element is 1, and the other elements are 0,c g,i Is c g The ith element of (2) is obtained by substituting equation (10) into equation (8), and the signal model when the array element amplitude-phase error exists is as follows:
c) Signal model when array element coupling exists
Similar to the situation of amplitude and phase errors, the array element coupling error is also a multiplicative error, and the array element coupling matrix is set as C c ，A c (θ) is the array prevalence matrix containing array element coupling errors, and the signal model of equation (8) can be rewritten as:
y(t)＝A c (θ)x(t)+n(t)＝C c A(θ)x(t)+n(t) (14)
is provided with C c,ij Is C c I.e. the coupling coefficients of array element i and array element j. When the distance between two array elements is greater than a certain distance, the coupling effect of the array elements can be ignored, and m is set max To ignore the proximity coefficient of the coupling effect,
for the coupling coefficient vector, we can obtain:
in the formula (d) i And d j The positions of array element i and array element j are respectively, and when m =0, c is easy to obtain c,0 =1. Document [7]]A general model is given for the structure of the uniform linear array, but the model is not suitable for the relatively prime array with larger difference in structure, and the example is given for more intuitively explaining the difference between the array element coupling matrix of the prime array and the array element coupling matrix of the uniform linear array. Provided with a co-prime array of the CACIS type, in common
The array coefficient M =2, N =3, p =2, and the basic array element spacing is lambda/2,m max =2, array element coupling matrix C c Comprises the following steps:
for uniform linear array with array element number and basic array element spacing equal to each other, when m is max When =2, its array element coupling matrix
Comprises the following steps:
comparing the formula (16) with the formula (17), it can be seen that the array element coupling matrixes of the uniform linear array and the co-prime array with the same array element number have obvious difference. The even linear array has the same array element spacing and its array element coupling matrix
The array element coupling matrix of the co-prime array has no Toeplitz property due to the sparsity of the array element positions, the uncertainty of the positions of the coupling coefficients is strong, and the positions of the non-zero elements can only be determined by the formula (15).
Similar to the processing method when the array element amplitude-phase error exists, the signal model under the existence of the array element coupling can be obtained as follows:
in the formula (I), the compound is shown in the specification,
to formula (19) about c c Taking the derivative, we can get:
in the formula，
Is composed of
The (c) th column of (a),
G ci,kj is G ci The kth row and jth column of (c) can result in:
as can be seen from the comprehensive equations (7), (13) and (18), for a relatively prime array, when only an array element position error, an array element amplitude phase error or an array element coupling error exists, a generalized array error signal model can be obtained as follows:
y(t)＝A(θ)x(t)+Q t c+n(t) (22)
in the formula, Q t And c differ by the form of the error. The generalized error parameter vector c can be separated from the array prevalence matrix containing errors by using the formula (22), so that the array prevalence matrix is degraded to the array prevalence matrix A (theta) under the error-free condition, and the subsequent DOA estimation algorithm can conveniently carry out DOA estimation and error parameter estimation under the array error condition.
2) SBL array correction algorithm based on Bessel prior
a) Probability density distribution of parameters
Based on a generalized array signal error model given by the formula (22), a space domain grid is set as
G is the number of grids, and x (t) is a space domain sparse model
The received signal may be sparsely represented as:
let the received signal y (t) obey the mean value
Variance of
A complex Gaussian distribution of (i.e.
Can obtain the product
The probability density function of (a) is:
in the formula (I), the compound is shown in the specification,
let the noise variance parameter α 0 The gamma distributions for parameters a and b have a probability density function of:
is provided with
Obeying a zero mean, a priori complex gaussian distribution with variance Γ = diag (γ), γ = [ γ ] 1 ,...,γ G ]Is a non-negative vector and is a non-negative vector,
the conditional probability density function of (a) is:
in the formula (I), the compound is shown in the specification,
is composed of
The ith element of (1). Let the hyper-parameter γ obey the gamma distribution of the parameters ε and η, and the probability density function is:
for array element position error parameter c p Since the value is random, it can be approximated to gaussian noise that interferes with the position information. Therefore set up c p Obeying a zero mean with a variance of σ -1 The probability density function is:
error parameter c different from array element position p Amplitude-phase error parameter c g And array element coupling error c c The cause of (c) is more complex and complex, and is difficult to describe by a probability distribution model, so c is directly described g And c c As model parameters. FIG. 1 shows a Bayesian structure model diagram representing the relationship among parameters.
As can be seen from fig. 1, the received data Y is characterized by a signal parameter, a noise parameter, and an error parameter, wherein the noise variance parameters a and b are known and not updated, and the other variables are unknown variables. In addition, to reduce the effect of lattice mismatch, the present invention uses a lattice
As an unknown hyper-parameter and updating, compared with the method used in the previous section, the processing method reduces the complexity of the model and the correlation influence among the parameters, and is more suitable for the grid loss under the array error conditionAnd (5) matching a DOA estimation problem.
b) Sparse Bayesian learning
Is provided with
Obeying an a posteriori complex Gaussian distribution with mean μ (t) and variance Σ, i.e.
Where the expressions for μ (t) and Σ are:
in the formula, the joint probability density function obtained from the formulas (24) to (28) is:
according to the Bayesian formula, the posterior probability density function can be obtained as follows:
in the formula (I), the compound is shown in the specification,
it can be seen that p (Y) is independent of the hyper-parameter
And model parameters xi = [ epsilon, eta, sigma ]]. From document [8]It can be seen that the maximum posterior estimate and the xi optimal solution of Θ are:
in the formula (I), the compound is shown in the specification,
in the formula (I), the compound is shown in the specification,
show about
Expectation of a posteriori probability density function, used hereinafter<·>A simplified representation is made. The solution process for the unknown parameters is described below.
(1) Updating the error parameter c p And σ
For the position error parameter c p Equation (33) is equivalent to:
by simplifying the formula (35), the following can be obtained:
with respect to c of formula (36) p,k Derivation, we can obtain:
order to
The following can be obtained:
degenerate reaction of formula (38) to c p,k The ith iteration c is obtained p,k The estimated values of (c) are:
by combining formula (6), formula (29) and formula (30), the compounds are obtained
And
are respectively:
in the formulae (40) to (42),
derivation of the above equation uses the multiplication equation a of the trace H Ba＝tr(Baa H )。
For parameter σ, equation (33) is equivalent to:
by expanding equation (43), we can obtain:
where const is a σ -independent term, for (44) a derivative is taken about σ and the derivative is made 0, the estimate of σ for the ith iteration can be obtained as:
when array element coupling errors or amplitude phase errors exist, the solution thought of error parameters is unchanged, and the formula (39) is rewritten as follows:
the derivation process of the formula (40) to the formula (42) is independent of the error type. Thus, for the other two errors, it will be
Is replaced by
That is, the method of the following formula (40) to formula (42) can be used to obtain
And
wherein the superscript g/c denotes either g or c, i.e.
Or
The same applies hereinafter.
(2) Updating the noise parameter alpha 0
For the noise parameter α 0 Equation (33) is equivalent to:
by developing equation (48), we can obtain:
wherein const is equal to alpha 0 The irrelevant item. To formula (49) about alpha 0 Derivation and let the derivative be 0, the ith iteration alpha can be obtained 0 The estimated values of (c) are:
(3) Updating signal parameters gamma, epsilon and eta
For the hyperparameter γ, equation (33) is equivalent to:
by developing equation (51), we can obtain:
wherein const is equal to gamma k Independent term, μ k (t) is the kth element of μ (t), Σ kk Is the kth row and kth column of Σ. With respect to gamma in the formula (52) k The derivative is obtained and is 0, and the ith iteration gamma can be obtained k The estimated values of (c) are:
for the parameters ε and η, equation (33) is equivalent to:
by simplifying equation (54), it is possible to obtain:
deriving equation (55) for ε and η, respectively, and making the derivative 0, one can obtain:
in equation (56), ψ (ε) is a digamma function representing the derivative of ln Γ (ε) at ε, and the updated value ε at the ith iteration can be obtained by solving equation (56) (i) . The updated value of η at the ith iteration is given directly by equation (57).
c) Grid mismatch root finding processing
Because the grid mismatching phenomenon only occurs in the adjacent grids of the true arrival angle, updating the grids far away from the true arrival angle does not have great influence on the result. Therefore, in order to improve the algorithm efficiency, the grid position to be updated is selected first before the grid parameters are updated. Because the number of the signals is generally unknown, considering that the algorithm is not converged in the initial iteration process, the grids where all spectral peaks of the reconstructed signals are located are selected as the grids to be updated, and the selected grid number set is set to be g (i) The unselected grids are not updated.
For grid variables
Equation (34) is equivalent to:
equation (58) is developed and simplified to obtain:
wherein const is and
the non-related items are,
is composed of
The (c) th column of (a),
for the intermediate variable k e g (i) . With respect to v in the formula (59) k Taking the derivative, we can get:
in the formula (I), the compound is shown in the specification,
combining formula (60) and formula (61), one can obtain:
in the formula, T k And F k Is an intermediate variable, F kj Is F k The jth element of (1), T k And F k Are respectively:
due to d 1 =0, equation (62) may be expressed in the form of matrix multiplication:
formula (65) relates to v k And, v k Has the highest order of
Coefficient of
Since the co-prime array is usually satisfied
I.e., v in the formula (65) k Is greater than the coefficient matrix W k Dimension (d) of (a). In practical application, in order to facilitate root finding of a polynomial, the dimension of the coefficient matrix needs to be expanded to be equal to that of the coefficient matrix
Is provided with
D 1 N element D of 1n Is composed of
D 2 J element D of 2j Is composed of
Expanded coefficient matrix
The jth element of (1)
Comprises the following steps:
in addition, for
The polynomial root of the order is obtained
And (4) screening solutions according to the requirement. Due to the fact that
Can convert | v i And | =1 is used as a limiting condition to screen the solution. The ith iteration v can be obtained by synthesis k The estimated values of (c) are:
in the formula (I), the compound is shown in the specification,
is expressed by a coefficient of
Root of a polynomial of (a) to obtain
Can be reversely solved
Comprises the following steps:
due to the fact that
Has uncertainty in the solution result, in order to avoid
The calculated deviation of (a) causes the result to diverge
The reliable range of the algorithm is restrained to ensure that the algorithm has stronger convergence. Setting the true angular distance of arrival grid of the signal when grid mismatch exists
Recently, it is available:
the meaning of formula (69) is: when in use
Distance grid
At the latest, select
As
The update value of (2), the effect of the grid mismatch can be reduced; when in use
Out of grid
When the grid estimate deviates from the true DOA position, to enhance the convergence rate, let
The value of (a) is kept unchanged from the i-1 iteration, and is not accepted
As
The update value of (2).
d) Algorithm steps
When the iteration is terminated, let P = [ ] 1 ,...,P G ]For the reconstructed signal power spectrum, we can obtain:
searching the position of the spectral peak of P, setting the grid number set of the spectral peak as g, and then estimating the angle of arrival of the signal e Comprises the following steps:
in the practical application, the DOA estimation under the array error condition by utilizing the BR-SBAC algorithm is roughly divided into three steps of parameter initialization, parameter iteration and DOA estimation, wherein the DOA estimation refers to selecting a spectrum peak from a reconstructed signal spectrum, and a grid where the spectrum peak is located is used as the DOA estimation value. For easy understanding, the steps of array element position error correction by using the BR-SBAC algorithm are given in Table 1, and other array error correction methods are the same.
TABLE 1 array element position error correction procedure by using BR-SBAC algorithm
4) Comparative analysis of method effects and other methods
Because different array errors have certain independence, simulation experiments are respectively carried out on the array element coupling error condition, the array amplitude-phase error condition and the array element position error condition, and the performance of the BR-SBAC algorithm under the conditions of different signal-to-noise ratios and sampling points is further analyzed through the simulation experiments. The array used in the simulation adopts CACIS type coprime array, and has 10 array elements, wherein the position [0,3,6,9,12] d of the subarray 1, and the position [0,5,10,15,20,25] d, d of the subarray 2 is half of the wavelength of the incoming wave signal. The simulation experiment does not analyze the estimation precision of the error parameters independently, and the performance of the algorithm is represented by the DOA estimation effect. The simulation software is MATLAB v.2014a, the computer system is Windows 8.1, and the CPU is i3-2120,8G memory.
(1) Array coupling error condition
To compare the performance of the algorithm, the SBAC algorithm under the coupling error condition [7] is adopted]、CMA-ES[9]Algorithm, OGSBI Algorithm [10]]And CRLB under array element coupling condition for comparison. Maximum number of iterations itermax =500 for BR-SBAC, OGSBI and SBAC, and iteration termination threshold τ =1 × 10 -4 Initial value of noise
γ (0) ε of =0,BR-SBAC algorithm (0) ＝0，η (0) Parameter and initial value settings of the cma-ES algorithm of =1,9]The grid range of the sparse reconstruction algorithm is [ -90 DEG, 90 DEG ]]. In the experiment, the grid interval of the SBAC is represented by r =1 ° and r =4 ° and is represented by SBAC (r = 1) and SBAC (r = 4), and the grid interval of the BR-SBAC and the OGSBI algorithm is r =4 °.
Firstly, a simulation experiment is carried out under the conditions of fixed sampling points and variable signal-to-noise ratio. 3 mutually independent far-field narrow-band signals are arranged, the arrival angles are respectively [11 degrees +2 delta theta, -25 degrees-3 delta theta, 52 degrees +2 delta theta ], and delta theta is randomly taken within the range of-5 degrees to 5 degrees, so that the influence of angle prior information on the performance of the algorithm is eliminated. The variation range of the signal-to-noise ratio is set to be-6 dB to 10dB, the step length is set to be 2dB, the sampling point is fixed to be 50, each signal-to-noise ratio is subjected to 500 Monte Carlo experiments, the success rate of DOA estimation, RMSE when the estimation is successful are calculated, and the operation time of each algorithm is recorded, and the result is shown in figure 2.
As can be seen from fig. 2 (a): (1) with the increase of the signal-to-noise ratio, the RMSE of the other three algorithms except the OGSBI algorithm is reduced to different degrees, the RMSE reduction amplitude of the BR-SBAC algorithm is the largest, and the improvement of DOA estimation precision along with the signal-to-noise ratio is relatively slow due to the influence of parameter setting and an initial value of the CMA-ES algorithm; (2) when the signal-to-noise ratio is greater than-3 dB, the RMSE of the BR-SBAC algorithm is lower than that of the four comparison algorithms, and gradually approaches to the CRLB along with the increase of the signal-to-noise ratio, so that higher DOA estimation precision is achieved; (3) compared with an SBAC (r = 4) algorithm, although the grid intervals are the same, the BR-SBAC algorithm is obviously improved in DOA estimation precision, and the effectiveness of the used Bessel prior sparse Bayes model and the grid mismatch root-finding processing mode is verified.
As can be seen from fig. 2 (b): under the condition of different signal-to-noise ratios, the success rates of the BR-SBAC algorithm and the SBAC (r = 1) algorithm are close to 100% and higher than those of other algorithms, and the detection success rate of the OGSBI algorithm applied to the conventional situation is always lower.
As can be seen from fig. 2 (c): (1) under different signal-to-noise ratios, the operation time of the BR-SBAC algorithm is shorter than that of the CMA-ES algorithm and the SBAC (r = 1) algorithm which are better represented in fig. 2 (a) and (b), while the operation time of the OGSBI and SBAC (r = 4) methods is relatively poor in DOA estimation accuracy and success rate although less; (2) and the operation time of the BR-SBAC algorithm has a more obvious descending trend along with the increase of the signal-to-noise ratio, when the signal-to-noise ratio is greater than 6dB, the operation time is close to that of the OGSBI algorithm, and because the operation amount of each iteration of the BR-SBAC algorithm is the same, the convergence rate of the BR-SBAC algorithm is accelerated under the condition of high signal-to-noise ratio.
Then, simulation experiments under the conditions of fixed signal-to-noise ratio and variable sampling points are carried out. Setting the variation range of sampling points to be 20-200, the step length to be 20, the signal-to-noise ratio to be fixed to be 0dB, conducting 500 Monte Carlo experiments on each sampling point, setting other experiment conditions and algorithm parameters to be the same as the condition of signal-to-noise ratio variation in FIG. 2, counting the success rate of DOA estimation, calculating RMSE when estimation is successful, and recording the operation time of each algorithm, wherein the result is shown in FIG. 3.
As can be seen from fig. 3 (a): with the increase of the number of sampling points, the DOA estimation precision of the other three algorithms except the OGSBI algorithm is improved to different degrees, and the BR-SBAC algorithm is always superior to the other algorithms and gradually approaches to the CRLB algorithm. Therefore, under the condition of array element coupling error, the dependence of the direction-finding precision of the BR-SBAC algorithm on the number of sampling points is low, and higher DOA estimation precision can be achieved when the number of sampling points is less.
As can be seen from fig. 3 (b): similar to the case of signal-to-noise ratio variation in fig. 1 (b), the success rate of BR-SBAC and SBAC (r = 1) is always close to 100%; although the CMA-ES algorithm has higher DOA estimation precision when the resolution is successful, the success rate of the resolution is only 85% -90%, which is not the same as the BR-SBAC algorithm; the success rate of the OGSBI algorithm under different sampling point conditions is still the lowest.
As can be seen from fig. 3 (c): (1) the operation time of BR-SBAC, SBAC (r = 1) and SBAC (r = 4) algorithms increases approximately linearly with the increase of the number of sampling points; (2) the operation time of the BR-SBAC algorithm is still lower than that of the SBAC (r = 1) algorithm by about 1s, and is also lower than that of the CMA-ES algorithm when the number of sampling points is less than 200, and although the operation time of the SBAC (r = 4) and the OGSBI algorithm is better than that of the BR-SBAC algorithm, the DOA estimation success rate and the accuracy are not ideal; (3) as the direction finding precision and the success rate of the BR-SBAC algorithm are less influenced by the number of sampling points, fewer sampling points can be adopted in practical application, and the negative influence of the linear increase of the operation time along with the number of the sampling points is reduced.
(2) Array element amplitude phase error condition
For comparing the performance of the algorithm, an SBAC algorithm [7], an S-TLS algorithm [11] and an OGSBI algorithm [10] under the condition of amplitude-phase errors and a CRLB under the condition of array element amplitude-phase errors are adopted for comparison. Parameter settings of BR-SBAC, OGSBI, and SBAC algorithms were the same as those of experiments under coupled conditions, and parameter settings of S-TLS algorithms were the same as those of literature [11]. In the experiment, the grid interval of the SBAC still adopts two cases of r =1 ° and r =4 °, and the grid interval of the BR-SBAC and the OGSBI algorithm is r =4 ° and the grid interval of the S-TLS algorithm is r =1 ° along the representation method of the SBAC (r = 1) and the SBAC (r = 4).
Firstly, a simulation experiment is carried out under the conditions of fixed sampling points and variable signal-to-noise ratio. 3 mutually independent far-field narrow-band signals are arranged, the arrival angles are respectively [11 degrees +2 delta theta, -25 degrees-3 delta theta, 52 degrees +2 delta theta ], and delta theta is randomly selected in the range of-5 degrees to 5 degrees. The variation range of the signal-to-noise ratio is set to be-6 dB to 10dB, the step length is set to be 2dB, the sampling point is fixed to be 50, each signal-to-noise ratio is subjected to 500 Monte Carlo experiments, the success rate of DOA estimation, RMSE when the estimation is successful are calculated, and the operation time of each algorithm is recorded, and the result is shown in figure 4.
As can be seen from fig. 4 (a): (1) the performance of the BR-SBAC and SBAC (r = 1) algorithms is improved along with the increase of the signal-to-noise ratio, the improvement range of the BR-SBAC algorithm is more obvious, and when the signal-to-noise ratio is larger than-4 dB, the RMSE of the BR-SBAC algorithm is lower than that of other four comparison algorithms and is gradually stabilized to be about 0.45; (2) the DOA estimation precision of the SBAC (r = 4) algorithm is poor, the DOA estimation precision is not as good as that of the OGSBI algorithm without error correction, and the SBAC algorithm has higher dependence on the sparse reconstruction grid width; (3) compared with the CRLB, the BR-SBAC algorithm has a certain difference, the RMSE average difference is 0.39, and the DOA estimation precision of the algorithm still has a further improvement space.
As can be seen from fig. 4 (b): the two algorithms of BR-SBAC and SBAC (r = 1) always keep higher success rate, and when the signal-to-noise ratio is larger than-6 dB, the success rate is close to 100%; the success rate of the S-TLS algorithm is kept above 90%; the success rate of the OGSBI algorithm under different sampling point conditions is still the lowest, and a satisfactory DOA estimation result is difficult to obtain.
As can be seen from fig. 4 (c): the operation time of BR-SBAC is reduced significantly and is less than that of SBAC (r = 1) algorithm when the SNR is greater than 2dB, and is close to that of S-TLS algorithm when the SNR is 10 dB. Therefore, when the signal-to-noise ratio is high, the BR-SBAC algorithm is more ideal in terms of operation time, and under the condition of low signal-to-noise ratio, although higher DOA estimation accuracy can be obtained, the operation efficiency still needs to be further improved.
And then carrying out simulation experiments under the conditions of fixed signal-to-noise ratio and variable sampling points. Setting the variation range of sampling points to be 20-200, the step length to be 20, the signal-to-noise ratio to be fixed to be 0dB, conducting 500 Monte Carlo experiments on each sampling point, setting other experiment conditions and algorithm parameters to be the same as the condition of signal-to-noise ratio variation in FIG. 4, counting the success rate of DOA estimation, calculating RMSE when estimation is successful, and recording the operation time of each algorithm, wherein the result is shown in FIG. 5.
As can be seen from fig. 5 (a): (1) under the condition of different sampling point numbers, the performance of the BR-SBAC algorithm is always superior to that of other four comparison algorithms, and when the sampling point number is more than 60, the RMSE is stabilized to be about 0.42; (2) the RMSE of the SBAC (r = 1) and S-TLS algorithms changes with the number of sampling points in a similar manner to the BR-SBAC algorithm, but the RMSE is 0.12 and 0.27 higher on average, respectively; (3) the RMSE of the BR-SBAC algorithm has a certain difference from the CRLB algorithm, the average RMSE is 0.34 higher, and although the RMSE is superior to other comparison algorithms, the RMSE still has a large promotion space.
As can be seen from fig. 5 (b): the success rate of BR-SBAC and SBAC (r = 1) algorithms is optimal, and approaches to 100% when the number of sampling points is more than 40, the success rate of OGSBI algorithm is the lowest, and although the success rate is obviously increased along with the increase of the number of sampling points, the success rate is still lower than that of other algorithms for correcting array element amplitude and phase errors. As can be seen from fig. 5 (c): the BR-SBAC algorithm is slightly higher than the SBAC (r = 1) algorithm in terms of operation time, and the operation time of the BR-SBAC algorithm and the SBAC algorithm still obviously increases with the increase of the number of sampling points.
(3) Array element position error condition
For comparing the performance of the algorithm, an SBAC algorithm [7], a Bi-affine algorithm [12] and an OGSBI algorithm [10] under the condition of array element position error and a CRLB under the condition of array element position error are used for comparison. The parameter settings of the BR-SBAC, OGSBI and SBAC algorithms are the same as the coupling error, the parameter settings of the Bi-affine algorithm are the same as the document [12], the grid interval of the SBAC in the experiment adopts two cases of r =1 ° and r =4 °, the representation methods of the SBAC (r = 1) and SBAC (r = 4) are followed, the grid interval of the BR-SBAC and OGSBI algorithms is r =4 °, and the grid interval of the Bi-affine algorithm is r =1 °.
Firstly, a simulation experiment is carried out under the conditions of fixed sampling points and variable signal-to-noise ratio. 3 mutually independent far-field narrow-band signals are arranged, the arrival angles are respectively [11 degrees +2 delta theta, -25 degrees-3 delta theta, 52 degrees +2 delta theta ], and delta theta is randomly selected in the range of-5 degrees to 5 degrees. The variation range of the signal-to-noise ratio is set to be-6 dB to 10dB, the step length is set to be 2dB, the sampling point is fixed to be 50, each signal-to-noise ratio is subjected to 500 Monte Carlo experiments, the success rate of DOA estimation, RMSE when the estimation is successful are calculated, and the operation time of each algorithm is recorded, and the result is shown in FIG. 6.
As can be seen from fig. 6 (a): (1) in the range of the signal-to-noise ratio of-6 to 0dB, the DOA estimation precision of the BR-SBAC algorithm is obviously improved along with the increase of the signal-to-noise ratio, when the signal-to-noise ratio is greater than-2 dB, the RMSE of the BR-SBAC algorithm is lower than that of the other four comparison algorithms and is gradually stabilized to about 0.41, and the higher DOA estimation precision is achieved; (2) compared with the CRLB, the RMSE of the BR-SBAC algorithm has an average difference of 0.33, and the difference is directly related to the position error magnitude in the simulation condition. Therefore, compared with the BR-SBAC algorithm in the prior art, the method has higher DOA estimation precision, but when the position error of the array element is larger, the performance of the algorithm still has a space for further improvement.
As can be seen from fig. 6 (b): the success rate of BR-SBAC under the condition of array element position error is reduced, but the success rate of BR-SBAC is still 90% at least and is higher than that of the SBAC (r = 1) algorithm with the RMSE being the closest, and the success rate is gradually increased and finally approaches to 100% as the signal-to-noise ratio is increased.
As can be seen from fig. 6 (c): (1) the operation time of the five algorithms is obviously distinguished, the operation time of the BR-SBAC algorithm is only higher than that of an SBAC (r = 4) algorithm and an OGSBI algorithm with poor DOA estimation precision and success rate, and the operation time of the BR-SBAC algorithm still has a reduction trend along with the increase of the signal-to-noise ratio; (2) although the Bi-affine algorithm has higher DOA estimation precision and success rate under the condition of low signal-to-noise ratio, the operation time is longest in the five algorithms and is close to one time of the BR-SBAC algorithm.
And then carrying out simulation experiments under the conditions of fixed signal-to-noise ratio and variable sampling points. Setting the variation range of sampling points to be 20-200, the step length to be 20, the signal-to-noise ratio to be fixed to be 0dB, conducting 500 Monte Carlo experiments on each sampling point, setting other experiment conditions and algorithm parameters to be the same as the condition of signal-to-noise ratio variation in FIG. 6, counting the success rate of DOA estimation, calculating RMSE when estimation is successful, and recording the operation time of each algorithm, wherein the result is shown in FIG. 7.
As can be seen from fig. 7 (a): (1) similar to the situation that amplitude and phase errors exist, under the condition of different sampling points, the RMSE of the BR-SBAC algorithm has a certain difference compared with the CRLB, but is lower than other four comparison algorithms, and compared with the existing algorithm, the direction-finding precision is higher; (2) the direction finding precision of the BR-SBAC algorithm is not greatly influenced by the number of sampling points, and when the number of the sampling points is more than 40, the RMSE of the algorithm can be gradually stabilized to about 0.44.
As can be seen from fig. 7 (b): (1) in general, the DOA estimation success rate of the five algorithms is increased to different degrees along with the increase of the number of sampling points, wherein the success rate of 90% can be achieved by the BR-SBAC, the SBAC (r = 1) and the Bi-affine algorithm when the number of the sampling points is more than 20; (2) in contrast, the success rate of the OGSBI and SBAC (r = 4) algorithms is low and the effect is not ideal.
As can be seen from fig. 7 (c): (1) the operation time of the BR-SBAC algorithm is still less than that of the SBAC (r = 1) algorithm and the Bi-affine algorithm, and the Bi-affine algorithm is not as good as that of the BR-SBAC algorithm in terms of RMSE and operation time although the success rate is high; (2) under the condition of position error, the operation time of the BR-SBAC algorithm is relatively slow along with the increasing amplitude of the number of sampling points, and the SBAC (r = 1) algorithm still has a relatively obvious linear rising trend, mainly because when the number of the sampling points increases, the convergence speed of the BR-SBAC algorithm is fast, and the influence of the increasing of the number of the sampling points on the operation time is partially counteracted.
References:
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Claims (2)
1. A new method for estimating a relatively prime array DOA under an array error condition is characterized by comprising the following steps:
general signal model for co-prime arrays in the presence of array errors:
when array position errors exist, the output signal model of the obtained co-prime array is as follows:
y(t)＝A p (θ)x(t)+n(t) (1)
in the formula, let Y (t) be the t-th column of the array output Y, x (t) be the incoming wave signal, t e [ L ∈ []，A p (theta) is an array prevalence matrix containing array element position errors; is provided with
Is a co-prime array
Array element position error vector of individual array element, d 0 The distance of the basic array elements is = lambda/2, and A can be obtained p The expression of (θ) is:
in DOA estimation, the relative position difference of the array elements is usually considered, and the theoretical solution set of the position error of the array elements can be obtained as
Coefficient of performance
Is composed of
The vector composed of 1 is selected to enhance the uniqueness of the solutionSelecting the first array element as a reference array element, and calculating relative position errors of other array elements and the first array element, i.e.
Is provided with
For the relative array element position error vector to be estimated, C p ＝diag([0,c p ])，A p1 (θ)＝A(θ)diag([j2πd 0 sinθ 1 /λ 0 ,...,j2πd 0 sinθ G /λ 0 ]) And A (theta) is the array popular matrix under the condition of no error, and when the position error of the array element is small, the array element is aligned with A p (theta) in
Performing a first-order taylor expansion to obtain:
A p (θ)＝A(θ)+C p A p1 (θ) (3)
in the formula:
in the formula (I), the compound is shown in the specification,
is a and c p An independent parameter matrix; to formula (5) about c p Taking the derivative, we can get:
in the formula (I), the compound is shown in the specification,
is composed of
The (c) th column of (a),
only the (i +1 ) th element is 1, and the other elements are 0,c p,i Is c p The ith element of (1); substituting equation (4) into equation (1), the signal model when the array element position error exists is:
a) Signal model for existence of array element amplitude phase error
When array element amplitude phase errors exist, the output signal model of the co-prime array is as follows:
y(t)＝A g (θ)x(t)+n(t)＝C g A(θ)x(t)+n(t) (8)
in the formula (I), the compound is shown in the specification,
and
are respectively co-prime arrays
Amplitude gain error and initial phase error of individual array elements, A g (θ) is the array prevalence matrix containing the magnitude-phase error, expressed as:
similar to the case of array element position error, the theoretical solution set of amplitude and phase error is C ge ＝b 0 C g Coefficient of
And b is 0 Not equal to 0, in order to enhance the uniqueness of the solution, the first array element is selected as a reference array element, and the relative error between other array elements and the first array element, namely alpha, is calculated 1 ＝1，
To C in formula (8) g A (theta) x (t) is developed to obtain:
in the formula (I), the compound is shown in the specification,
is provided with
With respect to c of formula (11) g Taking the derivative, we can get:
in the formula (I), the compound is shown in the specification,
is composed of
The (c) th column of (a),
only the (i +1 ) th element is 1, and the other elements are 0,c g,i Is c g The (i) th element of (a),by substituting equation (10) into equation (8), the signal model when the array element amplitude phase error exists can be obtained as follows:
b) Signal model when array element coupling exists
Similar to the situation of amplitude and phase errors, the array element coupling error is also a multiplicative error, and the array element coupling matrix is set as C c ，A c (θ) is the array prevalence matrix containing array element coupling errors, and the signal model of equation (8) can be rewritten as:
y(t)＝A c (θ)x(t)+n(t)＝C c A(θ)x(t)+n(t) (14)
is provided with C c,ij Is C c I.e. the coupling coefficients of array element i and array element j; when the distance between two array elements is greater than a certain distance, the coupling effect of the array elements can be ignored, and m is set max To ignore the proximity coefficient of the coupling effect,
for the coupling coefficient vector, we can obtain:
in the formula, d i And d j The positions of array element i and array element j are respectively, and when m =0, c is easy to obtain c,0 =1; provided with a co-prime array of the CACIS type, in common
The array coefficient M =2, N =3, p =2, and the basic array element spacing is lambda/2,m max =2, array element coupling matrix C c Comprises the following steps:
for uniform linear array with array elements having the same spacing as the basic array elements, when m max When =2, its array element coupling matrix
Comprises the following steps:
compared with the formula (16) and the formula (17), the uniform linear array and the co-prime array with the same array element number have obvious difference in the array element coupling matrix; the even linear array has the same array element spacing and its array element coupling matrix
The array element coupling matrix of the co-prime array has no Toeplitz property due to the sparsity of the array element positions, the uncertainty of the positions of the coupling coefficients is strong, and the positions of non-zero elements can be determined only by the formula (15);
similar to the processing method when the array element amplitude-phase error exists, the signal model under the existence of the array element coupling can be obtained as follows:
in the formula (I), the compound is shown in the specification,
to formula (19) about c c Taking the derivative, we can get:
in the formula，
Is composed of
The (c) th column of (a),
G ci,kj is G ci Row k, column j, the following are obtained:
as can be seen from the comprehensive equations (7), (13) and (18), for a relatively prime array, when only an array element position error, an array element amplitude phase error or an array element coupling error exists, a generalized array error signal model can be obtained as follows:
y(t)＝A(θ)x(t)+Q t c+n(t) (22)
in the formula, Q t And c differ by the form of the error; the generalized error parameter vector c can be separated from the array popular matrix containing errors by using the formula (22), so that the array popular matrix is degraded into an array popular matrix A (theta) under the error-free condition, and the subsequent DOA estimation algorithm can conveniently estimate the DOA and the error parameters under the array error condition;
further comprising:
SBL array correction algorithm based on Bessel prior:
a) Probability density distribution of parameters
Based on a generalized array signal error model given by the formula (22), a space domain grid is set as
G is the number of grids, and x (t) is a space domain sparse model
Receiving a signalSparsely representable as:
let the received signal y (t) obey the mean value
Variance of
A complex Gaussian distribution of (i.e.
Can obtain the product
The probability density function of (a) is:
in the formula (24), the reaction mixture is,
let the noise variance parameter α 0 The gamma distribution, subject to parameters a and b, has a probability density function of:
is provided with
Obeying a zero mean, a complex prior gaussian distribution with variance Γ = diag (γ), γ = [ γ = 1 ,...,γ G ]Is a non-negative vector and is a non-negative vector,
the conditional probability density function of (a) is:
in the formula (26), the reaction mixture is,
is composed of
The ith element of (2); let the hyper-parameter γ obey the gamma distribution of the parameters ε and η, and the probability density function is:
for array element position error parameter c p Because the value is random, the value can be approximate to Gaussian noise interfering with position information; therefore set up c p Obeying a zero mean with a variance of σ -1 The probability density function is:
error parameter c different from array element position p Amplitude-phase error parameter c g And array element coupling error c c The cause of (c) is more complex and complex, and is difficult to describe by a probability distribution model, so c is directly described g And c c As model parameters;
the received data Y is characterized by a signal parameter, a noise parameter and an error parameter together, wherein the noise variance parameters a and b are known and are not updated, and other variables are unknown variables; in addition, to reduce the effect of lattice mismatch, the lattice is used
The unknown hyper-parameter is used as an unknown hyper-parameter and is updated, the processing mode reduces the complexity of the model and the correlation influence among the parameters, and the method is more suitable for the grid mismatch DOA estimation problem under the array error condition;
b) Sparse Bayesian learning
Is provided with
Obeying an a posteriori complex Gaussian distribution with mean μ (t) and variance Σ, i.e.
Where the expressions for μ (t) and Σ are:
in equation (30), the joint probability density function obtained from equations (24) to (28) is:
according to the Bayesian formula, the posterior probability density function can be obtained as follows:
in the formula (32), the compound represented by the formula (32),
it can be seen that p (Y) is independent of the hyper-parameter
And model parameters xi = [ epsilon, eta, sigma ]](ii) a The maximum a posteriori estimate of Θ and the xi optimal solution are:
in the formula (33), the reaction mixture,
in the formula (34), the reaction mixture is,
show about
Expectation of posterior probability density function<·>Carrying out simplified representation; the solving process of the unknown parameters is as follows;
(1) Updating the error parameter c p And σ
For the position error parameter c p Equation (33) is equivalent to:
by simplifying the formula (35), the following can be obtained:
pair formula (36) about c p,k Derivation, we can obtain:
order to
The following can be obtained:
degenerate reaction of formula (38) to c p,k The ith iteration c p,k The estimated values of (c) are:
by combining formula (6), formula (29) and formula (30), the compounds are obtained
And
are respectively:
in the formulae (40) to (42),
derivation of the above equation uses the multiplication equation a of the trace H Ba＝tr(Baa H )；
For parameter σ, equation (33) is equivalent to:
by developing equation (43), we can obtain:
in equation (44), const is a σ -independent term, and by taking the derivative of equation (44) with respect to σ and making the derivative 0, the estimated value of σ in the ith iteration can be obtained as:
when array element coupling errors or amplitude phase errors exist, the solution thought of error parameters is unchanged, and the formula (39) is rewritten as follows:
the derivation process of the formulas (40) to (42) is independent of the error type; thus, for the other two errors, it will be
Is replaced by
That is, the method of the following formula (40) to formula (42) can be used to obtain
And
wherein the superscript g/c denotes either g or c, i.e.
Or
(2) Updating the noise parameter alpha 0
For the noise parameter α 0 Equation (33) is equivalent to:
by developing equation (48), we can obtain:
wherein const is equal to alpha 0 An unrelated item; to formula (49) about alpha 0 Derivation and let the derivative be 0, the ith iteration alpha can be obtained 0 The estimated values of (c) are:
(3) Updating signal parameters gamma, epsilon and eta
For the hyperparameter γ, equation (33) is equivalent to:
by developing equation (51), we can obtain:
wherein const is equal to gamma k Independent term, μ k (t) is the kth element of μ (t), Σ kk Is the kth row and the kth column of Σ; pair formula (52) about gamma k The derivative is obtained and is 0, and the ith iteration gamma can be obtained k The estimated values of (c) are:
for parameters ε and η, equation (33) is equivalent to:
by simplifying equation (54), it is possible to obtain:
deriving equation (55) for ε and η, respectively, and making the derivative 0, one can obtain:
in equation (56), ψ (ε) is a digamma function representing the derivative of ln Γ (ε) at ε, and the updated value ε at the ith iteration can be obtained by solving equation (56) (i) (ii) a The updated value of η at the ith iteration is given directly by equation (57).
2. The new method of relatively prime array DOA estimation under array error conditions of claim 1, further comprising:
a grid mismatch root-finding algorithm of the co-prime array;
because the grid mismatching phenomenon only occurs in the adjacent grids of the real arrival angle, updating the grids far away from the real arrival angle does not have great influence on the result; therefore, in order to improve the algorithm efficiency, before the grid parameters are updated, the grid position to be updated is selected firstly; because the number of the signals is generally unknown, considering that the algorithm is not converged in the initial iteration process, the grids where all spectral peaks of the reconstructed signals are located are selected as the grids to be updated, and the selected grid number set is set to be g (i) The unselected grids are not updated;
for grid variables
Equation (34) is equivalent to:
equation (58) is developed and simplified to obtain:
wherein const is and
the non-related items are,
is composed of
The (c) th column of (a),
for the intermediate variable k e g (i) (ii) a With respect to v in the formula (59) k Taking the derivative, we can get:
in the formula (I), the compound is shown in the specification,
combining formula (60) and formula (61), one can obtain:
in the formula, T k And F k Is an intermediate variable, F kj Is F k The jth element of (1), T k And F k Are respectively:
due to d 1 =0, equation (62) can be expressed in the form of matrix multiplication:
formula (65) relates to v k And, v k Has the highest order of
Coefficient of
Since a relatively prime array is usually satisfied
I.e., v in the formula (65) k Is greater than the coefficient matrix W k Dimension of (c); in practical application, in order to facilitate root finding of a polynomial, the dimension of a coefficient matrix needs to be expanded to be equal to that of the coefficient matrix
Is provided with
D 1 N element D of 1n Is composed of
D 2 J element D of 2j Is composed of
Expanded coefficient matrix
The jth element of (1)
Comprises the following steps:
in addition, for
The polynomial root of the order is obtained
Individual solutions, which need to be screened; due to the fact that
Can convert | v i Screening solutions by using | =1 as a limiting condition; the ith iteration v can be obtained by synthesis k The estimated values of (c) are:
in the formula (I), the compound is shown in the specification,
is expressed by a coefficient of
Root of a polynomial of (a) to obtain
Can be reversely solved
Comprises the following steps:
due to the fact that
Has a solution result ofCertainty, to avoid
The calculated deviation of (b) causes the result to diverge
The reliable range of the algorithm is restrained to ensure that the algorithm has stronger convergence; setting the true angular distance of arrival grid of the signal when grid mismatch exists
Recently, it is available:
the meaning of formula (69) is: when in use
Distance grid
At the latest, select
As
The update value of (2), the effect of the grid mismatch can be reduced; when the temperature is higher than the set temperature
Out of grid
When the grid estimation value deviates from the position of the true DOA, the grid estimation value is controlled to enhance the convergence rate
The value of (a) is kept unchanged from the i-1 iteration, and is not accepted
As
An updated value of (d);
algorithm steps
When the iteration is terminated, let P = [ ] 1 ,...,P G ]For the reconstructed signal power spectrum, we can obtain:
searching the position of the spectral peak of P, setting the grid number set of the spectral peak as g, and then estimating the angle of arrival of the signal e Comprises the following steps:
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| CN110927661A (en) * | 2019-11-22 | 2020-03-27 | 重庆邮电大学 | Single-basis expansion co-prime array MIMO radar DOA estimation method based on MUSIC algorithm |
| CN111948599B (en) * | 2020-08-14 | 2022-08-19 | 电子科技大学 | High-resolution positioning method for coherent signals under influence of angle-dependent mutual coupling |
| CN112904270B (en) * | 2021-01-15 | 2022-06-21 | 电子科技大学 | Direction-of-arrival estimation method based on fitting model under array model error |
| CN112881972B (en) * | 2021-01-15 | 2022-06-21 | 电子科技大学 | Direction-of-arrival estimation method based on neural network under array model error |
| CN113253250B (en) * | 2021-02-05 | 2024-07-02 | 中国人民解放军空军工程大学 | Coprime array robust self-adaptive beam forming method based on matrix reconstruction |
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