CN107817465A - DOA estimation method based on non-grid compressed sensing under super-Gaussian noise background - Google Patents

Abstract

The invention discloses a DOA estimation method based on gridless compressed sensing under the background of super-Gaussian noise. signal, and according to the noise distribution type measured in the actual environment, select the norm of l _p to constrain, and then determine the function expression to solve the noise-free signal x, the recovery problem of the noise-free signal x is described as the atomic norm minimization problem, take Semidefinite programming theory can be used to solve the problem; then the original solution can be obtained through the ADMM algorithm, and the dual solution can be obtained Finally, the DOA is solved to make it closer to the angle support set of the original signal, so as to complete the DOA estimation. The present invention uses the statistical characteristics of the noise to effectively constrain the noise item with a suitable norm, and finally uses the relationship between the original solution and the dual solution to obtain the angle support set of the original signal, thereby realizing efficient and accurate DOA estimation.

Description

DOA Estimation Method Based on Meshless Compressive Sensing under Super Gaussian Noise Background

technical field

The invention belongs to the direction-of-arrival (DOA) estimation technology of Grid-less Compressive Sensing (Grid-less Compressive Sensing, GCS), in particular, a method based on Grid-less Compressive Sensing under the background of super-Gaussian noise. DOA estimation method.

Background technique

As we all know, direction of arrival estimation plays an irreplaceable role in communication, voice, radar, sonar and many other aspects. Due to its unique advantages, compressive sensing technology has been widely used in spatial spectrum estimation problems. For DOA estimation, compressive sensing technology utilizes the sparsity of the signal in the space domain, and only needs a small amount of observation data to realize the reconstruction of the signal, and the signal to noise ratio (Signal to Noise Rate, SNR) is low, the source When the correlation is high, the algorithm is still robust. However, traditional super-resolution algorithms, such as the Multiple Signal Classification (MUSIC) algorithm based on the signal subspace and the Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) algorithm, etc., do not have This advantage. After research and development in recent years, the sparse signal reconstruction algorithm can achieve super-resolution DOA estimation even in the case of a small sample or a single snapshot.

In many practical applications, the parameter space of a signal is continuous. To solve this problem, the traditional compressive sensing algorithm first discretizes the continuous parameter space, divides the entire parameter space into a finite number of grids, and assumes that the parameters of the signal can be represented by some grid points. Although an overly dense grid can improve the estimation accuracy, it will also cause the correlation between adjacent atoms in the base dictionary to be too strong, which will reduce the constraint isocapacitance (RIP) of the dictionary, thereby reducing the reconstruction of compressed sensing. performance. In addition, the actual parameter values of the signal may not fall on the discrete grid, and the assumed transformation basis cannot represent the sparse signal. The researchers call this phenomenon Basis Mismatch.

In response to this phenomenon, the researchers found the best convex heuristic method to solve the underdetermined linear inverse problem - using the atomic norm to characterize the signal characteristics, that is, directly looking for the fewest atoms in the continuous parameter space to represent the given signal, There is no need to discretize the sparse domain, and the concept of gridless compressed sensing, or continuous compressive sensing (Continuous Compressive Sensing, CCS) is proposed. The parameter space is no longer divided into a finite number of grids, but directly regarded as a continuous variable. Tang et al. proposed in the literature (1. Tang Gong-guo, Bhaskar BN, ShahP, et al. Compressed sensing off the grid [J]. IEEE Transactions on Information Theory, 2013, 59(11): 7465-7490), when The spacing between signal frequencies is at least When , only O(K·logK·logM) samples are needed to reconstruct the frequency of the signal, where ε is a small constant, K is the number of signals, and M is the length of the signal. The solution of the atomic norm depends on the semi-definite programming (Semi-definite Programming, SDP) problem, and the semi-definite programming problem is a non-smooth convex optimization problem. When the size of the antenna array is large, the convex optimization method has problems such as high complexity and slow convergence speed. For this reason, many scholars adopt a better approximate solution algorithm - ADMM algorithm.

The ADMM algorithm has been widely used in the field of compressed sensing, such as image processing, channel estimation, wireless sensor network, etc. These studies still rely on grid partitioning. In recent years, related research has been extended to the field of gridless compressed sensing, such as radar imaging, weighted atomic norm minimization, etc.

At present, most of the DOA estimation algorithms based on compressed sensing sparse reconstruction are researched in the background of Gaussian noise. However, some noises in nature are often non-Gaussian and cannot be described by Gaussian distribution. Super-Gaussian noise is a type of non-Gaussian noise. Low-frequency atmospheric noise, speech signals, biomedical signals, and other impulsive noises are super-Gaussian. The probability density function of super-Gaussian noise has significant spikes and tailing phenomena, and the noise signal also has a few outliers with relatively large modulus values. This characteristic makes the statistical characteristics of this kind of non-Gaussian process significantly deviate from the Gaussian distribution. If the Gaussian distribution model is still used to describe these noises, the performance of the DOA estimation method originally designed based on the Gaussian noise mathematical model will decline.

Since the theory of gridless compressed sensing was proposed relatively late, in the existing research, only in the background of non-Gaussian noise, compressed sensing theory was not used for DOA estimation, such as literature (3.Wen-Jun Zeng, H.C.So , Lei Huang.lp-MUSIC:Robust Direction-of-Arrival Estimator for Impulsive NoiseEnvironments[J].IEEE Transactions on Signal Processing.Vol.61,No.17,Sep.2013,pp.4296-4308.); and in Under the background of non-Gaussian noise, using traditional compressive sensing technology to estimate DOA by dividing grid points, such as (4. Fei Sun, DOA estimation based on compressive sensing under non-Gaussian noise environment [D], Dalian Maritime University, 2015 .); and DOA estimation in Gaussian white noise background using mesh-free compressed sensing theory, such as (4.Zai Yang, Lihua Xie.Enhancing Sparsity and Resolution via Reweighted AtomicNorm Minimization[J].IEEE Transactions on Signal Processing , Vol.64, No.4, Feb.2016, pp.995-1006.). At present, there is no DOA estimation technology based on mesh-free compressive sensing theory in the super-Gaussian noise environment. Therefore, it is of great significance to study the DOA estimation method based on the mesh-free compressive sensing theory in the super-Gaussian noise environment.

Contents of the invention

The purpose of the present invention is to provide a DOA estimation method based on gridless compressed sensing under the background of super-Gaussian noise, so as to realize efficient and accurate DOA estimation.

The technical solution that realizes the object of the present invention is: a DOA estimation method based on gridless compressed sensing under the background of super-Gaussian noise, the steps are as follows:

(1) Determine the antenna array signal model, that is, receive signals through a linear antenna array with M array elements and equal spacing between each array element; the signal model received by the antenna array is expressed as

Among them, y indicates that the antenna array receives M noise-containing signals; the noise-free signal x is represented by an M-dimensional observation vector; is the M-dimensional steering vector, then the direction matrix s represents K source signals, w is M-dimensional zero-mean, super-Gaussian noise signal with variance σ ² ;

(2) According to the noise distribution type measured in the actual environment, select the norm of l _p to constrain, 1≤p<2;

(3) Determine the function expression for solving the noise-free signal x. The recovery problem of the noise-free signal x is described as an atomic norm minimization problem, which can be solved by using semidefinite programming theory. The functional expression of the atomic norm minimization problem The formula is:

Among them, st is the abbreviation of subject to, which indicates the constraint condition of the objective function; τ is a regular term, and the value is ||·|| _p represents the l _p norm of the vector or matrix, is the _p -th power of the lp norm; in addition, u is a vector of M dimension, T(u) represents the Toeplitz matrix of u, and its dimension is M×M; (·) ^H represents the conjugate transposition of a vector or matrix;

(4) The ADMM algorithm obtains the original solution, and the convex optimization toolkit can be used to solve the atomic norm minimization problem, or to reduce the complexity of the solution, the function expression in step (3) can be written as the Lagrangian function Form, and use ADMM-based first-order algorithm to solve the problem, get the estimate of the noise-free signal, that is, the original solution of the optimization problem

(5) Find the dual solution is the estimated value of the dual variable z of x; the function expression of the dual problem of the atomic norm minimization problem is

in is the dual norm of the atomic norm; due to the strong duality between the primal and dual problems, are the optimal solutions of the original solution and the dual solution, respectively, and the relationship between them is

(6) Solve the DOA, divide the grid points in the range of [-90°, 90°], and find The modulus of is 1, that is The modulus is The angle corresponding to the point at time is used as the estimated value of DOA; because Satisfy

see The maximum value of is equal to Using this method to solve, make it closer to the angle support set of the original signal, so as to complete the DOA estimation.

Compared with the prior art, the present invention has significant advantages: (1) By using the statistical characteristics of noise, the noise item is effectively constrained by using a suitable norm, and finally the original signal is obtained by using the relationship between the original solution and the dual solution set of angular supports. The simulation results in Fig. 3 show that due to the successful use of the statistical properties of super-noise, the DOA estimation has a high accuracy. (2) It is relatively easy to implement and has a good application prospect.

The present invention will be described in further detail below in conjunction with the accompanying drawings.

Description of drawings

Figure 1 is a uniform linear array receiving signal model.

Fig. 2 is a flow chart of the present invention.

Fig. 3 is an effect diagram of DOA estimation using different norms to constrain noise items under the background of super-Gaussian noise in the present invention.

Detailed ways

In conjunction with FIG. 1 , the antenna array signal model of the present invention is described: the gridless compressed sensing technology is applied to DOA estimation. Consider such a problem: there are K targets with unknown angles randomly distributed in the region θ∈[-90°,90°]. In order to determine their angles, the present invention receives signals through a linear antenna array (Uniform Linear Array, ULA) with the number of elements M, and then realizes DOA estimation.

The present invention is based on a uniform linear array, wherein each array element is an omnidirectional antenna, assuming that the number of array elements is M, and the distance between array elements is d. Assume that K narrowband far-field point source signals are incident on the antenna array from directions θ _k , k=1,...,K, that is, θ _k is the signal incident angle. Then at a certain moment, the signal received by the antenna array is expressed as

Where y=[y1,...,y _M ] ^T represents the M×1 dimensional column vector composed of the data received by M antenna elements at this moment, and x=[x ₁ ,...,x _M ] ^T represents the noise-free Signal, ( ) ^T represents the transpose of a vector or matrix. is an M×K dimensional direction matrix, K is the number of sources, is the steering vector corresponding to the phase difference ω _k , and ω _k =2πdsin(θ _k )/λ represents the phase difference between two adjacent array elements. λ is the signal wavelength, S=[s ₁ ,…,s _k ] ^T is a K×1-dimensional signal vector composed of K signals s ₁ ,…,s _k , w=[w ₁ ,…,w _M ] ^T is An M×1 dimensional vector composed of noise on each antenna element. It is assumed that the signal is statistically independent from the noise of the array elements, and the noise of each array element is also independent of each other.

In conjunction with Fig. 2, the DOA estimation method based on meshless compressed sensing under the background of super-Gaussian noise of the present invention is based on the theory of meshless compressed sensing, and the _lp norm is used to constrain the super-Gaussian noise in the received signal, through Solve the atomic norm minimization problem and its dual problem, so as to realize the estimation of DOA. In summary, in the field of mesh-free compressed sensing, the DOA estimation problem can be decomposed into two sub-problems: solving the atomic norm minimization problem, and obtaining the estimated value of the original signal Solve the dual of the atomic norm minimization problem to obtain DOA estimates. These two sub-problems can be decomposed into the following six steps.

Specific steps are as follows:

Step 1, determine the antenna array signal model. A uniform linear array (Uniform linear array, ULA) is adopted, in which each array element is an omnidirectional antenna, assuming that the number of array elements is M, and the distance between array elements is d. Assume that K narrowband far-field point source signals are incident on the antenna array from directions θ _k ∈ [-90°, 90°], k=1,...,K, that is, θ _k is the signal incident angle. Then at a certain moment, the signal received by the antenna array is expressed as

Where y=[y1,...,y _M ] ^T represents the M×1 dimensional column vector composed of the data received by M antenna elements at this moment, and x=[x ₁ ,...,x _M ] ^T represents the noise-free Signal, ( ) ^T represents the transpose of a vector or matrix. is the steering vector corresponding to the phase difference ω _k , ω _k =2πdsin(θ _k )/λ represents the phase difference between two adjacent array elements, A=[a(θ ₁ ),a(θ ₂ ),… , a(θ _K )] is an M×K dimensional direction matrix. λ is the signal wavelength, S=[s ₁ ,…,s _k ] ^T is a K×1-dimensional signal vector composed of K signals s ₁ ,…,s _k , w=[w ₁ ,…,w _M ] ^T is An M×1 dimensional vector composed of noise on each antenna element. It is assumed that the signal is statistically independent from the noise of the array elements, and the noise of each array element is also independent of each other;

A super-Gaussian noise signal with σ ² .

The super-Gaussian noise signal in step 1 obeys the generalized Gaussian distribution, and the probability density function of the generalized Gaussian distribution is:

Where w is a function variable, m represents its mean value, Γ( ) is the gamma function, α reflects the width of the peak of the generalized Gaussian probability density function; if the variance of x is limited to 1, α is expressed as β controls the shape of the generalized Gaussian probability density function, which is related to the sharpness of the probability density function. The smaller the β value, the sharper the density function;

The value of β determines the type of source signal: when the mean value of x is 0 and the variance is 1, if β=2, the signal x is a signal of standard Gaussian distribution; if 0<β<2, x is a super-Gaussian signal, where β =1, the signal x is a Laplacian distribution signal; if β>2, x is a sub-Gaussian signal; if β=+∞, the signal x converges to a uniform distribution U(-α,α) according to probability.

In the simulation experiment, according to the above-mentioned generalized Gaussian probability density function, we used the literature (6. Zhu Xiaoling, Jiang Hao, Research and Implementation of Pseudorandom Numbers with Arbitrary Probability Distribution [J], Computer Technology and Development, Vol.17, No. 12, Dec.2007, pp.116-118, 168.) The rejection method proposed to generate a super-Gaussian noise signal. Among them, super Gaussian noise is generated by setting β=0.5.

Step 2, according to the noise distribution type measured in the actual environment, select the norm of l _p for constraint, 1≤p<2.

Aiming at the DOA estimation problem based on gridless compressed sensing under super-Gaussian noise environment, the present invention proposes a method of using l _p norm constraint for optimizing the noise item of the objective function. The traditional DOA estimation method assumes that the background noise is Gaussian white noise, so the _l2 norm is used to constrain the noise term; and under the background of super-Gaussian noise, the same method has the problem of performance degradation; for this, an application proposed by the present invention The meshless compressed sensing DOA estimation method under the background of super-Gaussian noise effectively constrains the noise item by using the appropriate l _{p norm} by using the statistical characteristics of the noise; 2) The norm is constrained.

Since some noise signals have strong super-Gaussian properties, the norm of lp (0< _p <1) is required to be constrained. The original problem is no longer a convex optimization problem, but a non-convex _lp norm minimization problem, it is difficult to obtain a global minimum. Therefore, in the present invention, the l _p (0<p<1) norm will not be considered, and only the l _p (1≤p<2) norm is used to constrain the super-Gaussian noise.

Step 3: Determine the function expression for solving the noise-free signal x. The recovery problem of the noise-free signal x is described as an atomic norm minimization problem, which can be solved by using semidefinite programming theory. According to the literature (2.B.N.Bhaskar,G.Tang,B.Recht.Atomic norm denoising with applications to line spectral estimation[J].IEEE Transactions on Signal Processing,2013,61(23):5987-5999), the atomic norm The number minimization problem can be expressed as

in tr(·) represents the trace of the matrix.

Therefore, the function expression for minimizing the atomic norm is:

Among them, st is the abbreviation of subject to, which indicates the constraint condition of the objective function; τ is a regular term, and the value is ||·|| _p represents the l _p norm of the vector or matrix, is the _pth power of the lp norm; in addition, u is an M-dimensional vector, T(u) represents the Toeplitz matrix of u, and its dimension is M×M; (·) ^H represents the conjugate transposition of a vector or matrix. T(u) represents the Toeplitz matrix of u:

u _j represents the jth value of u. The function expression can obtain the solution of parameters x, u, t through the SeDuMi toolkit (8. JFSturm, "Using SeDuMi 1.02, a MATLAB toolbox for optimization oversymmetric cones," Optimization Methods and Software, vol.11-12, pp .625-653, 1999.);

In gridless compressed sensing technology, the parameter space is no longer divided into a finite number of grids, but directly regarded as a continuous variable. make is a set of atoms, in DOA estimation problem, the set can be written as its convex hull is a compact set that is centrally symmetric with respect to the origin and contains the origin as an interior point. a is set Atoms in , thus which is The elements in are extreme point of . Convex hull The norm defined by the scaling function is the atomic norm, using means that there is

where inf is the infimum (or the greatest lower bound) of the function. For the atomic norm, the atoms in the atomic set are the basic units used to construct any signal, then the convex hull The low-dimensional plane of also corresponds to a signal composed of very few corresponding atoms, so the atomic norm is actually given to the collection Added sparse constraints. This constraint will set the It is regarded as an infinite dictionary describing continuously changing parameters, and at the same time adding sparse constraints does not introduce discretized representation, which avoids the problem of base mismatch. x is a linear combination of the corresponding atoms. The atomic norm has semi-definite programming (Semi-definite programming, SDP) properties, and the linear semi-definite programming method can be used to solve the atomic norm. According to Caratheodory's lemma, any positive semi-definite Toeplitz matrix can be decomposed by Vandermonde, thus converting the atomic norm minimization problem into a semi-definite programming problem. Therefore, it is necessary to use semidefinite programming theory to solve it. We can use some existing convex optimization tools, such as SeDuMi, etc., to solve the above problem and obtain the estimate of the received noise-free signal

In the literature (2.BNBhaskar,G.Tang,B.Recht.Atomic norm denoising with applications to line spectral estimation[J].IEEE Transactions on Signal Processing,2013,61(23):5987-5999), the original solution and the dual solution The relationship between , as well as the upper and lower bounds of the regularization term τ, is the result of the derivation of the optimization model based on l ₂ norm constraints. But this result is no longer valid in the optimization model with l _p (1≤p<2) norm constraints.

Under the condition of using the l ₂ norm constraint, when , that is, in the direction of arrival, The value of is always equal to the constant τ. However, under the l _p (1≤p<2) norm constraint, Although there will be a peak in the direction of arrival, this value is no longer a constant. In practical situations, the number of signal sources is often unknown. When the signal-to-noise ratio is low, spurious peaks will appear on the spatial spectrum estimated by DOA, sometimes even higher than the peak in the direction of arrival. At this point, the signal source and the spurious peak will be difficult to distinguish. Our upper and lower bounds on the regularization term, and the original solution and the dual solution The relationship among them is deduced to make it suitable for the optimization model constrained by super-Gaussian noise background and l _p (1≤p<2) norm.

The process of solving the upper and lower bounds of the regular term τ in step 3 is as follows:

1) The upper bound of the regular term τ

The dual norm of the atomic norm of the variable z is denoted as:

The maximum modulus value of the polynomial Zn _is defined as

Assuming that the variables w ₁ ,…,w _N satisfy the generalized Gaussian distribution, the probability density function of the generalized Gaussian distribution is:

The values of the parameters are as follows: β=0.5, m=0,

The variable z satisfies z=|w| ^p-2 ·w, namely

The probability density function of z is thus

Therefore

in, The detailed derivation process is as follows:

where Γ(v,z) is an incomplete gamma function defined as follows:

Set parameter δ=2log(N), get

Among them, the standard deviation of the noise variable w is σ, and the standard deviation of the variable z is σ ^p-1 , from formulas (3.1) and (3.7)

Set N=4πnlog(n), then upper bound writing

When n≥3, according to claim 1, the upper bound of the regular term τ is upper bound times;

Among them, when p=1, That is, the upper bound of the regular term τ is:

2) The lower bound of the regular term τ

Let F be the distribution function of variables u ₁ ,…,u _N , and let M _n =max(u ₁ ,…,u _N ), then the distribution function of zero-mean hyper-Gaussian variables is

When n≥5, the lower bound of the regular term τ is

Among them, when p=1, That is, the upper bound of the regular term τ is

Step 4, the ADMM algorithm obtains the original solution, and uses convex optimization tools (such as the SeDuMi toolkit) to solve the atomic norm minimization problem, or in order to reduce the complexity of the solution, write the function expression in step 3 as Lagrang The form of the daily function, and use the first-order algorithm based on ADMM to solve it, and get the estimate of the noise-free signal, that is, the original solution (estimated value) of the optimization problem

The algorithm introduces the Lagrangian multiplier Λ, and makes According to (9. Sun Zhiqiang, Bai Shengjian, Zheng Yongbin, Liu Wei (translation), Introduction to Optimization (Fourth Edition) [M], Electronic Industry Press, pp.380-392), the optimization objective function expression in step three, It can be transformed into the form of Lagrangian function:

Where ||·|| _F represents the Frobenius norm of the vector or matrix, and <·> represents the inner product of the vector or matrix, then (·) ^H represents the conjugate transpose of a vector or matrix; ρ represents the penalty term of the Lagrangian function, which can control the convergence rate of the function. By deriving the matrix or vector, the parameters (t, u, x, Z, Λ) are iteratively updated until the algorithm converges.

l represents the lth iteration.

Step five, get the dual solution is the estimate of the dual variable z of x. For each norm, there is a corresponding dual norm. Compared with the original norm, the dual norm usually has some useful structures and properties, so it is widely used in the analysis and application of many specific problems. The dual norm of the atomic norm is equivalent to the atomic set The support set function of

in Represents the real part of the inner product z ^H x, sup is the supremum (or least upper bound) of the function, is the dual norm of the atomic norm. Due to the strong duality between the primal and dual problems, this means that are the optimal solutions to the primal and dual problems, respectively.

The function expression of the dual problem of atomic norm minimization problem is

in is the dual norm of the atomic norm; due to the strong duality between the primal and dual problems, are the optimal solutions of the original solution and the dual solution, respectively, and the relationship between them is

The solution process of the dual function expression in step five is as follows:

The functional expression for the atomic norm minimization problem is rewritten as:

And transform it into the form of Lagrangian function:

Denote the dual function by g(z):

Maximize g(z), then the third term is equal to zero, as shown in formula (5.5).

The original solution in step five and the dual solution The process of solving the relationship is as follows:

function exist The minimum value is obtained at , if the constant α∈(0,1), for the variable x, which is

is a convex function, according to the properties of convex functions so

For convenience, let Then the right-hand formula of (1) becomes

make f(α) satisfies the condition of Taylor expansion under the condition of α→0, so

Let α→0, and substitute the above formula into formula (1) and (2), we get

order variable Among them, represents the dot product of vectors, that is, for vectors The new vector obtained after taking the absolute value of each element in the power of p-2, and The dot product of is equal to Therefore, formula (3) can be transformed into

According to the definition of atomic norm and dual norm,

where inf is the infimum or maximum lower bound of the function, sup is the supremum or minimum upper bound of the function, according to the relationship between formula (5) and the upper and lower bounds of the function, formula (4) can be organized as

Through (6), it is concluded that if and only if hour, get the minimum value;

Finally, under the condition of p=2, τ satisfies the relation Still holds.

Step 6, solve the DOA, divide the grid points in the range of [-90°, 90°], and find The modulus of is 1, that is The modulus is The angle corresponding to the point at time is used as the estimated value of DOA; because Satisfy

The maximum value of is equal to Using this method to solve, make it closer to the angle support set of the original signal, so as to complete the DOA estimation.

In step six, the specific process of DOA solution is as follows:

In the DOA estimation problem, θ _i ∈ [-90°, -90°+Δθ,...,90°], Δθ represents the resolution of the angle; the steering vector form of the vector a is because Satisfy

That is, in this question The maximum value of is equal to You can search within the range of [-90°,90°], that is, get With the real part of the inner product of each vector a, find The modulus is The angle corresponding to the point at time is used as the estimated value of DOA. The angle resolution Δθ value can be set according to the needs. In the simulation experiment, Δθ can be accurate to 0.0001° or even higher. Such a high resolution makes the error between the actual parameters and the estimated value small enough to Implement super-resolution DOA estimation. Solve by this method, make it closer to the angle support set of the original signal, and complete the DOA estimation.

In step six, we use the derived original solution and the dual solution The relationship between and obtain the angle support set of the original signal. The present invention successfully utilizes the statistical characteristics of super-Gaussian noise. Compared with the traditional scheme using _l2 norm constraints, it reduces the error of DOA estimation and increases the probability of successful DOA estimation. Under the same conditions A higher DOA estimation accuracy is obtained. In a nutshell, the present invention first obtains the original solution by solving the optimization objective function and the dual solution Then solve the dual problem of optimizing the objective function to obtain the estimated value of DOA.

It is worth mentioning that, according to the traditional compressive sensing theory for DOA estimation, it is necessary to divide the grid in the parameter space in advance, and assume that the signal parameters are exactly on the grid points, and at the same time choose a suitable sparse transformation basis. In fact, the signal parameters are usually not exactly on the pre-divided grid points, which will cause the mismatch of the basis; if the grid division is too fine, so that the signal parameters completely fall on the grid points, it will cause the base dictionary The correlation between adjacent atoms is too strong. The above two factors will lead to the problem of reduced performance of compressed sensing reconstruction.

However, applying the meshless compressed sensing theory to DOA estimation does not involve grid division and selection of sparse transformation bases in the modeling process, and there is no problem that the performance of compressed sensing reconstruction will be reduced due to the above two factors. The problem reflects the superiority of the mesh-free compressive sensing theory.

The performance analysis of the present invention is carried out in conjunction with Fig. 3 . In order to test the performance of the present invention, we have carried out the following simulation experiments, firstly, the simulation environment is explained: a linear antenna array is selected, the number of array elements is M=64, and the number of information sources is K=4. In the simulation process, take ρ=2. All simulation experiments were repeated 100 times, and the results were averaged.

The performance of the present invention is tested as follows: under the background of super Gaussian noise, different _lp norms are used to constrain the noise term. As mentioned above, super-Gaussian noise is suitable to be constrained by l _p (1≤p<2) norm. Then, under the same signal-to-noise ratio, by comparing the estimated value and the real value of DOA under different norm constraints, it is shown that the high-precision estimation of the direction of arrival can be realized by using the appropriate l _p norm. Here, in the background of super Gaussian noise, the superiority of the DOA estimation method using the l _p (1≤p<2) norm to constrain the noise term is verified;

This scheme adopts l _1.2 norm and l _1.5 norm respectively. The signal-to-noise ratio of the input signal is 5dB, β=1.5, and the four direction-of-arrival angles are randomly assumed to be -59.35°, 12.70°, 23.52°, and 68.70°, respectively.

Fig. 3 is the spatial spectrum estimated by DOA in the first simulation experiment, which shows the validity of l _p (1≤p<2) norm constraint. Using l _1.2 and l _1.5 norm-constrained DOA estimation methods, the results are -59.57°, 12.55°, 23.89°, 68.59°, -59.23°, 12.46°, 23.88°, 68.54°, and the average error is 0.21 °, 0.22°. The spatial spectrum can reach its peak at the corresponding direction of arrival. Therefore, under the background of super-Gaussian noise, using l _1.2 and l _1.5 norms can effectively constrain the noise term and successfully estimate DOA.

Claims (6)

1. A DOA estimation method based on mesh-free compressed sensing under a super Gaussian noise background is characterized by comprising the following steps:

(1) determining an antenna array signal model, namely receiving signals through a linear antenna array with an array element number of M and equal array element spacing; the signal model received by the antenna array is expressed as

Wherein y representsThe antenna array receives M signals containing noise; representing a noise-free signal x by an M-dimensional observation vector;is a steering vector of M dimensions, then the direction matrixs represents K source signals, w is the zero mean of the M dimensions, and the variance is sigma²The super gaussian noise signal of (a);

(2) according to the noise distribution type measured by the actual environment, selecting l_pThe norm of (1) is restricted, and p is more than or equal to 1 and less than 2;

(3) determining a function expression for solving the noiseless signal x, describing the problem of recovering the noiseless signal x as an atomic norm minimization problem, and solving the noiseless signal x by adopting a semi-definite programming theory method, wherein the function expression of the atomic norm minimization problem is as follows:

wherein s.t. is an abbreviation for subject to, representing a constraint of the objective function; tau is a regularization term and takes a value of||·||_pRepresenting a vector or matrix of l_pThe norm of the number of the first-order-of-arrival,is 1_pThe norm to the power p; further u is a vector of dimension M, T (u) denotes the Toeplitz matrix of u, the dimension of which is M × M; (.)^HRepresents a conjugate transpose of a vector or matrix;

(4) solving an original solution by using an ADMM algorithm, solving the atomic norm minimization problem by using a convex optimization toolkit, or writing a function expression in the step (3) into a Lagrangian function form to reduce the complexity of the solution, and using first-order calculation based on the ADMMSolving to obtain an estimate of the noise-free signal, i.e. the original solution of the optimization problem

(5) Solving a dual solution Is an estimate of the dual variable z of x; the functional expression of the dual problem of the atomic norm minimization problem is

WhereinIs a dual norm of the atomic norm; due to the strong duality between the original and duality problems, respectively an optimal solution of an original solution and a dual solution, and the relationship between the two solutions is

(6) Solving DOA at [ -90 DEG, 90 DEG ]]Dividing grid points in the range of (1) and searchingHas a modulus of 1, i.e.Is a mold ofTaking the angle corresponding to the time point as a DOA estimated value; due to the fact thatSatisfy the requirement of

Is equal toThe solution is carried out by the method, so that the solution is closer to the angle support set of the original signal, and the DOA estimation is completed.

2. The DOA estimation method based on mesh-free compressed sensing under the ultrahigh Gaussian noise background according to claim 1, characterized in that: the superss noise signal in the step (1) obeys generalized Gaussian distribution, and the probability density function of the generalized Gaussian distribution is as follows:

where w is a function variable, m represents its mean value, and Γ () is a gamma function，α reflects the width of the peak of the generalized Gaussian probability density function, and if the variance of x is limited to 1, α is expressed asβ controls the shape of the generalized Gaussian probability density function, and is related to the sharpness degree of the probability density function, and the smaller the β value is, the sharper the density function is;

the value of β determines the type of source signal, where β is 2 for a standard gaussian distribution when the mean value of x is 0 and the variance is 1, where x is a super gaussian distribution when 0 < β < 2, where β is 1, where x is a Laplacian distribution, where β > 2, x is a sub-gaussian distribution, and where β + ∞, where x converges probabilistically to a uniform distribution U (- α, α).

3. The DOA estimation method based on mesh-free compressed sensing under the ultrahigh Gaussian noise background according to claim 1, characterized in that: the solving process of the upper bound and the lower bound of the regularization term tau in the step (3) is as follows:

1) upper bound of regularization term τ

The dual norm of the variable z atomic norm is noted as:

polynomial Z_nIs defined as the maximum modulus value of

Assume variable w₁,…,w_NSatisfying generalized Gaussian distribution, the probability density function of the generalized Gaussian distribution is as follows:

the parameters are as follows, beta is 0.5, m is 0,

the variable z satisfies z ═ w-^p-2W, i.e.

Whereby the probability density function of z is

Thus, can obtain

Wherein,the detailed derivation procedure is as follows:

where Γ (v, z) is an incomplete gamma function, defined as follows:

setting parameter delta to 2log (N) to obtain

Where the standard deviation of the noise variable w is σ, the standard deviation of the variable z is σ^p-1Obtained from the formulae (3.1) (3.7)

If N is set to 4 π nlog (N), thenUpper bound writing of

When n ≧ 3, the upper bound of the regularization term τ is, as known from claim 1Of the upper boundDoubling;

wherein, when p is 1,i.e. the upper bound of the regularization term τ is

2) Lower bound of regularization term τ

Let F be a variable u₁,…,u_NIs given, while ordering M_n＝max(u₁,…,u_N) Then the distribution function of the zero-mean super Gaussian variable is

When n is more than or equal to 5, the lower bound of the regular term tau is

Wherein, when p is 1,i.e. the upper bound of the regularization term τ is

4. The DOA estimation method based on mesh-free compressed sensing under the ultrahigh Gaussian noise background according to claim 1, characterized in that: the solving process of the dual function expression in the step (5) is as follows:

the functional expression of the atomic norm minimization problem is rewritten as:

and converts it to the form of the lagrange function:

the dual function is represented by g (z):

maximizing g (z), the third term is equal to zero, as shown in equation (5.5).

5. The DOA estimation method based on mesh-free compressed sensing under the ultrahigh Gaussian noise background according to claim 1, characterized in that: original solution in step (5)And dual solutionThe solving process of the relationship is as follows:

function(s)In thattaking the minimum value, if let the constant α ∈ (0,1), for the variable x,namely, it is

||·||_AIs a convex function, according to the nature of the convex functionTherefore, it is not only easy to use

To show convenience, orderThe right expression of the formula (1) is

Order tof (α) satisfies the Taylor expansion condition under the condition of α → 0, so

let alpha → 0, substitute the above formula for the formulas (1) and (2) to obtain

Let variable quantityIn which a represents the dot product of vectors, i.e. subtended quantitiesA new vector obtained by taking the absolute value of p-2 power of each element in the vector, anda dot product of, equal toTherefore, the formula (3) can be changed into

According to the definition of atomic norm and dual norm,

wherein inf is the infinitesimal or maximum lower bound of the function, sup is the infinitesimal or minimum upper bound of the function, and the formula (4) can be arranged as

From (6), if and only if When the temperature of the water is higher than the set temperature,obtaining a minimum value;

finally, τ satisfies the relationship under the condition that p is 2The same is true.

6. The DOA estimation method based on mesh-free compressed sensing under the ultrahigh Gaussian noise background according to claim 1, wherein the DOA solving process in the step (6) is as follows:

in the DOA estimation problem, θ_i∈[-90°,-90°+Δθ,…,90°]Δ θ represents the resolution of the angle; the vector a is in the form of a steering vectorDue to the fact thatSatisfy the requirement of

That is, in this problemIs equal toAt [ -90 DEG, 90 DEG ]]Is searched within the range of (1), i.e. to obtainWith the real part of the inner product of each vector a, findIs a mold ofThe angle corresponding to the time point is used as a DOA estimated value, and the DOA estimated value is solved by the method so that the DOA estimated value is closer to an angle support set of an original signal to finish DOA estimation.