CN107817465B - DOA estimation method based on non-grid compressed sensing under super-Gaussian noise background - Google Patents
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Abstract
The invention discloses a DOA estimation method based on non-grid compressed sensing under the background of hypersonic noise, which comprises the steps of firstly 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, and according to the actual conditionNoise distribution type of environment measurement, selectionpThe norm of the non-noise signal x is constrained, then a function expression for solving the non-noise signal x is determined, the problem of recovering the non-noise signal x is described as an atomic norm minimization problem, and the non-noise signal x can be solved by adopting a semi-definite programming theoretical method; then, the original solution is obtained through the ADMM algorithm, and the dual solution is obtainedAnd finally, solving the DOA to enable the DOA to be closer to an angle support set of the original signal, thereby finishing DOA estimation. According to the DOA estimation method, the noise item is effectively constrained by adopting a proper norm through the statistical characteristic of noise, and finally, the angle support set of the original signal is obtained through the relation between the original solution and the dual solution, so that the DOA estimation with high efficiency and accuracy is realized.
Description
Technical Field
The invention belongs to a Direction-of-arrival (DOA) estimation technology of Grid-less Compressive Sensing (GCS), in particular to a DOA estimation method based on Grid-less Compressive Sensing under a super Gaussian noise background.
Background
It is well known that direction of arrival estimation has irreplaceable roles in many aspects of communication, speech, radar, sonar, etc. The compressed sensing technology has been widely applied to the spatial spectrum estimation problem due to its unique advantages. For DOA estimation, the compressed sensing technology utilizes the sparsity of signals in a space domain, only a small amount of observation data is needed to reconstruct the signals, and the algorithm still has robustness on the occasions of low Signal-to-Noise Rate (SNR) and high information source correlation. However, conventional super-resolution algorithms, such as Multiple Signal Classification (MUSIC) algorithm based on Signal subspace and rotation invariant technology Estimation (ESPRIT) algorithm, do not have this advantage. Through research and development in recent years, the sparse signal reconstruction algorithm can realize super-resolution DOA estimation under the condition of small samples and even single snapshot.
In many practical applications, the parameter space of the signal is continuous. To solve this problem, the conventional compressed sensing algorithm firstly discretizes the continuous parameter space, divides the whole parameter space into a finite number of grids, and assumes that the parameters of the signal can be represented by some grid points. Although the accuracy of estimation can be improved by the grid which is too dense, the correlation between adjacent atoms in the base dictionary is too strong, so that the constraint isocynapticity (RIP) of the dictionary is reduced, and the reconstruction performance of compressed sensing is reduced. In addition, the actual parameter values of the signal are likely not to fall on a discrete grid, and the assumed transform Basis is not representative of a sparse signal, which is called Basis Mismatch by researchers.
For the phenomenon, researchers find an optimal convex heuristic method for solving the underdetermined linear inverse problem, namely, the atomic norm is adopted to characterize the signal, that is, the least atoms are directly searched in a Continuous parameter space to represent a given signal without discretizing a sparse domain, and a concept of meshless Compressive Sensing or Continuous domain Compressive Sensing (CCS) is proposed. The parameter space is no longer divided into a finite number of meshes, but is directly viewed as one continuous variable. Tang et al in the literature (1.Tang Gong-guo, Bhaskar B N, ShahP, et al]IEEE Transactions on Information Theory,2013,59(11):7465-7490) proposes that the intervals between signal frequencies are at leastThen, only O (K · logK · logM) samples are needed to reconstruct the frequency of the signal, wherein epsilon is a small constant, K is the number of the signals,m is the length of the signal. The solution of the atomic norm relies on the Semi-definite Programming (SDP) problem, which is a non-smooth convex optimization problem. When the antenna array is large in scale, the convex optimization method has the problems of high complexity, low convergence speed and the like. To this end, many scholars employ a better approximate solution algorithm, the ADMM algorithm.
The ADMM algorithm has been widely used in the compressed sensing field, such as image processing, channel estimation, wireless sensor network, etc. These studies still rely on lattice partitioning. In recent years, related research has been extended to the field of meshless compressed sensing, such as radar imaging, weighted atomic norm minimization, and the like.
At present, DOA estimation algorithm based on compressed sensing sparse reconstruction is mostly researched under the background of Gaussian noise. Some noises in nature tend to be non-gaussian and cannot be described by gaussian distributions. And super gaussian noise is one class of non-gaussian noise. Low frequency atmospheric noise, speech signals, biomedical signals and other impulse noise are of a super gaussian nature. The probability density function of the super-Gaussian noise has obvious spike pulse and tailing phenomena, and a few abnormal values with larger module values can also appear in the noise signal. This characteristic significantly deviates the statistical properties of such non-gaussian processes from gaussian distributions, and if the gaussian distribution model is still used to describe the noise, the performance of the DOA estimation method originally designed based on the gaussian noise mathematical model will be degraded.
Since the time proposed by the mesh-free compressed sensing theory is relatively late, in the existing research, the DOA estimation is not performed by using the compressed sensing theory only under the non-gaussian noise background, such as documents (3.Wen-Jun Zeng, h.c.so, Lei huang.lp-MUSIC: Robust Direction-of-Arrival estimation for impulse noise environments [ J ]. IEEE Transactions on Signal processing.vol.61, No.17, sep.2013, pp.4296-4308.); and documents that perform DOA estimation by grid-partitioning using conventional compressed sensing techniques on a non-gaussian noise background, such as (4. DOA estimation based on compressed sensing [ D ] in a grand fly, non-gaussian noise environment, university of maritime, 2015.); and DOA estimation on a Gaussian white noise background using the theory of meshless compressed sensing, such as (4.Zai Yang, Lihua Xie. enhancing spark and Resolution via weighted AtomicNorm Minimization [ J ]. IEEE Transactions on Signal Processing, Vol.64, No.4, Feb.2016, pp.995-1006.). At present, no DOA estimation technology based on the mesh-free compressed sensing theory under the ultrahigh Gaussian noise environment is found. Therefore, the DOA estimation method based on the mesh-free compressed sensing theory has very important significance in the research of the ultra-high Gaussian noise environment.
Disclosure of Invention
The invention aims to provide a DOA estimation method based on non-grid compressed sensing under a super Gaussian noise background, and the DOA estimation method is efficient and accurate.
The technical solution for realizing the purpose of the invention is as follows: a DOA estimation method based on mesh-free compressed sensing under a super Gaussian noise background comprises 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 represents that the antenna array receives M noisy signals; 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 lpThe 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 lpThe norm of the number of the first-order-of-arrival,is 1pThe norm to the power p; further u is a vector of dimension M, T (u) represents a Toeplitz matrix of u, the dimension of which is 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 in order to reduce the complexity of the solution, and solving the problem by using a first-order algorithm based on the ADMM to obtain an estimation of a noise-free signal, namely 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
See thatIs 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.
Compared with the prior art, the invention has the following remarkable advantages: (1) and finally, an angle support set of the original signal is obtained by utilizing the relation between the original solution and the dual solution. The simulation results of fig. 3 show that the accuracy of DOA estimation is high due to the successful use of the statistical properties of the super-noise. (2) The method is relatively easy to realize and has good application prospect.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
Fig. 1 is a model of a uniform linear array received signal.
Fig. 2 is a flow chart of the present invention.
Fig. 3 is a diagram of the DOA estimation effect of the present invention under the super gaussian noise background and with different norms to noise term constraints.
Detailed Description
The antenna array signal model of the present invention is explained with reference to fig. 1: a mesh-free compressed sensing technique is applied to DOA estimation. Consider one such problem: targets with unknown K angles are randomly distributed in the region θ e [ -90 °,90 ° ]. In order to determine their angles, the present invention receives signals through a Uniform linear antenna array (ULA) with an array element number M, and further implements DOA estimation.
The invention is based on a uniform linear array, wherein each array element is an omnidirectional antenna, the number of the array elements is assumed to be M, and the distance between the array elements is d. Suppose K narrow-band far-field point source signals are respectively from the direction thetakK1, …, K, incident on the array, i.e. θkIs the signal angle of incidence. At a certain moment the signal received by the antenna array is represented as
Wherein y is [ y1, …, yM]TAn M × 1-dimensional column vector representing a data composition received by M antenna elements at the time, x ═ x1,…,xM]TRepresenting a noise-free signal, (.)TRepresenting a transpose of a vector or matrix.Is a direction matrix with dimension of M multiplied by K, K is the number of information sources,is corresponding to the phase difference omegakOf the steering vector, ωk＝2πdsin(θk) And/lambda represents the phase difference between two adjacent array elements. λ is signal wavelength, S ═ S1,…,sk]TFor K signals s1,…,skA composed K × 1-dimensional signal vector, w ═ w1,…,wM]TAn M x 1 dimensional vector of noise on each antenna element. The signals and the array element noises are assumed to be statistically independent, and the array element noises are also mutually independent.
With reference to fig. 2, the DOA estimation method based on non-grid compressed sensing under the hypersonic noise background of the invention is based on the non-grid compressed sensing theory and adopts lpThe norm constrains the hypersonic noise in the received signal, and the estimation of DOA is realized by solving the atomic norm minimization problem and the dual problem thereof. In summary, in the domain of mesh-free compressed sensing, the DOA estimation problem can be decomposed into two sub-problems: solving the atomic norm minimization problem to obtain the estimated value of the original signalAnd solving the dual problem of the atomic norm minimization problem to obtain the DOA estimated value. These two subproblems can be decomposed into the following six steps in total.
The method comprises the following specific steps:
step one, determining an antenna array signal model. A Uniform Linear Array (ULA) is used, in which each element is an omni-directional antenna, assuming that the number of elements is M and the distance between the elements is d. FalseSetting K narrow-band far-field point source signals from direction thetak∈[-90°,90°]K1, …, K, incident on the array, i.e. θkIs the signal angle of incidence. At a certain moment the signal received by the antenna array is represented as
Wherein y is [ y1, …, yM]TAn M × 1-dimensional column vector representing a data composition received by M antenna elements at the time, x ═ x1,…,xM]TRepresenting a noise-free signal, (.)TRepresenting a transpose of a vector or matrix.Is corresponding to the phase difference omegakOf the steering vector, ωk＝2πdsin(θk) Where/λ represents the phase difference between two adjacent array elements, and A ═ a (θ)1),a(θ2),…,a(θK)]Is a direction matrix of dimension M × K. λ is signal wavelength, S ═ S1,…,sk]TFor K signals s1,…,skA composed K × 1-dimensional signal vector, w ═ w1,…,wM]TAn M x 1 dimensional vector of noise on each antenna element. Assuming that the signal and the array element noise are statistically independent, and the array element noises are also mutually independent;
is σ2Of the super gaussian noise signal.
The superss noise signal in the first step 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, Γ (·) is a gamma function,alpha reflects the width of the peak of the generalized Gaussian probability density function; if x is restrictedVariance is 1 and alpha is expressed asBeta 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 beta value is, the sharper the density function is;
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 is 2, the signal x is a signal of a standard gaussian distribution; if beta is more than 0 and less than 2, x is a super Gaussian signal, wherein when beta is 1, the signal x is a Laplacian distributed signal; if beta is more than 2, x is a sub-Gaussian signal; if β ∞, the signal x converges probabilistically to a uniform distribution U (- α, α).
In simulation experiments, we generated the super-gaussian noise signal according to the generalized gaussian probability density function described above, using the rounding method proposed in the literature (6. zhuiling, zingahao, random probability distribution, pseudo-random number research and implementation [ J ], computer technology and development, vol.17, No.12, dec.2007, pp.116-118,168.). Where super gaussian noise is generated by setting β to 0.5.
Step two, according to the noise distribution type measured by the actual environment, selecting lpThe norm of (1) is restricted, and p is more than or equal to 1 and less than 2.
Aiming at the DOA estimation problem based on non-grid compressed sensing in the super-Gaussian noise environment, the invention provides a method for optimizing the noise item of the objective function by adopting lpNorm constraining method. The traditional DOA estimation method assumes that the background noise is white Gaussian noise, so l is adopted2A norm constrained noise term; in the ultrahigh Gaussian noise background, the same method has the problem of performance degradation; in contrast, the grid-free compressed sensing DOA estimation method applied to the ultrahigh Gaussian noise background provided by the invention adopts proper l to the noise item by utilizing the statistical characteristic of the noisepCarrying out effective constraint on the norm; hypersonic noise is suitably employedpAnd (p is more than 0 and less than 2) norm for constraint.
Since some noise signals have strong supersonicity, l is requiredpThe norm of (0 < p < 1) is restricted, the original problem is not a convex optimization problem any more, butA non-convexpNorm minimization problem, it is difficult to obtain global minima. Therefore, in the present invention, l will not be consideredpFor the case of norm (0 < p < 1), only l is used for the Gaussian noisepAnd (p is more than or equal to 1 and less than 2) carrying out constraint.
And step three, determining a function expression for solving the noise-free signal x, describing the problem of recovering the noise-free signal x as an atomic norm minimization problem, and solving by adopting a semi-definite programming theory method. According to the document (2. B.N.Bhaskan, G.Tang, B.Recht.atomic norm differentiating with applications to line spectral estimation [ J ]. IEEE Transactions on Signal Processing,2013,61(23):5987-
Whereintr (-) denotes the trace of the matrix.
Thus, the functional expression for this atomic norm minimization is:
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 lpThe norm of the number of the first-order-of-arrival,is 1pThe norm to the power p; further u is a vector of dimension M, T (u) represents a Toeplitz matrix of u, the dimension of which is M; (.)HRepresenting the conjugate transpose of a vector or matrix. T (u) Toeplitz matrix representing u:
ujthe jth value of u is indicated. The functional expression can be solved by SeDuMi toolkit to obtain the solution of the parameters x, u, t (8.J.F. Sturm, "Using SeDuMi 1.02, A MATLAB toolbox for Optimization overmetrics conjugates," Optimization Methods and Software, vol.11-12, pp.625-653,1999.);
in the mesh-free compressed sensing technique, the parameter space is no longer divided into a finite number of meshes, but is directly viewed as one continuous variable. Order toIs an atomic set, which can be written asConvex hullIs a centrosymmetric compact with respect to the origin and contains the origin as the interior point. a is a setOf (b) therebyNamely, it isAll the elements in (A) areThe extreme point of (c). At the moment, the convex hullIs defined as an atomic norm, usingIndicates that there is
Where inf is the infimum (or maximum lower bound) of the function. For the atomic norm, where atoms in the set of atoms are the basic units for constructing any one signal, then the convex hullAlso corresponding to a signal consisting of a few corresponding atoms, and thus an atomic normIs actually a given setThe sparsity constraint is increased. This constraint will assembleThe method is regarded as an infinite dictionary for describing continuously-changing parameters, and meanwhile, discretization representation is not introduced when sparse constraint is added, so that the problem of mismatch of bases is avoided. x is the linear combination of the corresponding atoms. The atomic norm has a Semi-definite programming (SDP) property, and the atomic norm can be solved by adopting a linear Semi-definite programming method. According to the Caratheodory theorem, any semi-definite Toeplitz matrix can be subjected to Vandermonde decomposition, so that the atomic norm minimization problem is converted into a semi-definite planning problem. Therefore, it is necessary to apply the semi-definite programming theory to solve. The problem can be solved by using some existing convex optimization tools, such as SeDuMi, and the like, so as to obtain the estimation of the received noiseless signal
In the literature (2. B.N.Bhaskan, G.Tang, B.Recht.atomic approximation with applications to line separation [ J.]IEEE Transactions on Signal Processing,2013,61(23):5987-And dual solutionThe relationship between, and the upper and lower bounds of the regularization term τ are based on l2The derivation of the norm-constrained optimization model results. But this result is obtained byp(p is more than or equal to 1 and less than 2) norm constraint is not applicable to the optimization model.
In the use of2Under the condition of norm constraint, whenWhen the wave is directed, i.e. in the direction of arrival,always equal to the constant τ. However, when using lp(p is more than or equal to 1 and less than 2) under the norm constraint,although a peak occurs 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, a spurious peak appears on the spatial spectrum of the DOA estimate, sometimes even higher than the peak in the direction of arrival. At this time, the signal source and the hetero-peak will be difficult to distinguish. Our upper and lower bounds on the regularization term, and the original solutionAnd dual solutionThe relationship between the two is derived to be suitable for the background and the sum of the Gaussian noisep(p is more than or equal to 1 and less than 2) norm constraint optimization model.
The solving process of the upper and lower bounds of the regularization term τ in step three is as follows:
1) upper bound of regularization term τ
The dual norm of the variable z atomic norm is noted as:
polynomial ZnIs defined as the maximum modulus value of
Assume variable w1,…,wNSatisfying generalized Gaussian distribution, the probability density function of the generalized Gaussian distribution is as follows:
the values of the parameters are as follows: β 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 the parameter delta 2log (N) to obtain
Where the standard deviation of the noise variable w is σ, the standard deviation of the variable z is σp-1Obtained from the formulae (3.1) and (3.7)
Setting N 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,…,uNIs given, while ordering Mn＝max(u1,…,uN) 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
Step four, the ADMM algorithm obtains an original solution, a convex optimization tool (such as SeDuMi toolkit) is used for solving the atomic norm minimization problem, or in order to reduce the complexity of the solution, the function expression in the step three is written into a Lagrangian function form, the first-order algorithm based on the ADMM is used for solving, and the estimation of a noise-free signal, namely the original solution (estimation value) of the optimization problem is obtained
The algorithm introduces a Lagrange multiplier Lambda and ordersThe leading theory (fourth edition) is optimized according to (9. SundaeQiang, Bai Sheng, Zheng Yongbin, Liu Wei (translation) [ M]Electronic industry publishers, pp.380-392), the expression of the optimized objective function in step three can be converted into the form of lagrangian function:
wherein | · | purpleFRepresenting the Frobenius norm of a vector or matrix,<·>representing the inner product of a vector or matrix, then(·)HRepresents a conjugate transpose of a vector or matrix; ρ represents a penalty term of the lagrangian function, which can control the convergence rate of the function. And (3) repeatedly and iteratively updating the parameters (t, u, x, Z, Lambda) by a matrix or vector derivation method until the algorithm converges.
l denotes the l-th iteration.
Step five, solving dual solution Is an estimate of the dual variable z of x. For each norm there is a corresponding dual norm, which typically has some useful structure and properties compared to the original norm, and is therefore widely used for analysis and application of many specific problems. The dual norm of an atomic norm is equivalent to the atomic setSupporting set of functions, i.e.
WhereinRepresents the inner product zHThe real part of x, sup is the supremum (or minimum upper bound) of the function,is the dual norm of the atomic norm. This means that there is a strong duality between the original and duality problemsRespectively, are the optimal solutions to the original and dual problems.
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
In the step five, the solution process of the dual function expression 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):
let g (z) maximize, then the third term equals zero, as shown in equation (5.5).
Original solution in step fiveAnd dual solutionThe solving process of the relationship is as follows:
function(s)In thatTaking the minimum value, if let the constant alpha be (0,1), forAs far as the variable x is concerned,namely, it is
Is 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 (alpha) satisfies the Taylor expansion condition under the condition of alpha → 0, so
Let alpha → 0, substitute the above formula for the formulas (1) and (2) to obtain
Let variable quantityIn which a represents a vectorDot product, i.e. vectorA 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 ifWhen 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.
Step six, solving DOA at [ -90 degrees, 90 degrees ]]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.
In the sixth step, the specific process of solving the DOA is as follows:
in the DOA estimation problem, θi∈[-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 toCan be at [ -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 point in time is used as the DOA estimated value. The value of the angular resolution Δ θ can be set completely as required, and in simulation experiments, Δ θ can be accurate to 0.0001 ° or even higher, and such high resolution makes the error between the actual parameter and the estimated value small enough to realize super-resolution DOA estimation. The DOA estimation is completed by solving through the method and enabling the DOA estimation to be closer to an angle support set of an original signal.
In step six, we use the derived original solutionAnd dual solutionAnd obtaining the angle support set of the original signal according to the relation between the original signal and the reference signal. The invention successfully utilizes the statistical characteristic of the super-Gaussian noise, and compared with the traditional method which adopts l2The norm constraint scheme reduces the error of DOA estimation, increases the probability of successful DOA estimation, and obtains higher DOA estimation precision under the same condition. In summary, the present invention first solves the optimization objective function to obtain the original solutionAnd dual solutionThen theAnd solving the dual problem of the optimization objective function to obtain the DOA estimation value.
It is worth mentioning that, the DOA estimation is performed according to the conventional compressed sensing theory, the grid needs to be divided in the parameter space in advance, and it is assumed that the signal parameter is exactly on the grid point, and meanwhile, a proper sparse transformation base needs to be selected. In practice, the signal parameters will not normally be exactly on the pre-divided grid points, which causes a mismatch of the basis; if the meshing is too fine, so that the signal parameters fall completely on the mesh points, the correlation between the atoms adjacent to the base dictionary is too strong. Both of the above factors lead to a problem of reduced performance of compressed perceptual reconstruction.
However, the DOA estimation is carried out by applying the non-grid compressed sensing theory, the problems of grid division and sparse transformation base selection are not involved in the modeling process, the problem of reduction of the compressed sensing reconstruction performance caused by the two factors is solved, and the superiority of the non-grid compressed sensing theory is reflected.
The performance analysis of the present invention was performed in conjunction with fig. 3. To test the performance of the present invention, we performed the following simulation experiments, first describing the simulation environment: a linear antenna array is selected, the number of array elements M is 64, and the number of information sources K is 4. In the simulation process, ρ is taken to be 2. All simulation experiments were repeated 100 times and the results averaged.
The performance of the invention was tested as follows: using different l in a superss noise backgroundpThe norm constrains the noise term. As previously mentioned, superss noise is suitably employedpAnd (p is more than or equal to 1 and less than 2) norm for constraint. Then, under the condition of the same signal-to-noise ratio, by comparing the estimated value and the true value of DOA under the constraint of different norms, the fact that the proper l is adopted is illustratedpNorm, can realize the high-accuracy estimation of the direction of arrival. Here, it is verified that in a very Gaussian noise background,/, is usedp(p is more than or equal to 1 and less than 2) the superiority of the DOA estimation method of norm to noise term constraint;
the scheme respectively adopts l1.2Norm,/, of1.5And (4) norm. The signal-to-noise ratio of the input signal is 5dB, beta is 1.5, and four directions of arrival are randomly assumedThe angles are-59.35 °,12.70 °,23.52 °,68.70 °, respectively.
FIG. 3 is a spatial spectrum of DOA estimation in a first simulation experiment, showing lp(p is more than or equal to 1 and less than 2) validity of norm constraint. By means of1.2，l1.5Two norm-constrained DOA estimation methods resulted in-59.57 °,12.55 °,23.89 °,68.59 °, -59.23 °,12.46 °,23.88 °,68.54 °, respectively, with average errors of 0.21 °,0.22 °. The spatial spectrum can peak at the corresponding direction of arrival angle. Thus, in a very Gaussian noise background, l is used1.2，l1.5The two norms can both carry out effective constraint on the noise term and successfully estimate the 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 represents that the antenna array receives M noisy signals; 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); wherein theta iskIs the signal angle of incidence; d is the array element spacing of the antenna array; λ is the source signal wavelength;
(2) according to the noise distribution type measured by the actual environment, selecting lpThe 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 lpThe norm of the number of the first-order-of-arrival,is 1pThe norm to the power p; further u is a vector of dimension M, T (u) represents a Toeplitz matrix of u, the dimension of which is 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 in order to reduce the complexity of the solution, and solving by using a first-order algorithm based on the ADMM to obtain the estimation of a noise-free signal, namely 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 DOA estimation is completed; a is an atom in set A.
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, Γ (·) is a gamma function,alpha reflects the width of the peak of the generalized Gaussian probability density function; if the variance of the constraint x is 1, α is expressed asBeta 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 beta value is, the sharper the density function is;
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 is 2, the signal x is a signal of a standard gaussian distribution; if beta is more than 0 and less than 2, x is a super Gaussian signal, wherein when beta is 1, the signal x is a Laplacian distributed signal; if beta is more than 2, x is a sub-Gaussian signal; if β ∞, the signal 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 ZnIs defined as the maximum modulus value ofd is the array element spacing of the antenna array; λ is the source signal wavelength;
assume variable w1,…,wNSatisfying generalized Gaussian distribution, the probability density function of the generalized Gaussian distribution is as follows:
the values of the parameters are as follows: β 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 the parameter delta 2log (N) to obtain
Where the standard deviation of the noise variable w is σ, the standard deviation of the variable z is σp-1Obtained from the formulae (3.1) (3.7)
Setting N 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,…,uNIs given, while ordering Mn＝max(u1,…,uN) 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):
let g (z) maximize, then the third term equals zero.
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 the constant alpha belongs to (0,1), pairIn the case of the variable x, for example,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 (5.1) is
Order tof (alpha) satisfies the Taylor expansion condition under the condition of alpha → 0, so
Let alpha → 0, substitute the above formula for the formula (5.1), (5.2) to obtain
Let variable quantityWherein, watchIndicating dot products of vectors, i.e. vectors of vectorsA 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 (5.3) 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 (5.4) can be arranged into
From the formula (5.6), if and only ifWhen the temperature of the water is higher than the set temperature,the minimum value is obtained.
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 DOA estimation problemIn, thetai∈[-90°,-90°+Δθ,…,90°]Δ θ represents the resolution of the angle; the vector a is in the form of a steering vectorWherein d is the array element spacing of the antenna array; λ is the source signal wavelength; due 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.
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