CN114415110A - Direct positioning method for nonnegative sparse Bayesian learning - Google Patents

Direct positioning method for nonnegative sparse Bayesian learning Download PDF

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CN114415110A
CN114415110A CN202210023737.XA CN202210023737A CN114415110A CN 114415110 A CN114415110 A CN 114415110A CN 202210023737 A CN202210023737 A CN 202210023737A CN 114415110 A CN114415110 A CN 114415110A
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王伶
王秋平
谢坚
张兆林
张翔
宫延云
韩闯
陶明亮
粟嘉
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    • G—PHYSICS
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Abstract

The invention provides a direct positioning method of nonnegative sparse Bayesian learning, which is characterized in that covariance moment vectors are quantized into column vectors, and modeling derivation is carried out by utilizing the advantage of covariance matrix focusing energy, so that the virtual effective aperture of an array is effectively enlarged, and the algorithm resolution and positioning precision are improved; by utilizing the characteristic that the signal power value is nonnegative, a nonnegative sparse Bayesian learning algorithm is provided, the algorithm is not influenced by relevant signals, and meanwhile, the robustness of the algorithm can be ensured under the conditions of low signal-to-noise ratio and small snapshot number. The method avoids the problem of energy dispersion in the traditional modeling method, effectively enlarges the virtual effective aperture of the array by utilizing the advantage of covariance matrix focusing energy, and improves the algorithm resolution and positioning precision; the algorithm is not influenced by relevant signals, and meanwhile, the robustness of the algorithm can be guaranteed under the conditions of low signal-to-noise ratio and small snapshot number.

Description

Direct positioning method for nonnegative sparse Bayesian learning
Technical Field
The invention relates to the field of radiation source positioning, in particular to a signal direct positioning method.
Background
A review of the current state of the art shows that the direct localization problem is a typical highly non-convex non-linear parameter estimation problem, which is made even more problematic by the presence of co-channel multi-radiation sources. In recent years, an expert scholars converts an original non-convex continuous optimization problem into a discrete linear sparse coding problem by carrying out grid discretization on a radiation source parameter space and utilizing the sparsity of real parameters in the grid space, and then decodes the discrete linear sparse coding problem by a sparse recovery algorithm to obtain an approximate solution of a multi-source direct positioning problem. Research shows that under certain conditions, the multi-source direct positioning algorithm based on sparse representation breaks through the framework of the traditional multi-source direct positioning method and can obtain more excellent estimation performance.
However, it can be seen from the summary and analysis of the existing research that although scholars at home and abroad propose a lot of important direct positioning methods, the direct positioning method based on sparse representation in the existing literature still has the following disadvantages: (1) in general, the array received data is expressed as a steering vector matrix and a signal amplitude value when a signal reaches the array face and noise generated in the working process of array elements, and the existing algorithm directly utilizes the model to carry out sparse Bayesian learning modeling and has certain limitation; (2) the traditional direct positioning algorithm for sparse Bayesian learning assumes Gaussian distribution on signals, and is simple and easy to implement, but the Gaussian distribution has unobvious representation on the sparse characteristics of the signals, and is not enough for describing space sparse signals.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a direct positioning method of nonnegative sparse Bayesian learning. In order to overcome the defects of the existing sparse representation direct positioning technology, the invention provides a nonnegative sparse Bayesian learning direct positioning method by taking the direct positioning technology as a core, abandons the view that the traditional sparse Bayesian positioning method directly uses received data for modeling, quantizes covariance moment vectors into column vectors, utilizes the advantage of covariance matrix focusing energy for modeling derivation, effectively enlarges the virtual effective aperture of the array, and improves algorithm resolution and positioning accuracy; by utilizing the characteristic that the signal power value is nonnegative, a nonnegative sparse Bayesian learning algorithm is provided, the algorithm is not influenced by relevant signals, and meanwhile, the robustness of the algorithm can be ensured under the conditions of low signal-to-noise ratio and small snapshot number.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1, constructing a space sparse model for directly positioning covariance vectors:
step 1.1: establishing an array receiving data model: n receiving stations are arranged in a two-dimensional plane, Q fixed signal radiation sources are arranged at the far field of the receiving stations, and each signal radiation source is positioned at pq＝[xq,yq]T,q＝1,2,…,Q，pqPosition of the q-th radiation source, xqIs the abscissa, y, of the radiation sourceqThe signal radiated by each radiation source is a mutually-correlated narrow-band signal, and the N receiving stations receive the signal radiated by the radiation source and transmit data to the central processing station to realize the estimation of the position of the radiation source;
the data modeling of the K snapshots received by the nth receiving station is represented as:
rn＝Ansn+wn (1)
the number of array elements of each receiving station is M, the sampling fast beat number is K,
Figure BDA0003463616920000021
for the K-snapshot data received by the receiving station,
Figure BDA0003463616920000022
is an array flow pattern matrix, an(pq) The guide vector of the q radiation source reaching the nth observation station is determined by the relative position relationship between the target radiation source and the receiving station,
Figure BDA0003463616920000023
for the envelope of the signal received by the receiving station,
Figure BDA0003463616920000024
additive noise for the nth receiving station;
the signal source and noise are independent vectors that are uncorrelated with each other, and the covariance matrix of the array output is expressed as:
Figure BDA0003463616920000025
Figure BDA0003463616920000026
for the power value of the received signal, Q is 1,2, …, Q,
Figure BDA0003463616920000027
to describe the magnitude of the noise level, I, for the noise powerMAn identity matrix of dimension M × M;
the covariance matrix is elongated into vectors with:
Figure BDA0003463616920000028
Figure BDA0003463616920000029
expressed as the product of the KR,
Figure BDA00034636169200000210
is a vector of the signal power of the signal source,
Figure BDA00034636169200000211
and is
Figure BDA00034636169200000212
Wherein only the m component is 1, the rest elements are 0, and a new array flow pattern is defined by formula (3)
Figure BDA00034636169200000213
Wherein
Figure BDA00034636169200000214
Contains Q new guide vectors, the Q new guide vector is
Figure BDA00034636169200000215
Wherein
Figure BDA00034636169200000216
The degree of freedom of the virtual array is improved by the expression of the guide vector, and subsequent simulation shows that the positioning resolution is higher and the positioning performance is better; in the direct localization problem, the number Q of radiation sources is often unknown, and the localization target is to utilize the observed data covariance vector ynLocating the spatial positions of the Q radiation sources, N being 1,2, …, N;
step 1.2: constructing a space sparse model of the covariance vector;
in order to convert the source positioning problem into a sparse representation problem, which is inspired by the sparse reconstruction basic theory, the invention establishes a covariance vector as a model with sparse airspace, divides an interested space range into grid points, and totally G grids, wherein each grid is represented as the position of a potential radiation source, the divided grids are small enough, and the position of a real radiation source is on a grid point;
arranging corresponding samples of a space spectrum in a G multiplied by 1 vector, wherein G is far larger than the signal source number Q, the space signal vector is sparse, and under an ideal condition, most elements of the space signal vector are close to 0, and only Q elements have large difference with zero elements, so that the position estimation of a radiation source is obtained by adopting a sparse recovery method;
the radiation source is located at Q grid points, and under single sample fast shooting, formula (3) of the covariance vector is expressed as:
Figure BDA0003463616920000031
wherein phinIs composed of
Figure BDA0003463616920000032
A steering vector matrix extending to the whole grid space, an overcomplete array flow pattern matrix of known sparse representation, G is the grid point number of the division grid points, x is a non-negative signal power sparse vector, and M is assumed<<G、Q<<G, because the target radiation source is only positioned on a plurality of grid points, x is expressed as a space sparse power signal, only the corresponding grid point position with the signal source has a numerical value, and other elements are all 0;
according to the concept of sparse signals, the x vector of the known signal power is a sparse vector with sparsity of Q, wherein only Q nonzero elements exist, and other elements are zero elements, wherein the Q nonzero elements correspond to the positions of real radiation sources;
step 2, non-negative Laplace sparse prior distribution hypothesis of the signals:
(2.1) covariance vector a priori assumptions;
the prior distribution of covariance vector data is expressed by equation (4) as:
Figure BDA0003463616920000033
wherein,
Figure BDA0003463616920000034
k is the sampled fast beat number, since x is a non-negative vector, equation (5) is convenient to be expressed in real value, so the whole problem is put into the real value operation for discussion, and when the incident signal is circularly symmetric gaussian distribution, equation (5) is converted into the following real value gaussian distribution:
Figure BDA0003463616920000035
wherein,
Figure BDA0003463616920000036
and is
Figure BDA0003463616920000037
(2.2) an a priori assumption of signal power;
modeling x as the laplacian prior distribution of:
Figure BDA0003463616920000038
since x is a non-negative vector, the adjustment equation (7) is rewritten as:
Figure BDA0003463616920000041
in order to solve the problem that prior distribution in a formula (8) is not conjugated with conditional distribution of observed data, a hierarchical non-negative Laplace prior is provided; first, a first layer of non-negative laplacian priors is established as the following non-negative laplacian priors:
Figure BDA0003463616920000042
wherein N is+(xg|0,γg)＝2N(xg|0,γg),xg≥0，N+(xg|0,γg) Is a non-negative Gaussian probability density function with a mean value of 0, gammagIs a sparse enhancement hyper-parameter, and gamma is assumed to ensure sparsity of the probability distribution of xgFor the exponential prior distribution, G ═ 1,2, …, G, i.e., the second layer prior distribution is:
Figure BDA0003463616920000043
lambda is a hyper-parameter, two layers of prior distribution, namely an expression (9) and an expression (10), are synthesized, and the probability distribution density function of x is obtained as follows:
Figure BDA0003463616920000044
wherein the hyperparameter λ obeys the following gaussian distribution:
p(λ；ν)＝Γ(λ|ν,ν) (12)
nu is a normal number approaching to 0, and the edge distribution of x obtained in the way is Laplace distribution;
(2.3) noise power prior assumption;
to simplify the derivation process, assume
Figure BDA0003463616920000045
For no information prior, i.e.:
Figure BDA0003463616920000046
step 3, deducing nonnegative sparse Bayesian learning parameters based on Laplace prior distribution;
the direct positioning algorithm based on sparse Bayesian learning mainly derives posterior probability density according to a Bayesian rule according to a received data matrix and assumed probability distribution, and obtains the position of a target radiation source by maximizing a posterior probability density function; the method comprises the following specific steps:
and (3) obtaining a posterior probability density function by the prior probability hypothesis of the step (2) as follows:
Figure BDA0003463616920000051
the position of the non-0 element in the sparse signal x that maximizes the posterior probability density function of equation (14) is the position of the target radiation source.
In order to maximize the posterior probability density function of the formula, the invention adopts an EM algorithm: e, calculating the expectation of the complete likelihood logarithm; m, maximizing an expectation value by using an algorithm;
(3.1) the E step is expressed as the following formula:
Figure BDA0003463616920000052
wherein
Figure BDA0003463616920000053
Express the calculation obeys
Figure BDA0003463616920000054
The posterior probability of x is calculated as:
Figure BDA00034636169200000516
not less than the mean value mu of the matrix elements which are all greater than or equal to the symbolnSum variance ΣnRespectively as follows:
Figure BDA0003463616920000055
covariance matrix Λ ═ diag (α), and
Figure BDA0003463616920000056
representing the variance of the signal;
(3.2) in the M step, calculating the mean value of the posterior probability density function of gamma based on the mean value and the variance,
Figure BDA0003463616920000057
and maximize the mean to get γgThe updating expressions of (1) are as follows:
Figure BDA0003463616920000058
wherein,
Figure BDA0003463616920000059
further sum of λ
Figure BDA00034636169200000510
A posterior probability density function of
Figure BDA00034636169200000511
And maximize it, resulting in updated lambda and
Figure BDA00034636169200000512
the formula of (a):
Figure BDA00034636169200000513
Figure BDA00034636169200000514
wherein,
Figure BDA00034636169200000515
Figure BDA0003463616920000061
and simplify
Figure BDA0003463616920000062
Is composed of
Figure BDA0003463616920000063
Wherein xgThe respective mathematical expressions of the first order parameter and the second order parameter are as follows:
Figure BDA0003463616920000064
Figure BDA0003463616920000065
wherein, mug＝μ[g],
Figure BDA0003463616920000066
Error function
Figure BDA0003463616920000067
When the condition is satisfied
Figure BDA0003463616920000068
When there is<xg>→ 0 and
Figure BDA0003463616920000069
the approximation simplifies the calculation complexity of the algorithm, and the sparse Bayes method is used in the field of direct positioning; establishing a space sparse signal model; receiving the covariance of the data to perform vectorization operation; the Laplace distribution of the signal power is assumed, and the assumption is more consistent with a sparse scene, so that the positioning accuracy is higher.
To obtain gammagλ and
Figure BDA00034636169200000610
after the expression is updated, alternate iteration is performed until the sparse signal converges to a stable value and is not changed any more, and the position of the nonzero value of the sparse signal at the moment is the position of the target radiation source.
The method has the advantages that the covariance vector of the array received data is directly modeled, the problem of energy dispersion in the traditional modeling method is avoided, the advantage of covariance matrix focusing energy is utilized, the virtual effective aperture of the array is effectively enlarged, and the algorithm resolution and the positioning precision are improved; meanwhile, the characteristic that the signal power value is nonnegative is utilized, a nonnegative sparse Bayesian learning algorithm is provided, the algorithm is not influenced by relevant signals, and meanwhile, the robustness of the algorithm can be guaranteed under the conditions of low signal-to-noise ratio and small snapshot number. Fig. 6 to 8 in the examples demonstrate the above-described point.
Drawings
FIG. 1 is a flow chart of a positioning implementation of the method of the present invention.
Fig. 2 is a schematic view of a positioning scenario.
Fig. 3 is a schematic representation of spatial sparseness.
Fig. 4 is a schematic diagram of a spatial sparse representation of a signal.
FIG. 5 is a diagram illustrating the relationship between parameters.
FIG. 6 is a plot of the peak of the spectra located by the method of the present invention.
Fig. 7 is a graph of RMSE as a function of signal to noise ratio for the method of the present invention and a conventional positioning algorithm.
FIG. 8 is a graph of RMSE curves for the method of the present invention and a conventional positioning algorithm as a function of snapshot number.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings.
1. Constructing a space sparse model for directly positioning the covariance vector of the received data;
2. assuming the distribution of the signals to be a non-negative laplacian sparse prior distribution;
3. non-negative sparse Bayesian learning parameters based on Laplace prior distribution are deduced.
Example (b): the method comprises the following specific steps:
the method comprises the following steps of constructing a space sparse model for directly positioning a covariance vector of received data:
(1) establishing an array receiving data model: setting N receiving stations in two-dimensional plane, far field p at receiving stationq＝[xq,yq]TAnd Q is 1,2, …, wherein, Q is a fixed signal radiation source, each radiation source radiates signals which are mutually related narrow-band signals, and the N receiving stations receive the radiation source radiation signals and transmit data to the central processing station to realize the estimation of the radiation source position.
Then the data for the K snapshots received by the nth receiving station can be modeled as
rn＝Ansn+wn (23)
Setting the number of array elements of each receiving station as M, the sampling fast beat number as K,
Figure BDA0003463616920000071
for the K-snapshot data received by the receiving station,
Figure BDA0003463616920000072
is an array flow pattern matrix, an(pq) The guide vector of the q radiation source reaching the nth observation station is mainly determined by the relative position relationship between the target radiation source and the receiving station,
Figure BDA0003463616920000073
for the envelope of the signal received by the receiving station,
Figure BDA0003463616920000074
additive noise for the nth receiving station.
Assuming that the signal source and noise are independent vectors that are uncorrelated with each other, the covariance matrix of the array output can be expressed as:
Figure BDA0003463616920000075
Figure BDA0003463616920000076
in order to receive the power value of the signal,
Figure BDA0003463616920000077
to describe the magnitude of the noise level, I, for the noise powerMIs an identity matrix of dimension M x M.
Elongating the covariance matrix into vectors, having
Figure BDA0003463616920000078
Figure BDA0003463616920000081
Expressed as the product of the KR,
Figure BDA0003463616920000082
is a vector of the signal power of the signal source,
Figure BDA0003463616920000083
and is
Figure BDA0003463616920000084
Only the mth component of (1) and the remaining elements are all 0. From the above formula, a new array flow pattern is defined as
Figure BDA0003463616920000085
Wherein
Figure BDA0003463616920000086
Contains Q new guide vectors, the Q new guide vector is
Figure BDA0003463616920000087
Wherein
Figure BDA0003463616920000088
Is the product of Kronecker.
(2) Constructing a space sparse model of the covariance vector: in order to convert the source positioning problem into a sparse representation problem, which is inspired by the sparse reconstruction basic theory, the method of the invention considers that a covariance vector is established as a model with sparse airspace, a space range of interest is divided into grid points, G grids are provided in total, each grid is represented as the position of a potential radiation source, and if the divided grids are small enough, the position of a real radiation source is on or near a grid point, as shown in fig. 3.
The corresponding samples of the spatial spectrum are arranged in a G multiplied by 1 vector, because G is far larger than the signal source number Q, the spatial signal vector is sparse, ideally, most elements of the spatial signal vector are close to 0, and only Q elements have larger difference with zero elements.
Assuming that the radiation source (gray point) is located exactly at Q grid points, the covariance vector (25) can be expressed as:
Figure BDA0003463616920000089
wherein phinIs composed of
Figure BDA00034636169200000810
A steering vector matrix extending to the whole grid space, an overcomplete array flow pattern matrix of known sparse representation, G is the grid point number of the division grid points, x is a non-negative signal power sparse vector, and M is assumed<<G、Q<<And G, because the target radiation source is only positioned on a plurality of grid points, x is expressed as a space sparse power signal, a numerical value is only positioned at a position corresponding to the grid point with the signal source, and other elements are all 0.
From the concept of sparse signals, it can be known that a signal power x vector is a sparse vector with sparsity of Q, where there are only Q nonzero elements, and all other elements are zero elements, where the Q nonzero elements correspond to the positions of real radiation sources, as shown in fig. 4.
Assuming the distribution of the signals to be non-negative Laplace sparse prior distribution:
(1) covariance vector a priori assumptions.
From equation (26), the prior distribution of covariance vector data can be expressed as:
Figure BDA00034636169200000811
wherein,
Figure BDA00034636169200000812
since x is a non-negative vector, equation (27) is convenient to represent in real value, and therefore, will be the entire problemPut into discussion in real-valued operation. When the incident signal is circularly symmetric gaussian, (27) can be converted to a real-valued gaussian as follows:
Figure BDA0003463616920000091
wherein,
Figure BDA0003463616920000092
and is
Figure BDA0003463616920000093
(2) A priori assumption of signal power.
Modeling x as the laplacian prior distribution of:
Figure BDA0003463616920000094
since x is a non-negative vector, the adjustment equation (29) is rewritten as:
Figure BDA0003463616920000095
in order to solve the problem that the prior distribution in the formula (30) is not conjugate with the conditional distribution of the observed data, a hierarchical non-negative laplacian prior is provided. First, establishing a first layer prior of non-negative Laplace priors as the following non-negative Gaussian priors
Figure BDA0003463616920000096
Wherein N is+(xg|0,γg)＝2N(xg|0,γg),xg≥0，N+(xg|0,γg) Is a non-negative Gaussian probability density function with a mean value of 0, gammagIs a sparse enhancement hyper-parameter, and gamma is assumed to ensure sparsity of the probability distribution of xgG is 1,2, …, G is an exponential prior distribution, i.e. the second layer prior distribution is:
Figure BDA0003463616920000097
λ is a hyper-parameter, and a probability distribution density function of x can be obtained by combining the two layers of prior distributions, namely the formula (31) and the formula (32):
Figure BDA0003463616920000098
wherein the hyperparameter λ obeys the following Gaussian distribution
p(λ；ν)＝Γ(λ|ν,ν) (34)
ν is a normal number approaching 0, and the edge distribution of x obtained in this way is laplace distribution.
(3) Noise power a priori assumption.
To simplify the derivation process, the algorithm assumes
Figure BDA0003463616920000101
For no information prior, i.e.:
Figure BDA0003463616920000102
the probability distribution of the parameters needed in the direct localization model has been established so far, and the interrelation among the parameters is shown in fig. 5.
Deducing nonnegative sparse Bayesian learning parameters based on Laplace prior distribution:
(1) and (4) expressing an objective function.
By the prior probability hypothesis, a posterior probability density function is obtained as:
Figure BDA0003463616920000103
the position of the non-0 element in the sparse signal x where the objective function is maximized is the position of the target radiation source. In order to maximize the posterior probability density function of the formula, the method adopts an EM algorithm: e, mainly calculating the expectation of the complete likelihood logarithm; the M steps maximize the expectation using an algorithm.
(2) Step E is expressed as the following formula:
Figure BDA0003463616920000104
wherein
Figure BDA0003463616920000105
Express the calculation obeys
Figure BDA0003463616920000106
The posterior probability of x is calculated:
Figure BDA0003463616920000107
the matrix elements are more than or equal to the symbols, and the mean value and the variance are respectively as follows:
Figure BDA0003463616920000108
covariance matrix Λ ═ diag (α), and
Figure BDA0003463616920000109
representing the variance of the signal.
(3) In the M step, based on the above mean and variance, the mean of the posterior probability density function of gamma can be calculated,
Figure BDA00034636169200001010
and maximize it to give gammagThe updating expressions of (1) are as follows:
Figure BDA00034636169200001011
wherein,
Figure BDA00034636169200001012
further sum of λ
Figure BDA0003463616920000111
A posterior probability density function of
Figure BDA0003463616920000112
And maximize it, resulting in updated lambda and
Figure BDA0003463616920000113
the formula of (a):
Figure BDA0003463616920000114
Figure BDA0003463616920000115
wherein,
Figure BDA0003463616920000116
Figure BDA0003463616920000117
and can be simplified
Figure BDA0003463616920000118
Is composed of
Figure BDA0003463616920000119
Wherein xgThe respective mathematical expressions of the first order parameter and the second order parameter are as follows:
Figure BDA00034636169200001110
Figure BDA00034636169200001111
wherein, mug＝μ[g],
Figure BDA00034636169200001112
Error function
Figure BDA00034636169200001113
When the condition is satisfied
Figure BDA00034636169200001114
When there is<xg>→ 0 and
Figure BDA00034636169200001115
such an approximation simplifies the computational complexity of the present algorithm.
And performing alternate iterative solution on the mean value and the variance of the hyper-parameter and the sparse signal by solving the formula to finally obtain the recovery of the space sparse signal, and judging the space position of the target radiation source by judging the position corresponding to the larger power value of the sparse signal.
Example (b):
5 static uniform linear array receiving stations with 8 array elements are adopted, the distance between adjacent sensors of each receiving station is half wavelength, and as shown in figure 2, target radiation sources are respectively positioned at p0＝[-1.2,1.2]T(km)p1＝[0,0]T(km) assuming cross-correlation between the two target radiation sources, the 5 stations are each located at u1＝(-5,-5)T(km)、u2＝(-3.5,-5)T(km)、u3＝(-2,-5)T(km)、u4＝(-0.5,-5)T(km)、u5＝(1,-5)T(km)。
Setting SNR to 10dB, each observation station sampling the received signal 64 times with fast beat number K, and obtaining a positioning spectrum peak chart of the method of the present invention as shown in fig. 6, as can be seen from fig. 6, the bits corresponding to the spectrum peakDevice for placing
Figure BDA0003463616920000121
Figure BDA0003463616920000122
With set radiation source position p0And p1The method has the advantages that the method is consistent, the positioning accuracy is verified, the positioning spectrum peak is sharp, and the resolution ratio is high; FIG. 7 is an RMSE curve diagram of the method of the present invention and the conventional positioning algorithm varying with the signal-to-noise ratio, and FIG. 8 is an RMSE curve diagram of the method of the present invention and the conventional positioning algorithm varying with the snapshot number, as is apparent from the figure, when positioning the relevant signal source, both the conventional maximum likelihood type direct positioning method and the subspace type direct positioning algorithm fail, and the error level remains at a higher level, while the mean square error value of the method of the present invention is much lower, the positioning accuracy is high, and the positioning error can reach a lower level at low signal-to-noise ratio and small snapshot, and the algorithm is robust; meanwhile, compared with the traditional Sparse Bayesian (SBL) direct positioning algorithm, the method provided by the invention has higher positioning precision.

Claims (2)

1. A direct positioning method of nonnegative sparse Bayesian learning is characterized by comprising the following steps:
step 1, constructing a space sparse model for directly positioning covariance vectors:
step 1.1: establishing an array receiving data model: n receiving stations are arranged in a two-dimensional plane, Q fixed signal radiation sources are arranged at the far field of the receiving stations, and each signal radiation source is positioned at pq＝[xq,yq]T,q＝1,2,…,Q，pqPosition of the q-th radiation source, xqIs the abscissa, y, of the radiation sourceqThe signal radiated by each radiation source is a mutually-correlated narrow-band signal, and the N receiving stations receive the signal radiated by the radiation source and transmit data to the central processing station to realize the estimation of the position of the radiation source;
the data modeling of the K snapshots received by the nth receiving station is represented as:
rn＝Ansn+wn (1)
the number of array elements of each receiving station is M, the sampling fast beat number is K,
Figure FDA0003463616910000011
for the K-snapshot data received by the receiving station,
Figure FDA0003463616910000012
is an array flow pattern matrix, an(pq) The guide vector of the q radiation source reaching the nth observation station is determined by the relative position relationship between the target radiation source and the receiving station,
Figure FDA0003463616910000013
for the envelope of the signal received by the receiving station,
Figure FDA0003463616910000014
additive noise for the nth receiving station;
the signal source and noise are independent vectors that are uncorrelated with each other, and the covariance matrix of the array output is expressed as:
Figure FDA0003463616910000015
Figure FDA0003463616910000016
for the power value of the received signal, Q is 1,2, …, Q,
Figure FDA0003463616910000017
to describe the magnitude of the noise level, I, for the noise powerMAn identity matrix of dimension M × M;
the covariance matrix is elongated into vectors with:
Figure FDA0003463616910000018
Figure FDA0003463616910000019
expressed as the product of the KR,
Figure FDA00034636169100000110
is a vector of the signal power of the signal source,
Figure FDA00034636169100000111
and is
Figure FDA00034636169100000112
Wherein only the m component is 1, the rest elements are 0, and a new array flow pattern is defined by formula (3)
Figure FDA00034636169100000113
Wherein
Figure FDA00034636169100000114
Contains Q new guide vectors, the Q new guide vector is
Figure FDA00034636169100000115
Wherein
Figure FDA00034636169100000116
Is the product of Kronecker;
in the direct localization problem, the number Q of radiation sources is often unknown, and the localization target is to utilize the observed data covariance vector ynLocating the spatial positions of the Q radiation sources, N being 1,2, …, N;
step 1.2: constructing a space sparse model of the covariance vector;
establishing a covariance vector as a model with sparse airspace, dividing an interested space range into grid points, wherein G grids are provided in total, each grid is represented as the position of a potential radiation source, the divided grids are small enough, and the position of a real radiation source is on a grid point;
arranging corresponding samples of a space spectrum in a G multiplied by 1 vector, wherein G is far larger than the signal source number Q, the space signal vector is sparse, and under an ideal condition, most elements of the space signal vector are close to 0, and only Q elements have large difference with zero elements, so that the position estimation of a radiation source is obtained by adopting a sparse recovery method;
the radiation source is located at Q grid points, and under single sample fast shooting, formula (3) of the covariance vector is expressed as:
Figure FDA0003463616910000021
wherein phinIs composed of
Figure FDA0003463616910000022
A steering vector matrix extending to the whole grid space, an overcomplete array flow pattern matrix of known sparse representation, G is the grid point number of the division grid points, x is a non-negative signal power sparse vector, and M is assumed<<G、Q<<G, because the target radiation source is only positioned on a plurality of grid points, x is expressed as a space sparse power signal, only the corresponding grid point position with the signal source has a numerical value, and other elements are all 0;
according to the concept of sparse signals, the x vector of the known signal power is a sparse vector with sparsity of Q, wherein only Q nonzero elements exist, and other elements are zero elements, wherein the Q nonzero elements correspond to the positions of real radiation sources;
step 2, non-negative Laplace sparse prior distribution hypothesis of the signals:
(2.1) covariance vector a priori assumptions;
the prior distribution of covariance vector data is expressed by equation (4) as:
Figure FDA0003463616910000023
wherein,
Figure FDA0003463616910000024
k is the sampled fast beat number, since x is a non-negative vector, equation (5) is convenient to be expressed in real value, so the whole problem is put into the real value operation for discussion, and when the incident signal is circularly symmetric gaussian distribution, equation (5) is converted into the following real value gaussian distribution:
Figure FDA0003463616910000025
wherein,
Figure FDA0003463616910000026
and is
Figure FDA0003463616910000027
(2.2) an a priori assumption of signal power;
modeling x as the laplacian prior distribution of:
Figure FDA0003463616910000031
since x is a non-negative vector, the adjustment equation (7) is rewritten as:
Figure FDA0003463616910000032
in order to solve the problem that prior distribution in a formula (8) is not conjugated with conditional distribution of observed data, a hierarchical non-negative Laplace prior is provided; first, a first layer of non-negative laplacian priors is established as the following non-negative laplacian priors:
Figure FDA0003463616910000033
wherein N is+(xg|0,γg)＝2N(xg|0,γg),xg≥0，N+(xg|0,γg) Is a non-negative Gaussian probability density function with a mean value of 0, gammagIs a sparse enhancement hyper-parameter, and gamma is assumed to ensure sparsity of the probability distribution of xgFor the exponential prior distribution, G ═ 1,2, …, G, i.e., the second layer prior distribution is:
Figure FDA0003463616910000034
lambda is a hyper-parameter, two layers of prior distribution, namely an expression (9) and an expression (10), are synthesized, and the probability distribution density function of x is obtained as follows:
Figure FDA0003463616910000035
wherein the hyperparameter λ obeys the following gaussian distribution:
p(λ；ν)＝Γ(λ|ν,ν) (12)
nu is a normal number approaching to 0, and the edge distribution of x obtained in the way is Laplace distribution;
(2.3) noise power prior assumption;
to simplify the derivation process, assume
Figure FDA0003463616910000036
For no information prior, i.e.:
Figure FDA0003463616910000037
step 3, deducing nonnegative sparse Bayesian learning parameters based on Laplace prior distribution;
the direct positioning algorithm based on sparse Bayesian learning mainly derives posterior probability density according to a Bayesian rule according to a received data matrix and assumed probability distribution, and obtains the position of a target radiation source by maximizing a posterior probability density function; the method comprises the following specific steps:
and (3) obtaining a posterior probability density function by the prior probability hypothesis of the step (2) as follows:
Figure FDA0003463616910000041
the position of the non-0 element in the sparse signal x that maximizes the posterior probability density function of equation (14) is the position of the target radiation source.
2. The direct localization method of nonnegative sparse bayesian learning according to claim 1, characterized in that:
to maximize the formula a posteriori probability density function, the EM algorithm is used: e, calculating the expectation of the complete likelihood logarithm; m, maximizing an expectation value by using an algorithm;
(3.1) the E step is expressed as the following formula:
Figure FDA0003463616910000042
wherein
Figure FDA0003463616910000043
Express the calculation obeys
Figure FDA0003463616910000044
The posterior probability of x is calculated as:
Figure FDA0003463616910000045
not less than the mean value mu of the matrix elements which are all greater than or equal to the symbolnSum variance ΣnRespectively as follows:
Figure FDA0003463616910000046
covariance matrix Λ ═ diag (α), and
Figure FDA0003463616910000047
representing the variance of the signal;
(3.2) in the M step, calculating the mean value of the posterior probability density function of gamma based on the mean value and the variance,
Figure FDA0003463616910000048
and maximize the mean to get γgThe updating expressions of (1) are as follows:
Figure FDA0003463616910000049
wherein,
Figure FDA00034636169100000410
calculating lambda sum
Figure FDA00034636169100000411
A posterior probability density function of
Figure FDA00034636169100000412
And maximize it, resulting in updated lambda and
Figure FDA00034636169100000413
the formula of (a):
Figure FDA00034636169100000414
Figure FDA0003463616910000051
wherein,
Figure FDA0003463616910000052
Figure FDA0003463616910000053
and simplify
Figure FDA0003463616910000054
Is composed of
Figure FDA0003463616910000055
Wherein xgThe respective mathematical expressions of the first order parameter and the second order parameter are as follows:
Figure FDA0003463616910000056
Figure FDA0003463616910000057
wherein, mug＝μ[g],
Figure FDA0003463616910000058
Error function
Figure FDA0003463616910000059
When the condition is satisfied
Figure FDA00034636169100000510
When there is
Figure FDA00034636169100000511
And
Figure FDA00034636169100000512
(ii) present; building space rarityA sparse signal model; receiving the covariance of the data to perform vectorization operation;
to obtain gammagλ and
Figure FDA00034636169100000513
after the expression is updated, alternate iteration is performed until the sparse signal converges to a stable value and is not changed any more, and the position of the nonzero value of the sparse signal at the moment is the position of the target radiation source.
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