1. A method for estimating angle and frequency parameters of a noncoherent distributed broadband source is characterized by comprising the following steps:

s1, acquiring far-field incoherent distribution broadband source signals through the receiving array, estimating covariance of the incoherent distribution broadband source signals and vectorizing the incoherent distribution broadband source signals;

s2, establishing a vectorized incoherent distribution broadband source signal covariance model, and simultaneously reducing the dimension of the directional derivative matrix and the vectorized incoherent distribution broadband source signal covariance estimated in the step S1 according to the row repeatability and the singularity of the directional derivative matrix in the covariance model;

s3, formulating the vectorized incoherent distribution broadband source signal covariance model in the step S2 into a rank minimization problem, and solving the rank minimization problem by combining an alternating direction multiplier method and an iterative weighted kernel norm algorithm to obtain an estimator of an angle-frequency joint distribution matrix;

and S4, estimating key parameters of angle and frequency distribution of the incoherent distribution broadband source by combining a discrete grid estimator according to the estimation quantity of the angle-frequency joint distribution matrix obtained in the step S3.

2. The method of claim 1, wherein in step S1, a receiving array comprising L array elements is first set, and the array elements are spaced apart by a half wavelength corresponding to the maximum frequency of the source signal; if K far-field incoherent distribution broadband source signals are incident to the receiving array, the output signals of the receiving array at the time t are as follows:

where t is 1, 2.., Q, where Q is the fast beat number of signal samples; f and Γ are the frequency and frequency range of the source signal, respectively; theta and theta are the azimuth angle and the angular range of the source signal propagating in space, respectively; dS_k(f) Is a measure of the frequency spectrum of the kth source signal; gamma ray_k(θ, t) is the complex gain of the k-th source signal propagating in the angular direction;is the directional derivative of the receiving array with respect to (θ, f), j being the imaginary unit, (. DEG)^TTo transpose, τ_l(θ) is the delay difference between the source signal propagating to the L-th array element relative to the reference array element (the first array element is usually set as the reference array element), L is 1, 2. n (t) is white Gaussian noise.

3. The method according to claim 2, wherein in step S1, according to the output signal x (t), the covariance of the output signal of the receiving array is obtained as:

wherein E (-) represents desired; (.)^HRepresents a conjugate transpose; r_sAndcovariance of the source signal and white gaussian noise, respectively; the estimated amount of R is Is Gaussian white noise variance, I is unit vector, isPerforming eigenvalue decomposition to obtainThe estimator is set toThe minimum eigenvalue of (d); the covariance estimator of the incoherent distributed broadband source signal obtained by equation (2) is:

vectorizedComprises the following steps:

where vec (-) denotes stacking the matrix column by column into a column vector,here, theIs a set of complex numbers.

4. The method of claim 3, wherein the step S2 comprises the following steps:

s2.1, according to the receiving array output signal x (t) in the step S1, establishing a vectorized incoherent distribution broadband source signal covariance model r_sAnd further constructing a directional derivative matrix

S2.2, using directional derivative matricesOf the row repetition and singularity, while on the directional derivative matrixAnd the vectored incoherent distributed broadband source signal covariance estimated in step S1And (5) performing dimensionality reduction.

5. The method according to claim 4, wherein in step S2.1, firstly, for the incoherent distributed broadband source, if the signals of different sources are uncorrelated, the complex gains from different angles in the same source are incoherent, and the signals of different frequencies in the source signal are incoherent; therefore, according to the x (t) model, the covariance of the incoherent distribution broadband source signal is obtained as follows:

in the formulaa_θ，fIn simplified form, the directional derivative a (θ, f); propagating the angular distribution of complex gain, p, for the kth incoherent distributed broadband source signal_k(f) Is the kth incoherent branchFrequency distribution of broadband source signals;the angle-frequency joint distribution of K incoherent distribution broadband source signals is obtained;

secondly, the integral expression of the covariance formula (5) of the incoherent distribution broadband source signal is replaced by a summation expression; by P_kWhen theta is equal to theta₁，θ₂，...，θ_MAnd f ═ f₁，f₂，...，f_NWhen is p_kDiscretization of (theta, f), M and N being the number of discretizations of the angular range theta and the frequency range Γ, respectively, where P_k(θ_m，f_n)＝p_k(θ_m，f_n) Representation matrix P_kHas a value of p for the (m, n) -th element of (a)_k(θ_m，f_n) (ii) a While P represents P_θ，fThe discretization of (a) is carried out,here, theIs a set of real numbers, thenP represents an angle-frequency joint distribution matrix of K incoherent distribution broadband sources;

from the discretization, the covariance formula (5) of the incoherent distributed broadband source signal can be rewritten as:

in the formulaAre respectively A_θ，fAnd p_θ，fAt discrete points (theta)_m，f_n) A value of (d);

vectorised R_sCan be expressed as:

in the formulaIs thatVectorization of (a); p is a radical of_θ，f＝vec(P)，For vectorization of the matrix P, the following is specified:

finally, according to vectorized R_sModel equation (7), the directional derivative matrix to be constructed can be derived Here, theThe complex number set is as follows:

6. the method of claim 5, wherein the method comprises estimating the angular and frequency parameters of the incoherent distributed broadband sourceIn step S2.2, first, the point is at a discrete point (θ)_m，f_n) The directional derivative a (theta) of_m，f_n) The elements of the vector are in an equal ratio sequence such thatWhere the same element is present, then the directional derivative matrixThe same rows exist; the directional derivative matrix can thus be deletedIn-line while removing estimated vectored incoherent distributed broadband source signal covarianceTo lower the corresponding row inAnddimension (d);

second, the directional derivative matrixIs singular and can be pairedAndperforming secondary dimensionality reduction; the process comprises the following steps:

s2.2.1, pairMaking singular valuesDecomposition, i.e.∑_A、U_AAnd U_AAre respectivelyPerforming singular value diagonal matrix, left singular matrix and right singular matrix of SVD;

s2.2.2, reserving singular value principal components; setting a singular value principal component proportion epsilon close to 1 when the diagonal matrix sigma of the singular value_AWhen the ratio of the sum of the first (q +1) singular values on the diagonal to the sum of all singular values is greater than epsilon for the first time, the first q larger singular values are retained, and the remaining smaller singular values are deleted, i.e. the singular value diagonal matrix of the principal component is changed intoUpdating left singular matrix and right singular matrix corresponding to principal component into

S2.2.3 matrix of direction guide numbersSum-vectorized incoherent distributed wideband source signal covariancePerforming secondary dimensionality reduction to orderOrder to

7. The method of claim 6, wherein the step S3 comprises the following steps:

s3.1, according to the incoherent distribution broadband source signal covariance model R vectorized in the step S2_sThe angle-frequency joint distribution matrix P has a low-rank attribute, and a vectorized incoherent distribution broadband source signal covariance model formula (7) is converted into a rank minimization problem;

and S3.2, solving the rank minimization problem by combining an alternating direction multiplier method and an iterative weighted kernel norm algorithm, and estimating an angle-frequency joint distribution matrix P.

8. The method according to claim 7, wherein in step S3.1, the angle-frequency joint distribution matrix P of the quantities to be estimated is first expressed as two quantities with angle and frequency uncorrelated, and is then expressed asWherein p is_k(θ)＝[p_k(θ₁)，...，p_k(θ_M)]^T，p_k(f)＝[p_k(f₁)，...，p_k(f_N)]^TRespectively discrete vectors of the angular and frequency distributions,

since the rank of the vector is 1, the matrix P is known_kIs also 1, such that the rank of P is less than or equal to K; while the source number K is usually smaller than the discretization numbers M and N, the angle-frequency joint distribution matrix P has a low rank property;

according to the low-rank attribute of the angle-frequency joint distribution matrix P, a vectorized incoherent distribution broadband source signal covariance model formula (7) is converted into a rank minimization problem, and the specific steps are as follows:

where rank () represents the rank function;

by estimation of quantitiesSubstitute r_sEquation (10) is updated to the lagrangian soft constraint form:

where λ is the Lagrangian multiplier, usually λ ∈ (0, 1); kernel norm P (| non-conducting phosphor)_*For rank (P) non-convex substitution,denotes the sum of singular values of P with a nuclear norm P, where σ_iPerforming the ith singular value of SVD for P; i | · | purple wind₂Is a 2 norm.

9. The method for estimating angle and frequency parameters of an incoherent distributed broadband source according to claim 8, wherein in step S3.2, formula (11) can only obtain a sub-optimal solution of the angle-frequency joint distribution matrix P, and further optimizes the rank minimization problem by combining the alternating direction multiplier method and the iterative weighted norm algorithm to obtain a global optimal solution of the angle-frequency joint distribution matrix P, which is specifically as follows:

s3.2.1, writing equation (11) as a distributed minimization problem according to the alternating direction multiplier method:

in the formulaIs an unknown quantity introduced; equation (12) is written in augmented Lagrangian form:

in the formulaAnd β are bivariate and penalty parameters in Alternating Direction Multiplier Method (ADMM), respectively;<·>representing an inner product operator; i | · | purple wind_FRepresents the F norm;

the alternating direction multiplier method is divided into four steps of iteration:

s3.2.1.1, update P:

in the formula P^q，Z^q，Y^q，β^qRespectively representing the estimated values of P, Z, Y and beta in the q-th iteration;

s3.2.1.2, update Z:

s3.2.1.3, update Y:

Y^q+1＝Y^q+β^q(P^q+1-Z^q+1)； (16)

s3.2.1.4, update β:

β^q+1＝min(β_max，ξβ^q)； (17)

β_maxis the maximum value of beta set in the iterative process, xi is the updating operator, xi > 1;

s3.2.2, in order to obtain the optimal solution of P, an iterative weighted kernel norm (IRNN) algorithm is used for solving the minimization subproblem of the formula (14) about P, and the solving process is as follows:

consider a non-convex substitution of the nuclear norm:

in the formula (I), the compound is shown in the specification,is a Lipschitz continuous derivative function; g_δ(x) For a non-convex substitution function, when g_δ(x)＝1-e^-x/δThe rank function can be better approximated; in the iteration process, setting delta to be a large value, taking the delta to be 100-500, and enabling the delta to be delta/rho in each iteration, wherein the rho is larger than 1;

update equation (18) to the IRNN minimized form:

where μ > L (f) ensures convergence, L (f) isLipschitz constant of;is g_δ(x) A derivative of (a);

the rank minimization problem obtains the optimal solution as:

wherein G is^q＝U∑V^TIs G^qSVD of (1);is a generalized soft-boundary operator that is,

s3.2.3, the minimization subproblem of the formula (15) about Z has a closed solution, and the solving process is as follows:

the F norm of the matrix in simplified equation (15) is 2 norm of the vector, and the minimization subproblem of Z is simplified as:

the closed solution is:

Z^q+1by vec (Z)^q+1) Vectorization reverse operation recombination is carried out to obtain;

therefore, the variable P is initialized according to equation (10)⁰，Z⁰＝P⁰，Y⁰＝0，β⁰(ii) a Setting of lambda, beta_maxξ, δ, ρ, μ; setting d as an iteration stop threshold; the 4 iteration variables P, Z, Y and beta in the ADMM are updated iteratively according to the formula (20), the formula (22), the formula (16) and the formula (17), and when the iteration converges, the | | | P is enabled^q+1-P^q||_F/||P^q||_FWhen d is less than d, P is output as the estimator of the angle-frequency joint distribution matrix

10. The method of claim 9, wherein in step S4, the estimation is performed according to the estimation of the angle-frequency joint distribution matrix obtained in step S3Estimating key parameters of incoherent distributed broadband sources, including central DOA of angular distribution, using a discrete mesh estimator based on power spectral density moment estimationAnd angular spreadCenter frequency of frequency distributionSum frequency bandwidthThe method comprises the following specific steps:

in the form of bandwidthDefined as the standard deviation of the frequency distributionDoubling; theta_kAnd Γ_kRespectively the interested angle and frequency range of the kth incoherent distributed broadband source signal; for both angle and frequency distributions, when the distribution is gaussian or laplace, η ═ 1; when the distribution is uniform, η is 3.

Background

The broadband signal has the characteristics of high resolution, large information amount, strong anti-interference capability and the like, and is widely applied to the fields of radar, sonar, wireless communication and the like. The angle of arrival (DOA) estimation for most broadband sources is based on the point source assumption. However, the angular distribution caused by multipath or scattering propagation of the source signal is not negligible. The DOA estimation methods for most distributed sources are based on narrow-band assumptions. For distributed broadband sources, as the frequency bandwidth increases, the DOA estimation bias based on the narrowband hypothesis method also increases. Therefore, the DOA estimation problem of distributed broadband sources requires much attention and research.

In recent years, some DOA estimation methods for distributed broadband sources have been proposed. A Maximum Likelihood (ML) method estimates an angle parameter by reconstructing a shape of an angle distribution. Covariance Matching (CM) method, a frequency domain snapshot based matching method. However, these two methods have a large computational burden because the multidimensional search dimension increases with the number of sources. A parameter polynomial method, an autoregressive model method based on signal covariance time delay. These three methods all assume that the frequency distribution of the broadband signal of the distributed broadband source is known and the frequency range is the same. In practice, the wideband signal may have different frequency ranges.

Some researchers also propose a new method of combining a fractional Fourier transform (FRFT) algorithm with a Distributed Source Parameter Estimator (DSPE) algorithm, namely FRFT-DSPE method (refer to Yu J, Zhang L, Liu K. coherent Distributed Wireless LFM Source Localization [ J ]. IEEE Signal Processing Letters,2014,22(4): 504-508.). Although this method does not require the knowledge of the frequency distribution, as with the above method, a parameterized model of the known angular distribution is required, and the exact angular distribution model in practical applications is often unknown; meanwhile, the method can estimate the angular distribution of the source, but cannot effectively estimate the frequency distribution, and in fact, the frequency is also an important characteristic parameter of the target source. There are also scholars applying the Sparse Bayesian Learning (SBL) method to distributed broadband sources, which, although it does not require a distribution model with known angles and frequencies, requires that the distributed broadband sources must be sparse in the spatial domain. In practice the sparsity of the distributed sources decreases with angular spread.

The existing angle distribution estimation method of the distributed broadband source has the problems that a parameterized model and frequency distribution of a known angle are needed and the calculation complexity is high, and the angle distribution estimation of the incoherent distributed broadband source in a complex scene is difficult to meet. It is therefore desirable to provide a method that reduces computational complexity without the need for known angle and frequency distribution models, and that simultaneously estimates the angle and frequency distributions of a broadband distributed source.

Disclosure of Invention

The invention aims to solve the problems that the existing angle distribution estimation method of the distributed broadband source needs a parameterized model of a known angle and frequency distribution and has high calculation complexity, and the angle and frequency parameter estimation method of the incoherent distributed broadband source is developed based on low-rank matrix recovery.

The purpose of the invention is realized by at least one of the following technical solutions.

A method for estimating angle and frequency parameters of a noncoherent distributed broadband source comprises the following steps:

s1, acquiring far-field incoherent distribution broadband source signals through the receiving array, estimating covariance of the incoherent distribution broadband source signals and vectorizing the incoherent distribution broadband source signals;

s2, establishing a vectorized incoherent distribution broadband source signal covariance model, and simultaneously reducing the dimension of the directional derivative matrix and the vectorized incoherent distribution broadband source signal covariance estimated in the step S1 according to the row repeatability and the singularity of the directional derivative matrix in the covariance model so as to reduce the complexity of the subsequent calculation process;

s3, formulating the vectorized incoherent distribution broadband source signal covariance model in the step S2 into a rank minimization problem, and solving the rank minimization problem by combining an alternating direction multiplier (ADMM) algorithm and an iterative weighted kernel norm (IRNN) algorithm to obtain an estimator of an angle-frequency joint distribution matrix;

and S4, estimating key parameters of angle and frequency distribution of the incoherent distribution broadband source by combining a discrete grid estimator according to the estimation quantity of the angle-frequency joint distribution matrix obtained in the step S3.

Further, in step S1, a receiving array including L array elements is first set, and the array element spacing is a half-wavelength corresponding to the maximum frequency of the source signal; if K far-field incoherent distribution broadband source signals are incident to the receiving array, the output signals of the receiving array at the time t are as follows:

where t is 1, 2.., Q, where Q is the fast beat number of signal samples; f and f are the frequency and frequency range of the source signal, respectively; theta and theta are the azimuth angle and the angular range of the source signal propagating in space, respectively; dS_k(f) Is a measure of the frequency spectrum of the kth source signal; gamma ray_k(θ, t) is the complex gain of the k-th source signal propagating in the angular direction;is the directional derivative of the receiving array with respect to (θ, f), j being the imaginary unit, (. DEG)^TTo transpose, τ_l(θ) is the delay difference between the source signal propagating to the L-th array element relative to the reference array element (the first array element is usually set as the reference array element), L is 1, 2. n (t) is white Gaussian noise;

and obtaining the covariance of the output signals of the receiving array according to the output signals x (t):

wherein E (-) represents desired; (.)^HRepresents a conjugate transpose; r_sAndcovariance of the source signal and white gaussian noise, respectively; the estimated amount of R is Is Gaussian white noise variance, I is unit vector, isPerforming eigenvalue decomposition to obtainThe estimator is set toThe minimum eigenvalue of (d); the covariance estimator of the incoherent distributed broadband source signal obtained by equation (2) is:

vectorizedComprises the following steps:

where vec (-) denotes stacking the matrix column by column into a column vector,here, theIs a set of complex numbers.

Further, the specific steps of step S2 are as follows:

s2.1, according to the receiving array output signal x (t) in the step S1, establishing a vectorized incoherent distribution broadband source signal covariance model r_sAnd further constructing a directional derivative matrix

S2.2, using directional derivative matricesOf the row repetition and singularity, while on the directional derivative matrixAnd the vectored incoherent distributed broadband source signal covariance estimated in step S1And (5) performing dimensionality reduction.

4. A method for estimating angle and frequency parameters of an incoherent distributed broadband source according to claim 3, characterized in that in step S2.1, first, for an incoherent distributed broadband source, it is assumed that signals of different sources are uncorrelated, complex gains from different angles in the same source are incoherent, and signals of different frequencies in the source signal are incoherent; therefore, according to the x (t) model, the covariance of the incoherent distribution broadband source signal can be obtained as follows:

in the formulaa_θ,fIn simplified form, the directional derivative a (θ, f); propagating the angular distribution of complex gain, p, for the kth incoherent distributed broadband source signal_k(f) Frequency distribution of a kth incoherent distributed broadband source signal;the angle-frequency joint distribution of K incoherent distribution broadband source signals is obtained;

secondly, the integral expression of the covariance formula (5) of the incoherent distribution broadband source signal is replaced by a summation expression; by P_kWhen theta is equal to theta₁,θ₂,...,θ_MAnd f ═ f₁,f₂,...,f_NWhen is p_kDiscretization of (theta, f), M and N being the number of discretizations of the angular range theta and the frequency range Γ, respectively, where P_k(θ_m,f_n)＝p_k(θ_m,f_n) Representation matrix P_kHas a value of p for the (m, n) -th element of (a)_k(θ_m,f_n) (ii) a While P represents P_θ,fThe discretization of (a) is carried out,here, theIs a set of real numbers, thenP represents an angle-frequency joint distribution matrix of K incoherent distribution broadband sources; p is an unknown quantity to be estimated, and the solution of P is the key for estimating angle and frequency parameters;

from the discretization, the covariance formula (5) of the incoherent distributed broadband source signal can be rewritten as:

in the formulaAre respectively A_θ，fAnd p_θ，fAt discrete points (theta)_m，f_n) A value of (d);

vectorised R_sCan be expressed as:

in the formulaIs thatVectorization of (a); p is a radical of_θ，f＝vec(P)，For vectorization of the matrix P, the following is specified:

finally, according to vectorized R_sModel equation (7), the directional derivative matrix to be constructed can be derived Here, theThe complex number set is as follows:

further, in step S2.2, first, since at the discrete point (θ)_m，f_n) The directional derivative a (theta) of_m,f_n) The elements of the vector are in an equal ratio sequence such thatWhere the same element is present, then the directional derivative matrixThe same rows exist; the directional derivative matrix can thus be deletedIn-line while removing estimated vectored incoherent distributed broadband source signal covarianceTo lower the corresponding row inAnddimension (d);

second, the directional derivative matrixIs singular and can be pairedAndperforming secondary dimensionality reduction; the process comprises the following steps:

s2.2.1, pairSingular Value Decomposition (SVD), i.e.∑_A、U_AAnd U_AAre respectivelyPerforming singular value diagonal matrix, left singular matrix and right singular matrix of SVD;

s2.2.2, reserving singular value principal components; setting a singular value principal component proportion epsilon close to 1 when the diagonal matrix sigma of the singular value_AWhen the ratio of the sum of the first (q +1) singular values on the diagonal to the sum of all singular values is greater than epsilon for the first time, the first q larger singular values are retained, and the remaining smaller singular values are deleted, i.e. the singular value diagonal matrix of the principal component is changed intoUpdating left singular matrix and right singular matrix corresponding to principal component into

S2.2.3 matrix of direction guide numbersSum-vectorized incoherent distributed wideband source signal covariancePerforming secondary dimensionality reduction to orderOrder to

Further, the specific steps of step S3 are as follows:

s3.1, according to the incoherent distribution broadband source signal covariance model R vectorized in the step S2_sThe angle-frequency joint distribution matrix P has low rank property, and the vectorized incoherent distribution is wideThe covariance model with source signals is formulated as a rank minimization problem (7);

and S3.2, solving a rank minimization problem by combining an alternating direction multiplier (ADMM) algorithm and an iterative weighted kernel norm (IRNN) algorithm, and estimating an angle-frequency joint distribution matrix P.

Further, in step S3.1, first, since the angle and the frequency are two independent quantities, the angle-frequency joint distribution matrix P of the quantity to be estimated can be expressed asWherein p is_k(θ)＝[p_k(θ₁)，...，p_k(θ_M)]^T，p_k(f)＝[p_k(f₁)，...,p_k(f_N)]^TRespectively discrete vectors of the angular and frequency distributions,

since the rank of the vector is 1, the matrix P is known_kIs also 1, such that the rank of P is less than or equal to K; while the source number K is usually smaller than the discretization numbers M and N, the angle-frequency joint distribution matrix P has a low rank property;

according to the low-rank attribute of the angle-frequency joint distribution matrix P, a vectorized incoherent distribution broadband source signal covariance model formula (7) is converted into a rank minimization problem, and the specific steps are as follows:

where rank () represents the rank function;

the rank minimization problem is an NP difficult problem, and with the development of a low-rank matrix recovery theory, a rank function can be replaced by a nuclear norm; by estimation of quantitiesSubstitute r_sEquation (10) is updated to the lagrangian soft constraint form:

where λ is the Lagrangian multiplier, usually λ ∈ (0, 1); kernel norm P (| non-conducting phosphor)_*For rank (P) non-convex substitution,denotes the sum of singular values of P with a nuclear norm P, where σ_iPerforming the ith singular value of SVD for P; II-₂Is a 2 norm.

Further, in step S3.2, the formula (11) can only obtain a suboptimal solution of the angle-frequency joint distribution matrix P, and the rank minimization problem is further optimized by combining the alternating direction multiplier (ADMM) and the iterative weighted norm (IRNN) algorithm to obtain a global optimal solution of the angle-frequency joint distribution matrix P, which is specifically as follows:

s3.2.1, writing equation (11) as a distributed minimization problem according to the Alternating Direction Multiplier Method (ADMM):

in the formulaIs an unknown quantity introduced; equation (12) is written in augmented Lagrangian form:

in the formulaAnd β are bivariate and penalty parameters in Alternating Direction Multiplier Method (ADMM), respectively;<·>representing an inner product operator; II-_FRepresents the F norm;

the Alternating Direction Multiplier Method (ADMM) is divided into four iterations:

s3.2.1.1, update P:

in the formula p^q，Z^q，Y^q，β^qRespectively representing the estimated values of P, Z, Y and beta in the q-th iteration;

s3.2.1.2, update Z:

s3.2.1.3, update Y:

Y^q+1＝Y^q+β^q(P^q+1-Z^q+1)； (42)

s3.2.1.4, update β:

β^q+1＝min(β_max,ξβ^q)； (43)

β_maxis the maximum value of xi set in the iterative process, xi is an update operator, xi is more than 1;

s3.2.2, in order to obtain the optimal solution of P, an iterative weighted kernel norm (IRNN) algorithm is used for solving the minimization subproblem of the formula (14) about P, and the solving process is as follows:

consider a non-convex substitution of the nuclear norm:

in the formula (I), the compound is shown in the specification,is a Lipschitz continuous derivative function; g_δ(x) For a non-convex substitution function, when g_δ(x)＝1-e^-x/δThe rank function can be better approximated; in the iteration process, delta is set to start with a large value, the value of delta is 100-500, and the value of delta is delta/rho, rho is larger than 1 in each iteration, so that the iteration is prevented from being locally minimized due to the fact that the initial delta is too small;

update equation (18) to the IRNN minimized form:

where μ > L (f) ensures convergence, L (f) isLipschitz constant of;is g_δ(x) A derivative of (a);

the rank minimization problem obtains the optimal solution as:

wherein G is^q＝U∑V^TIs G^qSVD of (1);is a generalized soft-boundary operator that is,

s3.2.3, the minimization subproblem of the formula (15) about Z has a closed solution, and the solving process is as follows:

the F norm of the matrix in simplified equation (15) is 2 norm of the vector, and the minimization subproblem of Z is simplified as:

the closed solution is:

Z^q+1by vec (Z)^q+1) Vectorization reverse operation recombination is carried out to obtain;

therefore, the variable P is initialized according to equation (10)⁰，Z⁰＝P⁰，Y⁰＝0，β⁰(ii) a Setting of lambda, beta_maxξ, δ, ρ, μ; setting d as an iteration stop threshold; the 4 iteration variables P, Z, Y and beta in the ADMM are updated iteratively according to the formula (20), the formula (22), the formula (16) and the formula (17), and when the iteration converges, the | | | P is enabled^q+1-P^q||_F/||P^q||_FWhen d is less than d, P is output as the estimator of the angle-frequency joint distribution matrix

Further, in step S4, the estimation amount of the angle-frequency joint distribution matrix obtained in step S3 is used as the basisUsing discrete grid estimators based on power spectral density moment estimation (reference Shahbazpanahi S, Valee S, Gershman A B.A covariance fixing approach to parametric localization of multiple coherent distributed sources [ M]IEEE Press, 2004)), key parameters of the incoherent distributed broadband source are estimated, including the central DOA of the angular distributionAnd angular spreadCenter frequency of frequency distributionSum frequency bandwidthThe method comprises the following specific steps:

in the form of bandwidthDefined as the standard deviation of the frequency distributionDoubling; theta_kAnd Γ_kRespectively the interested angle and frequency range of the kth incoherent distributed broadband source signal; for both angle and frequency distributions, when the distribution is gaussian or laplace, η ═ 1; when the distribution is uniform, η is 3.

Compared with the existing angle distribution estimation method of the incoherent distribution broadband source, the method has the following advantages:

(1) the method can directly estimate the angle distribution of the source without knowing the angle distribution parameterized model of the incoherent distribution broadband source.

(2) The invention can estimate the frequency distribution at the same time without knowing the frequency distribution of the incoherent distribution broadband source signal.

(3) The invention does not need multidimensional searching and large-scale signal reconstruction, and greatly reduces the complexity.

Drawings

FIG. 1 is a signal model diagram of a distributed broadband source of the present invention;

FIG. 2 is a flow chart of a parameter estimation method of the present invention;

FIG. 3 is a graph of an angle-frequency joint distribution matrix simulated in practice by the present invention;

FIG. 4 is a graph of an angle-frequency joint distribution matrix estimated by simulation according to the present invention;

FIG. 5 is a graph of RMSE as a function of signal-to-noise ratio for the central DOA estimation of the present invention;

FIG. 6 is a graph of RMSE as a function of signal to noise ratio for the angular spread estimation of the present invention;

FIG. 7 is a graph of RMSE as a function of signal to noise ratio for the center frequency estimation of the present invention;

FIG. 8 is a graph of RMSE as a function of signal to noise ratio for the frequency bandwidth estimation of the present invention;

FIG. 9 is a graph of the runtime of the method of the present invention as a function of the number of array elements.

Detailed Description

The present invention is described in further detail below with reference to examples, which illustrate one mode of use of the present invention, but are not limited thereto.

Example (b):

a method for estimating angle and frequency parameters of a non-coherent distributed broadband source, as shown in fig. 2, includes the following steps:

s1, acquiring far-field incoherent distribution broadband source signals through the receiving array, estimating the covariance of the incoherent distribution broadband source signals and vectorizing the incoherent distribution broadband source signals, and the method comprises the following specific steps:

FIG. 1 is a signal model of a distributed broadband source (abbreviated as 'DW source') according to the present invention, first setting a receiving array including L array elements, where the array element spacing is a half wavelength corresponding to the maximum frequency of the signal; supposing that K far-field incoherent distribution broadband source signals are incident to a receiving array, theta_0k，Central DOA and angular spread of the angular distribution of the kth source, respectively;

the output signal of the receiving array at time t is:

where t is 1, 2.., Q, where Q is the fast beat number of signal samples; f and Γ are the frequency and frequency range of the source signal, respectively; theta and theta are the azimuth angle and the angular range of the source signal propagating in space, respectively; dS_k(f) Is a measure of the frequency spectrum of the kth source signal; gamma ray_k(θ, t) is the complex gain of the k-th source signal propagating in the angular direction;is the directional derivative of the receiving array with respect to (θ, f), j being the imaginary unit, (. DEG)^TTo transpose, τ_l(θ) is the delay difference between the source signal propagating to the L-th array element relative to the reference array element (the first array element is usually set as the reference array element), L is 1, 2. n (t) is white Gaussian noise;

the covariance of the output signal is estimated as:

in the formula (·)^HRepresenting a conjugate transpose.

The covariance estimator of the incoherent distributed broadband source signal obtained by equation (2) is:

in the formulaIs Gaussian white noise variance, I is unit vector, isPerforming eigenvalue decomposition to obtainThe estimator is set toThe minimum eigenvalue of (d);

vectorizedComprises the following steps:

where vec (-) denotes stacking the matrix column by column into a column vector,

s2, establishing a vectorized incoherent distribution broadband source signal covariance model, and simultaneously carrying out directional derivative matrix and the vectorized incoherent distribution broadband source signal covariance estimated in the step S1 according to the row repeatability and singularity of the directional derivative matrix in the covariance modelReducing the dimension to reduce the complexity of the subsequent calculation process, and specifically comprising the following steps of:

s2.1, establishing a vectorized incoherent distribution broadband source signal covariance model r_sConstructing a directional derivative matrixWherein r is_sIs shown as

Discretizing the angular range theta and the frequency range gamma to [ theta [ theta ] ]₁,θ₂，...，θ_M]And [ f₁，f₂，...，f_N]Where M and N are the angular range and frequency, respectivelyThe number of range discretizations; in the formulaIs thatAt discrete points (theta)_m，f_n) A value of (d);a_θ，fin simplified form, the directional derivative a (θ, f);is the angular-frequency joint distribution p of K incoherent distributed broadband source signals_θ，fAt discrete points (theta)_m,f_n) A value of (d); p is a radical of_θ,f＝vec(P)，The vectorization of the angle-frequency joint distribution matrix P of the incoherent distribution broadband source signal is specifically as follows:

here, theThe unknown quantity to be estimated is obtained, and the solution of P is the key for estimating angle and frequency parameters;

constructing directional derivative matricesHere, theThe complex number set is as follows:

s2.2, using directional derivative matricesOf the row repetition and singularity, while on the directional derivative matrixAnd the vectored incoherent distributed broadband source signal covariance estimated in step S1Performing dimensionality reduction;

first, the directional derivative matrix is deletedIn repeated rows whileTo lower the corresponding row inAnddimension (d);

secondly, toAndperforming secondary dimensionality reduction; the process comprises the following steps:

s2.2.1, pairSingular Value Decomposition (SVD), i.e.∑_A、U_AAnd U_AAre respectivelyPerforming singular value diagonal matrix, left singular matrix and right singular matrix of SVD;

s2.2.2, reserving singular value principal components; setting a singular value principal component proportion epsilon close to 1 when the diagonal matrix sigma of the singular value_AWhen the ratio of the sum of the first (q +1) singular values on the diagonal to the sum of all singular values is greater than epsilon for the first time, the first q larger singular values are retained, and the remaining smaller singular values are deleted, i.e. the singular value diagonal matrix of the principal component is changed intoUpdating left singular matrix and right singular matrix corresponding to principal component into

S2.2.3 matrix of direction guide numbersSum-vectorized incoherent distributed wideband source signal covariancePerforming secondary dimensionality reduction to orderOrder to

S3, converting the vectorized incoherent distribution broadband source signal covariance r in the step S2_sThe model is formulated into a rank minimization problem, the rank minimization problem is solved by combining an ADMM algorithm and an IRNN algorithm, and thus the estimation quantity of the angle-frequency joint distribution matrix is obtained, and the method specifically comprises the following steps:

s3.1, according to the low-rank property of the angle-frequency joint distribution matrix P, r can be obtained_sThe model is formulated as a rank minimization problem, specifically as follows:

where rank () represents the rank function;

s3.2, solving a rank minimization problem by combining an alternating direction multiplier (ADMM) algorithm and an iterative weighted kernel norm (IRNN) algorithm, and estimating an angle-frequency joint distribution matrix P;

equation (8) can be written as a distributed minimization problem according to the ADMM method

Where λ is the Lagrangian multiplier, usually λ ∈ (0, 1); kernel norm P (| non-conducting phosphor)_*For rank (P) non-convex substitution, hereDenotes the sum of singular values of P with a nuclear norm P, where σ_iPerforming the ith singular value of SVD for P; II-₂Is a 2 norm;is an unknown quantity introduced; equation (9) can be written in the form of augmented lagrange:

in the formulaAnd β are the bivariate and penalty parameters in the ADMM method, respectively;<·>representing an inner product operator; II-_FRepresents the F norm;

the ADMM method is divided into four steps of iteration:

s3.2.1, updating the matrix P to be estimated:

U∑V^T＝G^q； (13)

wherein q represents the number of iterations; in formula (11), g_δ(x)＝1-e^-x/δIs a substitute function for the kernel norm in the IRNN method;representing the derivative of the function on x. Here, δ setting starts with a large value, let ρ > 1, and δ is made δ/ρ for each iteration, avoiding that the iteration falls into local minimization because the initial δ is too small; in the formula (12)μ > L (f) ensures iterative convergence, L (f) isLipschitz constant. U Σ V in formula (13)^TIs G^qSVD of (1);

s3.2.2, updating unknown quantity Z:

Z^q+1can pass vec (Z)^q+1) Vectorization reverse operation recombination is carried out to obtain;

s3.2.3, updating double-variation vector Y:

Y^q+1＝Y^q+β^q(P^q+1-Z^q+1)； (17)

s3.2.4, updating penalty parameter beta:

β^q+1＝min(β_max,ξβ^q)； (18)

β_maxis the maximum value of beta in the iterative process, xi is an updating operator, xi is more than 1;

therefore, the variable P is initialized according to equation (8)⁰，Z⁰＝P⁰，Y⁰＝0，β⁰(ii) a Setting of lambda, beta_maxξ, δ, ρ, μ; setting d as an iteration stop threshold; the 4 iteration variables P, Z, Y and beta in the ADMM are iteratively updated according to the formula (15) -the formula (18), and when the iteration converges, the I P is obtained^q+1-P^q||_F/||P^q||_FWhen d is less than d, P is output as the estimator of the angle-frequency joint distribution matrix

S4, estimating quantity according to the angle-frequency joint distribution matrix obtained in the step S3Using discrete grid estimators based on power spectral density moment estimation (reference Shahbazpanahi S, Valee S, Gershman A B.A covariance fixing approach to parametric localization of multiple coherent distributed sources [ M]IEEE Press, 2004)), key parameters of the incoherent distributed broadband source are estimated, including the central DOA of the angular distributionAnd angular spreadCenter frequency of frequency distributionSum frequency bandwidthThe method comprises the following specific steps:

in the form of bandwidthDefined as the standard deviation of the frequency distributionDoubling; theta_kAnd Γ_kRespectively the interested angle and frequency range of the kth incoherent distributed broadband source signal; for both angle and frequency distributions, when the distribution is gaussian or laplace, η ═ 1; when the distribution is uniform, η is 3.

In this embodiment, the effect of the present invention can be further illustrated by the following simulation results, and the simulation experiment conditions are as follows:

two incoherent distributed broadband sources with different angle and frequency distributions in the far field are incident on an array with the array element number L equal to 30. Wherein the angle of the first source follows a Gaussian distribution and the frequency follows a uniform distribution; both the angle and the frequency of the second source are subject to a uniform distribution. The angular-frequency joint distribution of the two sources is then as follows:

where p is_k(θ, f) is the angle-frequency joint distribution of the kth source, which can be composed as the angle-frequency joint distribution of the K incoherent distributed broadband source signalsSetting the center DOA and angular spread of the angular distribution of the first source toThe center frequency and bandwidth of the frequency distribution are (f)₀₁，B₁) -350 Hz,100 Hz; the angular distribution of the second source is parametrically set to The frequency distribution parameter is set to (f)₀₂,B₂) (150Hz,100 Hz). The observation ranges of the angle and the frequency are set to [20 °,50 ° ], respectively]，Γ＝[90Hz,420Hz]. The discrete resolution of the angle and frequency of the joint distribution matrix is set to 1 ° and 10Hz, respectively. Fast beat number Q of 10⁴The SNR is 10 dB. For the iterative procedure, initializing δ to 100 and updating by δ to δ/2; the initialization penalty parameter is beta is 0.3, the updating parameter is xi is 1.2, beta_max＝10⁵(ii) a Setting lambda to 0.3 and mu to 10³，ε＝1-10⁶The simultaneous iteration stop threshold d is 10^-4。

For example, fig. 3 shows a simulated actual angle-frequency joint distribution matrix, fig. 4 is a joint distribution matrix estimated by the method of the present invention, and comparing the two figures shows that the joint distribution matrix estimated by the method of the present invention has better consistency with the reality. Embodying the feasibility of the invention to estimate the angular-frequency joint distribution of an incoherent distributed broadband source without knowing the angular and frequency distributions.

2.1 the estimated performance of the key parameters varies with the signal to noise ratio:

in order to show that the method has higher parameter estimation precision, the key parameters of the joint distribution are further estimated according to the estimated angle-frequency joint distribution, and the performance of parameter estimation is compared with the conventional FRFT-DSPE method and the SBL method. In the simulation, the array element number L is set to 20, the fast beat number Q is set to 100, the SNR changes from-10 dB to 10dB, and other parameters remain the same as the above example. The variation of RMSE with signal-to-noise ratio SNR of the parameter estimation of the method of the invention was analyzed by performing 100 Monte Carlo experiments. The estimated performance of the central DOA and the angular spread of the angular distribution is shown in fig. 5 and 6, respectively. As can be seen from the figure, the angular distribution parameter estimation performance of the method approaches the lower boundary of Cramer-Lo; and meanwhile, the method has higher estimation precision than the existing method, especially when the signal to noise ratio is low. Because the FRFT-DSPE method does not estimate the frequency parameters of the distributed broadband source, the method of the invention compares the performance of frequency parameter estimation with the SBL method. As shown in fig. 7 and 8, compared with the SBL method, the method of the present invention also has better estimation accuracy in frequency parameter estimation.

2.2 the computational complexity varies with the number of array elements:

in order to show that the method has lower computational complexity, the applicant analyzes the change of the running time of the method along with the array element number in a simulation experiment. Keeping other parameters unchanged, setting the signal-to-noise ratio SNR to 10dB, and increasing the array element number L from 10 to 50. As shown in fig. 9, the method of the present invention has lower computational complexity compared to the FRFT-DSPE method requiring multi-dimensional search and the SBL method requiring large-scale signal reconstruction.

S5, the frequency distribution and the parameter information estimated through the steps S3 and S4 are used for determining the signal type of the source target, and the target can be further identified; the estimated angular distribution and parametric information are used for target positioning, and the target position can be determined.

The above-described examples of the present invention are intended to illustrate the calculation procedure and calculation performance of the present invention in detail, and are not intended to limit the embodiments of the present invention. Other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principles of the invention are intended to be included within the scope of the invention.