CN109116338B - Broadband convex optimization DOA estimation method based on fourth-order cumulant - Google Patents

Abstract

A broadband DOA estimation algorithm based on fourth-order cumulant uses a broadband signal source to estimate DOA, combines the compressive sensing principle to decompose the broadband signal source into a plurality of sub-bands, estimates the sub-bands one by one to obtain the spatial spectrum of each sub-band, and then solves the arithmetic mean value to obtain more accurate DOADOA estimation of (a). For each sub-band component, the fourth-order cumulant is adopted to construct a signal subspace, so that the purpose of suppressing a noise item is achieved. The compressed sensing principle is embodied in the invention that the DOA estimation problem is converted into an optimization problem, and QUOTE is applied to each snapshot component of the signal

The norm constructs a vector with the same sparsity as the original signal and regularizes the parameter QUOTE

It is embedded into the objective function to achieve a balance of sparsity conditions and data solutions. Finally, the final DOA estimation can be done by optimizing the objective function. Compared with the prior art, the DOA estimation method and the DOA estimation device inhibit noise items and improve DOA estimation precision on the premise of ensuring algorithm complexity.

Description

Broadband convex optimization DOA estimation method based on fourth-order cumulant

Technical Field

The invention relates to a broadband convex optimization DOA estimation method based on fourth-order cumulant, and belongs to the technical field of underwater acoustic signal processing.

Background

Direction of arrival (DOA) refers to the Direction angle of a spatial signal reaching a reference array element, and is an important concept in spatial spectrum estimation. In the field of underwater acoustic signal processing, the sensor array estimates each parameter of a signal of interest, determines the position parameter of the signal source, and has important significance for underwater target surveying, obstacle identification and underwater communication. The direction of arrival is one of the most important parameters.

The traditional DOA estimation method mainly includes a beam forming method and various signal subspace algorithms. The resolution of the beamforming method for the estimation of the angle of arrival is limited by the existence of the Rayleigh limit, and the subsequent signal subspace class algorithm enables the DOA estimation with high resolution. The MUSIC algorithm is one of the most classical signal subspace algorithms, and the main idea is to perform singular value decomposition on a covariance matrix of a received signal to estimate a signal subspace and a noise subspace, and construct a spatial spectrum by utilizing the orthogonality of the signal subspace and the noise subspace so as to estimate a wave arrival angle.

The compressive sensing technology is an important research direction in the field of current signal processing, and the breakthrough heuristic of the compressive sensing technology on the signal processing research is that the Nyquist sampling theorem is broken. It is pointed out that if the signal satisfies the sparsity condition in a certain domain, it can be sampled at a frequency much lower than nyquist sampling and the signal can be reconstructed with a great probability. In underwater array signal processing, along with the increase of the number of sampling points, the requirements on instrument equipment are also more severe, so that the high-resolution identification of underwater signals is quite expensive. And the proposal of the compressed sensing technology provides a thought for alleviating the contradiction.

At present, in an algorithm for trying to apply the compressed sensing principle to array signal processing, a narrowband signal is mostly used as an information source signal, while a relatively wideband information source has a larger information amount and stronger anti-interference capability, and the application of the algorithm in DOA estimation will bring higher resolution. For P far-field wideband signals incident on a spatial array with M array elements, the signal received by the mth array element can be expressed as:

wherein a is_pRepresenting the response of the m-th array element to the p-th signal source, τ_m(θ_pAnd f) represents the time delay for the p signal source to reach the m array element. n is_m(t) represents the noise component on the mth array element. The signal is time domain sampled with a fast beat number of T.

The matrix form of the above equation is expressed as:

Y＝AS+N

where S is a matrix in the spatio-temporal dimensionThe signal s (t) in the time dimension does not generally satisfy the sparsity condition, but can be considered spatially sparse in view of the limited number of signal sources. Therefore, the time samples of the signal s can be applied l in the spatial dimension₂Norm, i.e.:

then vector

The sparsity of (a) corresponds to the sparsity of the spatial spectrum one by one. Then to the vector

Application of l₁Norm, then the objective function becomes:

wherein the Frobenius norm is defined as: | Y-AS | non-conducting phosphor_f ²＝||v c (Y-AS)||₂ ². The constraint problem can be solved using a second order cone method.

For K signal sources, there is a large amount of redundancy in the signal time samples at l₁In the SVD method, the array received signals are selected to be subjected to dimensionality reduction, and the signal matrix Y after dimensionality reduction is used_SVAs an estimate of the signal subspace. The objective function to be optimized is:

firstly, singular value decomposition is carried out on the array receiving signal Y to obtain: y ═ ULV^H. Reduced Y_SVMost of the energy of the original signal is reserved, and is represented as: y is_SV＝YVD_K＝ULD_KWherein D is_K＝[I_K 0]^H，I_KDenotes an identity matrix of K × K, and 0 denotes a zero matrix of K × (T-K). Most of the existing algorithms select pairs when estimating signal subspaceThe signal subspace is constructed either from the original signal or from the covariance matrix of the received signals using the array. E.g. the classical narrow-band signal DOA estimate l mentioned above₁The SVD algorithm, which considers the received signal as an estimate of the subspace after performing the dimensionality reduction operation, although the dimensionality reduction operation reduces the amount of computation considerably, it suppresses the noise term only slightly as an estimate of the original received signal. In the practical application process and under the current large trend of signal processing, the DOA estimation by using the broadband signal as the information source has wider prospect.

In the ideal case, the second moment is sufficient to express the statistical properties of the signal; however, for noisy signals, the covariance matrix method is not satisfactory for suppressing noise. And the use of higher order statistics may solve this problem. Since the higher order cumulant is insensitive to the gaussian process, the noise can be completely suppressed theoretically.

Disclosure of Invention

In order to solve the problems, the invention provides a broadband convex optimization DOA estimation method based on fourth-order cumulant, which solves the problems that the existing sparse reconstruction DOA estimation algorithm cannot inhibit noise items and has large calculation amount, and for achieving the purpose, the invention provides a broadband convex optimization DOA estimation algorithm based on fourth-order cumulant, and the method comprises the following specific steps:

1) adopting a broadband signal with stronger information carrying capacity by using an incident signal;

2) performing discrete Fourier transform on the array receiving signal, and decomposing the broadband signal into a plurality of narrowband components with non-overlapping frequencies;

3) for P far-field wideband signals incident on a spatial array with M array elements, the signal received by the mth array element can be expressed as:

the matrix form is represented as:

Y＝AS+N

estimation of fourth order cumulant matrix of array output signals

Can be divided into mutually orthogonal signal subspaces

And noise subspace

The wideband signal DOA estimation problem can be represented by a redundant dictionary as:

the peak value of the S is the position of the signal source.

As a further improvement of the invention, S in the redundant dictionary expression of step three in claim 1 is obtained by solving a convex optimization problem, that is,

wherein the Frobenius norm is defined as:

and is provided with

As a further development of the invention, the estimation of the signal subspace in step three in claim 1

Can be determined by estimating the fourth order cumulant matrix of the array output signal

Singular value decomposition from the previous p²The eigenvector corresponding to the larger eigenvalue is expanded into a signal subspace, and the rest eigenvaluesThe corresponding eigenvectors are expanded into noise subspaces to obtain orthogonal projection estimates of mutually orthogonal signal subspaces and noise subspaces

And

as described above

Estimated from the limited number of sampling points, so that the fourth-order cumulant matrix of the output signals of the array

Expressed as:

compared with the traditional DOA estimation algorithm based on narrow-band signals, the DOA estimation method based on the fourth-order cumulant broadband convex optimization has the advantages that the broadband signals are used as transmitting signals and carry more original position information, the fourth-order cumulant matrix of array output signals is used for constructing a sparse representation model, and noise items can be effectively suppressed. And then singular value decomposition is carried out on the cumulant matrix to simplify the model, so that the data scale is reduced, and the noise is further suppressed.

Drawings

FIG. 1 shows a DOA estimation algorithm l of a classical narrowband source₁-a schematic flow diagram of the SVD algorithm;

FIG. 2 is a schematic flow chart of a fourth-order cumulant-based DOA estimation algorithm for broadband convex optimization according to the present invention;

FIG. 3 is a comparison of the effect of the present invention and DOA estimation based on covariance method;

Detailed Description

The technical scheme of the invention is explained in detail in the following with the accompanying drawings:

the invention provides a four-order cumulant-based broadband convex optimization DOA estimation method, which solves the problems that the existing sparse reconstruction DOA estimation algorithm cannot inhibit noise terms and has large calculation amount.

FIG. 1 shows a conventional DOA estimation algorithm l for narrowband signals₁-operation flow of the SVD algorithm.

For P far-field narrow-band signals incident on a spatial array with M array elements, the signal received by the mth array element can be expressed as:

wherein a is_pRepresenting the response of the m-th array element to the p-th signal source, τ_m(θ_p) Representing the time delay for the p-th signal source to reach the m-th array element. n is_m(t) represents the noise component on the mth array element. The signal is time domain sampled with a fast beat number of T.

For each array receiving Y observation vectors of signals: y (t)₁),y(t₂),…,y(t_T) Written in matrix form:

Y＝AS+N

given that there are K transmit signals, the matrix Y is a K-dimensional space, so only the basis of the subspace needs to be preserved to estimate the matrix a, which consists of sparse column vectors. Consider now the dimensionality reduction operation on signal Y.

First, singular value decomposition is performed on Y:

Y＝ULV^H

the reduced signal retains most of the energy of the original signal and is represented as Y_SVIs an M × K matrix. And Y is_SVIs obtained by the following formula: y is_SV＝ULD_K＝YVD_KWherein D is_K＝[I_K,0]^H,I_KIs a K identity matrix, and 0 represents a zero matrix of K (T-K) dimension. Likewise, let S_SV＝SVD_K，N_SV＝NVD_KThe method comprises the following steps:

Y_SV＝AS_SV+N_SV

then by minimizing the objective function: | | Y_SV-AS_SV||_f ²To obtain an optimized data solution. Wherein the Frobenius norm is defined as:

||Y_SV-AS_SV||_f ²＝||vec(Y_SV-AS_SV)||₂ ²

considering the sparse characteristic of the signal in the spatial dimension, consider the signal S after the dimension reduction_SVApplying l in the singular vector dimension₂Norm to obtain vector

Namely:

then the vector at that time

Corresponding to the sparsity of the spatial spectrum one by one. For vector

Application of l₁The norm may be sparsified. Now, considering the perfection of the objective function, and constraining the sparsity condition with the regularization parameter λ, the objective function becomes:

the spatial spectrum of the signal is obtained by finding the value that minimizes the above-mentioned objective function.

Consider now the use of second order cone planning to minimize the above objective function.

Linearize the objective function, which can be rewritten as:

in view of the vector

Not negative, therefore the constraint

Can be written as:

1^Hr≤q,r＝[r₁,r₂,…r_N]

where 1 represents the entire column vector of dimension N × 1. z is a radical of formula_k＝y^SV(k)-As^SV(k)

Let z_k＝y^SV(k)-As^SV(k) Then the objective function is expressed as a second order cone form:

min(p+λq)

subject to||z₁ ^H,z₂ ^H,…z_K ^H||₂ ²≤p

1^Hr≤q,r＝[r₁,r₂,…r_N]

and solving the optimization problem to obtain the spatial spectrum of the signal.

Fig. 2 shows a flow chart of the DOA estimation algorithm based on the fourth-order cumulants and based on the broadband convex optimization of the present invention.

For P far-field wideband signals incident on a spatial array with M array elements, the signal received by the mth array element can be expressed as:

wherein a is_pTo representResponse of mth array element to pth signal source, tau_m,pRepresenting the time delay for the p-th signal source to reach the m-th array element. n is_m(t) represents the noise component on the mth array element. Considering the use of multiple frequency channels to separate the signal sources, the above equation is written in matrix form:

Y＝AS+N

wherein the matrix Y ═ Y (v)₁),y(v₂),…,y(v_F)]Is an M x F matrix, where F is the number of frequency channels. It is formed by connecting observation vectors of different frequencies in series, wherein y (v)_i)＝[y₁(v_i),y₂(v_i),…,y_M(v_i)]A is the array manifold, S is the incident signal, N is composed of noise components of different frequencies: n ═ N (v)₁),n(v₂),…,n(v_F)]

The fourth order cumulant of each subband component of the received signal is considered:

consider now the construction of a signal subspace in the fourth order cumulant of the signal. The obtained fourth-order cumulant is subjected to singular value decomposition, including,

wherein

The matrix V is composed of eigenvectors corresponding to the eigenvalues.

Selecting P from L²The larger eigenvalue, with its corresponding eigenvector, estimates the signal subspace, and the projection of the signal subspace can be estimated as:

accordingly, the noise subspace

From the rest (MF)²-P²The feature vector corresponding to the feature value is formed, and the signal model can be described by the following form:

now in

For a signal subspace and constraining the sparsity condition with a regularization parameter λ, an objective function can be defined:

the spatial spectrum of the signal is obtained by minimizing the objective function. The optimization problem can be realized by second-order cone programming with the same principle as l₁-SVD algorithm.

Fig. 3 compares the DOA estimation effect using the method of the present invention with that using the conventional compressed sensing method. It can be seen that the angular orientation of arrival estimated using the method of the present invention is closer to the actual value. The method based on the fourth-order cumulant effectively finishes the suppression of the noise term, so that the DOA estimation of the incoming wave signal is more accurate.

The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention in any way, but any modifications or equivalent variations made according to the technical spirit of the present invention are within the scope of the present invention as claimed.

Claims (3)

1. A broadband convex optimization DOA estimation algorithm based on fourth-order cumulant comprises the following specific steps:

the method comprises the following steps: using a broadband signal with stronger information carrying capacity as an incident signal;

step two: performing discrete Fourier transform on the array receiving signal, and decomposing the broadband signal into a plurality of narrowband components with non-overlapping frequencies;

step three: for P far-field wideband signals incident on a spatial array with M array elements, the signal received by the mth array element can be expressed as:

the matrix form is represented as:

Y＝AS+N

estimation of fourth order cumulant matrix of array output signals

Can be divided into mutually orthogonal signal subspaces

And noise subspace

The wideband signal DOA estimation problem can be represented by a redundant dictionary as:

the peak value of the S is the position of the signal source.

2. A fourth-order cumulant-based wideband convex optimization DOA estimation algorithm further characterized in that S in the redundant dictionary expression of step three in claim 1 is obtained by solving a convex optimization problem, i.e.,

wherein the Frobenius norm is defined as:

and is

。

3. A fourth order cumulant-based DOA estimation algorithm with broadband convex optimization, further characterized by the estimation of the signal subspace of step three in claim 1

Can be determined by estimating the fourth order cumulant matrix of the array output signal

Singular value decomposition from the previous p²The eigenvectors corresponding to the larger eigenvalues span a signal subspace, and the eigenvectors corresponding to the other eigenvalues span a noise subspace, so as to obtain orthogonal projection estimation of the signal subspace and the noise subspace which are orthogonal to each other

And

as described above

Estimated from the limited number of sampling points, so that the fourth-order cumulant matrix of the output signals of the array

Expressed as:

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