阅读说明：本技术 一种角度与频率联合估计的增广doa矩阵方法 (Angle and frequency joint estimation augmented DOA matrix method ) 是由 戴祥瑞 张小飞 于 2019-11-21 设计创作，主要内容包括：本发明公开了一种角度与频率联合估计的增广DOA矩阵方法,以阵列信号处理为背景讨论了单时延阵列信号接收系统中的角度与频率联合估计问题,在传统DOA矩阵法的基础之上,提出并探讨了一种基于单时延阵列信号接收系统的增广DOA矩阵法。利用了该信号系统的接收数据的自相关和互相关矩阵,构建一个增广DOA矩阵,然后通过对DOA矩阵的特征分解,直接获得待估的信号方向矢量和信号方向元素,并由此得到待估信号的DOA角度和频率估计。对比传统的DOA矩阵方法,本发明完全利用了信号接收系统的接收数据的自相关矩阵和互相关矩阵,所以具有更好的角度与频率估计性能。本发明不需要空间谱搜索,具有较低的算法复杂度,得到的DOA估计角度和频率估计能实现自动配对。(The invention discloses an angle and frequency joint estimation augmented DOA matrix method, which discusses the problem of angle and frequency joint estimation in a single time delay array signal receiving system by taking array signal processing as the background. The method comprises the steps of constructing an augmented DOA matrix by utilizing an autocorrelation matrix and a cross-correlation matrix of received data of the signal system, directly obtaining a signal direction vector and a signal direction element to be estimated through characteristic decomposition of the DOA matrix, and obtaining DOA angle and frequency estimation of the signal to be estimated. Compared with the traditional DOA matrix method, the method completely utilizes the autocorrelation matrix and the cross-correlation matrix of the received data of the signal receiving system, so that the method has better angle and frequency estimation performance. The method does not need space spectrum search, has lower algorithm complexity, and can realize automatic pairing of the obtained DOA estimation angle and frequency estimation.)

1. An angle and frequency joint estimation augmented DOA matrix method is characterized by comprising the following steps:

step 1, a linear array is arranged in space, when K uncorrelated narrow-band co-carrier signals enter the linear array, estimation of an autocorrelation matrix of the linear array receiving signals is solvedAdding a delay output tau to a received signal of a linear array, and solving an estimate of an autocorrelation matrix of the received signal after adding the delay output

Step 2, estimating the autocorrelation matrix of the linear array received signal

Step 3, estimating according to the cross correlation matrix

And 4, performing characteristic decomposition on the expanded DOA matrix R' to obtain a characteristic value and a characteristic vector, obtaining frequency estimation according to the characteristic value, and obtaining DOA angle estimation according to the characteristic vector.

2. The DOA matrix method for angle and frequency joint estimation according to claim 1, wherein the estimation of the autocorrelation matrix in step 1

wherein N is fast beat number, x (t) represents the receiving signal of the linear array at time t, y (t) represents the receiving signal of the linear array after delay output is added at time t ·)^HRepresenting a matrix conjugate transpose.

3. The DOA matrix method for angle and frequency joint estimation according to claim 1, wherein the matrix of step 2 is the DOA matrix

wherein the content of the first and second substances,

4. The method for amplifying the DOA matrix for joint angle and frequency estimation according to claim 1, wherein the formula of the extended DOA matrix R' in step 3 is as follows:

wherein R is₁And R₂Each of which represents a matrix of the image data,

5. The method for amplifying the DOA matrix of the joint estimation of the angle and the frequency according to the claim 1, wherein the specific process of the step 4 is as follows:

performing characteristic decomposition on the expanded DOA matrix R' to obtain a matrix A_EAnd a gamma-ray that is different from the gamma-ray,

Γ＝diag{exp(-j2πf₁τ),exp(-j2πf₂τ),…,exp(-j2πf_Kτ)}

Wherein f is_kRepresenting the carrier frequency, theta_kRepresenting the included angle between the kth narrowband same carrier signal and the linear array, c representing the light speed, tau representing delay output, K being the number of the narrowband same carrier signals, and j representing an imaginary unit;

obtaining the frequency f according to the characteristic value in the gamma_kEstimation of (2):

wherein λ is_kRepresenting the kth characteristic value;

according to A_EThe definition of (1) is divided into A and A Γ^-1Two parts, after feature decomposition, the estimation of the two parts is respectively

Technical Field

The invention relates to a signal source positioning method under a sensor array, in particular to an angle and frequency joint estimation DOA matrix augmentation method, and belongs to the technical field of array signal processing.

Background

The traditional DOA matrix method constructs a DOA matrix according to the properties of the covariance matrix. By means of characteristic decomposition of the DOA matrix, a signal direction vector and a signal direction element to be estimated can be directly obtained, and signal parameters can be estimated accordingly, so that polynomial search is completely avoided, the operation amount is small, but the DOA angle and frequency joint estimation performance is low because the autocorrelation information and the cross correlation information of the array received signals are not completely utilized.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: the DOA matrix augmentation method based on angle and frequency joint estimation is provided, the problem of two-dimensional DOA estimation under a single-delay sensor array receiving system is solved, and the DOA matrix augmentation method based on angle and frequency joint estimation has high estimation performance.

The invention adopts the following technical scheme for solving the technical problems:

an angle and frequency joint estimation augmented DOA matrix method comprises the following steps:

step 1, a linear array is arranged in space, when K uncorrelated narrow-band co-carrier signals enter the linear array, estimation of an autocorrelation matrix of the linear array receiving signals is solved

Adding a delay output tau to a received signal of a linear array, and solving an estimate of an autocorrelation matrix of the received signal after adding the delay output

Estimation of cross-correlation matrix for solving linear array received signal and delayed output received signal

And estimating the cross-correlation matrix of the delayed output received signal and the linear array received signal

Step 2, estimating the autocorrelation matrix of the linear array received signalDecomposing the characteristic value and removing the noise influence to obtain a matrix

Similarly, estimation of the autocorrelation matrix of the received signal to which the delay output is added

Decomposing the characteristic value and removing the noise influence to obtain a matrix

Step 3, estimating according to the cross correlation matrix

And

and a matrix

And

definition matrix R₁And R₂And constructing an extended DOA matrix

And 4, performing characteristic decomposition on the expanded DOA matrix R' to obtain a characteristic value and a characteristic vector, obtaining frequency estimation according to the characteristic value, and obtaining DOA angle estimation according to the characteristic vector.

As a preferred scheme of the invention, the estimation of the autocorrelation matrix in the step 1

And

and estimation of cross-correlation matrix

And

the following formula is obtained:

wherein N is fast beat number, x (t) represents the receiving signal of the linear array at time t, y (t) represents the receiving signal of the linear array after delay output is added at time t ·)^HRepresenting a matrix conjugate transpose.

As a preferred embodiment of the present invention, the matrix of step 2

And

the formula of (1) is as follows:

wherein the content of the first and second substances,

an estimate of an autocorrelation matrix representing a linear array of received signals,

representing an estimate of the autocorrelation matrix of the linear array received signal after addition of the delayed output,

represents an estimate of the variance of additive white gaussian noise and I represents the identity matrix.

As a preferred embodiment of the present invention, the formula of the extended DOA matrix R' in step 3 is as follows:

wherein R is₁And R₂Each of which represents a matrix of the image data,

an estimate of a cross-correlation matrix representing the linear array received signal and the delayed output added received signal,

representing an estimate of a cross-correlation matrix of the delayed output received signal with the linear array received signal,

and

each of which represents a matrix of the image data,

representing a matrix conjugate transpose.

As a preferred embodiment of the present invention, the specific process of step 4 is:

performing characteristic decomposition on the expanded DOA matrix R' to obtain a matrix A_EAnd a gamma-ray that is different from the gamma-ray,

A＝[a(f₁,θ₁),a(f₂,θ₂),…,a(f_K,θ_K)]represents a direction matrix and has

Γ＝diag{exp(-j2πf₁τ),exp(-j2πf₂τ),…,exp(-j2πf_Kτ)}

Wherein f is_kRepresenting the carrier frequency, theta_kRepresenting the included angle between the kth narrowband same carrier signal and the linear array, c representing the light speed, tau representing delay output, K being the number of the narrowband same carrier signals, and j representing an imaginary unit;

obtaining the frequency f according to the characteristic value in the gamma_kEstimation of (2):

wherein λ is_kRepresenting the kth characteristic value;

according to A_EThe definition of (1) is divided into A and A Γ^-1Two parts, after feature decomposition, the estimation of the two parts is respectively

And

for matrix

Normalizing a certain column to make the first term of the column be 1, taking a phase angle of the normalized column, estimating the phase difference between the narrow-band common-carrier signal and the linear array according to the phase angle, and finally obtaining the estimation of the DOA angle by using a least square method

Compared with the prior art, the invention adopting the technical scheme has the following technical effects:

1. the DOA matrix augmentation method provided by the invention keeps the advantages that the traditional DOA matrix method can completely avoid polynomial search and has small calculation amount, and simultaneously, the method completely utilizes the autocorrelation information and the cross-correlation information of array receiving signals to construct an augmented DOA matrix and improve the joint estimation performance of DOA angles and frequencies.

2. The DOA angle and the frequency estimated by the method can realize automatic pairing.

3. The method has lower complexity.

Drawings

FIG. 1 is a topological diagram of an array structure of the present invention.

Fig. 2 is a diagram of a signal receiving system for the method of the present invention.

FIG. 3 is a scatter plot for the method of the present invention.

FIG. 4 is a graph comparing the angular RMSE performance of the inventive method and the conventional DOA matrix method under different SNR conditions.

FIG. 5 is a graph comparing the frequency RMSE performance of the inventive method and the conventional DOA matrix method under different SNR conditions.

FIG. 6 is a graph comparing the angle RMSE performance of the method of the present invention and the conventional DOA matrix method under different snapshot conditions.

FIG. 7 is a graph comparing the frequency RMSE performance of the method of the present invention and the conventional DOA matrix method under different snapshot conditions.

Detailed Description

Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.

The symbols represent: used in the invention (·)^TRepresentation matrix transposition, (.)^HRepresenting the conjugate transpose of the matrix (.)^*Representing the conjugate of the matrix, the capital letter X representing the matrix, the lower case letter X (·) representing the vector, I representing the identity matrix, diag (v) representing the diagonal matrix made up of the elements in v, E [ ·]Indicating the expectation of the matrix, and angle (·) indicates the phase angle operation.

Data model

The signal receiving array consists of a non-uniform linear array of one of the sensors shown in FIG. 1, the array having M sensors with the M-th sensor spaced d from the first sensor_m(M ═ 1, …, M), where d₁0. Suppose that there are K uncorrelated narrowband co-carrier signals s in space_k(t) (K is 1. ltoreq. K. ltoreq.K) is incident on the array at an angle theta to the array_kCarrier frequency of f_k. So the received signal of the mth sensor is:

wherein n is_k(t) is zero mean, variance σ²C is the speed of light. To estimate the frequency of the signal, a delay output τ is added to the received signal of the sensor, as shown in FIG. 2, and it is assumed that 0 < 2 τ < 1/max (f)_k). The output signal after adding the delay τ is therefore:

writing the output signal in vector/matrix form, i.e.

x(t)＝As(t)+n(t)

y(t)＝As(t-τ)+n(t-τ)＝AΓs(t)+n(t-τ)

Wherein x (t) ═ x₁(t),x₂(t),…,x_M(t)]^T，y(t)＝[y₁(t),y₂(t),…,y_M(t)]^T，s(t)＝[s₁(t),s₂(t),…,s_K(t)]^T，n(t)＝[n₁(t),n₂(t),…,n_M(t)]^T。A＝[a(f₁,θ₁),a(f₂,θ₂),…,a(f_K,θ_K)]Represents a direction matrix and has

Γ＝diag{exp(-j2πf₁τ),exp(-j2πf₂τ),…,exp(-j2πf_Kτ)}

Second, method derivation

The received data x (t) has an autocorrelation matrix R_xxThe expression is

R_xx＝E[x(t)x^H(t)]＝AΨA^H+σ²I

Where Ψ ═ E [ s (t) s^H(t)]Is a covariance matrix, σ, of the signal source²Is the variance of additive white gaussian noise.

The autocorrelation matrix of the received data y (t) is R_yyThe expression is

R_yy＝E[y(t)y^H(t)]＝AΓΨΓ^HA^H+σ²I

＝AΨΓΓ^HA^H+σ²I

＝AΨA^H+σ²I

Considering the independence of noise itself and independent of signal, let the cross-correlation matrix of y (t) and x (t) be R_yxThen, then

R_yx＝E[y(t)x^H(t)]＝AΓΨA^H

Similarly, the cross-correlation matrix of x (t) and y (t) is

R_xy＝E[x(t)y^H(t)]＝AΓ^HΨA＝AΓ^-1ΨA

To R_xxPerforming eigenvalue decomposition (EVD) to let ε₁,…,ε_KIs a matrix R_xxUnder the assumption of white noise, the noise variance σ can be obtained by averaging the M-K small eigenvalues²Is estimated. Then, by removing the influence of noise, it is possible to obtain

The same can be obtained

C_yy＝AΨA^H＝R_yy-σ²I

Definition of

We define

Thus, it is possible to provide

R₁＝A_EΨA^H

R₂＝A_EΓΨA^H

According to the idea of DOA matrix method, the following DOA matrix can be defined

Wherein

If A and Ψ full rank, Γ, do not have identical diagonal elements, then the K non-zero eigenvectors of the DOA matrix R' are equal to the K diagonal elements in Γ, and the eigenvectors for these values are equal to the corresponding signal direction vectors, i.e., the

R′A_E＝A_EΓ

The matrix A can be obtained by performing characteristic decomposition on the DOA matrix R_EAnd Γ. From the eigenvalues in Γ, the frequency f can be derived_kEstimation of (2):

according to A_EBy definition of (A) and (A) Γ we divide it into^-1Two parts, after feature decomposition, the estimation of the two parts is respectively

And

estimate outAnd

then is provided with

And (c) in a certain column a, performing DOA angle estimation on the direction matrix by using the Vandermonde characteristics of the direction matrix. The direction vector a is firstly normalized, so that the initial term is 1. And (b) taking angle (a), estimating the phase difference between the arrays, and finally estimating the DOA angle by using a least square method. Because of the fact that

Can therefore obtain

u_k＝-angle(a(θ_k))

＝[0,2πd₂f_ksinθ_k/c,…,2πd_Mf_ksinθ_k/c]^T

Least squares fit toWherein

Wherein

For estimation of frequency, e₁＝sinθ_k. Thus e can be estimated by least squares_k

So that the angle is estimated as

In the same way, we can get from

To obtain

Thus DOA angle theta_kIs estimated by

The method comprises the following steps:

[1]estimation of autocorrelation and cross-correlation matrices for received data x (t) and y (t)

And

[2]removing noise influence on the autocorrelation matrix to obtain

And

[3]definition of R₁And R₂And constructing an augmented DOA matrix

[4] And decomposing the characteristic value of R', and respectively obtaining the frequency and the DOA angle estimation according to the characteristic value and the characteristic vector.

Third, method analysis and simulation

The DOA angle estimation method of the invention is subjected to complexity analysis to obtain the complexity O {4M } of the autocorrelation and cross-correlation matrix²N, wherein N represents the fast beat number of the received signal; computingHas a complexity of O {5M }³}; computing

Has a complexity of O {4M }³}; the complexity of characteristic decomposition of R' is O {8M³}. The total complexity of the calculation algorithm is O {4M²N+17M³}。

The method of the invention completely utilizes the autocorrelation information and the cross-correlation information of the array received data to construct an augmented DOA matrix, and the traditional DOA matrix method does not completely utilize the autocorrelation information and the cross-correlation information of the array received data, so that the method of the invention has higher DOA angle and frequency estimation performance than the traditional DOA matrix method.

And (3) simulation results:

three narrow-band signals (theta) in a far field of space are assumed₁,f₁)＝(10°,6MHz)，(θ₂,f₂) ═ 20 °,8MHz) and (θ₃,f₃) Incident on the array of figure 1 at (30 °,10MHz) the signals are uncorrelated. The DOA angle and frequency estimation performance is evaluated using 1000 monte carlo simulations, defining the Root Mean Square Error (RMSE) expression as follows:

wherein

Andrepresents the parameter estimation result theta of the kth information source in the ith Monte Carlo simulation_kAnd f_kRepresenting the true value of the parameter for the kth source.

Fig. 3 shows the scatter distribution diagram of the algorithm of the present invention, with the simulation parameters M-12, N-500, K-3 and SNR-10 dB. The DOA angle (theta) versus frequency (frequency) of the source is evident from the figure.

Fig. 4 and 5 show graphs of angle versus frequency estimation performance and comparison to CRB performance for conventional DOA matrix algorithms and augmented DOA matrix algorithms as a function of signal-to-noise ratio (SNR) under the same conditions. The simulation parameters are set as array element number M of the array being 12 and snapshot number N being 500.

As can be seen from fig. 4 and 5, the augmented DOA matrix method has high angle and frequency estimation performance.

Fig. 6 and 7 show performance graphs of angle and frequency estimation performance of the conventional DOA matrix algorithm and the augmented DOA matrix algorithm as a function of a snapshot under the same SNR, which is set to 10 dB.

As can be seen from fig. 6 and 7, as the number of fast beats increases, the performance of both algorithms is improved, and the angle and frequency estimation performance of the augmented DOA matrix method is significantly better than that of the conventional DOA matrix method.

The above embodiments are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modifications made on the basis of the technical scheme according to the technical idea of the present invention fall within the protection scope of the present invention.