CN115248413A - Off-grid signal direction-of-arrival estimation method suitable for non-uniform linear array - Google Patents

Abstract

The invention discloses a method for estimating the direction of arrival of off-grid signals suitable for a non-uniform linear array, which comprises the following steps: s1, acquiring an initial estimation value of a signal direction of arrival on a candidate grid by using a beam forming method according to single snapshot array element data; s2, calculating a complex value signal; s3, separating corresponding signal data from the received data of the array; s4, multiplying the conjugate of the previous element and the next element in the signal data to obtain a construction vector, and extracting the phase of each element of the construction vector; s5, performing ambiguity resolution on the phase of the extracted construction vector, and acquiring a closed expression of the signal direction of arrival according to the ambiguity resolution phase; s6, calculating the arrival direction of the updating signal; and S7, judging whether the signal wave arrival direction is converged or not, or whether the cycle number reaches a preset threshold or not, and if not, returning to the step S2. The method disclosed by the invention can be suitable for both uniform linear arrays and non-uniform linear arrays, can realize high-precision estimation of the signal direction of arrival, and is low in calculation complexity.

Description

A Direction of Arrival Estimation Method for Off-grid Signals Applicable to Non-Uniform Linear Arrays

technical field

The invention relates to the technical field of array signal processing, in particular to a method for estimating the direction of arrival of off-grid signals suitable for non-uniform linear arrays.

Background technique

Direction of arrival (DOA) refers to the direction of arrival of space signals, that is, the direction angle at which each signal arrives at the reference element of the array. The direction of arrival is an important concept in the theory of spatial spectrum estimation, and the spatial spectrum is an important concept in array signal processing. The spatial spectrum represents the energy distribution of the signal in all directions in space. Direction of arrival estimation is a key technology for practical engineering applications such as target positioning, detection, and identification. It is widely used in military and national economic applications such as radar, communication, radio astronomy, geophysics, speech recognition, sonar, and medical imaging.

The traditional direction of arrival estimation algorithm searches the signal direction of arrival by uniformly discretizing the angle space domain. This algorithm is also known as the grid signal direction of arrival estimation algorithm, and its precondition is to assume that the direction of arrival of the signal is exactly located on the discretized grid. However, in practical applications, the direction of arrival of the signal may not fall on the pre-divided grid. At this time, the grid mismatch problem will be caused, resulting in inaccurate estimation of the direction of arrival, and a sharp drop in the estimation accuracy of the algorithm. In order to solve the grid mismatch problem, a direction-of-arrival estimation algorithm for off-grid signals is proposed.

With the continuous research on DOA estimation algorithms for off-grid signals, a variety of DOA estimation algorithms for off-grid signals have been proposed, mainly including off-grid sparse Bayesian algorithms, algorithms based on phase deviation search and fractional Fu Liye coefficient interpolation algorithm. Among them, the off-grid sparse Bayesian algorithm applies the first-order Taylor expansion to the real incoming wave direction, and estimates the off-grid offset as a hyperparameter; the algorithm based on phase deviation search obtains the signal incoming wave through discrete Fourier transform The initial rough estimation of the direction, and then use the phase rotation method to correct the initial rough estimation; the fractional Fourier coefficient interpolation algorithm obtains high-precision Direction of arrival estimation.

However, the existing DOA estimation algorithms for off-grid signals generally have high computational complexity, slow convergence rate, and long running time of the algorithm, while most DOA estimation algorithms with low computational complexity use Fourier The coefficient interpolation method is only applicable to uniform linear arrays, and cannot be applied to non-uniform linear arrays which are more widely used in practice.

SUMMARY OF THE INVENTION

In order to solve some or all of the technical problems in the above-mentioned prior art, the present invention provides a method for estimating the direction of arrival of off-grid signals suitable for non-uniform linear arrays.

Technical scheme of the present invention is as follows:

A method for estimating the direction of arrival of off-grid signals suitable for non-uniform linear arrays is provided, and the method includes:

Step S1, according to the single snapshot array metadata, use the beamforming method to obtain the initial estimated value of the direction of arrival of the signal on the candidate grid;

Step S2, calculating the complex-valued signal according to the direction of arrival of the signal;

Step S3, according to the direction of arrival of the signal and the complex-valued signal, separate the corresponding signal data from the received data of the array;

Step S4, multiplying the conjugate of the previous element and the next element in the signal data to obtain a construction vector, and extracting the phase of each element of the construction vector;

Step S5, performing defuzzification on the phase of the extracted construction vector, and obtaining a closed-form expression of signal direction of arrival according to the defuzzification phase;

Step S6, calculating and updating the signal direction of arrival according to the closed expression of the signal direction of arrival;

Step S7, judging whether the signal direction of arrival has converged, or whether the number of cycles reaches a preset threshold, if yes, use the currently calculated signal direction of arrival as the final estimated value, if not, return to step S2.

In some possible implementations, it is set that: N antennas form a non-uniform array, the array element position c _n is an integer multiple of the unit array element spacing, n=1,2,...,N, the unit array element spacing d Set to half the wavelength, K far-field narrowband signals are incident on the array, and the direction of arrival is θ=[θ ₁ ,...,θ _K ], θ ₁ ,...,θ _K represent the first to the first The direction of arrival of K signals, the received data y of the array is expressed as y=As+m, s=[s ₁ ,...,s _K ] ^T represents the vector of K complex-valued deterministic signals, and m represents N× 1-dimensional Gaussian white noise, A represents the array manifold matrix, A=[a(θ ₁ ),a(θ ₂ ),...,a(θ _K )], a(θ _k ) represents the array steering vector,

中的M个均匀采样网格点，得到

网格间隔为

M表示网格数目；Consider angle search area

M uniformly sampled grid points in , get

The grid interval is

M represents the number of grids;

Set at the same time: the direction of arrival of the incident signal does not fall on the pre-divided discrete grid points;

Use beamforming to obtain an initial estimate of the direction of arrival of a signal on a candidate grid, including:

Define a normalized M×N-dimensional beamforming matrix F, the (p,q)th element of matrix F is

According to the defined matrix F, the spatial spectrum x=|Fy| after beamforming is obtained;

基于K个峰值对应的位置，利用公式

确定对应的信号波达方向的初始估计值

According to the spatial spectrum x, the positions corresponding to the K peaks are obtained

Based on the positions corresponding to the K peaks, using the formula

Determine the initial estimate of the corresponding signal direction of arrival

In some possible implementations, the complex-valued signal is calculated according to the direction of arrival of the signal, including:

Construct the problem of estimating complex-valued signals according to the direction of arrival of the signal;

The least square method is used to solve the problem of estimating complex-valued signals, and the closed-form solution of complex-valued signals is obtained.

In some possible implementations, the problem of estimating a complex-valued signal is:

The closed-form solution of the complex-valued signal is:

s=(A ^H A) ^-1 A ^H y

Among them, s represents a complex-valued signal, and A ^H represents the conjugate transpose matrix of A.

In some possible implementations, the following formula is used to separate the corresponding signal data from the received data of the array;

Among them, y _k represents the k-th signal data, s _i represents the i-th complex-valued signal, and a(θ _i ) represents the array steering vector corresponding to the i-th signal.

In some possible implementations, the following formulas are used to determine the elements of the signal data;

Among them, y _{k, n} represents the nth element of the kth signal, |s _k | represents the amplitude of the kth signal, φ _k represents the phase of the kth signal, ε _n represents the zero mean corresponding to the nth element Gaussian white noise;

Define the construction vector corresponding to the kth signal as r _k ∈ C ^N-1 , C represents a complex number set, and each element of the construction vector r _k is determined by the following formula;

表示y_k,n的共轭。Among them, r _k,n represents the nth element of the construction vector r _k ,

Indicates the conjugate of y _k,n .

In some possible implementations, the phase of the extracted construction vector is defuzzified using the following formula;

表示构造向量r_k的第n个元素的相位，ψ_n表示通过四舍五入得到的整数，

round(x)表示对x进行四舍五入；where g _n (r _k ) represents the deambiguation phase of the nth element of the construction vector r _k ,

Represents the phase of the nth element of the construction vector r _k , ψ _n represents the integer obtained by rounding,

round(x) means rounding x;

The deblurring phase of the construction vector is expressed as:

ε表示有色高斯噪声向量，ε＝[ε₂-ε₁,ε₃-ε₂,...,ε_N-ε_N-1]^T。Among them, g(r _k ) represents the deblurring phase of the construction vector r _k , g ₁ (r _k ), g ₂ (r _k ),…,g _(N-1) (r _k ) represent the deblurring phase vector g( r _k ), the N-1 elements of

ε represents a colored Gaussian noise vector, ε=[ε ₂ -ε ₁ ,ε ₃ -ε ₂ ,...,ε _N -ε _N-1 ] ^T .

In some possible implementation manners, the obtaining a closed-form expression of the direction of arrival of the signal according to the deambiguity phase includes:

Construct the objective function:

Minimize the objective function to obtain a closed-form expression for the signal direction of arrival;

where Q represents the covariance matrix of the colored Gaussian noise vector ε.

In some possible implementation manners, the following formula is used to calculate the signal direction of arrival;

The main advantages of the technical solution of the present invention are as follows:

The off-grid signal direction of arrival estimation method applicable to non-uniform linear arrays of the present invention first obtains the initial estimated value of the signal direction of arrival on the candidate grid through the beamforming method, and then receives the array according to the signal direction of arrival The data is processed, the corresponding signal data and construction vector are obtained, and the phase of the construction vector is defuzzified, and the closed-form expression of the signal direction of arrival is determined according to the defuzzification phase, which can reduce the computational complexity of the signal direction of arrival estimation process. The convergence speed is improved, and it can be applied to the estimation solution of the signal direction of arrival of the uniform linear array and the non-uniform linear array at the same time.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to these drawings without creative efforts.

FIG. 1 is a flow chart of a method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays according to an embodiment of the present invention;

Fig. 2 is the schematic diagram of a kind of uniform linear array provided by the present invention;

Fig. 3 is a schematic diagram of the change curve of the mean square error of the direction of arrival estimated value obtained by using different algorithms according to the example 1 of the present invention;

Figure 4 is a schematic diagram of a non-uniform linear array provided by the present invention;

Fig. 5 is a schematic diagram of the change curve of the mean square error of the direction of arrival estimation value obtained by using different algorithms in the case of a single signal provided by Example 2 of the present invention as a function of the signal-to-noise ratio;

FIG. 6 is a schematic diagram of the change curve of the mean square error of the direction of arrival estimation value obtained by using different algorithms in the case of multiple signals provided by Example 3 of the present invention as a function of the signal-to-noise ratio.

Detailed ways

In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be clearly and completely described below in conjunction with specific embodiments of the present invention and corresponding drawings. Apparently, the described embodiments are only some of the embodiments of the present invention, but not all of them. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts fall within the protection scope of the present invention.

The technical solutions provided by the embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings.

Referring to Fig. 1, an embodiment of the present invention provides a method for estimating the direction of arrival of off-grid signals suitable for non-uniform linear arrays. The method includes the following steps:

Step S1, according to the single snapshot array metadata, use the beamforming method to obtain the initial estimated value of the direction of arrival of the signal on the candidate grid;

Step S2, calculating the complex-valued signal according to the direction of arrival of the signal;

Step S3, according to the direction of arrival of the signal and the complex-valued signal, separate the corresponding signal data from the received data of the array;

Step S4, multiplying the conjugate of the previous element and the next element in the signal data to obtain a construction vector, and extracting the phase of each element of the construction vector;

Step S5, performing defuzzification on the phase of the extracted construction vector, and obtaining a closed-form expression of signal direction of arrival according to the defuzzification phase;

Step S6, calculating and updating the signal direction of arrival according to the closed expression of the signal direction of arrival;

Step S7, judging whether the signal direction of arrival has converged, or whether the number of cycles reaches a preset threshold, if yes, use the currently calculated signal direction of arrival as the final estimated value, if not, return to step S2.

An off-grid signal direction of arrival estimation method suitable for non-uniform linear arrays provided by an embodiment of the present invention first obtains the initial estimated value of the direction of arrival of the signal on the candidate grid through the beamforming method, and then according to the direction of arrival of the signal Process the received data of the array, obtain the corresponding signal data and construction vector, and defuzzify the phase of the construction vector, and determine the closed-form expression of the signal direction of arrival according to the defuzzification phase, which can reduce the complexity of the signal direction of arrival estimation process. Computational complexity, improved convergence speed, and can be applied to the estimation solution of the signal direction of arrival of uniform linear array and non-uniform linear array at the same time.

The steps and principles of the method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays provided by an embodiment of the present invention are described in detail below:

Step S1, according to the single-snapshot array metadata, the beamforming method is used to obtain the initial estimated value of the direction of arrival of the signal on the candidate grid.

A表示阵列流形矩阵，A＝[a(θ₁),a(θ₂),...,a(θ_K)]，a(θ_k)表示阵列导向矢量，

Specifically, set: N antennas form a non-uniform array, the array element position c _n is an integer multiple of the unit array element spacing, n=1,2,...,N, and the unit array element spacing d is set to half the wavelength , K far-field narrowband signals are incident on the array, and the direction of arrival is θ=[θ ₁ ,...,θ _K ], θ ₁ ,...,θ _K represent the arrival of the first to Kth signals direction, the received data y of the array is expressed as y=As+m, s=[s ₁ ,...,s _K ] ^T represents the vector of K complex-valued deterministic signals, m represents N×1-dimensional Gaussian white noise, The noise power is

A represents the array manifold matrix, A=[a(θ ₁ ),a(θ ₂ ),...,a(θ _K )], a(θ _k ) represents the array steering vector,

中的M个均匀采样网格点，得到

网格间隔为

M表示网格数目，同时设定：入射信号的波达方向没有落在预先划分好的离散网格点上。Consider angle search area

M uniformly sampled grid points in , get

The grid interval is

M represents the number of grids, and it is set at the same time that the direction of arrival of the incident signal does not fall on the pre-divided discrete grid points.

Based on the above settings, the beamforming method is used to obtain the initial estimated value of the direction of arrival of the signal on the candidate grid, including the following steps:

Define a normalized M×N-dimensional beamforming matrix F, the (p,q)th element of matrix F is

According to the defined matrix F, the spatial spectrum x=|Fy| after beamforming is obtained;

基于K个峰值对应的位置，利用公式

确定对应的信号波达方向的初始估计值

According to the spatial spectrum x, the positions corresponding to the K peaks are obtained

Based on the positions corresponding to the K peaks, using the formula

Determine the initial estimate of the corresponding signal direction of arrival

Step S2, calculating the complex-valued signal according to the direction of arrival of the signal.

In an embodiment of the present invention, according to the direction of arrival of the signal, the complex-valued signal can be calculated using the least square method.

Specifically, according to the direction of arrival of the signal, the least square method is used to calculate the complex-valued signal, including the following steps:

Construct the problem of estimating complex-valued signals according to the direction of arrival of the signal;

The least square method is used to solve the problem of estimating complex-valued signals, and the closed-form solution of complex-valued signals is obtained.

In an embodiment of the present invention, the problem of estimating a complex-valued signal can be formulated as:

The least squares method can be used to solve the minimization problem in the constructed problem of estimating complex-valued signals, and then obtain the closed-form solution of complex-valued signals.

Specifically, the least squares method solves the problem of estimating the complex-valued signal constructed, and the closed-form solution of the complex-valued signal can be obtained as:

s=(A ^H A) ^-1 A ^H y

Among them, s represents a complex-valued signal, and A ^H represents the conjugate transpose matrix of A.

Step S3, according to the direction of arrival of the signal and the complex-valued signal, separate the corresponding signal data from the received data of the array.

In order to estimate the angle of arrival of the signal, that is, to estimate the direction of arrival of the signal, it is necessary to separate the corresponding signal from the received data of the array.

Specifically, taking the need to separate the kth signal from the received data of the array as an example, in an embodiment of the present invention, the following formula can be used to separate the corresponding signal data from the received data of the array:

Among them, y _k represents the k-th signal data, s _i represents the i-th complex-valued signal, and a(θ _i ) represents the array steering vector corresponding to the i-th signal.

Further, based on the setting of the above step S1 and the above signal data separation formula, the kth signal data y _k can be expressed as:

Among them, |s _k | represents the magnitude of the k-th signal, and φ _k represents the phase of the k-th signal.

Step S4, multiplying the conjugate of the previous element and the next element in the signal data to obtain a construction vector, and extracting the phase of each element of the construction vector.

Since in the actual application process, when the signal-to-noise ratio is usually large, taking the kth signal as an example, in the case of a large signal-to-noise ratio, the elements of the kth signal data can be approximately expressed as:

Among them, y _k,n represents the n-th element of the k-th signal, ε _n represents the zero-mean Gaussian white noise corresponding to the n-th element, and the variance of ε _n is

Further, the construction vector corresponding to the kth signal is defined as r _k ∈ C ^N-1 , and C represents a complex number set. Since the elements in the construction vector are obtained by multiplying the conjugate of the previous element in the signal data with the next element, for this reason, in an embodiment of the present invention, based on the above setting, the construction vector has N-1 elements, and the construction Each element of vector r _k can be determined using the following formula:

表示y_k,n的共轭。Among them, r _k,n represents the nth element of the construction vector r _k ,

Indicates the conjugate of y _k,n .

表示相位

的第1个至第N-1个元素，则第i个元素

可以记作

其中，e_i表示(N-1)×1维列向量，且列向量的第i个位置为1，其余位置均为0，∠(·)表示提取相位。Further, set the phase of the construction vector r _k as

Indicates the phase

The 1st to N-1th elements of , then the i-th element

can be recorded as

Among them, e _i represents a (N-1)×1-dimensional column vector, and the i-th position of the column vector is 1, and the rest of the positions are 0, and ∠(·) represents the extraction phase.

In step S5, the phase of the extracted construction vector is defuzzified, and a closed-form expression of the direction of arrival of the signal is obtained according to the defuzzified phase.

Considering that the phase representation requires a modulo 2π, when the element spacing of the non-uniform linear array is greater than half the wavelength, the corresponding phase will be blurred because the number of periods is unknown. To this end, in an embodiment of the present invention, the phase of the extracted construction vector is defuzzified to overcome the above-mentioned problem.

Specifically, taking the kth signal as an example, the following formula can be used to defuzzify the phase of the extracted construction vector:

表示构造向量r_k的第n个元素的相位，ψ_n表示通过四舍五入得到的整数，

round(x)表示对x进行四舍五入。where g _n (r _k ) represents the deambiguation phase of the nth element of the construction vector r _k ,

Represents the phase of the nth element of the construction vector r _k , ψ _n represents the integer obtained by rounding,

round(x) means rounding x.

Further, based on the aforementioned settings and formulas, the defuzzification phase of the construction vector r _k can be expressed as:

ε表示有色高斯噪声向量，ε＝[ε₂-ε₁,ε₃-ε₂,...,ε_N-ε_N-1]^T。Among them, g(r _k ) represents the deblurring phase of the construction vector r _k , g ₁ (r _k ), g ₂ (r _k ),…,g _(N-1) (r _k ) represent the deblurring phase vector g( r _k ), the N-1 elements of

ε represents a colored Gaussian noise vector, ε=[ε ₂ -ε ₁ ,ε ₃ -ε ₂ ,...,ε _N -ε _N-1 ] ^T .

According to the above expression of the deambiguation phase of the construction vector, it is necessary to estimate the direction of arrival of the signal under the colored Gaussian noise. For this reason, in an embodiment of the present invention, according to the deambiguity phase, a weighted least square method can be used to calculate and obtain a closed-form expression of the direction of arrival of the signal.

Specifically, taking the kth signal as an example, the closed-form expression of the direction of arrival of the signal is obtained according to the deambiguity phase, including the following steps:

Construct the objective function:

Minimize the objective function to obtain a closed-form expression for the signal direction of arrival;

where Q represents the covariance matrix of the colored Gaussian noise vector ε,

能够得到πsinθ_k的最大似然估计，获取信号波达方向闭式表达式。Objective function constructed by minimizing

The maximum likelihood estimation of πsinθ _k can be obtained, and the closed expression of signal direction of arrival can be obtained.

In an embodiment of the present invention, by minimizing the objective function, a closed-form expression of the signal direction of arrival can be obtained, and the closed-form expression of the signal direction of arrival can specifically be expressed as:

Step S6, calculate and update the signal direction of arrival according to the closed expression of the signal direction of arrival.

In an embodiment of the present invention, the signal direction of arrival can be calculated according to the obtained closed-form expression of the signal direction of arrival.

Specifically, taking the kth signal as an example, based on the above-mentioned closed expression of signal direction of arrival, the following formula is used to calculate the signal direction of arrival:

Step S7, judging whether the signal direction of arrival has converged, or whether the number of cycles reaches a preset threshold, if yes, use the currently calculated signal direction of arrival as the final estimated value, if not, return to step S2.

In an embodiment of the present invention, the convergence condition and the preset threshold value of the number of cycles may be specifically set according to the estimation accuracy of the direction of arrival of the signal that is actually required.

The beneficial effect of the method for estimating the direction of arrival of the off-grid signal applicable to the non-uniform linear array provided by an embodiment of the present invention will be described below with reference to specific examples.

Example 1

Referring to Figure 2, in Example 1, 10 antennas are used to form a uniform linear array, the position of the array element is in units of half the wavelength of the signal source, the incident angle of the signal is 30.5°, and the number of snapshots is 1.

In this example 1, the off-grid sparse Bayesian algorithm of the prior art, the algorithm based on phase deviation search, the fractional Fourier sparse interpolation algorithm and the off-grid signal DOA estimation method provided by an embodiment of the present invention are respectively used to carry out Estimation of the signal direction of arrival, and using the mean square error of the estimated value of the direction of arrival of the signal as a measure, the mean square error of the estimated value of the direction of arrival obtained by different algorithms as shown in Figure 3 varies with the signal-to-noise ratio Schematic diagram of the curve, where the simulation results are the statistical results of 200 Monte Carlo experiments.

It can be seen that the mean square error corresponding to the off-grid sparse Bayesian algorithm cannot reach the Cramereau bound. When the signal-to-noise ratio is greater than 5dB, the mean square error corresponding to the algorithm based on phase deviation search can reach the Cramerot bound. When the signal-to-noise ratio is greater than 10dB, the mean square error corresponding to the fractional Fourier coefficient interpolation algorithm and the method for estimating the direction of arrival of the off-grid signal provided by an embodiment of the present invention can both reach the Cramereau boundary, but the algorithm based on the phase deviation search And the fractional Fourier coefficient interpolation algorithm can only be applied to the uniform linear array.

Example 2

Referring to Figure 4, in Example 2, 10 antennas are used to form a non-uniform linear array, the position of the array element is in units of half the wavelength of the signal source, the incident angle of the signal is 30.5°, and the number of snapshots is 1.

In this example 2, the off-grid sparse Bayesian algorithm of the prior art and the off-grid signal direction of arrival estimation method provided by an embodiment of the present invention are respectively used to estimate the signal direction of arrival, and the estimated value of the signal direction of arrival is used The mean square error of the mean square error is used as a measure standard, and the schematic diagram of the change curve of the mean square error of the DOA estimation value obtained by different algorithms as shown in Figure 5 with the signal-to-noise ratio is obtained, where the simulation results are the results of 200 Monte Carlo experiments statistical results.

It can be seen that the mean square error corresponding to the off-grid sparse Bayesian algorithm cannot reach the Cramereau bound. The error can reach the Cramereau bound.

Example 3

Referring to Figure 4, in Example 3, 10 antennas are also used to form a non-uniform linear array. The position of the array element is in units of half the wavelength of the signal source. There are 3 signals. ° and 60.5°, the number of snapshots is 1.

In this example 3, the off-grid sparse Bayesian algorithm of the prior art and the off-grid signal direction of arrival estimation method provided by an embodiment of the present invention are respectively used to estimate the signal direction of arrival, and the estimated value of the signal direction of arrival is used The mean square error of the mean square error is used as a measure standard, and the schematic diagram of the change curve of the mean square error of the direction of arrival estimation value obtained by different algorithms as shown in Figure 6 with the signal-to-noise ratio is obtained, where the simulation results are the results of 200 Monte Carlo experiments statistical results.

It can be seen that in the case of the three signals, the mean square error corresponding to the off-grid sparse Bayesian algorithm cannot reach the Cramereau boundary. The mean square errors of the three signals corresponding to the direction estimation method can reach the Cramereau bound.

Further, the computational complexity of the off-grid sparse Bayesian algorithm, the algorithm based on phase deviation search, the fractional Fourier sparse interpolation algorithm and the off-grid signal direction of arrival estimation method provided by an embodiment of the present invention After analysis, the computational complexity shown in the following table can be obtained;

algorithm Computational complexity Off-grid Sparse Bayesian Algorithm O(HNM²) Algorithm Based on Phase Deviation Search O(Nlog₂N+N+DKN) Fractional Fourier Coefficient Interpolation Algorithm O(Nlog_zN) Method for estimating the direction of arrival of off-grid signals according to the present invention O(K²H+N)

In the above table, H represents the number of cycles of the algorithm, N represents the number of antennas in the array, D represents the number of search grids, M represents the number of grid points, and K represents the number of signals.

It can be seen that the calculation complexity of the off-grid signal DOA estimation method provided by an embodiment of the present invention is less than that of the off-grid sparse Bayesian algorithm and the algorithm based on phase deviation search, although the calculation of the fractional Fourier coefficient interpolation algorithm is complicated The accuracy is lower than the method for estimating the direction of arrival of off-grid signals provided by an embodiment of the present invention, but it can only be applied to the case of uniform linear arrays.

It can be seen that the off-grid signal direction of arrival estimation method provided by an embodiment of the present invention can be applied to both uniform and non-uniform linear arrays, and can realize high-precision estimation of signal direction of arrival, and the computational complexity is relatively low. Low.

It should be noted that in this article, relative terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply these No such actual relationship or order exists between entities or operations. Moreover, the terms "comprising", "comprising" or any other variation thereof are intended to encompass a non-exclusive inclusion such that a process, method, article or device that includes a list of elements includes not only those elements, but also includes not explicitly listed or other elements inherent to such a process, method, article or apparatus. In addition, the terms "front", "rear", "left", "right", "upper", and "lower" herein refer to the placement states shown in the drawings.

Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: it can still be Modifications are made to the technical solutions described in the foregoing embodiments, or equivalent replacements are made to some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims (9)

1. A method for estimating the direction of arrival of off-grid signals suitable for non-uniform linear arrays, characterized in that it comprises: Step S1, according to the single snapshot array metadata, use the beamforming method to obtain the initial estimated value of the direction of arrival of the signal on the candidate grid; Step S2, calculating the complex-valued signal according to the direction of arrival of the signal; Step S3, according to the direction of arrival of the signal and the complex-valued signal, separate the corresponding signal data from the received data of the array; Step S4, multiplying the conjugate of the previous element and the next element in the signal data to obtain a construction vector, and extracting the phase of each element of the construction vector; Step S5, performing defuzzification on the phase of the extracted construction vector, and obtaining a closed-form expression of signal direction of arrival according to the defuzzification phase; Step S6, calculating and updating the signal direction of arrival according to the closed expression of the signal direction of arrival; Step S7, judging whether the signal direction of arrival has converged, or whether the number of cycles reaches a preset threshold, if yes, use the currently calculated signal direction of arrival as the final estimated value, if not, return to step S2. 2. The method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays according to claim 1, wherein it is set that: N antennas form a non-uniform array, and the array element position c _is the unit array element spacing Integer multiples of , n=1,2,...,N, the unit element spacing d is set to half of the wavelength, K far-field narrowband signals are incident on the array, and the direction of arrival is θ=[θ ₁ ,... ,θ _K ], θ ₁ ,...,θ _K represent the directions of arrival of the first to Kth signals respectively, and the received data y of the array is expressed as y=As+m, s=[s ₁ ,.. .,s _K ] ^T represents the vector of K complex-valued deterministic signals, m represents N×1-dimensional Gaussian white noise, A represents the array manifold matrix, A=[a(θ ₁ ),a(θ ₂ ),. ..,a(θ _K )], a(θ _k ) represents the array steering vector,

Consider angle search area

M uniformly sampled grid points in , get

The grid interval is

M represents the number of grids; Set at the same time: the direction of arrival of the incident signal does not fall on the pre-divided discrete grid points; Use beamforming to obtain an initial estimate of the direction of arrival of a signal on a candidate grid, including: Define a normalized M×N-dimensional beamforming matrix F, the (p,q)th element of matrix F is

According to the defined matrix F, the spatial spectrum x=|Fy| after beamforming is obtained; According to the spatial spectrum x, the positions corresponding to the K peaks are obtained

Based on the positions corresponding to the K peaks, using the formula

Determine the initial estimate of the corresponding signal direction of arrival

3. The off-grid signal direction of arrival estimation method suitable for non-uniform linear arrays according to claim 1 or 2, wherein calculating the complex-valued signal according to the signal direction of arrival includes: Construct the problem of estimating complex-valued signals according to the direction of arrival of the signal; The least square method is used to solve the problem of estimating complex-valued signals, and the closed-form solution of complex-valued signals is obtained. 4. The method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays according to claim 3, wherein the problem of estimating complex-valued signals is:

The closed-form solution of the complex-valued signal is: s=(A ^H A) ^-1 A ^H y Among them, s represents a complex-valued signal, and A ^H represents the conjugate transpose matrix of A. 5. The method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays according to any one of claims 2-4, wherein the corresponding signal data is separated from the received data of the array by using the following formula ;

Among them, y _k represents the k-th signal data, s _i represents the i-th complex-valued signal, and a(θ _i ) represents the array steering vector corresponding to the i-th signal. 6. The off-grid signal DOA estimation method applicable to non-uniform linear arrays according to claim 5, wherein the following formula is used to determine the elements of the signal data;

Among them, y _{k, n} represents the nth element of the kth signal, |s _k | represents the amplitude of the kth signal, φ _k represents the phase of the kth signal, ε _n represents the zero mean corresponding to the nth element Gaussian white noise; Define the construction vector corresponding to the kth signal as r _k ∈ C ^N-1 , C represents a complex number set, and each element of the construction vector r _k is determined by the following formula;

Among them, r _k,n represents the nth element of the construction vector r _k ,

Indicates the conjugate of y _k,n . 7. The method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays according to claim 6, wherein the following formula is used to defuzzify the phase of the extracted construction vector;

where g _n (r _k ) represents the deambiguation phase of the nth element of the construction vector r _k ,

Represents the phase of the nth element of the construction vector r _k , ψ _n represents the integer obtained by rounding,

round(x) means rounding x; The deblurring phase of the construction vector is expressed as:

Among them, g(r _k ) represents the deblurring phase of the construction vector r _k , g ₁ (r _k ), g ₂ (r _k ),…,g _(N-1) (r _k ) represent the deblurring phase vector g( r _k ), the N-1 elements of

ε represents a colored Gaussian noise vector, ε=[ε ₂ -ε ₁ ,ε ₃ -ε ₂ ,...,ε _N -ε _N-1 ] ^T . 8. The off-grid signal DOA estimation method suitable for non-uniform linear arrays according to claim 7, wherein the closed-form expression of the signal DOA obtained according to the deambiguation phase includes: Construct the objective function:

Minimize the objective function to obtain a closed-form expression for the signal direction of arrival; where Q represents the covariance matrix of the colored Gaussian noise vector ε. 9. The method for estimating the direction of arrival of off-grid signals applicable to non-uniform linear arrays according to claim 8, wherein the direction of arrival of signals is calculated using the following formula;

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