Array super-resolution direction finding method based on the inner product of scanning direction vector and characteristic vector

Abstract

The present invention relates to an array super-resolution direction finding method based on the inner product of a scanning direction vector and a characteristic vector. After obtaining the characteristic vectors of a signal subspace and a noise subspace from an array covariance matrix, firstly, the inner product modulus of the scanning direction vector and all the characteristic vectors is calculated; secondly, a sub-objective function is constructed based on the inner product modulus of the scanning direction vector and the characteristic vector of the signal subspace; thirdly, a sub-objective function is constructed based on the inner product modulus of the scanning direction vector and the characteristic vector of the noise subspace; fourthly, a new super-resolution direction finding objective function is constructed by combining the above two sub-objective functions; fifthly, a spatial spectrum is formed, from which candidate angles of the incident direction of the signal are estimated; sixthly, the wrongly estimated candidate angles are eliminated to obtain the estimated value of the incident direction angle of the signal. The present invention comprehensively utilizes the orthogonality of the signal subspace and the noise subspace, and the equivalence of the signal direction vector and the signal subspace, to construct a new super-resolution direction finding objective function, and can obtain an ultra-high direction finding resolution.

Description

Technical Field

[0001] The present invention relates to the field of direction finding in array signal processing, and in particular, can obtain ultra-high signal direction finding resolution by comprehensively utilizing the equivalence relationship between a signal direction vector and a signal subspace, the orthogonal relationship between a signal subspace and a noise subspace, and reasonably designing a new objective function. Background Art

[0002] Direction finding is one of the main research directions of array signal processing. Early radars performed direction finding of aerial targets by mechanically rotating antennas, and then phased array radars performed direction finding of aerial targets by changing the phase of phase shifters, with low direction finding resolution. In 1959, Capon proposed an adaptive beamformer by minimizing the average power of the array output under the condition that the gain in the main lobe direction was constant, which significantly improved the array direction finding resolution and entered the era of high-resolution direction finding. In 1986, Schmidt proposed the multiple signal classification method (MUSIC) by using the orthogonal relationship between the signal subspace and the noise subspace. In 1986, Roy et al. proposed the rotation invariant signal parameter estimation method (ESPRIT) by using the rotation invariant characteristics of the array geometry, entering the era of super-resolution direction finding. Since then, the technical progress has been small. The existing direction finding methods do not make full use of the orthogonal relationship between the signal subspace and the noise subspace, the equivalent relationship between the signal direction vector and the signal subspace, the direction finding objective function is relatively simple, the success rate of estimating adjacent signals is low, and there are certain limitations on improving the direction finding resolution.

[0003] In view of the above analysis, it is necessary to study new methods with ultra-high direction finding resolution to further improve the direction finding success rate and resolution based on super-resolution direction finding. Summary of the invention

[0004] The technology of the present invention solves the problem: fully utilizing the orthogonal relationship between the signal subspace and the noise subspace, and the equivalent relationship between the signal direction vector and the signal subspace, overcoming the shortcoming that the direction finding objective function of the prior art is relatively simple, and providing an array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector. Based on the equivalent relationship between the signal direction vector and the signal subspace, and the orthogonal relationship between the signal subspace and the noise subspace, by calculating the inner product modulus value of the scanning direction vector and all the characteristic vectors of the signal subspace, a weighted-product and other operations are used to construct the first sub-objective function; by calculating the inner product modulus value of the scanning direction vector and all the characteristic vectors of the noise subspace, a weighted-product-inverse operation is used to construct the second sub-objective function; and then a product operation is used to construct the final super-resolution direction finding objective function; a spatial spectrum is formed by angle scanning, from which the candidate angles of the signal incident direction are estimated; the candidate angles of the signal incident direction that are wrongly estimated are eliminated, and the angle estimation value of the signal incident direction is obtained. The method of the present invention further improves the signal direction finding success rate and resolution based on super-resolution direction finding, and is particularly advantageous under conditions of low signal-to-noise ratio, small snapshots, multiple signals and adjacent signals.

[0005] The objective of the present invention is achieved through the following technical solutions:

[0006] The present invention provides an array super-resolution direction finding method based on the inner product of a scanning direction vector and a characteristic vector, comprising the following steps:

[0007] Step 1, based on the orthogonality of the signal subspace and the noise subspace of the array covariance matrix, and the equivalence of the signal direction vector and the signal subspace, the inner product of the scanning direction vector and all the eigenvectors of the signal subspace is calculated and the modulus value is taken, and the inner product of the scanning direction vector and all the eigenvectors of the noise subspace is calculated and the modulus value is taken;

[0008] Step 2: Based on the property that the signal direction vector belongs to the signal subspace and other scanning direction vectors do not completely belong to or do not belong to the signal subspace, a sub-objective function is constructed by multiplication operation based on the inner product modulus value of the scanning direction vector and the signal subspace feature vector; when the scanning direction vector is equal to the signal direction vector, the sub-objective function obtains a sharp maximum value;

[0009] Step 3, using the property that the signal direction vector does not belong to the noise subspace and other scanning direction vectors belong to or do not completely belong to the noise subspace, the sub-objective function is constructed by the inner product modulus of the scanning direction vector and the noise subspace feature vector, using the product and reciprocal operation; when the scanning direction vector is equal to the signal direction vector, the objective function obtains a sharp maximum value;

[0010] Step 4: For the sub-objective function constructed based on the inner product modulus value of the scanning direction vector and the signal subspace feature vector in step 2 and the sub-objective function constructed based on the inner product modulus value of the scanning direction vector and the noise subspace feature vector in step 3, a new super-resolution direction finding objective function is constructed by multiplication operation; when the scanning direction vector is equal to the signal direction vector, the objective function obtains a sharper maximum value, which is conducive to obtaining the signal incident direction angle;

[0011] Step 5: Within the angle range of the incident direction, calculate the value of the super-resolution direction-finding objective function constructed in step 4 by angle scanning to form a spatial spectrum; estimate the candidate angles of the signal incident direction by searching for the maximum value of the spatial spectrum.

[0012] Step 6: Using the orthogonality between the signal direction vector and the noise subspace, the candidate angles of the signal incident direction that are mis-estimated are eliminated to obtain the estimated value of the angle of the signal incident direction.

[0013] Furthermore, in the array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector, step 1 comprises the following steps:

[0014] Step 11: Decompose the M-element array covariance matrix R to obtain the signal subspace E s =[e1,e2,…,e L ] and the noise subspace E n =[e L+1 ,e L+2 ,…,e M ], where e i is the eigenvector corresponding to the i-th largest eigenvalue of R, i = 1, 2, ..., M, L is the number of far-field uncorrelated narrowband signals (hereinafter referred to as signals) received by the M-element array, and L is less than M;

[0015] Step 12: Generate a scanning direction vector a(θ) corresponding to the incident direction angle θ according to the M-element array structure, the scanning range of θ is -90°≤θ≤90°, and normalize it, that is, ||a(θ)||2=1, ||||2 is the vector l2 norm operator;

[0016] Step 13: Signal subspace E based on array covariance matrix s and the noise subspace E n Orthogonality, signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s , where θ1, θ2, …, θ L is the incident direction angle of L signals. When the scanning direction angle θ is respectively aligned with the signal incident direction angles θ1, θ2, ..., θ LWhen they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L );Calculate the scanning direction vector a(θ) and all the characteristic vectors e1, e2,…, e in the signal subspace L The inner product value of and modulus value:

[0017] v i (θ)=|a H (θ)e i |,i=1,2,…,L,-90°≤θ≤90°;

[0018] Calculate the scanning direction vector a(θ) and all characteristic vectors e in the noise subspace L+1 ,e L+2 ,…,e M The inner product value of and modulus value:

[0019] v i (θ)=|a H (θ)e i |,i=L+1,L+2,…,M,-90°≤θ≤90°;

[0020] || is the modulo operator.

[0021] The step 2 comprises:

[0022] Based on the signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s For the signal incident angles θ1, θ2, …, θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L ), when i=1,2,…,L and l=1,2,…,L, a(θ l ) is each e i The linear combination of i (θ l )=|a H (θ l ) i | Larger; based on signal subspace E s and the noise subspace E n Orthogonality, for other incident angles θ, the scanning direction vector a(θ) is not completely in the signal subspace Es Even in the noise subspace E n When i=1,2,…,L, a(θ) is not the value of each e i The linear combination of i (θ)=|a H (θ)e i |Small or even 0, to highlight v i (θ l )=|a H (θ l ) i |(i=1,2,…,L and l=1,2,…,L) and the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2,…, θ L When they are equal, a sharp peak is obtained, and the sub-objective function F is constructed by product operation. s (θ):

[0023]

[0024] Among them, v i (θ)=|a H (θ)e i |,i=1,2,…,L, r is the weight factor, which is a positive number greater than or equal to 1, and is generally 2. Reasonable adjustment of the weight factor r can further improve the resolution of array direction finding.

[0025] For the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L ), so the corresponding F s (θ1), F s (θ2),…,F s (θ L ) obtains a maximum value and has a sharp peak; for other incident direction angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n Medium, F s (θ) value becomes smaller or even 0. The sub-objective function constructed in this way ensures that the signal incident direction angles θ1, θ2, ..., θ L , F s (θ1), F s (θ2),…,Fs (θ L ) always has a maximum value and a sharp peak, so for -90°≤θ≤90°, F s The angles corresponding to the L maximum values ​​of (θ) are the angles of the signal incident direction θ1, θ2, ..., θ L .

[0026] The step 3 comprises:

[0027] Based on the signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s The equivalence of the signal subspace E s and the noise subspace E n Orthogonality, for the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When i=L+1,L+2,…,M and l=1,2,…,L, the inner product modulus value v i (θ l )=|a H (θ l ) i | is 0; for other incident angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n When i=L+1,L+2,…,M, the inner product modulus value v i (θ)=|a H (θ)e i |Not 0 or even larger, to highlight v i (θ l )=|a H (θ l ) i |(i=L+1,L+2,…,M and l=1,2,…,L) is 0, and the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2,…, θ L When they are equal, a sharp peak is obtained, and the sub-objective function F is constructed by product and reciprocal operation. n (θ):

[0028]

[0029] Among them, v i (θ)=|a H (θ)e i|,i=L+1,L+2,…,M, w is a weight factor, which can be a positive number greater than 0, and is generally 1; reasonable adjustment of the weight factor w can further improve the resolution of array direction finding.

[0030] For the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s In the noise subspace E n Orthogonal, F n (θ1), F n (θ2),…,F n (θ L ) obtains a maximum value and has a sharp peak; for other incident direction angles θ, the scanning direction vector a(θ) is in or not completely in the noise subspace E n Medium, F n (θ) value becomes smaller or even 0. This construction of the objective function ensures that for the signal incident direction angles θ1, θ2, ..., θ L , F n (θ1), F n (θ2),…,F n (θ L ) always has a maximum value and a sharp peak, so for -90°≤θ≤90°, F n The angles corresponding to the L maximum values ​​of (θ) are the angles of the signal incident direction θ1, θ2, ..., θ L .

[0031] The step 4 comprises:

[0032] The sub-objective function F is constructed based on the inner product modulus of the scanning direction vector in step 2 and the signal subspace feature vector. s (θ), the sub-objective function F constructed by the inner product modulus of the scanning direction vector in step 3 and the characteristic vector of the noise subspace n (θ), for the signal incident direction angles θ1, θ2, ..., θ L , F s (θ1), F s (θ2),…,F s (θ L ) and F n (θ1), F n (θ2),…,F n (θ L ) always has a maximum value, which is when the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2, ..., θ L When they are equal, there is a sharper peak. The product operation is used to construct the following super-resolution direction finding objective function:

[0033] F(θ)=F s (θ)F n (θ), -90°≤θ≤90°;

[0034] The result of the operation is:

[0035]

[0036] The super-resolution direction finding objective function constructed in this way ensures that the incident angles θ1, θ2, ..., θ L , F(θ1), F(θ2),…, F(θ L ) is more prominent, which is beneficial to further improve the resolution of array direction finding.

[0037] Reasonable adjustment of the weight factors r and w can further improve the resolution of array direction finding.

[0038] The step 5 comprises:

[0039] For each angle value of θ scanned in the range of (-90°, 90°), the super-resolution direction finding objective function F(θ) is substituted and calculated to form a spatial spectrum F(θ). The angles corresponding to the L maximum values ​​of the spatial spectrum F(θ) are the angles of the signal incident direction θ1, θ2, ..., θ L By searching for the maximum value of the spatial spectrum, we can get the signal incident direction angles θ1, θ2, …, θ L .

[0040] In engineering applications, the array covariance matrix R is estimated from sample data, which contains certain random errors, resulting in the signal subspace E s and the noise subspace E n There are also certain random errors, so the L maximum values ​​of the above spatial spectrum F(θ) will not only have errors, but also redundant L f By searching the maximum value of the spatial spectrum, we can estimate L+L f Candidate angles of the signal incident direction

[0041] The step 6 comprises:

[0042] Step 6 estimates the signal incident direction L+L f Candidate angles There are signals incident at L angles, L f is the number of pseudo peaks, L f There is no signal incident at an angle; if is an estimate of the angle of the incident signal direction, and the noise subspace E nAll eigenvectors in are orthogonal if It is not an estimate of the angle of the incident direction of the signal. and the noise subspace E n All eigenvectors in are not orthogonal, calculate the discriminant function value:

[0043]

[0044] For the above L+L f The discriminant function values ​​are arranged in ascending order, with the largest L f The scanning direction vector of the candidate angle corresponding to the value and the noise subspace E n Non-orthogonal means that the candidate angles of the signal incident direction are estimated incorrectly, so they are eliminated; the remaining L candidate angles are the estimated values ​​of the signal incident direction angles.

[0045] In order to reduce the number of pseudo peaks L f , modify the above super-resolution direction finding objective function to:

[0046]

[0047] Among them, max{} is the element maximum operator, and ε is a small positive number, for example, it can be taken as 10 -100 .

[0048] The advantages of the present invention over the prior art are as follows: as can be seen from the technical solution provided by the present invention, the array received data covariance matrix is ​​eigen-decomposed to obtain the eigenvectors of the signal subspace and the noise subspace; the equivalent relationship between the signal direction vector and the signal subspace, the orthogonal relationship between the signal direction vector and the noise subspace, and the inner product modulus value of the scanning direction vector and the eigenvector are utilized to construct a new super-resolution direction finding objective function, the candidate angles of the signal incident direction are estimated from the formed spatial spectrum, and the discriminant function is further used to eliminate the wrongly estimated candidate angles of the signal incident direction, which has an ultra-high direction finding resolution, especially in low signal-to-noise ratio, small snapshot, multiple signal sources and adjacent signal sources.

[0049] (1) A more complete super-resolution direction finding objective function is constructed by making full use of the orthogonal relationship between the signal subspace and the noise subspace, and the equivalent relationship between the signal direction vector and the signal subspace.

[0050] (2) In view of the relatively simple problem of the direction finding objective function in the prior art, the inner product modulus of the scanning direction vector and all the characteristic vectors in the signal subspace is calculated, and the first objective function is constructed by weighting-product and other operations. The second objective function is constructed by calculating the inner product modulus of the scanning direction vector and all the characteristic vectors in the noise subspace, and the weighting-product-inverse operation is used. The final super-resolution direction finding objective function is constructed by the product operation. On the basis of super-resolution direction finding, the success rate and resolution of signal direction finding are further improved.

[0051] (3) The secondary direction finding approach is adopted. The candidate angles of the signal incident direction are first estimated from the formed spatial spectrum. Then, the discriminant function is used to eliminate the wrong candidate angles of the signal incident direction, thereby reducing the direction finding error. This is especially advantageous in low signal-to-noise ratio, small snapshots, multiple signals, and adjacent signals. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other accompanying drawings can be obtained based on these accompanying drawings without paying creative work.

[0053] Figure 1 A flow chart of an array super-resolution direction finding method based on the inner product of a scanning direction vector and a feature vector provided by an embodiment of the present invention;

[0054] Figure 2 A schematic diagram of an array signal receiving model provided by an embodiment of the present invention;

[0055] Figure 3 The comparison results of the method (F(θ)) proposed in the present invention and the MUSIC method under 500 Monte Carlo tests provided in the embodiment of the present invention are as follows: Figure 3 A is the comparison result of the estimated success rate with the change of signal-to-noise ratio. Figure 3 B is the comparison result of the estimated root mean square error curve as the signal-to-noise ratio changes. DETAILED DESCRIPTION

[0056] The embodiment of the present invention performs eigendecomposition on the array received data covariance matrix to obtain a signal subspace and a noise subspace; utilizes the equivalence relationship between the signal direction vector and the signal subspace to construct an objective function of the inner product value of the scanning direction vector and the signal subspace feature vector; utilizes the orthogonal relationship between the signal direction vector and the noise subspace to construct an objective function of the inner product value of the scanning direction vector and the noise subspace feature vector; comprehensively utilizes the above two objective functions to construct a new super-resolution direction finding objective function; estimates the candidate angles of the signal incident direction from the formed spatial spectrum, and further utilizes the discriminant function to eliminate the candidate angles of the signal incident direction that are wrongly estimated, which has an ultra-high direction finding resolution, especially has advantages in low signal-to-noise ratio, small snapshots, multiple signal sources and adjacent signal sources.

[0057] like Figure 1 As shown, the present invention mainly comprises the following steps:

[0058] Step 1, based on the orthogonality of the signal subspace and the noise subspace of the array covariance matrix, and the equivalence of the signal direction vector and the signal subspace, the inner product of the scanning direction vector and all the eigenvectors of the signal subspace is calculated and the modulus value is taken, and the inner product of the scanning direction vector and all the eigenvectors of the noise subspace is calculated and the modulus value is taken;

[0059] Step 2: Based on the property that the signal direction vector belongs to the signal subspace and other scanning direction vectors do not completely belong to or do not belong to the signal subspace, a sub-objective function is constructed by multiplication operation based on the inner product modulus value of the scanning direction vector and the signal subspace feature vector; when the scanning direction vector is equal to the signal direction vector, the sub-objective function obtains a sharp maximum value;

[0060] Step 3, using the property that the signal direction vector does not belong to the noise subspace and other scanning direction vectors belong to or do not completely belong to the noise subspace, the sub-objective function is constructed by the inner product modulus of the scanning direction vector and the noise subspace feature vector, using the product and reciprocal operation; when the scanning direction vector is equal to the signal direction vector, the objective function obtains a sharp maximum value;

[0061] Step 4: For the sub-objective function constructed based on the inner product modulus value of the scanning direction vector and the signal subspace feature vector in step 2 and the sub-objective function constructed based on the inner product modulus value of the scanning direction vector and the noise subspace feature vector in step 3, a new super-resolution direction finding objective function is constructed by multiplication operation; when the scanning direction vector is equal to the signal direction vector, the objective function obtains a sharper maximum value, which is conducive to obtaining the signal incident direction angle;

[0062] Step 5: within the angle range of the incident direction, calculate the value of the super-resolution direction finding objective function constructed in step 4 by angle scanning to form a spatial spectrum; estimate the candidate angles of the signal incident direction by searching for the maximum value of the spatial spectrum;

[0063] Step 6: Using the orthogonality between the signal direction vector and the noise subspace, the candidate angles of the signal incident direction that are mis-estimated are eliminated to obtain the estimated value of the angle of the signal incident direction.

[0064] Compared with the existing super-resolution direction finding method, the above scheme of the present invention performs eigendecomposition on the array received data covariance matrix to obtain the signal subspace and the noise subspace; utilizes the equivalent relationship between the signal direction vector and the signal subspace to construct a sub-objective function based on the inner product value of the scanning direction vector and the characteristic vector of the signal subspace; utilizes the orthogonal relationship between the signal direction vector and the noise subspace to construct a sub-objective function based on the inner product value of the scanning direction vector and the characteristic vector of the noise subspace; comprehensively utilizes the above two objective functions to construct a new super-resolution direction finding objective function; estimates the candidate angles of the signal incident direction from the formed spatial spectrum, and further utilizes the discriminant function to eliminate the candidate angles of the signal incident direction that are estimated incorrectly, which has an ultra-high direction finding resolution, especially has advantages in low signal-to-noise ratio, small snapshots, multiple sources and adjacent sources.

[0065] For ease of understanding, we first introduce the multiple signal classification method (MUSIC), then introduce the preprocessing, and then give a detailed explanation of the above six steps.

[0066] The examples of the present invention are applicable to any type of array, including linear array, circular array, conformal array, etc., and the applicable wave arrival directions include one-dimensional azimuth angle, one-dimensional elevation angle, two-dimensional azimuth angle and elevation angle. Figure 2 The linear array is discussed, and the specific array signal model is as follows:

[0067] Consider the M-element linear array receiving space with different incident angles θ1, θ2, …, θ L The first array element on the right is set as the reference array element, and the spacings of the other array elements from right to left relative to the reference array element are d1, d2, ..., d M-1 Since the incident direction angles of each signal are different, the plane wavefronts of each signal are different, the time delays of reaching each array element relative to reaching the reference array element are also different, and the signal direction vectors formed are also different. Then the received data of the array at observation time k (called the kth snapshot data received by the array) is expressed as:

[0068] x(k)=x s (k)+x n (k);

[0069] Among them, x s (k) and x n (k) represent signal and noise respectively, s l (k) is the waveform of the lth signal, each signal sl (k) are all zero mean and uncorrelated, a l is the direction vector of the lth signal, x n (k) is additive independent and identically distributed zero-mean white noise, each signal s l (k) is uncorrelated with the noise of each array element.

[0070] The M×M dimensional covariance matrix R of the array received signal vector is:

[0071]

[0072] Based on the assumptions, R s and A are both of rank L, so AR s A H is a Hermitian positive semidefinite matrix of rank L, whose L nonzero positive eigenvalues ​​are arranged in order μ1≥μ2≥…≥μ L > 0. R is a Hermitian positive definite matrix, and its M non-zero positive eigenvalues ​​are arranged in order to satisfy:

[0073]

[0074] The corresponding eigenvectors are e1, e2, …, e L ,e L+1 ,…,e M ,but

[0075]

[0076] For all l'>L, the characteristic decomposition property yields:

[0077]

[0078] therefore,

[0079] AR s A H e l' =0,l'>L,

[0080] This means:

[0081] a H (θ l ) l' =0,l=1,2,…,L,l'=L+1,L+2,…,M.

[0082] The above formula shows that the eigenvector corresponding to the smallest eigenvalue is orthogonal to the signal direction vector. The eigenvectors e1, e2, …, e corresponding to the L largest eigenvalues ​​are L A subspace is formed by the eigenvectors e corresponding to the remaining small eigenvalues ​​with equal ML. L+1,e L+2 ,…,e M Span into another subspace. Since these two subspaces are orthogonal, we know that the former is related to the signal and is called the signal subspace, denoted by E s ; The latter is the complement of the signal subspace, called the noise subspace, denoted by E n .

[0083] Create the following function:

[0084]

[0085] When θ is scanned, its L peaks correspond to the angles of the signal incident direction.

[0086] In practical engineering, the ideal array covariance matrix R is difficult to obtain, and we can only use the array sample covariance matrix Instead of R, the M×M dimensional covariance matrix of the array samples for:

[0087]

[0088] Wherein, K is the number of sample data snapshots received by the array.

[0089] The purpose of the present invention is to use the equivalent relationship between the signal direction vector and the signal subspace to construct a sub-objective function based on the inner product value of the scanning direction vector and the signal subspace feature vector; use the orthogonal relationship between the signal direction vector and the noise subspace to construct a sub-objective function based on the inner product value of the scanning direction vector and the noise subspace feature vector; comprehensively use the above two sub-objective functions to construct a new super-resolution direction finding objective function, so that the super-resolution direction finding objective function obtains a sharper maximum value at the angle of the signal incident direction; estimate the candidate angles of the signal incident direction from the formed spatial spectrum, and further use the discriminant function to eliminate the candidate angles of the signal incident direction that are wrongly estimated, so as to have an ultra-high direction finding resolution, especially in low signal-to-noise ratio, small snapshot, multiple sources and adjacent sources. It is implemented in the following five steps.

[0090] Step 1:

[0091] Step 11: Decompose the M-element array covariance matrix R to obtain the signal subspace E s =[e1,e2,…,e L ] and the noise subspace E n =[e L+1 ,e L+2 ,…,e M ], where e i is the eigenvector corresponding to the i-th largest eigenvalue of R, i = 1, 2, ..., M, L is the number of far-field uncorrelated narrowband signals (hereinafter referred to as signals) received by the M-element array, and L is less than M;

[0092] Step 12: Generate a scanning direction vector a(θ) corresponding to the incident direction angle θ according to the M-element array structure, the scanning range of θ is -90°≤θ≤90°, and normalize it, that is, ||a(θ)||2=1, || ||2 is the vector l2 norm operator;

[0093] Step 13: Signal subspace E based on array covariance matrix s and the noise subspace E n Orthogonality, signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s , where θ1, θ2, …, θ L is the incident direction angle of L signals. When the scanning direction angle θ is respectively aligned with the signal incident direction angles θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L );Calculate the scanning direction vector a(θ) and all the characteristic vectors e1, e2,…, e in the signal subspace L The inner product value of and modulus value:

[0094] v i (θ)=|a H (θ)e i |,i=1,2,…,L,-90°≤θ≤90°;

[0095] Calculate the scanning direction vector a(θ) and all characteristic vectors e in the noise subspace L+1 ,e L+2 ,…,e M The inner product value of and modulus value:

[0096] v i (θ)=|a H (θ)e i |,i=L+1,L+2,…,M,-90°≤θ≤90°;

[0097] | | is the modulo operator.

[0098] Step 2:

[0099] Based on the signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s For the signal incident angles θ1, θ2, …, θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E sWhen the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L ), when i=1,2,…,L and l=1,2,…,L, a(θ l ) is each e i The linear combination of i (θ l )=|a H (θ l ) i | Larger; based on signal subspace E s and the noise subspace E n Orthogonality, for other incident angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n When i=1,2,…,L, a(θ) is not the value of each e i The linear combination of i (θ)=|a H (θ)e i |Small or even 0, to highlight v i (θ l )=|a H (θ l ) i | has a maximum value effect, and the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2, ..., θ L When they are equal, a sharp peak is obtained, and the sub-objective function F is constructed by product operation. s (θ):

[0100]

[0101] Among them, v i (θ)=|a H (θ)e i |,i=1,2,…,L, r is the weight factor, which is a positive number greater than or equal to 1, and is generally 2. Reasonable adjustment of the weight factor r can further improve the resolution of array direction finding.

[0102] For the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2, ..., θ LWhen they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L ), so the corresponding F s (θ1), F s (θ2),…,F s (θ L ) obtains a maximum value and has a sharp peak; for other incident direction angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n Medium, F s (θ) value becomes smaller or even 0. The sub-objective function constructed in this way ensures that the signal incident direction angles θ1, θ2, ..., θ L , F s (θ1), F s (θ2),…,F s (θ L ) always has a maximum value and a sharp peak, so for -90°≤θ≤90°, F s The angles corresponding to the L maximum values ​​of (θ) are the angles of the signal incident direction θ1, θ2, ..., θ L .

[0103] Step 3:

[0104] Based on the signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s The equivalence of the signal subspace E s and the noise subspace E n Orthogonality, for the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When i=L+1,L+2,…,M and l=1,2,…,L, the inner product modulus value v i (θ l )=|a H (θ l ) i | is 0; for other incident angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n When i=L+1,L+2,…,M, the inner product modulus value v i (θ)=|a H (θ)e i |Not 0 or even larger, to highlight v i (θ l )=|aH (θ l ) i |(i=L+1,L+2,…,M and l=1,2,…,L) is 0, and the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2,…, θ L When they are equal, a sharp peak is obtained, and the sub-objective function F is constructed by product and reciprocal operation. n (θ):

[0105]

[0106] Among them, v i (θ)=|a H (θ)e i |,i=L+1,L+2,…,M, w is a weight factor, which can be a positive number greater than 0, and is generally 1; reasonable adjustment of the weight factor w can further improve the resolution of array direction finding.

[0107] For the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s In the noise subspace E n Orthogonal, F n (θ1), F n (θ2),…,F n (θ L ) obtains a maximum value and has a sharp peak; for other incident direction angles θ, the scanning direction vector a(θ) is in or not completely in the noise subspace E n Medium, F n (θ) value becomes smaller or even 0. This construction of the objective function ensures that for the signal incident direction angles θ1, θ2, ..., θ L , F n (θ1), F n (θ2),…,F n (θ L ) always has a maximum value and a sharp peak, so for -90°≤θ≤90°, F n The angles corresponding to the L maximum values ​​of (θ) are the angles of the signal incident direction θ1, θ2, ..., θ L .

[0108] Step 4:

[0109] The sub-objective function F is constructed based on the inner product modulus of the scanning direction vector in step 2 and the signal subspace feature vector. s (θ), the sub-objective function F constructed by the inner product modulus of the scanning direction vector in step 3 and the characteristic vector of the noise subspace n(θ), for the signal incident direction angles θ1, θ2, ..., θ L , F s (θ1), F s (θ2),…,F s (θ L ) and F n (θ1), F n (θ2),…,F n (θ L ) always has a maximum value, which is when the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2, ..., θ L When they are equal, there is a sharper peak. The product operation is used to construct the following super-resolution direction finding objective function:

[0110] F(θ)=F s (θ)F n (θ), -90°≤θ≤90°,

[0111] The result of the operation is:

[0112]

[0113] The super-resolution direction finding objective function constructed in this way ensures that the incident angles θ1, θ2, ..., θ L , F(θ1), F(θ2),…, F(θ L ) is more prominent, which is beneficial to further improve the resolution of array direction finding.

[0114] Reasonable adjustment of the weight factors r and w can further improve the resolution of array direction finding.

[0115] Step 5:

[0116] For each angle value of θ scanned in the range of (-90°, 90°), the super-resolution direction finding objective function F(θ) is substituted and calculated to form a spatial spectrum F(θ). The angles corresponding to the L maximum values ​​of the spatial spectrum F(θ) are the angles of the signal incident direction θ1, θ2, ..., θ L By searching for the maximum value of the spatial spectrum, we can get the signal incident direction angles θ1, θ2, …, θ L .

[0117] In engineering applications, the array covariance matrix R is estimated from sample data, which contains certain random errors, resulting in the signal subspace E s and the noise subspace E n There are also certain random errors, so the L maximum values ​​of the above spatial spectrum F(θ) will not only have errors, but also redundant L f By searching for the maximum value of the spatial spectrum, we estimate L+Lf Candidate angles of the signal incident direction

[0118] Step 6:

[0119] Step 6 estimates the signal incident direction L+L f Candidate angles There are signals incident at L angles, L f There is no signal incident at an angle; if is an estimate of the angle of the incident direction of the signal, and the noise subspace E n All eigenvectors in are orthogonal if It is not an estimate of the angle of the incident direction of the signal. and the noise subspace E n All eigenvectors in are not orthogonal, calculate the discriminant function value:

[0120]

[0121] For the above L+L f The discriminant function values ​​are arranged in ascending order, with the largest L f The scanning direction vector of the candidate angle corresponding to the value and the noise subspace E n Non-orthogonal means that the candidate angles of the signal incident direction are estimated incorrectly, so they are eliminated; the remaining L candidate angles are the estimated values ​​of the signal incident direction angles.

[0122] In order to reduce the number of pseudo peaks L f , modify the above super-resolution direction finding objective function to:

[0123]

[0124] Among them, max{} is the element maximum operator, and ε is a small positive number, for example, it can be taken as 10 -100 .

[0125] like Figure 2 For simplicity, Figure 2 Only a schematic diagram of a narrowband far-field signal in the receiving space of an M-element linear array is given. The angle between the incident direction of the signal and the array normal is θ, and the signal is considered to be incident on each array element in the form of a plane wave. The first array element on the right is set as the reference array element, d1, d2, …, d M-1 is the distance between other array elements and the reference array element.

[0126] Figure 3 The comparison results of the method (F(θ)) proposed in the present invention and the MUSIC method under 500 Monte Carlo tests provided in the embodiment of the present invention are as follows: Figure 3A is the comparison result of the estimated success rate with the change of signal-to-noise ratio. Figure 3 B is the comparison result of the estimated root mean square error curve with the change of signal-to-noise ratio. The experimental conditions are: 10-element uniform linear array, element spacing is half wavelength, number of snapshots is 30, incident angles of 2 unrelated sources are 10° and 13°, the absolute error of the first estimation at each angle is considered successful within 1.5°, the signal-to-noise ratio ranges from -15dB to 10dB, and the weight factors are r=2 and w=1. Figure 3 It shows that under the conditions of low signal-to-noise ratio, small snapshots, multiple signals and adjacent signals, the estimation success rate of the method proposed in the present invention is significantly higher than that of the MUSIC method, and the estimated root mean square error of the method proposed in the present invention is significantly lower than that of the MUSIC method, which is more advantageous.

[0127] Through the description of the above implementation methods, those skilled in the art can clearly understand that the above embodiments can be implemented by software, or by means of software plus necessary general hardware platforms. Based on such understanding, the technical solutions of the above embodiments can be embodied in the form of software products, which can be stored in a non-volatile storage medium (which can be a CD-ROM, a USB flash drive, a mobile hard disk, etc.), including several instructions for enabling a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in the various embodiments of the present invention.

[0128] The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. An array super-resolution direction finding method based on the inner product of a scanning direction vector and a characteristic vector, characterized in that: The steps include: Step 1, based on the orthogonality of the signal subspace and the noise subspace of the array covariance matrix, and the equivalence of the signal direction vector and the signal subspace, the inner product of the scanning direction vector and all the eigenvectors of the signal subspace is calculated and the modulus value is taken, and the inner product of the scanning direction vector and all the eigenvectors of the noise subspace is calculated and the modulus value is taken; Step 2: Based on the property that the signal direction vector belongs to the signal subspace and other scanning direction vectors do not completely belong to or do not belong to the signal subspace, a sub-objective function is constructed by multiplication operation based on the inner product modulus value of the scanning direction vector and the signal subspace feature vector; when the scanning direction vector is equal to the signal direction vector, the sub-objective function obtains a sharp maximum value; Step 3, using the property that the signal direction vector does not belong to the noise subspace and other scanning direction vectors belong to or do not completely belong to the noise subspace, the sub-objective function is constructed by the inner product modulus of the scanning direction vector and the noise subspace feature vector, using the product and reciprocal operation; when the scanning direction vector is equal to the signal direction vector, the objective function obtains a sharp maximum value; Step 4: For the sub-objective function constructed based on the inner product modulus value of the scanning direction vector and the signal subspace feature vector in step 2 and the sub-objective function constructed based on the inner product modulus value of the scanning direction vector and the noise subspace feature vector in step 3, a new super-resolution direction finding objective function is constructed by multiplication operation; when the scanning direction vector is equal to the signal direction vector, the objective function obtains a sharper maximum value, which is conducive to obtaining the signal incident direction angle; Step 5: within the angle range of the incident direction, calculate the value of the super-resolution direction finding objective function constructed in step 4 by angle scanning to form a spatial spectrum; estimate the candidate angles of the signal incident direction by searching for the maximum value of the spatial spectrum; Step 6: Using the orthogonality between the signal direction vector and the noise subspace, the candidate angles of the signal incident direction that are mis-estimated are eliminated to obtain the estimated value of the angle of the signal incident direction.

2. The array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector according to claim 1 is characterized in that: The step 1 comprises the following steps: Step 11: Decompose the M-element array covariance matrix R to obtain the signal subspace E s =[e1,e2,…,e L ] and the noise subspace E n =[e L+1 ,e L+2 ,…,e M ], where e i is the eigenvector corresponding to the i-th largest eigenvalue of R, i = 1, 2, ..., M, L is the number of far-field uncorrelated narrowband signals received by the M-element array, L is less than M; Step 12: Generate a scanning direction vector a(θ) corresponding to the incident direction angle θ according to the M-element array structure, the scanning range of θ is -90°≤θ≤90°, and normalize it, that is, ||a(θ)||2=1, || ||2 is the vector l2 norm operator; Step 13: Signal subspace E based on array covariance matrix s and the noise subspace E n Orthogonality, signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s , where θ1, θ2, ..., θ L is the incident direction angle of L signals. When the scanning direction angle θ is respectively aligned with the signal incident direction angles θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L );Calculate the scanning direction vector a(θ) and all the characteristic vectors e1, e2,…, e in the signal subspace L The inner product value of and modulus value: v i (θ)=|a H (i)e i |,i=1,2,…,L,-90°≤θ≤90°; Calculate the scanning direction vector a(θ) and all characteristic vectors e in the noise subspace L+1 ,e L+2 ,…,e M The inner product value of and modulus value: v i (θ)=|a H (i)e i |,i=L+1,L+2,…,M,-90°≤θ≤90°; || is the modulo operator.

3. The array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector according to claim 1 is characterized in that: In step 2: Based on the signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s For the signal incident angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L ), when i=1,2,…,L and l=1,2,…,L, a(θ l ) is each e i The linear combination of i (θ l )=|a H (θ l ) i | Larger; based on signal subspace E s and the noise subspace E n Orthogonality, for other incident angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n When i=1,2,…,L, a(θ) is not the value of each e i The linear combination of i (θ)=|a H (θ)e i |Small or even 0, to highlight v i (θ l )=|a H (θ l ) i | has a maximum value effect, and the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2, ..., θ L When they are equal, a sharp peak is obtained, and the sub-objective function F is constructed by product operation. s (θ): Among them, v i (θ)=|a H (θ)e i |,i=1,2,…,L, r is the weight factor, which is a positive number greater than or equal to 1; For the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When the scanning direction angle θ is respectively aligned with the signal incident direction angle θ1, θ2, ..., θ L When they are equal, the scanning direction vector a(θ) is equal to the signal direction vectors a(θ1), a(θ2), ..., a(θ L ), so the corresponding F s (θ1), F s (θ2),…,F s (θ L ) obtains a maximum value and has a sharp peak; for other incident direction angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n Medium, F s (θ) value becomes smaller or even 0. The sub-objective function constructed in this way ensures that for the signal incident direction angles θ1, θ2, ..., θ L , F s (θ1), F s (θ2),…,F s (θ L ) always has a maximum value and a sharp peak. For -90°≤θ≤90°, F s The angles corresponding to the L maximum values ​​of (θ) are the angles of the signal incident direction θ1, θ2, ..., θ L .

4. The array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector according to claim 3 is characterized in that: In step 3: Based on the signal direction vectors a(θ1), a(θ2), …, a(θ L ) and the signal subspace E s The equivalence of the signal subspace E s and the noise subspace E n Orthogonality, for the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s When i=L+1,L+2,…,M and l=1,2,…,L, the inner product modulus value v i (θ l )=|a H (θ l ) i | is 0; for other incident angles θ, the scanning direction vector a(θ) is not completely in the signal subspace E s Even in the noise subspace E n When i=L+1,L+2,…,M, the inner product modulus value v i (θ)=|a H (θ)e i |Not 0 or even larger, to highlight v i (θ l )=|a H (θ l ) i | is 0, and the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2, ..., θ L When they are equal, a sharp peak is obtained, and the sub-objective function F is constructed by product and reciprocal operation. n (θ): Among them, v i (θ)=|a H (θ)e i |,i=L+1,L+2,…,M, w is the weight factor, which is a positive number greater than 0; For the signal incident direction angles θ1, θ2, ..., θ L , signal direction vectors a(θ1), a(θ2), …, a(θ L ) in the signal subspace E s In the noise subspace E n Orthogonal, F n (θ1), F n (θ2),…,F n (θ L ) obtains a maximum value and has a sharp peak; for other incident direction angles θ, the scanning direction vector a(θ) is in or not completely in the noise subspace E n Medium, F n (θ) value becomes smaller or even 0, so the objective function is constructed to ensure that for the signal incident direction angles θ1, θ2, ..., θ L , F n (θ1), F n (θ2),…,F n (θ L ) always has a maximum value and a sharp peak, so for -90°≤θ≤90°, F n The angles corresponding to the L maximum values ​​of (θ) are the angles of the signal incident direction θ1, θ2, ..., θ L .

5. The array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector according to claim 1, characterized in that: In step 4, The sub-objective function F is constructed based on the inner product modulus of the scanning direction vector in step 2 and the signal subspace feature vector. s (θ), the sub-objective function F constructed by the inner product modulus of the scanning direction vector in step 3 and the characteristic vector of the noise subspace n (θ), for the signal incident direction angles θ1, θ2, ..., θ L , F s (θ1), F s (θ2),…,F s (θ L ) and F n (θ1), F n (θ2),…,F n (θ L ) always has a maximum value, which is when the scanning direction angle θ is respectively related to the signal incident direction angle θ1, θ2, ..., θ L When they are equal, there is a sharper peak. The product operation is used to construct the following super-resolution direction finding objective function: F(θ)=F s (i)F n (θ), -90°≤θ≤90°, The result of the operation is:

6. The array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector according to claim 1, characterized in that: In step 5, For each angle value of θ scanned in the range of (-90°, 90°), the super-resolution direction finding objective function F(θ) is substituted and calculated to form a spatial spectrum graph F(θ). The angles corresponding to the L maximum values ​​of the spatial spectrum graph F(θ) are the angles of the signal incident direction θ1, θ2, ..., θ L , by searching the maximum value of the spatial spectrum, the signal incident direction angles θ1, θ2, ..., θ L ; For the random error of the array covariance matrix, there will be an extra number of pseudo peaks L f Maximum value, by searching the maximum value of the spatial spectrum, estimate L+L f Candidate angles of the signal incident direction 7. The array super-resolution direction finding method based on the inner product of the scanning direction vector and the characteristic vector according to claim 6, characterized in that: In step 6, Step 6 estimates the signal incident direction L+L f Candidate angles There are signals incident at L angles, L f is the number of pseudo peaks, L f There is no signal incident at an angle; if is an estimate of the angle of the incident signal direction, and the noise subspace E n All eigenvectors in are orthogonal if It is not an estimate of the angle of the incident direction of the signal. and the noise subspace E n All eigenvectors in are not orthogonal, calculate the discriminant function value: For the above L+L f The discriminant function values ​​are arranged in ascending order, with the largest L f The scanning direction vector of the candidate angle corresponding to the value and the noise subspace E n Non-orthogonal means that the candidate angles of the signal incident direction are estimated incorrectly, so they are eliminated; the remaining L candidate angles are the estimated values ​​of the signal incident direction angles; In order to reduce the number of pseudo peaks L f , the super-resolution direction finding objective function is modified as follows: -90°≤θ≤90° Among them, max{} is the element-maximum operator, and ε is a small positive number.

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