CN109239648A - Spectrum correlation subspaces direction-finding method based on symmetrical cycle frequency - Google Patents
Spectrum correlation subspaces direction-finding method based on symmetrical cycle frequency Download PDFInfo
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- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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Abstract
The invention discloses the spectrum correlation subspaces direction-finding methods based on symmetrical cycle frequency, Cyclic Autocorrelation Function is established according to the cyclostationarity for receiving signal, utilize the weak dependence between the Cyclic Autocorrelation Function of symmetrical cycle frequency, construction circulation cross-correlation matrix, make the precision of direction finding better than existing spectrum correlation subspaces direction-finding method, and there is higher direction finding correct probability.
Description
Technical Field
The invention relates to the technical field of electronic information, in particular to a spectrum correlation subspace direction finding method based on symmetric cycle frequency.
Background
The array signal processing technology can be used for processing signals received by an array so as to measure the directions of arrival (direction finding for short) of a plurality of signals, and is widely applied to the fields of radars, sonars, communication systems, smart antennas and the like.
Among the existing various direction finding methods, the interferometer direction finding method has good direction finding performance for a single signal, but the method fails when multiple signals are in the same frequency, and is greatly limited in application. The traditional subspace type direction finding method has good direction finding performance on same-frequency multi-signal, but the direction finding freedom degree is limited by the number of antenna array elements, the more the number of signals is, the larger the number of antenna array elements is, the larger the volume of a radio receiver is, and the specific application is limited; the spectrum correlation subspace direction finding method based on the signal cyclostationarity characteristic has the advantages that the signal selection characteristic is achieved, the antenna direction finding freedom degree is expanded, meanwhile, the anti-interference capability of an algorithm is improved, and therefore the method is widely applied. However, the existing spectrum correlation subspace algorithm fails under the condition that the second-order cycle statistics of signals are consistent.
An effective solution to the problems in the related art has not been proposed yet.
Disclosure of Invention
Aiming at the technical problems in the related art, the invention provides a spectrum correlation subspace direction finding method based on symmetric cycle frequency, which comprises the following steps:
step 1: initialization processing: initializing and storing technical parameters;
the technical parameters comprise array element number M of the array, array element position md, signal propagation speed c, signal carrier frequency f and signal sampling frequency fsAzimuth angle theta, division interval delta theta of azimuth angle theta, sampling snapshot times T0Signal cycle frequency a, signal symmetric cycle frequency-a, number K of signals with cycle frequency a, and self-correlation time delay tau0；
Step 2: determining a discrete value set of the azimuth angle of the signal and a guide vector set corresponding to the discrete value set according to the azimuth angle theta and the division interval delta theta of the azimuth angle theta in the step 1;
and step 3: determining a time-domain sample vector of the array received signal: the receiving device samples the incident signals received by all array elements of the array, thereby determining a time domain sampling vector of the array received signals:
x(t)＝[x1(t),x2(t),...,xM(t)]
wherein: t is the time of the analog-to-digital conversion, i.e. the time-domain sampling time of the received signal, T1, 20M is the number of array elements, T0Is the number of snapshots;
and 4, step 4: determining a circular cross-correlation matrix of the sample data of the signal received by the array element, comprising:
step 4-1, calculating each vector x in the time domain sampling vectors in step 3 according to the signal cycle frequency α set in step 1i(t), i ═ 1, 2.., cyclic autocorrelation function of M
Step 4-2: according to the autocorrelation time delay tau set in the step 10Determining corresponding cyclic autocorrelation valuesAndwherein: 1,2,. M;
corresponding each array elementComposition matrixAnd combining each array elementCorrespond toComposition matrix
Step 4-3: according to step 4-2Determining its corresponding cyclic cross-correlation matrix X (α), X (- α);
step 4-4, constructing a circular cross correlation matrix psi (α) of the array receiving sample data according to X (α), X (- α) in the step 4-3;
and 5: determining a pseudo-spectrum of each steering vector in the set of steering vectors and the noise subspace, comprising:
step 5-1, respectively carrying out characteristic value decomposition on psi (α) in step 4 to determine a noise subspace thereof;
step 5-2, determining each steering vector a (α, theta) in the set of steering vectors in step 2i) With the pseudospectrum P (α, theta) of the noise subspace in step 5-1i)；
Step 6, determining the direction of arrival of the signal, in step 5, the pseudo-spectra P (α, theta)i) The maximum value in each pseudo spectrum is searched, a guide vector is corresponding to the maximum value in each pseudo spectrum, and the azimuth angle corresponding to the guide vector is taken as the arrival direction of the measured signal.
Further, in step 1, the signal cycle frequency a and the number K of signals are determined by a signal parameter estimation method, wherein: autocorrelation delay tau0Is taken as0＝[1 2 3 4 5 6 7 8]。
Further, in step 2: determining a discrete value set of the azimuth angle of the signal according to the azimuth angle theta and the division interval delta theta of the azimuth angle theta in the step 1 and a guide vector set corresponding to the discrete value set specifically include:
step 2-1, dividing the interval delta theta according to the azimuth angle theta in the step 1, and uniformly dividing the azimuth angle theta into NθEach discrete value being θiA set of (a);
step 2-2, corresponding to each discrete value thetaiα, determining an array steering vector a (α, θ)i) Wherein: 1,2, Nθ。
Further, in step 2: according to each discrete value thetaiDetermining an array steering vector a (α, θ)i) Comprises the following steps:
a(α,θi)＝[a1(α,θi)，a2(α,θi),L aM(α,θi)]
wherein,τm(i)＝dmsin(θi) C is the time difference of the signal arriving at the m-th array element, c is the propagation speed of the signal, i is 1,2θM is 1,2, wherein M is the number of array elements, and α is the cycle frequency of the kth signal.
Further, in step 4, the array element receives a cyclic autocorrelation function of signal sample dataAndthe calculation method is as follows:
wherein < · > represents the time averaging operation.
Further, the circular cross-correlation matrix X (α) and X (- α) in step 4 are calculated as follows:
wherein α is the cycle frequency of the signal and- α is the symmetric cycle frequency corresponding to α.
Further, the calculation method of the circular cross-correlation matrix Ψ (α) for constructing the array received sample data in step 4 is as follows:
Ψ(α)＝X(α)+I·X(-α)·I
wherein: i is an M × M switching matrix.
Further, the noise subspace is determined in step 5 by decomposing the eigenvalues of the cyclic cross-correlation matrix Ψ (α) of the array received sample data to obtain corresponding eigenvalues and eigenvectors according to the eigenvaluesThe decomposition of the characteristic value is carried out,
wherein: u shapek＝[uk1,uk2,...,ukM]，uk1,uk2,...,ukMIn order to be the left feature vector,vk1,vk2,...,vkMis the right eigenvector, sigma ═ diag { λk1,λk2,...λkM}，λk1＞λk2＞...＞λkMIs the corresponding characteristic value;
according to subspace theory, λk(K+1),λk(K+2),...,λkMThe space formed by the corresponding eigenvector is the noise subspace Ukn＝[uk(K+1),uk(K+2),...,ukM]。
Further, in step 5, each steering vector a in the set of steering vectors in step 2 is determined (α, theta)i) With the pseudospectrum P (α, theta) of the noise subspace in step 5-1i) The calculation method is as follows:
P(α,θi)＝20glg(||a(α,θi)||/||Ukna(α,θi)||)
where | · | | represents modulo.
The invention has the beneficial effects that: the spectrum correlation subspace direction finding method based on the symmetric cyclic frequency establishes a cyclic autocorrelation function according to the cyclostationarity of a received signal, and utilizes the weak correlation between the cyclic autocorrelation functions of the symmetric cyclic frequency to construct a cyclic cross-correlation matrix, so that the direction finding precision is superior to that of the existing spectrum correlation subspace direction finding method, and the method has higher direction finding correct probability.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without creative efforts.
FIG. 1 is a flow chart of the working principle of the spectrum correlation subspace direction finding method based on the symmetric cyclic frequency according to the invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments that can be derived by one of ordinary skill in the art from the embodiments given herein are intended to be within the scope of the present invention.
According to the embodiment of the invention, a spectrum correlation subspace direction finding method based on symmetric cyclic frequency is provided, a uniform linear array of 16 array elements is adopted, the distance between adjacent array elements is d equal to 37.5 meters, a straight line where the array elements are located is selected as an x axis, and a reference coordinate system is established by taking the first array element from the left as an origin. Two BPSK modulation signals with carrier frequency of 40MHz and code rate of 4MHz/s are transmitted to the uniform linear array from the directions of 1.43 degrees and 5.74 degrees, and the receiving device collects all array element receiving signals.
The specific process comprises the following steps:
step 1: initialization processing: initializing and storing technical parameters;
the technical parameters comprise array element number M of the array, array element position md, signal propagation speed c, signal carrier frequency f and signal sampling frequency fsAzimuth angle theta, division interval delta theta of azimuth angle theta, sampling snapshot times T0Signal cycle frequency a, signal symmetric cycle frequency-a, number K of signals with cycle frequency a, and self-correlation time delay tau0(ii) a Specifically, determining a signal cycle frequency a and a signal number K by a signal parameter estimation method;
in this embodiment: the number M of array elements of the initialized array is 16, the rectangular coordinate of the position of the array elements is md, wherein d is 37.5, M is 0,1, 15, and the propagation speed c of the signal is 3 × 108m/s, signal carrier frequency f 40MHz, signal sampling frequency fs320MHz, azimuth angle θ e [ -90 °,90 °]The division interval Δ θ is 0.1 °, and the number N of divisionsh180 °/Δ θ +1 1801, 4e6 as the signal cycle frequency α, 4e6 as the symmetrical cycle frequency- α,2 as the number of signals K, and the switching matrix IM×MSelf-correlation time delay tau0＝[1 2 3 4 5 6 7 8]Number of snapshots T0＝2000。
Step 2: determining a discrete value set of the azimuth angle of the signal and a guide vector set corresponding to the discrete value set according to the azimuth angle theta and the division interval delta theta of the azimuth angle theta in the step 1;
in this embodiment: according to each discrete value thetaiDetermining an array steering vector a (α, θ)i) Comprises the following steps:
a(α,θi)＝[a1(α,θi)，a2(α,θi),L aM(α,θi)]
wherein,τm(i)＝dmsin(θi) C is the time difference of the signal arriving at the m-th array element, c is the propagation speed of the signal, i is 1,2θM is 1,2, wherein M is the number of array elements, and α is the cycle frequency of the kth signal.
In this embodiment: firstly, uniformly dividing the azimuth angle theta into 1801 discrete values theta according to the division interval of the azimuth angle theta of 0.1 degree in the step 1i1, 1801, 1, x 0.1 degrees;
secondly, corresponding to each discrete value thetai,αDetermining the direction of arrival of the signal from the direction of arrival thetaiArray steering vector a at incidence (α, θ)i) Array steering vector a (α, θ)i) Are determined by the following formula:
wherein m is 0,1, 15, i is 1,2, 1801, d is the array element spacing, c is the propagation speed of the signal, α is the signal cycle frequency, the corresponding direction of arrival is 1.4 °, and the first 8 elements of the steering vector are:
1.0000+0.0000i,0.9969-0.0785i,0.9877-0.1564i,0.9724-0.2334i
0.9511-0.3090i,0.9239-0.3827i,0.8910-0.4540i,0.8526-0.5225i
corresponding to a direction of arrival of 5.7 °, the first 8 elements of its steering vector are:
1.0000+0.0000i,0.9511-0.3090i,0.8090-0.5878i,0.5878-0.8090i
0.3090-0.9511i,0.0000-1.0000i,-0.3090-0.9511i,-0.5878-0.8090i
and step 3: determining a time-domain sample vector of the array received signal: the receiving device samples the incident signals received by all array elements of the array, thereby determining a time domain sampling vector of the array received signals:
x(t)＝[x1(t),x2(t),...,xM(t)]
wherein: t is the time of the analog-to-digital conversion, i.e. the time-domain sampling time of the received signal, T1, 20M is the number of array elements, T0Is the number of snapshots;
the first 8 elements of the time-domain sampling of the first array element receiving signal are respectively:
-0.3878+0.2340i,-0.3972+0.6042i,0.2163-0.1836i,-0.2488+0.1669i,
-0.2183+0.0431i,-0.3023-0.5051i,0.1275+0.1100i,-0.3059+0.0095i
the first 8 elements of the time domain sampling of the last array element receiving signal are respectively:
-1.6073-0.4407i,1.9497-0.1891i,0.9620-0.3590i,-1.3549+0.0334i,
-1.9182+0.2205i,-1.4811-0.0252i,-0.9528-0.1640i,0.9736+0.3686i
and 4, step 4: determining a circular cross-correlation matrix of the sample data of the signal received by the array element, comprising:
step 4-1:calculating each vector x in the time domain sampling vectors of step 3 according to the signal cycle frequency α set in step 1i(t), i ═ 1, 2.., cyclic autocorrelation function of M
Step 4-2: according to the autocorrelation time delay tau set in the step 10Determining corresponding cyclic autocorrelation valuesAndwherein: 1,2,. M;
corresponding each array elementComposition matrixAnd corresponding each array elementComposition matrix
Step 4-3: according to step 4-2Determining its corresponding cyclic cross-correlation matrix X (α), X (- α);
step 4-4, constructing a circular cross correlation matrix psi (α) of the array receiving sample data according to X (α), X (- α) in the step 4-3;
in this embodiment: step 4, the array element receives the circular autocorrelation function of the signal sample dataAndthe calculation method is as follows:
wherein < · > represents the time averaging operation.
In the embodiment, the calculation modes of the circular cross-correlation matrixes X (α) and X (- α) in the step 4 are as follows:
wherein α is the cycle frequency of the signal and- α is the symmetric cycle frequency corresponding to α.
In the embodiment, the calculation method for constructing the circular cross-correlation matrix Ψ (α) of the array received sample data in step 4 is as follows:
Ψ(α)＝X(α)+I·X(-α)·I
wherein: i is an M × M switching matrix.
Specifically, determining a cyclic cross-correlation matrix of the sample data of the array element received signal in step 4: the first 8 elements of the first row of the corresponding cyclic cross-correlation matrix are:
0.0750,0.0715-0.0124i,0.0653-0.0249i,0.0572-0.0332i,
0.0414-0.0389i,0.0343-0.0463i,0.0237-0.0450i,0.0092-0.0365i
the first 8 elements of the last row of the corresponding cyclic cross-correlation matrix are:
0.0283-0.0029i,0.0197-0.0062i,0.0086-0.0015i,0.0072-0.0011i
-0.0028+0.0092i,-0.0005+0.0130i,0.0024+0.0238i,0.0029+0.0350i
and 5: determining a pseudo-spectrum of each steering vector in the set of steering vectors and the noise subspace, comprising:
step 5-1, respectively carrying out characteristic value decomposition on psi (α) in step 4 to determine a noise subspace thereof;
step 5-2, determining each steering vector a (α, theta) in the set of steering vectors in step 2i) With the pseudospectrum P (α, theta) of the noise subspace in step 5-1i)；
In the embodiment, the noise subspace is determined in the step 5 by decomposing the eigenvalue of the circular cross-correlation matrix Ψ (α) of the array received sample data to obtain the corresponding eigenvalue and eigenvector according to the eigenvalue and eigenvectorThe decomposition of the characteristic value is carried out,
wherein: u shapek＝[uk1,uk2,...,ukM]，uk1,uk2,...,ukMIn order to be the left feature vector,vk1,vk2,...,vkMis the right eigenvector, sigma ═ diag { λk1,λk2,...λkM}，λk1＞λk2＞...＞λkMIs the corresponding characteristic value;
according to subspace theory, λk(K+1),λk(K+2),...,λkMNull formed by corresponding feature vectorIndirectly is the noise subspace Ukn＝[uk(K+1),uk(K+2),...,ukM]。
In this embodiment, in step 5, each steering vector a (α, theta) in the steering vector set in step 2 is determinedi) With the pseudospectrum P (α, theta) of the noise subspace in step 5-1i) The calculation method is as follows:
P(α,θi)＝20glg(||a(α,θi)||/||Ukna(α,θi)||)
where | · | | represents modulo.
Specifically, the method comprises the following steps: determining the pseudo-spectrum of each guide vector and noise subspace in the guide vector set in the step 5: the directions of arrival-90 °, -89.9 °, -89.8 °, -89.7 ° correspond to pseudo-spectral values of 0.009685, 0.009685, 0.009682, 0.009679, for a total of 1801 pseudo-spectral values being determined.
Step 6, determining the direction of arrival of the signal, in step 5, the pseudo-spectra P (α, theta)i) Searching the maximum value, wherein the maximum value in each pseudo spectrum corresponds to a guide vector, and the azimuth angle corresponding to the guide vector is the measured signal arrival direction;
in this embodiment, the pseudo-spectral value P (α, θ) determined in step 5i) I 1, 2.. multidot.1801 searches for K2 maximum peaks, which are equal to 1.865, and the direction of arrival of the steering vector corresponding to the maximum peak is 6.1 °, that is, the direction of arrival of the measured signal;
the next largest peak is equal to 1.836, and the direction of arrival of the steering vector corresponding to the next largest peak is 1.5 °, i.e., the direction of arrival of the measured signal.
In the embodiment, the root mean square error of the signal direction of arrival measured by the method is tested, the measurement results of 500 independent tests are counted, the signal-to-noise ratio is-4 dB to 10dB, and under the condition of stepping by 2dB, the root mean square error of the signal direction of arrival measured by the method and the conventional spectrum correlation subspace method is shown in the following table 1;
table 1: error performance comparison of direction of arrival estimation methods
The method is used for counting the measurement results of 500 independent tests through the correct probability test of the measured signal direction of arrival, the signal-to-noise ratio is-10 dB to 4dB, and under the condition of stepping by 2dB, the method is used for one-time correct estimation when the absolute value of the direction-finding error is less than 2 degrees. The correct probability pairs of the signal direction of arrival determined by the method of the present invention and the existing spectral correlation subspace method are shown in table 2 below;
table 2: correct probability performance comparison of direction of arrival estimation methods
Therefore, the method can accurately determine the arrival direction of the signal; compared with the existing spectrum correlation subspace method, the method has the advantages that the root mean square error between the measured signal direction of arrival and the actual signal direction of arrival is smaller, and the correct probability is higher than that of the existing spectrum correlation subspace method under the condition that the absolute value of the direction-finding error is smaller than 2 degrees and is correct estimation.
Therefore, by means of the technical scheme, the spectrum correlation subspace direction finding method based on the symmetric cyclic frequency establishes the cyclic autocorrelation function according to the cyclostationary characteristic of the received signal, and constructs the cyclic cross correlation matrix by utilizing the weak correlation among the cyclic autocorrelation functions of the symmetric cyclic frequency, so that the direction finding precision is superior to that of the existing spectrum correlation subspace direction finding method, and the higher direction finding correct probability is achieved.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (9)
1. The method for finding the direction of the spectrum related subspace based on the symmetric cyclic frequency is characterized by comprising the following steps of:
step 1: initialization processing: initializing and storing technical parameters;
the technical parameters comprise array element number M of the array, array element position md, signal propagation speed c, signal carrier frequency f and signal sampling frequency fsAzimuth angle theta, division interval delta theta of azimuth angle theta, sampling snapshot times T0Signal cycle frequency a, signal symmetric cycle frequency-a, cycle frequency aNumber of signals K, autocorrelation delay tau0；
Step 2: determining a discrete value set of the azimuth angle of the signal and a guide vector set corresponding to the discrete value set according to the azimuth angle theta and the division interval delta theta of the azimuth angle theta in the step 1;
and step 3: determining a time-domain sample vector of the array received signal: the receiving device samples the incident signals received by all array elements of the array, thereby determining a time domain sampling vector of the array received signals:
x(t)＝[x1(t),x2(t),...,xM(t)]
wherein: t is the time of the analog-to-digital conversion, i.e. the time-domain sampling time of the received signal, T1, 20M is the number of array elements, T0Is the number of snapshots;
and 4, step 4: determining a circular cross-correlation matrix of the sample data of the signal received by the array element, comprising:
step 4-1, calculating each vector x in the time domain sampling vectors in step 3 according to the signal cycle frequency α set in step 1i(t), i ═ 1, 2.., cyclic autocorrelation function of M
Step 4-2: according to the autocorrelation time delay tau set in the step 10Determining corresponding cyclic autocorrelation valuesAndwherein: 1,2,. M;
corresponding each array elementComposition matrixAnd corresponding each array elementComposition matrix
Step 4-3: according to step 4-2Determining its corresponding cyclic cross-correlation matrix X (α), X (- α);
step 4-4, constructing a circular cross correlation matrix psi (α) of the array receiving sample data according to X (α), X (- α) in the step 4-3;
and 5: determining a pseudo-spectrum of each steering vector in the set of steering vectors and the noise subspace, comprising:
step 5-1, respectively carrying out characteristic value decomposition on psi (α) in step 4 to determine a noise subspace thereof;
step 5-2, determining each steering vector a (α, theta) in the set of steering vectors in step 2i) With the pseudospectrum P (α, theta) of the noise subspace in step 5-1i)；
Step 6, determining the direction of arrival of the signal, in step 5, the pseudo-spectra P (α, theta)i) The maximum value in each pseudo spectrum is searched, a guide vector is corresponding to the maximum value in each pseudo spectrum, and the azimuth angle corresponding to the guide vector is taken as the arrival direction of the measured signal.
2. The method according to claim 1, wherein the signal cycle frequency a and the number of signals K are determined by a signal parameter estimation method in step 1, wherein: autocorrelation delay tau0Is taken as0＝[1 2 3 4 5 6 7 8]。
3. The method for spectral correlation subspace direction finding based on symmetric cyclic frequencies as claimed in claim 1, wherein in step 2: determining a discrete value set of the azimuth angle of the signal according to the azimuth angle theta and the division interval delta theta of the azimuth angle theta in the step 1 and a guide vector set corresponding to the discrete value set specifically include:
step 2-1, dividing the interval delta theta according to the azimuth angle theta in the step 1, and uniformly dividing the azimuth angle theta into NθEach discrete value being θiA set of (a);
step 2-2, corresponding to each discrete value thetaiα, determining an array steering vector a (α, θ)i) Wherein: 1,2, Nθ。
4. The method for spectral correlation subspace direction finding based on symmetric cyclic frequencies of claim 3, wherein in step 2: according to each discrete value thetaiDetermining an array steering vector a (α, θ)i) Comprises the following steps:
a(α,θi)＝[a1(α,θi)，a2(α,θi),L aM(α,θi)]
wherein, am(θi)＝exp(j2πατm(i))，τm(i)＝dmsin(θi) C is the time difference of the signal arriving at the m-th array element, c is the propagation speed of the signal, i is 1,2θM is 1,2, wherein M is the number of array elements, and α is the cycle frequency of the kth signal.
5. The symmetric cyclic frequency based spectrum correlation subspace direction finding method according to claim 1, wherein said array element receives the cyclic autocorrelation function of the signal sample data in step 4Andthe calculation method is as follows:
wherein < · > represents the time averaging operation.
6. The method for spectral correlation subspace direction finding based on symmetric cyclic frequencies as claimed in claim 1, wherein the cyclic cross correlation matrices X (α) and X (- α) in step 4 are calculated by:
wherein α is the cycle frequency of the signal and- α is the symmetric cycle frequency corresponding to α.
7. The method of claim 1, wherein the computation of the circular cross-correlation matrix Ψ (α) for constructing the sample data for array reception in step 4 is as follows:
Ψ(α)＝X(α)+I·X(-α)·I
wherein: i is an M × M switching matrix.
8. The method according to claim 1, wherein the noise subspace is determined in step 5 by decomposing the eigenvalues of the cyclic cross-correlation matrix Ψ (α) of the sample data received by the array to obtain corresponding eigenvalues and eigenvectors according to the methodThe decomposition of the characteristic value is carried out,
wherein: u shapek＝[uk1,uk2,...,ukM]，uk1,uk2,...,ukMIn order to be the left feature vector,vk1,vk2,...,vkMis the right eigenvector, sigma ═ diag { λk1,λk2,...λkM}，λk1＞λk2＞...＞λkMIs the corresponding characteristic value;
according to subspace theory, λk(K+1),λk(K+2),...,λkMThe space formed by the corresponding eigenvector is the noise subspace Ukn＝[uk(K+1),uk(K+2),...,ukM]。
9. The method of claim 8, wherein in step 5, each steering vector a (α, θ) in the set of steering vectors in step 2 is determinedi) With the pseudospectrum P (α, theta) of the noise subspace in step 5-1i) The calculation method is as follows:
P(α,θi)＝20glg(||a(α,θi)||/||Ukna(α,θi)||)
where | · | | represents modulo.
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