CN108287333B - Main lobe anti-interference method combining JADE and CLEAN - Google Patents

Main lobe anti-interference method combining JADE and CLEAN Download PDF

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CN108287333B
CN108287333B CN201810222485.7A CN201810222485A CN108287333B CN 108287333 B CN108287333 B CN 108287333B CN 201810222485 A CN201810222485 A CN 201810222485A CN 108287333 B CN108287333 B CN 108287333B
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CN108287333A (en
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崔国龙
葛萌萌
陈芳香
时巧
余显祥
孔令讲
杨晓波
易伟
张天贤
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Chengdu Duopu Exploration Technology Co ltd
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University of Electronic Science and Technology of China
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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 a main lobe anti-interference method combining JADE and CLEAN, belongs to the technical field of radar anti-interference, and particularly relates to a blind source separation main lobe interference resisting technology. Firstly, estimating a steering vector and a waveform of an interference signal by using a JADE algorithm, and further reconstructing an interference array signal; then calculating and reconstructing the MUSIC spectrum of the interference array signal to obtain the MUSIC spectrum only containing the interference signal; and finally, canceling the MUSIC spectrum of the interference signal on the MUSIC spectrum of the received signal by utilizing a CLEAN algorithm in a space domain, thereby obtaining the estimation of the direction of arrival of the target signal. Simulation results show that the method can well complete interference suppression and estimate the target DOA.

Description

Main lobe anti-interference method combining JADE and CLEAN
Technical Field
The invention belongs to the technical field of radar anti-interference, and particularly relates to a blind source separation main lobe interference resisting technology.
Background
In modern electronic warfare, it has become a serious task for radar designers to improve the anti-interference performance of radar. In order to improve the survivability of the radar in a complex electromagnetic interference environment, various anti-interference measures such as ultralow sidelobe, sidelobe concealment, sidelobe cancellation and the like are adopted at present. However, when an interference signal enters the radar antenna from the main lobe, the detection performance of the radar is seriously influenced, and the traditional side lobe anti-interference measures cannot effectively interfere with the main lobe. Therefore, the method has important theoretical value and practical significance for ensuring that the radar correctly detects and tracks the target in a complex electromagnetic environment and improving the anti-interference capability of the main lobe of the radar.
Blind source separation technology is a signal processing technology developed in the last 80 th century, and is a process of extracting and recovering source signals which cannot be directly observed from a plurality of observed mixed signals. The technology has attracted extensive attention and application research in the aspects of wireless communication, biomedicine, voice signal processing and the like, and has good application prospect in the radar anti-interference technology. The literature [ Gaoming Huang, Lvxi Yang, Zhenya He.B.B.B. Source Processing used for radar anti-jamming, 2003 International Conference on neural Networks & Signal Processing, 1382-. In documents [ Wang, Zhang, Li wavelet, etc.. FastICA application and radar anti-mainlobe interference algorithm research [ J ].2015,31(4):497-453], separation of interference signals and target echo signals is realized by utilizing a FastICA algorithm, but the problems also exist.
Disclosure of Invention
Aiming at the defects in the background technology, the invention provides a main lobe anti-interference method combining a JADE (feature matrix joint diagonalization) blind source separation algorithm and a CLEAN algorithm. Firstly, estimating a steering vector and a waveform of an interference signal by using a JADE algorithm, and further reconstructing an interference array signal; then calculating the MUSIC spectrum of the received signal to obtain the MUSIC spectrum of the received signal, and calculating the reconstructed MUSIC spectrum of the interference array signal; and finally, canceling the strong interference signal on the MUSIC spectrum of the received signal through a space domain CLEAN algorithm, thereby obtaining the direction of arrival (DOA) estimation of the target signal. The simulation shows the effectiveness of the method, the algorithm does not need prior information of interference signals, various types of interference can be suppressed, and the method has universal applicability.
The invention provides a JADE and CLEAN combined main lobe anti-interference method, which comprises the following steps:
step 1: it is assumed that M target signals and P high-power interference signals which are mutually independent exist in a space, the difference of the arrival directions of the target signals and the interference signals is within the angle range of a main lobe, and the target signals and the interference signals simultaneously enter a space array, wherein the array consists of L array elements, the target signals and the interference signals are all far-field narrow-band signals, and the t-th array element receives signals:
Figure BDA0001600364990000021
wherein s ism(t), M is 1,2, …, M is the mth target signal, θTmIs the direction of arrival, DOA angle, J, of the mth target signalp(t), P is 1,2, …, P is the P-th interference signal, θJpIs the DOA angle of the arrival direction of the p-th interference signal, n (T) represents the noise signal at the time T, T represents the total number of samples, d represents the array element interval, lambda is the working wavelength,
Figure BDA0001600364990000022
then the antenna array receives the signal as:
X(t)＝[x1(t),x2(t),…,xL(t)]T,t＝1,2,…,T (2)
wherein, (.)TRepresenting a transpose operator, T representing a total number of samples;
step 2: calculating the MUSIC spectrum of the antenna array receiving signal X (t):
step 2-1: computing a spatial correlation matrix of a received signal
Figure BDA0001600364990000023
Figure BDA0001600364990000024
Wherein, (.)HRepresenting a conjugate transpose operator;
step 2-2: for correlation matrix
Figure BDA0001600364990000025
Decomposing the characteristic values, arranging the characteristic values according to a monotone increasing sequence, and arranging the characteristic vectors u corresponding to the last L-M-P characteristic valuesM+P+1,uM+P+2,…,uLForming a matrix G:
G＝[uM+P+1uM+P+2… uL](4)
step 2-3: the MUSIC spectral formula of x (t) is:
Figure BDA0001600364990000026
wherein the content of the first and second substances,
Figure RE-GDA0001663466430000031
is an array guide vector, d is an array element interval, and lambda is a working wavelength; since the direction of arrival information of the target source is unknown, the spatial angle theta is divided
Figure RE-GDA0001663466430000032
K represents the number of grids divided by space angle, and the function 1/a is calculated in sequenceH(θ)GGHa (theta) value to obtain a MUSIC spectrum P of X (t)X(θ)；
And step 3: separating the target signal and the interference signal by adopting a blind source separation algorithm of feature matrix joint diagonalization (JADE):
step 3-1: pre-whitening the received signal x (t) to obtain a whitened signal z (t), that is:
Z(t)＝WX(t) (6)
wherein W is a whitening matrix;
step 3-2: obtaining the fourth-order cumulant matrix Q of the whitening signal Z (t)z：
Figure BDA0001600364990000031
Wherein E [. C]Denotes an averaging operation, zi(t) i, j, k, l in the ith row of the whitening signal Z (t) are 1-M + P; to QzDecomposing the characteristic value to obtainTo the first M + P maximum eigenvalues lambda1,λ2,…,λM+PAnd its corresponding feature vector v1,v2,…,vM+PWherein v isiI is 1,2, …, M + P is (M + P)2X 1-dimensional column vectors, thus resulting in an object matrix { M } requiring approximate joint diagonalization1,M2,…,MM+P}; wherein Vec (M)i)＝λiviI 1,2, …, M + P, Vec (·) denotes a vectorization operator, i.e., a column vector of a matrix is arranged into column vectors in the order of arrangement in the matrix;
step 3-3: using unitary matrix V pair { M1,M2,…,MM+PPerforming approximate joint diagonalization;
step 3-4: obtaining a separation signal and an array flow pattern estimation:
Figure BDA0001600364990000032
wherein, W#Is the pseudo-inverse of the whitening matrix W; y (t) is a separate signal including an estimate of the waveform of the target signal
Figure BDA0001600364990000033
And interference signal waveform estimation
Figure BDA0001600364990000034
Figure BDA0001600364990000035
Array flow pattern estimation including target signal for array flow pattern estimation
Figure BDA0001600364990000036
Array flow pattern estimation with interference signal
Figure BDA0001600364990000037
And 4, step 4: suppose that the interference signal waveform estimated in step 3 is
Figure BDA0001600364990000038
The interference array flows as
Figure BDA0001600364990000039
The interference contribution in the reconstructed received signal is:
Figure BDA00016003649900000310
wherein the content of the first and second substances,
Figure BDA00016003649900000311
the estimated interference component;
and 5: calculating the MUSIC spectrum of the target by using a CLEAN algorithm in a space domain:
step 5-1: calculating the MUSIC spectrum of the interference component estimated in the step 4 according to the method in the step 2 to obtain the MUSIC spectrum P of the interference componentJ(θ)；
Step 5-2: to obtain the MUSIC spectrum of the target, the cost function needs to be minimized:
Figure BDA00016003649900000312
using P obtained in step 5-1J(theta) and P obtained in step 2X(theta) calculating the weight coefficient
Figure BDA00016003649900000313
Figure BDA0001600364990000041
Step 5-3: using the weight coefficients obtained in step 4-2
Figure BDA0001600364990000042
The MUSIC spectrum of the target was calculated as:
Figure BDA0001600364990000043
wherein, PT(theta) is the target MUSIC spectrum after interference suppression; before finding outThe M maximum values correspond to θ, which is the target direction of arrival.
The invention has the advantages that
The invention provides a main lobe anti-interference method combining JADE and CLEAN, compared with the existing interference suppression algorithm, the method does not need to know the prior information of interference, can be suitable for various types of interference, and realizes DOA estimation on a target.
Firstly, estimating a steering vector and a waveform of an interference signal by using a JADE algorithm, and further reconstructing an interference array signal; then calculating and reconstructing the MUSIC spectrum of the interference array signal to obtain the MUSIC spectrum only containing the interference signal; and finally, canceling the MUSIC spectrum of the interference signal on the MUSIC spectrum of the received signal by utilizing a CLEAN algorithm in a space domain, thereby obtaining the estimation of the direction of arrival of the target signal. Simulation results show that the method can well complete interference suppression and estimate the target DOA.
Drawings
FIG. 1 is a flow chart of the method
FIG. 2 is a flow chart of the algorithm for finding unitary matrix V by joint diagonalization
FIG. 3 is the MUSIC spectrum of a received signal under main lobe interference
FIG. 4 is a waveform diagram of a target signal and an interference signal
FIG. 5 is a waveform diagram of target signal and interference signal estimated by JADE
FIG. 6 is the MUSIC spectrum of the reconstructed interference signal
FIG. 7 shows the MUSIC spectrum of the target signal obtained after interference suppression
Detailed Description
Step 1:
it is assumed that M target signals and P high-power interference signals which are mutually independent exist in a space, the difference of the arrival directions of the target signals and the interference signals is within the angle range of a main lobe, and the target signals and the interference signals simultaneously enter a space array, wherein the array consists of L array elements, the target signals and the interference signals are all far-field narrow-band signals, and the t-th array element receives signals:
Figure BDA0001600364990000051
wherein s ism(t), M is 1,2, …, M is the mth target signal, θTmFor the direction of arrival of the corresponding target signal, Jp(t), P is 1,2, …, P is the pth interference signal, θJpAnd n (T) represents a noise signal at the time T, T represents the total number of samples, d represents the array element interval, and lambda is the working wavelength.
Then the antenna array receives the signal as:
X(t)＝[x1(t),x2(t),…,xL(t)]T,t＝1,2,…,T (10)
wherein, (.)TDenotes the transpose operator and T denotes the total number of samples.
Step 2: calculating the MUSIC spectrum of the antenna array receiving signal X (t):
step 2-1: computing a spatial correlation matrix of a received signal
Figure BDA0001600364990000052
Figure BDA0001600364990000053
Wherein, (.)HRepresenting a conjugate transpose operator.
Step 2-2: for correlation matrix
Figure BDA0001600364990000054
Decomposing the characteristic values, arranging the characteristic values according to a monotone increasing sequence, and arranging the characteristic vectors u corresponding to the last L-M-P characteristic valuesM+P+1,uM+P+2,…,uLForming a matrix G:
G＝[uM+P+1uM+P+2… uL](12)
step 2-3: the MUSIC spectral formula of x (t) is:
Figure BDA0001600364990000055
wherein the content of the first and second substances,
Figure BDA0001600364990000056
and d is the array guide vector, d is the array element interval, and lambda is the working wavelength. Since the direction of arrival information of the target source is unknown, the spatial angle theta is divided
Figure BDA0001600364990000057
K represents the number of grids divided by space angle, and the function 1/a is calculated in sequenceH(θ)GGHa (theta) value to obtain a MUSIC spectrum P of X (t)X(θ)。
And step 3: separating the target signal and the interference signal by adopting a blind source separation algorithm of feature matrix joint diagonalization (JADE):
step 3-1: pre-whitening the received signal x (t) to obtain a whitened signal z (t), that is:
Z(t)＝WX(t) (14)
wherein the content of the first and second substances,
Figure BDA0001600364990000058
in order to whiten the matrix, the matrix is,
Figure BDA0001600364990000059
Umax＝[u1u2… uM+P]， λ1,λ2,…,λM+Pfor the correlation matrix in step 2-1
Figure BDA0001600364990000067
The first M + P eigenvalues, u1,u2,…,uM+PIs its corresponding characteristic vector.
Step 3-2: obtaining the fourth-order cumulant matrix Q of the whitening signal Z (t)z：
Figure BDA0001600364990000061
Wherein E [. C]Denotes an averaging operation, zi(t) i, j, k, l of the ith row of the whitened signal Z (t), respectively1 to M + P. To QzDecomposing the eigenvalue to obtain the first M + P maximum eigenvalues lambda1,λ2,…,λM+PAnd its corresponding feature vector v1,v2,…,vM+PWherein v isiI is 1,2, …, M + P is (M + P)2X 1-dimensional column vectors, thus resulting in an object matrix { M } requiring approximate joint diagonalization1,M2,…,MM+P}. Wherein Vec (M)i)＝λiviI 1,2, …, M + P, Vec (·) denotes a vectorization operator, i.e., a column vector of a matrix is arranged into column vectors in the order of arrangement in the matrix.
Step 3-3: finding a unitary matrix V pair { M }1,M2,…,MM+PPerforming joint diagonalization, which comprises the following specific steps:
step 3-3-1: given an initial matrix V ═ IM+P，IM+PRepresents an (M + P) × (M + P) -dimensional identity matrix, and M + P object matrices M in step 3-2nN is 1,2, …, M + P, threshold ρ.
Step 3-3-2: for matrix
Figure BDA0001600364990000062
Decomposing the eigenvalue to obtain the eigenvector [ x, y, z ] corresponding to the maximum eigenvalue]TWherein h (M)n)＝[mii-mjjmij+mjii(mji-mij)]，mijRepresentation matrix MnThe ith row and the jth column of elements,
Figure BDA0001600364990000063
step 3-3-3: using [ x, y, z ] obtained in 3-3-2]TC, s is calculated as follows:
Figure BDA0001600364990000064
where c, s are elements in a Givens rotation matrix G, G(i,j,c,s)The (i, i), (i, j), (j, i), (j, j) th elements of the representation matrix are respectively
Figure BDA0001600364990000065
The other elements are the same as the unit array, and a matrix G is obtained according to c and s(i,j,c,s)。
Step 3-3-4: judging whether s is greater than or equal to rho, and if so, performing the step 3-3-5; if not, the obtained V is the unitary matrix V.
Step 3-3-5: updating matrix V ═ VG(i,j,c,s)And an object matrix
Figure BDA0001600364990000066
n is 1,2, …, M + P until i, j has traversed 1-M + P. The algorithm flow is shown in fig. 2.
Step 3-4: obtaining a separation signal and an array flow pattern estimation:
Figure BDA0001600364990000071
wherein, W#Is the pseudo-inverse of the whitening matrix W; y (t) is a separate signal including an estimate of the waveform of the target signal
Figure BDA0001600364990000072
And interference signal waveform estimation
Figure BDA0001600364990000073
Figure BDA0001600364990000074
Array flow pattern estimation including target signal for array flow pattern estimation
Figure BDA0001600364990000075
Array flow pattern estimation with interference signal
Figure BDA0001600364990000076
And 4, step 4: suppose that the interference signal waveform estimated in step 3 is
Figure BDA0001600364990000077
The interference array flows as
Figure BDA0001600364990000078
The interference contribution in the reconstructed received signal is:
Figure BDA0001600364990000079
wherein the content of the first and second substances,
Figure BDA00016003649900000710
is the estimated interference component.
And 5: calculating the MUSIC spectrum of the target by using a CLEAN algorithm in a space domain:
step 5-1: calculating the MUSIC spectrum of the interference component estimated in the step 3 according to the method in the step 2 to obtain the MUSIC spectrum P of the interference componentJ(θ)。
Step 5-2: to obtain the MUSIC spectrum of the target, the cost function needs to be minimized:
Figure BDA00016003649900000711
suppose the MUSIC spectrum of the received signal obtained in step 1 is PX(θ) by PJ(theta) and PX(theta) calculating the weight coefficient
Figure BDA00016003649900000712
Figure BDA00016003649900000713
Step 5-3: using the weight coefficients obtained in step 4-2
Figure BDA00016003649900000714
The MUSIC spectrum of the target was calculated as:
Figure BDA00016003649900000715
wherein, PTAnd (theta) is the target MUSIC spectrum after interference suppression. Find the first M maximumsThe values are each associated with θ, which is the direction of arrival of the target.
Simulation verification and analysis
Simulation parameters:
here, simulation verification is performed by taking the example that two targets and two interferences exist in the space. Assuming that the target signals are respectively a chirp signal and a complex sine signal, the expressions are respectively as follows:
Figure BDA00016003649900000716
s2(t)＝exp(j2π(f0t)),0＜t≤τp(22)
wherein, the chirp rate K of the chirp signal is B/taupWhere B is 10MHz as the working bandwidth and pulse width τ p10 mus, carrier frequency f0＝1GHz。
The interference signal considers spectrum dispersion (SMSP) interference and noise amplitude modulation interference, and the expressions are respectively as follows:
Figure BDA0001600364990000081
J2(t)＝(U0+Un(t))exp(j(2πfjt+φ)) (24)
wherein j issmsp(t)＝exp(j2πf0t+jπk′t2),k′＝nK,t∈[0,τp/n]And k 'is the frequency modulation slope of the interference signal, the value of k' is n times of the frequency modulation slope of the radar transmission signal, n is the number of interference sub-pulses, and n is set to be 5. U shapen(t) is white Gaussian noise with mean value of zero and variance of 1, fj1GHz is the interference carrier frequency and phi is 0,2 pi) a uniformly distributed random variable.
Let two target signals be respectively located at thetaT1＝30°,θT244.5 °, DOA of the two interference signals is θJ1＝30.4°,θJ245 deg. is equal to. The signal-to-noise ratio of the two target signals is SNR1＝SNR210dB, the dry-to-noise ratio is JNR160dB and JNR2＝70dB。
Simulation analysis:
as can be seen from fig. 3, on the MUSIC spectrum, the two interfering signal amplitudes are much higher than the target amplitude, resulting in difficulty in target DOA estimation. As can be seen from fig. 4 and 5, the JADE blind source separation algorithm has a good separation performance, and can effectively separate the target signal from the interference signal. The interference component is reconstructed by using the interference signal waveform and the interference guide vector obtained by JADE, and the MUSIC spectrum is calculated and shown in figure 6, so that the DOA of the interference signal can be accurately estimated. The estimated interfering MUSIC spectrum is cancelled from the MUSIC spectrum of the received signal by using the CLEAN algorithm to obtain the MUSIC spectrum of the target shown in fig. 7, and it can be seen from fig. 7 that the interference is suppressed and the DOA of the target can be well estimated. The effectiveness of the present invention is illustrated by the above results.

Claims (1)

1. A main lobe anti-interference method combining JADE and CLEAN comprises the following steps:
step 1: it is assumed that M target signals and P high-power interference signals which are mutually independent exist in a space, the difference of the arrival directions of the target signals and the interference signals is within the angle range of a main lobe, and the target signals and the interference signals simultaneously enter a space array, wherein the array consists of L array elements, the target signals and the interference signals are all far-field narrow-band signals, and the t-th array element receives signals:
Figure FDA0002265553070000011
wherein s ism(t), M is 1,2, …, M is the mth target signal, θTmIs the direction of arrival, DOA angle, J, of the mth target signalp(t), P is 1,2, …, P is the P-th interference signal, θJpIs the direction of arrival, i.e. DOA angle, n, of the p-th interfering signall(T) represents the noise signal at time T, T represents the total number of samples, d represents the array element spacing, λ is the operating wavelength,
Figure FDA0002265553070000012
then the antenna array receives the signal as:
X(t)＝[x1(t),x2(t),…,xL(t)]T,t＝1,2,…,T (2)
wherein, (.)TRepresenting a transpose operator, T representing a total number of samples;
step 2: calculating the MUSIC spectrum of the antenna array receiving signal X (t):
step 2-1: computing a spatial correlation matrix of a received signal
Figure FDA0002265553070000013
Figure FDA0002265553070000014
Wherein, (.)HRepresenting a conjugate transpose operator;
step 2-2: for correlation matrix
Figure FDA0002265553070000015
Decomposing the characteristic values, arranging the characteristic values according to a monotone increasing sequence, and arranging the characteristic vectors u corresponding to the last L-M-P characteristic valuesM+P+1,uM+P+2,…,uLForming a matrix G:
G＝[uM+P+1uM+P+2…uL](4)
step 2-3: the MUSIC spectral formula of x (t) is:
Figure FDA0002265553070000016
wherein the content of the first and second substances,
Figure FDA0002265553070000017
is an array guide vector, d is an array element interval, and lambda is a working wavelength; since the direction of arrival information of the target source is unknown, the spatial angle theta is divided
Figure FDA0002265553070000018
K represents the number of grids divided by space angle, and the function 1/a is calculated in sequenceH(θ)GGHa (theta) value to obtain a MUSIC spectrum P of X (t)X(θ)；
And step 3: separating a target signal and an interference signal by adopting a blind source separation algorithm of feature matrix joint diagonalization:
step 3-1: pre-whitening the received signal x (t) to obtain a whitened signal z (t), that is:
Z(t)＝WX(t) (6)
wherein W is a whitening matrix;
step 3-2: obtaining the fourth-order cumulant matrix Q of the whitening signal Z (t)z：
Figure FDA0002265553070000021
Wherein E [. C]Denotes an averaging operation, zi(t) i, j, k, l in the ith row of the whitening signal Z (t) are 1-M + P; to QzDecomposing the eigenvalue to obtain the first M + P maximum eigenvalues lambda1,λ2,…,λM+PAnd its corresponding feature vector v1,v2,…,vM+PWherein v isiI is 1,2, …, M + P is (M + P)2X 1-dimensional column vectors, thus resulting in an object matrix { M } requiring approximate joint diagonalization1,M2,…,MM+P}; wherein Vec (M)i)＝λiviI 1,2, …, M + P, Vec (·) denotes a vectorization operator, i.e., a column vector of a matrix is arranged into column vectors in the order of arrangement in the matrix;
step 3-3: using unitary matrix V pair { M1,M2,…,MM+PPerforming approximate joint diagonalization;
step 3-4: obtaining a separation signal and an array flow pattern estimation:
Figure FDA0002265553070000022
wherein, W#Is the pseudo-inverse of the whitening matrix W; y (t) is given byOff-signal, including target signal waveform estimation
Figure FDA0002265553070000023
And interference signal waveform estimation
Figure FDA0002265553070000024
Figure FDA0002265553070000025
Array flow pattern estimation including target signal for array flow pattern estimation
Figure FDA0002265553070000026
Array flow pattern estimation with interference signal
Figure FDA0002265553070000027
And 4, step 4: suppose that the interference signal waveform estimated in step 3 is
Figure FDA0002265553070000028
The interference array flows as
Figure FDA0002265553070000029
The interference component in the reconstructed received signal is:
Figure FDA00022655530700000210
wherein the content of the first and second substances,
Figure FDA00022655530700000211
the estimated interference component;
and 5: calculating the MUSIC spectrum of the target by using a CLEAN algorithm in a space domain:
step 5-1: calculating the MUSIC spectrum of the interference component estimated in the step 4 according to the method in the step 2 to obtain the MUSIC spectrum P of the interference componentJ(θ)；
Step 5-2: is composed ofTo obtain the MUSIC spectrum of the target, the cost function needs to be minimized:
Figure FDA0002265553070000031
using P obtained in step 5-1J(theta) and P obtained in step 2X(theta) calculating the weight coefficient
Figure FDA0002265553070000032
Figure FDA0002265553070000033
Step 5-3: using the weight coefficients obtained in step 4-2
Figure FDA0002265553070000034
The MUSIC spectrum of the target was calculated as:
Figure FDA0002265553070000035
wherein, PT(theta) is the target MUSIC spectrum after interference suppression; finding out theta corresponding to the first M maximum values respectively, namely the direction of arrival of the target.
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