CN110212966B - Antenna mutual coupling correction method based on importance resampling under coherent source condition - Google Patents

Antenna mutual coupling correction method based on importance resampling under coherent source condition Download PDF

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CN110212966B
CN110212966B CN201910502481.9A CN201910502481A CN110212966B CN 110212966 B CN110212966 B CN 110212966B CN 201910502481 A CN201910502481 A CN 201910502481A CN 110212966 B CN110212966 B CN 110212966B
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mutual coupling
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CN110212966A (en
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侯煜冠
高荷福
顾村锋
毛兴鹏
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Harbin Institute of Technology
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B17/00—Monitoring; Testing
    • H04B17/20—Monitoring; Testing of receivers
    • H04B17/21—Monitoring; Testing of receivers for calibration; for correcting measurements
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
    • H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
    • H04B7/0842—Weighted combining
    • H04B7/0848—Joint weighting
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
    • H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
    • H04B7/0842—Weighted combining
    • H04B7/086—Weighted combining using weights depending on external parameters, e.g. direction of arrival [DOA], predetermined weights or beamforming
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation

Abstract

An antenna mutual coupling correction method based on importance resampling under coherent source condition belongs to the technical field of signal arrival direction estimation. The method solves the problem of low accuracy of DOA estimation of the traditional MUSIC algorithm due to the influence of mutual coupling among array elements. The invention estimates the equivalent cross coupling matrix after the space smoothing by an importance resampling method and estimates the DOA of the target signal by using a new MUSIC space spectrum. Compared with the traditional MUSIC algorithm and the general mutual coupling matrix reconstruction MUSIC algorithm, the importance resampling method has better performance, can estimate the DOA of the target signal more accurately, can improve the estimation precision by about 30-40 percent, and can be used for antenna mutual coupling correction under the condition of coherent source. The method can be applied to the technical field of signal arrival direction estimation.

Description

Antenna mutual coupling correction method based on importance resampling under coherent source condition
Technical Field
The invention belongs to the technical field of signal arrival direction estimation, and particularly relates to an antenna mutual coupling correction method based on importance resampling under a coherent source condition.
Background
The Multiple Signal Classification (MUSIC) algorithm is used as a classical Direction-Of-Arrival (DOA) estimation method, and has a good estimation performance. When the target signal is a coherent signal, the spatial smoothing technique is required to process the signal, and then the MUSIC algorithm is used for estimation, so that the DOA of the signal can be better estimated.
However, under the condition that the antenna array has mutual coupling, since the array mutual coupling coefficient matrix is unknown, the DOA estimation performance of the conventional MUSIC algorithm is greatly affected, especially for the case that the target signal is a coherent signal and needs to be processed by using a spatial smoothing technique.
Disclosure of Invention
The invention aims to solve the problem of low DOA estimation precision of the traditional MUSIC algorithm due to the influence of mutual coupling among array elements, and provides an antenna mutual coupling correction method based on importance resampling under a coherent source condition.
The technical scheme adopted by the invention for solving the technical problems is as follows: an antenna mutual coupling correction method based on importance resampling under coherent source conditions comprises the following steps:
the method comprises the following steps: for a normalized linear array system comprising N array elements, calculating a receiving signal X (t) of the array system under a mutual coupling condition;
step two, processing the received signal X (t) of the array system by utilizing the space smoothing technology to obtain a forward space smoothing covariance matrix RfAnd backward spatial smoothing covariance matrix Rb；
Step three, utilizing a forward space smoothing covariance matrix RfAnd backward spatial smoothing covariance matrix RbComputing a forward and backward smoothed covariance matrix Rfb；
Step four: smoothing covariance matrix R for forward and backward directionsfbPerforming characteristic decomposition to obtain a noise subspace; forming spatial spectrum P of MUSIC algorithm using noise subspaceMU(θ) according to PMU(θ) completing a first estimation of the target signal DOA at the peak value;
step five: reconstructing the equivalent mutual coupling matrix by using the first estimated target signal DOA to obtain a reconstructed equivalent mutual coupling matrix
Figure BDA0002090690870000021
And equivalent cross-coupling matrix using reconstruction
Figure BDA0002090690870000022
Forming a MUSIC algorithm space spectrum;
step six: processing the reconstructed equivalent cross coupling matrix by using an importance resampling method to obtain a new equivalent cross coupling matrix; and forming a spatial spectrum of the MUSIC algorithm by using the new equivalent cross coupling matrix and the noise subspace to obtain the final target signal DOA estimation.
The invention has the beneficial effects that: the invention relates to an antenna mutual coupling correction method based on importance resampling under a coherent source condition. Compared with the traditional MUSIC algorithm and the general mutual coupling matrix reconstruction MUSIC algorithm, the importance resampling method has better performance, can estimate the DOA of the target signal more accurately, can improve the estimation precision by about 30-40 percent, and can be used for antenna mutual coupling correction under the condition of coherent source.
Drawings
FIG. 1 is a schematic diagram of the forward smoothing array element distribution of the present invention;
FIG. 2 is a graph of the results of DOA estimation using position resampling in a first simulation;
FIG. 3 is a graph of the results of DOA estimation using numerical resampling in a first simulation;
FIG. 4 is a graph of the results of DOA estimation using numerical and positional resampling in a first simulation;
FIG. 5 is a diagram of the results of DOA estimation using a zero resampling method in a first simulation;
FIG. 6 is a graph of the results of DOA estimation using the CCE-FBS-MUSIC method in a first simulation;
FIG. 7 is a graph showing the results of DOA estimation using the FBS-MUSIC method in the first simulation;
FIG. 8 is a graph of the results of DOA estimation using position resampling in a second simulation;
FIG. 9 is a graph of the results of DOA estimation using numerical resampling in a second simulation;
FIG. 10 is a graph of the results of DOA estimation using numerical and positional resampling in a second simulation;
FIG. 11 is a graph of the results of DOA estimation using a zero resampling method in a second simulation;
FIG. 12 is a graph of the results of DOA estimation using the CCE-FBS-MUSIC method in a second simulation;
FIG. 13 is a graph showing the results of DOA estimation using the FBS-MUSIC method in a second simulation;
FIG. 14 is a graph of the results of DOA estimation using position resampling in a third simulation;
FIG. 15 is a graph of the results of DOA estimation using numerical resampling in a third simulation;
FIG. 16 is a graph of the results of DOA estimation using numerical and positional resampling in a third simulation;
FIG. 17 is a diagram of the results of DOA estimation using a zero resampling method in a third simulation;
FIG. 18 is a graph of the results of DOA estimation using the CCE-FBS-MUSIC method in a third simulation;
FIG. 19 is a graph showing the results of DOA estimation using the FBS-MUSIC method in a third simulation;
FIG. 20 is a graph of the results of DOA estimation using the position resampling method in a fourth simulation;
FIG. 21 is a graph of the results of DOA estimation using numerical resampling in a fourth simulation;
FIG. 22 is a graph of the results of DOA estimation using numerical and positional resampling in a fourth simulation;
FIG. 23 is a diagram of the results of DOA estimation using a zero resampling method in a fourth simulation;
FIG. 24 is a graph of the results of DOA estimation using the CCE-FBS-MUSIC method in a fourth simulation;
FIG. 25 is a graph showing the results of DOA estimation using the FBS-MUSIC method in the fourth simulation;
FIG. 26 is a graph of the results of DOA estimation using the position resampling method in a fifth simulation;
FIG. 27 is a graph of the results of DOA estimation using numerical resampling in a fifth simulation;
FIG. 28 is a graph of the results of DOA estimation using numerical and positional resampling in a fifth simulation;
FIG. 29 is a diagram of the results of DOA estimation using a zero resampling method in a fifth simulation;
FIG. 30 is a graph of the results of DOA estimation using the CCE-FBS-MUSIC method in a fifth simulation;
FIG. 31 is a graph showing the results of DOA estimation using the FBS-MUSIC method in the fifth simulation;
FIG. 32 is a graph of the results of DOA estimation using the position resampling method in a sixth simulation;
FIG. 33 is a graph of the results of DOA estimation using numerical resampling in a sixth simulation;
FIG. 34 is a graph of the results of DOA estimation in a sixth simulation using a numerical and positional resampling method;
FIG. 35 is a diagram of the results of DOA estimation using a zero resampling method in a sixth simulation;
FIG. 36 is a graph showing the results of DOA estimation using the CCE-FBS-MUSIC method in a sixth simulation;
FIG. 37 is a graph showing the results of DOA estimation using the FBS-MUSIC method in a sixth simulation;
in the figure: SNR represents the signal-to-noise ratio.
Detailed Description
The first specific implementation way is as follows: the method for correcting mutual coupling of antennas based on importance resampling in the coherent source condition includes the following steps:
the method comprises the following steps: for a normalized linear array system comprising N array elements, calculating a receiving signal X (t) of the array system under a mutual coupling condition;
step two, processing the received signal X (t) of the array system by utilizing the space smoothing technology to obtain a forward space smoothing covariance matrix RfAnd backward spatial smoothing covariance matrix Rb；
Step three, utilizing a forward space smoothing covariance matrix RfAnd backward spatial smoothing covariance matrix RbComputing a forward and backward smoothed covariance matrix Rfb；
Step four: smoothing covariance matrix R for forward and backward directionsfbPerforming characteristic decomposition to obtain a noise subspace; forming spatial spectrum P of MUSIC algorithm using noise subspaceMU(θ) according to PMUThe peak value of (θ) completes a first estimation of the target signal DOA;
step five: reconstructing the equivalent mutual coupling matrix by using the first estimated target signal DOA to obtain a reconstructed equivalent mutual coupling matrix
Figure BDA0002090690870000041
And equivalent cross-coupling matrix using reconstruction
Figure BDA0002090690870000042
Forming a MUSIC algorithm spatial spectrum;
step six: processing the reconstructed equivalent cross coupling matrix by using an importance resampling method to obtain a new equivalent cross coupling matrix; and forming a spatial spectrum of the MUSIC algorithm by using the new equivalent cross coupling matrix and the noise subspace to obtain the final target signal DOA estimation.
The second embodiment is as follows: the first difference between the present embodiment and the specific embodiment is: the specific process of the step one is as follows:
for a normalized linear array system comprising N array elements, the distance between every two adjacent array elements is d;
the mutual coupling coefficient between adjacent array elements is almost the same and after increasing the array element spacing, the mutual coupling coefficient decreases and the mutual coupling coefficient between two sufficiently distant array elements is zero.
The mutual coupling coefficient between adjacent array elements is expressed by the Toeplitz matrix C as:
Figure BDA0002090690870000043
as can be seen from the structure of the matrix C, the unknown elements of the mutual coupling matrix C can be completely composed of the elements C in the first row of the matrix ═ 1, C1,…,cN-1]Determining;
wherein: c. C1Representing the mutual coupling coefficient between the 1 st and 2 nd array elements, cN-2Representing the mutual coupling coefficient between the 1 st and the N-1 st array elements, cN-1Representing the mutual coupling coefficient between the 1 st array element and the Nth array element;
suppose that at time t, an incoming wave has K narrow-band coherent signal sources, which are respectively denoted as s1(t)，s2(t)，…，sK(t) the incoming wave directions of the K narrow-band coherent signal sources are theta1，θ2，…，θKAnd the wavelengths of the K narrow-band coherent signal sources are lambda;
each array element in the array system has an independent and identically distributed variance of
Figure BDA0002090690870000044
The white gaussian noise, under the mutual coupling condition, the received signal x (t) of the array system is represented as:
X(t)＝CAS(t)+ξ(t) (2)
wherein:
X(t)＝[x1(t),x2(t),…,xN(t)]T (3)
x1(t) represents the received signal of the 1 st array element, x2(t) represents the received signal of the 2 nd array element, xN(T) represents the received signal of the Nth array element, and the superscript T represents the transposition of the matrix;
the expressions for the intermediate variables S (t) and ξ (t) are:
S(t)＝[s1(t),s2(t),…,sK(t)]T (4)
ξ(t)＝[ξ1(t),ξ2(t),…,ξN(t)]T (5)
wherein: xin(t), N is 1,2, …, N is the variance of the independent homodistribution of
Figure BDA0002090690870000051
White gaussian noise of (1);
the expression for the intermediate variable a is:
A＝[a(θ1),a(θ2),…,a(θK)] (6)
wherein: the kth column vector a (θ) in the intermediate variable Ak) The expression of (c) is:
a(θk)＝[1 β(θk) … β(θk)N-1]T,k＝1,2,…,K (7)
Figure BDA0002090690870000052
β(θk) Is an intermediate variable, j is an imaginary unit, θkIs the incoming wave direction of the kth narrow-band coherent signal source.
Because the received signal is a narrow-band coherent source, the signal needs to be subjected to spatial smoothing first, and then DOA estimation is carried out by using an MUSIC algorithm;
the third concrete implementation mode: the second embodiment is different from the first embodiment in that: the specific process of the second step is as follows:
for forward spatial smoothing, as shown in fig. 1, regarding the linear array system as L sub-arrays, where the number of array elements of each sub-array is M, starting from the first 1 st array element, the 1 st to mth array elements form the 1 st sub-array, the 2 nd to M +1 th array elements form the 2 nd sub-array, and so on, the N-M +1 th to nth array elements form the L-th sub-array, and the number L of sub-arrays is N-M + 1;
the received signal X of the 1 st subarray that is forward spatially smoothed1The expression of (t) is:
X1(t)＝C1(AS(t)+ξ(t)) (9)
wherein X1(t) and the intermediate variable matrix C1Are respectively:
X1(t)＝[x1(t),x2(t),…,xM(t)]T (10)
Figure BDA0002090690870000061
received signal X of forward spatially smoothed 2 nd sub-array2The expression of (t) is:
X2(t)＝C2(AS(t)+ξ(t)) (12)
intermediate variable matrix C2The expression of (a) is:
Figure BDA0002090690870000062
then C is1＝(C·E1)H，C2＝(C·E2)H
Figure BDA0002090690870000063
Wherein: the superscript H represents the conjugate transpose of the matrix;
Figure BDA0002090690870000064
the received signal X of the ith forward spatially smoothed sub-arrayi(t) is:
Xi(t)＝(Di-1E1)HC(AS(t)+ξ(t)),i＝1,2,...,L (16)
wherein: di-1Represents i-1 multiplications by D;
received signal X of ith forward spatially smoothed sub-arrayi(t) autocovariance matrix RiComprises the following steps:
Figure BDA0002090690870000071
Figure BDA0002090690870000072
wherein: i is a unit matrix, an intermediate variable RS＝E[S(t)SH(t)]，E[·]Represents a desire;
averaging the autocovariance matrix of each subarray receiving signal of the forward space smoothing to obtain a forward space smoothing covariance matrix RfComprises the following steps:
Figure BDA0002090690870000073
according to the characteristics of forward and backward space smooth subarray division, the intermediate variable matrix B is made into
Figure BDA0002090690870000074
The received signal Y of the 1 st subarray that is spatially smoothed backwards1(t) is expressed as:
Y1(t)＝BXL(t) (19)
wherein: xL(t) the received signal of the lth sub-array representing forward spatial smoothing;
for the ith subarray smoothed in the backward space, the received signal expression is:
Yi(t)＝BXL-i+1(t) (20)
wherein: xL-i+1(t) the received signal of the L-i +1 th subarray representing forward spatial smoothing;
received signal Y of ith sub-array smoothed to back spacei(t) autocovariance matrix Ri' is:
Figure BDA0002090690870000075
averaging the autocovariance matrix of each subarray receiving signal with backward space smoothing to obtain a backward space smoothing covariance matrix Rb：
Figure BDA0002090690870000076
The fourth concrete implementation mode is as follows: the third difference between the present embodiment and the specific embodiment is that: the specific process of the third step is as follows:
Figure BDA0002090690870000081
the fifth concrete implementation mode: the fourth difference between the present embodiment and the specific embodiment is that: the specific process of the step four is as follows:
to extract the cross-coupling coefficient matrix, an equivalent cross-coupling matrix C is assumedfbThen equation (23) is expressed as:
Figure BDA0002090690870000082
wherein the content of the first and second substances,
Figure BDA0002090690870000083
is a noise-induced spatial smoothing covariance matrix error;
smoothing the covariance matrix R in the forward and backward directionsfbWritten as in equation (25):
Figure BDA0002090690870000084
wherein: i' is 1,2, …, K, …, M, M stands for RfbThe number of eigenvalues of (d); lambda1,λ2,…,λK,λK+1,…,λMAre all RfbA characteristic value of (a), and1＞λ2＞…＞λK≥λK+1≥…≥λM，e1,e2,…,eK,eK+1,…,eMare each lambda1,λ2,…,λK,λK+1,…,λMA corresponding feature vector;
Λs＝diag{λ1,…,λK}，Λn＝diag{λK+1,…,λM}，Esa matrix of intermediate variables of K columns, EsColumn 1 of (A) as e1，EsColumn 2 of (A) is e2By analogy, EsIs listed as eK；EnIs an M-K column intermediate variable matrix, EnColumn 1 of (A) as eK+1By analogy, EnIs the Mth column of (e) as eM；
Assuming the known number of interference sources, C is used at high SNR and known mutual coupling coefficientfb(a(θ1))1,Cfb(a(θ2))1,…,Cfb(a(θK))1The spanned subspace is used as the signal subspace EsAccording to the MUSIC algorithm, EsFormed signal subspace and EnThe formed noise subspaces are orthogonal, then
Figure BDA0002090690870000085
Figure BDA0002090690870000086
Wherein, PMU(theta) is the spatial spectrum of the MUSIC algorithm, theta represents the azimuth range of the entire spatial spectrum,
Figure BDA0002090690870000091
every time theta equals thetakWhen the temperature of the water is higher than the set temperature,
Figure BDA0002090690870000092
is approximately 0, PMUThe peak of (θ) will coincide with the true DOA estimate. However, the device is not suitable for use in a kitchenAnd, due to the equivalent mutual coupling matrix CfbIs unknown, so the DOA estimation performance is affected when using the forward and backward smoothing MUSIC algorithm under non-mutual coupling conditions. Cfb(a(θ))1Represents CfbAnd (a (theta))1Multiplication.
The sixth specific implementation mode: the fifth embodiment is different from the fifth embodiment in that: the concrete process of the step five is as follows:
the embodiment introduces a reconstruction method of an equivalent cross-coupling matrix, which can improve the DOA estimation performance of the forward and backward smoothing MUSIC algorithm. Since the mutual coupling coefficient is considered to be zero when the array elements are far enough apart;
suppose that
Figure BDA0002090690870000093
Is an equivalent cross-coupling matrix CfbThe reconstruction result of (2) is the equivalent cross-coupling matrix CfbThe expression of (c) is:
Figure BDA0002090690870000094
Figure BDA0002090690870000095
is composed of a vector
Figure BDA0002090690870000096
Extended Toeplitz matrix, wherein
Figure BDA0002090690870000097
Figure BDA0002090690870000098
Are all elements in a vector, an
Figure BDA0002090690870000099
Figure BDA00020906908700000910
Is a vector
Figure BDA00020906908700000911
The number of non-zero mutual coupling coefficients; delta CfbError of the cross coupling coefficient matrix caused by the spatial smoothing technique; then the
Cfb(a(θ))1＝(Cfb+ΔCfb)(a(θ))1-ΔCfb(a(θ))1
Figure BDA00020906908700000912
Wherein: (a (theta))1＝[1 β(θ) … β(θ)M-1]T，
Figure BDA00020906908700000913
T[(a(θ))1]Is that
Figure BDA00020906908700000914
And T [ a (theta) ]1]The specific expression of (a) is as follows:
T[(a(θ))1]＝T1[(a(θ))1]+T2[(a(θ))1] (30)
intermediate variable T1[(a(θ))1]And T2[(a(θ))1]Are respectively:
Figure BDA00020906908700000915
Figure BDA00020906908700000916
wherein:
Figure BDA00020906908700000917
representative (a (theta))1I of (1)0+j0-a value of 1 element,
Figure BDA00020906908700000918
represents
Figure BDA0002090690870000101
Middle (i)0Line j (th)0The value of the element of the column,
Figure BDA0002090690870000102
representative (a (theta))1I of (1)0-j0A value of +1 elements of the number,
Figure BDA0002090690870000103
represent
Figure BDA0002090690870000104
Middle (i)0Line j (th)0The element values of the columns;
according to equation (26), assume that the obtained signal direction is θkK is 1,2, …, K, then
Figure BDA0002090690870000105
Wherein:
Figure BDA0002090690870000106
is the error of the feature vector caused by the spatial smoothing technique,
Figure BDA0002090690870000107
is the error of the feature vector caused by noise,
Figure BDA0002090690870000108
then
Figure BDA0002090690870000109
Figure BDA00020906908700001010
Wherein:
Figure BDA00020906908700001011
is a non-zero error caused by spatial smoothing techniques,
Figure BDA00020906908700001012
is a non-zero error caused by noise;
define the coefficient matrix Q as:
Figure BDA00020906908700001013
Figure BDA00020906908700001014
wherein: 0 is a zero vector, -qbiassmooth-qbiasnoise+qbiasprePartly the error vector, qbias, caused by noise and spatial smoothing techniquessmoothIs a variable quantity
Figure BDA00020906908700001015
Error vector of (2), qbiasnoiseIs a variable quantity
Figure BDA00020906908700001016
Error vector of (2), qbiaspreIs a variable quantity
Figure BDA00020906908700001017
The error vector of (2);
suppose that
Figure BDA00020906908700001020
And is
Figure BDA00020906908700001018
Figure BDA00020906908700001019
Then the least squares solution of equation (36)
Figure BDA0002090690870000111
The expression of (a) is:
Figure BDA0002090690870000112
wherein: ()#Represents a pseudo-inverse, due to
Figure BDA0002090690870000113
Will be provided with
Figure BDA0002090690870000114
Is recorded as a vector
Figure BDA0002090690870000115
Is estimated by
Figure BDA0002090690870000116
Is estimated by
Figure BDA0002090690870000117
The expression of (c) is:
Figure BDA0002090690870000118
Figure BDA0002090690870000119
wherein: qerror is an error vector; and the expression of the error vector qerror is:
Figure BDA00020906908700001110
and also
Figure BDA00020906908700001111
According to obtaining
Figure BDA00020906908700001112
Generating Toeplitz matrices
Figure BDA00020906908700001113
Will be provided with
Figure BDA00020906908700001114
As equivalent mutual coupling matrix CfbEstimate of reconstruction result
Figure BDA00020906908700001115
Obtaining a reconstructed equivalent mutual coupling matrix according to the formula (27)
Figure BDA00020906908700001116
The MUSIC algorithm space spectrum PC-MU(θ)：
Figure BDA00020906908700001117
The seventh embodiment: the sixth embodiment is different from the sixth embodiment in that: the concrete process of the sixth step is as follows:
step six, utilizing importance resampling function pair
Figure BDA00020906908700001118
Resampling to obtain new
Figure BDA00020906908700001119
Step six and two, new
Figure BDA00020906908700001120
Expansion into a new Toeplitz matrix
Figure BDA00020906908700001121
By using newToeplitz matrix
Figure BDA00020906908700001122
Obtaining a new MUSIC algorithm space spectrum
Figure BDA00020906908700001123
If the new MUSIC algorithm space spectrum
Figure BDA00020906908700001124
The MUSIC algorithm space spectrum P of the step five of the peak ratioC-MUPeak value of (theta) is high and
Figure BDA00020906908700001125
if the maximization constraint of formula (43) is satisfied, the new one will be
Figure BDA00020906908700001126
Is assigned to
Figure BDA00020906908700001127
Will be provided with
Figure BDA00020906908700001128
Assign to PC-MU(theta) and use of
Figure BDA00020906908700001129
Theta corresponding to the peak value ofkMean (theta)k) Updating is carried out;
if not, then,
Figure BDA00020906908700001130
PC-MU(theta) and mean (theta)k) Keeping the original shape;
the maximization constraint conditions are as follows:
Figure BDA0002090690870000121
mean(θk) MUSIC algorithm space spectrum P representing step fiveC-MUPeak value of (theta)Corresponding to thetakThe average value of delta theta and delta xi are both given thresholds; intermediate variables
Figure BDA0002090690870000122
Expressed as:
Figure BDA0002090690870000123
wherein:
Figure BDA0002090690870000124
is that
Figure BDA0002090690870000125
Taking the direction of the minimum value, K is 1,2, …, K;
sixthly, repeating the processes of the step six one and the step six two until the iteration times reach M, stopping iteration, and obtaining the final result
Figure BDA0002090690870000126
And
Figure BDA0002090690870000127
output according to the output
Figure BDA0002090690870000128
The peak value of (c) results in a final target signal DOA estimate.
In the second iteration, mean (θ) in the constraint is maximizedk) Is the first iteration's MUSIC algorithm spatial spectrum
Figure BDA0002090690870000129
Theta corresponding to the peak value ofkAnd step five, MUSIC algorithm space spectrum PC-MUTheta corresponding to peak value of (theta)kThe average value of (1) and so on;
in the second iteration, if the new space spectrum of MUSIC algorithm obtained in the first iteration is obtained
Figure BDA00020906908700001210
The MUSIC algorithm space spectrum P of the step fiveC-MUPeak value of (theta) is high and
Figure BDA00020906908700001211
satisfies the maximization constraint of equation (43), the new one obtained for the first iteration in the second iteration
Figure BDA00020906908700001212
Updating, and comparing the MUSIC algorithm spatial spectrum obtained by the second iteration with the MUSIC algorithm spatial spectrum obtained by the first iteration;
otherwise, for step five in the second iteration
Figure BDA00020906908700001213
Updating, and comparing the space spectrum of the MUSIC algorithm obtained by the second iteration with the space spectrum of the MUSIC algorithm in the fifth step
Stopping iteration until reaching the iteration times M, and obtaining the final product
Figure BDA00020906908700001214
And
Figure BDA00020906908700001215
output according to the output
Figure BDA00020906908700001216
The peak value of (c) results in a final target signal DOA estimate.
The derivation process of the maximization constraint condition is as follows:
equivalent cross coupling matrix CfbCan be expressed as the following minimization problem
Figure BDA0002090690870000131
Where K is 1,2, …, K. Order to
Figure BDA0002090690870000132
Equation (49) can be written as
Figure BDA0002090690870000133
From the formula (26), it is found that
Figure BDA0002090690870000134
At the same time order
Figure BDA0002090690870000135
Wherein
Figure BDA0002090690870000136
Is that
Figure BDA0002090690870000137
Taking the direction of the minimum value. Due to the target direction thetakIs unknown, so the minimization problem of equation (51) can be written as
Figure BDA0002090690870000138
Where Δ θ is a given threshold. To reduce the computational complexity of the formula (54) search, the reconstructed cross-coupling coefficient matrix is subjected to
Figure BDA0002090690870000141
Adding a limit condition to obtain a new optimization problem
Figure BDA0002090690870000142
Where ξ is the given threshold.
Albeit firstInformation checking target direction thetakUnknown, but the mean of multiple estimates can be used as the true target direction. Furthermore, it is assumed that the array output noise is Gaussian distributed, i.e. the error vector qerror follows a Gaussian distribution, resulting in a cross-coupling vector of references
Figure BDA0002090690870000143
So to reduce the amount of calculation, the above-obtained method can be utilized
Figure BDA0002090690870000144
To estimate an equivalent mutual coupling matrix Cfb. Further, the maximization problem of equation (55) can be written as
Figure BDA0002090690870000145
The specific implementation mode is eight: the seventh embodiment is different from the seventh embodiment in that: the expression of the importance resampling function is:
position resampling
Since the mutual coupling coefficients are arranged in descending order in the true mutual coupling coefficient matrix, the mutual coupling coefficient vector is initially estimated
Figure BDA0002090690870000146
The internal elements are not so arranged, so can be aligned with
Figure BDA0002090690870000147
Inner element rearrangement, i.e. position resampling.
Figure BDA0002090690870000151
Wherein:
Figure BDA0002090690870000152
represents the first
Figure BDA0002090690870000153
The number of sub-iterations is,
Figure BDA0002090690870000154
represents from
Figure BDA0002090690870000155
Before selection of
Figure BDA0002090690870000156
A set of the elements that are to be combined,
Figure BDA0002090690870000157
representing the selected front
Figure BDA0002090690870000158
The number of the elements is one,
Figure BDA0002090690870000159
is represented in
Figure BDA00020906908700001510
Before to be selected
Figure BDA00020906908700001511
The absolute values of the individual elements are arranged in descending order. After sorting the elements
Figure BDA00020906908700001512
As step six one
Figure BDA00020906908700001513
The specific implementation method nine: the seventh embodiment is different from the seventh embodiment in that: the expression of the importance resampling function is:
numerical resampling
Since the error vector qerror is gaussian distributed, the mutual coupling coefficient vector can be estimated at the beginning
Figure BDA00020906908700001514
Is added with a small elementThe random vector of (a) is fine-tuned in the hope of obtaining better results. This process is referred to as numerical resampling.
Figure BDA00020906908700001515
In the formula:
Figure BDA00020906908700001516
represents the first
Figure BDA00020906908700001517
The corresponding random vector is iterated a second time. Will obtain
Figure BDA00020906908700001518
As new step six one
Figure BDA00020906908700001519
The specific implementation mode is ten: the seventh embodiment is different from the seventh embodiment in that: the expression of the importance resampling function is:
location and value resampling
Based on the position resampling and the numerical resampling, a position and numerical resampling function is proposed that combines the two.
Figure BDA00020906908700001520
Wherein:
Figure BDA00020906908700001521
represents the first
Figure BDA00020906908700001522
The number of sub-iterations is,
Figure BDA00020906908700001523
represents the first
Figure BDA00020906908700001524
The random vector corresponding to the sub-iteration,
Figure BDA00020906908700001525
represents
Figure BDA00020906908700001526
Each element of (1) is respectively connected with
Figure BDA00020906908700001527
The corresponding elements in (A) are made and formed into a set, i.e.
Figure BDA00020906908700001528
To (1)
Figure BDA00020906908700001529
An element and
Figure BDA00020906908700001530
to (1)
Figure BDA0002090690870000161
The sum of the elements is made by the elements,
Figure BDA0002090690870000162
representing after making a sum
Figure BDA0002090690870000163
And (6) sorting. After sorting the elements
Figure BDA0002090690870000164
As step six one
Figure BDA0002090690870000165
The concrete implementation mode eleven: the seventh embodiment is different from the seventh embodiment in that: the expression of the importance resampling function is:
zero-setting resampling
Due to the distance between two array elementsWhen sufficiently far away, the mutual coupling coefficient is zero. While initially estimating the cross-coupling coefficient vector
Figure BDA0002090690870000166
Is non-zero, so can
Figure BDA0002090690870000167
Is zero in the hope of obtaining better results. This process is referred to as zero resampling.
Figure BDA0002090690870000168
Wherein:
Figure BDA0002090690870000169
represents the first
Figure BDA00020906908700001610
The number of sub-iterations is,
Figure BDA00020906908700001611
standing order
Figure BDA00020906908700001612
To (1)
Figure BDA00020906908700001613
Each element is 0. Order to
Figure BDA00020906908700001614
To (1)
Figure BDA00020906908700001615
After each element is 0, new
Figure BDA00020906908700001616
Will be new
Figure BDA00020906908700001617
As new step six one
Figure BDA00020906908700001618
Simulation analysis
According to the particle filtering algorithm, the importance resampling function is used for initial estimation
Figure BDA00020906908700001619
The process is performed to propose the importance resampling method framework in table 1 using the optimization problem of equation (43).
TABLE 1 importance resampling method framework
Figure BDA00020906908700001620
Figure BDA0002090690870000171
Consider a radar system whose receive antenna array is a uniform linear array of 10 identical antennas and the distance d between adjacent elements is half the wavelength. The signal noise is additive white gaussian noise. In simulation, two algorithms are selected for comparison with a MUSIC method based on importance resampling cross coupling coefficient matrix reconstruction, namely a MUSIC algorithm based on a spatial smoothing technology (FBS-MUSIC for short) and a MUSIC algorithm based on a formula (27) for cross coupling coefficient matrix reconstruction (CCE-FBS-MUSIC for short). According to the MUSIC algorithm based on the importance resampling cross coupling coefficient matrix reconstruction, a position resampling method is abbreviated as PR, a numerical value resampling method is abbreviated as NR, a numerical value and position resampling method is abbreviated as NPR, and a zero resetting resampling method is abbreviated as ZFR according to the difference of importance resampling functions. Six simulation experiments are carried out altogether, the number of snapshots of signal sampling is 512, the signal-to-noise ratio (SNR) variation range of the signal is-10 dB to 30dB, the step length is 0.4dB, and the DOA estimation process is repeated 10 times per step to obtain a plurality of estimation results. The simulation considers the conditions of single signal, multiple signals, different array element mutual coupling coefficients and the like.
In the first simulation, a single signal is considered, the direction of which is 0 °. For the array cross-coupling coefficient matrix C in equation (1), let C0＝1，c1＝0.63+j0.62，c2＝0.43+j0.43，c3＝0.25+j0.25，c4＝0.13+j0.13，c50.05+ j0.05, the remaining elements in the matrix are zero. The results of DOA estimation for PR, NR, NPR, ZFR, CCE-FBS-MUSIC, FBS-MUSIC are shown in FIGS. 2 to 6.
In a second simulation, three signals were considered, the directions of which were-10 °, 10 ° and 30 °. The array cross-coupling coefficient matrix C is the same as in the first simulation. The results of DOA estimation of PR, NR, NPR, ZFR, CCE-FBS-MUSIC, FBS-MUSIC are shown in FIGS. 8 to 13.
In a third simulation, a single signal is considered, the direction of which is 0 °. For the array cross-coupling coefficient matrix C in equation (1), let C0＝1，c1＝0.45+j0.45，c2＝0.27+j0.27，c3＝0.18+j0.18，c4＝0.1+j0.1，c50.03+ j0.03, the remaining elements in the matrix are zero. The DOA estimation results of PR, NRNPR, ZFR, CCE-FBS-MUSIC, FBS-MUSIC are shown in FIGS. 14 to 19.
In a fourth simulation, three signals were considered, the directions of the signals being-10 °, 10 ° and 30 °. The array cross-coupling coefficient matrix C is the same as in the third simulation. The results of DOA estimation of PR, NR, NPR, ZFR, CCE-FBS-MUSIC, FBS-MUSIC are shown in FIGS. 20 to 25.
In a fifth simulation, considering a single signal, the direction of the signal is varied, ranging from 20 ° to-20 °. The array cross-coupling coefficient matrix C is the same as in the first simulation. The results of DOA estimation of PR, NR, NPR, ZFR, CCE-FBS-MUSIC, FBS-MUSIC are shown in FIGS. 26 to 31.
In a sixth simulation, three signals were considered, the direction of which was varied over a range of (0 °, 20 °, 40 °) to (-20 °, 0 °, 20 °). The array cross-coupling coefficient matrix C is the same as in the first simulation. The DOA estimation results of PR, NR, NPR, ZFR, CCE-FBS-MUSIC, FBS-MUSIC are shown in FIGS. 32 to 37.
From the six simulation results, it can be seen that the magnitude of the array cross-coupling coefficient has a great influence on the DOA estimation result of the MUSIC algorithm. And the CCE-FBS-MUSIC algorithm for reconstructing the mutual coupling coefficient matrix is better than the DOA estimation performance of the FBS-MUSIC algorithm, the DOA estimation result of the MUSIC algorithm reconstructed based on the importance resampling mutual coupling coefficient matrix is more accurate than the CCE-FBS-MUSIC algorithm and the FBS-MUSIC algorithm, and the estimation result of the NPR algorithm is slightly better than the other three types of estimation results in the four importance resampling functions.
The above-described calculation examples of the present invention are merely to explain the calculation model and the calculation flow of the present invention in detail, and are not intended to limit the embodiments of the present invention. It will be apparent to those skilled in the art that other variations and modifications of the present invention can be made based on the above description, and it is not intended to be exhaustive or to limit the invention to the precise form disclosed, and all such modifications and variations are possible and contemplated as falling within the scope of the invention.

Claims (11)

1. An antenna mutual coupling correction method based on importance resampling under coherent source condition is characterized by comprising the following steps:
the method comprises the following steps: for a normalized linear array system comprising N array elements, calculating a receiving signal X (t) of the array system under a mutual coupling condition;
step two, processing the received signal X (t) of the array system by utilizing the space smoothing technology to obtain a forward space smoothing covariance matrix RfAnd backward spatial smoothing covariance matrix Rb；
Step three, utilizing a forward space smoothing covariance matrix RfAnd backward spatial smoothing covariance matrix RbComputing a forward and backward smoothed covariance matrix Rfb；
Step four: smoothing covariance matrix R for forward and backward directionsfbPerforming characteristic decomposition to obtain a noise subspace; forming spatial spectrum P of MUSIC algorithm using noise subspaceMU(θ) according to PMUPeak completion of (theta) to target signal DOAA first estimation;
step five: reconstructing the equivalent mutual coupling matrix by using the first estimated target signal DOA to obtain a reconstructed equivalent mutual coupling matrix
Figure FDA0003570136020000011
And equivalent cross-coupling matrix using reconstruction
Figure FDA0003570136020000012
Forming a MUSIC algorithm space spectrum;
step six: processing the reconstructed equivalent cross coupling matrix by using an importance resampling method to obtain a new equivalent cross coupling matrix; and forming a spatial spectrum of the MUSIC algorithm by using the new equivalent cross coupling matrix and the noise subspace to obtain the final target signal DOA estimation.
2. The method according to claim 1, wherein the first step comprises a specific process of:
for a normalized linear array system comprising N array elements, the distance between every two adjacent array elements is d;
then the mutual coupling coefficient between adjacent array elements is expressed by the Toeplitz matrix C as:
Figure FDA0003570136020000013
wherein: c. C1Representing the mutual coupling coefficient between the 1 st and 2 nd array elements, cN-2Representing the mutual coupling coefficient between the 1 st and the N-1 st array elements, cN-1Representing the mutual coupling coefficient between the 1 st array element and the Nth array element;
suppose that at time t, an incoming wave has K narrow-band coherent signal sources, which are respectively denoted as s1(t)， s2(t)，…，sK(t) the incoming wave directions of the K narrow-band coherent signal sources are theta1，θ2，…，θKAnd the wavelengths of the K narrow-band coherent signal sources are lambda;
each array element in the array system has independent and identically distributed variance of
Figure FDA0003570136020000021
The white gaussian noise, under the mutual coupling condition, the received signal x (t) of the array system is represented as:
X(t)＝CAS(t)+ξ(t) (2)
wherein:
X(t)＝[x1(t),x2(t),…,xN(t)]T (3)
x1(t) represents the received signal of the 1 st array element, x2(t) a received signal representing the 2 nd array element, xN(T) represents the received signal of the Nth array element, and the superscript T represents the transposition of the matrix;
the expressions for the intermediate variables S (t) and ξ (t) are:
S(t)＝[s1(t),s2(t),…,sK(t)]T (4)
ξ(t)＝[ξ1(t),ξ2(t),…,ξN(t)]T (5)
wherein: xin(t), N is 1,2, …, N is the variance of the independent homodistribution of
Figure FDA0003570136020000022
White gaussian noise of (1);
the expression for the intermediate variable a is:
A＝[a(θ1),a(θ2),…,a(θK)] (6)
wherein: the kth column vector a (θ) in the intermediate variable Ak) The expression of (a) is:
a(θk)＝[1 β(θk) … β(θk)N-1]T,k＝1,2,…,K (7)
Figure FDA0003570136020000023
β(θk) Is an intermediate variable, j is an imaginary unit, θkIs the incoming wave direction of the kth narrow-band coherent signal source.
3. The method according to claim 2, wherein the second step comprises the following specific steps:
for forward spatial smoothing, regarding a linear array system as L sub-arrays, if the number of array elements of each sub-array is M, starting from the first 1 st array element, the 1 st to mth array elements form the 1 st sub-array, the 2 nd to mth +1 array elements form the 2 nd sub-array, and so on, the N-M +1 th to nth array elements form the L-th sub-array, and then the number L of sub-arrays is equal to N-M + 1;
the received signal X of the 1 st subarray that is forward spatially smoothed1The expression of (t) is:
X1(t)＝C1(AS(t)+ξ(t)) (9)
wherein, X1(t) and the intermediate variable matrix C1Are respectively:
X1(t)＝[x1(t),x2(t),…,xM(t)]T (10)
Figure FDA0003570136020000031
received signal X of forward spatially smoothed 2 nd sub-array2The expression of (t) is:
X2(t)＝C2(AS(t)+ξ(t)) (12)
intermediate variable matrix C2The expression of (a) is:
Figure FDA0003570136020000032
then C1＝(C·E1)H，C2＝(C·E2)H，
Figure FDA0003570136020000033
Wherein: the superscript H represents the conjugate transpose of the matrix;
Figure FDA0003570136020000041
the received signal X of the ith forward spatially smoothed sub-arrayi(t) is:
Xi(t)＝(Di-1E1)HC(AS(t)+ξ(t)),i＝1,2,…,L (16)
wherein: di-1Represents i-1 multiplications by D;
received signal X of ith forward spatially smoothed sub-arrayi(t) autocovariance matrix RiComprises the following steps:
Figure FDA0003570136020000042
wherein: i is a unit matrix, an intermediate variable RS＝E[S(t)SH(t)]，E[·]Represents a desire;
averaging the autocovariance matrix of each subarray receiving signal of the forward space smoothing to obtain a forward space smoothing covariance matrix RfComprises the following steps:
Figure FDA0003570136020000043
let the intermediate variable matrix B be
Figure FDA0003570136020000044
Then it is empty to the rearReceived signal Y of 1 st subarray with smooth interval1(t) is expressed as:
Y1(t)＝BXL(t) (19)
wherein: xL(t) the received signal of the lth sub-array representing forward spatial smoothing;
for the ith subarray smoothed in the backward space, the received signal expression is:
Yi(t)＝BXL-i+1(t) (20)
wherein: xL-i+1(t) the received signal of the L-i +1 th subarray representing forward spatial smoothing;
received signal Y of ith sub-array smoothed to back spacei(t) autocovariance matrix Ri' is:
Figure FDA0003570136020000045
averaging the autocovariance matrix of each subarray receiving signal with backward space smoothing to obtain a backward space smoothing covariance matrix Rb：
Figure FDA0003570136020000051
。
4. The method for correcting mutual coupling of antennas based on importance resampling under coherent source conditions according to claim 3, wherein the specific process of the third step is as follows:
Figure FDA0003570136020000052
。
5. the method for correcting mutual coupling of antennas based on importance resampling under coherent source conditions according to claim 4, wherein the specific process of step four is as follows:
suppose an equivalent cross-coupling matrix CfbThen equation (23) is expressed as:
Figure FDA0003570136020000053
wherein the content of the first and second substances,
Figure FDA0003570136020000054
is a noise-induced spatial smoothing covariance matrix error;
smoothing the covariance matrix R in the forward and backward directionsfbWritten as in equation (25):
Figure FDA0003570136020000055
wherein: i ═ 1,2, …, K, …, M' represents RfbThe number of eigenvalues of (d); lambda1,λ2,…,λK,λK+1,…,λM′Are all RfbA characteristic value of (a), and1＞λ2＞…＞λK≥λK+1≥…≥λM′，e1,e2,…,eK,eK+1,…,eM′are each lambda1,λ2,…,λK,λK+1,…,λM′A corresponding feature vector;
Λs＝diag{λ1,…,λK}，Λn＝diag{λK+1,…,λM′}，Esis a K-column intermediate variable matrix, EsRepresenting a signal subspace, EsColumn 1 of (A) as e1，EsColumn 2 of (A) is e2By analogy, EsIs listed as eK；EnIs an intermediate variable matrix of M' -K columns, EnRepresenting a noise subspace, EnColumn 1 of (A) as eK+1By analogy, EnIs M' -K ofM′；
By Cfb(a(θ1))1,Cfb(a(θ2))1,…,Cfb(a(θK))1The spanned subspace is used as the signal subspace EsEstimation of (2), signal subspace EsAnd noise subspace EnIs orthogonal to
Figure FDA0003570136020000061
Figure FDA0003570136020000062
Wherein, PMU(θ) is the spatial spectrum of the MUSIC algorithm, θ represents the azimuth range of the entire spatial spectrum, (a (θ)k))1＝[1 β(θk) … β(θk)M-1]T，(a(θ))1＝[1 β(θ) … β(θ)M-1]T，
Figure FDA0003570136020000063
Cfb(a(θk))1Represents CfbAnd (a (theta)k))1Multiplication of Cfb(a(θ))1Represents CfbAnd (a (theta))1Multiplication.
6. The method for correcting mutual coupling of antennas based on importance resampling under coherent source conditions according to claim 5, wherein the concrete process of the fifth step is as follows:
suppose that
Figure FDA00035701360200000617
Is an equivalent cross-coupling matrix CfbThe reconstruction result of (2) is the equivalent cross coupling matrix CfbThe expression of (a) is:
Figure FDA0003570136020000064
Figure FDA0003570136020000065
is composed of a vector
Figure FDA0003570136020000066
Extended Toeplitz matrix, wherein
Figure FDA0003570136020000067
Figure FDA0003570136020000068
Are all elements in a vector, and
Figure FDA0003570136020000069
Figure FDA00035701360200000610
is a vector
Figure FDA00035701360200000611
The number of non-zero mutual coupling coefficients; delta CfbError of the cross coupling coefficient matrix caused by the spatial smoothing technique; then
Figure FDA00035701360200000612
Wherein: (a (theta))1＝[1 β(θ) … β(θ)M-1]T，
Figure FDA00035701360200000613
T[(a(θ))1]Is that
Figure FDA00035701360200000614
And T [ a (theta) ]1]The specific expression of (a) is as follows:
T[(a(θ))1]＝T1[(a(θ))1]+T2[(a(θ))1] (30)
intermediate variable T1[(a(θ))1]And T2[(a(θ))1]Are respectively:
Figure FDA00035701360200000615
Figure FDA00035701360200000616
wherein:
Figure FDA0003570136020000071
representative (a (theta))1I of (1)0+j0-a value of 1 element,
Figure FDA0003570136020000072
represents T1[(a(θ))1]Middle (i)0Line j (th)0The value of the element of the column,
Figure FDA0003570136020000073
representative (a (theta))1I of (1)0-j0A value of +1 elements of the number,
Figure FDA0003570136020000074
represents T2[(a(θ))1]Middle (i)0Line j (th)0The element values of the columns;
according to the formula (26), the obtained incoming wave direction of the kth narrow-band coherent signal source is assumed to be thetakK is 1,2, …, K, then
Figure FDA0003570136020000075
Wherein:
Figure FDA0003570136020000076
is the error of the feature vector caused by the spatial smoothing technique,
Figure FDA0003570136020000077
is the error of the feature vector caused by noise,
Figure FDA0003570136020000078
then
Figure FDA0003570136020000079
Wherein:
Figure FDA00035701360200000710
is a non-zero error caused by spatial smoothing techniques,
Figure FDA00035701360200000711
is a non-zero error caused by noise;
define the coefficient matrix Q as:
Figure FDA00035701360200000712
Figure FDA00035701360200000713
wherein: 0 is a zero vector, -qbiassmooth-qbiasnoise+qbiasprePartly the error vector, qbias, caused by noise and spatial smoothing techniquessmoothIs a variable quantity
Figure FDA00035701360200000714
Error vector of (2), qbiasnoiseIs a variable quantity
Figure FDA00035701360200000715
Error vector of (2), qbiaspreIs a variable quantity
Figure FDA00035701360200000716
The error vector of (2);
suppose that
Figure FDA00035701360200000721
And is
Figure FDA00035701360200000717
Figure FDA00035701360200000718
Then the least squares solution of equation (36)
Figure FDA00035701360200000719
The expression of (a) is:
Figure FDA00035701360200000720
wherein: ()#Represents a pseudo-inverse, due to
Figure FDA0003570136020000081
Will be provided with
Figure FDA0003570136020000082
Is recorded as a vector
Figure FDA0003570136020000083
Is estimated by
Figure FDA0003570136020000084
Is estimated by
Figure FDA0003570136020000085
The expression of (a) is:
Figure FDA0003570136020000086
wherein: qerror is an error vector; and the expression of the error vector qerror is:
Figure FDA0003570136020000087
and also
Figure FDA0003570136020000088
According to obtaining
Figure FDA0003570136020000089
Generating Toeplitz matrices
Figure FDA00035701360200000810
Will be provided with
Figure FDA00035701360200000811
As equivalent mutual coupling matrix CfbReconstructed result of (2)
Figure FDA00035701360200000812
Obtaining a reconstructed equivalent mutual coupling matrix according to the formula (27)
Figure FDA00035701360200000813
The MUSIC algorithm space spectrum PC-MU(θ)：
Figure FDA00035701360200000814
7. The method for correcting mutual coupling of antennas based on importance resampling under coherent source conditions according to claim 6, wherein the specific process of the sixth step is as follows:
step six, utilizing importance resampling function pair
Figure FDA00035701360200000815
Resampling to obtain new
Figure FDA00035701360200000816
Step six and two, new
Figure FDA00035701360200000817
Expansion into a new Toeplitz matrix
Figure FDA00035701360200000818
Using a new Toeplitz matrix
Figure FDA00035701360200000819
Obtaining a new MUSIC algorithm space spectrum
Figure FDA00035701360200000820
If the new MUSIC algorithm space spectrum
Figure FDA00035701360200000821
The MUSIC algorithm space spectrum P of the step fiveC-MUPeak value of (theta) is high and
Figure FDA00035701360200000822
if the maximization constraint of formula (43) is satisfied, the new one will be
Figure FDA00035701360200000823
Is assigned to
Figure FDA00035701360200000824
Will be provided with
Figure FDA00035701360200000825
Assign to PC-MU(theta) and use of
Figure FDA00035701360200000826
Theta corresponding to the peak value ofkMean (theta)k) Updating is carried out;
if not, then,
Figure FDA00035701360200000827
PC-MU(theta) and mean (theta)k) Keeping the original shape;
the maximization constraint conditions are as follows:
Figure FDA00035701360200000828
Figure FDA00035701360200000829
mean(θk) Space spectrum P of MUSIC algorithm representing step fiveC-MUTheta corresponding to peak value of (theta)kThe average value of delta theta and delta xi are both given thresholds; intermediate variables
Figure FDA0003570136020000091
Expressed as:
Figure FDA0003570136020000092
wherein:
Figure FDA0003570136020000093
is that
Figure FDA0003570136020000094
In the direction of the minima, K is 1,2, …, K,
Figure FDA0003570136020000095
sixthly, repeating the processes of the step six one and the step six two until the iteration times reach M, stopping iteration, and obtaining the final result
Figure FDA0003570136020000096
And
Figure FDA0003570136020000097
output according to the output
Figure FDA0003570136020000098
The peak value of (c) results in a final target signal DOA estimate.
8. The method according to claim 7, wherein the significance resampling-based antenna mutual coupling correction method under coherent source conditions is represented by the following expression:
Figure FDA0003570136020000099
wherein:
Figure FDA00035701360200000910
represents the first
Figure FDA00035701360200000911
The number of sub-iterations is,
Figure FDA00035701360200000912
represents from
Figure FDA00035701360200000913
Before selection of
Figure FDA00035701360200000914
A set of the elements that are to be combined,
Figure FDA00035701360200000915
representing the selected front
Figure FDA00035701360200000916
The number of the elements is one,
Figure FDA00035701360200000917
is represented in
Figure FDA00035701360200000918
Before to be selected
Figure FDA00035701360200000919
The absolute values of the individual elements are arranged in descending order.
9. The method according to claim 7, wherein the significance resampling-based antenna mutual coupling correction method under coherent source conditions is represented by the following expression:
Figure FDA00035701360200000920
in the formula:
Figure FDA00035701360200000923
represents the first
Figure FDA00035701360200000921
The corresponding random vector is iterated a second time.
10. The method according to claim 7, wherein the significance resampling-based antenna mutual coupling correction method under coherent source conditions is represented by the following expression:
Figure FDA00035701360200000922
wherein:
Figure FDA0003570136020000101
represents the first
Figure FDA0003570136020000102
The number of sub-iterations is,
Figure FDA0003570136020000103
represents the first
Figure FDA0003570136020000104
The random vector corresponding to the sub-iteration,
Figure FDA0003570136020000105
represents
Figure FDA0003570136020000106
Each element of (1) is respectively connected with
Figure FDA0003570136020000107
The corresponding elements in the group are combined to form a set,
Figure FDA0003570136020000108
representing after making a sum
Figure FDA0003570136020000109
And (6) sorting.
11. The method according to claim 7, wherein the significance resampling-based antenna mutual coupling correction method under coherent source conditions is represented by the following expression:
Figure FDA00035701360200001010
wherein:
Figure FDA00035701360200001011
represents the first
Figure FDA00035701360200001012
The number of sub-iterations is,
Figure FDA00035701360200001013
standing order
Figure FDA00035701360200001014
To (1)
Figure FDA00035701360200001015
Each element is 0.
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