CN113466899A - Navigation receiver beam forming method based on small fast beat number under high signal-to-noise ratio environment - Google Patents

Navigation receiver beam forming method based on small fast beat number under high signal-to-noise ratio environment Download PDF

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CN113466899A
CN113466899A CN202110927965.5A CN202110927965A CN113466899A CN 113466899 A CN113466899 A CN 113466899A CN 202110927965 A CN202110927965 A CN 202110927965A CN 113466899 A CN113466899 A CN 113466899A
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signal
covariance matrix
noise
interference
array
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CN113466899B (en
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滕云龙
郑植
元硕成
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University of Electronic Science and Technology of China
Yangtze River Delta Research Institute of UESTC Huzhou
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Yangtze River Delta Research Institute of UESTC Huzhou
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • 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
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/01—Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/13—Receivers
    • G01S19/21—Interference related issues ; Issues related to cross-correlation, spoofing or other methods of denial of service
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • 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
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/01—Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/13—Receivers
    • G01S19/35—Constructional details or hardware or software details of the signal processing chain
    • 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
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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  • Radar, Positioning & Navigation (AREA)
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Abstract

The invention discloses a navigation receiver beam forming method based on small fast beat number under the environment of high signal-to-noise ratio, which comprises the following steps: s1, calculating a received signal; s2, establishing an optimization target, and solving to obtain an optimal weight vector calculation formula; and S3, acquiring an accurate interference and noise covariance matrix, and substituting the accurate interference and noise covariance matrix into the optimal weight vector calculation formula obtained in S2. The method firstly removes the components of the expected signal (satellite signal) in the received signal of the navigation receiver array, so that the algorithm is suitable for the environment with high signal-to-noise ratio (practical application occasion), and then improves the accuracy of the covariance matrix under small snapshot data by using the covariance matrix estimation method, and finally obtains the beam forming algorithm which can effectively improve the performance of the navigation receiver in the practical application scene.

Description

Navigation receiver beam forming method based on small fast beat number under high signal-to-noise ratio environment
Technical Field
The invention belongs to the technical field of digital signal processing, and particularly relates to a navigation receiver beam forming method based on small fast beat number in a high signal-to-noise ratio environment of a satellite navigation receiver.
Background
With the formal network access of the last networking satellite of the third Beidou satellite, China comprehensively completes the constellation deployment of the Beidou global satellite navigation system, and the three-step walking strategy of the Beidou system is realized. In addition, the satellite navigation systems that are currently successfully put into operation include GPS in the united states, GLONASS in russia, Galileo in the european union, and a part of regional satellite navigation systems. Because the distance between the satellite and the ground receiver is far, the signal reaching the ground is very weak, and although the satellite navigation signal is transmitted in a spread spectrum mode, the receiver can obtain spread spectrum gain, so that the satellite navigation signal is not covered in noise, but is still easily influenced by an interference signal, and the receiver cannot normally work. Because the core of the navigation receiver is an antenna array, the application of the beam forming algorithm based on the array signal processing can enhance the navigation signal and simultaneously suppress the interference signal.
The beamforming algorithms are mostly based on Capon's criterion, with the goal of minimizing the power of the output signal while the desired signal passes through the beamformer undistorted, thereby achieving a high signal-to-noise ratio of the output signal. However, the direct application of this method to a navigation receiver has the following problems: firstly, the navigation receiver has higher requirement on the real-time performance of signal processing, so that the acquired snapshot number is less, the difference between a sample value and a true value of a signal covariance matrix is larger, and the performance of a beam forming algorithm is reduced.
At present, the covariance matrix estimation method can effectively estimate a signal covariance matrix under the condition of small snapshot, and the performance of a beam forming algorithm is improved. However, such methods are only suitable for low signal-to-noise ratio environments, and in practical situations, the power of the pilot signal after spreading is often higher than that of noise. In addition, the method for effectively removing the expected signal components in the covariance matrix does not always consider the situation based on small snapshot data, the navigation receiver has high real-time requirement on signal reception, and the acquired snapshot data is very limited. Therefore, the research is suitable for the wave beam forming algorithm based on the small fast beat number under the environment with high signal-to-noise ratio, and has important significance for improving the performance of the navigation receiver.
Disclosure of Invention
The invention aims to solve the problem that the performance of a navigation receiver is reduced when the snapshot number is limited in the high signal-to-noise ratio environment in the prior art, and provides a method for forming a navigation receiver beam based on the small snapshot number, which can improve the accuracy of a covariance matrix under small snapshot data by effectively combining an expected signal removing method and a covariance matrix estimation method, and finally obtain the navigation receiver beam forming method based on the small snapshot number in the high signal-to-noise ratio environment which can effectively improve the performance of the navigation receiver in an actual application scene.
The purpose of the invention is realized by the following technical scheme: a navigation receiver beam forming method based on small fast beat number under high signal-to-noise ratio environment includes the following steps:
s1, calculating a received signal;
s2, establishing an optimization target, and solving to obtain an optimal weight vector calculation formula;
and S3, acquiring an accurate interference and noise covariance matrix, and substituting the accurate interference and noise covariance matrix into the optimal weight vector calculation formula obtained in S2.
Further, the specific implementation method of S1 is as follows: for a uniform linear array with M array elements, assuming that signals received by the uniform linear array are narrow-band far-field signals, the signals are uncorrelated with each other, noise received by each array element is complex gaussian white noise, and the noise is also uncorrelated with the signals, at time t, the received signals of the array are expressed as:
x(t)＝xs(t)+xi(t)+xn(t) (1)
wherein ,xs(t)＝ss(t)as，xi(t)＝si(t)ai，xs(t)、xi(t)、xn(t) respectively representing a desired signal, an interference signal and noise, s (t) representing the waveform of the signal, a representing a guide vector of a corresponding signal, and for a uniform linear array with a known array structure, the guide vector of an array receiving signal is represented as follows according to a signal incoming direction theta:
Figure BDA0003209852640000021
where λ represents the signal wavelength and d represents the distance between adjacent array elements.
Further, the specific implementation method of S2 is as follows: passing complex weight vector w ═ w1,w2,…,wM]TAfter weighting the array receiving signals, the performance of the beam former is represented by the output signal to interference plus noise ratio (SINR):
Figure BDA0003209852640000022
wherein
Figure BDA0003209852640000023
To the desired signal power, Ri+nIs an interference plus noise covariance matrix;
in order to maximize the output signal-to-interference-and-noise ratio, according to the Capon criterion, the optimization target is obtained as follows:
Figure BDA0003209852640000024
subject to wHas＝1
solving to obtain the optimal weight vector of
Figure BDA0003209852640000025
Further, the specific implementation method of S3 is as follows: and removing a component of a desired signal in the array receiving signal by adopting a subspace projection method, and acquiring spatial information of an interference region by integrating an angle region where the interference signal is located because the array structure is known:
Figure BDA0003209852640000031
Θirepresenting a spatial angular region in which the interference signal is located;
carrying out characteristic value decomposition on psi, sorting characteristic values, selecting a characteristic vector corresponding to the maximum characteristic value to form a subspace P1By the use of P1Orthogonality of steering vectors with non-interfering signal regions, using projection matrices
Figure BDA0003209852640000032
The desired signal in the array receiving signal is effectively eliminated while the interference signal is not changed; the projected signal is represented as:
Figure BDA0003209852640000033
the sample covariance matrix for the desired signal component is expressed as:
Figure BDA0003209852640000034
k is the number of snapshots of the signal sample;
the processed sample covariance matrix is not an interference plus noise covariance matrix, and is specifically configured as an interference signal covariance matrix
Figure BDA0003209852640000035
And a part for processing noise
Figure BDA0003209852640000036
By adopting a covariance matrix estimation method with structural constraint, correcting a covariance matrix while improving the accuracy of the covariance matrix under small snapshot data to obtain an interference-plus-noise covariance matrix;
for a uniform linear array, a signal covariance matrix is a Toeplitz matrix, and the form of the Toeplitz structure covariance matrix is as follows:
Figure BDA0003209852640000037
wherein R represents a signal covariance matrix,
Figure BDA0003209852640000038
representative of the noise power, σ2As a lower boundary constraint for noise power, κMThe condition number is used as the upper boundary of the condition number of the covariance matrix estimated value and is used for ensuring that the covariance matrix estimated value is nonsingular; i is a unit matrix, λmax(M) is the maximum eigenvalue of M, λmin(M) is the minimum eigenvalue of M, TMThe representation matrix M has a Toeplitz structure;
the covariance matrix is obtained as the optimal estimated value by the following optimization problem:
Figure BDA0003209852640000041
Figure BDA0003209852640000042
representing an optimal estimate of the signal covariance matrix;
by (10), a more accurate interference plus noise covariance matrix is obtained by signal covariance matrix estimation under small snapshots; converting (10) further into the following form:
Figure BDA0003209852640000043
wherein ,
Figure BDA0003209852640000044
Figure BDA0003209852640000045
Figure BDA0003209852640000046
HMrepresents the Hermite matrix; a isiIs a set of complex, fitting matrices JiEnsuring that the matrix estimation value is a Toeplitz matrix; the optimization problem thus turns to finding a solution that lies in the convex set C1 and C2Distance at intersection
Figure BDA0003209852640000047
The nearest point M; solving by adopting a Dykstra projection algorithm, wherein a specific solving process is to project an objective function to two convex sets in sequence, and obtain an optimal solution of an optimization problem after respectively solving the optimization problem positioned in the two convex sets;
first, for convex set C1Obtaining:
Figure BDA0003209852640000048
the characteristic value decomposition is carried out on the Y,
Figure BDA0003209852640000049
and descending and arranging the characteristic values to obtain:
ΛY＝diag([d1,d2,…,dM]T) (16)
the optimal solution of equation (15) is expressed as:
Figure BDA0003209852640000051
wherein ,
Λ(u*)＝diag(λ*(u*))(18)
λ*(u*)＝[λ1(u),λ2(u),...,λM(u)]T(19)
λi(u)＝min(κMu,max(di,max(1,u))),i＝1,...M (20)
the choice of u involves an optimization problem:
Figure BDA0003209852640000052
wherein G (u) ═ max { hi(u)}；
When d isiWhen is greater than 1
Figure BDA0003209852640000053
When d isiWhen the temperature is less than or equal to 1
Figure BDA0003209852640000054
The optimal solution for equation (21) is:
Figure BDA0003209852640000061
for convex set C2Obtaining:
Figure BDA0003209852640000062
the optimal solution for equation (25) is:
Figure BDA0003209852640000063
and obtaining the optimal solution of covariance matrix estimation under the small snapshot data by iterative solution of the two convex set optimization problems, wherein the covariance matrix does not contain an expected signal component, and substituting the weight vector to obtain the optimal weight vector based on the small snapshot data under the high signal-to-noise ratio environment.
The invention has the beneficial effects that: the method firstly removes the components of the expected signal (satellite signal) in the received signal of the navigation receiver array, so that the algorithm is suitable for the environment with high signal-to-noise ratio (practical application occasion), and then improves the accuracy of the covariance matrix under small snapshot data by using the covariance matrix estimation method, and finally obtains the beam forming algorithm which can effectively improve the performance of the navigation receiver in the practical application scene.
Detailed Description
The technical solution of the present invention is further explained below.
The invention discloses a navigation receiver beam forming method based on small fast beat number under the environment of high signal-to-noise ratio, which comprises the following steps:
s1, calculating a received signal; the specific implementation method comprises the following steps: for a Uniform Linear Array (ULA) with M array elements, assuming that signals received by the ULA are narrow-band far-field signals, the signals are uncorrelated with each other, noise received by each array element is complex white gaussian noise, and the noise and the signals are also uncorrelated with each other, at time t, the received signals of the array are expressed as:
x(t)＝xs(t)+xi(t)+xn(t) (1)
wherein ,xs(t)＝ss(t)as，xi(t)＝si(t)ai，xs(t)、xi(t)、xn(t) respectively representing a desired signal, an interference signal and noise, s (t) representing the waveform of the signal, a representing a guide vector of a corresponding signal, and for a uniform linear array with a known array structure, the guide vector of an array receiving signal is represented as follows according to a signal incoming direction theta:
Figure BDA0003209852640000071
where λ represents the signal wavelength and d represents the distance between adjacent array elements.
S2, establishing an optimization target, and solving to obtain an optimal weight vector calculation formula; the specific implementation method comprises the following steps: passing complex weight vector w ═ w1,w2,…,wM]TAfter weighting the array receiving signals, the performance of the beam former is represented by the output signal to interference plus noise ratio (SINR):
Figure BDA0003209852640000072
wherein
Figure BDA0003209852640000073
To the desired signal power, Ri+nIs an interference plus noise covariance matrix;
in order to maximize the output signal-to-interference-and-noise ratio, according to the Capon criterion, the optimization target is obtained as follows:
Figure BDA0003209852640000074
subject to wHas＝1
solving to obtain the optimal weight vector of
Figure BDA0003209852640000075
It can be seen that on the premise that the desired signal steering vector is known, it is critical to obtain an accurate interference plus noise covariance matrix.
S3, acquiring an accurate interference and noise covariance matrix, and substituting the accurate interference and noise covariance matrix into the optimal weight vector calculation formula obtained in S2; the specific implementation method comprises the following steps: and removing a component of a desired signal in the array receiving signal by adopting a subspace projection method, and acquiring spatial information of an interference region by integrating an angle region where the interference signal is located because the array structure is known:
Figure BDA0003209852640000076
Θirepresenting a spatial angular region in which the interference signal is located;
carrying out characteristic value decomposition on psi, sorting characteristic values, selecting a characteristic vector corresponding to the maximum characteristic value to form a subspace P1By the use of P1Orthogonality of steering vectors with non-interfering signal regions, using projection matrices
Figure BDA0003209852640000077
The desired signal in the array receiving signal is effectively eliminated while the interference signal is not changed; the projected signal is represented as:
Figure BDA0003209852640000081
the sample covariance matrix for the desired signal component is expressed as:
Figure BDA0003209852640000082
k is the number of snapshots of the signal sample;
because the exact values of the sample covariance matrix and the signal covariance matrix differ significantly under small snapshots, the beamformer performance may still degrade significantly even if the desired signal components are eliminated. The processed sample covariance matrix is not an interference plus noise covariance matrix, and is specifically configured as an interference signal covariance matrix
Figure BDA0003209852640000083
And a part for processing noise
Figure BDA0003209852640000084
Therefore, the covariance matrix estimation method with structural constraint is adopted, the covariance matrix accuracy under small snapshot data is improved, and meanwhile the covariance matrix is corrected, so that an interference and noise covariance matrix is obtained;
for a uniform linear array, a signal covariance matrix is a Toeplitz matrix, and the form of the Toeplitz structure covariance matrix is as follows:
Figure BDA0003209852640000085
m represents a matrix whose specific structure/expression is the first equation, and the other conditions are all pairsIts constraints; wherein R represents a signal covariance matrix,
Figure BDA0003209852640000086
representative of the noise power, σ2As a lower boundary constraint for noise power, κMThe condition number is used as the upper boundary of the condition number of the covariance matrix estimated value and is used for ensuring that the covariance matrix estimated value is nonsingular; i is a unit matrix, λmax(M) is the maximum eigenvalue of M, λmin(M) is the minimum eigenvalue of M, TMThe representation matrix M has a Toeplitz structure;
the covariance matrix is obtained as the optimal estimated value by the following optimization problem:
Figure BDA0003209852640000091
Figure BDA0003209852640000092
representing an optimal estimate of the signal covariance matrix;
the specific idea is that under the structural constraint of a Toeplitz covariance matrix, the Frobenius norm of the difference value between the estimated value of the matrix and the sample covariance matrix is minimum, so that the estimated value is ensured to be close to the sample covariance matrix to the maximum extent while having a signal covariance matrix structure. The components of the desired signal in the sample covariance matrix have been eliminated due to the preamble processing, and the component structure of the matrix estimate is constrained in constraints to be the sum of the signal covariance matrix and the ideal noise covariance matrix. Therefore, through (10), a more accurate interference plus noise covariance matrix can be obtained through the sample covariance matrix estimation under the condition of small snapshot, so that the method is suitable for a high signal-to-noise ratio environment.
Converting (10) further into the following form:
Figure BDA0003209852640000093
wherein ,
Figure BDA0003209852640000094
Figure BDA0003209852640000095
Figure BDA0003209852640000096
HMrepresents the Hermitian Matrix (Hermitian Matrix); a isiIs a set of complex, fitting matrices JiEnsuring that the matrix estimation value is a Toeplitz matrix; the optimization problem thus turns to finding a solution that lies in the convex set C1 and C2Distance at intersection
Figure BDA0003209852640000097
The nearest point M; solving by adopting a Dykstra projection algorithm, wherein a specific solving process is to project an objective function to two convex sets in sequence, and obtain an optimal solution of an optimization problem after respectively solving the optimization problem positioned in the two convex sets;
first, for convex set C1Obtaining:
Figure BDA0003209852640000101
carrying out characteristic value decomposition on Y (Y has no specific meaning and is an independent variable used for representing the solution of the optimization problem),
Figure BDA0003209852640000102
and descending and arranging the characteristic values to obtain:
ΛY＝diag([d1,d2,...,dM]T) (16)
the optimal solution of equation (15) is expressed as:
Figure BDA0003209852640000103
wherein ,
Λ(u*)＝diag(λ*(u*)) (18)
λ*(u*)＝[λ1(u),λ2(u),…,λM(u)]T (19)
λi(u)＝min(κMu,max(di,max(1,u))),i＝1,...M (20)
the choice of u involves an optimization problem:
Figure BDA0003209852640000104
wherein G (u) ═ max { hi(u)}；
When d isiWhen is greater than 1
Figure BDA0003209852640000105
When d isiWhen the temperature is less than or equal to 1
Figure BDA0003209852640000106
The optimal solution for equation (21) is:
Figure BDA0003209852640000111
for convex set C2Obtaining:
Figure BDA0003209852640000112
the optimal solution for equation (25) is:
Figure BDA0003209852640000113
and obtaining the optimal solution of covariance matrix estimation under the small snapshot data by iterative solution of the two convex set optimization problems, wherein the covariance matrix does not contain an expected signal component, and substituting the weight vector to obtain the optimal weight vector based on the small snapshot data under the high signal-to-noise ratio environment.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (4)

1. The navigation receiver beam forming method based on the small fast beat number under the environment with high signal-to-noise ratio is characterized by comprising the following steps:
s1, calculating a received signal;
s2, establishing an optimization target, and solving to obtain an optimal weight vector calculation formula;
and S3, acquiring an accurate interference and noise covariance matrix, and substituting the accurate interference and noise covariance matrix into the optimal weight vector calculation formula obtained in S2.
2. The method as claimed in claim 1, wherein the S1 is implemented by using a method comprising: for a uniform linear array with M array elements, assuming that signals received by the uniform linear array are narrow-band far-field signals, the signals are uncorrelated with each other, noise received by each array element is complex gaussian white noise, and the noise is also uncorrelated with the signals, at time t, the received signals of the array are expressed as:
x(t)＝xs(t)+xi(t)+xn(t) (1)
wherein ,xs(t)＝ss(t)as，xi(t)＝si(t)ai，xs(t)、xi(t)、xn(t) respectively representing a desired signal, an interference signal and noise, s (t) representing the waveform of the signal, a representing a guide vector of a corresponding signal, and for a uniform linear array with a known array structure, the guide vector of an array receiving signal is represented as follows according to a signal incoming direction theta:
Figure FDA0003209852630000011
where λ represents the signal wavelength and d represents the distance between adjacent array elements.
3. The method as claimed in claim 2, wherein the S2 is implemented by: passing complex weight vector w ═ w1,w2,…,wM]TAfter weighting the array receiving signals, the performance of the beam former is represented by the output signal to interference plus noise ratio (SINR):
Figure FDA0003209852630000012
wherein
Figure FDA0003209852630000013
To the desired signal power, Ri+nIs an interference plus noise covariance matrix;
in order to maximize the output signal-to-interference-and-noise ratio, according to the Capon criterion, the optimization target is obtained as follows:
Figure FDA0003209852630000014
subject to wHas＝1
solving to obtain the optimal weight vector of
Figure FDA0003209852630000015
4. The method as claimed in claim 3, wherein the S3 is embodied as: and removing a component of a desired signal in the array receiving signal by adopting a subspace projection method, and acquiring spatial information of an interference region by integrating an angle region where the interference signal is located because the array structure is known:
Figure FDA0003209852630000021
Θirepresenting a spatial angular region in which the interference signal is located;
carrying out characteristic value decomposition on psi, sorting characteristic values, selecting a characteristic vector corresponding to the maximum characteristic value to form a subspace P1By the use of P1Orthogonality of steering vectors with non-interfering signal regions, using projection matrices
Figure FDA0003209852630000022
The desired signal in the array receiving signal is effectively eliminated while the interference signal is not changed; the projected signal is represented as:
Figure FDA0003209852630000023
Figure FDA0003209852630000024
Figure FDA0003209852630000025
the sample covariance matrix for the desired signal component is expressed as:
Figure FDA0003209852630000026
Figure FDA0003209852630000027
k is the number of snapshots of the signal sample;
the processed sample covariance matrix is not an interference plus noise covariance matrix, and is specifically configured as an interference signal covariance matrix
Figure FDA0003209852630000028
And a part for processing noise
Figure FDA0003209852630000029
By adopting a covariance matrix estimation method with structural constraint, correcting a covariance matrix while improving the accuracy of the covariance matrix under small snapshot data to obtain an interference-plus-noise covariance matrix;
for a uniform linear array, a signal covariance matrix is a Toeplitz matrix, and the form of the Toeplitz structure covariance matrix is as follows:
Figure FDA00032098526300000210
wherein R represents a signal covariance matrix,
Figure FDA00032098526300000211
representative of the noise power, σ2As a lower boundary constraint for noise power, κMAs a condition of covariance matrix estimationThe upper boundary of the number is used for ensuring that the covariance matrix estimated value is nonsingular; i is a unit matrix, λmax(M) is the maximum eigenvalue of M, λmin(M) is the minimum eigenvalue of M, TMThe representation matrix M has a Toeplitz structure;
the covariance matrix is obtained as the optimal estimated value by the following optimization problem:
Figure FDA0003209852630000031
Figure FDA0003209852630000032
representing an optimal estimate of the signal covariance matrix;
by (10), a more accurate interference plus noise covariance matrix is obtained by signal covariance matrix estimation under small snapshots; converting (10) further into the following form:
Figure FDA0003209852630000033
wherein ,
Figure FDA0003209852630000034
Figure FDA0003209852630000035
Figure FDA0003209852630000036
HMrepresents the Hermite matrix; a isiIs a set of complex, fitting matrices JiEnsuring that the matrix estimation value is a Toeplitz matrix; thus the optimization problem is shifted to finding the position in the convex setC1 and C2Distance at intersection
Figure FDA0003209852630000037
The nearest point M; solving by adopting a Dykstra projection algorithm, wherein a specific solving process is to project an objective function to two convex sets in sequence, and obtain an optimal solution of an optimization problem after respectively solving the optimization problem positioned in the two convex sets;
first, for convex set C1Obtaining:
Figure FDA0003209852630000038
the characteristic value decomposition is carried out on the Y,
Figure FDA0003209852630000041
and descending and arranging the characteristic values to obtain:
ΛY＝diag([d1,d2,...,dM]T) (16)
the optimal solution of equation (15) is expressed as:
Figure FDA0003209852630000042
wherein ,
Λ(u*)＝diag(λ*(u*)) (18)
λ*(u*)＝[λ1(u),λ2(u),...,λM(u)]T (19)
λi(u)＝min(κMu,max(di,max(1,u))),i＝1,…M (20)
the choice of u involves an optimization problem:
Figure FDA0003209852630000043
wherein G (u) ═ max { hi(u)}；
When d isiWhen is greater than 1
Figure FDA0003209852630000044
When d isiWhen the temperature is less than or equal to 1
Figure FDA0003209852630000045
The optimal solution for equation (21) is:
Figure FDA0003209852630000051
for convex set C2Obtaining:
Figure FDA0003209852630000052
the optimal solution for equation (25) is:
Figure FDA0003209852630000053
and obtaining the optimal solution of covariance matrix estimation under the small snapshot data by iterative solution of the two convex set optimization problems, wherein the covariance matrix does not contain an expected signal component, and substituting the weight vector to obtain the optimal weight vector based on the small snapshot data under the high signal-to-noise ratio environment.
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