CN113466899B - Navigation receiver beam forming method based on small snapshot number in high signal-to-noise ratio environment - Google Patents

Navigation receiver beam forming method based on small snapshot number in high signal-to-noise ratio environment Download PDF

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CN113466899B
CN113466899B CN202110927965.5A CN202110927965A CN113466899B CN 113466899 B CN113466899 B CN 113466899B CN 202110927965 A CN202110927965 A CN 202110927965A CN 113466899 B CN113466899 B CN 113466899B
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covariance matrix
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matrix
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CN113466899A (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 snapshot number in a high signal-to-noise ratio environment, 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; s3, acquiring an accurate interference plus noise covariance matrix, and bringing the accurate interference plus noise covariance matrix into the optimal weight vector calculation formula obtained in the S2. According to the invention, firstly, the component parts of expected signals (satellite signals) in the received signals of the navigation receiver array are removed, so that the algorithm is suitable for a high signal-to-noise ratio environment (practical application occasion), and then the accuracy of a covariance matrix under small snapshot data is improved by using a covariance matrix estimation method, and finally, the beam forming algorithm capable of effectively improving the performance of the navigation receiver in the practical application scene is obtained.

Description

Navigation receiver beam forming method based on small snapshot number in 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 snapshot numbers, which is applied to a satellite navigation receiver in a high signal-to-noise ratio environment.
Background
With the last networking satellite of the third Beidou, china comprehensively completes the constellation deployment of the global Beidou satellite navigation system, and realizes the strategy of 'three steps' of the Beidou system. In addition, satellite navigation systems that are currently being successfully put into operation include the GPS in the united states, GLONASS in russia, galileo in the european union, and partially regional satellite navigation systems. Because the satellite is far away from the ground receiver, the signal reaching the ground is very weak, although the satellite navigation signal is transmitted in a spread spectrum mode, the receiver can obtain the spread spectrum gain, so that the satellite navigation signal is not covered in noise, but is still easily influenced by the interference signal, and the receiver cannot work normally. Since the core of the navigation receiver is an antenna array, the application of the beamforming algorithm based on array signal processing can enhance the navigation signal and inhibit the interference signal.
The beamforming algorithm is mostly based on the Capon criterion, the goal of which is to minimize the power of the output signal while the desired signal passes through the beamformer without distortion, thus achieving a high signal-to-noise ratio of the output signal. However, this method directly applied to a navigation receiver has the following problems: (1) the navigation receiver has higher requirement on signal processing instantaneity, so that the obtained snapshot number is smaller, the sample value of the signal covariance matrix is larger than the true value, and the performance of the beam forming algorithm is reduced, and (2) the sample covariance matrix contains the expected signal components, so that when the navigation receiver is in a high signal-to-noise ratio environment, the expected signal is suppressed as an interference signal, and the performance of the beam forming algorithm is reduced.
At present, the existing covariance matrix estimation method can effectively estimate the signal covariance matrix under the condition of small snapshot, and improves the performance of a beam forming algorithm. However, such methods are only suitable for low signal-to-noise ratio environments, and in practical situations, the spread pilot signal power tends to be higher than the noise power. In addition, the method for effectively removing the desired signal components in the covariance matrix often does not consider the situation based on small snapshot data, and the navigation receiver has high requirement on the real-time performance of signal reception, and the obtained snapshot data is often very limited. Therefore, the research is suitable for the high signal-to-noise ratio environment, and the beamforming algorithm based on the small snapshot number 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 a high signal-to-noise ratio environment in the prior art, and provides a small snapshot number-based navigation receiver beam forming method in the high signal-to-noise ratio environment, which can improve the accuracy of a covariance matrix under small snapshot data and can effectively improve the performance of the navigation receiver in an actual application scene by effectively combining a desired signal removing method and a covariance matrix estimating method.
The aim of the invention is realized by the following technical scheme: a navigation receiver beam forming method based on small snapshot number in high signal-to-noise ratio environment comprises the following steps:
s1, calculating a received signal;
s2, establishing an optimization target, and solving to obtain an optimal weight vector calculation formula;
s3, acquiring an accurate interference plus noise covariance matrix, and bringing the accurate interference plus noise covariance matrix into the optimal weight vector calculation formula obtained in the S2.
Further, the specific implementation method of the S1 is as follows: for a uniform linear array with M array elements, assuming that the signals received by the uniform linear array are narrowband far-field signals, the signals are uncorrelated with each other, the noise received by each array element is complex Gaussian white noise, and the noise is uncorrelated with the signals, at the moment t, the received signals of the array are expressed as:
x(t)＝x s (t)+x i (t)+x n (t) (1)
wherein ,xs (t)＝s s (t)a s ，x i (t)＝s i (t)a i ，x s (t)、x i (t)、x n (t) represents a desired signal, an interference signal and noise, respectively, s (t) represents the waveform of the signal, a represents the steering vector of the corresponding signal, and for a uniform linear array of a known array structure, the steering vector of the array received signal is expressed as θ according to the signal:
Figure BDA0003209852640000021
where λ represents the signal wavelength and d represents the distance between adjacent array elements.
Further, the specific implementation method of the S2 is as follows: through complex weight vector w= [ w ] 1 ,w 2 ,…,w M ] T After weighting the array received signals, the beamformer performance is represented by the output signal-to-interference-and-noise ratio SINR:
Figure BDA0003209852640000022
wherein
Figure BDA0003209852640000023
To the desired signal power, R i+n A covariance matrix of interference plus noise;
in order to maximize the output signal-to-interference-and-noise ratio, according to the Capon criterion, the optimization objective is:
Figure BDA0003209852640000024
subject to w H a s ＝1
solving to obtain an optimal weight vector as
Figure BDA0003209852640000025
Further, the specific implementation method of the S3 is as follows: removing the components of the expected signals in the array received signals by adopting a subspace projection method, and obtaining the spatial information of the interference area by integrating the angular area where the interference signals are located because the array structure is known:
Figure BDA0003209852640000031
Θ i representing the spatial angle region where the interference signal is located;
decomposing the eigenvalue of the psi, sequencing the eigenvalue, selecting the eigenvector corresponding to the maximum eigenvalue to form a subspace P 1 By P 1 Orthogonalization with non-interfering signal region steering vectors using projection matrices
Figure BDA0003209852640000032
While the interference signal is not changed, the expected signal in the array receiving signal is effectively eliminated; the projected signal is expressed as:
Figure BDA0003209852640000033
the sample covariance matrix, which eliminates the desired signal component, is expressed as:
Figure BDA0003209852640000034
k is the number of beats of the signal sample;
the processed sample covariance matrix is not an interference plus noise covariance matrix, and is specifically formed as an interference signal covariance matrix
Figure BDA0003209852640000035
And a processing part for noise->
Figure BDA0003209852640000036
Adopting a covariance matrix estimation method with structural constraint, and 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, the signal covariance matrix is a Toeplitz matrix, and the Toeplitz structure covariance matrix is in the form of:
Figure BDA0003209852640000037
wherein R represents a signal covariance matrix,
Figure BDA0003209852640000038
representing noise power, sigma 2 Kappa as a lower boundary constraint for noise power M As the upper boundary of the condition number of the covariance matrix estimated value, the method is used for ensuring that the covariance matrix estimated value is non-singular; i is a unit matrix, lambda max (M) is the maximum eigenvalue of M, lambda min (M) is the minimum characteristic value of M, T M The representation matrix M has a Toeplitz structure;
obtaining an optimal estimated value by solving a covariance matrix according to the following optimization problem:
Figure BDA0003209852640000041
Figure BDA0003209852640000042
representing an optimal estimate of the signal covariance matrix;
through (10), under the condition of small snapshot, obtaining a more accurate interference plus noise covariance matrix through signal covariance matrix estimation; further converting (10) to the following form:
Figure BDA0003209852640000043
wherein ,
Figure BDA0003209852640000044
Figure BDA0003209852640000045
Figure BDA0003209852640000046
H M representing an hermite matrix; a, a i Is a complex number, coordinates with matrix J i Ensuring that the matrix estimation value is a Toeplitz matrix; the optimization problem is thus shifted to find the position in the convex set C 1 and C2 Intersection and distance
Figure BDA0003209852640000047
A nearest point M; the method comprises the steps of adopting Dykstra projection algorithm to solve, wherein the specific solving process is to sequentially project an objective function to two convex sets, and obtain an optimal solution after respectively solving the optimization problem of the two convex sets;
first, for convex set C 1 The method comprises the following steps of:
Figure BDA0003209852640000048
subjecting Y to special treatmentThe sign value is decomposed,
Figure BDA0003209852640000049
and the eigenvalues are arranged in descending order to obtain:
Λ Y ＝diag([d 1 ,d 2 ,…,d M ] T ) (16)
the optimal solution for equation (15) is expressed as:
Figure BDA0003209852640000051
wherein ,
Λ(u * )＝diag(λ * (u * ))(18)
λ * (u * )＝[λ 1 (u),λ 2 (u),...,λ M (u)] T (19)
λ i (u)＝min(κ M u,max(d i ,max(1,u))),i＝1,...M (20)
the choice of u involves an optimization problem:
Figure BDA0003209852640000052
wherein G (u) =max { h } i (u)}；
When d i At > 1
Figure BDA0003209852640000053
When d i When less than or equal to 1
Figure BDA0003209852640000054
The optimal solution for equation (21) is:
Figure BDA0003209852640000061
for convex set C 2 The method comprises the following steps of:
Figure BDA0003209852640000062
the optimal solution for equation (25) is:
Figure BDA0003209852640000063
and obtaining an optimal solution of covariance matrix estimation under the small snapshot data by carrying out iterative solution on the two convex set optimization problems, wherein the covariance matrix does not contain expected signal components, and substituting the expected signal components into a weight vector to obtain the optimal weight vector based on the small snapshot data under the high signal-to-noise ratio environment.
The beneficial effects of the invention are as follows: according to the invention, firstly, the component parts of expected signals (satellite signals) in the received signals of the navigation receiver array are removed, so that the algorithm is suitable for a high signal-to-noise ratio environment (practical application occasion), and then the accuracy of a covariance matrix under small snapshot data is improved by using a covariance matrix estimation method, and finally, the beam forming algorithm capable of effectively improving the performance of the navigation receiver in the practical application scene is obtained.
Detailed Description
The technical scheme of the invention is further described below.
The invention discloses a navigation receiver beam forming method based on small snapshot number in a high signal-to-noise ratio environment, 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 the signals received by the ULA are narrowband far-field signals, the signals are uncorrelated with each other, the noise received by each array element is complex gaussian white noise, and the noise is uncorrelated with the signals, then at time t, the received signals of the array are expressed as:
x(t)＝x s (t)+x i (t)+x n (t) (1)
wherein ,xs (t)＝s s (t)a s ，x i (t)＝s i (t)a i ，x s (t)、x i (t)、x n (t) represents a desired signal, an interference signal and noise, respectively, s (t) represents the waveform of the signal, a represents the steering vector of the corresponding signal, and for a uniform linear array of a known array structure, the steering vector of the array received signal is expressed as θ according to the signal:
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: through complex weight vector w= [ w ] 1 ,w 2 ,…,w M ] T After weighting the array received signals, the beamformer performance is represented by the output signal-to-interference-and-noise ratio SINR:
Figure BDA0003209852640000072
wherein
Figure BDA0003209852640000073
To the desired signal power, R i+n A covariance matrix of interference plus noise;
in order to maximize the output signal-to-interference-and-noise ratio, according to the Capon criterion, the optimization objective is:
Figure BDA0003209852640000074
subject to w H a s ＝1
solving to obtain an optimal weight vector as
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 plus noise covariance matrix, and bringing the accurate interference plus noise covariance matrix into an optimal weight vector calculation formula obtained in the S2; the specific implementation method comprises the following steps: removing the components of the expected signals in the array received signals by adopting a subspace projection method, and obtaining the spatial information of the interference area by integrating the angular area where the interference signals are located because the array structure is known:
Figure BDA0003209852640000076
Θ i representing the spatial angle region where the interference signal is located;
decomposing the eigenvalue of the psi, sequencing the eigenvalue, selecting the eigenvector corresponding to the maximum eigenvalue to form a subspace P 1 By P 1 Orthogonalization with non-interfering signal region steering vectors using projection matrices
Figure BDA0003209852640000077
While the interference signal is not changed, the expected signal in the array receiving signal is effectively eliminated; the projected signal is expressed as:
Figure BDA0003209852640000081
the sample covariance matrix, which eliminates the desired signal component, is expressed as:
Figure BDA0003209852640000082
k is the number of beats of the signal sample;
since the exact values of the sample covariance matrix and the signal covariance matrix differ significantly in the case of a small snapshot, beamforming is performed even if the desired signal component is eliminatedThe performance of the device is still significantly degraded. The processed sample covariance matrix is not an interference plus noise covariance matrix, and is specifically formed as an interference signal covariance matrix
Figure BDA0003209852640000083
And a processing part for noise->
Figure BDA0003209852640000084
Therefore, the covariance matrix estimation method with structural constraint is adopted, the covariance matrix is corrected while the accuracy of the covariance matrix under small snapshot data is improved, and the interference plus noise covariance matrix is obtained;
for a uniform linear array, the signal covariance matrix is a Toeplitz matrix, and the Toeplitz structure covariance matrix is in the form of:
Figure BDA0003209852640000085
m represents a matrix whose specific structure/expression is the first equation, the remaining conditions being constraints on it; wherein R represents a signal covariance matrix,
Figure BDA0003209852640000086
representing noise power, sigma 2 Kappa as a lower boundary constraint for noise power M As the upper boundary of the condition number of the covariance matrix estimated value, the method is used for ensuring that the covariance matrix estimated value is non-singular; i is a unit matrix, lambda max (M) is the maximum eigenvalue of M, lambda min (M) is the minimum characteristic value of M, T M The representation matrix M has a Toeplitz structure;
obtaining an optimal estimated value by solving a covariance matrix according to the following optimization problem:
Figure BDA0003209852640000091
Figure BDA0003209852640000092
representing an optimal estimate of the signal covariance matrix;
the method specifically aims at minimizing the Frobenius norm of the difference between the estimated value of the matrix and the sample covariance matrix under the structural constraint of the Toeplitz covariance matrix, so that the estimated value is ensured to approach the sample covariance matrix to the greatest extent while having a signal covariance matrix structure. Since the preamble processing has eliminated the component parts of the desired signal in the sample covariance matrix, and the component structure of the matrix estimate is constrained to the sum of the signal covariance matrix and the ideal noise covariance matrix in the constraint. Therefore, through (10), a more accurate interference plus noise covariance matrix can be estimated through a sample covariance matrix under the condition of small snapshot, so that the method is suitable for a high signal-to-noise ratio environment.
Further converting (10) to the following form:
Figure BDA0003209852640000093
wherein ,
Figure BDA0003209852640000094
Figure BDA0003209852640000095
Figure BDA0003209852640000096
H M represents Hermitian Matrix (Hermitian Matrix); a, a i Is a complex number, coordinates with matrix J i Ensuring that the matrix estimation value is a Toeplitz matrix; the optimization problem is thus shifted to find the position in the convex set C 1 and C2 Intersection and distance
Figure BDA0003209852640000097
A nearest point M; the method comprises the steps of adopting Dykstra projection algorithm to solve, wherein the specific solving process is to sequentially project an objective function to two convex sets, and obtain an optimal solution after respectively solving the optimization problem of the two convex sets;
first, for convex set C 1 The method comprises the following steps of:
Figure BDA0003209852640000101
y (Y has no specific meaning and is an independent variable used for representing the solution of the optimization problem) is subjected to eigenvalue decomposition,
Figure BDA0003209852640000102
and the eigenvalues are arranged in descending order to obtain:
Λ Y ＝diag([d 1 ,d 2 ,...,d M ] T ) (16)
the optimal solution for equation (15) is expressed as:
Figure BDA0003209852640000103
wherein ,
Λ(u * )＝diag(λ * (u * )) (18)
λ * (u * )＝[λ 1 (u),λ 2 (u),…,λ M (u)] T (19)
λ i (u)＝min(κ M u,max(d i ,max(1,u))),i＝1,...M (20)
the choice of u involves an optimization problem:
Figure BDA0003209852640000104
wherein G (u) =max { h } i (u)}；
When d i At > 1
Figure BDA0003209852640000105
When d i When less than or equal to 1
Figure BDA0003209852640000106
The optimal solution for equation (21) is:
Figure BDA0003209852640000111
for convex set C 2 The method comprises the following steps of:
Figure BDA0003209852640000112
the optimal solution for equation (25) is:
Figure BDA0003209852640000113
and obtaining an optimal solution of covariance matrix estimation under the small snapshot data by carrying out iterative solution on the two convex set optimization problems, wherein the covariance matrix does not contain expected signal components, and substituting the expected signal components into a weight vector to obtain the optimal weight vector based on the small snapshot data under the high signal-to-noise ratio environment.
Those of ordinary skill in the art will recognize that the embodiments described herein are for the purpose of aiding the reader in understanding the principles of the present invention and should be understood that the scope of the invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific modifications and combinations from the teachings of the present disclosure without departing from the spirit thereof, and such modifications and combinations remain within the scope of the present disclosure.

Claims (1)

1. The navigation receiver beam forming method based on the small snapshot number in the high signal-to-noise ratio environment is characterized by comprising the following steps:
s1, calculating a received signal; the specific implementation method comprises the following steps: for a uniform linear array with M array elements, assuming that the signals received by the uniform linear array are narrowband far-field signals, the signals are uncorrelated with each other, the noise received by each array element is complex Gaussian white noise, and the noise is uncorrelated with the signals, at the moment t, the received signals of the array are expressed as:
x(t)＝x s (t)+x i (t)+x n (t) (1)
wherein ,xs (t)＝s s (t)a s ，x i (t)＝s i (t)a i ，x s (t)、x i (t)、x n (t) represents a desired signal, an interference signal and noise, respectively, s (t) represents the waveform of the signal, a represents the steering vector of the corresponding signal, and for a uniform linear array of a known array structure, the steering vector of the array received signal is expressed as θ according to the signal:
Figure FDA0004127994980000011
wherein lambda 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: through complex weight vector w= [ w ] 1 ,w 2 ,…,w M ] T After weighting the array received signals, the beamformer performance is represented by the output signal-to-interference-and-noise ratio SINR:
Figure FDA0004127994980000012
wherein
Figure FDA0004127994980000016
To the desired signal power, R i+n A covariance matrix of interference plus noise;
in order to maximize the output signal-to-interference-and-noise ratio, according to the Capon criterion, the optimization objective is:
Figure FDA0004127994980000013
subject to w H a s ＝1
solving to obtain an optimal weight vector as
Figure FDA0004127994980000014
S3, acquiring an accurate interference plus noise covariance matrix, and bringing the accurate interference plus noise covariance matrix into an optimal weight vector calculation formula obtained in the S2; the specific implementation method comprises the following steps: removing the components of the expected signals in the array received signals by adopting a subspace projection method, and obtaining the spatial information of the interference area by integrating the angular area where the interference signals are located because the array structure is known:
Figure FDA0004127994980000015
Θ i representing the spatial angle region where the interference signal is located;
decomposing the eigenvalue of the psi, sequencing the eigenvalue, selecting the eigenvector corresponding to the maximum eigenvalue to form a subspace P 1 By P 1 Orthogonalization with non-interfering signal region steering vectors using projection matrices
Figure FDA0004127994980000021
While the interference signal is not changed, the expected signal in the array receiving signal is effectively eliminated; the projected signal is expressed as: />
Figure FDA0004127994980000022
The sample covariance matrix, which eliminates the desired signal component, is expressed as:
Figure FDA0004127994980000023
k is the number of beats of the signal sample;
the processed sample covariance matrix is not an interference plus noise covariance matrix, and is specifically formed as an interference signal covariance matrix
Figure FDA0004127994980000024
And a processing part for noise->
Figure FDA0004127994980000025
Adopting a covariance matrix estimation method with structural constraint, and 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, the signal covariance matrix is a Toeplitz matrix, and the Toeplitz structure covariance matrix is in the form of:
Figure FDA0004127994980000026
wherein R represents a signal covariance matrix,
Figure FDA0004127994980000027
representing noise power, sigma 2 Kappa as a lower boundary constraint for noise power M As the upper boundary of the condition number of the covariance matrix estimated value, the method is used for ensuring that the covariance matrix estimated value is non-singular; i is a unit matrix, lambda max (M) is the maximum eigenvalue of M, lambda min (M) is the minimum characteristic value of M, T M The representation matrix M has a Toeplitz structure;
obtaining an optimal estimated value by solving a covariance matrix according to the following optimization problem:
Figure FDA0004127994980000031
Figure FDA0004127994980000032
representing an optimal estimate of the signal covariance matrix;
through (10), under the condition of small snapshot, obtaining a more accurate interference plus noise covariance matrix through signal covariance matrix estimation; further converting (10) to the following form:
Figure FDA0004127994980000033
wherein ,
Figure FDA0004127994980000034
Figure FDA0004127994980000035
Figure FDA0004127994980000036
H M representing an hermite matrix; a, a i Is a complex number, coordinates with matrix J i Ensuring that the matrix estimation value is a Toeplitz matrix; the optimization problem is thus shifted to find the position in the convex set C 1 and C2 Intersection and distance
Figure FDA0004127994980000037
A nearest point M; the method comprises the steps of adopting Dykstra projection algorithm to solve, wherein the specific solving process is to sequentially project an objective function to two convex sets, and obtain an optimal solution after respectively solving the optimization problem of the two convex sets;
first, for convex set C 1 The method comprises the following steps of:
Figure FDA0004127994980000038
and decomposing the characteristic value of Y to obtain a characteristic value,
Figure FDA0004127994980000039
and the eigenvalues are arranged in descending order to obtain:
Λ Y ＝diag([d 1 ,d 2 ,...,d M ] T ) (16)
the optimal solution for equation (15) is expressed as:
Figure FDA0004127994980000041
wherein ,
Λ(u * )＝diag(λ * (u * )) (18)
λ * (u * )＝[λ 1 (u),λ 2 (u),...,λ M (u)] T (19)
λ i (u)＝min(κ M u,max(d i ,max(1,u))),i＝1,...M (20)
the choice of u involves an optimization problem:
Figure FDA0004127994980000042
wherein G (u) =max { h } i (u)}；
When d i At > 1
Figure FDA0004127994980000043
When d i When less than or equal to 1
Figure FDA0004127994980000044
The optimal solution for equation (21) is:
Figure FDA0004127994980000051
for convex set C 2 The method comprises the following steps of:
Figure FDA0004127994980000052
the optimal solution for equation (25) is:
Figure FDA0004127994980000053
and obtaining an optimal solution of covariance matrix estimation under the small snapshot data by carrying out iterative solution on the two convex set optimization problems, wherein the covariance matrix does not contain expected signal components, and substituting the expected signal components into a 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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