A space-time correction method for space-based distributed passive coherent reconnaissance based on external reference source
The space-time correction method for satellite platforms corrects phase and position errors using an external reference source, enhancing the accuracy and efficiency of satellite-based reconnaissance systems by directly calculating estimated positions and eliminating phase errors.
Patent Information
- Application Number
- CN202211038880.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-08-29
- Publication Date
- 2024-12-13
- Estimated Expiration
- 2042-08-29
AI Technical Summary
The space-based distributed passive coherent reconnaissance system suffers from reduced positioning accuracy due to position and phase errors in satellite platforms, which are not corrected effectively using traditional methods, especially when the baseline distances are large and far-field conditions are not applicable.
A space-time correction method using an external reference source to correct the positions and phases of satellite platforms by satellite platforms using an auxiliary correction method to correct the phase and position errors of satellite platforms.
The method allows for direct calculation of the estimated positions and phase corrections of all satellite platforms simultaneously, reducing calculation complexity and improving the reconnaissance system's performance by eliminating phase errors through directional derivative comparison.
Smart Images
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Abstract
Description
Technical Field
[0001] The invention relates to the field of satellite positioning, and specifically comprises a space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source. Background Art
[0002] The space-based distributed passive coherent reconnaissance system is to deploy multiple satellite-borne platforms in a dispersed manner in space, and achieve the performance of a large-aperture antenna by performing signal-level coherent fusion of the signals transmitted by the target signal source. The space-based distributed passive coherent reconnaissance system can locate and search for potential targets in a large area at all times of the day and in all weather conditions, providing an important guarantee for high-precision reconnaissance and positioning of ground radiation sources.
[0003] In the ideal model system, the space-based distributed passive coherent reconnaissance system has a very high accuracy in locating the target position. However, in the actual system, due to certain errors in the position of each array element and the inconsistency of the phase of each receiving channel, the space-based distributed coherent positioning capability is seriously reduced.
[0004] Therefore, how to correct the position error and phase error of each platform on board becomes the key to the performance of space-based distributed coherence. Since the distance between each platform of the baseline of the space-based distributed reconnaissance platform is large, the traditional far-field condition of electromagnetic waves is no longer used, and the position and phase error correction of each platform must be completed based on the near-field reference source correction method.
[0005] In the prior art, samples need to cover multiple arc segments, and there must be multiple detection samples in each arc segment in order to meet the specific implementation conditions of the technology, and the correction accuracy is low. Summary of the invention
[0006] In view of the above-mentioned deficiencies in the prior art, the present invention provides a space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source, which solves the problems of low correction accuracy and complex correction methods in the prior art.
[0007] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:
[0008] A space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source comprises the following steps:
[0009] S1. Select a satellite platform phase center as the phase reference platform of the distributed system, and the remaining satellite platforms as correction objects. Set M known position correction sources as auxiliary correction sources, where M≥4;
[0010] S2. Establish a model when the correction object receives the signal; make the correction object receive the signal transmitted by the mth auxiliary correction source, calculate the signal space correlation matrix corresponding to the mth auxiliary correction source and the eigenvector corresponding to the maximum eigenvalue of the signal space correlation matrix, and obtain the estimated directional derivative under the conditions of phase error and position error.
[0011] S3. Construct a multivariate nonlinear equation group containing only unknown positions according to the estimated directional derivatives under the conditions of phase error and position error, and obtain the estimated position of the correction object and the phase error of the correction object.
[0012] Furthermore, the specific implementation of step S1 is as follows:
[0013] Get the ideal position of the nth correction object as P n =[x n ,y n ,z n ]; Get the position of the mth auxiliary correction source as P m =[x m ,y m ,z m ]; the total number of correction objects is N, n = 1, 2…N; the total number of auxiliary correction sources is M, m = 1, 2,…,M.
[0014] Furthermore, the specific implementation of step S2 is as follows:
[0015] S2-1, calculating the difference Δr between the distance from the mth auxiliary correction source to other correction objects and the distance from the auxiliary correction source to the phase reference platform;
[0016] S2-2, according to the formula:
[0017] λ m =c / f m
[0018]
[0019] Get the wavelength λ of the signal emitted by the mth auxiliary correction source m , and the phase difference between the ideal position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform Among them, Δr 1 m represents the distance from the mth auxiliary correction source to the first correction object, θ m It represents the pitch angle of the mth auxiliary correction source with reference to the first correction object. represents the azimuth of the mth auxiliary correction source with respect to the first correction object, c represents the speed of light, and f m is the transmitted signal s m(t) carrier frequency, (x n ,y n ,z n ) represents the position of the nth calibration object, and π is the circumference of a circle;
[0020] S2-3, according to the formula:
[0021]
[0022] Get N×1 dimensional direction vector a m ; Where exp(·) represents the exponential with the natural constant e as the base, and j represents the imaginary unit;
[0023] S2-4, according to the formula:
[0024] x m (t) = a m ·s m (t)+n m (t)
[0025] Get the mth auxiliary correction source transmission signal s at time t m (t) is the N×1 dimensional vector x output by the calibration object m (t); where a m is an N×1 dimensional direction vector, n m (t) is an N×1-dimensional noise vector;
[0026] S2-5, according to the formula:
[0027]
[0028]
[0029] Get the position P m =[x m ,y m ,z m The signal data covariance matrix R of the auxiliary correction source emitted by m ; Where E[·] represents the statistical average of the data; is the signal power transmitted by the auxiliary correction source; 2 is the noise power; I is the N×N unit matrix; a m (x m ,y m ,z m ) represents the N×1 dimensional direction vector at the ideal position; Represents the N×1 dimensional direction vector a m (x m ,y m ,z m ) conjugate transpose; is the signal s emitted by the mth auxiliary correction source at time t m The conjugate transpose of (t); is an N×1 dimensional vector x m (t) conjugate transpose;
[0030] S2-6. Signal data covariance matrix R m Characteristic decomposition and according to the formula:
[0031]
[0032] a m (x m ,y m ,z m )=[1a m2 (x m ,y m ,z m )…a mN (x m ,y m ,z m )] T
[0033]
[0034]
[0035] Get the signal data covariance matrix R m Eigendecomposition results in, For R m The eigenvector corresponding to the maximum eigenvalue, [·] T represents transpose, represents a complex constant; a mN (x m ,y m ,z m ) represents the direction vector from the mth auxiliary correction source to the Nth correction object, is the Qth element in the eigenvector corresponding to the maximum eigenvalue, Q = N;
[0036] S2-7, according to the formula:
[0037]
[0038] The estimated directional derivative under the conditions of phase error and position error is obtained by the same method as steps S2-2 to S2-6. in N×1 dimensional direction vector for the estimated position of the correction object; The phase difference between the estimated position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform; is the phase error and position error condition The eigenvector corresponding to the maximum eigenvalue; is the first element in the eigenvector corresponding to the maximum eigenvalue under the conditions of phase error and position error.
[0039] Furthermore, the specific implementation of step S3 is as follows:
[0040] S3-1, according to the formula:
[0041]
[0042] The phase error and position error of the M-1 auxiliary correction source signals are combined to estimate the directional derivative under the condition of The directional derivative of the corresponding element in and the first auxiliary correction source In contrast, we obtain a multivariate nonlinear system of equations containing only unknown positions Where, β=2,3,…,M; For position P β =[x β ,y β ,z β The covariance matrix of the signal data emitted by the auxiliary correction source is The eigenvector corresponding to the maximum eigenvalue under the phase error and position error conditions; is the signal data covariance matrix The maximum eigenvalue under the phase error and position error conditions corresponds to the first element in the eigenvector; represents the azimuth angle of the βth auxiliary correction source with respect to the first phase reference platform; Δr 1 β To calculate the difference between the distance from the βth auxiliary correction source to other correction objects and the distance from the auxiliary correction source to the phase reference platform; θ β Indicates the pitch angle of the βth auxiliary correction source with the first correction object as reference; Get the latest phase error Γ, Γ 2 =Γ 3 …=Γ N ;
[0043] S3-2. Solving a multivariate nonlinear system of equations with only unknown positions using the trust region dogleg algorithm Get the estimated position of each calibration object Among them, 1≤i≤N;
[0044] S3-3. Estimated position of the calibration object Substitute the estimated directional derivative Get the phase error.
[0045] The beneficial effects of the present invention are as follows: the present invention directly performs phase correction on the position of the onboard platform of the space-based distributed satellite by introducing at least four auxiliary correction sources with known positions, and eliminates the phase error by using directional derivative comparison, so that the estimated positions of all onboard platforms can be directly calculated at one time, while also reducing the difficulty of calculation. BRIEF DESCRIPTION OF THE DRAWINGS
[0046] Figure 1 It is a flow chart of the present invention;
[0047] Figure 2 A model diagram of the correction object when receiving a signal;
[0048] Figure 3 It is a schematic diagram of the X-axis correction capability under the condition that the signal-to-noise ratio is 0 and there is no error in the position of the external radiation source;
[0049] Figure 4 It is a schematic diagram of the Y-axis correction capability under the condition that the signal-to-noise ratio is 0 and there is no error in the position of the external radiation source;
[0050] Figure 5 Schematic diagram of Z-axis correction capability under the condition of zero signal-to-noise ratio and no error in the position of the external radiation source;
[0051] Figure 6 Schematic diagram of phase error correction capability under the condition of signal-to-noise ratio of 0 and no error in the position of the external radiation source;
[0052] Figure 7 Schematic diagram of the X-axis correction capability when the signal-to-noise ratio is 10 and there is an error in the position of the external radiation source.
[0053] Figure 8 Schematic diagram of the Y-axis correction capability under the condition of a signal-to-noise ratio of 10 and an error in the position of the external radiation source.
[0054] Fig. 9 Schematic diagram of Z-axis correction capability under the condition of signal-to-noise ratio of 10 and error in the position of the external radiation source.
[0055] Fig.10 Schematic diagram of phase correction capability when the signal-to-noise ratio is 10 and there is an error in the position of the external radiation source. DETAILED DESCRIPTION
[0056] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.
[0057] like Figure 1 As shown, a space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source includes the following steps:
[0058] S1. Select a satellite platform phase center as the phase reference platform of the distributed system, and the remaining satellite platforms as correction objects. Set M known position correction sources as auxiliary correction sources, where M≥4;
[0059] S2. Establish a model when the correction object receives the signal; make the correction object receive the signal transmitted by the mth auxiliary correction source, calculate the signal space correlation matrix corresponding to the mth auxiliary correction source and the eigenvector corresponding to the maximum eigenvalue of the signal space correlation matrix, and obtain the estimated directional derivative under the conditions of phase error and position error.
[0060] S3. Construct a multivariate nonlinear equation group containing only unknown positions according to the estimated directional derivatives under the conditions of phase error and position error, and obtain the estimated position of the correction object and the phase error of the correction object.
[0061] The specific implementation of step S1 is as follows:
[0062] Get the ideal position of the nth correction object as P n =[x n ,y n ,z n ]; Get the position of the mth auxiliary correction source as P m =[x m ,y m ,z m ]; the total number of correction objects is N, n = 1, 2…N; the total number of auxiliary correction sources is M, m = 1, 2,…,M.
[0063] The specific implementation of step S2 is as follows:
[0064] S2-1, calculating the difference Δr between the distance from the mth auxiliary correction source to other correction objects and the distance from the auxiliary correction source to the phase reference platform;
[0065] S2-2, according to the formula:
[0066] λ m =c / f m
[0067]
[0068] Get the wavelength λ of the signal emitted by the mth auxiliary correction source m , and the phase difference between the ideal position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform Among them, Δr 1 m represents the distance from the mth auxiliary correction source to the first correction object, θ m It represents the pitch angle of the mth auxiliary correction source with reference to the first correction object. represents the azimuth of the mth auxiliary correction source with respect to the first correction object, c represents the speed of light, and f m is the transmitted signal s m (t) carrier frequency, (x n ,y n ,z n ) represents the position of the nth calibration object, and π is the circumference of a circle;
[0069] S2-3, according to the formula:
[0070]
[0071] Get N×1 dimensional direction vector a m ; Where exp(·) represents the exponential with the natural constant e as the base, and j represents the imaginary unit;
[0072] S2-4, according to the formula:
[0073] x m (t) = a m ·s m (t)+n m (t)
[0074] Get the mth auxiliary correction source transmission signal s at time t m (t) is the N×1 dimensional vector x output by the calibration object m (t); where a m is an N×1 dimensional direction vector, n m (t) is an N×1-dimensional noise vector;
[0075] S2-5, according to the formula:
[0076]
[0077]
[0078] Get the position P m =[x m ,y m ,z m The signal data covariance matrix R of the auxiliary correction source emitted by m ; Where E[·] represents the statistical average of the data; is the signal power transmitted by the auxiliary correction source; 2 is the noise power; I is the N×N unit matrix; am (x m ,y m ,z m ) represents the N×1 dimensional direction vector at the ideal position; Represents the N×1 dimensional direction vector a m (x m ,y m ,z m ) conjugate transpose; is the signal s emitted by the mth auxiliary correction source at time t m The conjugate transpose of (t); is an N×1 dimensional vector x m (t) conjugate transpose;
[0079] S2-6. Signal data covariance matrix R m Characteristic decomposition and according to the formula:
[0080]
[0081] a m (x m ,y m ,z m )=[1a m2 (x m ,y m ,z m )…a mN (x m ,y m ,z m )] T
[0082]
[0083]
[0084] Get the signal data covariance matrix R m Eigendecomposition results in, For R m The eigenvector corresponding to the maximum eigenvalue, [·] T represents transpose, represents a complex constant; a mN (x m ,y m ,z m ) represents the direction vector from the mth auxiliary correction source to the Nth correction object, is the Qth element in the eigenvector corresponding to the maximum eigenvalue, Q = N;
[0085] S2-7, according to the formula:
[0086]
[0087] The estimated directional derivative under the conditions of phase error and position error is obtained by the same method as steps S2-2 to S2-6. in N×1 dimensional direction vector for the estimated position of the correction object; The phase difference between the estimated position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform; is the phase error and position error condition The eigenvector corresponding to the maximum eigenvalue; is the first element in the eigenvector corresponding to the maximum eigenvalue under the conditions of phase error and position error.
[0088] The specific implementation of step S3 is as follows:
[0089] S3-1, according to the formula:
[0090]
[0091] The phase error and position error of the M-1 auxiliary correction source signals are combined to estimate the directional derivative under the condition of The directional derivative of the corresponding element in and the first auxiliary correction source In contrast, we obtain a multivariate nonlinear system of equations containing only unknown positions Where, β=2,3,…,M; For position P β =[x β ,y β ,z β The covariance matrix of the signal data emitted by the auxiliary correction source is The eigenvector corresponding to the maximum eigenvalue under the phase error and position error conditions; is the signal data covariance matrix The maximum eigenvalue under the phase error and position error conditions corresponds to the first element in the eigenvector; represents the azimuth angle of the βth auxiliary correction source with respect to the first phase reference platform; Δr 1 β To calculate the difference between the distance from the βth auxiliary correction source to other correction objects and the distance from the auxiliary correction source to the phase reference platform; θ β Indicates the pitch angle of the βth auxiliary correction source with the first correction object as reference; Get the latest phase error Γ, Γ 2 =Γ 3 …=Γ N ;
[0092] S3-2. Solving a multivariate nonlinear system of equations with only unknown positions using the trust region dogleg algorithm Get the estimated position of each calibration object Among them, 1≤i≤N;
[0093] S3-3. Estimated position of the calibration object Substitute the estimated directional derivative Get the phase error.
[0094] like Figure 2 As shown, θ represents the pitch angle from the auxiliary correction source to the correction object, Indicates the azimuth from the auxiliary correction source to the correction object; the x-axis, y-axis, and z-axis are the spatial coordinate system established with the correction object as the origin.
[0095] like Figure 3 , Figure 4 , Figure 5 and Figure 6 As shown in the figure, when the signal-to-noise ratio is 0 and the position error of the satellite platform is within 10 cm, the calibration accuracy of the position error is 10 when the phase error of the satellite platform is in the interval [-150°, 150°]. -4 m level, the correction accuracy of phase error reaches within 2°, and it has the ability to correct the position error and phase error of the satellite platform. This technology will greatly improve the reconnaissance performance of the space-based distributed passive coherent reconnaissance system.
[0096] like Figure 7 , Figure 8 , Fig. 9 and Fig.10 As shown in the figure, when the signal-to-noise ratio is 0 and the satellite platform position error is within 10 cm, when the satellite platform phase error is in the interval [-150°, 150°], the position of the external radiation source is added with a random error within [-3 cm, 3 cm]. Although the correction effect is reduced, the satellite platform position correction accuracy can reach 10 -3 At the cm level, the phase correction accuracy of the satellite-borne platform is within 10°, and it still has a certain performance correction capability.
[0097] In one embodiment of the present invention, M known position auxiliary sources are introduced to directly perform onboard platform position and phase correction on space-based distributed satellites, and the phase error is eliminated by using directional derivatives, so that the estimated true position of all onboard platforms can be directly calculated at one time, while also reducing the difficulty of calculation. In addition, the obtained position information can continue to be used to update the onboard platform phase error, achieving simultaneous correction of position error and phase error, where M≥4. At the same time, this technology meets the correction capability when there is a certain position error at the position of the external radiation source.
[0098] The position of the correction object at the next moment can be calculated based on the estimated position and estimated directional derivative of the correction object at the previous moment.
[0099] Γ=diag([Γ 1 ,Γ 2 ,…,Γ T ])=diag([exp(-jα 1 ),exp(-jα 2 ),…,exp(-jα T )])
[0100] The phase error exponential matrix Γ is obtained; where diag(·) represents a diagonal matrix; α T is the phase error of the Tth correction object; Γ T is the Tth element of the phase error exponentiation matrix Γ; T=N.
[0101] According to the formula:
[0102]
[0103]
[0104] Get the estimated directional derivative of the next position of the correction object; where, is the phase difference between the estimated position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform. The estimated position of the correction object at the next moment can be calculated based on the estimated directional derivative.
[0105] The present invention directly performs phase correction on the position of the onboard platform of the space-based distributed satellite by introducing at least four auxiliary correction sources with known positions, and eliminates the phase error by using directional derivative comparison, so that the estimated true positions of all onboard platforms can be directly calculated at one time, while also reducing the difficulty of calculation.
Claims
1. A space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source. It is characterized in that The following steps are involved: S1. Select a satellite platform phase center as the phase reference platform of the distributed system, and the remaining satellite platforms as correction objects. Set M known position correction sources as auxiliary correction sources, where M≥4; S2. Establish a model when the correction object receives the signal; make the correction object receive the signal transmitted by the mth auxiliary correction source, calculate the signal space correlation matrix corresponding to the mth auxiliary correction source and the eigenvector corresponding to the maximum eigenvalue of the signal space correlation matrix, and obtain the estimated directional derivative under the conditions of phase error and position error. S3. Construct a multivariate nonlinear equation group containing only unknown positions according to the estimated directional derivative under the conditions of phase error and position error to obtain the estimated position of the correction object and the phase error of the correction object. The specific implementation method is as follows: S3-1, according to the formula: The phase error and position error of the M-1 auxiliary correction source signals are combined to estimate the directional derivative under the condition of The directional derivative of the corresponding element in and the first auxiliary correction source In contrast, we obtain a multivariate nonlinear system of equations containing only unknown positions Where, β=2,3,…,M; For position P β =[x β ,y β ,z β The covariance matrix of the signal data emitted by the auxiliary correction source is The eigenvector corresponding to the maximum eigenvalue under the phase error and position error conditions; is the signal data covariance matrix The maximum eigenvalue under the phase error and position error conditions corresponds to the first element in the eigenvector; represents the azimuth angle of the βth auxiliary correction source with respect to the first phase reference platform; Δr 1 β To calculate the difference between the distance from the βth auxiliary correction source to other correction objects and the distance from the auxiliary correction source to the phase reference platform; θ β Indicates the pitch angle of the βth auxiliary correction source with the first correction object as reference; Get the latest phase error Γ, Γ 2 =Γ 3 …=Γ N ; j represents an imaginary unit; λ represents the wavelength of the signal emitted by the auxiliary correction source; N represents the total number of correction objects; S3-2. Solving a multivariate nonlinear system of equations with only unknown positions using the trust region dogleg algorithm Get the estimated position of each calibration object Among them, 1≤i≤N; S3-3. Estimated position of the calibration object Substitute the estimated directional derivative Get the phase error.
2. The space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source according to claim 1, It is characterized in that The specific implementation of step S1 is as follows: Get the ideal position of the nth correction object as P n =[x n ,y n ,z n ]; Get the position of the mth auxiliary correction source as P m =[x m ,y m ,z m ]; the total number of correction objects is N, n = 1, 2…N; the total number of auxiliary correction sources is M, m = 1, 2,…,M.
3. The space-time correction method for space-based distributed passive coherent reconnaissance based on an external reference source according to claim 2, It is characterized in that The specific implementation of step S2 is as follows: S2-1, calculating the difference Δr between the distance from the mth auxiliary correction source to other correction objects and the distance from the auxiliary correction source to the phase reference platform; S2-2, according to the formula: l m =c / f m Get the wavelength λ of the signal emitted by the mth auxiliary correction source m , and the phase difference between the ideal position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform Among them, Δr 1 m represents the distance from the mth auxiliary correction source to the first correction object, θ m It represents the pitch angle of the mth auxiliary correction source with reference to the first correction object. represents the azimuth of the mth auxiliary correction source with respect to the first correction object, c represents the speed of light, and f m is the transmitted signal s m (t) carrier frequency, (x n ,y n ,z n ) represents the position of the nth calibration object, and π is the circumference of a circle; S2-3, according to the formula: Get N×1 dimensional direction vector a m ; Where exp(·) represents the exponential with the natural constant e as the base, and j represents the imaginary unit; S2-4, according to the formula: x m (t)=a m ·s m (t)+n m (t) Get the mth auxiliary correction source transmission signal s at time t m (t) is the N×1 dimensional vector x output by the calibration object m (t); where a m is an N×1 dimensional direction vector, n m (t) is an N×1-dimensional noise vector; S2-5, according to the formula: Get the position P m =[x m ,y m ,z m The signal data covariance matrix R of the auxiliary correction source emitted by m ; Where E[·] represents the statistical average of the data; is the signal power transmitted by the auxiliary correction source; 2 is the noise power; I is the N×N unit matrix; a m (x m ,y m ,z m ) represents the N×1 dimensional direction vector at the ideal position; Represents the N×1 dimensional direction vector a m (x m ,y m ,z m ) conjugate transpose; is the signal s emitted by the mth auxiliary correction source at time t m The conjugate transpose of (t); is an N×1 dimensional vector x m (t) conjugate transpose; S2-6. Signal data covariance matrix R m Characteristic decomposition and according to the formula: a m (x m ,y m ,z m )=[1a m2 (x m ,y m ,z m )…a mN (x m ,y m ,z m )] T Get the signal data covariance matrix R m Eigendecomposition results in, For R m The eigenvector corresponding to the maximum eigenvalue, [·] T represents transpose, represents a complex constant; a mN (x m ,y m ,z m ) represents the direction vector from the mth auxiliary correction source to the Nth correction object, is the Qth element in the eigenvector corresponding to the maximum eigenvalue, Q = N; S2-7, according to the formula: The estimated directional derivative under the conditions of phase error and position error is obtained by the same method as steps S2-2 to S2-6. in N×1 dimensional direction vector for the estimated position of the correction object; The phase difference between the estimated position of the nth correction object under the mth auxiliary correction source and the reference phase reference platform; is the phase error and position error condition The eigenvector corresponding to the maximum eigenvalue; is the first element in the eigenvector corresponding to the maximum eigenvalue under the conditions of phase error and position error.