CN110376561B - Time domain dimension reduction multi-fast-beat iterative array element amplitude-phase error estimation method - Google Patents

Time domain dimension reduction multi-fast-beat iterative array element amplitude-phase error estimation method Download PDF

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CN110376561B
CN110376561B CN201910482669.1A CN201910482669A CN110376561B CN 110376561 B CN110376561 B CN 110376561B CN 201910482669 A CN201910482669 A CN 201910482669A CN 110376561 B CN110376561 B CN 110376561B
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clutter
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CN110376561A (en
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王彤
马欣
刘程
王萌
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Xidian University
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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
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/40—Means for monitoring or calibrating
    • 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
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
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Abstract

The invention belongs to the technical field of radars, and particularly relates to a time domain dimension reduction multi-fast-beat iterative array element amplitude-phase error estimation method, which comprises the following steps: step 1, according to echo data xlConstructing a base matrix; step 2, obtaining echo data z after dimension reduction by using a time domain dimension reduction methodK,lSum clutter representation basis matrix psiK,l(ii) a Step 3, according to the echo data z after the dimension reductionK,lObtaining the norm of echo data after dimension reduction; step 4, calculating the complex amplitude of the clutter data of each distance unit of the ith iteration
Figure DDA0002084354660000011
Step 5, calculating the optimal estimation e of the array element amplitude phase errors,opt. The method is an array element amplitude and phase error estimation method which has time domain dimension reduction, higher estimation precision and automatic convergence.

Description

Time domain dimension reduction multi-fast-beat iterative array element amplitude-phase error estimation method
Technical Field
The invention belongs to the technical field of radars, and particularly relates to a multi-fast-beat iterative array element amplitude-phase error estimation method for time domain dimension reduction.
Background
With the rapid development of information science and technology, defense has been in the information era. As a key device for winning modern information-based wars, the airborne early warning radar is regarded as an information acquisition interest device capable of controlling battlefield situations by the military of various countries and countries according to the unique strategic characteristics of the airborne early warning radar. The clutter suppression performance is a main factor influencing whether the airborne early warning radar can normally look down to work. The space-time adaptive processing STAP technology firstly proposed by Brennan and Reed can effectively inhibit ground clutter coupled at space time, and is a prototype of the STAP method.
Many of the STAP methods of the subsequent research are built on the basis of ideal models, and the STAP methods do not consider the systematic errors existing in the actual airborne radar system. Common error types of airborne radar systems include array element position error, mutual coupling effect, amplitude-phase error, directional diagram beam pointing error and the like, the system errors can influence clutter suppression performance of an ideal model-based STAP method, and the system errors can cause the problems that a self-adaptive algorithm cannot keep target gain and signal-to-noise-ratio is reduced. The amplitude-phase error of an array element in the system errors is usually generated due to inconsistent amplitude-phase characteristics among all receiving channels of the array antenna under the influence of the precision of hardware equipment and a processing process. The array element amplitude and phase error is an error independent of the direction, and the correction method thereof can be mainly divided into active correction and self-correction. Active correction is a method for off-line correcting array element errors by using an external precisely known auxiliary source, and the method can achieve better effect theoretically, but has higher performance requirement on the auxiliary source and increases the complexity of the system.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a time domain dimension reduction multi-fast-beat iterative array element amplitude and phase error estimation method. The technical problem to be solved by the invention is realized by the following technical scheme:
a time domain dimension reduction multi-fast-beat iterative array element amplitude-phase error estimation method comprises the following steps:
step 1: extracting echo data x of L distance units after pulse compression received by radarlWherein L is 1, 2.. times.l, and constructing a clutter representation base matrix ψ of the L distance unitsl,(l＝1,2,…,L)；
Step 2: echo data x by time domain dimension reduction methodlSum clutter representation basis matrix psilPerforming time domain dimension reduction processing to obtain echo data z after dimension reductionK,lSum clutter representation basis matrix psiK,lWherein L is 1,2,. and L;
and step 3: norm pK | | | Z of received data after initializing time domain dimensionality reductionK||FNoise level σ, iterative difference δKSigma, array element amplitude phase error es＝1N；
And 4, step 4: calculating the complex amplitude of each distance unit clutter data of the ith iteration
Figure BDA0002084354640000021
Figure BDA0002084354640000022
At the same time, according toComplex amplitude of clutter data for the L range bins
Figure BDA0002084354640000023
Calculating array element amplitude phase error es：
Figure BDA0002084354640000024
Wherein,
Figure BDA0002084354640000025
TKrepresenting the array element amplitude phase error tapered matrix subjected to time domain dimension reduction of KN multiplied by KN dimension, because the time domain dimension reduction does not influence the array element amplitude phase error vector, the matrix element amplitude phase error vector is subjected to the time domain dimension reduction
Figure BDA0002084354640000026
1KA vector of all dimensions Kx 1 being 1;
and 5: updating computations
Figure BDA0002084354640000031
Judging whether p is satisfiedKIs greater than sigma and deltaK> 0.01 σ: if yes, executing step 4; if not, the iteration process is ended; wherein ZK＝[zK,1 zK,2 … zK,L]The array element amplitude phase error estimated at the end of iteration is the optimal estimation e of the array element amplitude phase errors,opt。
In one embodiment of the present invention, the step 1 comprises:
spatial domain frequency fsThe expression of (a) is:
Figure BDA0002084354640000032
wherein d represents the array element spacing, λ represents the wavelength, θ represents the azimuth,
Figure BDA0002084354640000033
representing a pitch angle;
definition of
Figure BDA0002084354640000034
The calculation formula of the normalized spatial domain frequency is as follows:
Figure BDA0002084354640000035
wherein f issmIs the maximum spatial domain frequency;
doppler frequency fdThe expression of (a) is:
Figure BDA0002084354640000036
wherein, thetaαRepresenting antenna mounting angle, representing v-carrier speed;
definition of
Figure BDA0002084354640000037
The calculation formula of the normalized spatial domain frequency is as follows:
Figure BDA0002084354640000038
wherein f isrIs the pulse repetition frequency;
from this, the normalized Doppler frequency of the ith clutter scatterer can be obtained
Figure BDA0002084354640000039
Corresponding time domain steering vector
Figure BDA00020843546400000310
The calculation formula is as follows:
Figure BDA00020843546400000311
wherein i ∈ {1,2, …, NcDenotes the ith clutter scatterer, NcRepresenting the number of clutter scatterers in a range unit, M represents the number of transmit pulses in a CPI [ ·]TRepresenting a transpose;
normalized spatial frequency of ith clutter scatterer
Figure BDA00020843546400000312
The corresponding spatial steering vector is
Figure BDA00020843546400000313
The calculation formula is as follows:
Figure BDA00020843546400000314
wherein, N represents the array element number contained in the antenna array, i belongs to {1,2, …, Nc}；
The space-time steering vector si can be obtained from the time domain steering vector and the space domain steering vector of the ith clutter scatterer, and the expression is as follows:
Figure BDA0002084354640000041
wherein,
Figure BDA0002084354640000042
represents the Kronecker product;
MN is multiplied by N according to the formularNcSpace-time steering vector matrix S of the first distance unit of dimensionl：
Figure BDA0002084354640000043
Covariance matrix R due to the l-th range celllAnd SlThe expanded clutter subspaces are the same, and the specific construction mode of the clutter subspaces is as follows: [ U, Σ, V)]＝svd(Sl) Wherein svd (-) represents the singular value decomposition operation, and U and V are SlThe left and right singular vector matrices of (a), sigma is a singular value matrix:
Figure BDA0002084354640000044
wherein, sigma1＝diag(λ1 λ2 … λh) Whose diagonal elements are the singular values of a matrix and which satisfy lambda1≥λ2≥…≥λh≥0，h＝min{MN,NrNc}; number of normal clutter scatterers NrNcIs far larger than MN, so the MN is taken from h; according to sigma1The effective rank g can be estimated; the effective rank of the non-positive side matrix can be obtained by:
Figure BDA0002084354640000045
wherein eta ∈ [0,1 ]]The value of the effective rank g is that eta is more than or equal to eta0Is a minimum integer of [, ] n0The threshold value is close to 1, and the clutter subspace can be obtained by approximation through the value taking mode;
the effective rank of the positive side matrix can be obtained by:
Figure BDA0002084354640000046
wherein β is 2v/λ fr；
The feature vector corresponding to the large feature value belongs to the clutter subspace SubcNamely, the low-rank approximation of the clutter subspace can be realized by selecting the first g columns of the left singular vector matrix U:
Figure BDA0002084354640000047
ψlu (: 1: g), whereinlIs the clutter representation base matrix to be constructed,
Figure BDA0002084354640000048
representative matrix psilThe column space of (a).
In one embodiment of the present invention, the step 2 comprises:
echo data x of L distance units by using time domain dimension reduction methodlSum clutter representation basis matrix psilPerforming dimensionality reduction treatment, wherein L is 1,2,.., L; let Q be a dimension reduction matrix, since dimension reduction is only performed in the time domain here, and only K Doppler channels occupied by the main clutter are selected, the formula is:
Figure BDA0002084354640000051
wherein Q istIs a time domain dimension reduction matrix with dimension of M multiplied by K, M, K is the dimension before and after dimension reduction, INIs a unit array of dimension NxN;
actual received data x for airborne radarlPerforming time domain dimensionality reduction pretreatment to obtain dimensionality reduced data as follows: z is a radical ofK,l＝QHxl；
The space-time steering vector after dimensionality reduction is as follows: sz,l＝QHsl；
Clutter representation base matrix psi after dimension reductionz,l：[Uz,Σz,Vz]＝svd(Sz,l)，ψK,l＝Uz(:,1:gz) Wherein g iszFor effective rank after dimensionality reduction, Sz,l＝QHSlAnd representing a space-time steering vector matrix formed by the first distance unit after the time domain dimension reduction and all clutter scatterers of the corresponding fuzzy distance unit.
In one embodiment of the present invention, the step 3 comprises:
ZK＝[zK,1 zK,2 … zK,L]representing data obtained by time domain dimensionality reduction of L sampling units, with dimensions KN multiplied by L, zK,lThe data is the time domain dimensionality reduced data of the l-th sampling unit with KN multiplied by 1 dimensionality.
In one embodiment of the present invention, the step 4 comprises:
establishing a cost function:
Figure BDA0002084354640000052
wherein,
ZK＝[zK,1 zK,2 … zK,L]representing the data obtained by time domain dimensionality reduction of L sampling units from MN × L to KN × L, zK,lData after time domain dimensionality reduction for the l-th sampling unit of KN x 1 dimension, psiK,lRepresenting clutter base matrix after the time domain dimensionality reduction of the l-th sampling unit, with the dimensionality of KN multiplied by gz，
Figure BDA0002084354640000053
Representing clutter complex amplitude vector after time domain dimensionality reduction of the ith sampling unit to be estimated, wherein the dimensionality is gz×1，
TKRepresenting a time domain dimension reduction array element amplitude phase error tapering matrix of KN multiplied by KN dimension,
Figure BDA0002084354640000061
unfolding the above equation yields:
Figure BDA0002084354640000062
the Frobenius norm of the matrix can be changed to a 2 norm of the vector, and this property is:
Figure BDA0002084354640000063
wherein | · | purple sweet2Is the 2 norm of the vector, n is the number of columns of the matrix Γ;
further obtainable from the above formula:
Figure BDA0002084354640000064
wherein z isK,lSelecting time domain dimensionality-reduced data of a Doppler channel occupied by K main clutters of data received by an l-th sampling unit, wherein the dimensionality is KN multiplied by 1;
the above equation can be equivalent to L independent optimization problems, namely:
Figure BDA0002084354640000065
firstly, solving complex amplitude vector of clutter data after time domain dimension reduction
Figure BDA0002084354640000066
Hypothetical error tapering matrix TKKnown, we obtain:
Figure BDA0002084354640000067
because of the fact that
Figure BDA0002084354640000068
Is KN × gzDimensional and satisfies the relationship KN > gzFor this over-determined equation, a unique solution can be found:
Figure BDA0002084354640000069
the estimated clutter data after the time domain dimensionality reduction can be obtained by the following formula:
Figure BDA00020843546400000610
reconstructed time domain dimension reduction clutter data matrix of L sampling units
Figure BDA00020843546400000611
Can be expressed as:
Figure BDA00020843546400000612
according to what has been estimated
Figure BDA00020843546400000613
To solve the array element amplitude phase error vector es(ii) a Reducing the dimension of the time domain to clutter data matrix
Figure BDA00020843546400000614
Substituting into the initially established cost function yields:
Figure BDA00020843546400000615
the above formula is expanded as:
Figure BDA0002084354640000071
due to the fact that
Figure BDA0002084354640000072
The above equation can be further expanded as:
Figure BDA0002084354640000073
according to
Figure BDA0002084354640000074
The formula is further simplified, and the following can be obtained:
Figure BDA0002084354640000075
converting the above formula into a 2 norm form of a vector and sorting the vector, the expression of the final optimization function is:
Figure BDA0002084354640000076
solving the optimization function can obtain:
Figure BDA0002084354640000077
the invention has the beneficial effects that:
1. the invention utilizes the data of a plurality of distance units, and simulation experiments prove that the array element amplitude-phase errors estimated by the data of the plurality of distance units are more accurate than the array element amplitude-phase errors estimated by the data of a single distance unit, and the established cost function considers the condition that clutter distributions of different distance unit data are possibly different.
2. The invention utilizes a time domain dimension reduction method to reduce the dimension of the echo data and the clutter representation base matrix, and can effectively reduce the calculated amount of the algorithm.
3. The invention designs an iteration automatic stop criterion, can automatically converge, and avoids the situations of algorithm non-convergence and array element phase error estimation inaccuracy caused by insufficient iteration times, so that the estimation is more accurate.
The present invention will be described in further detail with reference to the accompanying drawings and examples.
Drawings
FIG. 1 is a block diagram of a flow chart of a method for estimating an amplitude-phase error of a multi-fast-beat iterative array with time domain dimension reduction according to an embodiment of the present invention;
fig. 2 is a relationship curve between an amplitude error RMSE and a sample number in the time-domain dimension-reduction multi-fast-beat iterative array element amplitude-phase error estimation method according to the embodiment of the present invention;
fig. 3 is a relationship curve between a phase error RMSE and a sample number in the time-domain dimension-reduction multi-fast-beat iterative array element amplitude-phase error estimation method according to the embodiment of the present invention;
fig. 4 is a curve of a relationship between an amplitude error RMSE and a number of pulses in the time-domain dimension-reduction multi-fast-beat iterative array element amplitude-phase error estimation method provided by the embodiment of the present invention;
fig. 5 is a curve of a relationship between a phase error RMSE and a number of pulses in the time-domain dimension-reduction multi-fast-beat iterative array element amplitude-phase error estimation method provided by the embodiment of the present invention;
FIG. 6 is a graph showing a relationship between an amplitude error RMSE and a noise-to-noise ratio in a time-domain dimension-reduction multi-fast-beat iterative array element amplitude-phase error estimation method according to an embodiment of the present invention;
fig. 7 is a relationship curve between a phase error RMSE and a noise-to-noise ratio in the time-domain dimension-reduction multi-fast-beat iterative array element amplitude-phase error estimation method provided by the embodiment of the present invention;
fig. 8 is a relationship curve of the number of pulses in the operation time domain in the method for estimating the amplitude-phase error of the multi-fast-beat iterative array element for time domain dimension reduction according to the embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to specific examples, but the embodiments of the present invention are not limited thereto.
A time domain dimension reduction multi-fast-beat iterative array element amplitude-phase error estimation method is shown in figure 1, and comprises the following steps:
step 1: extracting echo data x of L distance units after pulse compression received by radarlWherein L is 1, 2.. times.l, and constructing a clutter representation base matrix ψ of the L distance unitsl,(l＝1,2,…,L)；
Step 2: echo data x by time domain dimension reduction methodlSum clutter representation basis matrix psilPerforming time domain dimension reduction to obtain a reductionEcho data z after dimensionK,lSum clutter representation basis matrix psiK,lWherein L is 1,2,. and L;
and step 3: initializing norm p of received data after time domain dimensionality reductionK＝||ZK||FNoise level σ, iterative difference δKSigma, array element amplitude phase error es＝1N；
And 4, step 4: calculating the complex amplitude of each distance unit clutter data of the ith iteration
Figure BDA0002084354640000091
Figure BDA0002084354640000092
At the same time, the complex amplitude of the clutter data according to the L distance units
Figure BDA0002084354640000093
Calculating array element amplitude phase error es：
Figure BDA0002084354640000094
Wherein,
Figure BDA0002084354640000095
TKrepresenting the array element amplitude phase error tapered matrix subjected to time domain dimension reduction of KN multiplied by KN dimension, because the time domain dimension reduction does not influence the array element amplitude phase error vector, the matrix element amplitude phase error vector is subjected to the time domain dimension reduction
Figure BDA0002084354640000096
1KA vector of all dimensions Kx 1 being 1;
and 5: updating computations
Figure BDA0002084354640000097
Judging whether p is satisfiedKIs greater than sigma and deltaK> 0.01 σ: if yes, executing step 4; if not, the iteration process is ended; wherein ZK＝[zK,1 zK,2 … zK,L]Estimated at the end of the iterationThe array element amplitude phase error is the optimal estimation e of the array element amplitude phase errors,opt。
Specifically, after the amplitude-phase error of the array element is accurately estimated, the radar system can correct the error and adopt the STAP technology to carry out effective clutter suppression. The method is an array element amplitude and phase error estimation method based on airborne radar echo data, when only a single data sample is available, a good array element amplitude and phase error estimation effect can be obtained, and the array element amplitude and phase error estimation performance is improved along with the increase of the number of echo data samples.
Furthermore, the invention is an array element amplitude and phase error estimation method which has time domain dimension reduction, higher estimation precision and automatic convergence, and the dimension of the received echo data and the dimension of the clutter representation base matrix are reduced by using the time domain dimension reduction method, so that the calculated amount of the algorithm can be effectively reduced. By utilizing the data of a plurality of range units, the performance of array element amplitude-phase error estimation is better than that of the data of only one range unit, and the condition that clutter distributions of different range unit data are possibly different is considered by the established cost function, so that the method is not only suitable for the positive side array, but also suitable for a non-positive side array airborne radar system. By employing an iterative auto-stop criterion, the algorithm can automatically converge.
In one embodiment of the present invention, the step 1 comprises:
spatial domain frequency fsThe expression of (a) is:
Figure BDA0002084354640000101
wherein d represents the array element spacing, λ represents the wavelength, θ represents the azimuth,
Figure BDA0002084354640000102
representing a pitch angle;
definition of
Figure BDA0002084354640000103
The calculation formula of the normalized spatial domain frequency is as follows:
Figure BDA0002084354640000104
wherein f issmIs the maximum spatial domain frequency;
doppler frequency fdThe expression of (a) is:
Figure BDA0002084354640000105
wherein, thetaαRepresenting antenna mounting angle, representing v-carrier speed;
definition of
Figure BDA0002084354640000106
The calculation formula of the normalized spatial domain frequency is as follows:
Figure BDA0002084354640000107
wherein f isrIs the pulse repetition frequency;
from this, the normalized Doppler frequency of the ith clutter scatterer can be obtained
Figure BDA0002084354640000108
Corresponding time domain steering vector
Figure BDA0002084354640000111
The calculation formula is as follows:
Figure BDA0002084354640000112
wherein i ∈ {1,2, …, NcDenotes the ith clutter scatterer, NcRepresenting the number of clutter scatterers in a range unit, M represents the number of transmit pulses in a CPI [ ·]TRepresenting a transpose;
normalized spatial frequency of ith clutter scatterer
Figure BDA0002084354640000113
The corresponding spatial steering vector is
Figure BDA0002084354640000114
The calculation formula is as follows:
Figure BDA0002084354640000115
wherein, N represents the array element number contained in the antenna array, i belongs to {1,2, …, Nc}；
The space-time steering vector s of the ith clutter scatterer can be obtained from the time domain steering vector and the space domain steering vector of the ith clutter scattereriThe expression is as follows:
Figure BDA0002084354640000116
wherein,
Figure BDA0002084354640000117
represents the Kronecker product;
MN is multiplied by N according to the formularNcSpace-time steering vector matrix S of the first distance unit of dimensionl：
Figure BDA0002084354640000118
Covariance matrix R due to the l-th range celllAnd SlThe expanded clutter subspaces are the same, and the specific construction mode of the clutter subspaces is as follows: [ U, Σ, V)]＝svd(Sl) Wherein svd (-) represents the singular value decomposition operation, and U and V are SlThe left and right singular vector matrices of (a), sigma is a singular value matrix:
Figure BDA0002084354640000119
wherein, sigma1＝diag(λ1 λ2 … λh) Whose diagonal elements are the singular values of a matrix and which satisfy lambda1≥λ2≥…≥λh≥0，h＝min{MN,NrNc}; number of normal clutter scatterers NrNcIs far larger than MN, so the MN is taken from h; according to sigma1The effective rank g can be estimated; the effective rank of the non-positive side matrix can be obtained by:
Figure BDA00020843546400001110
wherein eta ∈ [0,1 ]]The value of the effective rank g is that eta is more than or equal to eta0Is smallest inInteger, η0The threshold value is close to 1, and the clutter subspace can be obtained by approximation through the value taking mode;
the effective rank of the positive side matrix can be obtained by:
Figure BDA0002084354640000121
wherein β is 2v/λ fr；
The feature vector corresponding to the large feature value belongs to the clutter subspace SubcNamely, the low-rank approximation of the clutter subspace can be realized by selecting the first g columns of the left singular vector matrix U:
Figure BDA0002084354640000122
ψlu (: 1: g), whereinlIs the clutter representation base matrix to be constructed,
Figure BDA0002084354640000123
representative matrix psilThe column space of (a).
In one embodiment of the present invention, the step 2 comprises:
echo data x of L distance units by using time domain dimension reduction methodlSum clutter representation basis matrix psilPerforming dimensionality reduction treatment, wherein L is 1,2,.., L; let Q be a dimension reduction matrix, since dimension reduction is only performed in the time domain here, and only K Doppler channels occupied by the main clutter are selected, the formula is:
Figure BDA0002084354640000124
wherein Q istIs a time domain dimension reduction matrix with dimension of M multiplied by K, M, K is the dimension before and after dimension reduction, INIs a unit array of dimension NxN;
actual received data x for airborne radarlPerforming time domain dimensionality reduction pretreatment to obtain dimensionality reduced data as follows: z is a radical ofK,l＝QHxl；
The space-time steering vector after dimensionality reduction is as follows: sz,l＝QHsl；
Clutter representation base matrix psi after dimension reductionz,l：[Uz,Σz,Vz]＝svd(Sz,l)，ψK,l＝Uz(:,1:gz)；
Wherein, gzFor effective rank after dimensionality reduction, Sz,l＝QHSlAnd representing a space-time steering vector matrix formed by the first distance unit after the time domain dimension reduction and all clutter scatterers of the corresponding fuzzy distance unit.
In one embodiment of the present invention, the step 3 comprises:
ZK＝[zK,1 zK,2 … zK,L]representing data obtained by time domain dimensionality reduction of L sampling units, with dimensions KN multiplied by L, zK,lThe data is the time domain dimensionality reduced data of the l-th sampling unit with KN multiplied by 1 dimensionality.
In one embodiment of the present invention, the step 4 comprises:
establishing a cost function:
Figure BDA0002084354640000131
wherein Z isK＝[zK,1 zK,2 … zK,L]Representing the data obtained by time domain dimensionality reduction of L sampling units from MN × L to KN × L, zK,lData after time domain dimensionality reduction for the l-th sampling unit of KN x 1 dimension, psiK,lRepresenting clutter base matrix after the time domain dimensionality reduction of the l-th sampling unit, with the dimensionality of KN multiplied by gz，
Figure BDA0002084354640000132
Representing clutter complex amplitude vector after time domain dimensionality reduction of the ith sampling unit to be estimated, wherein the dimensionality is gz×1，TKRepresenting the array element amplitude and phase error tapered matrix subjected to time domain dimension reduction of KN dimension, because the time domain dimension reduction does not influence the array element amplitude and phase error vector,
Figure BDA0002084354640000133
unfolding the above equation yields:
Figure BDA0002084354640000134
the Frobenius norm of the matrix can be changed to a 2 norm of the vector, and this property is:
Figure BDA0002084354640000135
wherein | · | purple sweet2Is the 2 norm of the vector, n is the number of columns of the matrix Γ;
further obtainable from the above formula:
Figure BDA0002084354640000136
wherein z isK,lSelecting time domain dimensionality-reduced data of a Doppler channel occupied by K main clutters of data received by an l-th sampling unit, wherein the dimensionality is KN multiplied by 1;
the above equation can be equivalent to L independent optimization problems, namely:
Figure BDA0002084354640000137
firstly, solving complex amplitude vector of clutter data after time domain dimension reduction
Figure BDA0002084354640000138
Hypothetical error tapering matrix TKKnown, we obtain:
Figure BDA0002084354640000139
because of the fact that
Figure BDA00020843546400001310
Is KN × gzDimensional and satisfies the relationship KN > gzFor this over-determined equation, a unique solution can be found:
Figure BDA00020843546400001311
the estimated clutter data after the time domain dimensionality reduction can be obtained by the following formula:
Figure BDA00020843546400001312
reconstructed time domain dimension reduction clutter data matrix of L sampling units
Figure BDA00020843546400001313
Can be expressed as:
Figure BDA0002084354640000141
according to what has been estimated
Figure BDA0002084354640000142
To solve the array element amplitude phase error vector es(ii) a Reducing the dimension of the time domain to clutter data matrix
Figure BDA0002084354640000143
Substituting into the initially established cost function yields:
Figure BDA0002084354640000144
the above formula is expanded as:
Figure BDA0002084354640000145
due to the fact that
Figure BDA0002084354640000146
The above equation can be further expanded as:
Figure BDA0002084354640000147
according to
Figure BDA0002084354640000148
The formula is further simplified, and the following can be obtained:
Figure BDA0002084354640000149
converting the above formula into a 2 norm form of a vector and sorting the vector, the expression of the final optimization function is:
Figure BDA00020843546400001410
solving the optimization function can obtain:
Figure BDA00020843546400001411
further, when the radar adopts a distance sampling frequency fs2MHz, wavelength lambda of 0.2m, pulse repetition frequency fr2500Hz, 6378km of the earth curvature radius R, 6km of the carrier height H, 125m/s of the carrier speed V, 10 antenna receiving channels, 0.5 times of the wavelength of the array element spacing, d/lambda less than or equal to 0.5, no grating lobe of an antenna directional diagram, 90 degrees of an included angle between the main beam direction and the array surface and a main beam pitch angle
Figure BDA00020843546400001412
Is 0 deg.. The noise to noise ratio is 50dB and the number of pulses is 128 as shown in fig. 2 and 3. In the single sample case, the amplitude error RMSE of the inventive method is-31.3195 and the phase error RMSE is 0.0238. It can be seen that the performance of array element magnitude-phase error estimation using multiple range gate data is better than that of a single sample. The array element amplitude and phase error estimation performance of the method is better and better along with the increase of the number of samples, and the performance of the method gradually tends to be stable when the number of samples is large. The method uses a time domain dimension reduction processing process, reduces the calculated amount, and simultaneously more accurately estimates the amplitude-phase error of the array element, and has obvious advantages.
Still further, when the noise-to-noise ratio is 50dB, the number of samples is 100, as shown in fig. 4 and 5. It can be seen that the method of the present invention has good performance when the number of pulses is small, because the method estimates the amplitude-phase error of the array element by fitting the received space-time data, and the doppler resolution does not have a great influence on the method. The method can accurately estimate the amplitude-phase error of the array element when the number of pulses is small, and the performance of the method is improved along with the increase of the number of pulses.
Further, when the number of pulses is 64, the number of samples is 100, and fig. 6 and 7 are evaluation results. It can be seen that when the noise-to-noise ratio is relatively low, the performance of the method of the present invention is degraded, and at this time, relatively high noise power may affect the clutter component in the echo data, which may result in the reduction of the amplitude-phase consistency of the clutter on different array elements, and may cause difficulty in the estimation process of the array element amplitude-phase error. With the increase of the noise-to-noise ratio, the estimation performance of the array element amplitude-phase error of the method is improved.
Further, when the noise-to-noise ratio is 50dB, the number of samples is 20, and fig. 8 is an evaluation result. It can be seen that the operation time of the method is less than that of the full-dimensional array element amplitude-phase error estimation method, and the effect of reducing the operation time of the method is more obvious along with the increase of the pulse number.
The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (4)

1. A multi-fast-beat iterative array element amplitude-phase error estimation method for time domain dimension reduction is characterized by comprising the following steps:
step 1: extracting echo data x of L distance units after pulse compression received by radarlWherein L is 1, 2.. times.l, and constructing a clutter representation base matrix ψ of the L distance unitsl,(l＝1,2,…,L)；
Step 2: echo data x by time domain dimension reduction methodlSum clutter representation basis matrix psilPerforming time domain dimension reduction processing to obtain echo data z after dimension reductionK,lSum clutter representation basis matrix psiK,lWherein L is 1,2,. and L; the above-mentionedThe time domain dimension reduction method is used for reducing the dimension of the echo data and the clutter representation base matrix so as to reduce the calculated amount;
and step 3: initializing norm p of received data after time domain dimensionality reductionK＝||ZK||FNoise level σ, iterative difference δKSigma, array element amplitude phase error es＝1N；
And 4, step 4: calculating the complex amplitude of each distance unit clutter data of the ith iteration
Figure FDA0002957154590000011
Figure FDA0002957154590000012
At the same time, the complex amplitude of the clutter data according to the L distance units
Figure FDA0002957154590000013
Calculating array element amplitude phase error es：
Figure FDA0002957154590000014
Wherein,
Figure FDA0002957154590000015
TKrepresenting the array element amplitude phase error tapered matrix subjected to time domain dimension reduction of KN multiplied by KN dimension, because the time domain dimension reduction does not influence the array element amplitude phase error vector, the matrix element amplitude phase error vector is subjected to the time domain dimension reduction
Figure FDA0002957154590000016
1KA vector of all dimensions Kx 1 being 1;
and 5: updating computations
Figure FDA0002957154590000017
Judging whether p is satisfiedKIs greater than sigma and deltaK> 0.01 σ: if yes, executing step 4; if not, the iteration process is ended; wherein ZK＝[zK,1 zK,2 … zK,L]The array element amplitude phase error estimated at the end of iteration is the optimal estimation e of the array element amplitude phase errors,opt；
The step 1 comprises the following steps:
spatial domain frequency fsThe expression of (a) is:
Figure FDA0002957154590000021
wherein d represents the array element spacing, λ represents the wavelength, θ represents the azimuth,
Figure FDA0002957154590000022
representing a pitch angle;
definition of
Figure FDA0002957154590000023
The calculation formula of the normalized spatial domain frequency is as follows:
Figure FDA0002957154590000024
wherein f issmIs the maximum spatial domain frequency;
doppler frequency fdThe expression of (a) is:
Figure FDA0002957154590000025
wherein, thetaαRepresenting antenna mounting angle, representing v-carrier speed;
definition of
Figure FDA0002957154590000026
The calculation formula of the normalized spatial domain frequency is as follows:
Figure FDA0002957154590000027
wherein f isrIs the pulse repetition frequency;
from this, the normalized Doppler frequency of the ith clutter scatterer can be obtained
Figure FDA0002957154590000028
Corresponding time domain steering vector
Figure FDA0002957154590000029
The calculation formula is as follows:
Figure FDA00029571545900000210
wherein i ∈ {1,2, …, NcDenotes the ith clutter scatterer, NcRepresenting the number of clutter scatterers in a range unit, M represents the number of transmit pulses in a CPI [ ·]TRepresenting a transpose;
normalized spatial frequency of ith clutter scatterer
Figure FDA00029571545900000211
The corresponding spatial steering vector is
Figure FDA00029571545900000212
The calculation formula is as follows:
Figure FDA00029571545900000213
wherein, N represents the array element number contained in the antenna array, i belongs to {1,2, …, Nc}；
The space-time steering vector s of the ith clutter scatterer can be obtained from the time domain steering vector and the space domain steering vector of the ith clutter scattereriThe expression is as follows:
Figure FDA00029571545900000214
wherein,
Figure FDA00029571545900000215
represents the Kronecker product;
MN is multiplied by N according to the formularNcSpace-time steering vector matrix S of the first distance unit of dimensionl：
Figure FDA0002957154590000031
Covariance matrix R due to the l-th range celllAnd SlThe expanded clutter subspaces are the same, and the specific construction mode of the clutter subspaces is as follows: [ U, Σ, V)]＝svd(Sl) Wherein svd (-) represents the singular value decomposition operation, and U and V are SlThe left and right singular vector matrices of (a), sigma is a singular value matrix:
Figure FDA0002957154590000032
wherein, sigma1＝diag(λ1 λ2 … λh) Whose diagonal elements are the singular values of a matrix and which satisfy lambda1≥λ2≥…≥λh≥0，h＝min{MN,NrNc}; number of normal clutter scatterers NrNcIs far larger than MN, so the MN is taken from h; according to sigma1The effective rank g can be estimated; the effective rank of the non-positive side matrix can be obtained by:
Figure FDA0002957154590000033
wherein eta ∈ [0,1 ]]The value of the effective rank g is that eta is more than or equal to eta0Is a minimum integer of [, ] n0The threshold value is close to 1, and the clutter subspace can be obtained by approximation through the value taking mode;
the effective rank of the positive side matrix can be obtained by:
Figure FDA0002957154590000034
wherein β is 2v/λ fr；
The feature vector corresponding to the large feature value belongs to the clutter subspace SubcNamely, the low-rank approximation of the clutter subspace can be realized by selecting the first g columns of the left singular vector matrix U:
Figure FDA0002957154590000035
ψlu (: 1: g), whereinlIs the clutter representation base matrix to be constructed,
Figure FDA0002957154590000036
representative matrix psilThe column space of (a).
2. The method for estimating the amplitude and phase errors of a multi-fast-beat iterative array element for time domain dimension reduction according to claim 1, wherein the step 2 comprises:
echo data x of L distance units by using time domain dimension reduction methodlSum clutter representation basis matrix psilPerforming dimensionality reduction treatment, wherein L is 1,2,.., L; let Q be a dimension reduction matrix, since dimension reduction is only performed in the time domain here, and only K Doppler channels occupied by the main clutter are selected, the formula is:
Figure FDA0002957154590000041
wherein Q istIs a time domain dimension reduction matrix with dimension of M multiplied by K, M, K is the dimension before and after dimension reduction, INIs a unit array of dimension NxN;
actual received data x for airborne radarlPerforming time domain dimensionality reduction pretreatment to obtain dimensionality reduced data as follows: z is a radical ofK,l＝QHxl；
The space-time steering vector after dimensionality reduction is as follows: sz,l＝QHsl；
Clutter representation base matrix psi after dimension reductionz,l：[Uz,Σz,Vz]＝svd(Sz,l)；ψK,l＝Uz(:,1:gz)；
Wherein, gzFor effective rank after dimensionality reduction, Sz,l＝QHSlAnd representing a space-time steering vector matrix formed by the first distance unit after the time domain dimension reduction and all clutter scatterers of the corresponding fuzzy distance unit.
3. The method for estimating the amplitude and phase errors of a multi-fast-beat iterative array element for time domain dimension reduction according to claim 1, wherein the step 3 comprises:
ZK＝[zK,1 zK,2 … zK,L]representing data obtained by time domain dimensionality reduction of L sampling units, with dimensions KN multiplied by L, zK,lThe data is the time domain dimensionality reduced data of the l-th sampling unit with KN multiplied by 1 dimensionality.
4. The method for estimating the amplitude and phase errors of a multi-fast-beat iterative array element for time domain dimension reduction according to claim 1, wherein the step 4 comprises:
establishing a cost function:
Figure FDA0002957154590000042
wherein Z isK＝[zK,1 zK,2 … zK,L]Representing the data obtained by time domain dimensionality reduction of L sampling units from MN × L to KN × L, zK,lData after time domain dimensionality reduction for the l-th sampling unit of KN x 1 dimension, psiK,lRepresenting clutter base matrix after the time domain dimensionality reduction of the l-th sampling unit, with the dimensionality of KN multiplied by gz，
Figure FDA0002957154590000043
Representing clutter complex amplitude vector after time domain dimensionality reduction of the ith sampling unit to be estimated, wherein the dimensionality is gz×1，TKRepresenting the array element amplitude phase error taper moment through time domain dimension reduction of KN multiplied by KN dimension,
Figure FDA0002957154590000051
unfolding the above equation yields:
Figure FDA0002957154590000052
the Frobenius norm of the matrix can be changed to a 2 norm of the vector, and this property is:
Figure FDA0002957154590000053
wherein | · | purple sweet2Is a 2 norm of a vector, n being the matrix ΓThe number of columns;
further obtainable from the above formula:
Figure FDA0002957154590000054
wherein z isK,lSelecting time domain dimensionality-reduced data of a Doppler channel occupied by K main clutters of data received by an l-th sampling unit, wherein the dimensionality is KN multiplied by 1;
the above equation can be equivalent to L independent optimization problems, namely:
Figure FDA0002957154590000055
firstly, solving complex amplitude vector of clutter data after time domain dimension reduction
Figure FDA0002957154590000056
Hypothetical error tapering matrix TKKnown, we obtain:
Figure FDA0002957154590000057
because of the fact that
Figure FDA0002957154590000058
Is KN × gzDimensional and satisfies the relationship KN > gzFor this over-determined equation, a unique solution can be found:
Figure FDA0002957154590000059
the estimated clutter data after the time domain dimensionality reduction can be obtained by the following formula:
Figure FDA00029571545900000510
reconstructed time domain dimension reduction clutter data matrix of L sampling units
Figure FDA00029571545900000511
Can be expressed as:
Figure FDA00029571545900000512
according to what has been estimated
Figure FDA00029571545900000513
To solve the array element amplitude phase error vector es(ii) a Reducing the dimension of the time domain to clutter data matrix
Figure FDA00029571545900000514
Substituting into the initially established cost function yields:
Figure FDA00029571545900000515
the above formula is expanded as:
Figure FDA00029571545900000516
due to the fact that
Figure FDA00029571545900000517
The above equation can be further expanded as:
Figure FDA0002957154590000061
according to
Figure FDA0002957154590000062
The formula is further simplified, and the following can be obtained:
Figure FDA0002957154590000063
converting the above formula into a 2 norm form of a vector and sorting the vector, the expression of the final optimization function is:
Figure FDA0002957154590000064
solving the optimization function can obtain:
Figure FDA0002957154590000065
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