CN112630784B - Plane array amplitude-phase error correction method based on convex optimization and neural network - Google Patents

Plane array amplitude-phase error correction method based on convex optimization and neural network Download PDF

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CN112630784B
CN112630784B CN202011411528.XA CN202011411528A CN112630784B CN 112630784 B CN112630784 B CN 112630784B CN 202011411528 A CN202011411528 A CN 202011411528A CN 112630784 B CN112630784 B CN 112630784B
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CN112630784A (en
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蒋荣欣
刘雪松
辜博轩
陈耀武
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Zhejiang University ZJU
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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
    • G01S15/00—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
    • G01S15/88—Sonar systems specially adapted for specific applications
    • G01S15/89—Sonar systems specially adapted for specific applications for mapping or imaging
    • 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
    • G01S7/52—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00
    • G01S7/52004—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
    • G01S7/52—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00
    • G01S7/539—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section

Abstract

The invention discloses a plane array amplitude-phase error correction method based on convex optimization and a neural network, which comprises the following steps: (1) Setting a correction sound source to be positioned in an unknown far-field direction relative to a sensor plane array, converting a direction-of-arrival estimation problem of the correction sound source into an optimization problem for solving a correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to primarily estimate the direction-of-arrival of the correction sound source; (2) Determining a corrected sound source position according to the primarily estimated direction of arrival of the step (1), acquiring beam intensity in the field range with the corrected sound source position as the center, and further estimating according to the beam intensity by utilizing a direction of arrival estimation model constructed based on a deep learning network to acquire the finally estimated direction of arrival of the corrected sound source; (3) The amplitude-phase error is estimated by a spatial matched filter based on the finally estimated direction of arrival of the corrected sound source. The method can accurately estimate and correct the amplitude and phase errors.

Description

Plane array amplitude-phase error correction method based on convex optimization and neural network
Technical Field
The invention relates to the technical fields of phased array three-dimensional imaging sonar systems, compressed sensing, convex optimization, neural networks, spatial filtering and the like, in particular to a plane array amplitude-phase error correction method based on convex optimization and the neural networks.
Background
The real-time three-dimensional sonar imaging technology is widely applied to the fields of underwater detection and the like in recent years. And the phased array three-dimensional imaging sonar system transmits an acoustic pulse signal, receives sonar echo signals through a large-scale planar array, and obtains a beam pattern through beam forming calculation. In phased array sonar beamforming algorithms, it is generally assumed that each sensor channel has consistent amplitude and phase characteristics. However, deviations in sensor position, inconsistencies in sensor and signal conditioning circuit performance, inter-channel coupling effects can lead to amplitude and phase errors in the received signal, resulting in increased sidelobe intensity in the beam pattern and a shift in focus direction. To solve this problem, it is necessary to correct the amplitude and phase of the sensor array reception model to compensate for the error.
The array amplitude-phase error correction method can be classified into active correction, which requires one or more correction sound sources whose accurate positions are known, and self-correction, which requires simultaneous estimation of the direction of arrival (DOA) of the correction sound source and the amplitude-phase error of the array, whose positions are unknown. The active correction method corrects the amplitude-phase characteristics of the array by a sound source with a known position, and mainly comprises a Maximum Likelihood Estimation (MLE) and a least square method (LS). The active correction method has higher estimation precision, but in practical application, the position of a correction source is difficult to accurately position, and the active correction method has great limitation in practical application, so that the array self-correction of the position of the unknown correction source is a main research direction. Array self-correction first requires DOA estimation, the estimation algorithm includes: spectrum searching, such as multiple signal classification (MUSIC) algorithms; signal parameter estimation rotation invariant technique (ESPRIT); a Toeplitz (TB) algorithm utilizing a data covariance matrix toeplitz structure; three-step iterative (TSI) algorithms, and the like. The DOA estimation method can be optimized by means of the auxiliary sensor array and the steering control. However, arrays of phased array sonar systems are often in underwater closed environments and methods that utilize auxiliary arrays and steering control devices are difficult to apply in practical scenarios.
Recently, a DOA estimation method based on Compressed Sensing (CS) has received a lot of attention. The CS method is a signal processing technology for reconstructing sparse signals by solving an underdetermined linear system, in DOA estimation, the CS method solves sound source signals through signal reconstruction received by a sensor, the solved solutions are on discrete grid points which are initially set, when a correction source is not on the grid points, the compressed sensing method has a plurality of solutions on the grid points which are close to the direction of the correction source, and the estimation precision is limited. In addition, the deep learning-based method is also applied to the DOA estimation and error self-correction fields, and the DOA and amplitude-phase errors can be estimated by using a Deep Neural Network (DNN) or a Convolutional Neural Network (CNN). However, the estimation accuracy of the neural network-based method is difficult to reach the conventional method due to the presence of a determined propagation model between the sampled signal and the acoustic source signal. After DOA estimation is completed, the amplitude-phase error is estimated by using a spatial matched filter based on the estimation result.
Disclosure of Invention
In view of the foregoing, an object of the present invention is to provide a planar array amplitude-phase error correction method based on convex optimization and neural network, for self-correction of large planar arrays.
In order to achieve the above purpose, the technical scheme provided by the invention is as follows:
a plane array amplitude-phase error correction method based on convex optimization and a neural network comprises the following steps:
(1) Setting a correction sound source to be positioned in an unknown far-field direction relative to a sensor plane array, converting a direction-of-arrival estimation problem of the correction sound source into an optimization problem for solving a correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to primarily estimate the direction-of-arrival of the correction sound source;
(2) Determining a corrected sound source position according to the primarily estimated direction of arrival of the step (1), acquiring beam intensity in the field range with the corrected sound source position as the center, and further estimating according to the beam intensity by utilizing a direction of arrival prediction model constructed based on a deep learning network to acquire the finally estimated direction of arrival of the corrected sound source;
(3) The amplitude-phase error is estimated by a spatial matched filter based on the finally estimated direction of arrival of the corrected sound source.
Compared with the prior art, the invention has the beneficial effects that at least the following steps are included:
according to the plane array amplitude-phase error correction method for the convex optimization and the neural network, the direction of arrival of the primarily estimated and corrected sound source is solved through the convex optimization method, then the direction of arrival estimation model constructed based on the deep learning network is utilized to accurately estimate according to the wave beam intensity so as to obtain the final direction of arrival of the corrected sound source, the accuracy of the direction of arrival of the corrected sound source is guaranteed through two times of estimation, and on the basis, the amplitude-phase error is estimated through a spatial matching filter, and the accuracy of the amplitude-phase error is improved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a planar array amplitude-phase error correction method based on convex optimization and a neural network provided by an embodiment of the invention;
FIG. 2 is a schematic diagram of a sensor planar array and a corrected acoustic source provided by an embodiment of the present invention;
FIG. 3 is a schematic diagram of a result of estimating DOA by the CVX method according to an embodiment of the present invention;
FIG. 4 is a beam pattern within a 5X 5 domain centered on the corrected sound source location provided by an embodiment of the present invention;
FIG. 5 is a block diagram of a deep learning network according to an embodiment of the present invention;
FIGS. 6 (a) and 6 (b) are graphs comparing actual and estimated phase errors of the present invention provided by embodiments of the present invention;
FIGS. 7 (a) and 7 (b) illustrate the Root Mean Square Error (RMSE) and standard deviation sigma of DOA and phase error provided by embodiments of the invention p Is a relationship diagram of (1);
fig. 8 (a) and 8 (b) are graphs of RMSE versus signal-to-noise ratio (SNR) for DOA and phase errors provided by embodiments of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the detailed description is presented by way of example only and is not intended to limit the scope of the invention.
In order to improve the accuracy of amplitude-phase error correction of a large planar array, the embodiment of the invention provides a planar array amplitude-phase error correction method based on convex optimization and a neural network, the proposed method requires a correction of the sound source at an unknown position in the far field. A dataset is generated using the amplitude phase error and noise that satisfy a Gaussian distribution. The correction source DOA is initially estimated using a convex optimization (CVX) method. The Deep Neural Network (DNN) is then trained to accurately estimate the correction source DOA. Finally, the spatial matched filter is utilized to estimate the amplitude and phase error based on the DOA estimation result.
Fig. 1 is a flowchart of a planar array amplitude-phase error correction method based on convex optimization and a neural network according to an embodiment of the present invention. As shown in fig. 1, the method for correcting the amplitude-phase error of the planar array comprises the following steps:
step 1, setting a correction sound source to be located in an unknown far-field direction relative to a sensor plane array.
In an embodiment, the correction sound source is located in an unknown far-field direction with respect to the sensor plane array, as shown in fig. 2, the correction sound source is disposed at (θ a ，θ b ) Far-field direction of the direction, for a planar array of mxn sensors, the array sampling signal is expressed as:
Figure BDA0002817172060000051
wherein x (m, n) represents the sampled signal acquired by the sensor position (m, n), d m ,d n Is the distance between the sensor position (m, n) and the reference sensor position, lambda is the acoustic wavelength, u 0 ＝sinθ a ，v 0 ＝sinθ b ，θ a ，θ b Indicating the far field direction, ζ, of the corrected sound source relative to the planar array of sensors g (m, n) is the gain error, subject to Gaussian distribution
Figure BDA0002817172060000052
ξ p (m, n) is the phase error, obeying Gaussian distribution +.>
Figure BDA0002817172060000053
Figure BDA0002817172060000054
And->
Figure BDA0002817172060000055
Represents the standard deviation, ε (m, n) is the noise of the sensor position (m, n);
the propagation model of the corrected sound source is expressed as:
x＝Ay+ε (2)
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002817172060000056
is a matrix of sampled signals, ">
Figure BDA0002817172060000057
Is a propagation matrix,/->
Figure BDA0002817172060000058
Is an acoustic signal matrix, epsilon is a noise matrix, and the elements in the propagation matrix a are given by:
Figure BDA0002817172060000059
where u=sinθ a ，v＝sinθ b (u, v) is divided into A X B directions within a range of (-1, -1).
And 2, converting the direction of arrival estimation problem of the corrected sound source into an optimization problem for solving the corrected sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to obtain the direction of arrival of the corrected sound source through preliminary estimation.
In this embodiment, the problem of estimating the direction of arrival of the corrected sound source is converted into an optimization problem of solving the corrected sound source signal by reconstructing the array sampling signal, expressed as:
Figure BDA00028171720600000510
mu represents a weight factor and is used to determine,
Figure BDA00028171720600000511
representing the square of the two norms, the optimization problem is passed through the Matlab convex optimization tool CVX, and +.>
Figure BDA00028171720600000512
Maximum corresponding estimated direction +.>
Figure BDA00028171720600000513
The direction of arrival of the corrected sound source obtained as a preliminary estimate.
And 3, further estimating the direction of arrival by using a direction of arrival estimation model constructed based on the deep learning network to obtain the final estimated direction of arrival of the corrected sound source.
When the direction of arrival (u 0 ,v 0 ) When the source direction (u) is not corrected in the A×B (u, v) directions 0 ,v 0 ) For (0.4617), as shown in FIG. 3, CVX estimated DOA Direction
Figure BDA0002817172060000061
Is (0.46). It follows that correcting the direction of arrival of the sound source with only convex optimized estimates is somewhat error-prone. In addition, it was found that the direction of arrival of the sound source was corrected +.>
Figure BDA0002817172060000062
Beam intensity in the surrounding direction is close +.>
Figure BDA0002817172060000063
Is used for the beam intensity of the beam.
In order to improve the estimation accuracy of the direction of arrival of the corrected sound source, the direction of arrival estimation model constructed based on the deep learning network is further used for further estimation on the basis of the step 2. The specific process is as follows:
and determining a corrected sound source position according to the primarily estimated direction of arrival, acquiring the beam intensity in the field range taking the corrected sound source position as the center, and further estimating according to the beam intensity by utilizing a direction of arrival estimation model constructed based on a deep learning network to obtain the finally estimated direction of arrival of the corrected sound source. In this example, the range of the field is (3 to 10) × (3 to 10) direction. In particular, the beam intensity in the 5×5 domain range centering on the corrected sound source position can be selected as shown in fig. 4.
In an embodiment, the method for constructing the direction of arrival estimation model includes:
constructing a beam intensity in a field range taking the position of a corrected sound source as the center, and a sample set taking the direction of arrival of the corrected sound source as a sample, correcting the random position of the sound source in a detection range, wherein the number of data sets is 1000;
constructing a deep learning network, wherein the deep learning network comprises at least one of a convolution layer and a full connection layer, and the activation function is a linear rectification function; specifically, the deep learning network may include a fully connected layer of 3 25 neurons, as shown in fig. 5, where the input of the deep learning network is the beam intensity in the area range with the corrected sound source position as the center, and the output is the direction of arrival of the corrected sound source;
and optimizing parameters of the deep learning network by using the sample set, and forming a direction-of-arrival estimation model by the determined parameters and the deep learning network after the optimization is finished.
And 4, estimating the amplitude-phase error through a spatial matched filter based on the finally estimated direction of arrival of the corrected sound source.
After the accurate direction of arrival of the corrected sound source is obtained in step 3, the amplitude-phase error can be estimated according to the spatial matched filter, and the specific process is as follows:
after determining the final estimated direction of arrival of the corrected sound source, the beam pattern B (t) of the estimated direction of arrival of the corrected sound source in each sample snapshot is expressed as:
Figure BDA0002817172060000071
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002817172060000072
is in relation to the final estimated direction of arrival +.>
Figure BDA0002817172060000073
The corresponding ideal propagation matrix, the superscript H represents the conjugate transpose, t is the sampling snapshot index, x (t) represents the array sampling signal when the sampling snapshot is t, t is [1, T]T represents the total number of sampling snapshots, the direction of arrival +.>
Figure BDA0002817172060000074
The response vector R of (2) is expressed as:
Figure BDA0002817172060000075
amplitude and phase errors
Figure BDA0002817172060000076
Obtained by the following formula:
Figure BDA0002817172060000077
wherein the symbols are
Figure BDA0002817172060000078
Representing division by element.
The deviation of the DOA estimate is Deltau and Deltav, then
Figure BDA0002817172060000079
Neglecting noise->
Figure BDA00028171720600000710
Figure BDA00028171720600000711
Can be expressed as:
Figure BDA00028171720600000712
wherein +. m And d n The larger the deviation of the phase estimation is, according to the least square method, for an equidistant array, when (m ref ,n ref ) Closest to the array element center, the estimated root mean square error is minimal. Thus, in an embodiment, the optimal sensor reference position is the midpoint of the planar array, i.e., the reference sensor position is selected at the rounding position ((M+1)/2, (N+1)/2).
After estimating the amplitude-phase error by adopting a spatial matched filter, carrying out normalization processing on the amplitude-phase error, wherein the normalization processing specifically comprises the following steps:
Figure BDA0002817172060000081
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002817172060000082
representing a reference sensor position (m ref ,n ref ) Is a phase error of the picture.
In a specific experimental example, for a planar array with 50 x 50 sensors and 0.5 lambda element spacing. The correction source is located in the direction (27.5 ° ). The amplitude and phase errors are respectively
Figure BDA0002817172060000083
And->
Figure BDA0002817172060000084
The signal-to-noise ratio (SNR) was 25dB. Sampling speedThe number of beats T is 1000. The parameter mu is set to 10 and the reference element positions (m, n) are (25, 25). And calculating the amplitude-phase error by using the convex optimization and neural network-based planar array amplitude-phase error correction method (CVX-DNN for short) in the steps 1 to 4.
The accuracy of the amplitude and phase error estimate is assessed by the following Root Mean Square Error (RMSE):
Figure BDA0002817172060000085
Figure BDA0002817172060000086
Figure BDA0002817172060000087
by CVX-DNN method, E d 4.4583X 10-5; eg is 0.0027; e (E) p Is 0.0017. The actual and estimated values of the phase error are shown in fig. 6 (a) and 6 (b). The actual and estimated values of the partial amplitude phase errors are shown in table 1.
TABLE 1 actual and estimated values of partial amplitude phase error
Figure BDA0002817172060000088
Figure BDA0002817172060000091
In this embodiment, in order to verify the performance of the CVX-DNN method under different experimental conditions, the estimation results of the CVX-DNN method with respect to the standard deviation of the amplitude and phase error and the SNR are considered and compared with the TSI and CVX methods. Since the amplitude error has little influence on DOA and phase error estimation, and the amplitude error RMSE of the CVX-DNN method under different experimental conditions is very close, the amplitude error standard deviation sigma g The constant was set to 0.2. In the CVX methodThe lattice spacing set in the (u, v) direction was 0.0005. 100 independent experiments were performed under each condition and the results averaged.
The estimated DOA and the RMSE of the phase error are compared to the standard deviation sigma of the phase error when the SNR is 25dB p The relationship of (a) and 7 (b) is shown in FIGS. 7 (a); when sigma is p At 0.5, the RMSE versus SNR curves for the estimated DOA and phase error are shown in fig. 8 (a) and 8 (b). The test result shows that the CVX-DNN method provided by the invention can accurately estimate and correct the amplitude and phase errors.
The foregoing detailed description of the preferred embodiments and advantages of the invention will be appreciated that the foregoing description is merely illustrative of the presently preferred embodiments of the invention, and that no changes, additions, substitutions and equivalents of those embodiments are intended to be included within the scope of the invention.

Claims (8)

1. A plane array amplitude-phase error correction method based on convex optimization and a neural network is characterized by comprising the following steps:
(1) Setting a correction sound source to be positioned in an unknown far-field direction relative to a sensor plane array, converting a direction-of-arrival estimation problem of the correction sound source into an optimization problem for solving a correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to primarily estimate the direction-of-arrival of the correction sound source;
(2) Determining a corrected sound source position according to the primarily estimated direction of arrival of the step (1), acquiring beam intensity in the field range with the corrected sound source position as the center, and further estimating according to the beam intensity by utilizing a direction of arrival estimation model constructed based on a deep learning network to acquire the finally estimated direction of arrival of the corrected sound source;
(3) The amplitude-phase error is estimated by a spatial matched filter based on the finally estimated direction of arrival of the corrected sound source.
2. The convex optimization and neural network-based planar array amplitude-phase error correction method of claim 1, wherein for a planar array of mxn sensors, the array sampling signal is represented as:
Figure FDA0002817172050000011
wherein x (m, n) represents the sampled signal acquired by the sensor position (m, n), d m ,d n Is the distance between the sensor position (m, n) and the reference sensor position, lambda is the acoustic wavelength, u 0 ＝sinθ a ，v 0 ＝sinθ b ，θ a ，θ b Indicating the far field direction, ζ, of the corrected sound source relative to the planar array of sensors g (m, n) is the gain error, subject to Gaussian distribution
Figure FDA0002817172050000012
ξ p (m, n) is the phase error, obeying Gaussian distribution +.>
Figure FDA0002817172050000013
Epsilon (m, n) is the noise of the sensor position (m, n);
the propagation model of the corrected sound source is expressed as:
x＝Ay+ε(2)
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0002817172050000021
is a matrix of sampled signals, ">
Figure FDA0002817172050000029
Is a propagation matrix,/->
Figure FDA00028171720500000210
Is an acoustic signal matrix, epsilon is a noise matrix, and the elements in the propagation matrix a are given by:
Figure FDA0002817172050000024
where u=sinθ a ，v＝sinθ b (u, v) is divided into A X B directions within a range of (-1, -1).
3. The convex optimization and neural network-based planar array amplitude-phase error correction method of claim 2, wherein the direction of arrival estimation problem of the corrected sound source is converted into an optimization problem for solving the corrected sound source signal by the reconstruction of the array sampling signal, expressed as:
Figure FDA0002817172050000025
mu represents a weight factor and is used to determine,
Figure FDA0002817172050000026
representing the square of the two norms, the optimization problem is passed through the convex optimization tool CVX and +.>
Figure FDA0002817172050000027
Maximum corresponding estimated direction +.>
Figure FDA0002817172050000028
The direction of arrival of the corrected sound source obtained as a preliminary estimate.
4. The convex optimization and neural network-based planar array amplitude-phase error correction method as set forth in claim 1, wherein the method for constructing the direction-of-arrival estimation model is as follows:
constructing a beam intensity in a field range with the corrected sound source position as a center, and correcting a direction of arrival of the sound source as a sample set of samples;
constructing a deep learning network, wherein the deep learning network comprises at least one of a convolution layer and a full connection layer, and the activation function is a linear rectification function;
and optimizing parameters of the deep learning network by using the sample set, and forming a direction-of-arrival estimation model by the determined parameters and the deep learning network after the optimization is finished.
5. The convex optimization and neural network-based planar array amplitude-phase error correction method according to claim 1 or 4, wherein the field range is (3-10) × (3-10) direction.
6. The convex optimization and neural network-based planar array amplitude-phase error correction method of claim 1, wherein in step (3), the process of estimating the amplitude-phase error from the spatial matched filter is:
after determining the final estimated direction of arrival of the corrected sound source, the beam pattern B (t) of the estimated direction of arrival of the corrected sound source in each sample snapshot is expressed as:
Figure FDA0002817172050000031
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0002817172050000032
is in relation to the final estimated direction of arrival +.>
Figure FDA0002817172050000033
The corresponding ideal propagation matrix, the superscript H represents the conjugate transpose, t is the sampling snapshot index, x (t) represents the array sampling signal when the sampling snapshot is t, t is [1, T]T represents the total number of sampling snapshots, the direction of arrival +.>
Figure FDA0002817172050000034
The response vector R of (2) is expressed as:
Figure FDA0002817172050000035
amplitude and phase errors
Figure FDA0002817172050000036
Obtained by the following formula:
Figure FDA0002817172050000037
wherein the symbols are
Figure FDA0002817172050000038
Representing division by element.
7. The convex optimization and neural network-based planar array amplitude-phase error correction method of claim 2, wherein the reference sensor position is selected at a rounding position ((m+1)/2, (n+1)/2).
8. The convex optimization and neural network-based planar array amplitude-phase error correction method as set forth in claim 7, wherein after the amplitude-phase error is estimated by using a spatial matched filter, the amplitude-phase error is normalized, specifically:
Figure FDA0002817172050000039
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA00028171720500000310
representing a reference sensor position (m ref ,n ref ) According to the least squares method, for an equally spaced array, when (m ref ,n ref ) Closest to the array element center, the estimated root mean square error is minimal.
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