CN112630784A - Planar array amplitude and phase error correction method based on convex optimization and neural network - Google Patents

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

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CN112630784A
CN112630784A CN202011411528.XA CN202011411528A CN112630784A CN 112630784 A CN112630784 A CN 112630784A CN 202011411528 A CN202011411528 A CN 202011411528A CN 112630784 A CN112630784 A CN 112630784A
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CN112630784B (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
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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
    • G01S7/52—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

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Abstract

The invention discloses a plane array amplitude and 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 the estimation problem of the direction of arrival of the correction sound source into an optimization problem of solving the correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to preliminarily estimate and obtain the direction of arrival of the correction sound source; (2) determining a corrected sound source position according to the direction of arrival preliminarily estimated in the step (1), acquiring beam intensity in a field range with the corrected sound source position as a center, and further estimating according to the beam intensity by using 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-phase error.

Description

Planar array amplitude and 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, compressive sensing, convex optimization, neural networks, spatial filtering and the like, in particular to a planar array amplitude-phase error correction method based on convex optimization and 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. The phased array three-dimensional imaging sonar system transmits an acoustic pulse signal, receives a sonar echo signal 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 uniform amplitude and phase characteristics. However, variations in sensor position, inconsistencies in performance of the sensor and signal conditioning circuitry, and cross-coupling effects between channels can lead to amplitude and phase errors in the received signal, resulting in increased sidelobe intensity and focus direction shifts in the beam pattern. To solve this problem, the amplitude and phase of the sensor array receive pattern need to be corrected to compensate for the error.
The array amplitude and phase error correction method can be divided into active correction and self-correction, wherein the active correction needs one or more correction sound sources with known accurate positions, and the self-correction sound source position is unknown, so that the direction of arrival (DOA) of the correction sound source and the amplitude and phase error of the array need to be estimated simultaneously. The active correction method corrects the amplitude-phase characteristics of the array through a sound source with a known position, and mainly comprises Maximum Likelihood Estimation (MLE) and least square method (LS). The active correction method has higher estimation accuracy, but is difficult to accurately position the position of the correction source in practical application, and the active correction method has great limitation in practical application, so that the array self-correction of unknown correction source position is the main research direction. Array self-correction first requires estimation of DOA, and the estimation algorithm includes: spectrum search, such as the multiple signal classification (MUSIC) algorithm; signal parameter estimation rotation invariant technology (ESPRIT); a Topritz (TB) algorithm using a data covariance matrix Topritz structure; a three-step iterative (TSI) algorithm, etc. The DOA estimation method can be optimized by using an auxiliary sensor array and a steering control device. However, the array of the phased array sonar system is generally in an underwater closed environment, and the method using the auxiliary array and the steering control device is difficult to be applied in a practical scene.
Recently, a DOA estimation method based on Compressed Sensing (CS) has received a wide attention. The CS method is a signal processing technique for reconstructing a sparse signal by solving an underdetermined linear system, and in DOA estimation, the CS method reconstructs and solves a sound source signal through a sensor reception signal, and the obtained solution is on initially set discrete grid points, whereas when a correction source is not on a grid point, a compressed sensing method has a plurality of solutions on grid points close to the direction of the correction source, and the estimation accuracy is also limited. In addition, the method based on deep learning is also applied to the field of DOA estimation and error self-correction, and DOA and amplitude and 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 achieve in the conventional method due to the existence of a certain propagation model between the sampled signal and the acoustic source signal. After DOA estimation is completed, the magnitude-phase error is estimated using a spatial matched filter based on the estimation result.
Disclosure of Invention
In view of the above, the present invention provides a planar array amplitude and phase error correction method based on convex optimization and neural network, for self-correction of large planar array.
In order to achieve the purpose of the invention, the technical scheme provided by the invention is as follows:
a planar array amplitude and 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 the estimation problem of the direction of arrival of the correction sound source into an optimization problem of solving the correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to preliminarily estimate and obtain the direction of arrival of the correction sound source;
(2) determining a corrected sound source position according to the direction of arrival preliminarily estimated in the step (1), acquiring beam intensity in a field range with the corrected sound source position as a center, and further estimating according to the beam intensity by using 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:
according to the plane array amplitude and phase error correction method for the convex optimization and neural network, the primary estimation is carried out through the convex optimization method to correct the direction of arrival of the sound source, then accurate estimation is carried out according to the beam intensity by using the direction of arrival estimation model constructed based on the deep learning network to obtain the final direction of arrival of the corrected sound source, accuracy of the direction of arrival of the corrected sound source is guaranteed through twice estimation, on the basis, the amplitude and phase error is estimated through the spatial matching filter, and accuracy of the amplitude and 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 used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flowchart of a planar array magnitude-phase error correction method based on convex optimization and a neural network according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a planar array of transducers and a collimated sound source provided by an embodiment of the present invention;
FIG. 3 is a diagram illustrating the DOA estimation result of the CVX method according to an embodiment of the present invention;
FIG. 4 is a beam pattern within a 5 × 5 domain centered on a corrected 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 an embodiment of the present invention;
FIGS. 7(a) and 7(b) provide Root Mean Square Error (RMSE) and standard deviation σ for DOA and phase errors for embodiments of the present inventionpA relationship diagram of (1);
fig. 8(a) and 8(b) are graphs of RMSE versus signal-to-noise ratio (SNR) for DOA and phase error provided by an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and examples. It should be understood that the detailed description and specific examples, while indicating the scope of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention.
In order to improve the accuracy of amplitude-phase error correction of a large-scale planar array, the embodiment of the invention provides a planar array amplitude-phase error correction method based on convex optimization and a neural network. A data set is generated using amplitude-phase errors and noise that satisfy a gaussian distribution. The correction source DOA is initially estimated using a convex optimization (CVX) method. A Deep Neural Network (DNN) is then trained to accurately estimate the correction source DOA. Finally, an amplitude-phase error is estimated using a spatial matched filter based on the DOA estimation result.
Fig. 1 is a flowchart of a planar array magnitude-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 and phase errors of the planar array comprises the following steps:
step 1, setting a correction sound source to be positioned in an unknown far-field direction relative to a sensor plane array.
In an embodiment, the correction acoustic source is positioned in an unknown far-field direction relative to the planar array of sensors, as shown in FIG. 2, the correction acoustic source being positioned at (θ) relative to the planar array of sensorsa，θb) Far field direction of direction, for a planar array of M × N sensors, the array sample signal is expressed as:
Figure BDA0002817172060000051
where x (m, n) represents the sampled signal collected at the sensor location (m, n), dm,dnIs the spacing between the sensor position (m, n) and the reference sensor position, λ is the acoustic wavelength, u0＝sinθa，v0＝sinθb，θa，θbRepresenting the far-field direction, ξ, of the correction sound source in relation to the planar array of sensorsg(m, n) is the gain error, following a Gaussian distribution
Figure BDA0002817172060000052
ξp(m, n) is the phase error, obeying a Gaussian distribution
Figure BDA0002817172060000053
Figure BDA0002817172060000054
And
Figure BDA0002817172060000055
represents the standard deviation, ε (m, n) is the noise of the sensor location (m, n);
the propagation model of the corrected sound source is expressed as:
x＝Ay+ε (2)
wherein the content of the first and second substances,
Figure BDA0002817172060000056
is a matrix of sampled signals that is,
Figure BDA0002817172060000057
is a matrix of the propagation of the signal,
Figure BDA0002817172060000058
is the acoustic signal matrix, epsilon is the noise matrix, and the elements in the propagation matrix a are given by:
Figure BDA0002817172060000059
wherein u is sin θa，v＝sinθb(u, v) is divided into A × B directions within (-1 to 1 ).
And 2, converting the estimation problem of the direction of arrival of the correction sound source into an optimization problem for solving the correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to preliminarily estimate the direction of arrival of the correction sound source.
In this embodiment, the estimation problem of the direction of arrival of the correction sound source is converted into an optimization problem that the correction sound source signal is solved by reconstructing the array sampling signal, and is expressed as:
Figure BDA00028171720600000510
μ represents a weight factor by which the weight of the sample,
Figure BDA00028171720600000511
representing the square of the two-norm, the optimization problem passes through the Matlab convex optimization tool CVX, and will
Figure BDA00028171720600000512
Maximum time 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 finally estimated direction of arrival of the corrected sound source.
When correcting the direction of arrival (u) of the sound source0,v0) When not in A × B (u, v) directions, the source direction (u) is corrected0,v0) Is (0.4617 ), as shown in FIG. 3, the DOA direction of the CVX estimate
Figure BDA0002817172060000061
Is (0.46 ). Thereby the device is provided withIt can be seen that correcting the direction of arrival of a sound source using only convex optimization estimates will have some error. In addition, it has been found that the direction of arrival of the sound source is corrected
Figure BDA0002817172060000062
Beam intensity in the peripheral direction approaches
Figure BDA0002817172060000063
The beam intensity of (a).
In order to improve the estimation accuracy of the direction of arrival of the corrected sound source, on the basis of the step 2, the estimation is further carried out by utilizing a direction of arrival estimation model constructed based on a deep learning network. The specific process is as follows:
and determining a corrected sound source position according to the preliminarily estimated direction of arrival, acquiring the beam intensity in the field range with the corrected sound source position as the center, and further estimating according to the beam intensity by using 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. In this embodiment, the domain ranges from (3-10) × (3-10) directions. Specifically, the beam intensity in the 5 × 5 domain range centered on the corrected sound source position may be selected as shown in fig. 4.
In the embodiment, the method for constructing the direction of arrival estimation model comprises the following steps:
constructing a sample set which takes the wave beam intensity in the field range with the position of the correction sound source as the center and the direction of arrival of the correction sound source as a sample, wherein the correction sound source is at the random position in the detection range, and the number of the data sets is 1000;
constructing a deep learning network which comprises at least one of a convolution layer and a full connection layer, wherein the activation function is a linear rectification function; specifically, the deep learning network may include 3 full-connected layers of 25 neurons, as shown in fig. 5, where the input of the deep learning network is the beam intensity in the domain range centered on the position of the correction sound source, and the output is the arrival direction of the correction sound source;
and optimizing parameters of the deep learning network by using the sample set, and after the optimization is finished, forming a direction of arrival estimation model by the determined parameters and the deep learning network.
And 4, estimating an amplitude-phase error through a spatial matching filter based on the finally estimated arrival direction 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 direction of arrival of the finally estimated correction sound source, the beam pattern b (t) of the direction of arrival of the correction sound source in each sampling snapshot is estimated as:
Figure BDA0002817172060000071
wherein the content of the first and second substances,
Figure BDA0002817172060000072
is the direction of arrival with the final estimate
Figure BDA0002817172060000073
The superscript H represents conjugate transposition, T is sampling snapshot index, x (T) represents array sampling signal when T sampling snapshot is carried out, and T belongs to [1, T ∈]T represents the total number of sampling snapshots and the direction of arrival
Figure BDA0002817172060000074
The response vector R of (a) is expressed as:
Figure BDA0002817172060000075
then amplitude phase error
Figure BDA0002817172060000076
Obtained by the following formula:
Figure BDA0002817172060000077
wherein, the symbol
Figure BDA0002817172060000078
Meaning divided by element.
The deviation of the DOA estimate is Δ u and Δ v, then
Figure BDA0002817172060000079
The noise is ignored and the noise is ignored,
Figure BDA00028171720600000710
Figure BDA00028171720600000711
can be expressed as:
Figure BDA00028171720600000712
wherein [ ] indicates multiplication by an element, it can be seen thatmAnd dnThe larger the deviation of the phase estimate, the larger the deviation, according to the least squares method, for an equally spaced array, when (m)ref,nref) The root mean square error of the estimate is the smallest when it is closest to the center of the array element. Thus, in an embodiment, the optimal sensor reference position is the midpoint of the planar array, i.e. the reference sensor position is chosen to be the rounded position of ((M +1)/2, (N + 1)/2).
After the amplitude-phase error is estimated by adopting a space matching filter, the amplitude-phase error is normalized, and the method specifically comprises the following steps:
Figure BDA0002817172060000081
wherein the content of the first and second substances,
Figure BDA0002817172060000082
indicating the reference sensor position (m)ref,nref) Amplitude-phase error of (2).
In the specific experimental example, for the test with 50X 50 transmissionA planar array of sensors and 0.5 lambda array element spacing. The calibration source is located in the direction (27.5 ° ). Amplitude and phase errors are respectively
Figure BDA0002817172060000083
And
Figure BDA0002817172060000084
the signal-to-noise ratio (SNR) is 25 dB. The number of sampling snapshots T is 1000. The parameter μ is set to 10 and the reference array element position (m, n) is (25, 25). And (4) 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-phase error estimate is evaluated by the following Root Mean Square Error (RMSE):
Figure BDA0002817172060000085
Figure BDA0002817172060000086
Figure BDA0002817172060000087
by the CVX-DNN method, Ed4.4583X 10-5; eg is 0.0027; epIs 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 error are shown in table 1.
TABLE 1 actual and estimated values of partial amplitude-phase errors
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-phase error and the SNR are considered and compared with the TSI method and the CVX method. Because 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 standard deviation sigma of the amplitude error isgThe constant is set to 0.2. In the CVX method, the lattice point interval set in the (u, v) direction is 0.0005. 100 independent experiments were performed under each condition and the results were averaged.
Estimated DOA and RMSE of phase error versus standard deviation σ of phase error when SNR is 25dBpThe relationship of (A) is shown in FIGS. 7(a) and 7 (b); when sigma ispAt 0.5, the estimated DOA and phase error RMSE versus SNR are shown in FIGS. 8(a) and 8 (b). The experimental result shows that the CVX-DNN method provided by the invention can accurately estimate and correct the amplitude and phase errors.
The above-mentioned embodiments are intended to illustrate the technical solutions and advantages of the present invention, and it should be understood that the above-mentioned embodiments are only the most preferred embodiments of the present invention, and are not intended to limit the present invention, and any modifications, additions, equivalents, etc. made within the scope of the principles of the present invention should be included in the scope of the present invention.

Claims (8)

1. A planar array amplitude and 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 the estimation problem of the direction of arrival of the correction sound source into an optimization problem of solving the correction sound source signal through array sampling signal reconstruction, and estimating and solving the optimization problem by adopting a convex optimization method to preliminarily estimate and obtain the direction of arrival of the correction sound source;
(2) determining a corrected sound source position according to the direction of arrival preliminarily estimated in the step (1), acquiring beam intensity in a field range with the corrected sound source position as a center, and further estimating according to the beam intensity by using 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 magnitude-phase error correction method of claim 1, wherein for a planar array of M x N sensors, the array sampling signal is expressed as:
Figure FDA0002817172050000011
where x (m, n) represents the sampled signal collected at the sensor location (m, n), dm,dnIs the spacing between the sensor position (m, n) and the reference sensor position, λ is the acoustic wavelength, u0＝sinθa，v0＝sinθb，θa，θbRepresenting the far-field direction, ξ, of the correction sound source in relation to the planar array of sensorsg(m, n) is the gain error, following a Gaussian distribution
Figure FDA0002817172050000012
ξp(m, n) is the phase error, obeying a Gaussian distribution
Figure FDA0002817172050000013
ε (m, n) is the noise of the sensor location (m, n);
the propagation model of the corrected sound source is expressed as:
x＝Ay+ε(2)
wherein the content of the first and second substances,
Figure FDA0002817172050000021
is a matrix of sampled signals that is,
Figure FDA0002817172050000029
is a matrix of the propagation of the signal,
Figure FDA00028171720500000210
is the acoustic signal matrix, epsilon is the noise matrix, and the elements in the propagation matrix a are given by:
Figure FDA0002817172050000024
wherein u is sin θa，v＝sinθb(u, v) is divided into A × B directions within (-1 to 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 correctional acoustic source is converted into an optimization problem for solving the correctional acoustic source signal through array sampling signal reconstruction, expressed as:
Figure FDA0002817172050000025
μ represents a weight factor by which the weight of the sample,
Figure FDA0002817172050000026
representing the square of the two-norm, the optimization problem passes through a convex optimization tool CVX, and will
Figure FDA0002817172050000027
Maximum time 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 of claim 1, wherein the direction of arrival estimation model is constructed by:
constructing a sample set which takes the position of a corrected sound source as a center, beam intensity in a field range and the direction of arrival of the corrected sound source as samples;
constructing a deep learning network which comprises at least one of a convolution layer and a full connection layer, wherein the activation function is a linear rectification function;
and optimizing parameters of the deep learning network by using the sample set, and after the optimization is finished, forming a direction of arrival estimation model by the determined parameters and the deep learning network.
5. The convex optimization and neural network-based planar array magnitude-phase error correction method as claimed in claim 1 or 4, wherein the domain range is (3-10) × (3-10) directions.
6. The convex optimization and neural network-based planar array amplitude-phase error correction method as claimed in claim 1, wherein in the step (3), the process of estimating the amplitude-phase error according to the spatial matched filter is as follows:
after determining the direction of arrival of the finally estimated correction sound source, the beam pattern b (t) of the direction of arrival of the correction sound source in each sampling snapshot is estimated as:
Figure FDA0002817172050000031
wherein the content of the first and second substances,
Figure FDA0002817172050000032
is the direction of arrival with the final estimate
Figure FDA0002817172050000033
The superscript H represents conjugate transposition, T is sampling snapshot index, x (T) represents array sampling signal when T sampling snapshot is carried out, and T belongs to [1, T ∈]T represents the total number of sampling snapshots,direction of arrival
Figure FDA0002817172050000034
The response vector R of (a) is expressed as:
Figure FDA0002817172050000035
then amplitude phase error
Figure FDA0002817172050000036
Obtained by the following formula:
Figure FDA0002817172050000037
wherein, the symbol
Figure FDA0002817172050000038
Meaning divided by element.
7. The convex optimization and neural network-based planar array magnitude-phase error correction method of claim 2, wherein the reference sensor position is selected at a rounding position of ((M +1)/2, (N + 1)/2).
8. The convex optimization and neural network-based planar array amplitude-phase error correction method as claimed in claim 7, wherein after the amplitude-phase error is estimated by using the spatial matched filter, the amplitude-phase error is normalized, specifically:
Figure FDA0002817172050000039
wherein the content of the first and second substances,
Figure FDA00028171720500000310
indicating the reference sensor position (m)ref,nref) The amplitude-phase error of (d) is calculated according to the least square method for an equally spaced array of (m)ref,nref) The root mean square error of the estimate is the smallest when it is closest to the center of the array element.
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