Method and system for array shape estimation and target detection during steering of towed line array sonar

Abstract

The invention provides a method and a system for array shape estimation and target detection during steering of towed line array sonar. The method comprises the following steps: S1, modeling a distorted towed line array into a parabola model; s2, parabola parameter values and other parameter values are set; s3, calculating the position coordinates of each array element and the corresponding position slope; s4, calculating to obtain a gradient vector of the target function; s5, judging whether the absolute value of the gradient vector of the target function is smaller than an estimation error threshold or not, and if yes, outputting a current parabola parameter value; if not, the parameter value of the parabola is updated again, and the steps S2-S4 are repeated. According to the method, the formation estimation problem is converted into a least square optimization problem, an iterative solving algorithm of the problem is given through a forward and backward propagation calculation method, only the shipborne or towed active sonar on the ship is used for transmitting pulse signals, an existing towed line array does not need to be transformed, the algorithm principle is simple, the calculation amount is small, and the method is suitable for large-scale popularization and application. And engineering realization is easy.

Description

Technical Field

[0001] The present invention relates to the field of sonar technology in underwater acoustic engineering, and in particular to a method and system for estimating the formation and detecting targets when a towed array sonar is turned, and in particular to a method and system for estimating the formation and detecting targets when a towed array sonar is turned using bow sound source calibration. Background Art

[0002] When the ship's towed platform turns and maneuvers, it will cause the formation of the towed array to be distorted, resulting in a sharp drop in the spatial processing gain of the towed array sonar. At the same time, the azimuth value of target detection will also have a large deviation, and in severe cases it will not work. In order to ensure that the towed array sonar still maintains the corresponding target detection and azimuth estimation performance under the condition of formation distortion, it is necessary to solve the problem of estimating the distorted formation.

[0003] Scholars at home and abroad have proposed some methods to solve this problem: for example, using the received acoustic data to estimate the formation through signal processing methods, mainly including gradient method, eigenvector method and time delay estimation method, but the amount of calculation is large, the application conditions are strict and the robustness is poor, which is difficult to meet the needs of engineering applications; the underwater state of the towed array is mechanically modeled, the boundary conditions are determined by the mechanical sensor data, and the equation is solved to obtain the estimated formation, but the underwater environment is complex, and the mechanical modeling method is also difficult to accurately grasp and verify, and the practicality is poor; the parabolic model and exponential model are used to perform three-dimensional modeling of the distorted formation, and the parameters are estimated using the arc length formula and the length between the heading sensors, but when the ship platform is turning and maneuvering, the surrounding magnetic field changes significantly due to the change of its own working conditions. The heading sensor may have a large inherent deviation under the influence of it, resulting in a sharp decline in the formation estimation ability, so the robustness of the algorithm is poor. In summary, the above methods are subject to certain limitations in practical applications and cannot achieve the desired effect.

[0004] In order to improve the performance of towed array sonar when a ship is turning and maneuvering, the present invention proposes a formation estimation method that utilizes a towed sound source at the bow of the towed array or an active sonar on the hull to transmit a high-intensity short pulse signal, and calibrates the distorted towed array when the ship is turning by using the time delay difference of the direct sound. Summary of the invention

[0005] In view of the defects in the prior art, the object of the present invention is to provide a method and system for array estimation and target detection when a towed array sonar is turned.

[0006] A method for estimating the formation and detecting a target when a towed array sonar is turned according to the present invention comprises:

[0007] Step S1: Modeling the distorted towed array as a parabola model and establishing a rectangular coordinate system to obtain a parabola expression;

[0008] Step S2: setting parabola parameter values ​​and other parameter values;

[0009] The other parameter values ​​include an iteration step size and an estimation error threshold;

[0010] Step S3: Calculate the position coordinates of each array element and the corresponding position slope according to the initial value of the parabola;

[0011] Step S4: Calculate the gradient vector of the objective function based on the position coordinates and position slope of the array element;

[0012] Step S5: Determine whether the absolute value of the gradient vector of the objective function is less than the estimation error threshold. If so, stop the iteration, output the current parabola parameter value, and execute step S6; if not, update the parabola parameter value according to the iteration step size and the objective function, and repeat steps S2-S4;

[0013] Step S6: Solve the coordinates of each array element according to the parabola parameter value, compensate during beamforming, and perform target detection.

[0014] Preferably, the step S1 comprises:

[0015] Assume that the towed line array contains N evenly distributed array elements, and the spacing between adjacent array elements is d. The distorted towed line array acoustic array segment is modeled. The position of the first array element is taken as the origin, and a rectangular coordinate system is established in the horizontal plane with its heading as the opposite direction of the X-axis. In this coordinate system, the parabolic model of the distorted array is expressed as:

[0016]

[0017] Where a and b are the parabola parameter values.

[0018] Preferably, the calculation process of the position coordinates of each array element and the corresponding position slope includes:

[0019] Assume the coordinates of the first element at the origin are:

[0020] (x1,y1)=(0,0)

[0021] With the help of the idea of ​​piecewise straight line approximation and forward propagation, the coordinates of each array element are

[0022] (x j ,y j )(j=2,…,N)

[0023] make It is obtained by the following formula:

[0024] k j-1 =ax j-1 +b

[0025]

[0026]

[0027] Among them, k j-1 is the position slope corresponding to the position coordinate of the array element.

[0028] Preferably, the step S3 further comprises:

[0029] τ j It represents the theoretical time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, expressed as τ j * represents the actual time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, and the array estimation problem is transformed into a least squares optimization problem:

[0030]

[0031] Among them, Ψ is the objective function.

[0032] Preferably, step S4 includes the following sub-steps:

[0033] Step S4.1: Based on the idea of ​​backward propagation, the parameters a and b are derived layer by layer from back to front. The formula is:

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040] in,

[0041] Step S4.2: Obtain the gradient vector of the objective function Ψ as

[0042]

[0043]

[0044]

[0045] Preferably, step S5 comprises:

[0046] Determine whether the gradient vector meets the conditions

[0047]

[0048] In the formula, △ th is the estimation error threshold;

[0049] If satisfied, the iteration stops and the current a and b values ​​are used as parameter estimates; if not satisfied, a and b are updated using the gradient vector:

[0050]

[0051]

[0052] Where μ is the iteration step size;

[0053] Repeat steps S2-S4 using the updated parameters a and b to obtain an estimated value of the distorted formation.

[0054] According to the present invention, a system for estimating the formation and detecting the target when a towed array sonar is turned comprises:

[0055] Module M1: Model the distorted towed array as a parabola model and establish a rectangular coordinate system to obtain a parabola expression;

[0056] Module M2: Set the parabola parameter values ​​and other parameter values;

[0057] The other parameter values ​​include an iteration step size and an estimation error threshold;

[0058] Module M3: Calculate the position coordinates of each array element and the corresponding position slope according to the initial value of the parabola;

[0059] Module M4: Calculate the gradient vector of the objective function based on the position coordinates and position slope of the array element;

[0060] Module M5: Determine whether the absolute value of the gradient vector of the objective function is less than the estimation error threshold. If so, stop the iteration, output the current parabola parameter value, and execute module M6; if not, update the parabola parameter value according to the iteration step size and the objective function, and repeat modules M2-M4;

[0061] Module M6: Calculate the coordinates of each array element according to the parabola parameter value, compensate during beamforming, and perform target detection.

[0062] Preferably, the module M1 comprises:

[0063] Assume that the towed line array contains N evenly distributed array elements, and the spacing between adjacent array elements is d. The distorted towed line array acoustic array segment is modeled. The position of the first array element is taken as the origin, and a rectangular coordinate system is established in the horizontal plane with its heading as the opposite direction of the X-axis. In this coordinate system, the parabolic model of the distorted array is expressed as:

[0064]

[0065] Where a and b are the parabola parameter values.

[0066] Preferably, the calculation process of the position coordinates of each array element and the corresponding position slope includes:

[0067] Assume the coordinates of the first element at the origin are:

[0068] (x1,y1)=(0,0)

[0069] With the help of the idea of ​​piecewise straight line approximation and forward propagation, the coordinates of each array element are

[0070] (x j ,y j )(j=2,…,N)

[0071] make It is obtained by the following formula:

[0072] k j-1 =ax j-1 +b

[0073]

[0074]

[0075] Among them, k j-1 is the position slope corresponding to the position coordinate of the array element.

[0076] Preferably, the module M3 further includes:

[0077] τ j It represents the theoretical time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, expressed as τ j * represents the actual time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, and the array estimation problem is transformed into a least squares optimization problem:

[0078]

[0079] Among them, Ψ is the objective function.

[0080] Preferably, the module M4 includes the following submodules:

[0081] Module M4.1: Based on the idea of ​​backward propagation, the parameters a and b are derived layer by layer from back to front. The formula is:

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] in,

[0089] Module M4.2: The gradient vector of the objective function Ψ is obtained as

[0090]

[0091]

[0092]

[0093] Preferably, the module M5 comprises:

[0094] Determine whether the gradient vector meets the conditions

[0095]

[0096] In the formula, △ th is the estimation error threshold;

[0097] If satisfied, the iteration stops and the current a and b values ​​are used as parameter estimates; if not satisfied, a and b are updated using the gradient vector:

[0098]

[0099]

[0100] Where μ is the iteration step size;

[0101] Repeat modules M2-M4 using the updated parameters a and b to obtain an estimate of the distorted formation.

[0102] Compared with the prior art, the present invention has the following beneficial effects:

[0103] 1. The formation estimation method of the distorted towed array when the ship turns provided by the present invention transforms the formation estimation problem into a least squares optimization problem, and provides an iterative solution algorithm for the problem through the forward and backward propagation calculation method; the method only utilizes the pulse signal emitted by the ship-borne or towed active sonar on the ship, and there is no need to modify the existing towed array. The algorithm principle is simple, the calculation amount is small, and it is easy to implement in engineering.

[0104] 2. The present invention establishes the distorted towed array as a parabolic model, performs real-time calibration on the received signals of each array element when the towed array sonar is turning and maneuvering, or uses the ship radiation noise at the bow of the towed array as a passive sound source with a known distance and azimuth, thereby realizing the formation estimation and target detection of the towed array sonar, and has good practicality.

[0105] Other beneficial effects of the present invention will be explained in the specific implementation manner through the introduction of specific technical features and technical solutions. Through the introduction of these technical features and technical solutions, those skilled in the art should be able to understand the beneficial technical effects brought about by the technical features and technical solutions. BRIEF DESCRIPTION OF THE DRAWINGS

[0106] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0107] Figure 1 The present invention is a flow chart of a method for estimating the formation and detecting targets when a towed array sonar calibrated with a bow sound source is turned.

[0108] Figure 2 FIG. 1 is a schematic diagram of theoretical and estimated array element positions in an embodiment of the present invention.

[0109] Figure 3 This is a diagram of target orientation detection results in one embodiment of the present invention. DETAILED DESCRIPTION

[0110] The present invention is described in detail below in conjunction with specific embodiments. The following embodiments will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can also be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

[0111] Reference Figure 1 As shown, a method for estimating the formation and detecting the target when the towed array sonar is turned includes the following steps:

[0112] Step 1: N array elements are arranged according to the half wavelength corresponding to the highest frequency of the processed signal, and the spacing between adjacent array elements is d. The distorted towed array acoustic array segment is modeled. The position of the first array element is taken as the origin, and a rectangular coordinate system is established in the horizontal plane with its heading as the opposite direction of the X-axis. In this coordinate system, the parabolic model of the distorted array can be expressed as:

[0113]

[0114] Step 2: Set the initial values ​​of parameters a and b, the initial value of the iteration step μ, and the estimation error threshold Δ th ;

[0115] Step 3: The coordinates of the first array element at the origin are (x1, y1) = (0, 0). With the help of the idea of ​​piecewise straight line approximation and forward propagation, the coordinates of each array element (x j ,y j )(j=2,…,N) is obtained from the following formulas:

[0116] k j-1 =ax j-1 +b (2)

[0117]

[0118]

[0119] Step 4: Take τ j It represents the theoretical time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, represents the actual time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, and the array estimation problem is transformed into a least squares optimization problem:

[0120]

[0121] Step 5, find the gradient vector of the objective function Ψ;

[0122] Step 501, refer to the idea of ​​backward propagation to derive the parameters a and b layer by layer from back to front, the formula is:

[0123]

[0124]

[0125]

[0126]

[0127]

[0128]

[0129] in,

[0130] Step 502, the gradient vector of the objective function Ψ is obtained as

[0131]

[0132]

[0133] Step 6: Determine whether the gradient vector meets the conditions If satisfied, stop the iteration, and the current a and b are the parameter estimates; if not satisfied, use the gradient vector to update a and b:

[0134]

[0135]

[0136] Repeat steps 2-6 with the updated parameters a and b to obtain the estimated value of the distorted formation.

[0137] Step 7: Calculate the coordinates of each array element according to the estimated distorted array and array element spacing, and perform compensation during beamforming to achieve target detection.

[0138] In one embodiment, the number of array elements in the base array segment of the towed array is 96, the array element spacing is 0.4 m, and the distortion model of the towed array is:

[0139] y=f(x)=0.016x 2 +0.01x,

[0140] It is known that the bow sound source is at 20° on the starboard side of the towed array, emitting a high signal-to-noise ratio pulse signal, and the stationary target is at 90°. The delay difference error of the array element receiving signal is added to the delay difference of the array element receiving signal, which follows the standard normal distribution N(0,1) (unit: ms). The estimated position of the array element obtained by the method of the present invention is compared with the theoretical position of the array element. Figure 2 As shown in Figure 2, under the condition of error, the estimation deviation of the array element position is within the acceptable range; after the formation distortion is compensated, the target detection result obtained by beamforming is compared with the theoretical target detection result. Figure 3 As shown, the target detection results are almost completely overlapped.

[0141] The formation estimation method of the distorted towed array when the ship turns provided by the present invention transforms the formation estimation problem into a least squares optimization problem, and provides an iterative solution algorithm for the problem through a forward and backward propagation calculation method; the method only utilizes the pulse signal emitted by the ship-borne or towed active sonar on the ship, and there is no need to modify the existing towed array; the algorithm principle is simple, the calculation amount is small, and it is easy to implement in engineering.

[0142] The present invention also provides a system for estimating the formation and detecting the target when a towed array sonar is turned. The system for estimating the formation and detecting the target when a towed array sonar is turned can be implemented by executing the process steps of the method for estimating the formation and detecting the target when a towed array sonar is turned, that is, those skilled in the art can understand the method for estimating the formation and detecting the target when a towed array sonar is turned as a preferred implementation of the system for estimating the formation and detecting the target when a towed array sonar is turned.

[0143] Specifically, a system for estimating the formation and detecting the target when a towed array sonar is turned includes:

[0144] Module M1: Model the distorted towed array as a parabola model and establish a rectangular coordinate system to obtain a parabola expression;

[0145] Module M2: Set the parabola parameter values ​​and other parameter values;

[0146] The other parameter values ​​include an iteration step size and an estimation error threshold;

[0147] Module M3: Calculate the position coordinates of each array element and the corresponding position slope according to the initial value of the parabola;

[0148] Module M4: Calculate the gradient vector of the objective function based on the position coordinates and position slope of the array element;

[0149] Module M5: Determine whether the absolute value of the gradient vector of the objective function is less than the estimation error threshold. If so, stop the iteration, output the current parabola parameter value, and execute module M6; if not, update the parabola parameter value according to the iteration step size and the objective function, and repeat modules M2-M4;

[0150] Module M6: Calculate the coordinates of each array element according to the parabola parameter value, compensate during beamforming, and perform target detection.

[0151] The module M1 comprises:

[0152] Assume that the towed line array contains N evenly distributed array elements, and the spacing between adjacent array elements is d. The distorted towed line array acoustic array segment is modeled. The position of the first array element is taken as the origin, and a rectangular coordinate system is established in the horizontal plane with its heading as the opposite direction of the X-axis. In this coordinate system, the parabolic model of the distorted array is expressed as:

[0153]

[0154] Where a and b are the parabola parameter values.

[0155] The calculation process of the position coordinates of each array element and the corresponding position slope includes:

[0156] Assume the coordinates of the first element at the origin are:

[0157] (x1,y1)=(0,0)

[0158] With the help of the idea of ​​piecewise straight line approximation and forward propagation, the coordinates of each array element are

[0159] (x j ,y j )(j=2,…,N)

[0160] make It is obtained by the following formula:

[0161] k j-1 =ax j-1 +b

[0162]

[0163]

[0164] Among them, k j-1 is the position slope corresponding to the position coordinate of the array element.

[0165] The module M3 also includes:

[0166] τ j It represents the theoretical time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, expressed as τ j * represents the actual time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, and the array estimation problem is transformed into a least squares optimization problem:

[0167]

[0168] Among them, Ψ is the objective function.

[0169] The module M4 includes the following submodules:

[0170] Module M4.1: Based on the idea of ​​backward propagation, the parameters a and b are derived layer by layer from back to front. The formula is:

[0171]

[0172]

[0173]

[0174]

[0175]

[0176]

[0177] in,

[0178] Module M4.2: The gradient vector of the objective function Ψ is obtained as

[0179]

[0180]

[0181]

[0182] The module M5 comprises:

[0183] Determine whether the gradient vector meets the conditions

[0184]

[0185] In the formula, △ th is the estimation error threshold;

[0186] If satisfied, the iteration stops and the current a and b values ​​are used as parameter estimates; if not satisfied, a and b are updated using the gradient vector:

[0187]

[0188]

[0189] Where μ is the iteration step size;

[0190] Repeat modules M2-M4 using the updated parameters a and b to obtain an estimate of the distorted formation.

[0191] Those skilled in the art know that, in addition to realizing the system and its various devices, modules, and units provided by the present invention in a purely computer-readable program code, it is entirely possible to realize the same functions in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers by logically programming the method steps. Therefore, the system and its various devices, modules, and units provided by the present invention can be considered as a hardware component, and the devices, modules, and units included therein for realizing various functions can also be regarded as structures within the hardware component; the devices, modules, and units for realizing various functions can also be regarded as both software modules for realizing the method and structures within the hardware component.

[0192] The above describes the specific embodiments of the present invention. It should be understood that the present invention is not limited to the above specific embodiments, and those skilled in the art can make various changes or modifications within the scope of the claims, which does not affect the essence of the present invention. In the absence of conflict, the embodiments of the present application and the features in the embodiments can be combined with each other arbitrarily.

Claims

1. A method for estimating the formation and detecting the target when a towed array sonar is turned, characterized in that: include: Step S1: Modeling the distorted towed array as a parabola model and establishing a rectangular coordinate system to obtain a parabola expression; Step S2: setting parabola parameter values ​​and other parameter values; The other parameter values ​​include an iteration step size and an estimation error threshold; Step S3: Calculate the position coordinates of each array element and the corresponding position slope according to the initial value of the parabola; Step S4: Calculate the gradient vector of the objective function based on the position coordinates and position slope of the array element; Step S5: Determine whether the absolute value of the gradient vector of the objective function is less than the estimation error threshold. If so, stop the iteration, output the current parabola parameter value, and execute step S6; if not, update the parabola parameter value according to the iteration step size and the objective function, and repeat steps S2-S4; Step S6: Solve the coordinates of each array element according to the parabola parameter value, compensate during beamforming, and perform target detection.

2. The method for estimating the formation and detecting the target when the towed array sonar is turned according to claim 1, characterized in that: The step S1 comprises: Assume that the towed line array contains N evenly distributed array elements, and the spacing between adjacent array elements is d. The distorted towed line array acoustic array segment is modeled. The position of the first array element is taken as the origin, and a rectangular coordinate system is established in the horizontal plane with its heading as the opposite direction of the X-axis. In this coordinate system, the parabolic model of the distorted array is expressed as: Where a and b are the parabola parameter values.

3. The method for estimating the formation and detecting the target when the towed array sonar is turned according to claim 2, characterized in that: The calculation process of the position coordinates of each array element and the corresponding position slope includes: Assume the coordinates of the first element at the origin are: (x1,y1)=(0,0) With the help of the idea of ​​piecewise straight line approximation and forward propagation, the coordinates of each array element are (x j ,y j )(j=2,…,N) make It is obtained by the following formula: k j-1 =ax j-1 +b Among them, k j-1 is the position slope corresponding to the position coordinate of the array element.

4. The method for estimating the formation and detecting the target when the towed array sonar is turned according to claim 3, characterized in that: The step S3 further comprises: τ j It represents the theoretical time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, expressed as τ j * represents the actual time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, and the array estimation problem is transformed into a least squares optimization problem: Among them, Ψ is the objective function.

5. The method for estimating the formation and detecting the target when the towed array sonar is turned according to claim 4, characterized in that: The step S4 comprises the following sub-steps: Step S4.1: Based on the idea of ​​backward propagation, the parameters a and b are derived layer by layer from back to front. The formula is: in, Step S4.2: Obtain the gradient vector of the objective function Ψ as 6. The method for estimating the formation and detecting the target when the towed array sonar is turned according to claim 5, characterized in that: The step S5 comprises: Determine whether the gradient vector meets the conditions In the formula, △ th is the estimation error threshold; If satisfied, the iteration stops and the current a and b values ​​are used as parameter estimates; if not satisfied, a and b are updated using the gradient vector: Where μ is the iteration step size; Repeat steps S2-S4 using the updated parameters a and b to obtain an estimated value of the distorted formation.

7. A system for estimating the formation and detecting the target when a towed array sonar is turned, characterized in that: include: Module M1: Model the distorted towed array as a parabola model and establish a rectangular coordinate system to obtain a parabola expression; Module M2: Set the parabola parameter values ​​and other parameter values; The other parameter values ​​include an iteration step size and an estimation error threshold; Module M3: Calculate the position coordinates of each array element and the corresponding position slope according to the initial value of the parabola; Module M4: Calculate the gradient vector of the objective function based on the position coordinates and position slope of the array element; Module M5: Determine whether the absolute value of the gradient vector of the objective function is less than the estimation error threshold. If so, stop the iteration, output the current parabola parameter value, and execute module M6; if not, update the parabola parameter value according to the iteration step size and the objective function, and repeat modules M2-M4; Module M6: Calculate the coordinates of each array element according to the parabola parameter value, compensate during beamforming, and perform target detection.

8. The system for estimating formation and detecting targets when towed array sonar is turned according to claim 7, characterized in that: The module M1 comprises: Assume that the towed line array contains N evenly distributed array elements, and the spacing between adjacent array elements is d. The distorted towed line array acoustic array segment is modeled. The position of the first array element is taken as the origin, and a rectangular coordinate system is established in the horizontal plane with its heading as the opposite direction of the X-axis. In this coordinate system, the parabolic model of the distorted array is expressed as: Where a and b are the parabola parameter values.

9. The system for estimating the formation and detecting the target when the towed array sonar is turned according to claim 8, characterized in that: The calculation process of the position coordinates of each array element and the corresponding position slope includes: Assume the coordinates of the first element at the origin are: (x1,y1)=(0,0) With the help of the idea of ​​piecewise straight line approximation and forward propagation, the coordinates of each array element are (x j ,y j )(j=2,…,N) make It is obtained by the following formula: k j-1 =ax j-1 +b Among them, k j-1 is the position slope corresponding to the position coordinate of the array element.

10. The system for estimating the formation and detecting the target when the towed array sonar is turned according to claim 9, characterized in that: The module M3 also includes: τ j It represents the theoretical time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, expressed as τ j * represents the actual time delay when the jth array element receives the high-frequency pulse relative to the first array element receiving the high-frequency pulse, and the array estimation problem is transformed into a least squares optimization problem: Among them, Ψ is the objective function.

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