A Fast Beamforming Method Based on Nested Non-Uniform Fourier Transform

Abstract

This invention discloses a fast beamforming method based on nested non-uniform Fourier transform, comprising the following steps: (1) pre-calculating relevant parameters of the nested non-uniform Fourier transform, including the fast Fourier transform length, the pre-interpolation coefficient matrix, and the post-interpolation coefficient matrix; (2) performing pre-interpolation on the array response signal in the frequency domain based on the pre-interpolation coefficient matrix to obtain the pre-interpolation matrix; (3) performing fast Fourier transform on the pre-interpolation matrix based on the fast Fourier transform length to achieve spatial domain transformation and obtain the transformed matrix; (4) interpolating the transformed matrix based on the post-interpolation coefficient matrix to obtain the post-interpolation matrix as the beamforming result. This method significantly reduces the computational burden of beamforming in a phased array three-dimensional camera sonar system while ensuring accuracy.

Description

Technical Field

[0001] This invention belongs to the technical field of phased array three-dimensional camera sonar systems, specifically relating to a fast beamforming method based on nested non-uniform Fourier transform. Background Technology

[0002] A phased array 3D camera sonar system uses a narrowband acoustic pulse to transmit across the entire underwater scene and utilizes a two-dimensional transducer array to receive backscattered echo signals from the scene, generating an array response signal. These array response signals are then processed using beamforming and real-time image processing techniques to obtain high-resolution underwater 3D images.

[0003] To improve image resolution while maintaining a large detection angle, two-dimensional transducer arrays typically have large apertures and contain hundreds or thousands of elements. Furthermore, the beamform obtained through signal processing often contains tens of thousands of beam directions, leading to a significant computational burden.

[0004] Traditional delay-summing beamforming methods adjust and accumulate the time delay of array element sampling signals according to the beam direction, accurately obtaining beam results in any direction. However, their enormous computational burden cannot guarantee real-time imaging. Frequency-domain direct beamforming methods can effectively reduce computational requirements, but their computational burden remains high. Therefore, some frequency-domain fast beamforming methods have been proposed, such as the Chirp-Z Transform (CZT) and the Nonuniform Fast Fourier Transform (NUFFT). These fast beamforming methods utilize the high computational efficiency of the Fast Fourier Transform in different ways, significantly reducing their computational burden compared to frequency-domain direct beamforming methods, but they still remain a design bottleneck for phased array 3D camera sonar systems.

[0005] The main computational burden of the two algorithms mentioned above lies in the Fast Fourier Transform (FFT). To ensure computational accuracy, both algorithms require performing a relatively long FFT, which includes a significant amount of redundant computation. This is particularly true for phased array 3D camera sonar systems, which require a 2D FFT, further exacerbating the computational burden of beamforming due to the longer FFT length. Summary of the Invention

[0006] In view of the above, the purpose of this invention is to provide a fast beamforming method based on nested non-uniform Fourier transform. This method is based on non-uniform fast Fourier transform, and to address its main computational burden (the relatively long fast Fourier transform), a second-order symmetric non-uniform fast Fourier transform is introduced to construct a fast beamforming method based on nested non-uniform Fourier transform. This can reduce the length of the fast Fourier transform in the calculation, thereby significantly reducing the computational burden.

[0007] To achieve the above-mentioned objectives, an embodiment provides a fast beamforming method based on nested non-uniform Fourier transform, characterized by comprising the following steps:

[0008] (1) Pre-calculate the relevant parameters of nested non-uniform Fourier transform, including the fast Fourier transform length, the pre-interpolation coefficient matrix, and the post-interpolation coefficient matrix;

[0009] (2) Based on the pre-interpolation coefficient matrix, the array response signal in the frequency domain is pre-interpolated to obtain the pre-interpolation matrix;

[0010] (3) Perform a fast Fourier transform on the pre-interpolation matrix based on the fast Fourier transform length to achieve spatial domain transformation and obtain the transformed matrix;

[0011] (4) Interpolate the transformed matrix based on the post-interpolation coefficient matrix to obtain the post-interpolation matrix as the beam result.

[0012] Preferably, the length of the Fast Fourier Transform is calculated using the following formula:

[0013]

[0014] Where, N f1 and N f2 The lengths of the Fast Fourier Transform in both directions are represented by Δ, and the beam spacing is represented by D. x and D y The array aperture represents the two directions, k represents the wavenumber, and P and Q represent the beam size of the two-dimensional arbitrary shape array.

[0015] Preferably, the pre-interpolation coefficient matrix and the post-interpolation coefficient matrix are calculated as follows:

[0016] Calculate the offset phase parameters, and then calculate the corrected coordinates of the two-dimensional array elements based on the offset phase parameters. Using the minimum-maximum criterion, calculate the interpolation coefficients *st* for the forward non-uniform Fourier transform based on the corrected coordinates of the two-dimensional array elements. j and scaling factor st p , among which, st j It is an n×PQ sparse matrix, st p It is a PQ×1 vector;

[0017] Calculate the beam correction direction and, using the minimax criterion, calculate the interpolation coefficients sr of the inverse non-uniform Fourier transform based on the beam correction direction. j and scaling factor sr p , among which, sr j It is a PQ×N f1 N f2 sparse matrix, sr p It is an N f1 N f2 A vector of size ×1;

[0018] Based on the interpolation coefficients st j Scaling factor st p Interpolation coefficients sr j and scaling factor sr p Calculate the pre-interpolation coefficient matrix It pre and the post-interpolation coefficient matrix It post , represented as:

[0019] It pre =st j '⊙sr p

[0020]

[0021] It post =sr j ⊙st p ⊙(om shift )

[0022] Among them, om shift Denotes the reverse offset matrix, st j ' indicates st j Cut into n×N f1 N f2 A sparse matrix, where ⊙ represents the dot product operation. Shift represents the vector product. x and shift y The offset phase parameters in the x and y directions are represented, j represents the imaginary unit, and p and q represent the (p,q)th beam.

[0023] Preferably, the offset phase parameter and the two-dimensional array element correction coordinates are calculated using the following formula:

[0024] shift x =kΔD x / 2

[0025] shift y =kΔD y / 2

[0026] Where Δ represents the beam spacing, and D x and D y The array aperture represents the two directions, and k represents the wavenumber;

[0027] om x =-kΔx n +shift x

[0028] om y =-kΔy n +shift y

[0029] Among them, om x and om y Represents the corrected coordinates in the x and y directions, (x n ,y n ) represents the coordinates of the nth element.

[0030] Preferably, the beam correction direction is calculated using the following formula:

[0031]

[0032] Among them, om p and om q Let P and Q represent the correction directions of the p-th and q-th beams, respectively, where P and Q represent the beam sizes of the two-dimensional array of arbitrary shapes.

[0033] Preferably, the array response signal in the frequency domain is pre-interpolated based on the pre-interpolation coefficient matrix to obtain the pre-interpolation matrix, including:

[0034] Calculate the pre-interpolation vector:

[0035] S n =s n ×It pre

[0036] Among them, S n It is a vector obtained after pre-interpolation, with a size of 1×N. f1 N f2 s n It represents the sampled data of the nth element, i.e., the array response signal. pre N represents the pre-interpolation coefficient matrix. f1 and N f2 Indicates the length of the Fast Fourier Transform in both directions;

[0037] Vector S n Rewritten as size N f1 ×N f2 The pre-interpolation matrix S mn .

[0038] Preferably, performing a Fast Fourier Transform (FFT) on the pre-interpolation matrix based on the FFT length to achieve a spatial domain transformation and obtain the transformed matrix includes:

[0039] For the pre-interpolation matrix S mn Execution length is N f1 and N f2 The two-dimensional fast Fourier transform yields the transformed matrix B. m .

[0040] Preferably, the transformed matrix is ​​interpolated based on the post-interpolation coefficient matrix to obtain the beamforming result, including:

[0041] Calculate the interpolation vector after calculation:

[0042] b = It post ×B m

[0043] Where b is the vector obtained after post-interpolation, with a size of PQ×1, It post B represents the post-interpolation coefficient matrix. m Let P and Q represent the transformed matrix, and let Q represent the beam size of the two-dimensional array of arbitrary shape.

[0044] The vector b is rewritten as a post-interpolation matrix B of size P×Q and used as the beamforming result.

[0045] Compared with the prior art, the beneficial effects of the present invention include at least the following:

[0046] This invention provides a fast beamforming method based on nested non-uniform Fourier transform, which fully utilizes the physical characteristics of phased array 3D camera sonar systems to minimize the length of the fast Fourier transform (depending on the array aperture and beam range). Compared to conventional Chirp-Z transform or non-uniform Fourier transform algorithms, the fast Fourier transform used in this invention can be reduced by more than 50% in length while maintaining consistent computational accuracy. Therefore, this invention can significantly reduce the computational burden of beamforming in phased array 3D camera sonar systems. Attached Figure Description

[0047] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0048] Figure 1 This is a flowchart illustrating the fast beamforming method based on nested non-uniform Fourier transform of the present invention.

[0049] Figure 2 A schematic diagram of the signal model of the transducer array of a phased array three-dimensional camera sonar system;

[0050] Figure 3 To compare the beam pattern results calculated using the method of this invention with those calculated using fast beamforming methods such as CZT transform and NUFFT transform, this method has the same calculation accuracy as CZT transform and NUFFT transform.

[0051] Figure 4 A comparison of the computational burden of the method of this invention with that of CZT transform and NUFFT transform under different array parameters shows that the method of this invention has the lowest computational burden. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of protection of this invention.

[0053] In a phased array 3D camera sonar system, a two-dimensional array of arbitrary shape containing N elements is located in the XOY plane of 3D space. The coordinates of the nth element can be represented as (x... n ,y n The beam direction is determined by two parameters θ. a and θ e This means that a total of P×Q beams need to be generated. For the (p,q)th beam, its beam direction is (θ). ap ,θ eq The beam signal can be represented as:

[0054]

[0055] Among them, s n Let τ(n,p,q) represent the sampled data of the nth element, k represent the wavenumber, and τ(n,p,q) represent the nth element for (θ). ap ,θ eq The phase shift in the direction of ) , where j represents the imaginary unit, and

[0056] τ(n,p,q)=x n sin(θ ap )+y n sin(θ eq )

[0057] In a specific example, suppose the transducer array in a particular phased array three-dimensional camera sonar system is a 25×25 two-dimensional uniform rectangular array, the signal center frequency is 150kHz, and the element spacing is λ / 2, where λ is the signal wavelength; for example... Figure 2As shown, the array center is located at the origin of the coordinate system. A total of 128×128 beams need to be calculated within the range of (-60°, 60°). The target is set at (0°, 0°), (18°, 0°), (-25°, -25°), and (-34.5°, -55°).

[0058] The fast beamforming method based on nested non-uniform Fourier transform provided in this embodiment performs fast beamforming on the transducer array in the above example, including the following steps:

[0059] S1 pre-computes the relevant parameters of the nested non-uniform Fourier transform, including the fast Fourier transform length, the pre-interpolation coefficient matrix, and the post-interpolation coefficient matrix.

[0060] In this embodiment, the length of the Fast Fourier Transform is first calculated, i.e.:

[0061]

[0062] Where, N f1 and N f2 The lengths of the Fast Fourier Transform in both directions are represented by Δ, and the beam spacing is represented by D. x and D y The array aperture represents the two directions, and P and Q represent the number of beams in the two directions.

[0063] Therefore, a two-dimensional fast Fourier transform of length 84×84 is performed in step S3.

[0064] Compared to the CZT transform, which requires two 152×152 two-dimensional fast Fourier transforms, and the NUFFT transform, which requires one 256×256 two-dimensional fast Fourier transform, this method greatly reduces the computational burden required for the fast Fourier transform.

[0065] Then, the calculation methods for the pre-interpolation coefficient matrix and the post-interpolation coefficient matrix are as follows:

[0066] (a) Calculate the offset phase parameters, and calculate the corrected coordinates of the two-dimensional array elements based on the offset phase parameters, i.e.:

[0067] shift x =kΔD x / 2 = 0.5101

[0068] shift y =kΔD y / 2 = 0.5101

[0069] om x =-kΔx n +shift x

[0070] om y =-kΔy n +shift y

[0071] Among them, shift x and shift y om represents the phase offset in the x and y directions. x and om y This represents the corrected coordinates in the x and y directions. In this embodiment, a two-dimensional uniform rectangular array is used, with the same corrected coordinates in the x and y directions, as shown in Table 1.

[0072] Table 1 Corrected coordinates of two-dimensional array elements

[0073]

[0074]

[0075] The interpolation coefficients st of the forward non-uniform Fourier transform are calculated based on the coordinates of the two-dimensional array elements using the minimum-maximum criterion. j and scaling factor st p , among which, st j It is an n×PQ sparse matrix, that is, a 625×16384 sparse matrix (containing 2500 non-zero elements), st p It is a PQ×1 vector, which is a 16384×1 vector.

[0076] (b) Calculate the beam correction direction, i.e.:

[0077]

[0078] Among them, om p and om q This indicates the correction direction for the p-th and q-th beams. In this embodiment, the observation range and number of beams are the same for both beam directions. p and om q The correction directions are the same, as shown in Table 2:

[0079] Table 2 Beam Correction Direction

[0080]

[0081]

[0082] The interpolation coefficients sr of the inverse non-uniform Fourier transform are calculated based on the beam correction direction using the minimum-maximum criterion. j and scaling factor sr p , among which, sr j It is a PQ×Nf1 N f2 The sparse matrix, i.e., a sparse matrix of 16384×7225, sr p It is an N f1 N f2 A vector of size ×1, that is, a vector of size 7056×1;

[0083] (c) Based on the interpolation coefficients st j Scaling factor st p Interpolation coefficients sr j and scaling factor sr p Calculate the pre-interpolation coefficient matrix It pre (size 2500×7056), and the post-interpolation coefficient matrix It post (Dimensions are 16384×7056), represented as:

[0084] It pre =st j '⊙sr p

[0085]

[0086] It post =sr j ⊙st p ⊙(om shift )

[0087] Among them, om shift Denotes the reverse offset matrix, st j ' indicates st j Cut into n×N f1 N f2 A sparse matrix of size 625 × 7056, where ⊙ represents the dot product operation. Shift represents the vector product. x and shift y The offset phase parameters in the x and y directions are represented, j represents the imaginary unit, and p and q represent the (p,q)th beam.

[0088] Step S1 only requires pre-calculation; the fast Fourier transform length, the pre-interpolation coefficient matrix, and the post-interpolation coefficient matrix can be stored in advance, without the need for real-time calculation.

[0089] S2, based on the pre-interpolation coefficient matrix, pre-interpolates the array response signal in the frequency domain to obtain the pre-interpolation matrix.

[0090] Specifically, calculate the pre-interpolation vector:

[0091] S n =s n ×It pre

[0092] Among them, S n It is a vector obtained after pre-interpolation, with a size of 1×N. f1 N f2 That is, 1×7056;

[0093] Vector S n Rewritten as size N f1 ×N f2 The pre-interpolation matrix S (i.e., 84×84) mn .

[0094] S3 performs a Fast Fourier Transform on the pre-interpolation matrix based on the Fast Fourier Transform length to achieve spatial domain transformation and obtain the transformed matrix.

[0095] Specifically, for the pre-interpolation matrix S mn Execution length is N f1 and N f2 The two-dimensional fast Fourier transform yields the transformed matrix B. m .

[0096] S4. Based on the post-interpolation coefficient matrix, the transformed matrix is ​​interpolated to obtain the post-interpolation matrix as the beam result.

[0097] Specifically, the interpolation vector is calculated as follows:

[0098] b = It post ×B m

[0099] Where b is the vector obtained after post-interpolation, with a size of PQ×1, i.e., 16384×1;

[0100] The vector b is rewritten as a post-interpolation matrix B of size P×Q (i.e., 128×128) and used as the beamforming result.

[0101] The beam pattern obtained by the method of this invention is as follows: Figure 3 As shown in (a), for comparison, Figure 3 (b) and Figure 3 Figure (c) shows the beammap results based on CZT transform and NUFFT transform.

[0102] Table 3 shows the intensity of the beam directions corresponding to the four scatterers obtained based on CZT transform, NUFFT transform, and the method of this invention in the embodiments. Table 4 shows the statistics of the real-time computational cost required by the three methods in this embodiment. Figure 4 A comparison of the real-time computational cost of three methods under different array sizes is presented.

[0103] Table 3 Beam Strength

[0104]

[0105]

[0106] Table 4

[0107] method Real-time computation CZT transform 3589800 NUFFT transform 5423680 The method proposed in this invention 2695290

[0108] As shown in Table 3, the present invention achieves computational accuracy comparable to CZT transform and NUFFT transform; as shown in Table 4, the amount of real-valued computation required by the present invention is significantly less than that of CZT transform and NUFFT transform. Therefore, the fast beamforming method based on nested non-uniform Fourier transform proposed in this invention can significantly reduce the computational burden of beamforming in phased array three-dimensional camera sonar systems while ensuring accuracy.

[0109] The specific embodiments described above illustrate the technical solution and beneficial effects of the present invention in detail. It should be understood that the above description is only the most preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A fast beamforming method based on nested non-uniform Fourier transform, characterized in that, Includes the following steps: (1) Pre-calculate the relevant parameters of the nested non-uniform Fourier transform, including the fast Fourier transform length, the pre-interpolation coefficient matrix, and the post-interpolation coefficient matrix; the pre-interpolation coefficient matrix and the post-interpolation coefficient matrix are calculated as follows: Calculate the offset phase parameters, and then calculate the corrected coordinates of the two-dimensional array elements based on the offset phase parameters. Using the minimum-maximum criterion, calculate the interpolation coefficients *st* for the forward non-uniform Fourier transform based on the corrected coordinates of the two-dimensional array elements. j and scaling factor st p , among which, st j It is an n×PQ sparse matrix, st p It is a PQ×1 vector; Calculate the beam correction direction and, using the minimax criterion, calculate the interpolation coefficients sr of the inverse non-uniform Fourier transform based on the beam correction direction. j and scaling factor sr p , among which, sr j It is a PQ×N f1 N f2 sparse matrix, sr p It is an N f1 N f2 A vector of size ×1; Based on the interpolation coefficients st j Scaling factor st p Interpolation coefficients sr j and scaling factor sr p Calculate the pre-interpolation coefficient matrix It pre and the post-interpolation coefficient matrix It post , represented as: It pre =st j ’⊙sr p It post =sr j ⊙st p ⊙(om shift ) Among them, om shift Denotes the reverse offset matrix, st j ' indicates st j Cut into n×N f1 N f2 A sparse matrix, where ⊙ represents the dot product operation. Shift represents the vector product. x and shift y The offset phase parameters in the x and y directions are represented, j represents the imaginary unit, and p and q represent the (p,q)th beam. (2) The array response signal in the frequency domain is pre-interpolated based on the pre-interpolation coefficient matrix to obtain the pre-interpolation matrix; (3) Perform a fast Fourier transform on the pre-interpolation matrix based on the fast Fourier transform length to achieve spatial domain transformation and obtain the transformed matrix; (4) Interpolate the transformed matrix based on the post-interpolation coefficient matrix to obtain the post-interpolation matrix as the beam result.

2. The fast beamforming method based on nested non-uniform Fourier transform according to claim 1, characterized in that, The length of the Fast Fourier Transform is calculated using the following formula: Where, N f1 and N f2 The lengths of the Fast Fourier Transform in both directions are represented by Δ, and the beam spacing is represented by D. x and D y The array aperture represents the two directions, k represents the wavenumber, and P and Q represent the beam size of the two-dimensional arbitrary shape array.

3. The fast beamforming method based on nested non-uniform Fourier transform according to claim 1, characterized in that, The offset phase parameter and the corrected coordinates of the two-dimensional array elements are calculated using the following formula: shift x =kΔD x / 2 shift y =kΔD y / 2 Where Δ represents the beam spacing, and D x and D y The array aperture represents the two directions, and k represents the wavenumber; about x =-kΔx n +shift x about y =-kΔy n +shift y Among them, om x and om y Represents the corrected coordinates in the x and y directions, (x n ,y n ) represents the coordinates of the nth element.

4. The fast beamforming method based on nested non-uniform Fourier transform according to claim 1, characterized in that, The beam correction direction is calculated using the following formula: Among them, om p and om q Let P and Q represent the correction directions of the p-th and q-th beams, respectively, where P and Q represent the beam sizes of the two-dimensional array of arbitrary shapes.

5. The fast beamforming method based on nested non-uniform Fourier transform according to claim 1, characterized in that, The array response signal in the frequency domain is pre-interpolated based on the pre-interpolation coefficient matrix to obtain the pre-interpolation matrix, which includes: Calculate the pre-interpolation vector: S n =s n ×It pre Among them, S n It is a vector obtained after pre-interpolation, with a size of 1×N. f1 N f2 s n It represents the sampled data of the nth element, i.e., the array response signal. pre N represents the pre-interpolation coefficient matrix. f1 and N f2 Indicates the length of the Fast Fourier Transform in both directions; Vector S n Rewritten as size N f1 ×N f2 The pre-interpolation matrix S mn .

6. The fast beamforming method based on nested non-uniform Fourier transform according to claim 1, characterized in that, Performing a Fast Fourier Transform (FFT) on the pre-interpolated matrix based on the FFT length achieves spatial domain transformation to obtain the transformed matrix, including: For the pre-interpolation matrix S mn Execution length is N f1 and N f2 The two-dimensional fast Fourier transform yields the transformed matrix B. m .

7. The fast beamforming method based on nested non-uniform Fourier transform according to claim 1, characterized in that, The beamforming results are obtained by interpolating the transformed matrix based on the post-interpolation coefficient matrix, including: Calculate the interpolation vector after calculation: b=It post ×B m Where b is the vector obtained after post-interpolation, with a size of PQ×1, It post B represents the post-interpolation coefficient matrix. m Let P and Q represent the transformed matrix, and let Q represent the beam size of the two-dimensional array of arbitrary shape. The vector b is rewritten as a post-interpolation matrix B of size P×Q and used as the beamforming result.

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