A method for optimizing the low sidelobe beam of a linear array based on overscan

Abstract

The present invention relates to a method for optimizing the low sidelobe beam of a linear array based on overscan. By introducing an additional delay into the weighting vector pointing to the end-fire direction, lower sidelobes can be obtained. The present invention also presents overscan beam optimization methods for both narrowband and broadband cases. For the narrowband case, the relationship between the sidelobe level and the overscan amount is polynomially fitted to obtain the overscan amount-sidelobe level relationship curve. From this curve, the overscan amount corresponding to the required sidelobe level can be easily determined, and then beamforming can be performed to obtain a low sidelobe beam with performance similar to that of the second-order cone optimization method. For the broadband case, a fitting polynomial of the overscan amount-spacing wavelength ratio (d / λ) can be obtained based on the narrowband case, and the corresponding overscan amounts can be brought into the beamforming process to obtain a low sidelobe beam. Compared with the beam optimization method based on second-order cone programming, the method of the present invention has a simpler process and more convenient calculation.

Description

Technical Field

[0001] The present invention belongs to the fields of acoustic array signal processing, sonar and radar technologies, and relates to a method for optimizing the low sidelobe beam of a linear array based on overscanning, mainly aiming at the beamforming method of a linear array. Background Art

[0002] The main indicators of the beam pattern include directivity, main lobe width, sidelobe level, robustness, etc. An ideal beamforming method is expected to have higher directivity, narrower main lobe width, lower sidelobes and better robustness. However, these indicators of the beam pattern restrict each other and it is impossible to achieve the optimal simultaneously. One can only find the best compromise among these conflicting indicators. Some existing sidelobe control beamforming methods with better effects mostly need to use complex algorithms to solve the sidelobe constraints. For example, the high-gain beamforming based on second-order cone programming mentioned in Document 1 "Optimized beamforming method for sensor arrays with arbitrary geometries and element directivities, Acta Acustica, 2005, vol. 30(3), p. 264-270" can obtain a robust low-sidelobe beam, but it needs to use second-order cone programming and the calculation is relatively complex. The patent with publication number CN 112099017A discloses a method for optimizing the low sidelobe beam of a circular array based on overscanning, which mainly aims at a circular array and only considers the narrowband case. Summary of the Invention

[0003] Technical Problems to be Solved

[0004] In order to avoid the deficiencies of the prior art, the present invention proposes a method for optimizing the low sidelobe beam of a linear array based on overscanning, aiming at the problem that some existing sidelobe constraint algorithms are relatively complex.

[0005] Technical Solution

[0006] A method for optimizing the low sidelobe beam of a linear array based on overscanning, characterized by the following steps:

[0007] Step 1: Given the overscanning amount, calculate the beam responses at different overscanning amounts:

[0008]

[0009] where: w ovs is the weighted vector of the overscanning method, the superscript H represents the complex conjugate transpose, P(θ) is the array manifold of the linear array, P(θ) = [p(θ1) p(θ2)... p(θ i )... p(θ D )], D is the number of horizontal angle divisions in the observation viewing area, p(θ i ) = [1, exp(jkdcos(θ i),..., exp(jk(M - 1)d cos(θ i ))] T ;

[0010] The

[0011] said p ovs = [1, exp(jkd cos(θ0 + σ)),..., exp(jk(M - 1)d cos(θ0 + σ))] T

[0012] where: k is the wave number, d is the element spacing, θ0 is the end - fire direction angle, σ is the over - scan amount, M is the number of elements, and the superscript T represents the transpose;

[0013] R n is the noise covariance matrix:

[0014] The th element is:

[0015] where: is the distance between the m - th and the th element;

[0016] Repeat step 1. When the main lobe of the beam is not in the end - fire direction, the over - scan amount is the limit over - scan amount:

[0017] γ = min{σ|B σ (θ0) dB ≠ 0}

[0018] where: B σ (θ0) dB is the normalized decibel representation of the beam response in the end - fire direction when the over - scan amount is σ;

[0019] Step 2, calculate the sidelobe levels at different over - scan amounts:

[0020] Within the determined limit over - scan amount range, calculate the sidelobe levels corresponding to different over - scan amounts:

[0021] SL σ = max{B σ (θ)…θ ∈ [- 180°, θ0 - θ ML / 2] ∪ [θ0 + θ ML / 2, 180°]}, σ ∈ [0, γ]

[0022] where: θ ML is the main lobe width, and θ0 is the end - fire direction;

[0023] Step 3: Give the fitting curves under narrowband and broadband:

[0024] 1. In the narrowband case:

[0025] According to the relationship between the overscan amount and the sidelobe level (σ - SL) obtained in Step 2, determine the fitting range of the overscan amount:

[0026] Γ = [0, σ S

[0027] where: σ S is the overscan amount corresponding to the minimum sidelobe level;

[0028] Fit the σ - SL relationship within the overscan amount fitting range:

[0029]

[0030] where: x represents the sidelobe level (unit: dB), N is the highest order of the fitting polynomial, a n is the coefficient of the nth - order polynomial;

[0031] 2. Under broadband conditions

[0032] Given the desired sidelobe level, according to the foregoing steps, obtain the overscan amount corresponding to the specified sidelobe level at a certain d / λ value, and then use a polynomial to fit the relationship between the overscan amount and d / λ:

[0033]

[0034] where: y represents d / λ, P is the highest order of the fitting polynomial, b p is the coefficient of the pth - order polynomial;

[0035] Step 4: Plot the beam pattern and calculate the beam performance:

[0036] 1. In the narrowband case

[0037] Determine the overscan amount according to the required sidelobe level, substitute the obtained overscan amount into the beamforming process in Step 1 to obtain the beam pattern under the constraint of the required sidelobe level, and calculate the directivity index and white noise gain of the obtained beam pattern:

[0038]

[0039] where: x0 is the required sidelobe level, and σ0 is the corresponding overscan amount;

[0040] The calculation methods of the directivity index and white noise gain are as follows:

[0041]

[0042] ​

[0043] where: p0 = [1, exp(jkd cosθ0),..., exp(jk(M - 1)d cosθ0)] T is the end - fire direction response vector of the linear array, and w ovs is the weighted vector of the over - scan method obtained by substituting σ0 into step 1, and R n is the noise covariance matrix;

[0044] 2. In the wide - band case

[0045] Substitute the over - scan amounts corresponding to different d / λ values under the specified desired sidelobe level obtained in step 3 into the beamforming in step 1 to obtain beam patterns at different d / λ values. And calculate the sidelobe level, directivity index, and white noise gain at different d / λ values.

[0046] Beneficial effects

[0047] A linear - array low - sidelobe beam optimization method based on over - scan proposed by the present invention can obtain lower sidelobes by introducing an additional delay in the weighted vector pointing to the end - fire direction. The present invention also gives over - scan beam optimization methods for both narrow - band and wide - band cases. For the narrow - band case, by fitting the relationship between the sidelobe level and the over - scan amount with a polynomial, an over - scan amount - sidelobe level relationship curve can be obtained. From this curve, the over - scan amount corresponding to the required sidelobe level can be conveniently determined, and then beamforming can be performed to obtain a low - sidelobe beam with performance similar to that of the second - order cone optimization method. For the wide - band case, a fitting polynomial of the over - scan amount - spacing - wavelength ratio (d / λ) can be obtained based on the narrow - band case, and by substituting the corresponding over - scan amounts into the beamforming process, a low - sidelobe beam can be obtained. Compared with the beam optimization method based on second - order cone programming, the method of the present invention has a simpler process and more convenient calculation.

[0048] The present invention proposes a new low - sidelobe beam optimization method for a linear array. By specifying the over - scan amount and introducing a certain delay in the end - fire direction of the linear array, lower sidelobes can be obtained. The over - scan methods proposed by the present invention for both narrow - band and wide - band cases can obtain beams with performance similar to that of the second - order cone optimization method under the same sidelobe constraints. Moreover, the present invention does not require complex algorithms in second - order cone programming. The corresponding relationships between the over - scan amount and the sidelobe level (narrow - band) for a given number of array elements and element spacing, and the corresponding relationship between the over - scan amount and the spacing - wavelength ratio for a given desired sidelobe level (wide - band) can be stored in a database in advance. According to the beam optimization requirements, real - time calling can be performed, which can greatly simplify the sidelobe constraint process. Description of the drawings

[0049] Figure 1 are beam patterns under different over - scan amounts;

[0050] Figure 2 Beam patterns when the beam is "unsplit" and "split" under different overscan amounts;

[0051] (a) 0°; (b) 10°; (c) 23°; (d) 25°.

[0052] Figure 3 Curve of sidelobe level varying with overscan amount;

[0053] Figure 4 Curve of overscan amount varying with sidelobe level after fitting;

[0054] Figure 5 Overscan amounts corresponding to each d / λ value and the results of 8th-order polynomial fitting when the desired sidelobe level is -20 dB;

[0055] Figure 6 Beam patterns obtained by the overscan method and the second-order cone optimization method when the overscan amount is 15.2° under the -25 dB sidelobe constraint;

[0056] Figure 7 Beam patterns obtained by the overscan method and the second-order cone optimization method when the desired sidelobe level is -20 dB;

[0057] (a) Overscan method; (b) Second-order cone method.

[0058] Figure 8 Curves of sidelobe level, directivity index, and white noise gain varying obtained by the overscan method and the second-order cone optimization method when the desired sidelobe level is -20 dB;

[0059] (a) Sidelobe level; (b) Directivity index; (c) White noise gain. Detailed implementation manner

[0060] The present invention will be further described in conjunction with embodiments and the accompanying drawings:

[0061] The specific steps of the present invention are as follows

[0062] Step 1, calculate the beam responses under different overscan amounts:

[0063] The direction response vector of the linear array in the θ direction is:

[0064] p(θ) = [1, exp(jkd cosθ),..., exp(jk(M - 1)d cosθ)] T

[0065] The direction response vector in the end-fire direction is:

[0066] p(θ0) = [1, exp(jkd cosθ0),..., exp(jk(M - 1)d cosθ0)] T

[0067] The direction response vector after applying a σ overscan amount is:

[0068] p ovs = [1, exp(jkd cos(θ0 + σ)), …, exp(jk(M - 1)d cos(θ0 + σ))] T

[0069] Where: k is the wave number, θ0 = 0°, d is the element spacing, M is the number of elements, and the superscript T represents the transpose;

[0070] Referring to the MVDR method, the weighted vector of the overscan method is:

[0071]

[0072] Where: R n is the noise covariance matrix:

[0073]

[0074] The th element is:

[0075]

[0076] Where: is the distance between the mth and the th element;

[0077] The beam response is:

[0078]

[0079] Where: P(θ) is the linear array manifold, P(θ) = [p(θ1) p(θ2) … p(θ i ) … p(θ D )], D is the number of horizontal angle divisions in the observation viewing area:

[0080] p(θ i ) = [1, exp(jkd cos(θ i )),..., exp(jk(M - 1)d cos(θ i ))] T ;

[0081] The beam patterns when the overscan amounts are 0°, 10°, and 16° are referred to Figure 1 .

[0082] Step 2. Determine the limit overscan amount γ:

[0083] In Step 1, continuously increase the overscan amount. When the main lobe of the beam is not in the end-fire direction (the beam "splits", see Figure 2 (c)(d)), the overscan method at this time is no longer applicable, and the overscan amount at this time is the limit overscan amount:

[0084] γ = min{σ|B σ (θ0) dB ≠ 0}

[0085] where: B σ (θ0) dB is the normalized decibel representation of the beam response in the end-fire direction when the overscan amount is σ;

[0086] B σ (θ0) dB = 20log 10 (B σ (θ0) / max(B σ (θ)))

[0087] The beam patterns at overscan amounts of 0°, 10°, 23°, and 25° are referred to Figure 2 , and at this time γ = 23°.

[0088] Step 3. Determine the sidelobe levels at different overscan amounts:

[0089] Within the range of the limit overscan amount determined in Step 2, calculate the sidelobe levels corresponding to different overscan amounts:

[0090] SL σ = max{B σ (θ)|θ ∈ [-180°, θ0 - θ ML / 2] ∪ [θ0 + θ ML / 2, 180°]}, σ ∈ [0, γ]

[0091] where: θ ML is the main lobe width, and θ0 is the end-fire direction;

[0092] The curve of the sidelobe level varying with the overscan amount is referred to Figure 3 .

[0093] Step 4. Give the fitting polynomials of the overscan amount - sidelobe level (narrowband) and the fitting polynomials of the overscan amount - spacing wavelength ratio (d / λ) (wideband) and the fitting curves:

[0094] 1. Under narrowband conditions

[0095] Figure 3 For the SL-σ curve, the fitting range is:

[0096] Γ = [0°, 17°]

[0097] Fit the σ-SL relationship within the overscan amount fitting range, and use an 8th-order polynomial for fitting:

[0098]

[0099] Where:

[0100]

[0101] According to this fitting polynomial, draw the curve of overscan amount versus sidelobe level (σ-SL curve), refer to Figure 4 .

[0102] 2. Under broadband conditions

[0103] Given the desired sidelobe level of -20 dB, take d / λ ∈ [0.1, 0.4]. When the element spacing d = 1 m, the corresponding frequency range is f ∈ [150 Hz, 600 Hz];

[0104] According to the aforementioned steps, the overscan amount corresponding to a sidelobe level of -20 dB at a certain d / λ value can be obtained. Furthermore, a polynomial can be used to fit the relationship between the overscan amount and d / λ, and an 8th-order polynomial is used for fitting:

[0105]

[0106] Where:

[0107]

[0108] The fitting result refers to Figure 5 .

[0109] Step 5. Plot the beam pattern and calculate the beam performance:

[0110] 1. Under narrowband conditions

[0111]

[0112] Substitute the required sidelobe level of -25 dB into the fitting polynomial to obtain σ0 = 15.2°. Substitute the obtained overscan amount into the beamforming operation in Step 1 to obtain the beam pattern with the required sidelobe level. At this time, the overscan optimized beam and the second-order cone optimized beam refer to Figure 6 .

[0113] The directivity index and white noise gain are calculated as follows:

[0114]

[0115]

[0116] Under the -25 dB sidelobe constraint, Figure 6 the sidelobe levels of the overscanned beam and the second-order cone beam are both -25 dB, meeting the constraint requirements. The directivity indices of the overscanned beam and the second-order cone beam are 17.17 dB and 17.65 dB respectively, and the white noise gains are -72.66 dB and -78.51 dB respectively. At this time, it can be considered that the performance of the overscanned beam is similar to that of the second-order cone beam.

[0117] 2. Under broadband conditions

[0118] Substitute the overscan amounts corresponding to different d / λ values obtained in step 4 under the given desired sidelobe level into the beamforming in step 1 to obtain beam patterns at different d / λ.

[0119] When the desired sidelobe level is -20 dB, the equal-sidelobe beam pattern obtained by the overscan method is referenced to Figure 7 (a). For comparison, when the desired sidelobe level is -20 dB, the equal-sidelobe beam pattern obtained by the second-order cone optimization method is referenced to Figure 7 (b).

[0120] Calculate the sidelobe level, directivity index, and white noise gain for the beams at different d / λ;

[0121] For the obtained beam patterns, the change curves of the sidelobe level, directivity index, and white noise gain are referenced to Figure 8 . For comparison, the performance indicators of the second-order cone optimization method are also plotted on the graph.

[0122] Figure 8 In (a), the sidelobe levels of the overscan method and the second-order cone optimization method are both -20 dB, meeting the constraint requirements. Figure 8 In (b) and (c), the directivity indices and white noise gains of the two methods are similar, and it can be approximately considered that they have similar performance.

[0123] The overscan-based linear array low-sidelobe beam optimization method proposed by the present invention has a simple process and convenient calculation. The corresponding relationships between the overscan amount and the sidelobe level (narrowband) under the given number of array elements and element spacing and the corresponding relationship between the overscan amount and the spacing-wavelength ratio (broadband) under the given desired sidelobe level can be stored in a database in advance. According to the beam optimization requirements, it can be called in real time, which can greatly simplify the sidelobe constraint process.

[0124] For both narrowband and broadband cases, the overscan method proposed by the present invention can obtain beams with performance similar to that of the second-order cone optimization method under the equal-sidelobe constraint, and does not require the complex algorithm of the second-order cone optimization method, simplifying the sidelobe constraint process.

Claims

1. A method for optimizing the low sidelobe beam of a linear array based on overscan, characterized in that The steps are as follows: Step 1: Given the oversampling amount, calculate the beam responses at different oversampling amounts: where: w ovs is the weighted vector of the overscan method, the superscript H represents the complex conjugate transpose, P(θ) is the linear array manifold, P(θ) = [p(θ1) p(θ2)…p(θ i )…p(θ D )], D is the number of horizontal angle divisions in the observation view area, p(θ i ) = [1, exp(jkdcos(θ i )),..., exp(jk(M - 1)dcos(θ i ))] T ; The said The said p ovs = [1, exp(jkdcos(θ0 + σ)),..., exp(jk(M - 1)dcos(θ0 + σ))] T where: k is the wave number, d is the element spacing, θ0 is the end-fire direction angle, σ is the oversampling amount, M is the number of elements, and the superscript T represents transpose; R n is the noise covariance matrix: Element No. is: Wherein: is the distance between the m-th and the th array elements; Repeat Step 1. The oversampling amount when the main lobe of the beam is not in the end-fire direction is the limiting oversampling amount: γ = min{σ | B σ (θ0) dB ≠ 0} Where: B σ (θ0) dB is the normalized decibel representation of the end-fire direction beam response when the overscan amount is σ; Step 2: Calculate the sidelobe levels at different oversampling amounts: Within the determined range of the limiting oversampling amount, calculate the sidelobe levels corresponding to different oversampling amounts: SL σ = max{B σ (θ) | θ ∈ [-180°, θ0 - θ ML / 2] ∪ [θ0 + θ ML / 2, 180°]}, σ ∈ [0, γ] Where: θ ML is the main lobe width, and θ0 is the end-fire direction; Step 3: Give the fitting curves under narrowband and broadband conditions:

1. Under narrowband conditions: According to the relationship between the oversampling amount and the sidelobe level (σ-SL) obtained in Step 2, determine the fitting range of the oversampling amount: Γ = [0, σ S ​ where: σ S is the overscan amount corresponding to the minimum sidelobe level; Fit the σ-SL relationship within the fitting range of the oversampling amount: Where: x represents the sidelobe level (unit: dB), N is the highest order of the fitting polynomial, and a n is the coefficient of the nth order polynomial; 2. Under broadband conditions Given the desired sidelobe level, following the aforementioned steps, obtain the oversampling amount corresponding to the specified sidelobe level at a certain d / λ value, and then use a polynomial to fit the relationship between the oversampling amount and d / λ: where: y represents d / λ, P is the highest order of the fitting polynomial, and b p is the coefficient of the p-th order polynomial; Step 4: Plot the beam pattern and calculate the beam performance:

1. Under narrowband conditions Determine the oversampling amount according to the required sidelobe level, substitute the obtained oversampling amount into the beamforming process in Step 1 to obtain the beam pattern under the constraint of the required sidelobe level, and calculate the directivity index and white noise gain of the obtained beam pattern: where: x0 is the required sidelobe level, and σ0 is the corresponding oversampling amount; The calculation methods of the directivity index and white noise gain are as follows: where: p0 = [1, exp(jkdcosθ0),..., exp(jk(M - 1)dcosθ0)] T is the end-fire direction response vector of the linear array, and w ovs is the weighted vector of the over-scanning method obtained by substituting σ0 into Step 1, and R n is the noise covariance matrix; 2. Under broadband conditions Substitute the oversampling amounts corresponding to different d / λ values at the specified desired sidelobe level obtained in Step 3 into the beamforming in Step 1 to obtain the beam patterns at different d / λ values, and calculate the sidelobe levels, directivity indices, and white noise gains at different d / λ values.

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