CN103903609B - A kind of circular array design of beamformer with constant beamwidth method - Google Patents

Abstract

The present invention relates to a kind of circular array design of beamformer with constant beamwidth method, propose a kind of simple and accurate design of beamformer with constant beamwidth method for uniform rings shape array.The method utilizes the characteristic of circular matrix, gives the accurate solution of ring array least squares error beam pattern synthtic price index, and the wave beam finally synthesized and least squares error are expressed simply as subcomponent superposition sum.When expecting that wave beam is converted to appropriate format, the closed expression of available beamformer with constant beamwidth weight vector.Instant invention overcomes prior art operation complicated and accurate not enough not: optimal solution be exactly represented as subcomponent superposition and, and have the expression formula of enclosed.

Description

A kind of circular array design of beamformer with constant beamwidth method

Technical field

The invention belongs to a kind of Beam-former method for designing, relate to a kind of circular array beamformer with constant beamwidth Method for designing, it is adaptable to the broadband target detection of circular array and broadband target Bearing Estimation, belongs to marine acoustics, array letter Number process and the field such as sonar technique.

Background technology

Circular array is simple due to formation, do not have port and starboard ambiguity and can all-round in the range of form substantially constant Wave beam, be therefore widely used in fields such as sonar, radar, communication and voice engineerings.Signal processing side about this array Method also emerges in an endless stream, and wherein Broadband Beamforming Method has been a great concern especially.

Owing to the wave beam of conventional arrays output changes with frequency, especially when signal is from other direction incidence of main lobe, Different frequency signals frequency spectrum will be distorted, and as having carried out low-pass filtering, is unfavorable for subsequent treatment.For solving this Problem, needs design beamformer with constant beamwidth, i.e. wave beam not with frequency shift, particularly ensures that main lobe is in given bandwidth Interior invariable.This type of method has been proposed that a lot, is based primarily upon two class thoughts: (1) utilizes Wave beam forming medium frequency and aperture Relation, invariable by changing the effective aperture wave beam that makes different frequency corresponding of array, typically have document 1 “Constant beamwidth receiving arrays for broad band sonar systems,Acustica, 1970, vol.23 (1), p.21-26 " disclosed combination battle array method, but the method requires that formation strictly meets expansion structure, basic matrix Size is relatively big, needs array element number many.(2) wave beam needing synthesis is made to approach, such as with expectation wave beam under certain criterion Document 2 " arbitrary geometry array wideband constant beamwidth Wave beam forming new method, acoustic journal, 2001, vol.26 (1), p.55-58 " Disclosed Bessel series method and document 3 " any sensor array time domain constant beam-width wave beam based on Second-order cone programming Formed, acoustic journal, 2005, vol.30 (4), p.309-316 " disclosed in convex optimization method.Although this kind of method is the suitableeest For any formation, but they fail to make full use of the characteristic of circular array, do not provide Exact.Document 4 " On the design of digital broadband beamformer for uniform circular array with frequency invariant characteristics,IEEE International Symposium on Circuits And Systems (ISCAS), 2002, vol.1, p.I-693-I-696 " it is specifically designed for ring array and discloses a kind of simple permanent Determining beam-width beam forming method, ultimate principle is to utilize phase place modal theory, not amount with frequency shift is separated, then Matching expectation wave beam, obtains required constant beam-width wave beam.Although the method is simple, but phase place modal theory relates to spatial domain adopts Sample and truncated series, the final result obtained is not accurate.

Summary of the invention

Solve the technical problem that

In place of the deficiencies in the prior art, the present invention proposes a kind of circular array beamformer with constant beamwidth Method for designing, solves prior art operation complicated and accurate not enough.

Technical scheme

A kind of circular array design of beamformer with constant beamwidth method, it is characterised in that step is as follows:

Step 1: determine that expectation wave beam is the two dimensional beam in horizontal extent, by circular array at reference frequency theory meter Obtaining, its expression formula is:Wherein M is element number of array, ()^*Represent and ask altogether Yoke, p_s(ka, φ) is the unit amplitude plane wave signal incident from direction φ that the s array element receives, ω_m(k_ra_r) for giving Fixed expectation weight vector element, and need to meetM=1,2 ..., M/2-1；

DescribedIn formula:a_rFor the radius of circle of reference annular array, k_r=2 π/ λ_r, λ_rRepresent the wavelength with reference to incident plane wave, φ_s=s β, β=2 π/M；

Described

Step 2: calculate expectation wave beam and synthesize the least mean-square error of wave beam: Wherein:And λ_mIt is respectively circular matrix ρ in reference frequency k_ra_rWith eigenvalue during frequency ka to be calculated,For Cyclic Moment Battle arrayEigenvalue, meet relation With

Described

Described

Described

In formula: J₀() is the 0th rank cylindricality Bessel function, Δ r_sr=2a_rSin (s β/2) and Δ r_s=2asin (s β/ 2) it is with reference to the distance between ring array and ring array to be studied each the m and m' sensor, s=| m-m ' | respectively；

Step 3: calculate beamformer with constant beamwidth weight vector in the frequency band range determined, the of weight vector M element expression is:

${\mathrm{\ω}}_m\left(ka\right)=\frac{{\stackrel{\‾}{\mathrm{\λ}}}_m}{{\mathrm{\λ}}_m}{\mathrm{\ω}}_m\left(k_r{a}_r\right)$

Composition weight vector is ω=[ω₀,ω₁,…,ω_M-1]^T, subscript T represents transposition；

Step 4: according to the multi-form of weight vector, uses following two method to synthesize final wave beam:

1, directly calculate:

2, Element space calculates:

Wherein w=V ω, V=[v₀,v₁,…,v_M-1], P (ka, φ)=[p₀(ka,φ),p₁(ka,φ),…,p_M-1(ka, φ)]^TFor array manifold vector, v_m=M^-1/2[1 e^imβ … e^i(M-1)mβ]^TFor circular matrix characteristic vector and meet relationKa is value in the frequency range of wave beam, and this frequency range is according to design accuracy, step 2 minimum obtained is all Side curve of error δ (ka) determines.

Beneficial effect

A kind of circular array design of beamformer with constant beamwidth method that the present invention proposes, for uniform rings shape battle array Row propose a kind of simple and accurate design of beamformer with constant beamwidth method.The method utilizes the characteristic of circular matrix, gives Go out the accurate solution of ring array least squares error beam pattern synthtic price index, and the wave beam finally synthesized and least square have been missed Difference is expressed simply as subcomponent superposition sum.When expecting that wave beam is converted to appropriate format, available constant beam-width wave beam shape Grow up to be a useful person the closed expression of weight vector.Instant invention overcomes prior art operation complicated and accurate not enough: optimal solution Be exactly represented as subcomponent superposition and, and have the expression formula of enclosed.

Have the beneficial effect that:

1. the present invention is without the biggest array aperture, and array number does not the most require a lot, simpler than the method for document 1.

2. The present invention gives the accurate solution making synthesis wave beam minimum with the square error of expectation wave beam, principle is not made Any approximation, is not artificially induced other error, more more reliable than the mode territory method of document 4.

3. when expectation wave beam be transformed to appropriate format, the present invention obtained beamformer with constant beamwidth weight vector and The closed expression of final error, and other document is all difficult to obtain such result.

Accompanying drawing explanation

Fig. 1: circular array schematic diagram.

Fig. 2: expectation beam pattern.

Fig. 3: expectation wave beam and the least squares error variation relation with frequency synthesizing wave beam.

The beam pattern of the constant beam-width broadband beams figure of Fig. 4: synthesis is with the change of frequency.

The Overlapping display of the constant beam-width broadband beams figure of Fig. 5: synthesis.

Fig. 6: the directional gain of the conventional constant beam-width wave beam with synthesis.

Detailed description of the invention

In conjunction with embodiment, accompanying drawing, the invention will be further described:

The present invention is applicable to the design of beamformer with constant beamwidth method of uniform rings shape array, gives uniform rings The accurate solution of the least squares error beam pattern synthtic price index of battle array, and expectation wave beam is converted to appropriate format, obtain constant The closed expression of beam-width beam shaper weight vector.Its process is:

(1) given expectation wave beam.Owing to circular array more attention is the performance of Wave beam forming in horizontal plane, so Expect the two dimensional beam that wave beam is only taken as in horizontal extent.Expect that wave beam is obtained in reference frequency Theoretical Calculation by circular array Arriving, its expression formula is:

$B_d(k_r{a}_r,\mathrm{\φ})=\underset{m=0}{\overset{M-1}{\Σ}}{\mathrm{\ω}}_m^{*}\left(k_r{a}_r\right)E_m(k_r{a}_r,\mathrm{\φ}),---\left(1\right)$

Wherein M is element number of array, ()^*Represent and ask conjugation, p_s(k_ra_r, φ) be m-th array element receive from direction φ Incident unit amplitude plane wave signal, expression formula is:

$p_s(k_r{a}_r,\mathrm{\φ})=e^{-{\mathrm{ik}}_r{a}_{r}cos(\mathrm{\φ}-{\mathrm{\φ}}_s)}---\left(2\right)$

In formulaa_rFor the radius of circle of reference annular array, k_r=2 π/λ_r, λ_rRepresent the ripple with reference to incident plane wave Long.Additionally have:

$E_m(k_r{a}_r,\mathrm{\φ})=\frac{1}{\sqrt{M}}\underset{s=0}{\overset{M-1}{\Σ}}{e}^{-ism\mathrm{\β}}\·p_s(k_r,a_r,\mathrm{\φ}),---\left(3\right)$

β=2 π/M, φ_s=s β.ω_m(k_ra_r) it is given expectation weight vector element, and need to meet

${\mathrm{\ω}}_{M-m}={(-1)}^m{\mathrm{\ω}}_m^{*},(m=1,2,...,M/2-1).$

(2) relation that least squares error changes is calculated with frequency, to determine applicable according to required synthesis precision Frequency band range.Expect that wave beam with the least squares error of synthesis wave beam is:

$\mathrm{\δ}\left(ka\right)=\underset{m=0}{\overset{M-1}{\Σ}}{\left|{\mathrm{\ω}}_m\left(k_r{a}_r\right)\right|}^2\[{\widehat{\mathrm{\λ}}}_m\left(k_r{a}_r\right)-{\left|{\stackrel{\‾}{\mathrm{\λ}}}_m\right|}^2/{\mathrm{\λ}}_m\left(ka\right)\]---\left(4\right)$

WhereinAnd λ_mIt is respectively circular matrix ρ in reference frequency k_ra_rWith eigenvalue during frequency ka to be calculated,For Circular matrixEigenvalue, be satisfied by relationλ_m=λ_M-mWithCalculating formula is followed successively by:

${\widehat{\mathrm{\λ}}}_m\left(k_r{a}_r\right)=\underset{s=0}{\overset{M-1}{\Σ}}{\mathrm{\ρ}}_s\left(k_r{a}_r\right)e^{ism\mathrm{\β}},{\mathrm{\ρ}}_s\left(k_r{a}_r\right)=J_0(k_r\·{\mathrm{\Δr}}_{sr}),---\left(5\right)$

${\mathrm{\λ}}_m\left(ka\right)=\underset{s=0}{\overset{M-1}{\Σ}}{\mathrm{\ρ}}_s\left(ka\right)e^{ism\mathrm{\β}},{\mathrm{\ρ}}_s\left(ka\right)=J_0(k\·{\mathrm{\Δr}}_s),---\left(6\right)$

${\stackrel{\‾}{\mathrm{\λ}}}_m=\underset{s=0}{\overset{M-1}{\Σ}}{\stackrel{\‾}{\mathrm{\ρ}}}_s{e}^{ism\mathrm{\β}},{\stackrel{\‾}{\mathrm{\ρ}}}_s=J_0\left(\sqrt{{\left(k_r{a}_r\right)}^2+{\left(ka\right)}^2-2k_r{a}_r\·ka\·cos\left(s\mathrm{\β}\right)}\right),---\left(7\right)$

J in formula₀() is the 0th rank cylindricality Bessel function, Δ r_sr=2a_rSin (s β/2) and Δ r_s=2asin (s β/2) It is with reference to the distance between ring array and ring array to be studied each the m and m' sensor, s=| m-m ' | respectively.

(3) beamformer with constant beamwidth weight vector in the frequency band range determined is calculated.Weight vector m-th unit The expression formula of element is:

${\mathrm{\ω}}_m\left(ka\right)=\frac{{\stackrel{\‾}{\mathrm{\λ}}}_m}{{\mathrm{\λ}}_m}{\mathrm{\ω}}_m\left(k_r{a}_r\right)---\left(8\right)$

Weight vector is ω=[ω₀,ω₁,…,ω_M-1]^T, subscript T represents transposition.

(4) synthesize final wave beam, according to the multi-form of weight vector, be divided into following two situation:

A) direct formula for calculating is:

$B(ka,\mathrm{\φ})={\mathrm{\ω}}^{H}E(ka,\mathrm{\φ})=\underset{m=0}{\overset{M-1}{\Σ}}{\mathrm{\ω}}_m^{*}{E}_m(ka,\mathrm{\φ})---\left(9\right)$

Wherein E=[E₀,E₁,…,E_M-1]^T, subscript H represents conjugate transpose；

B) Element space calculating formula is:

$B(ka,\mathrm{\φ})=w^{H}P(ka,\mathrm{\φ})=\underset{m=0}{\overset{M-1}{\Σ}}{w}_m^{*}{P}_m(ka,\mathrm{\φ})---\left(10\right)$

Wherein w=V ω, V=[v₀,v₁,…,v_M-1], P (ka, φ)=[p₀(ka,φ),p₁(ka,φ),…,p_M-1(ka, φ)]^TFor array manifold vector, v_m=M^-1/2[1 e^imβ … e^i(M-1)mβ]^TFor circular matrix characteristic vector and meet relationKa is value in the frequency range of wave beam, and this frequency range is according to design accuracy, step 2 minimum obtained is all Side curve of error δ (ka) determines.

Noting, a) result with b) two kinds of method synthesis is consistent, and first method is relatively simple, and second method is more It is beneficial to practical operation.

Specific embodiment is as follows:

(1) with reference to Fig. 1 and 2.This circular array comprises M equally distributed array element.Due to the more pass of circular array Note is the performance of Wave beam forming in horizontal plane, it is desirable to wave beam is only taken as the two dimensional beam in horizontal extent.Expect wave beam Being obtained in reference frequency Theoretical Calculation by circular array, its expression formula is:

$B_d(k_r{a}_r,\mathrm{\φ})=\underset{m=0}{\overset{M-1}{\Σ}}{\mathrm{\ω}}_m^{*}\left(k_r{a}_r\right)E_m(k_r{a}_r,\mathrm{\φ}),---\left(11\right)$

Wherein M is element number of array, ()^*Represent and ask conjugation, p_s(k_ra_r, φ) be m-th array element receive from direction φ Incident unit amplitude plane wave signal, expression formula is:

$p_s(k_r{a}_r,\mathrm{\φ})=e^{-{\mathrm{ik}}_r{a}_{r}cos(\mathrm{\φ}-{\mathrm{\φ}}_s)}---\left(12\right)$

In formulaa_rFor the radius of circle of reference annular array, k_r=2 π/λ_r, λ_rRepresent the ripple with reference to incident plane wave Long.Additionally have:

$E_m(k_r{a}_r,\mathrm{\φ})=\frac{1}{\sqrt{M}}\underset{s=0}{\overset{M-1}{\Σ}}{e}^{-ism\mathrm{\β}}\·p_s(k_r{a}_r,\mathrm{\φ}),---\left(13\right)$

β=2 π/M, φ_s=s β.ω_m(k_ra_r) it is given expectation weight vector element, and need to meet

${\mathrm{\ω}}_{M-m}={(-1)}^m{\mathrm{\ω}}_m^{*},(m=1,2,\mathrm{...},M/2-1).$

Assuming to expect that wave beam is calculated by 16 yuan of uniform rings battle arrays, reference frequency is k_ra_r=3, its weight vector element ω₀～ω₈It is listed in table 1, notices that other element is by relationDirectly obtain.The expectation ripple obtained by formula (11) The side lobe levels that is mainly characterized by of bundle figure is below-20dB, and the linear trend successively decreased.

Table 1

ω0

ω1

ω2

ω3

ω4

ω5

ω6

ω7

ω8

-0.0721

-O.O548i

-0.0365

0.0538i

0.1129

-O.3079i

-0.9794

4.2747i

25.9878

(2) with reference to Fig. 3.Calculate the relation that changes with frequency of least mean-square error, with true according to required synthesis precision The fixed frequency band range being suitable for.

Expect that wave beam is expressed as δ=< B with the square error of design wave beam_d(φ)-B(φ),B_d(φ)-B (φ) >, symbol <>representsIt is derived by further:

$\mathrm{\&delta;}\left(ka\right)=\underset{m=0}{\overset{M-1}{\&Sigma;}}{\left|{\mathrm{\&omega;}}_m\left(k_r{a}_r\right)\right|}^2\&lsqb;{\widehat{\mathrm{\&lambda;}}}_m\left(k_r{a}_r\right)-{\left|{\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_m\right|}^2/{\mathrm{\&lambda;}}_m\left(ka\right)\&rsqb;---\left(14\right)$

WhereinAnd λ_mIt is respectively circular matrix ρ in reference frequency k_ra_rWith eigenvalue during frequency ka to be calculated, For circular matrixEigenvalue, be satisfied by relationλ_m=λ_M-mWithCalculating formula is followed successively by:

${\widehat{\mathrm{\&lambda;}}}_m\left(k_r{a}_r\right)=\underset{s=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&rho;}}_s\left(k_r{a}_r\right)e^{ism\mathrm{\&beta;}},{\mathrm{\&rho;}}_s\left(k_r{a}_r\right)=J_0(k_r\&CenterDot;{\mathrm{\&Delta;r}}_{sr}),---\left(15\right)$

${\mathrm{\&lambda;}}_m\left(ka\right)=\underset{s=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&rho;}}_s\left(ka\right)e^{ism\mathrm{\&beta;}},{\mathrm{\&rho;}}_s\left(ka\right)=J_0(k\&CenterDot;{\mathrm{\&Delta;r}}_s),---\left(16\right)$

${\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_m=\underset{s=0}{\overset{M-1}{\&Sigma;}}{\stackrel{\&OverBar;}{\mathrm{\&rho;}}}_s{e}^{ism\mathrm{\&beta;}},{\stackrel{\&OverBar;}{\mathrm{\&rho;}}}_s=J_0\left(\sqrt{{\left(k_r{a}_r\right)}^2+{\left(ka\right)}^2-2k_r{a}_r\&CenterDot;ka\&CenterDot;cos\left(s\mathrm{\&beta;}\right)}\right),---\left(17\right)$

J in formula₀() is the 0th rank cylindricality Bessel function, Δ r_sr=2a_rSin (s β/2) and Δ r_s=2asin (s β/2) It is with reference to the distance between ring array and ring array to be studied each the m and m' sensor, s=| m-m ' | respectively.By Formula (14) is known, square error is solely dependent upon expectation weight vector and the eigenvalue of associated cyclic matrix, and equal to M sub-error Sum.

A minimum is had with frequency variation curve at ka=3, meaning by formula (14) calculated least squares error Now synthesis wave beam to be completely superposed with expectation wave beam.On minimizing right side, error raises with frequency and becomes big, left side then phase Instead.If ensureing that the error of synthesis is less thanThen suitably frequency band range is ka ∈ [1,6].

(3) beamformer with constant beamwidth weight vector in the frequency band range determined is calculated.Weight vector m-th unit The expression formula of element is:

${\mathrm{\&omega;}}_m\left(ka\right)=\frac{{\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_m}{{\mathrm{\&lambda;}}_m}{\mathrm{\&omega;}}_m\left(k_r{a}_r\right)---\left(18\right)$

Weight vector is ω=[ω₀,ω₁,…,ω_M-1]^T, subscript T represents transposition.

(4) synthesize final wave beam, according to the multi-form of weight vector, be divided into following two situation:

A) direct formula for calculating is:

$B(ka,\mathrm{\&phi;})={\mathrm{\&omega;}}^{H}E(ka,\mathrm{\&phi;})=\underset{m=0}{\overset{M-1}{\&Sigma;}}{\mathrm{\&omega;}}_m^{*}{E}_m(ka,\mathrm{\&phi;})---\left(19\right)$

Wherein E=[E₀,E₁,…,E_M-1]^T, subscript H represents conjugate transpose；

B) Element space calculating formula is:

$B(ka,\mathrm{\&phi;})=w^{H}P(ka,\mathrm{\&phi;})=\underset{m=0}{\overset{M-1}{\&Sigma;}}{w}_m^{*}{P}_m(ka,\mathrm{\&phi;})---\left(20\right)$

Wherein w=V ω, V=[v₀,v₁,…,v_M-1], P (ka, φ)=[p₀(ka,φ),p₁(ka,φ),…,p_M-1(ka, φ)]^TFor array manifold vector, v_m=M^-1/2[1 e^imβ … e^i(M-1)mβ]^TFor circular matrix characteristic vector and meet relationKa is value in the frequency range of wave beam, and this frequency range is according to design accuracy, step 2 minimum obtained is all Side curve of error δ (ka) determines.

Noting, a) result with b) two kinds of method synthesis is consistent, and first method is relatively simple, and second method is more It is beneficial to practical operation.

With reference to Fig. 4 and Fig. 5 and Fig. 6.Consider 16 yuan of uniform rings battle arrays, formula (19) wave beam synthesized is with expectation wave beam all Coincideing very well, their beam angle is constant, especially main lobe region.It is emphasized that obtain here is constant Beam-width beam is minimum with the square error of expectation wave beam in all getable wave beams.It addition, the finger of constant beam-width wave beam Directional index is all at about 12dB, much larger than conventional method in given frequency range.This means that the present invention is had There is the constant beam-width wave beam of super directivity, significant for the target detection probability and resolving power improving system.

Claims (1)

1. a circular array design of beamformer with constant beamwidth method, it is characterised in that step is as follows:

Step 1: determine that expectation wave beam is the two dimensional beam in horizontal extent, circular array obtain in reference frequency Theoretical Calculation Arriving, its expression formula is:Wherein M is element number of array, ()^*Represent and seek conjugation, p_n(k_ra_r, φ) and it is the unit amplitude plane wave signal incident from direction φ that receive of the n-th array element, ω_m(k_ra_r) it is given Expectation weight vector element, and need to meetM=1,2 ..., M/2-1；

DescribedIn formula:a_rFor the radius of circle of reference annular array, k_r=2 π/λ_r, λ_r Represent the wavelength with reference to incident plane wave, φ_n=n β, β=2 π/M；

Described

Step 2: calculate expectation wave beam and synthesize the least mean-square error of wave beam: Wherein:And λ_mIt is respectively circular matrix ρ in reference frequency k_ra_rWith eigenvalue during frequency ka to be calculated,For Cyclic Moment Battle arrayEigenvalue, meet relationλ_m=λ_M-mWith

Describedρ_s(k_ra_r)=J₀(k_r·Δr_sr)；

Describedρ_s(ka)=J₀(k·Δr_s)；

Described

In formula: J₀() is the 0th rank cylindricality Bessel function, Δ r_sr=2a_rSin (s β/2) and Δ r_s=2a sin (s β/2) point It not with reference to the distance between ring array and ring array to be studied each the m and m' sensor, s=| m-m ' |；

Step 3: calculate beamformer with constant beamwidth weight vector in the frequency band range determined, the m-th of weight vector Element expression is:

${\mathrm{\&omega;}}_m\left(ka\right)=\frac{{\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_m}{{\mathrm{\&lambda;}}_m}{\mathrm{\&omega;}}_m\left(k_r{a}_r\right)$

Composition weight vector is ω=[ω₀,ω₁,…,ω_M-1]^T, subscript T represents transposition；

Step 4: according to the multi-form of weight vector, uses following two method to synthesize final wave beam:

1, directly calculate:

2, Element space calculates:

Wherein w=V ω, V=[v₀,v₁,…,v_M-1], P (ka, φ)=[p₀(ka,φ),p₁(ka,φ),…,p_M-1(ka,φ)]^T For array manifold vector, v_m=M^-1/2[1 e^imβ … e^i(M-1)mβ]^TFor circular matrix characteristic vector and meet relationKa is value in the frequency range of wave beam, and this frequency range is according to design accuracy, step 2 minimum obtained is all Side curve of error δ (ka) determines.