CN106100608B - Weighted least-squares airspace matrix filter design method - Google Patents
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Abstract
The invention belongs to array signal processing technologies, are related to a kind of weighted least-squares airspace matrix filter design method, are related specifically to the airspace matrix filter design method of constant passband response error and stopband response.It is characterized in that obtaining constant passband response error and constant stopband response by way of to optimal airspace matrix filter weighting parameter iteration.And the response effect of passband and stopband can be adjusted by the proportionality coefficient λ of setting passband response error and stopband response.
Description
Technical field
The invention belongs to array signal processing technologies, are related to the data processing of sensor array, are related specifically to
Constant passband response error and the airspace matrix filter design method of stopband response.
Background technique
The application is project of national nature science fund project " airspace matrix filtering technique and its answering in Underwater acoustic signal processing
With research ", in the design method, least square, zeros constrained and zero response error constrained procedure of passband can be directly given
The solution of optimal airspace matrix filter.Stopband response, passband response error global restriction, bilateral stopband global response constrain airspace
Matrix filter only demand solution include 1 or 2 unknown number Solving Nonlinear Equation optimal L agrange multiplier, can be obtained most
The solution of excellent airspace matrix filter.And constant stopband response constrains airspace matrix filter since stopband to be limited responds or passband
The maximum value of response error is both less than certain particular constraints value, and the optimization problem established cannot directly give optimal solution, need
It is solved by complicated Optimum Theory and algorithm, calculates complexity, be unfavorable for the design of real-time empty domain matrix filter, particularly with
The case where broadband array signal is handled, and to design corresponding airspace matrix filter to multiple subbands is even more so.
The present invention designs airspace matrix filter, this airspace matrix filter design method by the way of response weighting
Constant passband response error and constant stopband response filter effect, design efficiency can be obtained by way of successive iteration
It is high.
Summary of the invention
It is constant the technical problem to be solved by the present invention is to generate passband response error by alternative manner, and stopband response misses
The constant airspace matrix filter of difference.
The technical scheme is that
Airspace matrix filter does Element space data processing before array data is for target Bearing Estimation.If for frequency
The airspace matrix filter of rate ω design is H (ω), is incident on array using target Bearing Estimation and Matched Field positioning information source
Mathematical model does data filtering processing.Array received far field plane wave, receiving array data are direction vector and information source product,
And it is superimposed ambient noise n (t, ω):
X (t, ω)=A (τ, ω) s (t, ω)+n (t, ω)
Wherein, A (τ, ω) is delay vector, and s (t, ω) is source signal, and n (t, ω) is ambient noise, and x (t, ω) is battle array
Column receive data.
It is filtered using the airspace matrix filter H (ω) that frequency is ω to array data is received, filtered output y (t,
ω) are as follows:
Y (t, ω)=H (ω) x (t, ω)=H (ω) A (τ, ω) s (t, ω)+H (ω) n (t, ω) (1)
Known array manifold matrix is A (τ, ω)={ a (φ, θ, ω) | φ ∈ Φ, θ ∈ Θ }, and Φ and Θ are right respectively here
It should be in horizontal and vertical azimuth coverage.The effect that airspace matrix filter generates enhancing to plane wave signal or inhibits is to pass through
The effect of direction vector is realized, whenWhen close to 0, illustrate filter to (φi,θi) direction frequency
Rate is that the plane wave signal of ω has stronger inhibiting effect.Conversely, working asEqual to 0, say
Bright filter is to (φi,θi) direction frequency be ω plane wave signal filtering after it is undistorted.For matrix norm square.Airspace
The design of matrix filter is by designing to different directions (φi,θi) response, realize to (φi,θi) bearing data
Undistorted response or inhibition.
For alignment array sensor, then direction vector a (φi,θi, ω) it is only related with direction θ.At this point, direction vector is a
(θi, ω), given frequency band of detection ω, a (θi, ω) and a (θ can be abbreviated asi).H (ω) is abbreviated as H.
Illustrate weighting type airspace matrix filter design method below for linear array.
Assuming that array elements number is N, passband airspace discretization number is P, and stopband airspace discretization number is S, θpDirection
Corresponding passband direction vector a (θp) response error e (θp) and θsThe corresponding stopband direction vector a (θ in directions) response e
(θs) be respectively as follows:
Wherein, ΘPAnd ΘSThe respectively value region of passband direction vector and stopband direction vector.
Utilize weighting coefficient w (θp) and w (θs) to passband response error e (θp) and stopband response e (θs) weighting, then it weights
After obtain weighting passband response error ew(θp) and weighting stopband response ew(θs) it is as follows:
The stopband global response (weighting) after passband global response error (weighting) and normalization after normalization is respectively as follows:
Wherein, P and S corresponds to the discretization number of pass band areas and stop band region, for that can adopt convenient for calculating and handling
With equally spaced mode discretization.
Using the passband global response error (weighting) and stopband global response (weighting) after normalization, airspace matrix is constructed
Filter design optimization problem, and it is as follows to stopband global response weighting coefficient λ
Optimization problem 1:
By the theory that optimizes it is found that as w (θp) when taking smaller value,To the contribution of J (H (ω)) compared with
It is small, conversely, being then affected to J (H (ω)).Similarly, w (θs) size also determineTribute to J (H (ω))
Offer size.
With w (θp) and w (θs) value increase,WithValue therewith
Increase, and optimization problem 1 needs to make J (H (ω)) minimum, then big w (θp) and w (θs) must obtain it is lesserWithTherefore, adjustment factor w (θ can be passed throughp) and w (θs) can be realized to objective function
Optimal value adjust, to adjust the response effect of airspace matrix filter.
It constructs Lagrange function and solves optimization problem 1:
Matrix is constructed in above formula:
VP=[a (θ1),…,a(θp),…,a(θP)],θp∈ΘP
VS=[a (θ1),…,a(θs),…,a(θS)],θs∈ΘS
In above formula, VPAnd VSCorrespond respectively to the matrix that passband direction vector and stopband direction vector are constructed.RP,1/2With
RS,1/2Correspond respectively to the diagonal matrix that passband response error weighting coefficient and stopband response weighting coefficient are constructed.
And it enables:
RP=diag [w (θ1),w(θ2),…,w(θP)]P×P,θp∈ΘP
RS=diag [w (θ1),w(θ2),…,w(θS)]S×S,θs∈ΘS
J (H (ω)) is asked about matrix H*The partial derivative of (ω), and order is zero, to obtain the solution of optimal filter.
It obtains
By adjusting λ, the ratio of adjustable passband response error and stopband response.It is logical after normalization as λ=1
It is identical with the passband response after normalization with response error.As λ > 1, the stopband response after normalization will reduce, and normalize
Passband response error afterwards will increase.Conversely, λ < 1, normalization stopband response obtained is higher than normalization passband response and misses
Difference.
Pass through the weighting coefficient w (θ to passband response errorp) and stopband response weighting coefficient w (θs) iteration, Ke Yishi
Existing passband response error and stopband response are less than or equal to the effect of certain steady state value.Independent iteration passband response error can be passed through
(or stopband response) weighting coefficient w (θp) (or w (θs)), obtain the constant binding effect of passband (stopband).It can also be with passband response
Iteration, realization passband response error and stopband respond while meeting constant together for error weighting coefficient and stopband response weighting coefficient
Constraint.
1, constant passband response error alternative manner
By alternative manner, passband response error is made to be less than or equal to constant binding occurrence.
Enable RS=I, making stopband response weighted value is all 1, namely iteration stopband does not respond weighted value.
Initial value:
w1(θp)=1, p=1 ..., P (8)
Kth time iteration:
RP,k=diag [wk(θ1,ω),wk(θ2,ω),…,wk(θP,ω)] (9)
Ek(θp, ω) and=Hk(ω)a(θp,ω)-a(θp, ω), p=1 ..., P (11)
wk+1(θp)=[βk(θp,ω)+ο]wk(θp) (13)
Wherein, Ek(θp, ω) and it is kth time iteration passband direction vector θpThe response error at place.βk(θp, ω) and it is that kth time changes
The Product-factor in generation.wk(θp) it is passband response error weighting coefficient used in kth time iteration.RP,kFor the passband of kth time iteration
Response error weighting coefficient matrix.Hk(ω) is the resulting airspace matrix filter of kth time iteration.ο is close to 0 in formula (13)
Constant value, it is therefore an objective to avoid wk(θsWhen)=0, wk+1(θs)=0.
Stopping criterion for iteration:
(a) k=K, algorithm terminate.Wherein, K is preset integer value；
(b)Algorithm terminates.Wherein,For the termination threshold value of setting；
(c)Algorithm terminates.Wherein,For the termination threshold of setting
Value.
Above-mentioned termination condition can optional one.
2, constant stopband responds alternative manner
By alternative manner, stopband response is made to be less than or equal to constant binding occurrence.
Enable RP=I, making passband response error weighted value is all 1, namely not iteration passband response error weighted value.
Initial value:
w1(θs)=1, s=1 ..., S (14)
Kth time iteration:
RS,k=diag [wk(θ1,ω),wk(θ2,ω),…,wk(θS,ω)] (15)
Ek(θs, ω) and=Hk(ω)a(θs, ω), s=1 ..., S (17)
wk+1(θs)=[βk(θs,ω)+ο]wk(θs) (19)
Wherein, Ek(θs, ω) and it is stopband θ in kth time iterationsLocate the corresponding filter response of direction vector, βk(θs, ω) be
The weighted product factor used in kth time iteration.RS,kStopband for kth time iteration responds weighting coefficient matrix.Hk(ω) is the
The resulting airspace matrix filter of k iteration.ο is the constant value close to 0 in formula (19), it is therefore an objective to avoid wk(θsWhen)=0,
wk+1(θs)=0.
Stopping criterion for iteration:
(a) k=K, algorithm terminate.Wherein, K is preset integer value；
(b)Algorithm terminates.Wherein,For the termination threshold value of setting；
(c)Algorithm terminates.Wherein,For the termination threshold of setting
Value.
Above-mentioned termination condition can optional one.
3, constant passband response error and constant stopband respond alternative manner
By alternative manner, passband response error is made to be less than or equal to steady state value, while it is constant to be less than or equal to stopband response
Value, namely constant passband and stopband response effect are generated simultaneously.
Initial value:
Kth time iteration:
Here, Hk(ω) is the resulting airspace matrix filter of kth time iteration, RP,kAnd RS,kRespectively kth time iteration institute
The passband and stopband weighted iteration coefficient diagonal matrix obtained.Ek(θp, ω) and Ek(θs, ω) and it is respectively passband response error and resistance
Band response.βk(θp, ω) and βk(θs, ω) and correspond respectively to the kth time passband of iteration and the Product-factor of stopband vector.wk(θp)
And wk(θs) correspond respectively to the passband and stopband weighting coefficient of kth time iteration.
ο is the constant value close to 0 in formula (25).
Stopping criterion for iteration:
(a) k=K, algorithm terminate.Wherein, K is preset integer value；
(b)Algorithm terminates.Wherein,For setting
Terminate threshold value；
(c)It calculates
Method terminates.Wherein,For the termination threshold value of setting.
Above-mentioned termination condition can optional one.
The present invention obtains constant passband response and misses by way of to optimal airspace matrix filter weighting parameter iteration
Poor and constant stopband response.And passband and resistance can be adjusted by the proportionality coefficient λ of setting passband response error and stopband response
The response effect of band.
Detailed description of the invention
Fig. 1 a indicates the weighting airspace matrix filter response of the passband response error of λ=1
Fig. 1 b indicates that the passband response error of λ=1 weights airspace matrix filter response error
Fig. 2 a indicates the weighting airspace matrix filter response of the stopband response error of λ=1
Fig. 2 b indicates that the stopband response error of λ=1 weights airspace matrix filter response error
Fig. 3 a indicates the passband response of λ=1 error and stopband response while weighting the response of airspace matrix filter
Fig. 3 b indicates the passband response of λ=1 error and stopband response while weighting airspace matrix filter response error
Fig. 4 a indicates the passband response of λ=8 error and stopband response while weighting the response of airspace matrix filter
Fig. 4 b indicates the passband response of λ=8 error and stopband response while weighting airspace matrix filter response error
Fig. 1 a, Fig. 1 b, in Fig. 2 a and Fig. 2 b, the corresponding array element number N=30 of designed filter, array element is equidistant,
Passband is [- 15 °, 15 °], and stopband is [- 90 °, -20 °) ∪ (20 °, 90 °], 0.1 ° of passband and stopband discrete sampling interval, needle
It is poised for battle half-wavelength Frequency Design airspace matrix filter.
It is given in Fig. 1 a and Fig. 1 b and passband response error is weighted, least square airspace obtained matrix filter
Design effect uses the iteration of formula (8)~formula (13), using stop criterion, and λ=1.The effect from figure is it is found that filter
Passband response error after 20 weighted iterations, tend to equal constant constant value.
Fig. 2 a and Fig. 2 b give to respond stopband and weight, resulting weighted least-squares airspace matrix filter design effect
Fruit.Using formula (14)~formula (19) iteration, stop criterion, and λ=1 are used.From the simulated effect in figure it is found that left and right
Stopband response all has constant constraint, selects the number of iterations for 20 times herein.
Fig. 3 a and Fig. 3 b give to passband response error and stopband response while weighting, and resulting weighted least-squares are empty
Domain matrix filter design effect.Using formula (20)~formula (25) iteration, stop criterion, and λ=1 are used.From figure
Simulated effect it is found that passband response error and left and right stopband response all have constant constraint, select herein the number of iterations for
20 times.
Fig. 4 a and Fig. 4 b give to passband response error and stopband response while weighting, and resulting weighted least-squares are empty
Domain matrix filter design effect.Using formula (20)~formula (25) iteration, stop criterion is used.It is lower in order to obtain
Stopband response, chooses λ=8, selects the number of iterations for 20 times.From the simulated effect in figure it is found that passband response error and left and right
Stopband response all has constant constraint, and it will be evident that stopband response has apparent reduction, while passband response error increases
Greatly.
Specific embodiment
Describe specific embodiments of the present invention in detail below in conjunction with scheme and attached drawing.
1, constant passband response error iteration
Step 1: the passband and stopband of setting airspace matrix filter, and to passband and stopband discretization, and setting direction
The initial weighting coefficients w of vector1(θp)=1, p=1 ..., P；
Step 2: being directed to look-in frequency ω, calculates a (θp), p=1 ..., P and a (θs), s=1 ..., S, and obtain VP=
[a(θ1),…,a(θP)],θp∈ΘPAnd VS=[a (θ1),…,a(θS)],θs∈ΘS；
Step 3: the ratio lambda of setting passband response error and stopband response.For example, λ=1 may be selected.
Step 4: selection Stopping criteria (a), (b) or (c)；Repeatedly using formula (9), (10), (11), (12), (13)
In generation, then stops iteration when iteration meets stop criterion condition.Assuming that amounting at this time iteration K times；
Step 5: it calculatesHK(ω) is most
Whole airspace matrix filter can be responded and response error observation filter device effect by filter.
2, constant stopband responds iteration
Step 1: the passband and stopband of setting airspace matrix filter, and to passband and stopband discretization, and setting direction
The initial weighting coefficients w of vector1(θs)=1, s=1 ..., S；
Step 2: being directed to look-in frequency ω, calculates a (θp), p=1 ..., P and a (θs), s=1 ..., S, and obtain VP=
[a(θ1),…,a(θP)],θp∈ΘPAnd VS=[a (θ1),…,a(θS)],θs∈ΘS；
Step 3: the ratio lambda of setting passband response error and stopband response.For example, λ=1 may be selected.
Step 4: selection Stopping criteria (a), (b) or (c)；Repeatedly using formula (14), (15), (16), (17), (18)
In generation, then stops iteration when iteration meets stop criterion condition.Assuming that amounting at this time iteration K times；
Step 5: it calculatesHK(ω) is final
Airspace matrix filter can be responded and response error observation filter device effect by filter.
3, constant passband response error and constant stopband respond iteration
Step 1: the passband and stopband of setting airspace matrix filter, and to passband and stopband discretization, and setting direction
The initial weighting coefficients w of vector1(θp)=1, p=1 ..., P, w1(θs)=1, s=1 ..., S；
Step 2: being directed to look-in frequency ω, calculates a (θp), p=1 ..., P and a (θs), s=1 ..., S, and obtain VP=
[a(θ1),…,a(θP)],θp∈ΘPAnd VS=[a (θ1),…,a(θS)],θs∈ΘS；
Step 3: the ratio lambda of setting passband response error and stopband response.For example, λ=1 may be selected.
Step 4: selection Stopping criteria (a), (b) or (c)；Repeatedly using formula (21), (22), (23), (24), (25)
In generation, then stops iteration when iteration meets stop criterion condition.Assuming that amounting at this time iteration K times；
Step 5: it calculatesHK(ω) i.e.
For final airspace matrix filter, can be responded and response error observation filter device effect by filter.
Claims (4)
1. a kind of weighted least-squares airspace matrix filter design method, which is characterized in that
If array elements number is N, passband airspace discretization number is P, and stopband airspace discretization number is S, θpDirection is corresponding logical
Band direction vector a (θp) response error e (θp) and θsThe corresponding stopband direction vector a (θ in directions) response e (θs) respectively
Are as follows:
Wherein, ΘPAnd ΘSThe respectively value region of passband direction vector and stopband direction vector；
Utilize weighting coefficient w (θp) and w (θs) to passband response error e (θp) and stopband response e (θs) weighting, then it is obtained after weighting
Weight passband response error ew(θp) and weighting stopband response ew(θs) it is as follows:
The stopband weighting global response after weighting passband global response error and normalization after normalization is respectively as follows:
Wherein, P and S corresponds to the discretization number of pass band areas and stop band region, using equally spaced mode discretization；
Using the weighting passband global response error and weighting stopband global response after normalization, construction airspace matrix filter is set
Count optimization problem:
Optimization problem 1:
As w (θp) when taking smaller value,It is smaller to the contribution of J (H (ω)), conversely, then to J's (H (ω))
It is affected；Similarly, w (θs) size also determineTo the contribution of J (H (ω))；
With w (θp) and w (θs) value increase,WithValue increase therewith,
And optimization problem 1 needs to make J (H (ω)) minimum, then big w (θp) and w (θs) must obtain it is lesserWithPass through adjustment factor w (θp) and w (θs) realize to the optimal value tune of objective function
Section, to adjust the response effect of airspace matrix filter；
It constructs Lagrange function and solves optimization problem 1:
Matrix is constructed in above formula:
VP=[a (θ1),…,a(θp),…,a(θP)],θp∈ΘP
VS=[a (θ1),…,a(θs),…,a(θS)],θs∈ΘS
In above formula, VPAnd VSCorrespond respectively to the matrix that passband direction vector and stopband direction vector are constructed；RP,1/2And RS,1/2
Correspond respectively to the diagonal matrix that passband response error weighting coefficient and stopband response weighting coefficient are constructed；
And enable: RP=diag [w (θ1),w(θ2),…,w(θP)]P×P,θp∈ΘP
RS=diag [w (θ1),w(θ2),…,w(θS)]S×S,θs∈ΘS
Partial derivative about matrix H * (ω) is asked to J (H (ω)), and order is zero, to obtain the solution of optimal filter；
It obtains
By adjusting λ, the ratio of passband response error and stopband response is adjusted；As λ=1, the passband response after normalization is missed
Difference is identical with the passband response after normalization；As λ > 1, the stopband response after normalization will reduce, and the passband after normalizing
Response error will increase；Conversely, λ < 1, normalization stopband response obtained is higher than normalization passband response error；
Respond weighting coefficient by independent iteration passband response error or stopband, obtain passband or stopband constant binding effect or
Iteration, realization passband response error and stopband respond simultaneously together for passband response error weighting coefficient and stopband response weighting coefficient
Meet constant constraint；
2. matrix filter design method in weighted least-squares airspace according to claim 1, which is characterized in that the perseverance
Determine passband response error alternative manner to comprise the concrete steps that:
Enable RS=I, making stopband response weighted value is all 1, namely iteration stopband does not respond weighted value；
Initial value:
w1(θp)=1, p=1 ..., P (8)
Kth time iteration:
RP,k=diag [wk(θ1,ω),wk(θ2,ω),…,wk(θP,ω)] (9)
Ek(θp, ω) and=Hk(ω)a(θp,ω)-a(θp, ω), p=1 ..., P (11)
wk+1(θp)=[βk(θp,ω)+ο]wk(θp) (13)
Wherein, Ek(θp, ω) and it is kth time iteration passband direction vector θpThe response error at place；βk(θp, ω) and it is kth time iteration
Product-factor；wk(θp) it is passband response error weighting coefficient used in kth time iteration；RP,kFor the passband response of kth time iteration
Error weighting coefficient matrix；Hk(ω) is the resulting airspace matrix filter of kth time iteration；In formula (13) ο be close to 0 it is normal
Numerical value, it is therefore an objective to avoid wk(θsWhen)=0, wk+1(θs)=0；
Stopping criterion for iteration:
(a) k=K, algorithm terminate；Wherein, K is preset integer value；
(b)Algorithm terminates；Wherein,For the termination threshold value of setting；
(c)Algorithm terminates；Wherein,For the termination threshold value of setting.
3. matrix filter design method in weighted least-squares airspace according to claim 1 or 2, which is characterized in that described
Constant stopband response alternative manner step be:
By alternative manner, stopband response is made to be less than or equal to constant binding occurrence；
Enable RP=I, making passband response error weighted value is all 1, namely not iteration passband response error weighted value；Initial value:
w1(θs)=1, s=1 ..., S (14)
Kth time iteration:
RS,k=diag [wk(θ1,ω),wk(θ2,ω),…,wk(θS,ω)] (15)
Ek(θs, ω) and=Hk(ω)a(θs, ω), s=1 ..., S (17)
wk+1(θs)=[βk(θs,ω)+ο]wk(θs) (19)
Wherein, Ek(θs, ω) and it is stopband θ in kth time iterationsLocate the corresponding filter response of direction vector, βk(θs, ω) and it is kth
The weighted product factor used in secondary iteration；RS,kStopband for kth time iteration responds weighting coefficient matrix；Hk(ω) is kth time
The resulting airspace matrix filter of iteration；ο is the constant value close to 0 in formula (19), it is therefore an objective to avoid wk(θsWhen)=0, wk+1
(θs)=0；
Stopping criterion for iteration:
(a) k=K, algorithm terminate；Wherein, K is preset integer value；
(b)Algorithm terminates；Wherein,For the termination threshold value of setting；
(c)Algorithm terminates；Wherein,For the termination threshold value of setting.
4. matrix filter design method in weighted least-squares airspace according to claim 1 or 2, which is characterized in that it is special
Sign is obtained constant by way of responding weighting parameter iteration to optimal airspace matrix filter passband response error and stopband
Passband response error and the response of constant stopband；It comprises the concrete steps that:
By alternative manner, passband response error is made to be less than or equal to steady state value, while stopband response being made to be less than or equal to steady state value,
Constant passband and stopband response effect are generated simultaneously；
Initial value:
Kth time iteration:
Here, Hk(ω) is the resulting airspace matrix filter of kth time iteration, RP,kAnd RS,kRespectively kth time iteration is resulting logical
Band and stopband weighted iteration coefficient diagonal matrix；Ek(θp, ω) and Ek(θs, ω) and it is respectively passband response error and stopband response；
βk(θp, ω) and βk(θs, ω) and correspond respectively to the kth time passband of iteration and the Product-factor of stopband vector；wk(θp) and wk(θs)
Correspond respectively to the passband and stopband weighting coefficient of kth time iteration；
ο is the constant value close to 0 in formula (25)；
Stopping criterion for iteration:
(a) k=K, algorithm terminate；Wherein, K is preset integer value；
(b)Algorithm terminates；Wherein,For the termination of setting
Threshold value；
(c)Algorithm is whole
Only；Wherein,For the termination threshold value of setting.
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| CN110211601B (en) * | 2019-05-21 | 2020-05-08 | 出门问问信息科技有限公司 | Method, device and system for acquiring parameter matrix of spatial filter |
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| CN103888104A (en) * | 2014-02-25 | 2014-06-25 | 广东省电信规划设计院有限公司 | Method and system for designing FIR digital filter |
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| CN103414504A (en) * | 2013-07-09 | 2013-11-27 | 西南交通大学 | Adaptive beamforming method based on convex combination |
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