CN106161303B - response error envelope weighting least square spatial matrix filtering design method - Google Patents
response error envelope weighting least square spatial matrix filtering design method Download PDFInfo
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Abstract
The invention belongs to the technical field of array signal processing, relates to data processing of a sensor array, and particularly relates to a method for designing a least square spatial matrix filter by using response error envelope weighting. The method is characterized in that a constant passband response error and a constant stop band response are obtained by iterating weight parameters of a least square spatial matrix filter. And the response effects of the pass band and the stop band can be adjusted by setting the proportion coefficient of the pass band response error and the stop band response.
Description
Technical Field
The invention belongs to the technical field of array signal processing, relates to data processing of a sensor array, and particularly relates to a least square spatial matrix filter design method using extreme point envelope weighting.
Background
The patent is subsidized by the national science foundation project 'airspace matrix filtering technology and application research thereof in underwater acoustic signal processing', and has the project number of 11374001.
The spatial domain matrix filtering technology is used for realizing the purposes of reserving passband signals and restraining strong interference by preprocessing received array data. The technology has wide application in the field of array signal processing, in particular to the field of underwater acoustic signal processing. The spatial matrix filter can be used for array data preprocessing of target orientation estimation and can also be used for array data preprocessing of matching field positioning. The core of the spatial matrix filtering technology is the filter design. In the design method of the spatial matrix filter, the least square, zero point constraint and passband zero response error constraint methods can directly provide the solution of the optimal spatial matrix filter. The optimal spatial matrix filter can be directly provided by the stop band response, the pass band response error total constraint and the bilateral stop band total response constraint spatial matrix filter, but the optimal spatial matrix filter contains 1 to 2 unknowns and needs to solve a nonlinear equation. The constant stop band response constrains the spatial matrix filter and the constant pass band response error spatial matrix filter to obtain a flat pass band error or stop band response, but the method cannot directly obtain an optimal solution, needs to solve by means of a complex optimization theory and algorithm, is complex in calculation and is not beneficial to design of a real-time spatial matrix filter. How to obtain a filter with constant stopband response and passband response error in a simplified manner is crucial to real-time array data processing. By using a least square spatial matrix filter response weighting mode, the optimal solution of the filter can be directly given, and meanwhile, the response weighting coefficient is iterated to obtain the constant stop band response or the constant pass band response error.
the invention designs the spatial domain matrix filter by adopting a response weighting mode, and the spatial domain matrix filter design method can obtain constant passband response error and constant stopband response filtering effect by a successive iteration mode. The weighting coefficient adopts the envelope form of the passband response error and the stopband response, and the design efficiency is high. \ A
Disclosure of Invention
The invention aims to solve the technical problem of generating a spatial matrix filter with constant passband response error and constant stopband response error by an iteration method.
The technical scheme of the invention is as follows:
And the spatial matrix filter performs array element domain data processing before the array data is used for target azimuth estimation. And setting a spatial matrix filter designed for the frequency omega as H (omega), and performing data filtering processing by using a mathematical model of target azimuth estimation and matching field positioning information source incidence array. The array receives far-field plane waves, the received array data is a product of a direction vector and a source, and environmental noise n (t, omega) is superposed:
x(t,ω)＝A(τ,ω)s(t,ω)+n(t,ω)
Where a (τ, ω) is a delay vector, s (t, ω) is a source signal, n (t, ω) is ambient noise, and x (t, ω) is array receive data.
Using frequency omegaSpatial matrix filterFiltering the received array data, the filtered output y (t, ω) being:
y(t,ω)＝H(ω)x(t,ω)＝H(ω)A(τ,ω)s(t,ω)+H(ω)n(t,ω) (1)
the array manifold matrix is known as a (ω) { a (Φ, θ, ω) | Φ ∈ Φ, θ ∈ Θ }, where Φ and Θ correspond to the horizontal and vertical azimuth ranges, respectively. The effect of the spatial domain matrix filter on enhancing or suppressing the plane wave signal is realized by the action on the direction vectorWhen close to 0, the filter pair is illustrated (phi)i,θi) The plane wave signal with the direction frequency omega has stronger inhibiting effect. On the contrary, whenEquals 0, illustrating the filter pair (phi)i,θi) And the plane wave signal with the direction frequency of omega is free from distortion after being filtered.Is the square of the matrix norm. The spatial matrix filter is designed by designing for different directions (phi)i,θi) Realizes the pair (phi)i,θi) Undistorted response or suppression of directional data.
for a linear array sensor, the direction vector a (phi)i,θiω) is only related to the direction θ. At this time, the direction vector is a (θ)iω), given a detection band ω, a (θ)iω) can be abbreviated as a (θ)i). H (ω) is abbreviated as H.
the weighted spatial matrix filter design method will be described below with respect to a linear array.
Assuming that the number of spatial discretizations is M, the direction vector of each azimuth is a (theta)m) M is 1, …, M, and the expected response vector is b (θ)m). To filter the matrixThe filter retains the signal of the pass band and filters the noise of the stop band, so that an ideal matrix filter should satisfy the following conditions:
Wherein Θ isP,ΘSRepresenting the set of passband and stopband spatial azimuth angles, respectively.
The error between the actual response and the expected response of the spatial matrix filter to the array signal is given by the following equation.
Using the error of the actual response and the expected response, a weighted optimization problem is constructed as follows.
Optimization problem 1:
Wherein, w (θ)m) Is the response weighting factor for each direction vector.
from the theory of optimization, when w (theta)m) When the value is smaller, the value is selected,the contribution to J (H) is small, and the influence on J (H) is large. With w (theta)m) The increase of the value of the additive is increased,the value of (A) increases, resulting in an increase in J (H). In order to obtain the optimal spatial matrix filter, it is necessary to obtain a balance between all response errors, i.e., a large w (θ)m) It is inevitable to obtain a matrix filter at θmA small response error value. Therefore, the optimal value of the objective function can be adjusted by adjusting the coefficient, so that the response effect of the spatial matrix filter can be adjusted.
The optimal solution for optimization problem 1 is:
In the formula:
A＝[a(θ1),…,a(θM)]
B＝[b(θ1),…,b(θM)]
R＝diag[w(θ1),w(θ2),…,w(θM)]M×M
Through iteration of the weighting coefficient matrix R, the effect that the passband response error and the stopband response are constant can be achieved.
Suppose that the kth filter matrix H is obtained over k-1 iterationskThen can pass through HkThe absolute value | E of the error between the desired response and the actual response of the filter at this time is obtainedk(θm)|＝|Hka(θm)-b(θm) 1, …, M, solving all local maximum values for the absolute value of the error, connecting the local maximum values by straight line segments, and using the corresponding values on the straight line segments as the iterative response azimuth thetamweight coefficient w ofk(θm). Here, assume that there are Q local maximum values in total, and the abscissa iscorresponding extreme value, i.e. ordinate, is
It should be noted here that, for the end point azimuth of the azimuth estimation, since the local maximum value does not generally appear at both ends, the weighting values to be used between the left end point of the detection azimuth and the 1 st local maximum value, and between the right end point of the detection azimuth and the last local maximum value need to be set specifically. Using the 1 st local maximum pointand 2 nd local maximum positionThe reverse extension line of the connecting line obtains the value (theta) of the straight line at the position of the left end point1,z1) Order (theta)1,max(|Ek(θ1)|,z1) Is a weighted starting point of the left end point, and is compared withAre connected to obtainThe weight coefficient of the interval. Similarly, the 1 st local maximum point is usedAnd 2 nd local maximum pointthe extension line of the connecting line between the two points obtains the value (theta) of the right end point on the straight lineM,zM) Order (theta)M,max(zM,|Ek(θM) |))) is a weighted starting point of the right end point and is compared withConnecting, taking the value on the corresponding connecting line asThe weighting values of (a).
The iteration of the weighting coefficient matrix R involves the weighting vector w (theta)m) As can be seen from the foregoing discussion, the weighting vector is related to the values of the filter for both the pass band response error and the stop band response.
Set up betak(θm) Is a weighted product coefficient in the k-th iteration process, andHere, αk(θm) Is thetamOn the corresponding response error envelope line segmentthe value of (a). Gamma (theta)m) The response proportion of the left and right stopband and passband response errors is set by the following method:
Wherein, thetaS1And ΘS2Is a set of left and right stopband spatial incidence azimuths. a, b and c are response ratios of a preset pass band, a preset left stop band and a preset right stop band.
By applying a response proportionality coefficient gamma (theta)m) Into the weighting factor w (theta)m) M1, …, M, after the algorithm is terminated, the difference between the left and right stopband responses and the passband response errorAndRespectively as follows:
The above equation is a difference in response given in dB. When a, b, and c are selected, the pass band response error of the spatial matrix filter is the same as the left and right stop band response values.
drawings
Fig. 1a shows a pass-band response error weighted spatial matrix filter (1 iteration, with a-b-c).
Fig. 1b shows a pass-band response error weighted spatial matrix filter (1 iteration, with a-b-c).
Figure 2a shows the filter left stopband response envelope weighting resulting from iteration 1.
Figure 2b shows the resulting filter right stop band response envelope weighting for iteration 1.
figure 2c shows the resulting filter passband response error envelope weighting for iteration 1.
Fig. 3a shows a passband response weighted spatial matrix filter (a ═ b ═ c ═ 1, iterated 7 times).
Fig. 3b shows a pass-band response error weighted spatial matrix filter (a ═ b ═ c ═ 1, iterated 7 times).
figure 4a shows the filter left stopband response envelope weighting resulting from the 7 th iteration.
Figure 4b shows the resulting filter right stop band response envelope weighting for the 7 th iteration.
figure 4c shows the resulting filter passband response error envelope weighting for the 7 th iteration.
Fig. 5a shows a pass-band response error weighted spatial matrix filter (a ═ b/2 ═ 4c ═ 1, iteration 7 times)
Fig. 5b shows a pass-band response error weighted spatial matrix filter (a ═ b/2 ═ 4c ═ 1, iteration 7 times)
in the figure, the number N of array elements corresponding to the designed filter is 30, the array elements are equally spaced, the passband is between-15 degrees and 15 degrees, the stopband is between-90 degrees and-20 degrees, U (20 degrees and 90 degrees), the discretization sampling interval between the passband and the stopband is 0.1 degrees, and the spatial matrix filter is designed aiming at half-wavelength frequency of the array.
Fig. 1a and 1b show the design effect of the least-squares spatial matrix filter obtained by weighting the response errors in the case where a is 1, b is c, and where the weighting coefficient matrix R is iterated only 1 time. FIG. 1a shows the filter responseFIG. 1b shows filter response errorThe design effect of the least square matrix filter is given in the figure, and the weighted least square matrix filter is obtained by taking the least square matrix filter as a starting point and utilizing envelope weighted iteration of local maximum values of response errors.
Fig. 2a, 2b and 2c show the pass band response error and left and right stop band responses of the least square matrix filter, and the weighting coefficient matrix R is constructed by using the connecting line of the pass band response error and the local maximum point of the stop band response.
fig. 3a and 3b show the matrix filter effect obtained after 7 iterations as a result of further iterations of fig. 3a and 3 b.
Fig. 4a, 4b and 4c are respective envelope weighting line segments used in the last iteration.
Fig. 5a and 5b show the matrix filter effect obtained after 7 iterations, where a ═ b/2 ═ 4c ═ 1. As can be seen from the effect in the figure, the left stopband response is 3dB higher than the passband response error, and the right stopband response is 6dB lower than the passband response error.
Detailed Description
The following detailed description of specific embodiments of the invention refers to the accompanying drawings.
The iterative algorithm based on the response error envelope weighting criterion is as follows:
Step 1: let k be 0, discretize the detection space and calculate a, B. Let w0(θs) Calculating an initial optimal spatial matrix filter (1)Setting response proportionality coefficients gamma (theta) of left stop band, pass band and right stop bandm)，
Step 2: calculation of Ek(θm)＝Hka(θm)-b(θm) M is 1 and M. Obtaining | Ek(θm) A local maximum point of | acquiring the abscissa of the local maximumand its corresponding ordinateFor example, in fig. 2a, 2b, and 2c, which are the passband response error and the left and right stopband responses, the line segment is the connection line between the corresponding local maxima.
And step 3: by usingAndAn extension line of a line connecting two points is calculated on the abscissa theta1Value z of1Set up (theta)1,max(|Ek(θ1)|,z1) Is an envelope weighting starting point.
And 4, step 4: by usingAndAn extension line of a line connecting two points is calculated on the abscissa thetaMValue z ofMSet up (theta)M,max(zM,|Ek(θM) |))) is the envelope weighting endpoint.
And 5: calculating (theta)1,max(|Ek(θ1)|,z1))、 (θM,max(zM,|Ek(θM) |)) Q +2 points, and take αk(θm) Is thetamThe values on the corresponding line segments.
Step 6: calculating the following formulas
wk+1(θm)＝βk(θm)γ(θm)wk(θm)
Rk+1＝diag[wk+1(θ1),wk+1(θ2),…,wk+1(θM)]
Hk+1＝BRk+1AH(ARk+1AH)-1
Wherein, betak(θm) Is the product factor of the kth iteration, wk+1(θm) Weighting factors, R, used for the kth iterationk+1Weighting factor matrix for the (k + 1) th iteration, Hk+1The spatial matrix filter obtained from the (k + 1) th iteration.
Judgment of Hk+1Whether one of the following termination conditions is satisfied:
(a) K +1 ═ K. At this time, the iteration is performed for K times, and the algorithm is terminated;
(b)After iteration, the difference value between the actual response and the expected response of the spatial matrix filter to all the azimuths is smaller than a constantThe algorithm is terminated;
(c)After iteration, the response error change rate of the spatial matrix filter to all the azimuths is smaller than a constant valueThe algorithm terminates.
And 7: if the iteration termination condition is satisfied, Hk+1namely the final spatial matrix filter. Otherwise, let k be k +1, repeat steps 2-6.
Claims (2)
1. A response error envelope weighting least square spatial matrix filter design method is characterized in that a weighting matrix is constructed by the pass band response error envelope of an optimal spatial matrix filter, and constant pass band response error and stop band response are obtained by using a parameter iteration mode; the method is characterized by comprising the following steps:
Step 1: calculating a ═ a (θ)1),…,a(θM)]、B＝[b(θ1),…,b(θM)]Setting left and right stop bands and channelResponse scale factor gamma (theta) with response errorm) Let k be 0; wherein, a (theta)m) For each azimuthal direction vector, b (θ)m) To expect a response vector, θmIn order to respond to the azimuth, M is the discretization number of the airspace, and k is the iteration number;
Step 2: calculation of Ek(θm)＝Hka(θm)-b(θm) M is 1, …, M; obtaining | Ek(θm) A local maximum point of | acquiring the abscissa of the local maximumand its corresponding ordinatewherein E isk(θm) To pass through HkThe error between the expected response and the actual response of the filter at this time, H, is obtainedkObtaining the kth filter matrix after k-1 iterations, wherein Q is the number of local maximums;
And step 3: by usingAndAn extension line of a line connecting two points is calculated on the abscissa theta1value z of1set up (theta)1,max(|Ek(θ1)|,z1) ) is an envelope weighting starting point;
And 4, step 4: by usingAndAn extension line of a line connecting two points is calculated on the abscissa thetaMValue z ofMSet up (theta)M,max(zM,|Ek(θM) |)) is an envelope weighting endpoint;
And 5: calculating (theta)1,max(|Ek(θ1)|,z1))、 (θM,max(zM,|Ek(θM) |)) Q +2 points, and take αk(θm) Is thetamValues on the corresponding line segments; order toIs the weighted product coefficient in the k iteration process;
Step 6: calculating the following formulas
wk+1(θm)＝βk(θm)γ(θm)wk(θm)
Rk+1＝diag[wk+1(θ1),wk+1(θ2),…,wk+1(θM)]
Hk+1＝BRk+1AH(ARk+1AH)-1
Wherein: w (theta)m) Is the response weighting factor for each direction vector; r is a weighting coefficient matrix; w is a0(θm)＝1；
judgment of Hk+1Whether one of the following termination conditions is satisfied:
(a) K +1 ═ K; at this time, iterating for K times, and terminating the algorithm; wherein K is 1 to K;
(b)After iteration, the difference value between the actual response and the expected response of the spatial matrix filter to all the azimuths is smaller than a constantThe algorithm is terminated;
(c)After iteration, the response error change rate of the spatial matrix filter to all the azimuths is smaller than a constant valuethe algorithm is terminated;
And 7: if the iteration termination condition is satisfied, Hk+1The final spatial domain matrix filter is obtained; otherwise, let k be k +1, repeat steps 2-6.
2. The design method of a response error envelope weighted least squares spatial matrix filter as claimed in claim 1 further characterized by adjusting the pass band response error, left and right stop band response scaling factor γ (θ)m) The adjustment of the difference value between the passband response error and the left and right stopband responses is realized;
By setting:
Wherein, thetaPRepresenting a set of passband spatial incident azimuths, ΘS1And ΘS2Is a set of left and right stopband spatial incidence azimuths; a, b and c are response ratios of a preset pass band, a preset left stop band and a preset right stop band;
The difference between the left and right stopband responses and the passband response error can be obtained as:
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