CN106026972B - Passband response error weights the response constraint airspace matrix filter design method of stopband zero - Google Patents

Abstract

The invention belongs to array signal processing technologies, are related to the data processing of sensor array, are related specifically to airspace matrix filter design method.It is characterized in that obtaining constant passband response error, while stopband response is zero by way of to optimal airspace matrix filter passband response error weighting parameter iteration.Passband response error weighting coefficient matrix is that the envelope by matrix filter in the response error of passband obtains.By way of to each passband response error weighting coefficient, the setting difference of each passband response error may be implemented.

Description

Design Method of Passband Response Error Weighted Stopband Zero Response Constrained Spatial Matrix Filter

technical field

The invention belongs to the technical field of array signal processing, relates to data processing of sensor arrays, and in particular relates to a design method of a space domain matrix filter.

Background technique

This patent is supported by the National Natural Science Foundation of China project "Spatial Domain Matrix Filtering Technology and Its Application Research in Underwater Acoustic Signal Processing", project number No.11374001.

The spatial domain matrix filter performs array element domain data processing before the array data is used for target orientation estimation. Let the spatial domain matrix filter designed for frequency ω be H(ω), and use the target orientation estimation and matching field to locate the mathematical model of the source incident to the array for data filtering. The array receives the far-field plane wave, and the receiving array data is the product of the direction vector and the source, and the environmental noise n(t,ω) is superimposed:

x(t,ω)=A(τ,ω)s(t,ω)+n(t,ω)

Among them, A(τ,ω) is the delay vector, s(t,ω) is the source signal, n(t,ω) is the environmental noise, x(t,ω) is the received data of the array.

Use the spatial domain matrix filter H(ω) with frequency ω to filter the receiving array data, and the filtered output y(t,ω) is:

y(t,ω)=H(ω)x(t,ω)=H(ω)A(τ,ω)s(t,ω)+H(ω)n(t,ω)

The known array manifold matrix is A(ω)={a(φ,θ,ω)|φ∈Φ,θ∈Θ}, where Φ and Θ correspond to the horizontal and vertical azimuth angle ranges, respectively. The enhancement or suppression effect of the spatial matrix filter on the plane wave signal is realized by the effect on the direction vector, when When it is close to 0, it means that the filter has a strong suppression effect on the plane wave signal with frequency ω in the (φ _i , θ _i ) direction. Conversely, when It is equal to 0, indicating that the filter has no distortion after filtering the plane wave signal with frequency ω in the (φ _i , θ _i ) direction. is the square of the matrix norm. The design of the spatial domain matrix filter is to realize the undistorted response or suppression of the data in the (φ _i , θ _i ) direction by designing the response values to different directions (φ _i , θ _i ). For a linear array sensor, the direction vector a(φ _i ,θ _i ,ω) is only related to the direction θ. At this time, the direction vector is a(θ _i ,ω), and when the detection frequency band ω is given, a(θ _i ,ω) can be abbreviated as a(θ _i ). H(ω) is abbreviated as H.

The spatial domain matrix filter design method is mainly realized by constructing the optimization problem for the passband response error and the stopband response. When there is strong interference in the detection azimuth, the suppression of strong interference can be achieved by setting the stop band for the detection azimuth and setting the stop band response to zero. For this design idea, there is a design method of zero-point constrained spatial domain matrix filter, but this method also has certain problems. The response error of the passband orientation is not completely zero, and the error of each passband orientation is not equal. , that is, in the data output by the spatial domain matrix filter, there is a certain distortion for the passband.

The present invention is aimed at this problem, and proposes a way of weighting the passband response error, so that the passband response error is less than or equal to a certain constant value in each passband orientation, and can be achieved by different weighting of the passband orientation The effect of the passband response error having different values on each tile. Wherein, the weighting coefficient of the passband response error is obtained through an iterative manner, wherein the envelope weighting of the passband response error is used.

Contents of the invention

The technical problem to be solved by the present invention is to generate a spatial matrix filter with zero stopband response under the condition that the passband response error is constant through the iterative method of the passband response error weighted matrix of the optimal spatial matrix filter. Wherein, the passband response error weighting matrix is obtained by weighting the passband response error envelope in each iteration process.

Technical scheme of the present invention is:

Assume that the matrices formed by the passband and stopband direction vectors obtained after discretization of the passband and stopband are:

V _P ＝[a(θ ₁ ),…,a(θ _p ),…,a(θ _P )],1≤p≤P,θ _p ∈Θ _P

V _S ＝[a(θ ₁ ),…,a(θ _s ),…,a(θ _S )],1≤s≤S,θ _s ∈Θ _S

where a(θ _p ) and a(θ _s ) are the p-th and s-th direction vectors after discretization of the passband and stopband, respectively, and P and S correspond to the number of discretized direction vectors in the passband and stopband intervals, respectively. V _P and V _S are the matrix formed by the passband direction vector and the stopband direction vector respectively. Θ _P and Θ _S are the value ranges of the passband direction vector and the stopband direction vector, respectively.

The response error E(θ _p ) of the spatial domain matrix filter to the passband direction vector is:

Let w(θ _p ) be the weighting coefficient of the passband direction vector response error, then the passband weighted overall response error of the spatial matrix filter is:

Among them, N is the number of array elements, and R _1/2 is a diagonal matrix formed by the square root of the weighted value of the passband direction vector.

The response of the spatial matrix filter to the stopband vector is:

Ha(θ _s ),s=1,…,S,θ _s ∈Θ _S

Construct the optimization problem, under the condition that the filter's response to the stopband direction vector is 0, find the weighted passband overall response error weight of the filter to be the smallest. Through such a design, complete suppression of strong interference azimuths can be achieved, and at the same time, the response errors of all azimuths in the passband tend to be the same.

Optimization problem:

The optimal solution can be obtained by using Lagrange multiplier method expression for:

Here, R=diag[w(θ ₁ ),w(θ ₂ ),...,w(θ _S )] _S×S is a diagonal matrix composed of passband response error weighting coefficients. I _N×N is an identity matrix, and its dimension is N×N. By iterating on R, the constant effect of the passband response error can be achieved.

weighted iterative algorithm

The constant response effect of the passband direction vector can be achieved through iteration. During the iteration process, the weighting matrix R is adjusted by enveloping the passband response error and obtaining it by weighting the envelope value. Now, aiming at the optimization problem, a spatial matrix filter with constant passband response error is designed.

Through the discretization of passband and stopband, V _P and V _S can be obtained, and the initial iterative spatial matrix filter can be obtained by formula (1) Wherein, an initial iterative weighting coefficient matrix of R ₀ =diag[1,1,...,1] _P×P is used.

After k iterations to obtain the filter matrix H _k , the absolute value of the passband response error can be obtained through H _k Find all the local maxima for the absolute value of the error, connect _{the local maxima with a straight line segment to find its envelope, and use the corresponding value on the straight line segment as the weight coefficient w k (} θ _p ). Here, it is assumed that there are Q local maxima, and the abscissa is The corresponding extremum, that is, the ordinate is

Since the local maxima usually do not appear at both ends, the weighting values used between the left end point of the detection azimuth and the first local maximum value, and between the right end point of the detection azimuth and the last maximum value need to be special. set up. Use the first local maximum point and the second local maximum position The reverse extension line of the connecting line, obtain the value (θ ₁ , z ₁ ) of the line at the position of the left end point, let (θ ₁ , max(|E _k (θ ₁ )|z ₁ )) be the weighted starting point of the left end point starting point, and with connect, get The weight coefficient of the interval. Similarly, use the penultimate local maximum point and the second last local maximum point The extension line between the connecting lines, get the value (θ _P ,z _P ) of the right end point on this line, let (θ _P ,max(z _P ,|E _k (θ _P )|)) be the weight of the right end point starting point, and with connected, and the value on the corresponding connection is taken as weighted value on .

The iteration of the weighting coefficient matrix R involves the passband response error weighting vector w(θ _p ) in it.

Set β _k (θ _p ) as the weighted product coefficient in the kth iteration process, and let Here, α _k (θ _p ) is the value of θ _p on the corresponding response error envelope segment.

Set γ(θ _p ) as the response scaling factor for each passband, set as follows:

γ(θ _p )=γ _i , θ _p ∈ _{Θ Pi} (2)

Among them, Θ _Pi is the set of incident azimuth angles in the azimuth space of the i-th passband, and the set response proportional coefficient is γ _i in the i-th passband.

The iterative method of the passband response error weighting coefficient matrix is determined by the following formula:

w _k+1 (θ _p )=β _k (θ _p )γ(θ _p )w _k (θ _p )

Through the above formula, a new weighting coefficient matrix R _k+1 =diag[w _k+1 (θ ₁ ), w _k+1 (θ ₂ ), . . . , w _k+1 (θ _P )] can be determined.

By bringing the response proportional coefficient γ(θ _p ) into the iteration of the weighting coefficient w(θ _p ), after the algorithm is terminated, the response difference of each passband orientation is:

where _γi and _γj correspond to the response scale coefficient values of the i-th and j-th passbands, respectively.

The above formula is the response difference given in dB. If γ(θ _p ) is chosen as a constant value, the passband response error of the spatial matrix filter is the same.

Description of drawings

Figure 1a shows the effect of the passband response weighted stopband zero response constrained spatial domain matrix filter (γ(θ _p )=1,θ _p ∈Θ _P1 ∪Θ _P2 ∪Θ _P3 ).

Figure 1b passband response error weighted stopband zero response constrained spatial domain matrix filter effect (γ(θ _p )=1,θ _p ∈Θ _P1 ∪Θ _P2 ∪Θ _P3 ).

Figure 2 The passband response error and its envelope corresponding to the first iteration (γ(θ _p )=1,θ _p ∈Θ _P1 ∪Θ _P2 ∪Θ _P3 ).

Figure 3a passband response weighted stopband zero response constrained spatial matrix filter effect (γ(θ _p )=1,θ _p ∈Θ _P1 ∪Θ _P2 ,γ(θ _p )=1/4,θ _p ∈Θ _P3 ) .

Figure 3b passband response error weighted stopband zero response constrained spatial matrix filter effect (γ(θ _p )=1,θ _p ∈Θ _P1 ∪Θ _P2 ,γ(θ _p )=1/4,θ _p ∈Θ _P3 ).

In the figure, the number of array elements corresponding to the designed filter is N=30, and the array elements are equally spaced.

Each passband is Θ _P1 = [-90°, -35°), Θ _P2 = (-25°, -5°), Θ _P3 = (5°, 90°], and the stop band is Θ _S = -30° ∪0°, the passband discretization sampling interval is 0.1°, and the spatial domain matrix filter is designed for the half-wavelength frequency of the array.

Figure 1a and Figure 1b show the case of γ(θ _p )=1, θ _p ∈ _{Θ P1} ∪Θ _P2 ∪Θ _P3 , using the passband response error envelope weighting, the obtained stopband zero response constrained spatial matrix The design effect of the filter, here the weighting coefficient matrix R is iterated 10 times in total. Figure 1a shows the filter response Figure 1b shows the filter response error The figure also shows the design effect of the matrix filter without iteration, represented by the curve.

Figure 2 shows the filter passband response error corresponding to the first iteration in Figure 1a and Figure 1b, which is given by the curve, and through the passband response error, the filter response envelope is obtained, which is given by the curve. And obtain the weighted iterative matrix R with the envelope.

Figure 3a and Figure 3b show γ(θ _p ) = 1, θ _p ∈ _{Θ P1} ∪Θ _P2 , γ(θ _p ) = 1/4, θ _p ∈ Θ _P3 , using the passband response error envelope Weighting, the design effect of the obtained stopband zero-response constrained spatial domain matrix filter, where the weighting coefficient matrix R is iterated 10 times in total. The difference from Figure 1a and Figure 1b is that the response error ratio coefficients of the 1st passband, 2nd passband, and 3rd passband are 1:1:1/4, therefore, the filter in the 1st passband and the 2nd passband The response error of the passband is 6dB lower than that of the third passband.

Detailed ways

The specific implementation examples of the present invention are described in detail below in conjunction with the scheme.

The iterative algorithm based on the matrix filter stopband response envelope weighting criterion is as follows:

Step 1: Let k=0, w ₀ (θ _p )=1. Discretize the detection space to obtain V _P , V _S , Θ _Pi , Θ _P . Using formula (1), calculate the initial optimal spatial domain matrix filter Set each passband response ratio γ(θ _p );

Step 2: Calculate Find the local maximum value point of |E _k (θ _p )|, and obtain the corresponding vertical coordinate of the abscissa of the local maximum value as

Step 3: Take advantage of and The extension line of the connecting line between two points, calculate the value z ₁ at the abscissa θ ₁ , and set (θ ₁ ,max(|E _k (θ ₁ )|,z ₁ )) as the starting point of envelope weighting.

Step 4: Leverage and The extension line of the connecting line between two points, calculate the value z _P at the abscissa θ _P , and set (θ _P ,max(z _P ,|E _k (θ _P )|)) as the end point of the envelope weighting.

Step 5: Calculate (θ ₁ , max(|E _k (θ ₁ )|, z ₁ )), (θ _P , max(z _P , |E _k (θ _P )|)) is a line segment between Q+2 points, and take α _k (θ _p ) as the value of θ _p on the corresponding line segment.

Step 6: Calculate the following formulas

w _k+1 (θ _p )=β _k (θ _p )γ(θ _p )w _k (θ _p )

R _k+1 ＝diag[w _k+1 (θ ₁ ),w _k+1 (θ ₂ ),…,w _k+1 (θ _P )]

where β _k (θ _p ) is the weighted multiplication factor used in the kth iteration. R _k+1 is the stopband response weighting coefficient matrix of the k+1th iteration. H _k+1 is the spatial domain matrix filter obtained by the k+1th iteration.

Determine whether H _k+1 satisfies one of the following termination conditions:

(a) k+1=K. At this time, iterate K times at this time, and the algorithm terminates;

(b) After iterations, the actual response error value of the spatial matrix filter to all azimuths in the passband is less than the constant Algorithm terminates;

(c) After iterations, the response error rate of the spatial matrix filter for all orientations in the passband is less than a constant value Algorithm terminated.

Step 7: If the iteration termination condition is satisfied, then H _k+1 is the final spatial matrix filter. Otherwise, set k=k+1 and repeat steps 2-6.

Claims (2)

1. A design method of a pass band response error weighted stop band zero response constraint spatial matrix filter is characterized by comprising the following steps,

step 1: let k equal to 0, w₀(θ_p) 1 is ═ 1; discretizing the detection space domain to obtain V_P，V_S，Θ_P(ii) a Calculating an initial optimal spatial matrix filter using equation (1)Setting respective pass band response ratios gamma (theta)_p)；

V_P＝[a(θ₁),…,a(θ_p),…,a(θ_P)],1≤p≤P,θ_p∈Θ_P

V_S＝[a(θ₁),…,a(θ_s),…,a(θ_S)],1≤s≤S,θ_s∈Θ_S

Wherein a (theta)_p)、a(θ_s) Respectively the P-th and S-th direction vectors after the discretization of the pass band and the stop band, wherein P and S respectively correspond to the discretization direction vector number of the pass band and the stop band interval, V_PAnd V_SAre matrices formed by pass band direction vectors and stop band direction vectors, theta_PAnd Θ_SThe value regions theta of the pass band direction vector and the stop band direction vector respectively₁,…,θ_PRespectively corresponding to the discretization direction of the pass band, theta₁Is the first discretization direction of the pass band, θ_PIs the last discretization direction of the pass band;

wherein, I_N×NIs a unit matrix with dimension N × N, N is the number of array elements,

R＝diag[w(θ₁),w(θ₂),…,w(θ_P)]_P×Pweighting coefficients w (theta) for passband response errors_p) P is 1, …, and an initial weighting factor w (θ) is set_p)＝w₀(θ_p) Each pass band response scaling factor γ (θ) is set to 1_p)；

Calculating an initial optimal spatial matrix filter using equation (1)Is marked as

Step 2: computingObtaining | E_k(θ_p) A local maximum point of | acquiring the abscissa of the local maximumCorresponding ordinate is

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(|E_k(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 theta_PValue z of_PSet up (theta)_P,max(z_P,|E_k(θ_P) |)) is an envelope weighting endpoint;

and 5: computing (θ_P,max(z_P,|E_k(θ_P) I))) togetherThe line segment between the Q +2 points is taken and α is taken_k(θ_p) Is theta_pValues on the corresponding line segments;

β_k(θ_p) Is a weighted product coefficient in the k-th iteration process, and

α_k(θ_p) Is theta_pValues on the corresponding response error envelope segments;

step 6: calculating the following formulas

w_k+1(θ_p)＝β_k(θ_p)γ(θ_p)w_k(θ_p)

R_k+1＝diag[w_k+1(θ₁),w_k+1(θ₂),…,w_k+1(θ_P)]

Wherein, β_k(θ_p) Is the weighted product factor used in the kth iteration; r_k+1A stopband response weighting coefficient matrix for the (k + 1) th iteration; h_k+1A spatial matrix filter obtained for the (k + 1) th iteration;

judgment of H_k+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 actual response error value of the spatial matrix filter to all the positions on the pass band is smaller than a constantThe algorithm is terminated;

(c)after iteration, the response error change rate of the spatial matrix filter to all the orientations on the pass band is smaller than a constant valueThe algorithm is terminated;

and 7: if the iteration termination condition is satisfied, H_k+1The final spatial domain matrix filter is obtained; otherwise, let k be k +1, repeat steps 2-6.

2. The pass-band response error weighted zero-response constrained spatial matrix filter design method according to claim 1, characterized in that the response difference of each pass-band orientation is adjusted by setting pass-band response proportionality coefficients:

setting gamma (theta)_p) The response proportionality coefficient for each pass band is set by:

wherein,is the ith pass band azimuth space incident azimuth angle set, and the set response proportionality coefficient is gamma on the ith pass band_i；

The iterative method of the passband response error weighting coefficient matrix is determined by:

w_k+1(θ_p)＝β_k(θ_p)γ(θ_p)w_k(θ_p)

by the above formula, a new weighting coefficient matrix R can be determined_k+1＝diag[w_k+1(θ₁),w_k+1(θ₂),…,w_k+1(θ_P)]；

By responding toCoefficient of proportionality gamma (theta)_p) Into the weighting factor w (theta)_p) After the algorithm is terminated, the response difference for each pass band position is:

wherein, γ_iAnd gamma_jResponse ratio coefficient values corresponding to the ith and jth passbands, respectively;

the above equation is a response difference given in dB; if gamma (theta) is selected_p) The passband response error of the spatial matrix filter is the same for a constant value.