CN103969626A - Wideband digital wave beam forming method based on all-pass type variable fractional delay filter - Google Patents

Abstract

The invention discloses a wideband digital wave beam forming method based on an all-pass type variable fractional delay filter. The problem that a needed filter in the prior art is large in order number and high in complexity is solved. The method comprises the steps that (1) according to radar system parameters, sampling rate fs and sampling time ts are determined; (2) according to the radar array geometric structure and the sampling time ts, a propagation time delay tau of each array element relative to a reference array element is calculated; (3) according to the sampling rate fs, an AD chip is selected to digitize analog signals received by the array elements; (4) according to the propagation time delay tau and the digitized received analog signals, the all-pass type variable fractional delay filter is designed; (5) the digitized received signals are filtered; (6) phase delay of the filtered signals is compensated for; (7) a window function is selected to conduct weighted summation on the filtered compensated signals to obtain a wave beam forming result. The wideband digital wave beam forming method can achieve wideband digital beam forming, saves hardware resources, improves instantaneity and can be used for wideband digital array radar.

Description

Based on the broadband digital beam forming method of the variable mark filtering wave by prolonging time of wildcard-filter style device

Technical field

The invention belongs to Radar Technology field, relate to wide band digital beam-forming, can be used for the processing of wideband digital array radar signal.

Background technology

The processing of array radar signal, it is exactly the diverse location in space by multiple sensor settings, composition sensor array, and utilize it that spacing wave is received and processed, be intended to extract interested signal and the characteristic information thereof that array receives, suppress interference and noise or uninterested signal simultaneously.It has, and wave beam control is flexible, processing gain is high, antijamming capability is strong and spatial resolution advantages of higher, obtains people's extensive concern and develops rapidly in fields such as radar, radio astronomy, sonar, communication, direction finding, seismology and medical diagnosiss.The research contents of Array Signal Processing mainly contains two classes: a class is Estimation of Spatial Spectrum, and it is by array received data, estimated signal arrival bearing; One class is that wave beam forms, and also referred to as airspace filter, its objective is the difference of antenna reception is sampled to signal, by certain weighting, improves as much as possible the intensity of the useful signal of array output, suppress simultaneously other to signal intensity.

At present, beam-forming technology is mostly for narrow band signal, but along with the range of application of Array Signal Processing is more and more wider, runs into the problem of many Wideband Signal Processing, as Sonar Signal, radar signal, seismic signal etc.Broadband signal can be similar to the simple summation that is equal to each narrow-band array model, if still utilize narrow-band beam formation processing technical finesse broadband signal, because not realizing different weightings according to different frequency components, cause beam pattern to distort, be embodied in beam position skew and beam-broadening, i.e. spatial dispersion and time dispersive problem.Existing Broadband Beamforming Method mainly contains frequency-domain and time-domain Broadband Beamforming Method.

Frequency domain Broadband Beamforming Method, is first broadband signal to be carried out to arrowband decomposition, then each arrowband component is done to corresponding beam weighting.Existing frequency domain wideband constant beamwidth Beamforming Method mainly utilizes the submatrix of different pore size to receive different signal frequency domain components, and compensation wave beam is with the variation of frequency.Due to submatrix Limited Number, although the wave beam that the method forms can keep constant beam-width in each frequency range, signal still has distortion in whole cps, and realizes more complicated.Meanwhile, utilizing discrete Fourier transformation, on frequency domain, each frequency component is carried out to narrow-band beam formation processing, is a kind of piece disposal route, and the phase place between piece and piece cannot be connected, and causes the phase place of output signal time domain waveform discontinuous.

Broadband beamforming in time domain method is conventionally to adopt tapped delay line group structure or FIR filter construction to realize broadband beams to form.The exponent number of the length of tapped delay line or FIR wave filter depends on the bandwidth of incoming signal: bandwidth is larger, and the length exponent number longer or FIR wave filter of tapped delay line is higher.When frequency is higher, bandwidth is when larger, need to the tapped delay line group of length or the FIR wave filter of higher-order number, speed of convergence is slower, calculated amount is larger, sample frequency realizes difficulty simultaneously, even cannot realize.In addition, in existing broadband digital beam forming method, each array element has different time delays with respect to reference array element, need to design different mark filtering wave by prolonging time devices for each array element, and design complexities is high, and task amount is large.

Summary of the invention

The object of the invention is to the problem existing in above-mentioned broadband digital beam forming method, propose a kind of broadband digital beam forming method based on the variable mark filtering wave by prolonging time of wildcard-filter style device, to reduce the exponent number of required wave filter, reduced design complexities.

For achieving the above object, the present invention includes following steps:

(1) determine sample rate f according to radar system parameter _s, obtain sampling time t _s=1/f _s;

(2) determine the time delays τ of each radar array element with respect to reference array element according to the geometry of array _n, and by time delay τ _nwith sampling time t _s, obtain the time delay of wave filter τ _qn=τ _n/ t _s, wherein, n represents array element index, for τ _qnthe result of round;

(3) select AD chip, the simulating signal that each radar array element is received changes into digital signal x _n, wherein, 1≤n≤N, N is total element number of array;

(4), according to radar system parameter, adopt iterative optimization method design wildcard-filter style mark filtering wave by prolonging time device;

(5) by digital signal x _n, be input to the variable mark filtering wave by prolonging time of wildcard-filter style device and carry out filtering as input variable with the time delay Δ of wave filter, obtain filtered signal vector

(6) give filtered signal vector be multiplied by phase compensating factor exp (j2 π sd/ λ), receive the phase delay of signal to compensate each array element, the signal vector obtaining after filtering compensation is wherein, exp () represents to ask the power of natural logarithm e, and j is imaginary unit, π represents the radian that semicircle is corresponding, the propagation vector that s is signal, and d represents the distance vector between n array element and reference array element, λ is signal wavelength;

(7) according to radar beam sidelobe level δ _lselect window function pair weighted sum, obtains wave beam and forms result wherein, W (n) is the discrete function value of selected window function, 1≤n≤N.

The present invention has the following advantages:

(1) save hardware resource.The variable mark filtering wave by prolonging time of wildcard-filter style device belongs to infinite impulse response filter, than limited impulse FIR wave filter, can realize identical filtering performance with filter order still less;

(2) simplified design complexities.The variable mark filtering wave by prolonging time of the wildcard-filter style device of design can be realized the arbitrary small number time delay in-0.5～0.5 scope, therefore, only need once design, and just may be used on whole radar array element;

(2) real-time.Real-time calculating or off-line without filter weights are downloaded.

Brief description of the drawings

Fig. 1 is general flow chart of the present invention;

Fig. 2 is the geometric model of radar array used in the present invention in cartesian coordinate system;

Fig. 3 is the sub-process figure of the variable mark filtering wave by prolonging time of wildcard-filter style device Optimization Solution in the present invention;

The geometric model figure of the linear radar array adopting when Fig. 4 is emulation of the present invention;

The radar signal oscillogram using when Fig. 5 is emulation of the present invention;

The group delay frequency characteristic figure of the variable mark filtering wave by prolonging time of the wildcard-filter style device designing when Fig. 6 is emulation of the present invention;

The group delay frequency characteristic Error Graph of the variable mark filtering wave by prolonging time of the wildcard-filter style device designing when Fig. 7 is emulation of the present invention;

The oscillogram of wide band digital beam-forming result when Fig. 8 is emulation of the present invention;

The Error Graph of wide band digital beam-forming result when Fig. 9 is emulation of the present invention.

Embodiment

With reference to Fig. 1, performing step of the present invention is as follows:

Step 1: according to radar system parameter, determine sample rate f _swith sampling time t _s:

(1.1) suppose that radar system carrier frequency is f _c, bandwidth is B, obtains according to bandpass sample theory:

$f_s=\frac{4\·f_c}{2n+1},$

Wherein, n gets and makes sample rate f _sbe not less than the arbitrary integer of bandwidth B;

(1.2) by sample rate f _sobtain the sampling time

Step 2: determine the time delays τ of each radar array element with respect to reference array element according to the geometry of array _n:

(2.1) geometric model in cartesian coordinate system with reference to Fig. 2 radar array, the position vector of determining n array element is p _n, the position vector of No. 0 array element of reference array element is p ₀, the propagation vector of radar signal is s, the spacing vector that obtains two array elements is d _n=p _n-p ₀;

(2.2) by the position vector p of n array element _nposition vector p with No. 0 array element ₀, obtain the range difference l=d of two array elements on the direction of propagation _ns;

(2.3) by range difference l and electromagnetic wave propagation speed c, obtain the propagation delay of n array element with respect to No. 0 reference array element ${\mathrm{\τ}}_n=\frac{l}{c}.$

Step 3: by propagation delay τ _nwith sampling time t _s, obtain the time delay of wave filter τ _qn=τ _n/ t _s, wherein, n represents array element index, for τ _qnthe result of round.

Step 4: select AD chip, the simulating signal that each radar array element is received changes into digital signal x _n, wherein, 1≤n≤N, N is total element number of array.

Step 5: according to radar system parameter, adopt iterative optimization method design wildcard-filter style mark filtering wave by prolonging time device, comprise design of filter optimization problem structure and solve, its step is as follows:

(5.1) with the optimization problem of weighted least mean square error criterion structure design of filter:

(5.1a) each filter coefficient is expressed as to the polynomial expression of wave filter time delay p:

$a_n\left(p\right)=\underset{m=0}{\overset{M}{\mathrm{\Σ}}}{b}_{\mathrm{nm}}{p}^m,1\≤n\≤L,$

Wherein, the exponent number that L is wave filter, the exponent number that M is fitting of a polynomial, b _nmbe n m the coefficient of polynomial fitting that filter tap is corresponding;

(5.1b) by polynomial expression a _n(p) obtain the z territory form of the unit impulse response of wildcard-filter style mark filtering wave by prolonging time device:

$\begin{array}{c}H(z,p)=\frac{a_L\left(p\right)+a_{L-1}\left(p\right)z^{-1}+...+a_1\left(p\right)z^{-(L-1)}+z^{-L}}{1+a_1\left(p\right)z^{-1}+...a_{L-1}\left(p\right)z^{-(L-1)}+a_L\left(p\right)z^{-L}}\\ =\frac{z^{-L}A(z^{-1},p)}{A(z,p)}\end{array}---<1>$

Wherein, denominator part the amplitude-frequency response perseverance of the wave filter representing due to formula <1> is 1, makes z=e ^{j ω}, formula <1> can be rewritten as:

H(e ^jω,p)＝e ^(jθ(ω,p))， <2>

In formula <2>, e ⁽⁾the power that represents to ask natural logarithm e, θ (ω, p) is the phase-frequency response of wave filter, is obtained by formula <1> and formula <2> simultaneous:

θ(ω,p)＝-Lω-2θ _A(ω,p)， <3>

Wherein, 0≤ω≤π is digital angular frequency, ${\mathrm{\&theta;}}_A(\mathrm{\&omega;},p)=\arctan \left(\frac{1+\underset{n=1}{\overset{L}{\mathrm{\&Sigma;}}}{a}_n\left(p\right)\cos \left(\mathrm{n\&omega;}\right)}{-\underset{n=1}{\overset{L}{\mathrm{\&Sigma;}}}{a}_n\left(p\right)\sin \left(\mathrm{n\&omega;}\right)}\right),$ Arctan () represents to negate tangent;

(5.1c) frequency response of desirable mark filtering wave by prolonging time device is H _d(e ^{j ω}, p)=e ^{(jD (ω, p))}, its phase-frequency response is D (ω, p)=-ω (I+p), it is the intrinsic integer delay of wave filter.The phase-frequency response θ (ω, p) that convolution <3> provides, obtains the phase-frequency response error of wave filter:

e ₁(ω,p)＝-Lω-2θ _A(ω,p)-D(ω,p)； <4>

(5.1d) with the optimization problem of weighted least mean square error criterion structure design of filter, obtain following cost function:

$J_1=\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}W(\mathrm{\&omega;},p){\left|e_1(\mathrm{\&omega;},p)\right|}^2\mathrm{dpd\&omega;},---<5>$

Wherein, W (ω, p) is the fixed weighting function in 0≤ω≤α π and-0.5≤p≤0.5 scope, and α is frequency band control parameter, meets the cost function that formula <5> provides represents phase-frequency response error e ₁(ω, p) first square, then weighted sum in 0≤ω≤α π and-0.5≤p≤0.5 scope;

(5.1e) by equation by cost function J ₁further change into:

$J_2=4\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}W(\mathrm{\&omega;},p){\left|\frac{e_2(\mathrm{\&omega;},p)}{A(e^{\mathrm{j\&omega;}},p)}\right|}^2\mathrm{dpd\&omega;},---<6>$

Wherein,

$e_2(\mathrm{\&omega;},p)=\underset{n=1}{\overset{L}{\mathrm{\&Sigma;}}}\underset{m=0}{\overset{M}{\mathrm{\&Sigma;}}}{b}_{\mathrm{nm}}{p}^m{c}_n(\mathrm{\&omega;},p)+\sin \left(\mathrm{\&beta;}(\mathrm{\&omega;},p)\right),$

c _n(ω,p)＝sin(β(ω,p)+nω)， $\mathrm{\&beta;}(\mathrm{\&omega;},p)=-\frac{1}{2}[D(\mathrm{\&omega;},p)+\mathrm{L\&omega;}];$

(5.1f) in order to construct the cost function of vector quantization, by coefficient of polynomial fitting b _nm, 1≤n≤L, 1≤m≤M, is write as vector form b=(b ₁₀, b ₁₁..., b _nM) ^t, and introduce vector

c(ω,p)＝(c ₁₀(ω,p)p ⁰,..,c _NM(ω,p)p ^M) ^t，

Wherein, () ^trepresent to ask vectorial transposition.The cost function that formula <6> provides can further be write as vector form:

$J_3=b^t{Q}_{1}b-{2g}_1^{t}b+d_1---<7>$

Wherein,

$Q_1=\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p)c(\mathrm{\&omega;},p)c{(\mathrm{\&omega;},p)}^t\mathrm{dpd\&omega;},$

$g_1=-\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p)c(\mathrm{\&omega;},p)\sin \left(\mathrm{\&beta;}(\mathrm{\&omega;},p)\right)\mathrm{dpd\&omega;},$

$d_1=\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p){\left|\sin \left(\mathrm{\&beta;}(\mathrm{\&omega;},p)\right)\right|}^2\mathrm{dpd\&omega;},$

In above formula, $\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p)=\frac{4W(\mathrm{\&omega;},p)}{{\left|A(e^{\mathrm{j\&omega;}},p)\right|}^2};$

(5.1g) by the cost function J after vector quantization ₃, as follows with the optimization problem of minimum weight mean-square error criteria structure design of filter:

$\underset{b}{\min }{J}_3=\underset{b}{\min }(b^t{Q}_{1}b-{2g}_1^{t}b+d_1)---<8>$

This is a quadratic form optimization problem, represents, taking coefficient of polynomial fitting vector b as optimized variable, to make cost function J ₃minimum, this problem needs iterative;

(5.2) optimization problem that iterative formula <8> provides:

With reference to Fig. 3, solution procedure is as follows:

(5.2a) initialization:

Determine the fitting of a polynomial exponent number M of filter order L and filter coefficient;

Determine the integral part in desirable mark filtering wave by prolonging time device group delay response D (ω, p)=-ω (I+p)

According to the sample rate f of selecting _swith radar signal bandwidth B, determine and optimize parameter α, requirement

Set primary iteration number of times k=1, vectorial initial value b to be optimized _k-1=0,0 represents null vector;

(5.2b) utilize the result b of the k-1 time iterative _k-1calculate the required weighted value of iterative the k time:

$W_{k-1}(\mathrm{\&omega;},p)=\frac{4\&CenterDot;W(\mathrm{\&omega;},p)}{{\left|A_{k-1}(e^{\mathrm{j\&omega;}},p)\right|}^2},$

Wherein w (ω, p) is in 0≤ω≤α π and-0.5≤p≤0.5 scope, the fixed weighting value of square error;

(5.2c) weighted value W step (5.2b) being obtained _k-1the result b of (ω, p) and k-1 step iterative _k-1, compute matrix Q _1k, vectorial g _1kwith numerical value d _1k;

(5.2d) solving equations Q _1kb _k=g _1k, obtain the result b of the k time iterative _k;

(5.2e) end condition judgement: determine error margin ε according to radar system parameter, if set up, stop iteration, obtain coefficient of polynomial fitting vector b _k; Otherwise k=k+1, jumps to step (5.2b) and carries out next round iterative.

Step 6: adopt the variable mark filtering wave by prolonging time of the wildcard-filter style device of design in step 5 to digitized reception signal x _ncarry out filtering:

(6.1) suppose that the coefficient of polynomial fitting vector being obtained by step (5.2) iterative is b _o, the filter coefficient can obtain amount of delay and be p by formula <9> time:

$a_n\left(p\right)=\underset{m=0}{\overset{M}{\mathrm{\&Sigma;}}}{b}_{\mathrm{nm}}{p}^m,1\&le;n\&le;L,---<9>$

Wherein, b _nmvectorial b _oelement;

(6.2) by digitized reception signal x _nwith Filter delay p, variable mark filtering wave by prolonging time device is inputted in-0.5≤p≤0.5 simultaneously, obtains filtered signal and is:

$\stackrel{\&OverBar;}{y\left(t\right)}=x(t-L)+\underset{l=1}{\overset{L}{\mathrm{\&Sigma;}}}{a}_n\left(p\right)x(t-l-L)-\underset{l=1}{\overset{L}{\mathrm{\&Sigma;}}}{a}_n\left(p\right)y(t-l),---<10>$

In formula <10>, 0≤t≤N _s, N _sit is the total sampling number being determined by radar system parameter;

By filtered signal obtaining filtered signal vector is () ^trepresent to ask vectorial transposition.

Step 7: compensate each radar array element with respect to the phase delay with reference to radar array element.

Give filtered signal vector be multiplied by phase compensating factor exp (j2 π sd/ λ), the signal vector obtaining after filtering compensation is wherein, exp () represents to ask the power of natural logarithm e, and j is imaginary unit, π represents the radian that semicircle is corresponding, the propagation vector that s is signal, and d represents the distance vector between n array element and reference array element, λ is signal wavelength.

Step 8: according to radar beam sidelobe level δ _lsignal vector after selecting window function to filtering compensation weighted sum, obtains wave beam and forms result wherein, W (n) is the discrete function value of selected window function, 1≤n≤N, and N is the total number of radar array element.

Effect of the present invention further illustrates by following emulation experiment:

1. experiment scene: consider certain wideband digital array radar, Fig. 4 has provided the geometric model of this array radar, its carrier frequency transmitting is 1GHz, bandwidth is 200MHz, time wide be 1 μ s, sampling rate is 400MHz, sampled point number N=400.The radar waveform that this emulation adopts is the linear FM signal based on above parameter, and signal waveform as shown in Figure 5.

2. emulation content:

Emulation 1, based on following simulation parameter: frequency band constraint factor α=0.8, filter order is 8, fitting of a polynomial exponent number is 5, designs a wildcard-filter style mark filtering wave by prolonging time device.Fig. 6 has provided the group delay frequency characteristic figure of the variable mark filtering wave by prolonging time of wildcard-filter style device.Fig. 7 has provided the group delay frequency characteristic Error Graph of the variable mark filtering wave by prolonging time of wildcard-filter style device.

Emulation 2, based on the wave filter of above-mentioned design, adopt the first sidelobe level be-Chebyshev window of 40dB carries out array weight, carries out wide band digital beam-forming.Fig. 8 is the oscillogram of the wide band digital beam-forming result that obtains based on this method.Fig. 9 has provided the Error Graph that adopts the wide band digital beam-forming result that the inventive method obtains.In Fig. 8, provide desirable wide band digital beam-forming result simultaneously, be used for verifying the validity of the inventive method.

3. analysis of simulation result:

As can be seen from Figure 6, the group delay frequency characteristic of the designed wave filter of the present invention is all very smooth in design frequency band.The group delay error figure that Fig. 7 provides shows, the group delay error of the mark filtering wave by prolonging time device of the present invention's design in whole mark reference time delay all lower than-50dB.

As can be seen from Figure 8, the wave beam obtaining based on the inventive method forms result and desirable wideband digital wave number formation result is overlapping in wave beam main lobe part.

Wave beam shown in Fig. 9 forms resultant error figure and shows, both errors are no more than 1 × 10 ^-3, understand furtherly both degree of closeness, verify the validity of the inventive method.

Claims (4)

1. the broadband digital beam forming method based on the variable mark filtering wave by prolonging time of wildcard-filter style device, comprises the steps:

(1) determine sample rate f according to radar system parameter _s, obtain sampling time t _s=1/f _s;

(2) determine the time delays τ of each radar array element with respect to reference array element according to the geometry of array _n, and by time delay τ _nwith sampling time t _s, obtain the time delay of wave filter τ _qn=τ _n/ t _s, wherein, n represents array element index, for τ _qnthe result of round;

(3) select AD chip, the simulating signal that each radar array element is received changes into digital signal x _n, wherein, 1≤n≤N, N is total element number of array;

(4), according to radar system parameter, adopt iterative optimization method design wildcard-filter style mark filtering wave by prolonging time device;

(5) by digital signal x _n, be input to the variable mark filtering wave by prolonging time of wildcard-filter style device and carry out filtering as input variable with the time delay Δ of wave filter, obtain filtered signal vector

(6) give filtered signal vector be multiplied by phase compensating factor exp (j2 π sd/ λ), receive the phase delay of signal to compensate each array element, the signal vector obtaining after filtering compensation is wherein, exp () represents to ask the power of natural logarithm e, and j is imaginary unit, π represents the radian that semicircle is corresponding, the propagation vector that s is signal, and d represents the distance vector between n array element and reference array element, λ is signal wavelength;

(7) according to radar beam sidelobe level δ _lselect window function pair weighted sum, obtains wave beam and forms result wherein, W (n) is the discrete function value of selected window function, 1≤n≤N.

2. the broadband digital beam forming method based on the variable mark filtering wave by prolonging time of wildcard-filter style device, wherein step (4) described according to radar system parameter, adopt iterative optimization method design wildcard-filter style mark filtering wave by prolonging time device, carry out as follows:

(4a) with the optimization problem of weighted least mean square error criterion structure design of filter;

(4b) the above-mentioned optimization problem of iterative.

3. the broadband digital beam forming method based on the variable mark filtering wave by prolonging time of wildcard-filter style device, the wherein described optimization problem with weighted least mean square error criterion structure design of filter of step (4a), carry out as follows:

(4a1) each filter coefficient is expressed as to the polynomial expression into wave filter time delay p:

$a_n\left(p\right)=\underset{m=0}{\overset{M}{\mathrm{\&Sigma;}}}{b}_{\mathrm{nm}}{p}^m,1\&le;n\&le;L,$

Wherein, L is the exponent number of variable mark filtering wave by prolonging time device, the exponent number that M is fitting of a polynomial, b _nmbe n m the coefficient of polynomial fitting that filter tap is corresponding;

(4a2) by polynomial expression a _n(p) and the phase-frequency response D (ω, p) of desirable mark filtering wave by prolonging time device=-ω (I+p), obtain the phase-frequency response error of wave filter:

e ₁(ω,p)＝-Lω-2θ _A(ω,p)-D(ω,p)，

Wherein, ${\mathrm{\&theta;}}_A(\mathrm{\&omega;},p)=\arctan \left(\frac{1+\underset{n=1}{\overset{L}{\mathrm{\&Sigma;}}}{a}_n\left(p\right)\cos \left(\mathrm{n\&omega;}\right)}{-\underset{n=1}{\overset{L}{\mathrm{\&Sigma;}}}{a}_n\left(p\right)\sin \left(\mathrm{n\&omega;}\right)}\right),$ ω is digital angular frequency, meets 0≤ω≤π, and arctan () represents to negate tangent, it is the intrinsic integer delay of wave filter;

(4a3) by phase-frequency response error e ₁(ω, p), obtains following cost function:

$J_1=\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}W(\mathrm{\&omega;},p){\left|e_1(\mathrm{\&omega;},p)\right|}^2\mathrm{dpd\&omega;},$

Wherein, W (ω, p) is the fixed weighting function in 0≤ω≤α π and-0.5≤p≤0.5 scope, and α is frequency band control parameter, meets $\frac{B}{f_s}\&le;\mathrm{\&alpha;}<1;$

(4a4) by equation by cost function J ₁further change into:

$J_2=4\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}W(\mathrm{\&omega;},p){\left|\frac{e_2(\mathrm{\&omega;},p)}{A(e^{\mathrm{j\&omega;}},p)}\right|}^2\mathrm{dpd\&omega;},$

Wherein,

$e_2(\mathrm{\&omega;},p)=\underset{n=1}{\overset{L}{\mathrm{\&Sigma;}}}\underset{m=0}{\overset{M}{\mathrm{\&Sigma;}}}{b}_{\mathrm{nm}}{p}^m{c}_n(\mathrm{\&omega;},p)+\sin \left(\mathrm{\&beta;}(\mathrm{\&omega;},p)\right),$

c _n(ω，p)＝sin(β(ω，p)+nω)， $\mathrm{\&beta;}(\mathrm{\&omega;},p)=-\frac{1}{2}[D(\mathrm{\&omega;},p)+\mathrm{L\&omega;}];$

(4a5) by the cost function J after transforming ₂vector quantization, obtains the cost function after vector quantization

Wherein,

b＝(b ₁₀,b ₁₁,...,b _NM) ^t，

$Q_1=\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p)c(\mathrm{\&omega;},p)c{(\mathrm{\&omega;},p)}^t\mathrm{dpd\&omega;},$

$g_1=-\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p)c(\mathrm{\&omega;},p)\sin \left(\mathrm{\&beta;}(\mathrm{\&omega;},p)\right)\mathrm{dpd\&omega;},$

$d_1=\underset{\mathrm{\&omega;}=0}{\overset{\mathrm{\&alpha;\&pi;}}{\&Integral;}}\underset{p=-0.5}{\overset{0.5}{\&Integral;}}\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p){\left|\sin \left(\mathrm{\&beta;}(\mathrm{\&omega;},p)\right)\right|}^2\mathrm{dpd\&omega;},$

$\stackrel{\&OverBar;}{W}(\mathrm{\&omega;},p)=\frac{4W(\mathrm{\&omega;},p)}{{\left|A(e^{\mathrm{j\&omega;}},p)\right|}^2},$

c(ω,p)＝(c ₁₀(ω,p)p ⁰,..,c _NM(ω,p)p ^M) ^t，

Wherein, b ^trepresent the transposition of vectorial b;

(4a6) by the cost function J after vector quantization ₃, as follows with the optimization problem of minimum weight mean-square error criteria structure design of filter:

$\underset{b}{\min }{J}_3=\underset{b}{\min }(b^t{Q}_{1}b-{2g}_1^{t}b+d_1)$

This is a quadratic form optimization problem, taking coefficient of polynomial fitting vector b as optimized variable, needs iterative.

4. the broadband digital beam forming method based on the variable mark filtering wave by prolonging time of wildcard-filter style device, the wherein described above-mentioned optimization problem of iterative of step (4b), carry out as follows:

(4b1) initialization:

Determine the fitting of a polynomial exponent number M of filter order L and filter coefficient;

Determine the integral part in desirable mark filtering wave by prolonging time device group delay response D (ω, p)=-ω (I+p)

According to the sample rate f of selecting _swith radar signal bandwidth B, determine and optimize parameter α, requirement

Set primary iteration number of times k=1, vectorial initial value b to be optimized _k-1=0,0 represents null vector;

(4b2) utilize the result b of the k-1 time iterative _k-1calculate the required weighted value of iterative the k time

$W_{k-1}(\mathrm{\&omega;},p)=\frac{4\&CenterDot;W(\mathrm{\&omega;},p)}{{\left|A_{k-1}(e^{\mathrm{j\&omega;}},p)\right|}^2},$

Wherein w (ω, p) is in 0≤ω≤α π and-0.5≤p≤0.5 scope, the fixed weighting value of square error;

(4b3) the weighted value W being obtained by step (4b2) _k-1(ω, p), compute matrix Q _1k, vectorial g _1kwith numerical value d _1k;

(4b4) solving equations Q _1kb _k=g _1kobtain the result b of the k time iterative _k;

(4b5) end condition judgement: according to radar system parameter, determine error margin ε, if set up, stop iteration, obtain coefficient of polynomial fitting vector b _k; Otherwise k=k+1, jumps to step (4b2) and carries out next round iterative.