CN104459667A - Sparse array DOA estimation method based on CLEAN - Google Patents

Abstract

The invention relates to a sparse array DOA estimation method based on CLEAN. The sparse array DOA estimation method based on CLEAN comprises the following steps that the number of targets is calculated according to the MDL rule; an amplitude square accumulation result of wave beams formed by all snapshot data is calculated; the calculation result is searched for the number of the wave beam corresponding to the maximum value; an accurate number of the wave beam is calculated through an equisignal angle measurement method; a target angle is calculated; whether all signals are detected or not is judged; detected target responses are removed through the CLEAN concept, and then iteration is conducted. By the adoption of the sparse array DOA estimation method based on CLEAN, the influence of action of a strong target and a high sidelobe on weak target angle estimation is well restrained, and a high angular accuracy is achieved; in addition, the consumed time is short, and time consumption is not sensitive to the number of array elements.

Description

A kind of thinned array direction of arrival DOA estimation method based on CLEAN

Technical field

The invention belongs to Radar Signal Processing Technology field, be specifically related to a kind of thinned array direction of arrival DOA estimation method based on CLEAN.

Background technology

In radar and communication system, array antenna application is extremely extensive.Up to the present, the antenna of uniform intervals distribution is one of simple and the most most widely used array antenna.But there is the possibility occurring graing lobe in the aerial array of uniform intervals distribution, for avoiding the appearance of graing lobe, usually requires that the spacing of aerial array is not more than the half of wavelength.Therefore, if go for higher angular resolution just need a lot of array element, this not only can increase the cost of antenna, and the mass data produced also can increase the burden of digital information processing system.

LS-SVM sparseness is carried out to equally distributed array, namely from uniform array, removes some array elements at random, such that array element is no longer regularly arranged obtains thinned array.Thinned array effectively can suppress the appearance of graing lobe, and can angular resolution be improved, but its peak side-lobe and average secondary lobe all higher, the interactional response of strong target and high secondary lobe likely can be greater than the response of weak signal target, occurs the possibility of decoy.

It is one of the focus in Array Signal Processing field that direction of arrival (DOA, Direction of Arrival) is estimated always, and accurate DOA estimates most important to subsequent treatment.The current research to DOA algorithm for estimating mainly concentrates on the outstanding algorithm of MUSIC and ESPRIT two kinds.

MUSIC (Multiple Signal Classification) algorithm has good resolving power, but this algorithm needs to carry out spectrum peak search in all possible angle, and operand is very big, and the interval of angle measurement accuracy and selected search angle is relevant.Root-MUSIC proposes on the basis of MUSIC, and it, without the need to carrying out spectrum peak search, directly obtains closed solutions.But, the strategy of Root-MUSIC algorithm asks equation root, and along with the increase of array number, the time of finding roots of complex functional equation can sharply increase, so can increase temporal burden at the occasion Root-MUSIC that array number is more.

Summary of the invention

The technical matters solved

In order to avoid the deficiencies in the prior art part, the present invention proposes a kind of thinned array direction of arrival DOA estimation method based on CLEAN, solve strong and weak target when coexisting, strong target and thinned array height secondary lobe act on response corresponding to the decoy of generation mutually and are greater than the problem that weak signal target responds.

Technical scheme

Based on a thinned array direction of arrival DOA estimation method of CLEAN, it is characterized in that step is as follows:

Step 1: utilize the target number in the thinned array signal of minimum description length MDL criterion calculating CLEAN;

During step 2:T>0, calculate the amplitude square accumulation result that all fast beat of data form wave beam otherwise end computation process; Wherein: y _sbe s fast beat of data x _sutilize the beam data that FFT is formed, y _s=FFT (x _s); L is fast umber of beats;

Step 3: search wave beam number corresponding to maximal value in amplitude square accumulation result P;

Step 4: employing waits signal angle-measuring method to calculate " accurate wave beam number " p _t,

$p_t=\frac{(p_1+p_2)}{2}+k\·\mathrm{sign}\·\min (\frac{|P\left(p_1\right)-P\left(p_2\right)|}{P\left(p_1\right)+P\left(p_2\right)|},0.5)$

Wherein, p ₁for the wave beam number that maximal value in P is corresponding, p ₂for the wave beam number of second largest value, k is the measuring angle by comparing amplitude slope that array is corresponding, and sign is sign bit;

$p_2=\left\{\begin{array}{cc}{p}_1-1,& P(p_1-1)>P(p_1+1)\\ p_1+1,& P(p_1-1)\≤P(p_1+1)\end{array}\right.$

Described $\mathrm{sign}=\left\{\begin{array}{cc}-1,& p_1<p_2\\ 1,& p_1\&GreaterEqual;p_2\end{array}\right.$

Step 5: calculate angle on target,

Wherein, λ is wavelength, and d is array element distance, and M is that FFT counts;

Step 6: application CLEAN thought removes the target response detected, T value is subtracted one simultaneously, goes to step 2 and carry out iteration.

Described thinned array random some array element of removing in uniform intervals array is formed, and uses array A _rrrepresent, wherein, the element that the array element of removal is corresponding is 0, and the element that the array element of reservation is corresponding is 1.

Described application CLEAN thought is removed the target response detected and is: the amplitude square that the amplitude square accumulation result P that all fast beat of data step 2 calculated forms wave beam is approximately a fast beat of data formation wave beam adds up, and first tries to achieve target T _mcorresponding amplitude

$A\left(T_m\right)=\frac{\mathrm{sqrt}\left(P_{m-1}\left(z_m\right)\right)}{\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{A}_{\mathrm{rr}}\left(n\right)e^{\mathrm{jknd}\sin {\mathrm{\&theta;}}_m}\&CenterDot;e^{-\frac{j2\mathrm{\&pi;}{z}_{m}n}{M}}}$

Wherein, P _m-1be the result of the m-1 time iteration, z _mfor P _m-1the wave beam number that middle maximal value is corresponding, θ _mfor step (f)

The target T calculated _mangle, N is array number, k=2 π/λ;

At P _m-1remove target T _mresponse

$P_m\left(q\right)=P_{m-1}\left(q\right)-{|\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}A\left(T_m\right)e^{\mathrm{jknd}\sin {\mathrm{\&theta;}}_m}{A}_{\mathrm{rr}}\left(n\right)\&CenterDot;e^{-\frac{j2\mathrm{\&pi;qn}}{M}}|}^2,q=0\&CenterDot;\&CenterDot;\&CenterDot;M-1.$

Beneficial effect

A kind of thinned array direction of arrival DOA estimation method based on CLEAN that the present invention proposes, comprises the following steps: (a) utilizes minimum description length (MDL) criterion to calculate target number; B () calculates the amplitude square accumulation result that all fast beat of data form wave beam; C () searches wave beam number corresponding to maximal value in result of calculation; D () utilization waits signal angle-measuring method to calculate " accurate wave beam number "; E () calculates angle on target; F () judges whether all signals detect all, if it is terminate, otherwise go to (g); G () application CLEAN thought removes the target response detected, go to (c) and carry out iteration.The method can be good at suppressing strong target and high secondary lobe exercising result on the impact of weak signal target angle estimation, reaches very high angle measurement accuracy, and meanwhile, the method is consuming time very short and consuming time insensitive to array number.

Because method of the present invention utilizes CLEAN thought can remove the impact of strong signal on Testing of Feeble Signals, and algorithm calculates fairly simple, therefore, it is possible to when strong and weak signals coexists, the direction of arrival of thinned array still can be estimated fast, accurately.

Accompanying drawing explanation

In institute's drawings attached, FAC represents the inventive method, and RM represents Root-MUSIC method,

Fig. 1 is the process flow diagram of the step that the inventive method is shown;

Fig. 2 is array number when being all 28, the directional diagram contrast schematic diagram of array element distance to be the Sparse Array (representing with SA in figure) of 0.5909 λ (λ is wavelength) and array element distance the be one dimension even linear array (representing with ULA in figure) of 0.5 λ.

Fig. 3 is this case method and the time comparison diagram of Root-MUSIC method relative to array number.

For this case method asks two targets, (SNR is 20dB to Fig. 4, and angle is 10 °; Another SNR is 40dB, and angle is 30 °) exemplary plot, wherein (a) is initial response diagram, and (b) is the response diagram of strong target, and (c) removes the response diagram that (b) obtain in (a).

Fig. 5 is this case method and the Root-MUSIC method comparison diagram relative to the Averaged Square Error of Multivariate root (RSME) of SNR, and number of targets is 2, and angle on target is respectively 10 ° and 30 °, and the SNR of two targets is equal, changes to 70dB from 10dB, is spaced apart 1dB.

Fig. 6 is when the SNR of two targets is identical (being 20dB), incident angle relatively time (one of them angle on target is fixed as 20 °, another angle on target changes to 26 ° from 20.15 °, be spaced apart 0.1 °), the schematic diagram of this case method and Root-MUSIC method performance comparison, wherein, a () is RMSE comparison diagram, b () is that angle contrast schemes (blue solid lines (angle-1) and dotted line (angle-2) represent the real angle of two targets, and red and green submeter represents the angle value that FAC and RM tries to achieve)).

Fig. 7 is that (target 1 is 20dB constantly when the SNR difference of two targets, target 2 is 25dB), incident angle relatively time (wherein target 1 is fixed as 20 °, target 2 changes to 26 ° from 20.15 °, be spaced apart 0.1 °), the schematic diagram of this case method and Root-MUSIC method performance comparison, wherein, a () is RMSE comparison diagram, b () is that angle contrast schemes (blue solid lines (angle-1) and dotted line (angle-2) represent the real angle of two targets, and red and green submeter represents the angle value that FAC and RM tries to achieve))

Embodiment

Now in conjunction with the embodiments, the invention will be further described for accompanying drawing:

Be described for one-dimensional array in content of the present invention.

For the one-dimensional linear array antenna that is made up of N number of array element, remove some array elements wherein at random and can obtain thinned array.For convenience of description, the one-dimension array A of N number of element is comprised with one _rrrepresent thinned array, wherein, the element that the array element of removal is corresponding is 0, and the element that the array element of reservation is corresponding is 1.Assuming that array element distance is d, have p incident target, direction is respectively θ _i(i=1 ... p), the noise that each array element receives is zero mean Gaussian white noise.Then receive signal can be expressed as:

X=(AS+N) A _rr(1) in formula, X is sampled data, and S is target vector, and N is noise vector, and A is steering vector matrix, is specially:

A=[a (θ ₁) a (θ ₂) ... a (θ _p)] in (2) formula, a (θ _i) be the steering vector of i-th signal, be specially:

$a\left({\mathrm{\&theta;}}_i\right)=[1,e^{\mathrm{jkd}\sin {\mathrm{\&theta;}}_i},\&CenterDot;\&CenterDot;\&CenterDot;,e^{\mathrm{jk}(N-1)d\sin {\mathrm{\&theta;}}_i}{]}^T---\left(3\right)$ In formula, k=2 π/λ, d are array element distance.

Refer now to Fig. 1 to describe according to thinned array Wave arrival direction estimating method of the present invention

First the inventive method needs the number judging target.Accurately determining DOA estimation of target number is very important, if target number is selected excessive, then there will be false target, if select too small, then can miss target.As shown in Figure 1, in step S101, in this algorithm, adopt minimum description length (MDL) criterion to calculate target number T.

When forming wave beam, summation being weighted to the sampled data of each array element, obtaining array and exporting (4)

Y=W ^hx=W ^h{ (AS+N) A _rr(4) wherein, W=[W ₁, W ₂... W _n] ^tfor the weights that each array element is corresponding.As W=a (θ _i) time, wave beam will point to θ _idirection.

Now,

$y=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}\{(\mathrm{AS}+N)\&CenterDot;A_{\mathrm{rr}}\}{e}^{-\mathrm{jknd}\sin {\mathrm{\&theta;}}_i}---\left(5\right)$

If quantize by formula (6),

$\sin \left({\mathrm{\&theta;}}_i\right)=\frac{\mathrm{\&lambda;q}}{\mathrm{Md}}---\left(6\right)$

Then formula (5) has just become the DFT expression formula of X (t), in reality, realizes through conventional FFT.So the sampled value that pair array obtains carries out M point FFT, just can obtain M received beam data, the relation of each wave beam number and corresponding beam position angle is such as formula shown in (6).

In step s 102, judge whether T is greater than zero, if be greater than zero, carry out step S103, otherwise terminate computation process

In step s 103, suppose that s fast beat of data is x _s, then Wave beam forming result is: y _s=FFT (x _s).Be the data of L for fast umber of beats, the accumulation result that all fast beat of data obtained forms the amplitude square of wave beam is:

$P=\underset{s=1}{\overset{L}{\mathrm{\&Sigma;}}}{\left|y_s\right|}^2---\left(7\right)$

Difference beam between two wave beams formed by FFT is approximated to direct ratio with the ratio of wave beam and the deviation of target equivalent signals axle, similar, first to ask square the amplitude of two wave beams, and then wave beam Σ He er bu tong Δ of suing for peace, Δ/Σ such as still to depart from signal shaft deviation to target is directly proportional.Therefore, after utilizing formula (7) to try to achieve the accumulation result of each wave beam, in P, the difference beam of adjacent two wave beams is linear relationship with the signal shaft such as departing from the ratio of wave beam and target.

In step S104, find out the wave beam p that maximal value in P is corresponding ₁, and the wave beam p of second largest value corresponding to this target ₂.

$p_2=\left\{\begin{array}{cc}{p}_1-1,& P(p_1-1)>P(p_1+1)\\ p_1+1,& P(p_1-1)\&le;P(p_1+1)\end{array}\right.---\left(8\right)$

In step S105, formula (9) is utilized to obtain " accurate wave beam number " p corresponding to this target _t.

$p_t=\frac{(p_1+p_2)}{2}+k\&CenterDot;\mathrm{sign}\&CenterDot;\min (\frac{|P\left(p_1\right)-P\left(p_2\right)|}{P\left(p_1\right)+P\left(p_2\right)|},0.5)---\left(9\right)$

Wherein, k is the measuring angle by comparing amplitude slope that array is corresponding, relevant to concrete array, utilizes formula (5), tries to achieve multi-beam response y, then gets P=|y| ², by p ₁and p ₂elect 0 as ... the value of arbitrary neighborhood in M-1, then by p _tbe taken as p ₁and p ₂between value, utilize formula (9) to obtain k.Sign is sign bit, and value is as follows:

$\mathrm{sign}=\left\{\begin{array}{cc}-1,& p_1<p_2\\ 1,& p_1\&GreaterEqual;p_2\end{array}\right.---\left(10\right)$

In step s 106, formula (6) is utilized angle corresponding to this target to be:

$\mathrm{\&theta;}=\arcsin \left(\frac{p_t\mathrm{\&lambda;}}{\mathrm{Md}}\right)---\left(11\right)$

Because the secondary lobe of thinned array is higher, the value corresponding in P of the decoy of generation is probably greater than real weak signal target value corresponding in P, thus maximum point in P likely corresponding be decoy.Therefore, when MDL has tried to achieve T target, in P, got a maximum T maximum point thought that target is incorrect.

In step s 107, in order to suppress the impact of high secondary lobe, the thought of application CLEAN, after often obtaining a target, removes the response of this target in result of calculation.Accurate way should be remove at each snap the impact that this target brings, but algorithm complex can be very large like this, and the occasion high at requirement of real-time does not reach system requirements.By emulation experiment, we find, the result of calculation of all snaps (i.e. P) are regarded as the result of calculation of a snap, and only to do the result that a clear operation obtains very close for the work result that obtains of a clear operation and each snap.Hypothetical target T _mat P _m-1in (result of the m-1 time iteration), corresponding maximum wave beam number is z _m, the angle of being tried to achieve by formula (11) is θ _m, then the amplitude of this target is:

$A\left(T_m\right)=\frac{\mathrm{sqrt}\left(P_{m-1}\left(z_m\right)\right)}{\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{A}_{\mathrm{rr}}\left(n\right)e^{\mathrm{jknd}\sin {\mathrm{\&theta;}}_m}\&CenterDot;e^{-\frac{j2\mathrm{\&pi;}{z}_{m}n}{M}}}---\left(12\right)$

At P _m-1remove target T _mresponse after, obtain:

$P_m\left(q\right)=P_{m-1}\left(q\right)-{|\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}A\left(T_m\right)e^{\mathrm{jknd}\sin {\mathrm{\&theta;}}_m}{A}_{\mathrm{rr}}\left(n\right)\&CenterDot;e^{-\frac{j2\mathrm{\&pi;qn}}{M}}|}^2,q=0\&CenterDot;\&CenterDot;\&CenterDot;M-1---\left(13\right)$

After execution of step S107, go to step S102 and carry out iteration.

For the validity of the inventive method is described, the inventive method and Root-MUSIC method are carried out emulation experiment contrast below.In subsequent content of the present invention, FAC represents the inventive method, and RM represents Root-MUSIC method.

Without loss of generality, we have selected thinned array A _rr=[1 10001100000100010 10 011011000111100100010001011001110110 00 11 1], array element distance is 0.5909 λ, and all number of times (MC) relating to Monte Carlo experiment are all 200 times.In experiment, institute's plus noise is additive white Gaussian noise, and fast umber of beats is 64, and counting of all FFT is 128 points.Except testing one, other experiments all use two targets carry out testing and added Taylor's window.Control methods is Averaged Square Error of Multivariate (RMSE) mainly, and computing formula corresponding to 2 targets is:

$\mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{MC}}\left(\underset{i=1}{\overset{\mathrm{MC}}{\mathrm{\&Sigma;}}}\frac{{({\widehat{\mathrm{\&theta;}}}_1-{\mathrm{\&theta;}}_1)}^2+{({\widehat{\mathrm{\&theta;}}}_2-{\mathrm{\&theta;}}_2)}^2}{2}\right)}---\left(14\right)$

Experiment one: the effective array number (A of thinned array adopted in this experiment _rrin be 1 number) be 28, be that the directional diagram of 28 even linear arrays of 0.5 λ contrasts by the directional diagram of this Sparse Array and array element distance, result is as shown in Figure 2.Can find out, in identical (effectively) array element situation, Sparse Array (red in corresponding diagram) can obtain narrower main lobe, but average secondary lobe and peak side-lobe are all higher.

Experiment two: for contrasting the time complexity of FAC and RM, choose the one dimension even linear array that array number is 6 to 60 (being spaced apart 1), the incident direction of two targets is respectively 20 ° and 40 °, and SNR is 20dB.Experimental result as shown in Figure 3.As seen from Figure 3 when array number is less than 20, RM is consuming time less, and when array number is more than or equal to 20, FAC is consuming time less; Can find out, when array number increases, the time of RM increases fast simultaneously, but the time of FAC increases very slow.Can illustrate that the time of FAC method is insensitive to array number, consuming time always little.

Experiment three: for verifying that the response when strong target secondary lobe is greater than the performance of FAC when weak signal target responds, in this experiment, the SNR of two targets elects 20dB and 40dB as respectively, and incident angle is respectively 10 ° and 30 °.The response diagram utilizing formula (7) to calculate is shown in Fig. 4 (a), can find out maximum of points corresponding be the target of 30 °, and response corresponding to 10 ° of targets is not an extreme point, because the side lobe response that 30 ° of targets are brought has exceeded response corresponding to 10 ° of targets.Response corresponding to 30 ° of targets (also i.e. formula (13) in Section 2) is as shown in Fig. 4 (b), in Fig. 4 (a), remove Fig. 4 (b) obtain Fig. 4 (c), can find out that response corresponding to 30 ° of targets is removed well, peak value corresponding to 10 ° of targets has manifested.The result that FAC tries to achieve is: 10.031 ° and 29.999 °, and the result that RM calculates is: 10.044 ° and 29.946 °.By this description of test, the situation that FAC is greater than weak signal target response at strong target side lobe response still can estimate the incident angle of all targets accurately.

Test four: two angle on targets and elect 10 ° and 30 ° as, SNR is equal, is all to change to 70dB from 10dB, is spaced apart 1dB, and experimental result as shown in Figure 5.As seen from Figure 5, when two target incident angle differences are comparatively large, and when SNR is identical, the precision of FAC and RM is all very high, and FAC is better than RM, and FAC and RM is insensitive to SNR change.

Experiment five: for checking when two target SNR identical, angle close to time, the performance of algorithm, the SNR of two targets elects 20dB as, and the incident angle of a target is fixed as 20 °, and the incident angle of another target changes to 26 ° from 20.15 °, be spaced apart 0.1 °, experimental result as shown in Figure 6.As can be seen from Fig. 6 (a), when SNR is identical, two target incident angles very close to time, the effect of RM and FAC is all poor, angle very close to time FAC effect better, along with the increase of angle, become RM effect gradually better, when angle difference is larger, the two effect is all fine.

In Fig. 6 (b), blue solid lines (angle-1) and dotted line (angle-2) represent the real angle of two targets, the angle that red and green FAC and RM of expression respectively tries to achieve.Can find out, when two angle on targets are close, FAC and RM is not very accurate.When two angle on target differences are larger, the angle that FAC and RM tries to achieve is all very accurate.

Experiment six: for checking when two target SNR are different, angle close to time, the performance of algorithm, the SNR of first aim is taken as 20dB, the SNR of second target is taken as 25dB, and the incident angle of first aim is fixed as 20 °, and the incident angle of second target changes to 26 ° from 20.15 °, be spaced apart 0.1 °, experimental result as shown in Figure 7.As can be seen from Fig. 7 (a), when the SNR of two targets is different, in most of the cases the result of FAC is better than the result of RM.As can be seen from Fig. 7 (b), when the SNR of two targets is different, two angle on targets relatively time, in most of the cases, FAC method almost can estimate that higher angle of SNR accurately, and that the value of another angle and RM estimate is basically identical; And two angles that RM estimates have comparatively big error.

Consider all emulation experiments, can illustrate, the inventive method is consuming time little and consuming time insensitive to array number, and the precision estimated angle on target is very high, can be good at suppressing the strong target of Sparse Array and high secondary lobe exercising result on the impact of weak signal target angle estimation.

Claims (3)

1., based on a thinned array direction of arrival DOA estimation method of CLEAN, it is characterized in that step is as follows:

Step 1: utilize the target number in the thinned array signal of minimum description length MDL criterion calculating CLEAN;

During step 2:T>0, calculate the amplitude square accumulation result that all fast beat of data form wave beam otherwise end computation process; Wherein: y _sbe s fast beat of data x _sutilize the beam data that FFT is formed, y _s=FFT (x _s); L is fast umber of beats;

Step 3: search wave beam number corresponding to maximal value in amplitude square accumulation result P;

Step 4: employing waits signal angle-measuring method to calculate " accurate wave beam number " p _t,

$p_t=\frac{(p_1+p_2)}{2}+k\&CenterDot;\mathrm{sign}\&CenterDot;\min (\frac{|P\left(p_1\right)-P\left(p_2\right)|}{|P\left(p_1\right)+P\left(p_2\right)|},0.5)$

Wherein, p ₁for the wave beam number that maximal value in P is corresponding, p ₂for the wave beam number of second largest value, k is the measuring angle by comparing amplitude slope that array is corresponding, and sign is sign bit;

$p_2=\left\{\begin{array}{cc}{p}_1-1,& P(p_1-1)>P(p_1+1)\\ p_1+1,& P(p_1-1)\&le;P(p_1+1)\end{array}\right.$

Described $\mathrm{sign}=\left\{\begin{array}{cc}-1,& p_1<p_2\\ 1,& p_1\&GreaterEqual;p_2\end{array}\right.;$

Step 5: calculate angle on target,

Wherein, λ is wavelength, and d is array element distance, and M is that FFT counts;

Step 6: application CLEAN thought removes the target response detected, T value is subtracted one simultaneously, goes to step 2 and carry out iteration.

2. according to claim 1 based on the thinned array direction of arrival DOA estimation method of CLEAN, it is characterized in that: described thinned array random some array element of removing in uniform intervals array is formed, and uses array A _rrrepresent, wherein, the element that the array element of removal is corresponding is 0, and the element that the array element of reservation is corresponding is 1.

3. according to claim 1 based on the thinned array direction of arrival DOA estimation method of CLEAN, it is characterized in that: described application CLEAN thought is removed the target response detected and is: the amplitude square that the amplitude square accumulation result P that all fast beat of data step 2 calculated forms wave beam is approximately a fast beat of data formation wave beam adds up, and first tries to achieve target T _mcorresponding amplitude

$A\left(T_m\right)=\frac{\mathrm{sqrt}\left(P_{m-1}\left(z_m\right)\right)}{\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{A}_{\mathrm{rr}}\left(n\right)e^{\mathrm{jknd}\sin {\mathrm{\&theta;}}_m}\&CenterDot;e^{-\frac{j2\mathrm{\&pi;}{z}_{m}n}{M}}}$

Wherein, P _m-1be the result of the m-1 time iteration, z _mfor P _m-1the wave beam number that middle maximal value is corresponding, θ _mfor the target T that step (f) calculates _mangle, N is array number, k=2 π/λ;

At P _m-1remove target T _mresponse

$P_m\left(q\right)=P_{m-1}\left(q\right)-{|\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}A\left(T_m\right)e^{\mathrm{jknd}\sin {\mathrm{\&theta;}}_m}{A}_{\mathrm{rr}}\left(n\right)\&CenterDot;e^{-\frac{j2\mathrm{\&pi;qn}}{M}}|}^2,q=0\&CenterDot;\&CenterDot;\&CenterDot;M-1.$