CN106154241B - Tough parallel factorial analysis algorithm under impulse noise environment - Google Patents
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
- G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
- G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
Abstract
The present invention relates to the tough parallel factorial analysis new algorithms under impulse noise environment, belong to computer application technology.The present invention improves the cost function based on TALS criterion in PARAFAC algorithm using MCC criterion, has derived suitable for the bistatic MIMO radar target component Combined estimator new algorithm under impulse noise environment.Algorithm can not only effectively inhibit the interference of impulse noise, have preferable estimated accuracy, and can be realized automatic matching.Emulation experiment shows under impulse noise and Gaussian noise environment, and compared with the PARAFAC algorithm based on TLAS criterion, MCC_PARAFAC algorithm all has good parameter Estimation performance, especially embodies better adaptability to the signal environment of mutation.
Description
Technical field
The present invention relates to the tough parallel factorial analysis new algorithms under impulse noise environment, belong to Computer Applied Technology neck
Domain.
Background technique
Bistatic MIMO (Multiple-Input Multiple-Output) radar is by MIMO technology and biradical land mine
A kind of new system radar combined up to technology[1-2].The wherein echo that bistatic relevant MIMO radar is received using receiving array
The coherence having between signal, and Signal separator, each battle array of emission array and receiving array are carried out by matched filter
First spacing is smaller and is centrally placed, and transmitting array element emits mutually orthogonal signal, while all transmitting receiving antennas are to phase
Same RCS value.The present invention mainly studies the Parameter Estimation Problem of bistatic relevant MIMO radar.
Target component estimation and positioning are an important contents of Radar Signal Processing.Document [3-5] have studied MUSIC,
The MIMO radars such as ESPRIT, dimensionality reduction Capon, propagation operator, the method based on fraction Fourier conversion and polynomial rooting ginseng
Number estimation method has preferable estimated accuracy, but is unable to the automatic matching of object of experiment parameter.Document [6-7] is based on
The three faces battle array model and ESPRIT method of PARAFAC estimates the transmitting-receiving angle of target and Doppler frequency, can be realized certainly
Dynamic pairing.But this two documents are all to carry out parameter Estimation under the premise of assuming that noise circumstance is white Gaussian noise.Especially
Ground, Zhang Jianyun are based on parallel factor analysis theory [6], and parameter is completed from the matrix that three linear least-squares iteration obtain and is estimated
Meter.However the algorithm only has preferable estimation performance under white Gaussian noise environment, and it is very sensitive to impulse noise, so that calculating
The performance of method decreases sharply under impulse noise environment.
However, theoretical research in recent years and actual measured results discovery, radar, sonar and wireless communication system are actually made an uproar
Contain a large amount of pulse repetitions in sound.In this case using Gaussian noise signal model be it is inappropriate, this noise like is more suitable
Alpha Stable distritation model is shared to describe [8-10].Since limited second moment is not present in Stable distritation noise,
Under impulse noise environment, the above-mentioned method for parameter estimation performance degradation based on second-order statistic even fails.
Summary of the invention
In recent years, measurement of the joint entropy as a kind of new stochastic variable local similarity, receives significant attention [11-
12].Principe etc. proves that joint entropy can induce a distance measure (CIM, Correntropy Induced Metric),
And propose maximal correlation entropy criterion (MCC, Maximum Correntropy Criterion) accordingly.It is quasi- different from traditional MSE
Then, MCC criterion embodies the adaptability to impulse noise circumstance.MCC criterion is applied under impulse noise environment by Principe
Channel blind equalization problem.Aimin Song solves the problems, such as the time delay estimadon [9] under Stable distritation noise using MCC criterion.Zhang Jin
MCC criterion is applied in projection approximation subspace tracking algorithm by phoenix.Emulation experiment shows above-mentioned algorithm to impulse noise circumstance
Adaptability[10]。
It is inspired by above-mentioned document, the present invention is using the target letter based on TALS criterion in MCC criterion modification PARAFAC algorithm
Number is allowed to be suitable for impulse noise environment, derives the PARAFAC algorithm (MCC_PARAFAC algorithm) based on MCC criterion, and will
The algorithm is applied in bistatic MIMO radar target component estimation, realizes the Combined estimator of target component, and can be automatic
Pairing.Emulation experiment shows that new algorithm proposed by the present invention is in impulse noise environment following table relative to TALS-PARAFAC algorithm
Reveal good robustness.
Process of the invention is described in detail below,
Signal model
Bistatic MIMO radar system structure used in the present invention is as shown in Figure 1.Within a transmitting pulse period, target
Scattering resonance state (RCS) remain unchanged, and the fluctuating between pulse and pulse is statistical iteration, and the RCS of different target
Fluctuation is incoherent.Transmitting and reception array element number are respectively M and N, and array element spacing is respectively dtAnd dr, in same distance point
It distinguishes on unit there are P target,Indicate radar emission angle corresponding to i-th of target and acceptance angle [6].Each transmitting battle array
Member emits mutually orthogonal phase-coded signal simultaneously, if first of pulse of m-th of array element transmitting is
sm.l(t)=sm(t '+lT), (1)
In formula, t and t ' respectively correspond slow time and fast time, and T indicates the pulse repetition period.smIt (t) is m-th of transmitting battle array
The baseband waveform of member.When then single goal is observed, n-th of received first of echo impulse of reception array element is
N=1 ... in formula, N, l=1 ..., L, τ are that the echo of target is delayed, wn,l(t) being is that standard S α S Stable distritation is made an uproar
Sound.ρliEmit scattering coefficient of the pulse in i-th of target for first.αni=2 π (n-1) drsinθi/ λ andIt is to receive steering vector and transmitting steering vector respectively.fdiFor Doppler's frequency of i-th of target
Rate.
Since the signal of each transmitting array element transmitting is mutually orthogonal, that is, meet:Wherein sq(t) and
sk(t) the transmitting signal of q-th and k-th transmitting array element is respectively indicated, * is conjugate operation.Emit the transmitting of array element using M
Signal carries out matched filtering to the received echo-signal of each reception array element respectively, and signal is separated, can be obtained in P mesh
In the case of mark, the filter output of the l times echo is
Wherein,B (θ)=[ar(θ1),…,ar
(θP)],cl(fd)=[ρl1 exp(j2πfd1Tl),…,ρlPexp(j2πfdPTl)], ⊙ is
Khatri-Rao product.
Available under P target conditions by formula (3), the filter output of L echo is
Wherein Y=[η1,η2,…,ηL] it is the output matrix that MN × L is tieed up.For P ×
The matrix vector of L dimension, it is the function (assuming that the scattering coefficient of target is known) of Doppler frequency.By formula (4) it is found that right
The estimation of the angle of departure of MIMO radar, acceptance angle and Doppler frequency can be converted into pairB (θ) and C (fd) 3 matrixes
Estimation.
Joint entropy
For two stochastic variables X and Y, joint entropy is defined as:
Vσ(X, Y)=E [kσ(X-Y)], (5)
Wherein,For kernel function, σ > 0 is the long parameter of core, and E [] is mathematic expectaion.Document[12]
It proves, a kind of degeneration that joint entropy can regard the Renyi quadratic entropy based on Parzen kernel estimates as indicates, and is able to reflect two
The similarity of a stochastic variable.In practical application, the joint probability density of stochastic variable X and Y are often unknown, can only be by limited
Observation dataEstimate the joint entropy of stochastic variable X and Y
It can be seen that joint entropy from the definition of joint entropy and contain gaussian kernel function, thus to significantly impulse
Non-Gaussian noise has good inhibiting effect.By document[12]It is found that Vσ(X, Y) has following two property:
Vσ(X, Y)=Vσ(Y, X), (7)
V when X=Y, in formula (8)σ(X, Y) obtains maximum value.
MCC-PARAFAC new algorithm
Parallel factor analysis (parallel factor, PARAFAC) is suggested first to be analyzed as data in physiology
Tool is mainly used for Chemical Measurement, spectroscopy and chromatography etc., is a kind of method of multidimensional data analysis.In recent years, believing
Number processing and the communications field, parallel factor technology is by extensive concern [13-15].Parallel transport is three faces battle array or multi-panel battle array
The general name of low-rank decomposition, it is, meet Kruskal under the conditions of parallel factor theoretical based on three linear decompositions that it, which handles three-dimensional data,
Model has unique identifiability, the matrix containing target component information can be obtained in a matrix decomposition, so that parameter
It being capable of automatic matching.
Consider matrixIt constitutes I × J × K and ties up three faces battle array X, then any one of its
Element can be decomposed into
A in formulai,f,bj,f,ck,fRespectively matrix A, the element of B, C, further, it is possible to obtain three matrixes, respectively IJ
The matrix X of × K1'=[A ⊙ B] CT, the matrix X of KJ × I2'=[B ⊙ C] AT, the matrix X of KI × J3'=[C ⊙ A] BT, respectively will
Noise matrix is added in three matrixes, then following expression formula can be obtained
X1=[A ⊙ B] CT+W1, (10)
X2=[B ⊙ C] AT+W2, (11)
X3=[C ⊙ A] BT+W3, (12)
Wherein ⊙ is Khatri-Rao product, W1, W2And W3It is noise.
Above three matrix alternately uses least square method to be iterated update, until algorithmic statement, step
It is as follows:
(1) optionally random matrix initializationWithIteration serial number k=1,2,3 ....
(2) willSubstitution formula (13), seeks its least square solution, obtains the kth time iterative estimate value of CSuch as formula
(114) shown in.
(3) willSubstitution formula (15) asks its least square solution, the kth time iterative estimate value of acquisitionSuch as formula
(16) shown in.
(4) willSubstitution formula (17), seeks its least square solution, obtains the kth time iterative estimate value of BSuch as formula
(18) it shown in, and calculatesIf | δk-δk-1| > ε, ε are error threshold, then repeatedly step (2)-
(4).If | δk-δk-1| < ε then goes to step (5)
(5) pass through above-mentioned iterative calculation, obtain the final estimated value of A, B and C With
It is well known that least-squares algorithm be based on second-order statistic, and impulsive noise be not present second moment, therefore
The method for parameter estimation performance degradation being iterated under impulse noise environment using least square method is even failed.
In order to improve the parameter Estimation performance of TALS-PARAFAC algorithm in impulse noise environment, the present invention is quasi- using MCC
Then the cost function (13) of the iteration in algorithm is improved
It, can be by maximization problem in order to solve formula (19)It is equivalent to minimization problem, then cost function is
Wherein
Similarly, the generation of maximal correlation entropy criterion proposed by the present invention is also used to the cost function in formula (15) and (17)
Valence function is replaced, so that the parallel factor analysis new algorithm based on MCC criterion under suitable impulse noise environment has been obtained,
Its step are as follows:
(1) optionally random matrix initializationWithIteration serial number k=1,2,3 ....
(2) willSubstitution formula (21) seeks maximal correlation entropy solution, obtains the kth time iterative estimate value of CSuch as formula
(22) shown in.
(3) willSubstitution formula (23) asks its least square solution, the kth time iterative estimate value of acquisitionSuch as formula
(24) shown in.
(4) willSubstitution formula (25), seeks its least square solution, obtains the kth time iterative estimate value of BSuch as formula
(26) it shown in, and calculatesWherein Xi..=BDi[A]CT+W1.If | δk-δk-1|>
ε, ε are error threshold, then repeatedly step (2)-(4).If | δk-δk-1| < ε then goes to step (5)
(5) pass through above-mentioned iterative calculation, obtain the final estimated value of A, B and CWith
Target component Combined estimator new method based on MCC-PARAFAC algorithm
The output of matched filterWith three faces battle array model characteristics, therefore it can be with Y along recipient
Set of slices Y in, the direction of the launch and snap direction1,Y2,Y3It indicates, wherein
In order to improve the parameter Estimation performance of TALS-PARAFAC algorithm in impulse noise environment, the present invention is quasi- using MCC
Then the cost function of the iteration in algorithm is improved, proposes the PARAFAC new algorithm based on MCC criterion, and by the calculation
Method is applied in bistatic MIMO radar target component estimation.
Specific steps are as follows:
(1) optionally random matrix initializationWithIteration serial number k=1,2,3 ....
(2) willSubstitution formula (28) seeks its maximal correlation entropy solution, and the kth time iteration for obtaining B (θ) is estimated
EvaluationAs shown in formula (29).
It, can be by maximization problem in order to solve formula (27)It is equivalent to minimization problem, then cost function
For
Wherein
(3) willSubstitution formula (30) asks its maximal correlation entropy solution, the kth time iterative estimate value of acquisitionAs shown in formula (31).
(4) willSubstitution formula (32) seeks its maximal correlation entropy solution, obtains C (fd) kth time iterative estimate valueAs shown in formula (33), and calculateWherein Dl[] indicates
The diagonal matrix formed by the l row element of matrix,If | δk-δk-1(ε is | > ε
Error threshold), then repeatedly step (2)-(4).If | δk-δk-1| < ε then goes to step (5)
(5) pass through above-mentioned iterative calculation, obtainB (θ) and C (fd) final estimated valueWith
And it enablesThe i-th column element of jth row of respectively 3 estimated matrix, by formula (30)-(32) to each
The method that column vector is averaging obtains(i=1 ..., P).Angle () indicates to take the phase angle operation of element.
Beneficial effects of the present invention:
The present invention improves the cost function based on TALS criterion in PARAFAC algorithm using MCC criterion, has derived and has been suitable for
Bistatic MIMO radar target component Combined estimator new algorithm under impulse noise environment.Algorithm can not only effectively inhibit impulse
The interference of noise has preferable estimated accuracy, and can be realized automatic matching.Emulation experiment show in impulse noise and
Under Gaussian noise environment, compared with the PARAFAC algorithm based on TLAS criterion, MCC_PARAFAC algorithm all has good ginseng
Number estimation performance, especially embodies better adaptability to the signal environment of mutation.
Detailed description of the invention
Fig. 1 is bistatic MIMO radar Array Model.
Fig. 2 (a) Doppler frequency parameter Estimation RMSE is with GSNR change curve；
Fig. 2 (b) DOD estimates RMSE with GSNR change curve；
Fig. 2 (c) DOA estimates RMSE with GSNR change curve；
Fig. 3 (a) Doppler frequency parameter Estimation RMSE is with noise characteristic index α change curve；
Fig. 3 (b) DOD estimates RMSE with noise characteristic index α change curve；
Fig. 3 (c) DOA estimates RMSE with noise characteristic index α change curve；
Fig. 4 is the relationship of MCC_PARAFAC algorithm performance Yu the long parameter σ of core；
The accuracy rate of Fig. 5 (a) Doppler-frequency estimation is with GSNR change curve；
The accuracy rate of Fig. 5 (b) DOD estimation is with GSNR change curve；
The accuracy rate of Fig. 5 (c) DOA estimation is with GSNR change curve；
The accuracy rate of Fig. 6 (a) Doppler frequency parameter Estimation is with noise characteristic index α change curve；
The accuracy rate of Fig. 6 (b) DOD estimation is with noise characteristic index α change curve；
The accuracy rate of Fig. 6 (c) DOA estimation is with noise characteristic index α change curve.
Specific embodiment
The present invention will be further described with reference to the accompanying drawing.
It is assumed that transmitting array element and reception array element number are respectively M=6 and N=8, there are 2 in bistatic MIMO radar far field
Target, i.e. P=2, the angle of departure and acceptance angle relative to transmitting array element and reception array element are respectively Doppler frequency parameter fd1=160Hz, fd2=100Hz, echo number L=100.Each transmitting array element
Emit mutually orthogonal Hadamard encoded signal, and the phase code number Q=256 in each repetition period.This section uses
Degree of the broad sense signal-to-noise ratio [16] (Generalized Signal-to-Noise Ratio, GSNR) as signal and impulse noise
Amount.The definition of broad sense signal-to-noise ratio is
In formula,Indicate that the power of signal, γ are the coefficients of dispersion of S α S distribution.Under the same conditions, with TALS-
PARAFAC algorithm[6]Middle method for parameter estimation is compared, and all simulation results are by 500 Monte-Carlo experiment systems
Meter obtains.
Experiment 1: in the experiment of this trifle, it is assumed that characteristic index α=1.4 of impulse noise, the model of broad sense signal-to-noise ratio GSNR
Enclosing is 0≤GSNR≤30.Fig. 2 (a)-(c) provides inventive algorithm and the root mean square of TALS-PARAFAC algorithm parameter estimation misses
Difference is with GSNR change curve.
By Fig. 2 (a) it can be found that as GSNR >=10dB, the parameter Estimation performance of inventive algorithm is substantially better than TALS-
PARAFAC algorithm.By the curve of Fig. 2 (b) it can be found that when GSNR >=12dB, inventive algorithm DOD parameter Estimation performance compares
Stablize, and the root-mean-square error of DOD parameter Estimation is significantly less than TALS-PARAFAC algorithm.Curve in Fig. 2 (c) is shown
Inventive algorithm has lower root-mean-square error compared to TALS-PARAFAC algorithm, about DOA estimation.
Therefore, by the emulation experiment of this trifle, it will be seen that the performance of MCC_PARAFAC algorithm is better than TALS_
PARAFAC algorithm.This is because second-order statistic is not present, so most based on second-order statistic under impulse noise environment
Small two multiplication algorithms performance degradation.And MCC_PARAFAC algorithm, maximal correlation entropy criterion is used as cost function, it can
The interference of good impulse noise mitigation, therefore there is preferable estimation performance.
Experiment 2: the relationship of parameter Estimation performance Yu impulse noise characteristic index α is had studied.Parameter setting is in this trifle,
The variation range of broad sense signal-to-noise ratio GSNR=15dB, the characteristic index α of impulse noise are 1≤α≤2.Fig. 3 (a)-(c) gives
The relationship of the RMSE and noise characteristic index α of the estimation of two kinds of algorithm parameters.
By Fig. 3 (a) it can be found that in 1≤α≤2, MCC-PARAFAC algorithm of the invention is to Doppler frequency estimation
RMSE be respectively less than TALS-PARAFAC algorithm, and as α >=1.3, it is more stable that inventive algorithm parameter Estimation obtains performance,
With more stable RMSE.The parameter Estimation performance of inventive algorithm is substantially better than it can be seen from the curve of Fig. 3 (b)
TALS-PARAFAC algorithm.As 1≤α≤1.3, inventive algorithm is reduced rapidly about the RMSE value of DOD parameter Estimation, and is become
In steady.Curve in Fig. 3 (c) shows inventive algorithm compared to TALS-PARAFAC algorithm, about DOA estimation have compared with
Low root-mean-square error.
Therefore, pass through the emulation experiment of this trifle, it can be appreciated that as α >=1.3 MCC_PARAFAC algorithm have compared with
Good performance.The impulse of the smaller noise of α is stronger, TALS_PARAFAC algorithm not to the inhibiting effect of impulse noise, so
When α is smaller, algorithm performance is poor, and as α=2, impulse noise is converted into Gaussian noise, so when α is close to 2, algorithm tool
There is estimation performance carefully.It can be seen that TALS_PARAFAC algorithm is more sensitive to impulse noise, it should in the environment of impulse noise
The parameter Estimation performance of algorithm is poor.Therefore, it can be seen from Fig. 2 and Fig. 3 under impulse noise environment MCC_PARAFAC
The parameter Estimation performance of algorithm is far superior to TALS_PARAFAC algorithm.
Experiment 3: this trifle has studied the pass of the root-mean-square error RMSE and the long σ of core of MCC_PARAFAC algorithm parameter estimation
System.In the experiment of this trifle, parameter is set as broad sense signal-to-noise ratio GSNR=15dB, characteristic index α=1.4 of impulse noise, core
The variation range of long parameter σ is 0.1≤σ≤2.From fig. 4, it can be seen that the performance of MCC_PARAFAC algorithm parameter estimation is by core
The influence of long parameter σ is little.
Experiment 4 has studied the accuracy rate of parameter Estimation and the relationship of broad sense signal-to-noise ratio GSNR and characteristic index α.Parameter Estimation
Accuracy rate PaIt may be defined asWherein D is true value,For estimated value.The P when multiple targetsa
The average value of accuracy rate is estimated for multiple target components, the present invention is the average value of two target accuracys rate.Fig. 5 (a)-(c) is aobvious
Show the accuracy rate of parameter Estimation with the change curve of GSNR.
Fig. 5 (a) shows the accuracy rate of Doppler frequency estimation and the relationship of broad sense signal-to-noise ratio, as can be seen from Fig. MCC-
The accuracy rate of PARAFAC algorithm is higher than TALS-PARAFAC algorithm.Fig. 5 (b) shows the accuracy rate of angle of departure DOD estimation with wide
The variation relation of adopted signal-to-noise ratio.Curve shows that inventive algorithm is calculated about the DOD accuracy rate estimated also above TALS-PARAFAC
Method.Fig. 5 (c) shows the accuracy rate of DOA estimation and the relationship of broad sense signal-to-noise ratio, as can be seen from Fig. MCC-PARAFAC algorithm
Accuracy rate be higher than TALS-PARAFAC algorithm, especially when broad sense signal-to-noise ratio be greater than 15dB when it is particularly evident.
Fig. 6 shows the accuracy rate of parameter Estimation with the change curve of noise characteristic index α.Fig. 6 (a) is it can be seen that when making an uproar
When acoustic signature index is less than 1.3, the pulse feature of noise is stronger, inventive algorithm about Doppler frequency estimation accuracy rate more
Significantly it is higher than TALS-PARAFAC algorithm, as noise characteristic index increases, the pulse feature of noise weakens, TALS-PARAFAC
The performance of algorithm increases, but still less than inventive algorithm.Fig. 6 (b) shows that the accuracy rate of DOD estimation refers to noise characteristic
Several relationships.As seen from the figure with the increase of noise characteristic index, the accuracy rate and TALS-PARAFAC of inventive algorithm
The gap of algorithm is also increasing, and the accuracy rate of this algorithm parameter estimation is apparently higher than TALS-PARAFAC algorithm.Equally, Fig. 6 (c)
Curve show DOA estimation accuracy rate and noise characteristic index relationship.It can be seen from this figure that as noise characteristic refers to
Several increases, the accuracy rate of inventive algorithm are higher and higher in TALS-PARAFAC algorithm.
Since MCC_PARAFAC algorithm considers the influence of impulse noise, using maximal correlation entropy criterion cost letter the most
Number is iterated.And TALS_PARAFAC algorithm is based on second moment, limited second moment is not present in impulse noise, therefore
TALS_PARAFAC algorithm performance under impulse noise environment can significantly degenerate.It can from Fig. 5 (a)-(c) and Fig. 6 (a)-(c)
MCC_PARAFAC algorithm ratio TALS_PARAFAC algorithm accuracy rate with higher out.
The present invention relates to bibliography
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For the ordinary skill in the art, specific embodiment is only exemplarily described the present invention,
Obviously the present invention specific implementation is not subject to the restrictions described above, as long as use the inventive concept and technical scheme of the present invention into
The improvement of capable various unsubstantialities, or not improved the conception and technical scheme of the invention are directly applied to other occasions
, it is within the scope of the present invention.
Claims (1)
1. the tough parallel factorial analysis algorithm under impulse noise environment, characterized by the following steps: (1) optionally with
Machine matrix initialisationWithIteration serial number k=1,2,3 ...,
(2) willSubstitution formula (28) seeks its maximal correlation entropy solution, obtains the kth time iterative estimate value of B (θ)As shown in formula (29),
In order to solve formula (27), by maximization problemIt is equivalent to minimization problem, then cost function is
WhereinIt is the long parameter of core, B (θ)=[ar(θ1),…,ar(θP)], Y1Indicate Y along receiving direction
Set of slices,
(3) willSubstitution formula (30) asks its maximal correlation entropy solution, the kth time iterative estimate value of acquisition
As shown in formula (31)
Wherein, Y2Indicate set of slices of the Y in the direction of the launch,
(4) willSubstitution formula (32) seeks its maximal correlation entropy solution, obtains C (fd) kth time iterative estimate valueAs shown in formula (33), and calculateWherein Dl[] indicates
The diagonal matrix formed by the l row element of matrix,If | δk-δk-1| > ε, ε
For error threshold, then repeatedly step (2)-(4), if | δk-δk-1| < ε then goes to step (5)
Wherein, Y3Indicate set of slices of the Y on snap direction,
(5) pass through above-mentioned iterative calculation, obtainB (θ) and C (fd) final estimated valueWithAnd it enables The i-th column element of jth row of respectively 3 estimated matrix, by formula (34)-(36) to each column vector
The method of averaging obtainsAngle () expression takes the phase angle operation of element,
Wherein, M indicates that transmitting array element number, N indicate to receive array element number, dtAnd drRespectively indicate transmitting array element spacing and reception
Array element spacing, T indicate the pulse repetition period.
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