CN106154241A - Tough parallel factorial analysis new algorithm under impulse noise environment - Google Patents

Abstract

The present invention relates to the tough parallel factorial analysis new algorithm under impulse noise environment, belong to Computer Applied Technology field.The present invention uses MCC criterion to improve cost function based on TALS criterion in PARAFAC algorithm, the bistatic MIMO radar target component Combined estimator new algorithm being applicable under impulse noise environment of having derived.Algorithm can not only effectively suppress the interference of impulse noise, has preferable estimated accuracy, and is capable of automatic matching.Emulation experiment shows, under impulse noise and Gaussian noise environment, compared with PARAFAC algorithm based on TLAS criterion, MCC_PARAFAC algorithm is respectively provided with good parameter estimation performance, especially the signal environment of sudden change is embodied more preferable adaptability.

Description

Tough parallel factorial analysis new algorithm under impulse noise environment

Technical field

The present invention relates to the tough parallel factorial analysis new algorithm under impulse noise environment, belong to Computer Applied Technology neck Territory.

Background technology

Bistatic MIMO (Multiple-Input Multiple-Output) radar is by MIMO technology and double-basis Rhizoma Anemones flaccidae A kind of new system radar that the technology of reaching combines^[1-2].The echo that the most bistatic relevant MIMO radar utilizes receiving array to receive The coherence having between signal, and carry out Signal separator by matched filtering device, each battle array of emission array and receiving array Unit's spacing is less and concentrates placement, launches array element and launches mutually orthogonal signal, and the most all of transmitting reception antenna is to having phase Same RCS value.The present invention mainly studies the Parameter Estimation Problem of bistatic relevant MIMO radar.

Target component estimation and location are important contents of Radar Signal Processing.Document [3-5] have studied MUSIC, The MIMO radar ginsengs such as ESPRIT, dimensionality reduction Capon, propagation operator, method based on fraction Fourier conversion and polynomial rooting Number estimation method, has a preferable estimated accuracy, but can not the automatic matching of object of experiment parameter.Document [6-7] based on Transmitting-receiving angle and the Doppler frequency of target are estimated by three battle array models and the ESPRIT method of PARAFAC, it is possible to realize certainly Dynamic pairing.But this two documents is all assuming that noise circumstance carries out parameter estimation on the premise of being white Gaussian noise.Especially Ground, Zhang Jianyun is theoretical [6] based on parallel factor analysis, completes parameter and estimate from the matrix that three linear least-squares iteration obtain Meter.But this algorithm only has under white Gaussian noise environment and preferably estimates performance, very sensitive to impulse noise to calculate The performance of method decreases sharply under impulse noise environment.

But, theoretical research in recent years and actual measured results find, the actual of radar, sonar and wireless communication system is made an uproar Containing a large amount of pulse repetition in sound.The signal model using Gaussian noise in this case is inappropriate, and this noise like is suitableeer Share Alpha Stable distritation model and describe [8-10].Owing to Stable distritation noise does not exist limited second moment, therefore, exist Under impulse noise environment, above-mentioned method for parameter estimation performance degradation based on second-order statistic even lost efficacy.

Summary of the invention

In recent years, joint entropy, as the tolerance of a kind of new stochastic variable local similarity, receives significant attention [11- 12].Principe etc. prove that joint entropy can induce a distance measure (CIM, Correntropy Induced Metric), And maximal correlation entropy criterion (MCC, Maximum Correntropy Criterion) is proposed accordingly.It is different from traditional MSE accurate 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 utilizes MCC criterion to solve the time delay estimadon problem [9] under Stable distritation noise.Zhang Jin MCC criterion is applied in projection approximation subspace tracking algorithm by phoenix.Emulation experiment shows that above-mentioned algorithm is to impulse noise circumstance Adaptability^[10]。

Being inspired by above-mentioned document, the present invention uses target letter based on TALS criterion in MCC criterion modification PARAFAC algorithm Number is allowed to be applicable to impulse noise environment, derives PARAFAC algorithm (MCC_PARAFAC algorithm) based on MCC criterion, and will This algorithm is applied in the estimation of bistatic MIMO radar target component, it is achieved that the Combined estimator of target component, and can be automatic Pairing.Emulation experiment shows, relative to TALS-PARAFAC algorithm, the new algorithm that the present invention proposes is in impulse noise environment following table Reveal good robustness.

The process of the present invention is described in detail below,

Signal model

Bistatic MIMO radar system structure used by the present invention is as shown in Figure 1.Launch in the pulse period at one, target Scattering resonance state (RCS) keep constant, and pulse and interpulse fluctuating are statistical iteration, and the RCS of different target Fluctuation is incoherent.Launching and receive array element number and be respectively M and N, array element distance is respectively d_tAnd d_r, divide in same distance Distinguish and on unit, there is P target,Represent the radar emission angle corresponding to i-th target and acceptance angle [6].Respectively launch battle array Unit launches mutually orthogonal phase-coded signal simultaneously, if the l pulse that m-th array element is launched is

s_m.l(t)=s_m(t '+lT), (1)

In formula, t and t ' the most corresponding slow time and fast time, T represents the pulse repetition period.s_mT () is that m-th launches battle array The baseband waveform of unit.Then during single goal observation, the n-th the l echo impulse receiving array element reception is

$y_{n,l}(t-\mathrm{\τ})=\underset{m=1}{\overset{M}{\Σ}}{\mathrm{\ρ}}_{li}{s}_{m,l}(t-\mathrm{\τ})\exp ({\mathrm{j\α}}_{ni}+{\mathrm{j\β}}_{mi})\exp \left(j2{\mathrm{\πf}}_{di}\right(t-\mathrm{\τ}\left)\right)+w_{n,l}\left(i\right),i=1,...,P,---\left(2\right)$

N=1 in formula ..., N, l=1 ..., L, τ are the echo time delay of target, w_n,lT () is for being that standard S α S Stable distritation is made an uproar Sound.ρ_liIt is the l and launches pulse scattering coefficient in i-th target.α_ni=2 π (n-1) d_rsinθ_i/ λ andIt is to receive steering vector and launch steering vector respectively.f_diDoppler's frequency for i-th target Rate.

The signal launched due to each transmitting array element is mutually orthogonal, the most satisfied:Wherein s_q(t) and s_kT () represents that q-th and kth launch the transmitting signal of array element respectively, * is conjugate operation.Utilize M the transmitting launching array element The echo-signal that each reception array element is received by signal respectively carries out matched filtering, is separated by signal, can get at P mesh In the case of mark, the wave filter of the l time echo is output as

Wherein,B (θ)=[a_r(θ₁),…,a_r (θ_P)],c_l(f_d)=[ρ_l1exp(j2πf_d1Tl),…,ρ_lPexp(j2πf_dPTl)], ⊙ is Khatri-Rao amasss.

Can be obtained under P target conditions by formula (3), the wave filter of L echo is output as

Wherein Y=[η₁,η₂,…,η_L] be MN × L dimension output matrix.For P × The matrix vector of L dimension, it is the function scattering coefficient of target (assume be known) of Doppler frequency.From formula (4), right It is right that the estimation of the angle of departure of MIMO radar, acceptance angle and Doppler frequency can be converted intoB (θ) and C (f_d) the estimating of 3 matrixes Meter.

Joint entropy

For two stochastic variable X and Y, its joint entropy is defined as:

V_σ(X, Y)=E [k_σ(X-Y)], (5)

Wherein,For kernel function, σ > 0 it is the long parameter of core, E [] is mathematic expectaion.Document^[12] Proving, joint entropy can be regarded a kind of degeneration of Renyi quadratic entropy based on Parzen kernel estimates as and represent, can reflect again two The similarity of individual stochastic variable.In actual application, the joint probability density of stochastic variable X with Y is the most unknown, can only be by limited Observed dataEstimate the joint entropy of stochastic variable X Yu Y

$V_{\mathrm{\σ}}(X,Y)=\frac{1}{N}\underset{i=1}{\overset{N}{\Σ}}{\mathrm{\κ}}_{\mathrm{\σ}}(x_i-y_i),---\left(6\right)$

From the definition of joint entropy it can be seen that joint entropy contains gaussian kernel function, thus to having significantly impulse Non-Gaussian noise has good inhibiting effect.By document^[12]Understand, V_σ(X, Y) has following two character:

V_σ(X, Y)=V_σ(Y, X), (7)

$0\≤V_{\mathrm{\σ}}(X,Y)\≤\frac{1}{\sqrt{2\mathrm{\π}}\mathrm{\σ}},---\left(8\right)$

During and if only if X=Y, the V in formula (8)_σ(X, Y) obtains maximum.

MCC-PARAFAC new algorithm

It is as data analysis in physiology that parallel factor analysis (parallel factor, PARAFAC) is first suggested Instrument, is mainly used in Chemical Measurement, spectroscopy and chromatography etc., is a kind of method of multidimensional data analysis.In recent years, at letter Number process and the communications field, parallel factor technology is by extensive concern [13-15].Parallel transport is three battle arrays or multiaspect battle array The general name of low-rank decomposition, it processes three-dimensional data is theoretical based on three linear decomposition, parallel factor under the conditions of meeting Kruskal Model has unique identifiability, can obtain the matrix containing target component information so that parameter in a matrix decomposition Can automatic matching.

Consider matrixConstitute I × J × K and tie up three battle arrays X, then its any one unit Element can be decomposed into

$\begin{array}{c}{x}_{i,j,k}=\underset{f=1}{\overset{F}{\Σ}}{a}_{i,f}{b}_{j,f}{c}_{k,f}\\ i=1,...,I,j=1,...J,k=1,...,K\end{array},---\left(9\right)$

A in formula_i,f,b_j,f,c_k,fBeing respectively matrix A, the element of B, C, further, it is possible to obtain three matrixes, respectively IJ The matrix X of × K₁'=[A ⊙ B] C^T, the matrix X of KJ × I₂'=[B ⊙ C] A^T, the matrix X of KI × J₃'=[C ⊙ A] B^T, respectively will Three matrixes add noise matrix, then can get table below and reach formula

X₁=[A ⊙ B] C^T+W₁, (10)

X₂=[B ⊙ C] A^T+W₂, (11)

X₃=[C ⊙ A] B^T+W₃, (12)

Wherein ⊙ is that Khatri-Rao amasss, W₁, W₂And W₃It is noise.

Above three matrix alternately uses least square method to be iterated updating, until algorithmic statement, its step As follows:

(1) optionally random matrix initializesWithIteration serial number k=1,2,3 ....

(2) willSubstitution formula (13), seeks its least square solution, it is thus achieved that the kth of C time iterative estimate valueSuch as formula (114) shown in.

(3) willSubstitution formula (15), seeks its least square solution, it is thus achieved that kth time iterative estimate valueSuch as formula (16) Shown in.

(4) willSubstitution formula (17), seeks its least square solution, it is thus achieved that the kth of B time iterative estimate valueSuch as formula (18) Shown in, and calculateIf | δ_k-δ_k-₁| > ε, ε be error threshold, then repeat step (2)-(4). If | δ_k-δ_k-1| < ε then goes to step (5)

(5) through above-mentioned iterative computation, the final estimated value of A, B and C is obtainedWith

It is known that least-squares algorithm is based on second-order statistic, and there is not second moment in impulsive noise, therefore exists The method for parameter estimation performance degradation using method of least square to be iterated under impulse noise environment even lost efficacy.

In order to improve the parameter estimation performance of TALS-PARAFAC algorithm in impulse noise environment, the present invention uses MCC accurate Then the cost function (13) of the iteration in algorithm is improved

In order to solve formula (19), can be by maximization problemIt is equivalent to minimization problem, then cost function is

Wherein

In like manner, the cost function in formula (15) and (17) is also adopted by the generation of the maximal correlation entropy criterion that the present invention proposes Valency function is replaced, thus has obtained the parallel factor analysis new algorithm based on MCC criterion under applicable impulse noise environment, Its step is as follows:

(1) optionally random matrix initializesWithIteration serial number k=1,2,3 ....

(2) willSubstitution formula (21), seeks maximal correlation entropy solution, it is thus achieved that the kth of C time iterative estimate valueSuch as formula (22) shown in.

(3) willSubstitution formula (23), seeks its least square solution, it is thus achieved that kth time iterative estimate valueSuch as formula (24) shown in.

(4) willSubstitution formula (25), seeks its least square solution, it is thus achieved that the kth of B time iterative estimate valueSuch as formula (26) Shown in, and calculateWherein X_i..=BD_i[A]C^T+W₁.If | δ_k-δ_k-1| > ε, ε be Error threshold, then repeat step (2)-(4).If | δ_k-δ_k-1| < ε then goes to step (5)

(5) through above-mentioned iterative computation, the final estimated value of A, B and C is obtainedWith

Target component Combined estimator new method based on MCC-PARAFAC algorithm

The output of matched filtering deviceHaving three battle array model characteristics, therefore it can be with Y along recipient To, set of slices Y launched on direction and snap direction₁,Y₂,Y₃Represent, wherein

In order to improve the parameter estimation performance of TALS-PARAFAC algorithm in impulse noise environment, the present invention uses MCC accurate Then the cost function of the iteration in algorithm is improved, it is proposed that PARAFAC new algorithm based on MCC criterion, and by this calculation Method is applied in the estimation of bistatic MIMO radar target component.

Concrete step is as follows:

(1) optionally random matrix initializesWithIteration serial number k=1,2,3 ....

(2) willSubstitution formula (28), seeks its maximal correlation entropy solution, it is thus achieved that the kth of B (θ) time iteration is estimated EvaluationAs shown in formula (29).

In order to solve formula (27), can be by maximization problemIt is equivalent to minimization problem, then cost function For

Wherein

(3) willSubstitution formula (30), seeks its maximal correlation entropy solution, it is thus achieved that kth time iterative estimate valueAs shown in formula (31).

(4) willSubstitution formula (32), seeks its maximal correlation entropy solution, it is thus achieved that C (f_d) kth time iterative estimate valueAs shown in formula (33), and calculateWherein D_l[] represent by The diagonal matrix that the l row element of matrix is formed,If | δ_k-δ_k-1| > (ε is for by mistake for ε Difference thresholding), then repeat step (2)-(4).If | δ_k-δ_k-1| < ε then goes to step (5)

(5) through above-mentioned iterative computation, obtainB (θ) and C (f_d) final estimated valueWith And makeIt is respectively jth row i-th column element of 3 estimated matrix, by formula (30)-(32) to each row The method that vector is averaging obtainsAngle () represents the phase angle computing taking element.

Beneficial effects of the present invention:

The present invention uses MCC criterion to improve cost function based on TALS criterion in PARAFAC algorithm, has derived and has been applicable to Bistatic MIMO radar target component Combined estimator new algorithm under impulse noise environment.Algorithm can not only effectively suppress impulse The interference of noise, has preferable estimated accuracy, and is capable of automatic matching.Emulation experiment shows, at impulse noise and Under Gaussian noise environment, compared with PARAFAC algorithm based on TLAS criterion, MCC_PARAFAC algorithm is respectively provided with well ginseng Number estimates performance, especially the signal environment of sudden change is embodied more preferable adaptability.

Accompanying drawing explanation

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 that RMSE is with GSNR change curve；

Fig. 2 (c) DOA estimates that RMSE is 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 that RMSE is with noise characteristic index α change curve；

Fig. 3 (c) DOA estimates that RMSE is with noise characteristic index α change curve；

Fig. 4 is the relation of MCC_PARAFAC algorithm performance and long parameter σ of core；

The accuracy rate of Fig. 5 (a) Doppler-frequency estimation is with GSNR change curve；

The accuracy rate that Fig. 5 (b) DOD estimates is with GSNR change curve；

The accuracy rate that Fig. 5 (c) DOA estimates 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 that Fig. 6 (b) DOD estimates is with noise characteristic index α change curve；

The accuracy rate that Fig. 6 (c) DOA estimates is with noise characteristic index α change curve.

Detailed description of the invention

The present invention will be further described below in conjunction with the accompanying drawings.

Being respectively M=6 and N=8 assuming that launch array element with receiving array element number, bistatic MIMO radar far field exists 2 Target, i.e. P=2, be respectively relative to the angle of departure launching array element and receive array element and acceptance angle Doppler frequency parameter f_d1=160Hz, f_d2=100Hz, echo number L=100.Respectively launch array element Launch phase code number Q=256 in mutually orthogonal Hadamard coding signal, and each repetition period.This section uses Broad sense signal to noise ratio [16] (Generalized Signal-to-Noise Ratio, GSNR) is as signal and the degree of impulse noise Amount.The definition of broad sense signal to noise ratio is

$GSNR=10lg({\mathrm{\&sigma;}}_s^2/\mathrm{\&gamma;}),$

In formula,Representing the power of signal, γ is the coefficient of dispersion of S α S distribution.At identical conditions, with TALS- PARAFAC algorithm^[6]Middle method for parameter estimation is contrasted, and all simulation results are by 500 Monte-Carlo experiment systems Meter obtains.

Experiment 1: in this trifle is tested, 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 the root-mean-square of inventive algorithm and the estimation of TALS-PARAFAC algorithm parameter by mistake Difference is with GSNR change curve.

By Fig. 2 (a) it is 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 is found that during GSNR >=12dB, inventive algorithm DOD parameter estimation Performance comparision Stablize, and the root-mean-square error of DOD parameter estimation is significantly less than TALS-PARAFAC algorithm.Curve table in Fig. 2 (c) understands Inventive algorithm, compared to TALS-PARAFAC algorithm, estimates have relatively low root-mean-square error about DOA.

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 under impulse noise environment, there is not second-order statistic, so based on second-order statistic Young waiter in a wineshop or an inn's multiplication algorithm performance degradation.And MCC_PARAFAC algorithm, have employed maximal correlation entropy criterion as cost function, it can Well the interference of impulse noise mitigation, therefore has and preferably estimates performance.

Experiment 2: have studied the relation of parameter estimation performance and impulse noise characteristic index α.In this trifle, parameter is set as, Broad sense signal to noise ratio GSNR=15dB, the excursion of characteristic index α of impulse noise is 1≤α≤2.Fig. 3 (a)-(c) gives The relation of the RMSE and noise characteristic index α of two kinds of algorithm parameter estimations.

By Fig. 3 (a) it is found that when 1≤α≤2, the MCC-PARAFAC algorithm of the present invention is to Doppler frequency estimation RMSE be respectively less than TALS-PARAFAC algorithm, and when α >=1.3, it is stable that inventive algorithm parameter estimation obtains Performance comparision, There is RMSE more smoothly.By the curve of Fig. 3 (b) it can be seen that the parameter estimation performance of inventive algorithm is substantially better than TALS-PARAFAC algorithm.When 1≤α≤1.3, inventive algorithm reduces rapidly about the RMSE value of DOD parameter estimation, and becomes In steadily.Curve table in Fig. 3 (c) understands that inventive algorithm, compared to TALS-PARAFAC algorithm, estimates have relatively about DOA Low root-mean-square error.

Therefore, by the emulation experiment of this trifle, it can be appreciated that when α >=1.3, MCC_PARAFAC algorithm has relatively Good performance.The impulse of the least noise of α is the strongest, TALS_PARAFAC algorithm not inhibitory action to impulse noise, so When α is less, algorithm performance is poor, and when α=2, impulse noise is converted into Gaussian noise, so when α is close to 2, algorithm has There is estimation performance carefully.Visible TALS_PARAFAC algorithm is more sensitive to impulse noise ratio, should in the environment of impulse noise The parameter estimation Performance comparision of algorithm is poor.Therefore, by Fig. 2 and Fig. 3 it can be seen that under impulse noise environment MCC_PARAFAC The parameter estimation performance of algorithm is far superior to TALS_PARAFAC algorithm.

Experiment 3: this trifle have studied root-mean-square error RMSE and the pass of the long σ of core that MCC_PARAFAC algorithm parameter is estimated System.This trifle experiment in, parameter be set as broad sense signal to noise ratio GSNR=15dB, characteristic index α=1.4 of impulse noise, core The excursion of long parameter σ is 0.1≤σ≤2.From fig. 4, it can be seen that the performance that MCC_PARAFAC algorithm parameter is estimated is by core The impact of long parameter σ is little.

Experiment 4 have studied accuracy rate and broad sense signal to noise ratio GSNR and the relation of characteristic index α of parameter estimation.Parameter estimation Accuracy rate P_aMay be defined asWherein D is actual value,For estimated value.The P when multiple target_a Estimate the meansigma methods of accuracy rate for multiple target components, the present invention is the meansigma methods of two target accuracys rate.Fig. 5 (a)-(c) shows Show the accuracy rate change curve with GSNR of parameter estimation.

Fig. 5 (a) shows the accuracy rate of Doppler frequency estimation and the relation of broad sense signal to noise ratio, MCC-as can be seen from Fig. The accuracy rate of PARAFAC algorithm is higher than TALS-PARAFAC algorithm.Fig. 5 (b) shows that accuracy rate that angle of departure DOD estimates is with extensively The variation relation of justice signal to noise ratio.Curve shows that the accuracy rate that inventive algorithm is estimated about DOD is calculated also above TALS-PARAFAC Method.Fig. 5 (c) shows the relation of accuracy rate that DOA estimates and broad sense signal to noise ratio, MCC-PARAFAC algorithm as can be seen from Fig. Accuracy rate higher than TALS-PARAFAC algorithm, especially particularly evident when broad sense signal to noise ratio is more than 15dB.

Fig. 6 shows the accuracy rate change curve with noise characteristic index α of parameter estimation.Fig. 6 (a) can be seen that when making an uproar Acoustic signature index less than 1.3 time, the pulse feature of noise is stronger, inventive algorithm about Doppler frequency estimation accuracy rate more Significantly higher than TALS-PARAFAC algorithm, along with 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 that DOD estimates refers to noise characteristic The relation of number.As seen from the figure along with the increase of noise characteristic index, the accuracy rate of inventive algorithm and TALS-PARAFAC The gap of algorithm is also increasing, and the accuracy rate that this algorithm parameter is estimated is apparently higher than TALS-PARAFAC algorithm.Equally, Fig. 6 (c) Curve table understand the relation of accuracy rate that DOA estimates and noise characteristic index.By this figure it can be seen that along with noise characteristic refers to The increase of number, the accuracy rate of inventive algorithm is increasingly higher than TALS-PARAFAC algorithm.

Owing to MCC_PARAFAC algorithm considers the impact of impulse noise, use maximal correlation entropy criterion cost letter the most Number is iterated.And TALS_PARAFAC algorithm is based on second moment, there is not limited second moment in impulse noise, therefore TALS_PARAFAC algorithm performance under impulse noise environment can significantly be degenerated.Can from Fig. 5 (a)-(c) and Fig. 6 (a)-(c) Go out MCC_PARAFAC algorithm, than TALS_PARAFAC algorithm, there is higher accuracy rate.

The list of references that the present invention relates to

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For the ordinary skill in the art, the present invention is simply exemplarily described by specific embodiment, Obviously the present invention implements and is not subject to the restrictions described above, as long as the method design that have employed the present invention is entered with technical scheme The improvement of various unsubstantialities of row, or the most improved design by the present invention and technical scheme directly apply to other occasion , all within protection scope of the present invention.

Claims (1)

1. the tough parallel factorial analysis new algorithm under impulse noise environment, it is characterised in that: comprise the steps: that (1) is optional Random matrix initializesWithIteration serial number k=1,2,3 ...,

(2) willSubstitution formula (28), seeks its maximal correlation entropy solution, it is thus achieved that the kth of B (θ) time iterative estimate valueAs shown in formula (29),

In order to solve formula (27), can be by maximization problemIt is equivalent to minimization problem, then cost function is

Wherein

(3) willSubstitution formula (30), seeks its maximal correlation entropy solution, it is thus achieved that kth time iterative estimate valueAs Shown in formula (31).

(4) willSubstitution formula (32), seeks its maximal correlation entropy solution, it is thus achieved that C (f_d) kth time iterative estimate valueAs shown in formula (33), and calculateWherein D_l[] represent by The diagonal matrix that the l row element of matrix is formed,If | δ_k-δ_k-1| > ε, ε be Error threshold, then repeat step (2)-(4).If | δ_k-δ_k-₁| < ε then goes to step (5)

(5) through above-mentioned iterative computation, obtainB (θ) and C (f_d) final estimated valueWithAnd make It is respectively jth row i-th column element of 3 estimated matrix, by formula (30)-(32) to each column vector The method being averaging obtains(i=1 ..., P), angle () represents the phase angle computing taking element,