CN109212467A - Deficient standing wave arrival direction estimating method under amplitude phase error based on partial collimation nested array - Google Patents

Deficient standing wave arrival direction estimating method under amplitude phase error based on partial collimation nested array Download PDF

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CN109212467A
CN109212467A CN201811083177.7A CN201811083177A CN109212467A CN 109212467 A CN109212467 A CN 109212467A CN 201811083177 A CN201811083177 A CN 201811083177A CN 109212467 A CN109212467 A CN 109212467A
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array
phase error
amplitude phase
submatrix
order cumulant
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CN109212467B (en
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田野
徐鹤
王彦茹
史佳欣
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Yanshan University
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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
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/74—Multi-channel systems specially adapted for direction-finding, i.e. having a single antenna system capable of giving simultaneous indications of the directions of different signals
    • 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
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/14—Systems for determining direction or deviation from predetermined direction

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Abstract

The invention discloses the deficient standing wave arrival direction estimating methods under a kind of amplitude phase error based on partial collimation nested array comprising following steps: step 1: selection array output data constructs fourth order cumulant vector；Step 2: combining threshold application judgement in the case where reference source works in due course and even multiplier function completes the estimation of array amplitude phase error；Step 3: using amplitude phase error estimated result amendment fourth order cumulant vector and Toeplitz non-singular matrix is obtained based on Subarray partition strategy；Step 4: Mutual coupling is completed by ESPRIT algorithm.2L is effectively estimated using 2L array element in the present invention2A information source, and preferable robustness is all had to array amplitude phase error, white Gaussian noise/coloured noise and unknown non-uniform noise, while effectively preventing the angle raster search of macrooperation amount.

Description

Deficient standing wave arrival direction estimating method under amplitude phase error based on partial collimation nested array
Technical field
The invention belongs to array signal processing fields, and in particular to partial collimation nested array is based under a kind of amplitude phase error Deficient standing wave arrival direction estimating method.
Background technique
Mutual coupling is an important research direction of array signal processing and plays in dual-use field Extensive effect.It is well known that for a uniform linear array with L array element, using classics such as MUSIC, ESPRIT L-1 information source can only be at most effectively estimated in Wave arrival direction estimating method.However in actual orientation of information source, often there is Information source number is greater than array number, that is, owes fixed situation, therefore such as how less array number estimates that more information source numbers have become letter The research hotspot of source positioning field and the extensive concern for having attracted domestic and international researcher.Solve the problems, such as a kind of this effective scheme It is that statistic or cumulant domain formation virtual array are listed in using nonuniform array, to owe with the completion of higher number of degrees of freedom, Mutual coupling under condition.Currently, common nonuniform array is classified as nested array and relatively prime array.
Based on nested array, P.Pal and P.P.Vaidyanathan are in 2010 in " Nested arrays:A novel Approach to array processing with enhanced degrees of freedom " propose space smoothing MUSIC method owes to determine the Mutual coupling in situation, could not use void when constructing non-singular matrix though this method can be realized All array elements of matroid column, so as to cause certain information loss.
Based on relatively prime array, S.Qi, Y.D.Zhang and M.G.Amin in 2015 in " Generalized coprime Array configurations for direction-of-arrival estimation " it proposes sparse reconstruct class and owes fixed Wave arrival direction estimating method, main problem existing for this method is that calculation amount is huge.
Simultaneously it is further noted that above-mentioned all methods are only applicable to ideal array received condition.However In practice due to the influence of local oscillator clock offset and sensor position error etc., array received data often exist centainly Amplitude and phase error, i.e. amplitude phase error.Under this background condition, has deficient standing wave arrival direction estimating method performance and can Appearance, which is decreased obviously, even fails.Therefore, effective standing wave arrival direction estimating method of owing under amplitude phase error is studied to raising reality Detection system performance is of great significance.
In fact, in order to reduce the influence of array amplitude phase error, Recent study scholars also propose in succession some to be had The method of effect, including S.G.Cao and Z.F.Ye et al. are in " A hadamard product based method for DOA Estimation and gain-phase error calibration " propose Hadamard Product method, J.Liu, W.D.Zhou et al. is in " Covariance matrix based fast smoothed sparse DOA estimation With partly calibrated array " the quick SL0 method etc. that proposes, but these methods are not suitable for owing condition Under Mutual coupling, while needing a large amount of angle raster search mostly, complexity is high, and very to non-gaussian white noise Sensitivity, noise robustness are weaker.
Summary of the invention
The present invention overcomes deficiency in the prior art, provide under a kind of amplitude phase error based on partial collimation nested array Standing wave arrival direction estimating method is owed, being used to solve existing Wave arrival direction estimating method, there are amplitude phase errors and information source number in array Problem of Failure when greater than array number (it is fixed to owe).
Technical scheme is as follows:
A kind of deficient standing wave arrival direction estimating method based on partial collimation nested array under amplitude phase error comprising following step It is rapid:
Step 1: receiving data according to given array time domain and information source be digital, is original to form maximum continuously and virtually aperture Then, array output data is selected, fourth order cumulant vector r is constructed；
Step 2: building threshold decision function f andAnd connect multiplier function F1, F2,Replace in reference source It is to complete array amplitude phase error in 2Q to estimate that multiplier function is adjudicated and connected to joint threshold application, which in sampling time section, under start and stop work； In preceding Q sampling time section, stop reference source work, when threshold decision function f > 0 andIt carries out algebraic operation and completes width Phase estimation error；When threshold decision function be unsatisfactory for f > 0 andStart reference source work, and in rear Q sampling time section It carries out algebraic operation and completes amplitude phase error estimation, wherein Q is time span, and Q > 0, unit is the second；
Step 3: utilizing amplitude phase error estimated result, corrects fourth order cumulant vector r, obtainsAnd it is based on Subarray partition Strategy obtains Toeplitz non-singular matrix；
Step 4: Mutual coupling is completed by ESPRIT algorithm.
Preferably, the step of fourth order cumulant vector r is constructed in step 1 is as follows:
S1: presetting the K information sources with non-zero quadravalence peak value and be incident on the nested array being made of 2L array element, Element position be 0d, 1d ..., Ld, 2Ld ..., L2D }, wherein label location is that the array element of { 0d, 1d, 2d ..., Ld } corresponds to Submatrix 1, label location are { 0d, Ld, 2Ld ..., L2D } array element correspond to submatrix 2, and label location is { 0d, 1d, Ld } Array element is calibrated, and d=λ/2, λ are carrier wavelength；
S2: setting x (t) as array time domain and receive data, then m, n, p, and the four of a array element observation data of q ∈ { 1 ..., 2L } Rank cumulant are as follows:
Wherein, function cum { g }, which is represented, carries out fourth order cumulant operation；PmAnd γmRespectively represent m (∈ { 1 ..., 2L }) The position of a array element and amplitude phase error, PnAnd γnThe position and amplitude phase error of n-th (∈ { 1 ..., 2L }) a array element are respectively represented,For γnConjugation；PpAnd γpRespectively represent the position and amplitude phase error of pth (∈ { 1 ..., 2L }) a array element；PqAnd γqPoint The position and amplitude phase error of q (∈ { 1 ..., 2L }) a array element are not represented,For γqConjugation；
xm(t) m ∈ { 1 ..., 2L } row data of x (t) are represented；
xn(t) the n-th ∈ { 1 ..., 2L } row data of x (t) are represented,For to xn(t) each component is conjugated Result after operation；
xp(t) pth ∈ { 1 ..., 2L } row data of x (t) are represented；
xq(t) q ∈ { 1 ..., 2L } row data of x (t) are represented,For to xq(t) each component is conjugated Result after operation；It is the fourth order cumulant of kth (1≤k≤K) a source signal, skRepresent answering for k-th of source signal Amplitude；
φk=2 π sin (θk)/λ；Wherein φkIndicate basic phase difference, θkEnter for kth (1≤k≤K) a source signal Firing angle degree；
S3: being based on nested array, to form maximum continuously and virtually aperture as principle, selects m, n, p, it is tired that q value forms quadravalence Accumulated amount vector r, is expressed as
R=FAc
Wherein, F is corresponding to m, n, p, the diagonal matrix of q array element amplitude phase error product；A=[a (θ1),…,a(θK)] Virtual guiding matrix is represented,It represents and corresponds to kth (1≤k ≤ K) a source signal virtual steering vector,L is the half of practical array number, 4L2+ 1 is shape At Virtual array number, d be array element spacing, meet d=λ/2, λ is carrier wavelength, φk=2 π sin (θk)/λ, wherein φkIt indicates Basic phase difference, θkFor the incident angle of kth (1≤k≤K) a source signal,It is kth (1≤k≤K) a source signal Fourth order cumulant, skThe complex magnitude of k-th of source signal is represented,Subscript T indicates transposition operation.
Preferably, the threshold value decision function described in step 2 is defined as
F=Q2- τ=| C2,1,1,1+C2,1,2,2|/2-τ
Wherein, f is the threshold value decision function corresponding to submatrix 1,For the threshold value decision function corresponding to submatrix 2,
And
For decision threshold set by user, | g | expression takes absolute value operation, It indicates to carry out summation operation to K element,It is the fourth order cumulant of kth (1≤k≤K) a source signal, skIt represents k-th The complex magnitude of source signal；
Even multiplier function is defined as
Wherein, F1And F2For company's multiplier function corresponding to submatrix 1,WithFor company's multiplier letter corresponding to submatrix 2 Number, and
It represents to k in bracket1A element carries out product calculation.
Preferably, the specific steps of array amplitude phase error estimation are completed in step 2 are as follows:
A) if f > 0 andThen the amplitude phase error of submatrix 1 and submatrix 2 is estimated as respectively
Wherein,WithRespectively kth1+ 1 and L+k2The amplitude phase error of+1 array element, F1And F2For corresponding to son Company's multiplier function of battle array 1,WithFor company's multiplier function corresponding to submatrix 2, Q2=| C2,1,1,1+C2,1,2,2|/2,Q3=(C1,1,1,1+C2,2,2,2+CL+1,L+1,L+1,L+1)/3；And
B) if f≤0 orThen start reference source transmitted reference signal, and in next sampling period based on above-mentioned Identical processing mode completes the estimation of array amplitude phase error.
Preferably, the specific steps of Toeplitz non-singular matrix are obtained in step 3 are as follows:
IfThe estimated value for representing F takes and obtains modified fourth order cumulant vector after inversely processing and be expressed as
Wherein,ForInverse matrix, A represents virtual guiding matrix,
It willIt is divided into 2L2+ 1 part obtains non-singular matrix
Wherein,0M×NRepresent M × N's Null matrix,Represent (2L2+1)×(2L2+ 1) unit matrix.
Provide that anti-diagonal matrix is
Form Toeplitz non-singular matrix
Preferably, m, n, p are selected, q value forms fourth order cumulant vector, the specific steps of which are as follows:
Wherein, r (l) represents first of element of fourth order cumulant vector.
Compared with prior art, the beneficial effects of the present invention are:
The method of the present invention not only has by combining using nested array and fourth order cumulant come array extending virtual aperture Better noise robustness, and 2L only is effectively estimated with 2L array element2A information source.
Method provided by the invention is in the case where a reference source works in due course, the judgement of use in conjunction threshold value and even multiplier function The estimation of array amplitude phase error is completed, without the prior information of noise variance and information source number, and solution procedure is only simple algebra Operation, complexity is low, runing time is short.
Method provided by the invention obtains Toeplitz non-singular matrix by Subarray partition, is based further on ESPRIT acquisition The closed solutions of Mutual coupling effectively prevent the angle raster search of macrooperation amount.
Detailed description of the invention
Fig. 1 is the structure chart for the partial collimation nesting sensor array that the present invention uses；
Fig. 2 is the deficient standing wave arrival direction estimating method under amplitude phase error proposed by the present invention based on partial collimation nested array Flow chart；
Fig. 3 is the deficient standing wave arrival direction estimating method under amplitude phase error proposed by the present invention based on partial collimation nested array It can estimate the result figure of maximum information source number；
Fig. 4 A is to determine Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error proposed by the present invention Root-mean-square error of the method under white Gaussian noise compares figure with signal-to-noise ratio variation；
Fig. 4 B is to determine Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error proposed by the present invention Root-mean-square error of the method under gauss heat source model compares figure with signal-to-noise ratio variation；
Fig. 4 C is to determine Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error proposed by the present invention Root-mean-square error of the method under unknown non-uniform noise compares figure with signal-to-noise ratio variation；
Fig. 5 is the deficient standing wave arrival direction estimating method under amplitude phase error proposed by the present invention based on partial collimation nested array Single average operating time with sampling number of snapshots variation compare figure.
Specific embodiment
Below in conjunction with attached drawing, the present invention will be described in detail.
Referring to Figures 1 and 2, steps are as follows for realization of the invention:
Step 1: it reasonably selects array output data and constructs fourth order cumulant vector；
Assuming that the K information sources with non-zero quadravalence peak value are incident on the nested array being made of 2L array element, such as Fig. 1 institute Show.Element position be 0d, 1d ..., Ld, 2Ld ..., L2D }, wherein label location is that the array element of { 0d, 1d ..., Ld } corresponds to Submatrix 1, label location are { 0d, Ld ..., L2D } array element correspond to submatrix 2, and label location be { 0d, 1d, Ld } array element Calibrated, d=λ/2, λ are carrier wavelength.If x (t) is that array time domain receives data, then m, n, p, q ∈ { 1 ..., 2L } a battle array The fourth order cumulant of member observation data can calculate are as follows:
Wherein, PmAnd γmThe position and amplitude phase error of m (∈ { 1 ..., 2L }) a array element are respectively represented,It is kth (1 ≤ k≤K) a source signal fourth order cumulant, φk=2 π sin (θk)/λ；
Subscript * is conjugate operation, and function cum { g }, which is represented, carries out fourth order cumulant operation.
Property based on nested array, selects suitable m, n, p, q value formed the fourth order cumulant that has the property that R is measured, is expressed as
R=FAc
Wherein, A=[a (θ1),…,a(θK)], F is corresponding to m, n, p, the diagonal matrix of q array element amplitude phase error product.
Subscript T indicates transposition operation.
The detailed forming process of fourth order cumulant vector r is
Step 2: combine threshold application judgement in the case where reference source works in due course and mutually missed with even multiplier function completion array width Difference estimation；
Define threshold value decision function are as follows:
F=Q2- τ=| C2,1,1,1+C2,1,2,2|/2-τ
Wherein,For decision threshold set by user, | g | expression takes absolute value operation.
Definition connects multiplier function are as follows:
If f > 0 andThen the amplitude phase error of submatrix 1 and submatrix 2 is estimated as respectively
Wherein, Q3=(C1,1,1,1+C2,2,2,2+CL+1,L+1,L+1,L+1)/3。
If f≤0 orThen start reference source transmitted reference signal, and is based on above-mentioned phase in next sampling period Same processing mode completes the estimation of array amplitude phase error.
Step 3: it is obtained using amplitude phase error estimated result amendment fourth order cumulant vector and based on Subarray partition strategy Toeplitz non-singular matrix；
IfThe estimated value for representing F takes and obtains modified fourth order cumulant vector after inversely processing and be expressed as
It willIt is divided into 2L2+ 1 part obtains non-singular matrix
Wherein,0M×NRepresent M × N's Null matrix,Represent (2L2+1)×(2L2+ 1) unit matrix.
Provide that anti-diagonal matrix is
Form Toeplitz non-singular matrix
In a specific embodiment of the present invention, Toeplitz non-singular matrixIt is represented by
Wherein,
Subscript H is conjugate transposition operation, diag () representation vector diagonalization operation.
It is rightCarry out Eigenvalues Decomposition acquisition (2L2+ 1) × K signal subspace Es thinks according to the core of ESPRIT algorithm Think, certainly exist a nonsingular value matrix T, meets
By the preceding 2L of Es2Capable and rear 2L2Row uses Es respectively1And Es2It indicates, then has
Wherein,SubscriptRepresent pseudo-inverse operation.
It is rightIt carries out Eigenvalues Decomposition and obtains characteristic value qk, k=1, K, K, then direction of arrival can be estimated as
Wherein, arg () representative takes angle to operate.
It is owed below by under analysis of simulation experiment amplitude phase error proposed by the invention based on partial collimation nested array The estimation performance and calculating validity of standing wave arrival direction estimating method.Simulation process, which passes through MATLAB software, to carry out, computer It is configured to Intel Intel Core i3 CPU, 3.4G dominant frequency, 4G memory.
Emulation experiment 1: 2L is effectively estimated with 2L array element to prove the proposed method of the present invention in the experiment2The energy of a information source Power.Total element number of array is 2L=4, and 8 information sources are incident on array by { -60 °, -43 °, -26 °, -9 °, 8 °, 25 °, 42 °, 59 ° }, Signal-to-noise ratio is 20 decibels, and sampling number of snapshots are 2000, array range error and phase error byWithIt generates, andξlAnd ζlFor the stochastic variable being evenly distributed in [- 0.5,0.5] range. Simulation result is as shown in Figure 3, it can be seen that there is only the deviations of very little for the result and true value of the method for the present invention estimation, it was demonstrated that The method of the present invention can use 2L array element really and 2L be effectively estimated2A information source.
Emulation experiment 2: the experiment is to analyze the proposed method of the present invention in white Gaussian noise/coloured noise and unknown non-homogeneous Mutual coupling performance under noise.Total element number of array is 2L=10, and two information sources are incident on battle array by { -15.2 °, 9.8 ° } Column, sampling number of snapshots are 2000, and signal-to-noise ratio changes to 20 decibels by 0 decibel, and array amplitude phase error is identical as emulation experiment 1.Figure 4A~Fig. 4 C is followed successively by the Mutual coupling root-mean-square error under white Gaussian noise, gauss heat source model and unknown non-uniform noise With the variation relation figure of signal-to-noise ratio, wherein gauss heat source model variance meets
Unknown non-uniform noise variance is
N=diag ([0.5 3.8 1.2 8.0 2.3 0.75 15 1.9 5.2 2.2]T)
The mentioned method performance of the present invention is substantially better than control methods it can be seen from simulation result, and mutually misses to array width Poor and various noises all have good robustness.
Emulation experiment 3: computation complexity of the experiment to evaluate the proposed method of the present invention.Total element number of array is 2L= 10, two information sources are incident on array by { -15.2 °, 9.8 ° }, and signal-to-noise ratio is fixed as 20 decibels, and sampling number of snapshots are transformed to by 200 2000, array amplitude phase error is identical as emulation experiment 1.Angle raster search to the HadamardProduct method compared Step-length is 0.1, and search range is [- 30 °, 30 °].The time used is emulated as shown in figure 5, it can be found that side provided by the present invention The runing time of method is due to without angle raster search and well below Hadamard Product method.
Above-mentioned the simulation experiment result sufficiently demonstrates the Wave arrival direction estimating method under phase error provided by the present invention Validity, robustness and high efficiency.
It is understood that the principle that embodiment of above is intended to be merely illustrative of the present and the exemplary implementation that uses Mode, however the present invention is not limited thereto.Those skilled in the art should understand that: it still can be to aforementioned implementation Technical solution documented by example is modified, or is equivalently replaced to part of or all technical features；And these are repaired Change or replaces, the range for technical solution of various embodiments of the present invention that it does not separate the essence of the corresponding technical solution.

Claims (6)

1. the deficient standing wave arrival direction estimating method under a kind of amplitude phase error based on partial collimation nested array, it is characterised in that: its Include the following steps:
Step 1: receiving data according to given array time domain and information source be digital, to form maximum continuously and virtually aperture as principle, selects Array output data is selected, fourth order cumulant vector r is constructed；
Step 2: building threshold decision function f andAnd connect multiplier function F1, F2,Replace start and stop in reference source It is that the estimation of array amplitude phase error is completed in 2Q that multiplier function is adjudicated and connected to the lower joint threshold application that works, which in sampling time section,；Preceding In Q sampling time section, stop reference source work, when threshold decision function f > 0 andAlgebraic operation completion width is carried out mutually to miss Difference estimation；When threshold decision function be unsatisfactory for f > 0 andStart reference source work, and is carried out in rear Q sampling time section Amplitude phase error estimation is completed in algebraic operation, and wherein Q is time span, and Q > 0, unit is the second；
Step 3: utilizing amplitude phase error estimated result, corrects fourth order cumulant vector r, obtainsAnd it is based on Subarray partition strategy Obtain Toeplitz non-singular matrix；
Step 4: Mutual coupling is completed by ESPRIT algorithm.
2. determining Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error according to claim 1 Method, it is characterised in that: the step of fourth order cumulant vector r is constructed in step 1 is as follows:
S1: presetting the K information sources with non-zero quadravalence peak value and be incident on the nested array being made of 2L array element, array element Position be 0d, 1d ..., Ld, 2Ld ..., L2D }, wherein label location is that the array element of { 0d, 1d, 2d ..., Ld } corresponds to submatrix 1, label location is { 0d, Ld, 2Ld ..., L2D } array element correspond to submatrix 2, and label location be { 0d, 1d, Ld } array element Calibrated, d=λ/2, λ are carrier wavelength；
S2: setting x (t) as array time domain and receive data, then m, n, p, and the quadravalence of a array element observation data of q ∈ { 1 ..., 2L } is tired Accumulated amount are as follows:
Wherein, function cum { g }, which is represented, carries out fourth order cumulant operation；PmAnd γmRespectively represent m (∈ { 1 ..., 2L }) a battle array The position of member and amplitude phase error, PnAnd γnThe position and amplitude phase error of n-th (∈ { 1 ..., 2L }) a array element are respectively represented,For γnConjugation；PpAnd γpRespectively represent the position and amplitude phase error of pth (∈ { 1 ..., 2L }) a array element；PqAnd γqIt respectively represents The position of q (∈ { 1 ..., 2L }) a array element and amplitude phase error,For γqConjugation；
xm(t) m ∈ { 1 ..., 2L } row data of x (t) are represented；
xn(t) the n-th ∈ { 1 ..., 2L } row data of x (t) are represented,For to xn(t) each component carries out conjugate operation Result afterwards；
xp(t) pth ∈ { 1 ..., 2L } row data of x (t) are represented；
xq(t) q ∈ { 1 ..., 2L } row data of x (t) are represented,For to xq(t) each component carries out conjugate operation Result afterwards；It is the fourth order cumulant of kth (1≤k≤K) a source signal, skRepresent the complex magnitude of k-th of source signal；
φk=2 π sin (θk)/λ；Wherein φkIndicate basic phase difference, θkFor the incidence angle of kth (1≤k≤K) a source signal Degree；
S3: being based on nested array, to form maximum continuously and virtually aperture as principle, selects m, n, p, q value forms fourth order cumulant Vector r, is expressed as
R=FAc
Wherein, F is corresponding to m, n, p, the diagonal matrix of q array element amplitude phase error product；A=[a (θ1),…,a(θK)] represent Virtual guiding matrix,It represents and corresponds to kth (1≤k≤K) The virtual steering vector of a source signal,L is the half of practical array number, 4L2+ 1 is to be formed Virtual array number, d are array element spacing, meet d=λ/2, λ is carrier wavelength, φk=2 π sin (θk)/λ, wherein φkIndicate basis Phase difference, θkFor the incident angle of kth (1≤k≤K) a source signal,It is the four of kth (1≤k≤K) a source signal Rank cumulant, skThe complex magnitude of k-th of source signal is represented,Subscript T indicates transposition operation.
3. determining Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error according to claim 2 Method, it is characterised in that: the threshold value decision function described in step 2 is defined as
F=Q2- τ=| C2,1,1,1+C2,1,2,2|/2-τ
Wherein, f is the threshold value decision function corresponding to submatrix 1,For the threshold value decision function corresponding to submatrix 2, Q2=| C2,1,1,1+C2,1,2,2|/2,And
For decision threshold set by user, | g | expression takes absolute value operation,Expression pair K element carries out summation operation,It is the fourth order cumulant of kth (1≤k≤K) a source signal, skRepresent k-th of information source letter Number complex magnitude；
The even multiplier function is defined as
Wherein, F1And F2For company's multiplier function corresponding to submatrix 1,WithFor company's multiplier function corresponding to submatrix 2, and
It represents to k in bracket1A element carries out product calculation.
4. determining Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error according to claim 3 Method, it is characterised in that: the specific steps of array amplitude phase error estimation are completed in step 2 are as follows:
A) if f > 0 andThen the amplitude phase error of submatrix 1 and submatrix 2 is estimated as respectively
Wherein,WithRespectively kth1+ 1 and L+k2The amplitude phase error of+1 array element, F1And F2For corresponding to submatrix 1 Even multiplier function,WithFor company's multiplier function corresponding to submatrix 2, Q2=| C2,1,1,1+C2,1,2,2|/2,Q3=(C1,1,1,1+C2,2,2,2+CL+1,L+1,L+1,L+1)/3；And
B) if f≤0 orThen start reference source transmitted reference signal, and in next sampling period based on above-mentioned identical Processing mode complete array amplitude phase error estimation.
5. determining Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error according to claim 4 Method, it is characterised in that: the specific steps of Toeplitz non-singular matrix are obtained in step 3 are as follows:
IfThe estimated value for representing F takes and obtains modified fourth order cumulant vector after inversely processing and be expressed as
Wherein,ForInverse matrix, A represents virtual guiding matrix,
It willIt is divided into 2L2+ 1 part obtains non-singular matrix
Wherein,0M×NRepresent the zero moment of M × N Battle array,Represent (2L2+1)×(2L2+ 1) unit matrix.
Provide that anti-diagonal matrix is
Form Toeplitz non-singular matrix
6. determining Mutual coupling side based on the deficient of partial collimation nested array under amplitude phase error according to claim 2 Method, it is characterised in that: selection m, n, p, q value form fourth order cumulant vector, the specific steps of which are as follows:
Wherein, r (l) represents first of element of fourth order cumulant vector.
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