CN105282067B - A kind of complex field blind source separation method - Google Patents
A kind of complex field blind source separation method Download PDFInfo
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Abstract
The invention discloses a kind of complex field blind source separation method, construct complex field objective matrix group, and carry out real symmetrization, the objective matrix group that the real value objective matrix reconfigured is formed, complex field Joint diagonalization problem is converted into real number field Joint diagonalization problem, can be used for solving the problems, such as complex field blind source separating；And compared with other are equally applicable to complex field algorithm, do not constrain and treat that Joint diagonalization objective matrix is hermitian symmetrical matrix or positive definite Hermitian matrix, there is extremely wide applicability；Using the alternating least-squares iteration algorithm based on Joint diagonalization least square cost function, the design feature for the objective matrix group that real value objective matrix forms is made full use of, to realize the Joint diagonalization of fresh target matrix group；Using alternately least-squares iteration Algorithm for Solving cost function, the estimate of aliasing matrix is obtained, realizes blind source separating, simulation results show the method for the present invention has higher convergence precision.
Description
Technical field
The invention belongs to blind signal processing technology field, is related to a kind of complex field blind source separation method.
Background technology
Blind source separating (Blind Source Separation, BSS) is also known as Blind Signal Separation (Blind Signal
Separation, BSS), it is widely used in wireless communication, radar, image, voice is biomedical, the field such as seismic wave detection,
It is field of signal processing hot research problem.
In order to realize blind source separating, in the case where the priori such as source signal and its aliasing parameter is unknown, arranging
Fuzzy and scale obscure it is acceptable under the premise of, many methods often utilize the aliasing signal received and the system based on source signal
Characteristic is counted, constructs one group of objective matrix with diagonalizable structure, by carrying out Joint diagonalization operation to objective matrix group,
Be referred to as the estimation of the aliasing matrix (or its inverse matrix) of " Joint diagonalization device ", and and then recover source signal accordingly, it is real
Existing blind source separating.
But existing Joint diagonalization algorithm, mostly with its limitation, it is embodied in constructed objective matrix
Many restrictions.For example, document[1]It is required that objective matrix is positive definite matrix.Document[2]-[3]It is required that at least one target square
Battle array is positive definite matrix, to construct whitening matrix accordingly, will be asked again after " Joint diagonalization device " prewhitening asked is unitary matrice
Solution.But since objective matrix group is obtained by statistical method, it is limited to the influence of sample number and noise, each mesh
The diagonalizable structure of matrix is marked in itself there are certain error, and pre-whitening operation is equivalent to sacrifice remaining objective matrix
Joint diagonalization precision has exchanged the stringent diagonalization of positive definite objective matrix for, introduces error.And this draw in the prewhitening stage
The error entered, can not be corrected in subsequent orthogonal Joint diagonalization algorithm, thus have impact on the overall performance of algorithm.
In view of error caused by the above-mentioned prewhitening stage, many scholars propose some whitening pretreatments that are not required successively
Non-orthogonal joint diagonalization algorithm, such as J-Di algorithms[4], QDIAG algorithms[5], WEDGE algorithms[6], ACDC algorithms[7], SVDJD calculations
Method[8], SeDJoCo algorithms[9], FAJD algorithms[10], CVFFDIAG algorithms[11]Deng.It is many to calculate in the algorithm of these excellent performances
Method assumes that objective matrix group is necessary for real value, it means that these real number field Joint diagonalization algorithms can only solve aliasing matrix
It is the blind source separating problem of real number value with source signal, it is many logical when blind source separating is applied to communication when some key areas
Letter signal is complex valued signals；In addition, when solving convolution mixed blind source separation problem, easier frequency domain side is usually used
Method, and the object to be separated on each Frequency point, often complex value, real number field Joint diagonalization algorithm can not solve such plural number
Domain problem.For the algorithm that can realize complex field Joint diagonalization, ACDC algorithms and SeDJoCo algorithms require target square
Battle array is hermitian symmetrical matrix or real symmetric matrix, and it is positive definite Hermitian matrix that SVDJD algorithms, which require objective matrix,.Obviously, to target
The limitation of matrix, greatly constrains the application range of above-mentioned algorithm.
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diagonalization and its application to blind source separation[J].IEEE
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The content of the invention
For above-mentioned problems of the prior art or defect, it is an object of the present invention to provide one kind can improve connection
The applicability of diagonalization algorithm is closed, effectively solves the problems, such as the complex field blind source separation method of complex field blind source separating.
To achieve these goals, the present invention adopts the following technical scheme that：
A kind of complex field blind source separation method, comprises the following steps：
Step 1：According to the T observation signal sample of observation signal x (t)Construct complex field objective matrix group
{Ck, k=1 ... K }, and
Ck=ADkAH, k=1 ... K (2)
Wherein, Dk, k=1 ... K is diagonal matrix, A ∈ FM×N(M >=N) is aliasing matrix, AHFor being total to for aliasing matrix A
Yoke transposed matrix；
Step 2：To the complex field objective matrix group { C of step 1 constructionk, k=1 ... K } real symmetrization is carried out, obtain
The objective matrix group that the real value objective matrix reconfigured is formed
Step 3：Utilize objective matrix groupBuild Joint diagonalization least square cost function；
Step 4：Utilize the Joint diagonalization least square cost that alternately least-squares iteration Algorithm for Solving step 3 obtains
Function, obtains the estimation of aliasing matrix AAccording toObtain the estimation of source signalRealize blind source separating, its
InRepresentInverse matrix, x (t) represent observation signal.
Specifically, the process of the step two includes：
The complex field objective matrix group { C constructed according to step 1k, k=1 ... K } according to formula (3) and (4) again structure
Make 2K matrix groupWith
Wherein, CkHRepresent CkAssociate matrix, Re (Dk) represent diagonal matrix DkReal part, Im (Dk) represent diagonal
Matrix DkImaginary part；
Remember that matrixing function is f (X)：
By formula (3) and formula (5), obtain
Wherein, fT(A) transposition for being f (A),
By formula (4) and (5), obtain
Wherein,
The objective matrix group being made of formula (6) and formula (7), the real value objective matrix that must be reconfigured
Wherein,K=1 ..., 2K,RepresentTransposition；With can Joint diagonalization structure,K=
1 ..., 2K is diagonal matrix；
Specifically, the process of the step three includes：
Build Joint diagonalization least square cost function J (H, Λ1,…,Λ2K,F)：
Wherein, H representing matrixesEstimated matrix, Λk, k=1 ..., 2K represents diagonal matrix groupK=1 ..., 2K's
Estimated matrix；Represent Frobenius norms；FTForEstimated matrix,RepresentTransposed matrix, and H=EF, E are
Generalized Permutation Matrix.
Specifically, the process of the step 4 includes：
Step 4.1:Remember iterative initial value
Step 4.2:In the m times iteration, Joint diagonalization least square cost function is：Jk(H(m-1),Λk(m),K=1 ..., 2K, on Λk(m) smallest point is sought, obtains diagonal matrix Λk
(m), the estimate of k=1 ..., 2K, wherein, H (m-1) represents the estimate for the H that the m-1 times iteration obtains, and F (m-1) is represented
The estimate for the F that the m-1 times iteration obtains；
Step 4.3：The diagonal matrix Λ that the estimate F (m-1) and step 4.2 of the F obtained using the m-1 times iteration is obtainedk
(m), k=1 ..., 2K, is asked so that Joint diagonalization least square cost function J (H (m), Λ1(m),…,Λ2K(m),Reach the H (m) during minimum value；
Step 4.4：The Λ that the H (m) and step 4.2 obtained using step 4.3 is obtainedk(m), k=1 ..., 2K, ask so that
J(H(m),Λ1(m),…,Λ2K(m),Reach the F (m) during minimum value；
Step 4.5：Judge whether algorithm restrains, that is, judge whether to meetAnd
Wherein δ is the threshold value of setting；If conditions are not met, return to step 4.2；If it is satisfied, then the estimation of final H and F are obtained, two
Person can be used as matrixEstimation, according toObtain the estimation of aliasing matrix AAccording toObtain source
The estimation of signalRealize blind source separating, whereinRepresentInverse matrix, x (t) represent observation signal.
Compared with prior art, the present invention has following technique effect：
1st, complex field objective matrix group is constructed, and carries out real symmetrization, the real value objective matrix reconfigured is formed
Objective matrix group, complex field Joint diagonalization problem is converted into real number field Joint diagonalization problem, can be used for solving multiple
Number field blind source separating problem；And compared with other are equally applicable to complex field algorithm, do not constrain and treat Joint diagonalization objective matrix
For hermitian symmetrical matrix or positive definite Hermitian matrix, there is extremely wide applicability.
2nd, using the alternating least-squares iteration algorithm based on Joint diagonalization least square cost function, reality is made full use of
It is worth the design feature for the objective matrix group that objective matrix is formed, to realize the Joint diagonalization of fresh target matrix group.
3rd, using alternately least-squares iteration Algorithm for Solving cost function, the estimate of aliasing matrix is obtained, realizes blind source
Separation, simulation results show the method for the present invention have a higher convergence precision.
4th, present invention eliminates pre-whitening operation, avoid and introduce albefaction error.
Brief description of the drawings
Fig. 1 is the flow chart of the method for the present invention；
Fig. 2 is that the overall situation refuses the performance curve that horizontal GRL changes with iterations of making an uproar in experiment one；
Fig. 3 is to test the performance curve that parameter JH changes with iterations in one
Fig. 4 is in testing two under different signal-to-noise ratio NER, GRL with iterations change curve；
Fig. 5 is tested under signal-to-noise ratio NER different in two, and be averaged change curves of the GRL with iterations；
Fig. 6 is to use distinct methods, and in the case where signal-to-noise ratio NER is 15dB, average GRL is bent with the change of iterations
Line；
Fig. 7 is to use distinct methods, and in the case where signal-to-noise ratio NER is 10dB, average GRL is bent with the change of iterations
Line；
Fig. 8 is the average GRL for testing 3 100 independent experiments with the change curve of iterations；
Fig. 9 is the planisphere of 4 source signals；
Figure 10 is the planisphere of 4 reception signals；
Figure 11 is the planisphere of 4 recovery signals；
Further details of explanation and illustration is done to the solution of the present invention with reference to the accompanying drawings and examples.
Embodiment
Above-mentioned technical proposal is deferred to, complex field blind source separation method of the invention, referring to Fig. 1, specifically includes following steps：
Step 1：Construct complex field objective matrix group.
The linear array being made of M array element, N number of narrow-band source are incided in linear array from different directions, the M Wei Guan that linear array receives
Surveying signal x (t) is：
X (t)=As (t)+n (t) (1)
Wherein, A ∈ FM×N(M >=N) is aliasing matrix, s (t)=[s1(t),…,sN(t)]TFor source signal vector, x (t)=
[x1(t),…,xM(t)]TFor observation signal, n (t)=[n1(t),…,nM(t)]TFor noise vector.
According to the T observation signal sample of observation signal x (t)Construct the objective matrix group that one group of total number is K
{Ck, k=1 ... K }, wherein each objective matrix is respectively provided with following diagonalizable structure：
Ck=ADkAH, k=1 ... K (2)
Wherein Dk, k=1 ... K is diagonal matrix, AHRepresent the associate matrix of aliasing matrix A.
The presence of limitation and noise in view of sample number, formula (2) described diagonal structure is only existing for approximation.Construction
The main method of objective matrix group has：Covariance matrix under time windows[1], the cross-correlation matrix under different time shifts[2], three
Section matrix under rank or higher order cumulant[12], spatial time-frequency matrix[13], or second feature function[14].The method of the present invention
It is not special to objective matrix available for solving the problems, such as most common complex field Joint diagonalization to realize complex field blind source separating
It is required that allow aliasing matrix A and diagonal matrix Dk, k=1 ... K is complex valued matrices, therefore the construction side of objective matrix group
The combination of above-mentioned one kind or any several method may be selected in method.
Step 2：Real symmetrization, the real value reconfigured are carried out to the complex field objective matrix group of step 1 construction
The objective matrix group that objective matrix is formedComplex field Joint diagonalization problem is converted into real number field Joint diagonalization problem.
The complex field objective matrix group { C constructed according to step 1k, k=1 ... K } according to formula (3) and (4) again structure
Make 2K matrix groupWith
Wherein, CkHRepresent CkAssociate matrix, Re (Dk) represent diagonal matrix DkReal part, Im (Dk) represent diagonal
Matrix DkImaginary part.
Remember that matrixing function is f (X)：
By formula (3) and formula (5), obtain
Wherein, fT(A) transposition for being f (A),
By formula (4) and formula (5), obtain
Wherein,
The objective matrix group being made of formula (6) and formula (7), the real value objective matrix that must be reconfigured
Matrix group { the C that above-mentioned conversion process forms the complex value objective matrix that K dimension is M × Mk, k=1 ... K } turn
Turn to the objective matrix group that 2K dimension is formed for real value objective matrix new 2M × 2MSent out by analyzing
Existing, new objective matrix groupWith following design feature：
(T1) objective matrix groupWith can Joint diagonalization structure, i.e.,K=1 ..., 2K is diagonal matrix, even go to
AmountThenWherein diag [] represents that it is diagonal to construct with vector
The diagonal matrix of line element.
(T2) objective matrix is real symmetric matrix, i.e.,K=1 ..., 2K.
(T3) in expression formula (8), matrixRepresentation it is as follows：
From formula (9), if obtainedPreceding M column elements, using its design feature, can be readily availableBelow
M column elements.
Step 3：The objective matrix group obtained using step 2Build Joint diagonalization least square cost function.
Complex field general goals matrix group is converted into real number field symmetric targets square by the matrixing process described in step 2
Battle array group, can be directed to formula (8), using the real number field Joint diagonalization algorithm of existing excellent performance, recover matrixRealize blind source
Separation.However, since existing real number field Joint diagonalization algorithm is not specifically for problem shown in formula (8), thus calculating
The parametric structures feature described in (T1)~(T3) can not be taken into full account in the optimization process of method and is used.In order to optimize
These design features are made full use of in journey as priori to improve the performance of algorithm, the present invention proposes diagonal based on joint
Change the alternating least-squares iteration algorithm of least square cost function, to realize the Joint diagonalization of fresh target matrix group.
The cost function form of common characterization Joint diagonalization degree mainly has：Information theory criterion cost function, F norms
Minimize criterion cost function and least square cost function.The Joint diagonalization least square used according to formula (8), the present invention
Cost function, is shown below：
Wherein, H representing matrixesEstimated matrix, HTRepresenting matrixTransposed matrixEstimated matrix, Λk, k=
1 ..., 2K represents diagonal matrix groupEstimated matrix, if diagonal matrix groupWithRepresent, thenI=1 ..., M is represented described in step 2 (T1)I=1 ..., M
Estimate；Represent Frobenius norms；
In formula (10), objective matrix groupIt is known, asks for H and Λk, k=1 ..., the estimation of 2K
Value so that H ΛkHT, k=1 ..., 2K is approached as much as possibleI.e. so that cost functionIt is small as far as possible,
It is the purpose of following steps of the process.By observing cost functionUnderstanding, it is biquadratic function for unknown matrix H, thus,
Computation complexity is higher, is not easy to ask for function smallest point.
For the ease of calculating, cost function is handled as follows：
In above formula, matrix F is introducedTForEstimated matrix so that cost function is by the biquadratic function on unknown matrix H
Deteriorate to respectively about H and FTQuadratic function, reduce solution difficulty.Moreover, understand H and F using the property of matrix decomposition
It is that essence is equal, i.e. H=EF, wherein E are Generalized Permutation Matrix, i.e. every a line in E is each to be shown and an only non-zero
Element.Moreover, H and FTObviously (T3) described matrix in should having rapid twoDesign feature, this design feature will be rear
It is used in continuous optimization process.
Step 4：Utilize alternately least-squares iteration Algorithm for Solving cost function.
Introduce a kind of alternating least-squares iteration algorithm based on gradient descent method, the alternative and iterative algorithm each round iteration
Three steps can be divided into, in each step, alternately estimate H and F, and diagonal matrix Λk, k=1 ..., 2K, search cost function J is most
Dot, and the design feature of involved each parameter is made full use of in the process.
Step 4.1:According to (T3) matrix in step 2Design feature, remember iterative initial value
Step 4.2:In the m times iteration, Joint diagonalization least square cost function is：Jk(H(m-1),Λk(m),K=1 ..., 2K, on Λk(m) smallest point is sought, obtains diagonal matrix Λk
(m), the estimate of k=1 ..., 2K, wherein, H (m-1) represents the estimate for the H that the m-1 times iteration obtains, and F (m-1) is represented
The estimate for the F that the m-1 times iteration obtains.Concrete methods of realizing is as follows：
To 2K Joint diagonalization least square cost function, respectively about Λk(m) M different diagonal entriesI=1 ..., M derivations, and it is zero to make derivative, obtains row vector described in step 2 (T1)Estimation
Wherein, (i, j) of matrix P a element pijFor：
Column vector qkI-th of element be：
In formula (13) and (14), hiThe i-th column vector of representing matrix H (m-1), parameter hi+M,hj,hj+MCan reasoning accordingly；fi
The i-th column vector of representing matrix F (m-1), fi,fi+M,fj,fj+MImplication can reasoning accordingly.So far,
Then diagonal matrix
Step 4.3：The diagonal matrix Λ that the estimate F (m-1) and step 4.2 of the F obtained using the m-1 times iteration is obtainedk
(m), k=1 ..., 2K, is asked so that Joint diagonalization least square cost function J (H (m), Λ1(m),…,Λ2K(m),Reach the H (m) during minimum value, concrete methods of realizing is as follows：
Remember H (m)=[H1(m),H2(m)], wherein H1(m) and H2(m) it is respectively the 1st row of H (m) to m column vector sum the
M+1 arranges 2M column vectors, andWherein,WithRespectively Gk(m) the 1st
To M row vectors and M+1 to 2M row vectors；
To Joint diagonalization least square cost function on H1(m) derivation and to make derivative be zero, due to the m times iteration
H2(m) not yet update, the iteration result H still taken turns using m-12(m-1), obtain：
Then by H1(m), according to (T3) matrix in step 2Structure it is special
Point, can obtain H2(m), and then H (m) is obtained.
Step 4.4：The Λ that the H (m) and step 4.2 obtained using step 4.3 is obtainedk(m), k=1 ..., 2K, ask so that
J(H(m),Λ1(m),…,Λ2K(m),Reach the F (m) during minimum value, implement
Method is as follows：
Remember F (m)=[F1(m),F2(m)], wherein F1(m) and F2(m) it is respectively the 1st row of F (m) to m column vector sum the
M+1 arranges 2M column vectors, andWhereinWithRespectively Lk(m)
1 arrives m column vector sum M+1 to 2M column vectors；
To Joint diagonalization least square cost function on F1(m) derivation and to make derivative be zero, in calculating process, by
In F2(m) not yet update, still using the iteration result F of the m-1 times2(m-1), obtain：
Then by F1(m), according to (T3) matrix in step 2Structure it is special
Point, can obtain F2(m), and then F (m) is obtained.
Step 4.5：Judge whether algorithm restrains, that is, judge whether to meetAndWherein δ is the threshold value of setting, is less positive number, is set to 0.01；If not, return to step
4.2；If it is, obtaining the estimation of final H and F, the two difference very little, can be used as matrixEstimation, according toObtain the estimation of aliasing matrix AAccording toObtain the estimation of source signalRealize blind source separating,
WhereinRepresentInverse matrix, x (t) represent observation signal.
Emulation experiment：
Two performance indicators are defined, first performance digit synbol is JH, is defined as：
Above-mentioned parameter JH is obviously consistent with the cost function value constructed by the method for the present invention, and the method for the present invention of reflection is repeatedly
The variation tendency of cost function value during generation, since parameter JH fails directly to embody the estimation performance to aliasing matrix,
Only it is applied in experiment one, to illustrate validity of the method for the present invention for the cost function shown in formula (11).
Second individual character energy index is known as the overall situation and refuses make an uproar horizontal (Global Rejection Level, GRL), mixed to weigh
Repeatedly Matrix Estimation valueDifference between actual value A, is defined as：
Wherein, gijRepresent matrixThe i-th row jth column element.
Experiment one：Utilize validity and convergence without noise targets matrix group verification algorithm.
20 independent experiments are run, that tests construction N × N every time is free of noise targets matrix group { Ck=A ΛkAH, k=
1 ... K }, wherein K=15, N=10, aliasing matrix A and diagonal matrix ΛkIt is the complex valued matrices randomly generated.What Fig. 2 was provided
It is that the overall situation refuses the performance curve that horizontal GRL changes with iterations of making an uproar, what abscissa represented is iterations, what ordinate represented
That the overall situation refuses the horizontal GRL that makes an uproar, in order to obtain this curve, it is necessary to each iteration (by m take turns iteration exemplified by) in, according to what is tried to achieve
Relational expression shown in H (m) convolutions (9)The aliasing matrix for obtaining current iteration is estimated
EvaluationAnd then the GRL of current iteration is tried to achieve according to formula (18).What Fig. 3 was provided is that the performance that parameter JH changes with iterations is bent
Line.Fig. 2 and Fig. 3 illustrate, having in noiseless, that is, objective matrix accurately can be in the case of Joint diagonalization structure, and the present invention calculates
Method reaches convergence with high precision.As shown in Figure 1, after algorithmic statement, aliasing Matrix Estimation valueDifference between actual value
Minimum, algorithm of the invention can effectively realize the estimation of aliasing matrix.
Experiment two：Made an uproar using the band of construction the rapidity and validity of objective matrix group verification algorithm.
Provide the target square formation group of N × N：
Ck=A ΛkAH+ΔCk(k=1 ..., K) (19)
Wherein K=15, N=10.Aliasing matrix A and diagonal matrix ΛkIt is the complex valued matrices randomly generated.Make an uproar to characterize
The intensity of sound matrix, will be free of noise section A ΛkAHWith noise section Δ CkRatio be expressed as NER：
In simulations, complex values noise matrix Δ C is randomly generatedk(k=1 ..., K) to meet NER=10dB respectively,
15dB, 20dB, 25dB.Under different signal-to-noise ratio NER values, be separately operable 20 independent experiments, every time independently realize GRL with
For the change curve of iterations as shown in figure 4, under difference NER values, average GRL is shown in Fig. 5 with the change curve of iterations,
Two figures show in the case of band is made an uproar, i.e. objective matrix be under non-critical diagonalizable structure situation carry algorithm still can be with
Degree of precision restrains.
CVFFDIAG methods are respectively adopted[11], FAJD methods[10]Distinguish with the method for the present invention (STBJD) in signal-to-noise ratio NER
For 15dB, in the case of 20dB, be separately operable 20 independent experiments, average GRL with iterations change curve such as Fig. 6 and figure
Shown in 7, the method for the present invention can be restrained compared with remaining two methods with degree of precision.
Experiment three：Verify validity of the method for the present invention when solving blind source separating problem.
Provide the complex value source signal of 4 zero-mean statistical iterations：s1(t)=sin (310 π t)+jcos (100 π t), s2(t)
=sin (180 π t)+jsin (400 π t), s3(t)=sin (20 π t) sin (600 π t)+jcos (20 π t) cos (600 π t), s4(t)
=sin [600 π t+6cos (120 π t)]+jcos (900 π t), hereinAssuming that 4 information sources are received by 4 array elements, aliasing
Channel parameter is the complex valued matrices A randomly generated, sample of signal number T=400, and the noise introduced in receive process is N=randn
(4,T)+jrandn(4,T).Signal-to-noise ratio settings are 10dB.Objective matrix receives the cross-correlation under signal difference time shift by asking for
Matrix obtains and selectes number K=8.Operation inventive algorithm realizes the estimation of aliasing matrix A, and recovers source signal.
The average GRL of 100 independent experiments is as shown in Figure 8 with the change curve of iterations.In 100 independent experiments
The 33rd experiment is randomly selected, checks the recovery effects of source signal：Fig. 9-Figure 11 represents the planisphere of 4 source signals respectively, 4
Receive the planisphere of signal, and the planisphere of 4 recovery signals.As seen from the figure, source signal be sufficiently mixed by aliasing matrix and
There are under larger noise situations, method of the invention remains to realize convergence and convergence error is smaller, and it is more smart to realize source signal
True blind separation.Meanwhile compare source signal shown in Fig. 9 with recovering signal shown in Figure 11, it can clearly find blind source separating field
Intrinsic arrangement ambiguity and scale ambiguity phenomenon.
Claims (4)
1. a kind of complex field blind source separation method, it is characterised in that comprise the following steps：
Step 1：According to the T observation signal sample of observation signal x (t)Construct complex field objective matrix group { Ck, k=
1 ... K }, and
Ck=ADkAH, k=1 ... K (2)
Wherein, Dk, k=1 ... K is diagonal matrix, A ∈ FM×N(M >=N) is aliasing matrix, AHFor the conjugate transposition of aliasing matrix A
Matrix；
Step 2：To the complex field objective matrix group { C of step 1 constructionk, k=1 ... K } real symmetrization is carried out, obtain structure again
The objective matrix group that the real value objective matrix made is formed
Step 3：Utilize objective matrix groupBuild Joint diagonalization least square cost function；
Step 4：Utilize the Joint diagonalization least square cost letter that alternately least-squares iteration Algorithm for Solving step 3 obtains
Number, obtains the estimation of aliasing matrix AAccording toObtain the estimation of source signalRealize blind source separating, whereinRepresentInverse matrix, x (t) represent observation signal.
2. complex field blind source separation method as claimed in claim 1, it is characterised in that the process of the step two includes：
The complex field objective matrix group { C constructed according to step 1k, k=1 ... K } according to formula (3) and (4) reconfigure 2K
Matrix groupWith
<mrow>
<msup>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
</mrow>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msup>
<mi>C</mi>
<mi>k</mi>
</msup>
<mo>+</mo>
<msup>
<mi>C</mi>
<mrow>
<mi>k</mi>
<mi>H</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<mi>Re</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>D</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
<msup>
<mi>A</mi>
<mi>H</mi>
</msup>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msup>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>j</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<msup>
<mi>C</mi>
<mi>k</mi>
</msup>
<mo>-</mo>
<msup>
<mi>C</mi>
<mrow>
<mi>k</mi>
<mi>H</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<mi>Im</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>D</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
<msup>
<mi>A</mi>
<mi>H</mi>
</msup>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, CkHRepresent CkAssociate matrix, Re (Dk) represent diagonal matrix DkReal part, Im (Dk) represent diagonal matrix Dk
Imaginary part；
Remember that matrixing function is f (X)：
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>Re</mi>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>Im</mi>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mi>Im</mi>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>Re</mi>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
By formula (3) and formula (5), obtain
<mrow>
<msup>
<mover>
<mi>C</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
</mrow>
</msup>
<mover>
<mo>=</mo>
<mi>&Delta;</mi>
</mover>
<mi>f</mi>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>Re</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>D</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mi>Re</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>D</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<msup>
<mi>f</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
</mrow>
</msup>
<msup>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, fT(A) transposition for being f (A),
By formula (4) and (5), obtain
<mrow>
<msup>
<mover>
<mi>C</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mover>
<mo>=</mo>
<mi>&Delta;</mi>
</mover>
<mi>f</mi>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>Im</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>D</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mi>Im</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>D</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<msup>
<mi>f</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
The objective matrix group being made of formula (6) and formula (7), the real value objective matrix that must be reconfigured
<mrow>
<msup>
<mover>
<mi>C</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msup>
<mo>=</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msup>
<msup>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mn>2</mn>
<mi>K</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, RepresentTransposition；With can Joint diagonalization structure,For
Diagonal matrix；
3. complex field blind source separation method as claimed in claim 2, it is characterised in that the process of the step three includes：
Build Joint diagonalization least square cost function J (H, Λ1,…,Λ2K,F)：
<mrow>
<mi>J</mi>
<mrow>
<mo>(</mo>
<mi>H</mi>
<mo>,</mo>
<msup>
<mi>&Lambda;</mi>
<mn>1</mn>
</msup>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msup>
<mi>&Lambda;</mi>
<mrow>
<mn>2</mn>
<mi>K</mi>
</mrow>
</msup>
<mo>,</mo>
<mi>F</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>2</mn>
<mi>K</mi>
</mrow>
</munderover>
<mo>|</mo>
<mo>|</mo>
<msup>
<mover>
<mi>C</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msup>
<mo>-</mo>
<msup>
<mi>H&Lambda;</mi>
<mi>k</mi>
</msup>
<msup>
<mi>F</mi>
<mi>T</mi>
</msup>
<mo>|</mo>
<msubsup>
<mo>|</mo>
<mi>F</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, H representing matrixesEstimated matrix, Λk, k=1 ..., 2K represents diagonal matrix groupEstimation square
Battle array；Represent Frobenius norms；FTForEstimated matrix,RepresentTransposed matrix, and H=EF, E put for broad sense
Change matrix.
4. complex field blind source separation method as claimed in claim 3, it is characterised in that the process of the step 4 includes：
Step 4.1:Remember iterative initial value
Step 4.2:In the m times iteration, Joint diagonalization least square cost function is：On Λk(m) smallest point is sought, is obtained
To diagonal matrix Λk(m), the estimate of k=1 ..., 2K, wherein, H (m-1) represents the estimate for the H that the m-1 times iteration obtains,
F (m-1) represents the estimate for the F that the m-1 times iteration obtains；
Step 4.3：The diagonal matrix Λ that the estimate F (m-1) and step 4.2 of the F obtained using the m-1 times iteration is obtainedk(m),k
=1 ..., 2K, is asked so that Joint diagonalization least square cost functionReach the H (m) during minimum value；
Step 4.4：The Λ that the H (m) and step 4.2 obtained using step 4.3 is obtainedk(m), k=1 ..., 2K, ask so thatReach the F (m) during minimum value；
Step 4.5：Judge whether algorithm restrains, that is, judge whether to meetAnd
Wherein δ is the threshold value of setting；If conditions are not met, return to step 4.2；If it is satisfied, then the estimation of final H and F are obtained, two
Person can be used as matrixEstimation, according toObtain the estimation of aliasing matrix AAccording toObtain source
The estimation of signalRealize blind source separating, whereinRepresentInverse matrix, x (t) represent observation signal.
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