CN110058192A - Atom norm mutual coupling DOA estimation method based on auxiliary array element - Google Patents
Atom norm mutual coupling DOA estimation method based on auxiliary array element Download PDFInfo
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- G—PHYSICS
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- 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
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- G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
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Abstract
The invention belongs to array signal processing fields, to propose the super-resolution DOA estimation technique scheme to far field narrowband information source, the present invention, atom norm mutual coupling DOA estimation method based on auxiliary array element, first, by analyzing the Toeplitz structure without the mutual coefficient matrix MCM in the case of auxiliary array element, construction selection matrix is constructed received data are truncated by symmetrical mode, so that remaining data has the mutual coupling coefficient of circulation；Then, by covering unknown the mutual coupling coefficient in information source part, the new steering vector of truncated data is obtained；Finally, according to the frame of atom norm minimum ANM, the Semidefinite Programming SDP for carrying out mutual coupling DOA estimation model is solved, the frequency content that x (t) is included is estimated using the Prony method for describing equidistant sampled data, then according to the corresponding relationship of frequency f and incidence angle θ, the Mutual coupling of the far field narrowband information source of input is completed.Present invention is mainly applied to array signal processing occasions.
Description
Technical field
The invention belongs to array signal processing fields, can be on continuous domain directly to the frequency of signal, phase according to atom norm
The unknown parameters such as position the characteristics of being estimated, under conditions of considering mutual coupling factor, ULA (Uniform Linear Array,
Even linear array) both ends increase auxiliary array element after, completed using atom norm parameter Estimation frame and far field narrow band signal surpassed
The DOA (Direction of Arrival, direction of arrival) of resolution ratio estimates.More particularly to the atom norm based on auxiliary array element
Mutual coupling DOA estimating system and method.
Background technique
DOA estimation has a wide range of applications under the scenes such as radar, sonar, channel detection, wireless communication[1].And two or
The super-resolution problem of multiple direction of arrival for closing on information source is all the research hotspot in array signal processing all the time, furthermore
How to overcome the mutual coupling effect between the antenna element of the array under actual conditions is another that DOA estimation is carried out to information source
Hot spot.
For the super-resolution estimation problem of direction of arrival, with compressive sensing theory[2,3]Proposition, be based on grid dividing
Rarefaction representation class method for parameter estimation[4,5]Breach MUSIC (Multiple Signal Classificaion, multiple signal
Classification) algorithm and ESPRIT (Estimation of Signal Parameters via Rotational Invariance
Techniques, the Signal parameter estimation of revolution space invariance) traditional DOA algorithm for estimating such as algorithm[1]Performance.In base
In the rarefaction representation class method for parameter estimation algorithm of grid dividing, the parameter space of estimative parameter is discretized as a system
The grid of column, these grids constitute an excessively complete dictionary, and parameter to be estimated only exists on a small number of grids with these grids,
It then treats and estimates the estimation problem of parameter and translated into the sparse Restoration model of parameter.But the sparse table based on grid dividing
Show class parameter Estimation class algorithm there are high calculation amount brought by grid dividing, be limited to RIP (Restricted Isometry
Property, limited equidistant condition) criterion and base the disadvantages of mismatching, this makes the resolution ratio of such method not high.
To solve the problems, such as that resolution ratio brought by the scatter estimation to continuous parameter is not high, Tang et al. exists first
The theory for the atom norm that Chandrasekaran et al. is proposed[6]On the basis of, in conjunction with Toeplitz (Teoplitz) matrix point
Solve theorem[7,8], propose the method for parameter estimation of gridless (mesh free) class ultrahigh resolution[9](hereinafter referred to as it is based on atom
The parameter estimation algorithm of norm).The technology path of parameter estimation algorithm based on atom norm includes four steps: 1) company of foundation
Signal model on continuous domain；2) construction of atom is carried out according to signal model；3) atom norm minimum problem is converted into phase
SDP (SemiDefinite Programming, the semi definite programming) problem answered, and solved；4) according to optimization in 3)
The result come completes last parameter Estimation.Because the parameter estimation algorithm based on atom norm directly continuous domain to parameter into
Row estimation, thus the characteristic with ultrahigh resolution.
In mutual coupling DOA estimation problem, due to the influence of mutual coupling, traditional MUSIC algorithm and ESPRIT algorithm can fail.
For this purpose, document [10,11] uses auxiliary array element to eliminate mutual coupling interference, MUSIC algorithm and ESPRIT algorithm are then used again
Carry out the parameter Estimation of MIMO radar.Architectural characteristic in view of mutual coefficient matrix and the rarefaction representation class based on grid dividing
The development of algorithm then takes L1_SVD (L1_Singular Value by being truncated to receiving signal in document [12]
Decomposition, L1- singular value decomposition) algorithm[4]Carry out DOA estimation；Document [13] is proposed to be estimated based on the sparse DOA of block
Calculating method；Document [14] proposes one kind and is based on after carrying out structure compensation to mutual coefficient matrix using auxiliary array element
The DOA estimation method of ESPRIT algorithm；Document [15] proposes for the first time carries out mutual coupling feelings using the algorithm based on atom norm
DOA estimation under condition.
[1]H.Krim and M.Viberg,“Two decades of array signal processing
research:The parametric approach,”IEEE Signal Process.Mag.,vol.13,no.4,pp.67-
94,Jul.1996.
[2]D.Donoho,“Superresolution via sparsity constraints,”SIAM Journal
on Mathematical Analysis,vol.23,pp.1309-1331,Sep.1992.
[3]E.Candes,“Compressive sampling,”in Proc.of the International
Congress of Mathematicians:Madrid,August 22-30,2006:invited lectures,2006,
pp.1433-1452.
[4]D.Malioutov,M.Cetin,and A.S.Willsky,“Asparse signal reconstruction
perspective for source localization with sensor arrays,”IEEE Transactions on
Signal Processing,vol.53,no.8,pp.3010-3022.
[5]M.M.Hyder and K.Mahata,“Direction-of-arrival estimation using a
mixed l2,0norm approximation,”IEEE Transactions on Signal Processing,vol.58,
no.9,pp.4646-4655,Sep.2010.
[6]V.Chandrasekaran,B.Recht,P.A.Parrilo,et al.The Convex algebraic
geometry of linear inverse problems[C].48th Annual Allerton Conference on
Communication,Control,and Computing(Allerton),Allerton,IL,2010:699-703.
[7]U.Grenander and G.“Toeplitz forms and their applications,”
Chelsea Publishing Co New York,1958.
[8]Z.Yang,L.Xie,and P.Stoica,“Vandermonde decomposition of multilevel
Toeplitz matrices with application to multidimensional super-resolution,”IEEE
Transactions on Information Theory,vol.PP,no.99,pp.1–1,2016
[9]G.Tang,B.N.Bhaskar,P.Shah,and B.Recht,“Compressed sensing off the
grid,”IEEE Transactions on Information Theory,vol.59,no.11,pp.7465-7490,
Nov.2013.
[10]Z.Zheng,J.Zhang,and Y.Wu,“Multi-target localization for bistatic
MIMO radar in the presence of unknown mutual coupling,”Journal of Systems
Engineering and Electronics,vol.23,no.5,pp.708-714,2012
[11]Z.Zheng,J.Zhang,and C.Niu,“Angle estimation of bistatic MIMO
radar in the presence of unknown mutual coupling,”in Proceedings of 2011IEEE
CIE International Conference on Radar,vol.1,2011,pp.55-58.
[12]J.Dai,D.Zhao,and X.Ji,“Asparse representation method for
DOAestimation with unknown mutual coupling,”IEEE Antennas and Wireless
Propagation Letters,vol.11,pp.1210-1213,2012.
[13]Q.Wang,T.Dou,H.Chen,W.Yan and W.Liu,“Effective Block Sparse
Representation Algorithm for DOA Estimation With Unknown Mutual Coupling,”
IEEE Communications Letters,vol.21,no.12,pp.2622-2625,Dec.2017.
[14]L.Hao and W.Ping,“DOA estimation in an antenna array with mutual
coupling based on ESPRIT,”Proc.International Workshop on Microwave and
Millimeter Wave Circuits and System Technology,pp.86-89,2013.
[15]Q.Wang,X.Wang,T.Dou,H.Chen and X.Wu,“Gridless Super-resolution
DOA estimation with Unknown Mutual Coupling,”submitted to IEEE
Int.Conf.Acoust.,Speech,Signal Process(ICASSP)2019..
Summary of the invention
In order to overcome the deficiencies of the prior art, the present invention is directed to study in the case where even linear array, there are when mutual coupling effect,
It is proposed the super-resolution DOA estimation technique scheme to far field narrowband information source.For this reason, the technical scheme adopted by the present invention is that being based on
The atom norm mutual coupling DOA estimation method of array element is assisted, firstly, by analysis without the mutual coupling coefficient square in the case of auxiliary array element
The Toeplitz structure of battle array MCM (Mutual Coupling Matrix), constructs construction selection matrix by symmetrical mode
Received data are truncated, so that remaining data has the mutual coupling coefficient of circulation；Then, by including by unknown the mutual coupling coefficient
Into information source part, the new steering vector of truncated data is obtained；Finally, according to atom norm minimum ANM (Atomic
Norm Minimization) frame, the Semidefinite Programming SDP for carrying out mutual coupling DOA estimation model is solved, using using exponential function
Linear combination estimate the frequency content that x (t) is included to describe the Prony method of equidistant sampled data, then according to frequency
The corresponding relationship of rate f and incidence angle θ complete the Mutual coupling of the far field narrowband information source of input.
Specifically:
1) signal model is established
For an even linear array ULA being made of M isotropic antenna (Uniform Linear Array), and have
N number of far field narrow band signal sn(t), n=1,2 ..., N is with angle, θ1,θ2,…,θNIt is incident on the ULA gusts, if between array element
Away from d=2/ λ；
When not considering mutual coupling factor, under the sampling of single snap, the ULA gusts of reception signal is
X (t)=A (θ) s (t)+n (t) (1)
Wherein x (t)=[x1(t),x2(t),…,xM(t)]TFor output signal of the M array element under this time sampling；For array manifold, steering vector Incoming signal is s (t)=[s1(t),s2(t),…,sN(t)]T；Noise vector is n (t)=[n1
(t),n2(t),…,nM(t)]T；
To make with down conversion convenient for analysis:
Work as θn∈[-90°,90°] (2)
Then x (t) is rewritten are as follows:
X (t)=As (t)+n (t) (3)
Whereina(fn)=[1, β (fn),…,β(fn)M-1]T, β (fn)=
exp(-j2πfn)；
Consider mutual coupling factor, single reception signal taken fastly are as follows:
X (t)=CAs (t)+n (t) (4)
Wherein
Correspondingly, consider mutual coupling factor, the receipt signal model in the case of T snap are as follows:
Wherein
2) atomic structre
Atomic structre in the case of single snap:
For the structure of mutual coefficient matrix C, between the row and row of preceding P-1 row to the mutual coupling coefficient between rear P-1 row
There is recurrence relations, use auxiliary array element as a result, to carry out " symmetrical expression supplement " to C, obtainWhereinIn coefficient c1、
cP-1To use the symmetry coefficient newly increased after auxiliary array element,
At this timeEach row between there is recurrence relations, then increase auxiliary array element after, M array element of former array
Output signal are as follows:
Wherein,
?In, steering vector is
Pass through observationStructure, obtain
WhereinFor scalar,
Further, by the way that H ' (f) is covered information source part, formula (8) is converted into
Wherein
For formula (11), it is without the reception data in the case of making an uproar
Wherein,It indicatesI-th column；It indicatesI-th of element；And | φ 'i|=1.
Then the atom in the case of single snap is defined as
Wherein [0,1] f ∈,And | φ ' |=1
Then the atom collection in the case of single snap is defined as
ThenAtom norm be defined as
Its semi definite programming SDP (SemiDefinite Programming) form of equal value is
Accordingly for how soon clap the case where, received signal is expressed as
Receiving matrix under its noise-free case is
WhereinIt isI-th column,And | | φi′||2=1
Atom in the case of then how soon clapping is defined as
Wherein [0,1] f ∈,||φ′||2=1
Atom collection in the case of then how soon clapping is defined as
ThenAtom norm be defined as
The SDP form of formula (21) equivalence is
3) SDP is solved
By formula (11), obtaining the optimization problem based on atom norm in the case of single snap is
Wherein τ is regularisation parameter；
By formula (16), obtaining SDP problem corresponding to formula (23) is
Accordingly for how soon clapping situation, the optimization problem based on atom norm is
By formula (22), the SDP problem of equal value for obtaining formula (25) is
By solving formula (24) (or formula (26), in the case of how soon clapping), Toeplitz matrix T (u) can be obtained；
4) Prony method seeks frequency
According to by step 1) to the T (u) found out in 3), it is used Prony method estimate frequency that x (t) is included at
Point, then according to the corresponding relationship of frequency f and incidence angle θ, see formula (2), that is, the wave for completing the far field narrowband information source of input reaches side
To estimation.
The features of the present invention and beneficial effect are:
Application in the quasi- DOA estimation for exploring atom norm there are mutual coupling of the present invention, according to mutual coupling situation
The lower structure for receiving signal is analyzed, and according to the theoretical frame of atom norm, completes atomic structre, SDP problem solving
With Frequency Estimation, the ultrahigh resolution for realizing the direction of arrival in even linear array to far field narrow band signal is estimated
Meter.
Detailed description of the invention:
Fig. 1 even linear array layout.
Fig. 2 increases mutual coupling effect schematic diagram before and after auxiliary array element.
(a) do not increase mutual coupling effect schematic diagram when auxiliary array element (by taking the array element of serial number 1 as an example).
(b) the mutual coupling effect schematic diagram after increasing auxiliary array element (by taking the array element of serial number 1 as an example).
Fig. 3 RMSEs (Root Mean Square Errors, root-mean-square error) is with SNR (Signal to Noise
Ratio, signal-to-noise ratio) change curve: where array number M=15, number of snapshots 16, array number N=3, and frequency to be estimated is
F1=0.15, f2=0.19, f3=0.80, the mutual coupling coefficient c1=0.5844-0.5476j, c2=0.2625+0.1414j, c3
=0.1163-0.1289j carries out Mcr=100 Monte Carlo simulation.
Fig. 4 RMSEs (Root Mean Square Errors, root-mean-square error) is with the change curve of number of snapshots: wherein,
Array number is M=15, array number N=3, and frequency to be estimated is f1=0.15, f2=0.19, f3=0.80, the mutual coupling coefficient c1
=0.5844-0.5476j, c2=0.2625+0.1414j, c3=0.1163-0.1289j, SNR are fixed as 10dB, carry out Mcr
=100 Monte Carlo simulations.
Specific embodiment
Aiming at the problem that present invention aperture loss present in document [15], propose first with auxiliary array element to mutual coupling system
Matrix number carries out structure compensation, and the algorithm of DOA estimation is then carried out using atom norm frame.
The technical scheme is that firstly, by analysis without the mutual coefficient matrix in the case of auxiliary array element
The Toeplitz structure of (Mutual Coupling Matrix, MCM), by symmetrical mode construct construction selection matrix come
Received data are truncated, so that remaining data has the mutual coupling coefficient of circulation.Then, by covering unknown the mutual coupling coefficient
In information source part, the new steering vector of truncated data is obtained.Finally, according to atom norm minimum (Atomic Norm
Minimization, ANM) frame, propose the mutual coupling DOA estimation model Semidefinite Programming (SDP) problem.In addition,
The simulating, verifying validity of the algorithm is carried out on matlab.
Present invention be described in more detail with specific example with reference to the accompanying drawing.
1.. signal model is established
Comparative diagram (1), it is assumed that there are an even linear array (Uniform Linear being made of M isotropic antenna
Array, ULA), and have N number of far field narrow band signal sn(t), (n=1,2 ..., N) with angle, θ1,θ2,…,θNIt is incident on the ULA
Battle array.If the spacing d=2/ λ between array element.
When not considering mutual coupling factor, under the sampling of single snap, the ULA gusts of reception signal is
X (t)=A (θ) s (t)+n (t) (1)
Wherein x (t)=[x1(t),x2(t),…,xM(t)]TFor output signal of the M array element under this time sampling；For array manifold, steering vector Incoming signal is s (t)=[s1(t),s2(t),…,sN(t)]T；Noise vector be n (t)=
[n1(t),n2(t),…,nM(t)]T。
To make with down conversion convenient for analysis:
Work as θn∈[-90°,90°] (2)
Then x (t) is rewritten are as follows:
X (t)=As (t)+n (t) (3)
Whereina(fn)=[1, β (fn),…,β(fn)M-1]T, β (fn)=
exp(-j2πfn)。
Consider mutual coupling factor, single reception signal taken fastly are as follows:
X (t)=CAs (t)+n (t) (4)
Wherein
Correspondingly, consider mutual coupling factor, how soon clap the receipt signal model under situation (T snap) are as follows:
Wherein
X (t)=[x1(t),x2(t),…,xM(t)]TWithIt is to believe in array respectively
The mathematical model of information received by the array element of array in the case where commonly indicating single snap in number processing and how soon clap.
Specifically:
X (t)=[x1(t),x2(t),…,xM(t)]TBe array M array element unitary sampling (or be single snap,
Single snapshot) under the data of vector form that receive, t therein desirable 1,2 ..., T successively indicate 1 sampling,
2 samplings ..., T sampling.Specifically, when exactly taking t=1, there is x (1)=[x1(1),x2(1),…,xM(1)]T, similarly
Have as t=2, x (2)=[x1(2),x2(2),…,xM(2)]T... ..., there is x (T)=[x as t=T1(T),x2(T),…,
xM(T)]T.The reception signal that this T times samples the array element under (or T snap) is just obtained altogetherThat is X is signal received by the array element of the expression array obtained under T sampling
Matrix.
2.. atomic structre
Atomic structre problem in the case of single snap is discussed first.
The structure of mutual coefficient matrix C is analyzed, finds the mutual coupling system of intermediate portion (preceding P-1 row is between rear P-1 row)
There is recurrence relations between several rows and row, for this purpose, being considered as auxiliary array element to carry out " symmetrical expression supplement " to C, obtainWhereinIn coefficient c1、cP-1For using the symmetry coefficient newly increased after auxiliary array element.It is noted that at this timeIt is each
There is recurrence relations between row.Increase the mutual coupling effect figure before and after assisting array element as shown in figure (2).
After then increasing auxiliary array element, the output signal of M array element of former array are as follows:
Wherein,
?In, steering vector is
Pass through observationStructure, can obtain
WhereinFor scalar,
Further, by the way that H ' (f) is covered information source part, formula (8) can be converted into
Wherein
For formula (11), it is without the reception data in the case of making an uproar
Wherein,It indicatesI-th column；It indicatesI-th of element；And | φ 'i|=1.
Then the atom in the case of single snap is defined as
Wherein [0,1] f ∈,And | φ ' |=1.
Then the atom collection in the case of single snap is defined as
ThenAtom norm be defined as
Its SDP (SemiDefinite Programming, semi definite programming) form of equal value is
Accordingly for how soon clap the case where, received signal is represented by
Receiving matrix under its noise-free case is
WhereinIt isI-th column,And | | φi′||2=1.
Atom in the case of then how soon clapping is defined as
Wherein [0,1] f ∈,||φ′||2=1.
Atom collection in the case of then how soon clapping is defined as
ThenAtom norm be defined as
The SDP form of formula (21) equivalence is
3..SDP it solves
By formula (11), obtaining the optimization problem based on atom norm in the case of single snap is
Wherein τ is regularisation parameter.
By formula (16), obtaining SDP problem corresponding to formula (23) is
Accordingly for how soon clapping situation, the optimization problem based on atom norm is
By formula (22), the SDP problem of equal value for obtaining formula (25) is
By solving formula (24) (or formula (26), in the case of how soon clapping), Toeplitz matrix T (u) can be obtained.
4..Prony method seeks frequency
According to the T (u) by finding out in 1 to 3, Prony method is used to it, and (note: Prony algorithm is the line using exponential function
Property combination the mathematical model of equidistant sampled data is described, using the frequency of least square method estimation Setting signal, amplitude and
The information such as phase) frequency content that x (t) is included can be estimated, then according to the corresponding relationship of f and θ (see formula (2)), i.e.,
Complete the Mutual coupling of the far field narrowband information source of input.
5.. simulating, verifying
For the performance for verifying the mutual coupling DOA algorithm for estimating based on atom norm proposed, following two groups of experiments have been carried out.
Wherein, the calculation formula of RMSE isWherein Mcr is Meng Teka
Lip river simulation times, N are number of source, fnFor frequency to be estimated,For under i-th Monte Carlo simulation to frequency fnEstimated value.
(comparative diagram (3)) in the first set of experiments have probed into the performance of the algorithm with the situation of change of signal-to-noise ratio, experiment
The result shows that the algorithm performance in the algorithm and document [15] improves with the raising of signal-to-noise ratio, and compared to document
[15] algorithm in, algorithm proposed by the present invention have better performance.
(comparative diagram (4)) in the second set of experiments have probed into the performance of the algorithm with the situation of change of number of snapshots, experiment
The result shows that the performance of the algorithm and the algorithm in document [15] can just reach preferable performance under less number of snapshots, but
With the increase of number of snapshots, RMSEs variation degree is unobvious, and algorithm generally proposed by the invention is at identical conditions
Performance it is always better than the algorithm in document [15].Illustrate that the algorithm has a good application prospect in the case where low snap.
Claims (2)
1. a kind of atom norm mutual coupling DOA estimation method based on auxiliary array element, characterized in that firstly, by analysis without auxiliary
The Toeplitz structure of the mutual coefficient matrix MCM (Mutual Coupling Matrix) to cheer in first situation, by symmetrical
Mode construct construction selection matrix and be truncated received data so that remaining data has the mutual coupling coefficient of circulation；Then,
By covering unknown the mutual coupling coefficient in information source part, the new steering vector of truncated data is obtained；Finally, according to atom
The frame of norm minimum ANM (Atomic Norm Minimization) carries out the Semidefinite Programming of mutual coupling DOA estimation model
SDP is solved, and estimating x (t) using the Prony method for describing equidistant sampled data with the linear combination of exponential function is included
Frequency content, then according to the corresponding relationship of frequency f and incidence angle θ, the wave for completing the far field narrowband information source of input reaches
Direction estimation.
2. the atom norm mutual coupling DOA estimation method as described in claim 1 based on auxiliary array element, characterized in that specifically:
1) signal model is established
For an even linear array ULA being made of M isotropic antenna (Uniform Linear Array), and have N number of
Far field narrow band signal sn(t), n=1,2 ..., N is with angle, θ1,θ2,…,θNIt is incident on the ULA gusts, if the spacing d between array element
=2/ λ；
When not considering mutual coupling factor, under the sampling of single snap, the ULA gusts of reception signal is
X (t)=A (θ) s (t)+n (t) (1)
Wherein x (t)=[x1(t),x2(t),…,xM(t)]TFor output signal of the M array element under this time sampling；For array manifold, steering vector Incoming signal is s (t)=[s1(t),s2(t),…,sN(t)]T；Noise vector is n (t)=[n1
(t),n2(t),…,nM(t)]T；
To make with down conversion convenient for analysis:
Then x (t) is rewritten are as follows:
X (t)=As (t)+n (t) (3)
Whereina(fn)=[1, β (fn),…,β(fn)M-1]T, β (fn)=exp (-
j2πfn)；
Consider mutual coupling factor, single reception signal taken fastly are as follows:
X (t)=CAs (t)+n (t) (4)
Wherein
Correspondingly, consider mutual coupling factor, the receipt signal model in the case of T snap are as follows:
Wherein
2) atomic structre
Atomic structre in the case of single snap:
For the structure of mutual coefficient matrix C, exist between the row and row of preceding P-1 row to the mutual coupling coefficient between rear P-1 row
Recurrence relation, use auxiliary array element to carry out " symmetrical expression supplement " to C as a result, obtainWhereinIn coefficient c1、cP-1
To use the symmetry coefficient newly increased after auxiliary array element,
At this timeEach row between there is recurrence relations, then increase auxiliary array element after, M array element of former array it is defeated
Signal out are as follows:
Wherein,
?In, steering vector is
Pass through observationStructure, obtain
WhereinFor scalar,
Further, by the way that H ' (f) is covered information source part, formula (8) is converted into
Wherein
For formula (11), it is without the reception data in the case of making an uproar
Wherein,It indicatesI-th column；It indicatesI-th of element；And
Then the atom in the case of single snap is defined as
Wherein [0,1] f ∈,And | φ ' |=1
Then the atom collection in the case of single snap is defined as
ThenAtom norm be defined as
Its semi definite programming SDP (SemiDefinite Programming) form of equal value is
Accordingly for how soon clap the case where, received signal is expressed as
Receiving matrix under its noise-free case is
WhereinIt isI-th column,And | | φi′||2=1
Atom in the case of then how soon clapping is defined as
Wherein [0,1] f ∈,||φ′||2=1
Atom collection in the case of then how soon clapping is defined as
ThenAtom norm be defined as
The SDP form of formula (21) equivalence is
3) SDP is solved
By formula (11), obtaining the optimization problem based on atom norm in the case of single snap is
Wherein τ is regularisation parameter；
By formula (16), obtaining SDP problem corresponding to formula (23) is
Accordingly for how soon clapping situation, the optimization problem based on atom norm is
By formula (22), the SDP problem of equal value for obtaining formula (25) is
By solving formula (24) (or formula (26), in the case of how soon clapping), Toeplitz matrix T (u) can be obtained；
4) Prony method seeks frequency
To the T (u) found out in 3), Prony method is used to estimate the frequency content that x (t) is included to it, so according to by step 1)
Afterwards according to the corresponding relationship of frequency f and incidence angle θ, formula (2) are seen, that is, the direction of arrival for completing the far field narrowband information source of input is estimated
Meter.
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