CN107800658B - Two-dimensional harmonic signal frequency estimation method - Google Patents

Two-dimensional harmonic signal frequency estimation method Download PDF

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CN107800658B
CN107800658B CN201711105306.3A CN201711105306A CN107800658B CN 107800658 B CN107800658 B CN 107800658B CN 201711105306 A CN201711105306 A CN 201711105306A CN 107800658 B CN107800658 B CN 107800658B
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杨世永
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Abstract

The invention relates to a letterThe field of signal processing, in particular to a two-dimensional harmonic signal frequency estimation method, which comprises the following steps: calculating covariance c (s, t); constructing a matrix F; decomposing the eigenvalue of the matrix F, and recording the obtained eigenvalue as lambda according to the magnitude sequence1,λ2,…,λKThe corresponding eigenvalue vector is denoted as e1,e2,…,eK(ii) a Constructing a matrix U; calculating a polynomial r (z); calculating an estimate of the first harmonic frequency component; constructing a matrix G; decomposing the eigenvalue of the matrix G, and recording the obtained eigenvalue as gamma from large to small1,γ2,…,γLThe corresponding eigenvalue vector is denoted d1,d2,…,dLConstructing matrix V, calculating polynomial S (z), calculating the estimated value of the second harmonic frequency component, constructing matrix H, decomposing the characteristic values of matrix H, and recording the obtained characteristic values as η1,η2,…,ηKLThe corresponding eigenvalue vector is denoted b1,b2,…,bKL(ii) a Constructing a matrix W; calculating an estimator Qk,l(ii) a An estimate of the two-dimensional harmonic frequency pair is calculated. The method can realize super-resolution frequency estimation of the two-dimensional harmonic signals and has the characteristics of high estimation precision and low calculation complexity.

Description

Two-dimensional harmonic signal frequency estimation method
Technical Field
The invention relates to the field of signal processing, in particular to a two-dimensional harmonic signal frequency estimation method.
Background
The problem of parameter estimation of two-dimensional harmonic signals is widely applied in many signal processing fields, and mainly is to estimate the frequency of two-dimensional harmonic signals from observation signals polluted by noise.
At present, the Estimation method of Two-dimensional harmonic Signal frequency mainly includes matrix enhancement and matrix beam method (y.hua, Estimating Two-dimensional frequencies by matrix enhancement and matrix beam, IEEE transformation on Signal Processing, vol.40, No.9, pp.2267-2279,1992.), Two-dimensional MUSIC method (j.w.endpoint, e.barnard, and c.w.i.pitch, Two-dimensional reconstruction using the MUSIC analysis, IEEE transformation on noise amplification, vol.42, No.10, pp.1386-1391,1994.), high-order statistical method (j.m.audio g.b.g. transmission frequencies and 2-simulation amplifying, vol.313, echo-noise, system, echo-noise, echo-3, echo-1391,1994, noise-noise, noise, IEEE Transaction on Signal Processing, vol.49, No.1, pp.237-245,2001.) and modified matrix bundle method (F.J.Chen, C.C.Fung, C.W.Kok, and S.KWong, Estimation of two-dimensional frequency using modified matrix Processing, IEEETransaction on Signal Processing, vol.22, No.2, pp.718-724,2007.). However, these methods have some disadvantages, such as low frequency estimation resolution or high computational complexity. Therefore, it is necessary to solve the above-mentioned technical problems.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, adapt to the practical needs and provide a two-dimensional harmonic signal frequency estimation method.
In order to realize the purpose of the invention, the technical scheme adopted by the invention is as follows:
a two-dimensional harmonic signal frequency estimation method is designed, and comprises the following steps:
1) calculating covariance c (s, t);
2) constructing a matrix F;
3) decomposing the eigenvalue of the matrix F, and recording the obtained eigenvalue as lambda from large to small1,λ2,…,λKThe corresponding eigenvalue vector is denoted as e1,e2,…,eK；
4) Constructing a matrix U;
5) calculating a polynomial r (z);
6) calculating an estimate of the first harmonic frequency component;
7) constructing a matrix G;
8) decomposing the eigenvalue of the matrix G, and recording the obtained eigenvalue as gamma from large to small1,γ2,…,γLThe corresponding eigenvalue vector is denoted d1,d2,…,dL；
9) Constructing a matrix V;
10) calculating a polynomial s (z);
11) calculating an estimate of the second harmonic frequency component;
12) constructing a matrix H;
13) the matrix H is subjected to eigenvalue decomposition, and the obtained eigenvalues are recorded in descending order η1,η2,…,ηKLThe corresponding eigenvalue vector is denoted b1,b2,…,bKL；
14) Constructing a matrix W;
15) calculating an estimator Qk,l；
16) An estimate of the two-dimensional harmonic frequency is calculated.
Setting the MN data measurement values of the two-dimensional harmonic signals as x (M, n), wherein M is 1,2, …, M; n is 1,2, …, N, the number of two-dimensional harmonic components is P, and for one harmonic component is more than 2P and less than 2P
Figure BDA0001464196210000031
And one is greater than 2P and less than
Figure BDA0001464196210000032
The integer L, covariance c (s, t) of (a) is calculated by:
Figure BDA0001464196210000033
wherein s is more than or equal to 1-K and less than or equal to K-1, t is more than or equal to 1-L and less than or equal to L-1 (·)*Representing a complex conjugate.
Constructing a K multiplied by K matrix F by using the covariance c (s,0) obtained by the calculation in the step 1, wherein s is more than or equal to 1 and is less than or equal to K-1, namely:
Figure BDA0001464196210000034
the method for constructing the matrix U comprises the following steps: using the feature vector e calculated in step 3P+1,eP+2,…,eKConstructing matrix U ═ eP+1,eP+2,…,eK]；
The method for constructing the matrix G comprises the following steps: constructing an L multiplied by L matrix G by using the covariance c (0, t) obtained by the calculation in the step 1, wherein t is more than or equal to 1 and is less than or equal to L-1, namely:
Figure BDA0001464196210000035
the method for constructing the matrix V comprises the following steps: using the feature vector d calculated in step 8P+1,dP+2,…,dLThe construction matrix V ═ dP+1,dP+2,…,dL]；
The method for constructing the matrix H comprises the following steps: constructing a KL multiplied by KL matrix H by using the covariance c (s, t) obtained by the calculation in the step 1, wherein s is more than or equal to 1 and is less than or equal to K-1, and t is more than or equal to 1 and is less than or equal to L-1, namely:
Figure BDA0001464196210000041
wherein
Figure BDA0001464196210000042
The method for constructing the matrix W comprises the following steps: using the feature vector b calculated in step 13P+1,bP+2,…,bKLThe construction matrix W ═ bP+1,bP+2,…,bKL]。
Let r (z) ═ 1, z, …, zK-1]TCalculating the polynomial R (z) ═ rT(z-1)UUHr (z), wherein (·)TShowing transposition, (.)HRepresenting a conjugate transpose.
The method for calculating the estimated value of the first harmonic frequency component comprises the following steps: all roots of R (z) ═ 0 were obtained, and the P roots located in the unit circle and closest to the unit circle were designated as r1,r2,…,rP. Calculating fk＝∠rkK is 1,2, …, P, wherein ∠ denotes argument calculation f1,f2,…,fPIs an estimate of the first harmonic frequency component.
The method for calculating the polynomial S (z) is as follows: let s (z) equal [1, z, …, zL-1]TCalculating the polynomial S (z) sT(z-1)VVHs(z)。
ComputingThe method for estimating the second harmonic frequency component is as follows: all roots of s (z) 0 are obtained, and P roots located in the unit circle and closest to the unit circle are designated as s1,s2,…,sP. Calculate gk＝∠sk，k＝1,2,…,P。g1,g2,…,gPIs an estimate of the second harmonic frequency component.
The calculated estimator Qk,lThe method comprises the following steps: using estimated value f of the first harmonic frequency component1,f2,…,fPAnd an estimate g of the second harmonic frequency component1,g2,…,gPStructure P2An individual vector ak,l(fk,gl),k,l＝1,2,…,P，
Figure BDA0001464196210000051
Wherein
Figure BDA0001464196210000052
Representing the Kronecker product operation. Computing
Figure BDA0001464196210000053
k,l＝1,2,…,P。
The method for calculating the estimated value of the two-dimensional harmonic frequency comprises the following steps: will P2A Qk,lArranging the first P Q in the sequence from big to smallk,lCorresponding vector ak,l(f) ofk,gl) Record as (omega)1q,ω2q)，q＝1,2,…,P。(ω1q,ω2q) Q is 1,2, …, and P is an estimate of the two-dimensional harmonic frequency
The invention has the beneficial effects that: the method can realize the super-resolution frequency estimation of the two-dimensional harmonic signal, has the characteristics of high estimation precision and low calculation complexity, and solves the technical problems of low resolution and high calculation complexity of the two-dimensional harmonic signal frequency estimation of the existing method.
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FIG. 1 is a schematic flow diagram of the process of the present invention;
Detailed Description
The invention is further illustrated with reference to the following figures and examples:
example 1: a two-dimensional harmonic signal frequency estimation method, see fig. 1, the method comprising the steps of:
1) calculate covariance c (s, t): setting the MN data measurement values of the two-dimensional harmonic signals as x (M, n), wherein M is 1,2, …, M; n is 1,2, …, N, the number of two-dimensional harmonic components is P, and for one harmonic component is more than 2P and less than 2P
Figure BDA0001464196210000054
And one is greater than 2P and less than
Figure BDA0001464196210000055
The integer L, covariance c (s, t) of (a) is calculated by:
Figure BDA0001464196210000056
wherein s is more than or equal to 1-K and less than or equal to K-1, t is more than or equal to 1-L and less than or equal to L-1 (·)*Representing a complex conjugate.
2) Constructing a matrix F: constructing a K multiplied by K matrix F by using the covariance c (s,0) obtained by the calculation in the step 1, wherein s is more than or equal to 1 and is more than or equal to K-1:
Figure BDA0001464196210000061
3) decomposing the eigenvalue of the matrix F, and recording the obtained eigenvalue as lambda from large to small1,λ2,…,λKThe corresponding eigenvalue vector is denoted as e1,e2,…,eK。
4) Constructing a matrix U: using the feature vector e calculated in step 3P+1,eP+2,…,eKConstructing matrix U ═ eP+1,eP+2,…,eK]。
5) Calculating a polynomial r (z): let r (z) ═ 1, z, …, zK-1]TCalculating a plurality of termsFormula (II)
R(z)＝rT(z-1)UUHr (z), wherein (·)TShowing transposition, (.)HRepresenting a conjugate transpose.
6) Calculating an estimate of the first harmonic frequency component: all roots of R (z) ═ 0 were obtained, and the P roots located in the unit circle and closest to the unit circle were designated as r1,r2,…,rP. Calculating fk＝∠rkK is 1,2, …, P, wherein ∠ denotes argument calculation f1,f2,…,fPIs an estimate of the first harmonic frequency component.
7) Constructing a matrix G: constructing an L multiplied by L matrix G by using the covariance c (0, t) obtained by the calculation in the step 1, wherein t is more than or equal to 1 and is less than or equal to L-1:
Figure BDA0001464196210000062
8) decomposing the eigenvalue of the matrix G, and recording the obtained eigenvalue as gamma from large to small1,γ2,…,γLThe corresponding eigenvalue vector is denoted d1,d2,…,dL。
9) Constructing a matrix V: using the feature vector b calculated in step 8P+1,bP+2,…,bKLThe construction matrix V ═ dP+1,dP+2,…,dL]。
10) Calculating a polynomial s (z): all roots of s (z) 0 are obtained, and P roots located in the unit circle and closest to the unit circle are designated as s1,s2,…,sP. Calculate gk＝∠sk，k＝1,2,…,P。g1,g2,…,gPIs an estimate of the second harmonic frequency component.
11) Calculating an estimate of the second harmonic frequency component: all roots of s (z) 0 are obtained, and P roots located in the unit circle and closest to the unit circle are designated as s1,s2,…,sP. Calculate gk＝∠sk，k＝1,2,…,P。g1,g2,…,gPFor frequency components of the second harmonicAnd (6) estimating the value.
12) Constructing a matrix H: constructing a KL multiplied by KL matrix H by using the covariance c (s, t) obtained by the calculation in the step 1, wherein s is more than or equal to 1 and is less than or equal to K-1 and t is more than or equal to 1 and is less than or equal to 1 and L-1:
Figure BDA0001464196210000071
wherein
Figure BDA0001464196210000072
13) The matrix H is subjected to eigenvalue decomposition, and the obtained eigenvalues are recorded in descending order η1,η2,…,ηKLThe corresponding eigenvalue vector is denoted b1,b2,…,bKL。
14) Constructing a matrix W: using the feature vector b calculated in step 13P+1,bP+2,…,bKLThe construction matrix W ═ bP+1,bP+2,…,bKL]。
15) Calculating an estimator Qk,l: using estimated value f of the first harmonic frequency component1,f2,…,fPAnd an estimate g of the second harmonic frequency component1,g2,…,gPStructure P2An individual vector ak,l(fk,gl),k,l＝1,2,…,P，
Figure BDA0001464196210000073
，
Wherein
Figure BDA0001464196210000081
Representing the Kronecker product operation. Computing
Figure BDA0001464196210000082
k,l＝1,2,…,P。
16) Calculating an estimate of the two-dimensional harmonic frequency: will P2A Qk,lArranging according to the sequence from big to smallThe first P of Qk,lCorresponding vector ak,l(f) ofk,gl) Record as (omega)1q,ω2q)，q＝1,2,…,P。(ω1q,ω2q) Q is 1,2, …, and P is an estimate of the two-dimensional harmonic frequency.
Although the embodiments of the present invention have been disclosed in the form of preferred embodiments, it is not limited thereto, and those skilled in the art will be able to easily understand the spirit of the present invention based on the above embodiments and make various extensions and changes without departing from the spirit of the present invention.

Claims (1)

1. A two-dimensional harmonic signal frequency estimation method is characterized in that: the method comprises the following steps:
1) calculating covariance c (s, t);
2) constructing a matrix F;
3) decomposing the eigenvalue of the matrix F, and recording the obtained eigenvalue as lambda from large to small1，λ2，…，λKThe corresponding eigenvalue vector is denoted as e1，e2，…，eK；
4) Constructing a matrix U;
5) calculating a polynomial r (z);
6) calculating an estimate of the first harmonic frequency component;
7) constructing a matrix G;
8) decomposing the eigenvalue of the matrix G, and recording the obtained eigenvalue as gamma from large to small1，γ2，…，γLThe corresponding eigenvalue vector is denoted d1，d2，…，dL；
9) Constructing a matrix V;
10) calculating a polynomial s (z);
11) calculating an estimate of the second harmonic frequency component;
12) constructing a matrix H;
13) the matrix H is subjected to eigenvalue decomposition, and the obtained eigenvalues are recorded in descending order η1，η2，…ηKLCorresponding toThe eigenvalue vector is denoted b1，b2，…，bKL；
14) Constructing a matrix W;
15) calculating an estimator Qk,l；
16) Calculating an estimated value of the two-dimensional harmonic frequency;
the method for calculating the covariance c (s, t) comprises the following steps: setting the MN data measurement values of the two-dimensional harmonic signals as x (M, n), wherein M is 1,2, …, M; n is 1,2, …, N, the number of two-dimensional harmonic components is P, and for one harmonic component is more than 2P and less than 2P
Figure FDA0002408699060000021
And one is greater than 2P and less than
Figure FDA0002408699060000022
The integer L, covariance c (s, t) of (a) is calculated by:
Figure FDA0002408699060000023
wherein s is more than or equal to 1-K and less than or equal to K-1, t is more than or equal to 1-L and less than or equal to L-1 (·)*Represents a complex conjugate;
the method for constructing the matrix F comprises the following steps: constructing a K matrix F using the covariance c (s,0), 1-K ≦ s ≦ K-1, i.e.:
Figure FDA0002408699060000024
the method for constructing the matrix U comprises the following steps: u ═ eP+1，eP+2，…，eK]；
The method for constructing the matrix G comprises the following steps: using the covariance c (0, t), 1-L ≦ t ≦ L-1, an L × L matrix G is constructed, i.e.:
Figure FDA0002408699060000025
the method for constructing the matrix V comprises the following steps: v ═ dP+1，dP+2，…，dL]；
The method for constructing the matrix H comprises the following steps: constructing a KL multiplied by KL matrix H by using the covariance c (s, t), s is more than or equal to 1-K and less than or equal to K-1, and t is more than or equal to 1-L and less than or equal to L-1, namely:
Figure FDA0002408699060000031
wherein
Figure FDA0002408699060000032
The method for constructing the matrix W comprises the following steps: w ═ bP+1，bP+2，…，bKL]；
The method for calculating the polynomial R (z) is as follows: let r (z) ═ 1, z, …, zK-1]TCalculating the polynomial R (z) ═ rT(z-1)UUHr (z), wherein (·)TShowing transposition, (.)HRepresents a conjugate transpose;
the method for calculating the estimated value of the first harmonic frequency component comprises the following steps: all roots of R (z) ═ 0 were obtained, and the P roots located in the unit circle and closest to the unit circle were designated as r1，r2，…，rPCalculating fk＝∠rkK is 1,2, …, P, wherein ∠ denotes argument computation, f1，f2，…，fPAn estimate of the first harmonic frequency component;
the method for calculating the polynomial S (z) is as follows: let s (z) equal [1, z, …, zL-1]TCalculating the polynomial S (z) sT(z-1)VVHs(z)；
The method of calculating the estimated value of the second harmonic frequency component is: all roots of S (z) 0 were obtained, P roots located in the unit circle and closest to the unit circle were designated as S1, S2, …, SP, and g was calculatedk＝∠sk，k＝1，2，…，P；g1，g2，…，gPIs an estimate of the second harmonic frequency component;
the calculated estimator Qk，lThe method comprises the following steps: using estimated value f of the first harmonic frequency component1，f2，…，fPAnd an estimate g of the second harmonic frequency component1，g2，…，gPStructure P2An individual vector
ak，i(fk，gl)，k，l＝1，2，…，P，
Figure FDA0002408699060000041
Wherein
Figure FDA0002408699060000042
Expressing the product operation and calculation of Kronecker
Figure FDA0002408699060000043
The method for calculating the estimated value of the two-dimensional harmonic frequency comprises the following steps: will P2A Qk,lArranging the first P Q in the sequence from big to smallk,lCorresponding vector ak,l(f) ofk,gl) Record as (omega)1q,ω2q)，q＝1，2，…，P，(ω1q，ω2q) Q is 1,2, …, and P is an estimate of the two-dimensional harmonic frequency.
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CN105044453A (en) * 2015-08-11 2015-11-11 杨世永 Harmonic signal frequency estimation method suitable for complex noise background
CN106291101A (en) * 2016-10-14 2017-01-04 九江学院 Harmonic frequency signal method of estimation in a kind of property taken advantage of with super-resolution and additive noise
CN106772227A (en) * 2017-01-12 2017-05-31 浙江大学 A kind of unmanned plane direction determining method based on the identification of vocal print multiple-harmonic

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