CN110749858A - Expanded co-prime matrix direction finding estimation method based on polynomial root finding - Google Patents

Expanded co-prime matrix direction finding estimation method based on polynomial root finding Download PDF

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CN110749858A
CN110749858A CN201910917061.7A CN201910917061A CN110749858A CN 110749858 A CN110749858 A CN 110749858A CN 201910917061 A CN201910917061 A CN 201910917061A CN 110749858 A CN110749858 A CN 110749858A
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叶长波
张小飞
沈金清
朱倍佐
汪云飞
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Nanjing University of Aeronautics and Astronautics
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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
    • G01S3/14—Systems for determining direction or deviation from predetermined direction
    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae

Abstract

The invention discloses an expanded co-prime matrix direction finding estimation method based on polynomial root solving, which comprises the following steps: establishing a mathematical model of an array signal of an expanded co-prime linear array; solving a covariance matrix according to the established mathematical model, and performing eigenvalue decomposition on the covariance matrix to obtain a signal subspace and a noise subspace; constructing an extraction matrix according to the relationship between the expanded co-prime linear arrays and the one-dimensional uniform linear arrays, and constructing a root-seeking polynomial according to the relationship between the extraction matrix and the noise subspace; carrying out root finding through the obtained root finding polynomial, and determining a root corresponding to the information source angle; and completing one-dimensional DOA estimation. By combining the advantages of the ROOT-MUSIC algorithm and the expanded co-prime linear array, the spatial degree of freedom can be effectively improved by utilizing the expanded co-prime linear array, and simultaneously, the ROOT-MUSIC algorithm does not need to search a spectral peak so as to reduce the complexity of the algorithm, thereby obtaining the one-dimensional DOA estimation with higher precision.

Description

Expanded co-prime matrix direction finding estimation method based on polynomial root finding
Technical Field
The invention belongs to the field of array signal processing, and particularly relates to an expanded co-prime array direction finding estimation method based on polynomial root finding.
Background
Early DOA estimation studies were based on uniform linear arrays, i.e., one-dimensional arrays in which the array elements are arranged on the same straight line at equal intervals, which have the advantage of simple structure, but because the array elements are located on the same straight line, only one-dimensional angle estimation can be performed. Although the traditional one-dimensional uniform linear array can carry out effective DOA estimation, certain defects exist. Taking the uniform line array as an example, the uniform line array including M array elements can only detect M-1 information sources at most, that is, DOF is equal to M, and the spatial degree of freedom is small. Meanwhile, in order to avoid the problem of angle ambiguity, the array element spacing of the conventional array usually needs to be less than or equal to half the wavelength of the received signal, but too close distance between array elements causes strong mutual coupling effect, thereby reducing the estimation accuracy. To solve these two problems, researchers have proposed the concept of sparse arrays. Common sparse arrays include minimum redundant arrays, nested arrays, co-prime arrays, and the like. The mutual prime array is composed of two uniform linear arrays with mutually prime array elements and superposed first array elements. If the array element numbers of the two sub-arrays are M and N respectively, the co-prime linear arrays with the array element total number of M + N-1 can obtain the spatial degree of freedom of O (MN), and the uniform linear arrays with the same array element number can only obtain the spatial degree of freedom of O (M + N), so that the co-prime structure greatly improves the detectable information source number of the array. Meanwhile, the array element spacing of two sub-arrays in the co-prime linear array is respectively N times and M times of half wavelength, and the array element spacing is far larger than that of the uniform linear array, so that the cross coupling effect between the array elements is effectively weakened. On the basis, the scholars successively put forward a plurality of optimized co-prime array structures, such as an enlarged co-prime linear array, a compressed array element pitch co-prime linear array, a sub-array shifted co-prime linear array and the like, and the optimized arrays improve the spatial freedom of the arrays or weaken the mutual coupling effect of the arrays to a certain extent.
In the traditional co-prime matrix DOA estimation algorithm, two broad categories of fuzzy solution methods and virtualization methods are used, the fuzzy solution methods are used for respectively carrying out DOA estimation on two sub-matrices, and the co-prime characteristics of the array elements of the sub-matrices are used for comparing estimation results, so that a unique DOA estimation value is obtained. However, although this deblurring method is easy to implement, the method of estimating two sub-arrays separately greatly reduces the spatial degree of freedom.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects in the prior art, the method for estimating the direction of the expanded co-prime matrix based on polynomial root solving is provided, the high complexity caused by the fact that a traditional MUSIC algorithm needs to search a spectral peak for one-dimensional angle estimation is reduced, the characteristic of high degree of freedom of the expanded co-prime matrix is fully utilized, and meanwhile the performance of signal angle parameter estimation is guaranteed.
The technical scheme is as follows: in order to achieve the above object, the present invention provides an expanded co-prime matrix direction finding estimation method based on polynomial root finding, comprising the following steps:
s1: establishing a mathematical model of an array signal of an expanded co-prime linear array;
s2: solving a covariance matrix according to the established mathematical model, and performing eigenvalue decomposition on the covariance matrix to obtain a signal subspace and a noise subspace;
s3: constructing an extraction matrix according to the relationship between the expanded co-prime linear arrays and the one-dimensional uniform linear arrays, and constructing a root-seeking polynomial according to the relationship between the extraction matrix and the noise subspace;
s4: performing root finding through the root finding polynomial obtained in the step S3, and determining a root corresponding to the information source angle;
s5: and completing one-dimensional DOA estimation.
Further, the establishing of the mathematical model in the step S1 specifically includes:
the expanded co-prime linear array is composed of two sub-arrays, namely a first array and a second array on the left side and the right side of an original point, the number of array elements of the first array and the second array is M and N respectively, the array element spacing is N lambda/2 and M lambda/2 respectively, M and N are prime numbers and lambda is a wavelength, the expanded co-prime arrays are overlapped at the original point, except the array elements at the original point, the positions of other array elements are not overlapped, and the array element spacing d is recorded1＝Nλ/2，d2Where M λ/2, the position of each array element on the extended reciprocal linear array can be expressed as
Ls＝{(-m1d1，0)，|m1＝0，1，2…M-1}∪{(0，m2d2)，|m2＝0，1，2…N-1}
The array elements at the original point are shared by the first array and the second array, and the total number of the array elements of the mutually prime linear array is M + N-1;
setting K independent information sources in the space domain, and recording the incidence angle of the K signal as thetakAnd K is 1, 2, …, K, the received signal of the ith second array is denoted as Xi＝AiS+Ni
Ai＝[ai(θ1)，ai(θ2)，…，ai(θk)]，Subarray 1 differs from subarray 2 only in that:
Figure BDA0002216415380000022
Aidirection matrix representing sub-array i, S ═ S1，s2，…，sK]TIs a source matrix, sk＝[sk(1)，sk(2)，…，sk(J)](K is 1, 2, …, K), J represents fast beat number,
Figure BDA0002216415380000023
the noise matrix for sub-array i.
Considering the direction matrix of the developed co-prime matrix for the whole process, a ═ a (θ)1)，a(θ2)，…，a(θk)]，
Figure BDA0002216415380000024
S is a source matrix, J represents a fast beat number, N belongs to C(M+N-1)×JA noise matrix of a co-prime matrix is expanded. The received signal may be denoted AS X ═ AS + N.
Further, the covariance matrix in step S2The specific solution of (a) is as follows:
obtaining covariance matrixes obtained by J snapshotsEstimation of (2):
the expression of the eigenvalue decomposition of the covariance matrix in step S2 is as follows:
Figure BDA0002216415380000032
wherein, USRepresenting a signal subspace, UNRepresenting a noise subspace, DSAnd DNEach represents a diagonal matrix.
Further, the construction of the extraction matrix in the step S3 is specifically: in the one-dimensional uniform linear array, a polynomial is defined based on a ROOT-MUSIC algorithm:
Figure BDA0002216415380000033
wherein u islIs the l-th eigenvector of the matrix R, and p1(z)＝[1，z，…，zN-1]T；
The position of the array element of the one-dimensional uniform linear array is expressed as Ls1＝{(-(M-1)d，…，0，…，(N-1)d)}
Defining A as the decimation matrix, Ls1The following relationship exists between Ls and
Ls＝ALs1
considering the developed co-prime array as the result of the extraction from a one-dimensional uniform linear array, the above relationship is reflected in p (z):
p(z)＝Ap1(z)。
further, the expression of the root polynomial in step S3 is as follows:
Figure BDA0002216415380000034
further, in the step S4, p is usedT(z-1) In place of pH(z) obtaining a root-seeking MUSIC polynomial, i.e.
(z) is a 2(2MN- (M + N)) degree polynomial whose roots are mirror pairs with respect to the unit circle, where the K roots with the largest magnitude
Figure BDA0002216415380000036
Gives an estimate of the direction of arrival.
Further, the one-dimensional DOA estimation expression in step S5 is as follows:
Figure BDA0002216415380000037
the basic design idea of the invention is as follows: determining a covariance matrix of a received signal by a mathematical model of an array signal, obtaining a signal subspace and a noise subspace by performing eigenvalue decomposition on the covariance matrix, then constructing an extraction matrix by the relationship of a co-prime matrix and a one-dimensional linear array, and finally constructing a root-seeking polynomial according to the relationship of the extraction matrix and the noise subspace, and performing root-seeking and information source angle estimation.
The method can effectively improve the spatial degree of freedom by utilizing the expanded co-prime linear array by combining the advantages of the ROOT-MUSIC algorithm and the expanded co-prime linear array, and simultaneously reduces the algorithm complexity because the ROOT-MUSIC algorithm does not need to carry out spectral peak search, thereby obtaining the one-dimensional DOA estimation with higher precision.
Has the advantages that: compared with the prior art, the invention provides a direction finding estimation method based on polynomial root finding in signal angle parameter estimation under the condition of multi-information-source incidence in an expanded co-prime linear array, and the method has the following advantages:
①, the characteristic of high degree of freedom of space for expanding the co-prime linear array can be fully utilized, and the angle estimation performance is effectively improved;
② the calculation complexity is lower than that of the traditional fuzzy MUSIC algorithm;
③ can be effectively used for one-dimensional DOA estimation, and simultaneously, an angle estimation result with higher precision is obtained;
④ the angle estimation performance of the algorithm of the present invention is superior to the deblurred MUSIC algorithm and the ESPRIT algorithm.
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FIG. 1 is an algorithm flow diagram of the algorithm of the present invention;
fig. 2 is a schematic view of a co-prime linear array;
fig. 3 is a schematic diagram of an unfolded co-prime linear array;
FIG. 4 is a comparison of the angle estimation performance of the algorithm of the present invention under different snapshot count conditions;
FIG. 5 is a comparison graph of the angle estimation performance of the algorithm of the present invention under different array element conditions;
FIG. 6 is a comparison graph of the algorithm of the present invention applied to uniform linear arrays, co-prime linear arrays and expanded co-prime linear arrays under the conditions of the same number of array elements and the same fast beat number;
FIG. 7 is a comparison graph of the angle estimation performance of the algorithm and the deblurred MUSIC algorithm, the deblurred ESPRIT algorithm and the one-dimensional full-array MUSIC algorithm under the conditions of the same array structure and the same fast beat number.
Detailed Description
The invention is further elucidated with reference to the drawings and the embodiments.
The invention provides an expanded co-prime array direction-finding estimation method based on polynomial root finding, wherein a figure 2 is a co-prime linear array, a figure 3 is an expanded co-prime linear array obtained based on the co-prime linear array, the expanded co-prime linear array is composed of sub-arrays formed by two uniform linear arrays and respectively marked as a sub-array 1 (a first array) and a sub-array 2 (a second array), the array element numbers of the sub-arrays are M and N respectively, the array element intervals are Nd and Md respectively, wherein d is half wavelength, and M and N are co-prime; the two sub-arrays both use the original point as a reference point, wherein the sub-array 1 and the sub-array 2 are respectively positioned at the left side and the right side of the original point. With reference to fig. 1 and 2, K sources are incident on the extended co-prime linear array in space, and one dimension of the direction of arrival is θkK is 1, 2,.. K, where θkRepresenting the elevation angle of the kth source.
In this embodiment, a method is performed on the basis of the developed linear array with mutual primeThe direction-finding estimation method based on polynomial root finding uses (·) for easy understandingTRepresentation matrix transposition, (.)HRepresenting the conjugate transpose of a matrix, the capital letter X representing the matrix, the lower case letter X (-) representing a vector, and the arg (-) representing the phase angle of a complex number, as shown in fig. 1, the method comprises the following steps:
s1: establishing a mathematical model of the array signal:
as shown in fig. 3, the array elements of the sub-array 1 and the sub-array 2 of the expanded co-prime linear array are M and N, respectively, and the array element spacing is N λ/2 and M λ/2, respectively, wherein M and N are prime numbers to each other, λ is the wavelength, the expanded co-prime arrays composed of the array 1 and the array 2 are overlapped at the origin, except the array element at the origin, the positions of other array elements are not overlapped with each other;
interval d of array elements1＝Nλ/2，d2When M λ/2, the position of each array element on the extended reciprocal linear array can be expressed as:
Ls＝{(-m1d1，0)，|m1＝0，1，2…M-1}∪{(0，m2d2)，|m2＝0，1，2…N-1}
the array elements at the origin may be shared by the two sub-arrays, so that the total number of array elements of the co-prime linear array is M + N-1, but degrees of freedom of o (mn) may be provided. Since there are K mutually independent sources in the space domain, the incident angle of the K signal is recorded as thetakAnd K is 1, 2, …, K, the received signal of the ith right sub-array can be represented as:
Xi＝AiS+Ni
Ai＝[ai(θ1)，ai(θ2)，…，ai(θk)]， subarray 1 differs from subarray 2 only in that:
Figure BDA0002216415380000052
Aidirection matrix representing sub-array i, S ═ S1，s2，…，sK]TIs a source matrix, sk＝[sk(1)，sk(2)，…，sk(J)](K is 1, 2, …, K), J represents fast beat number,
Figure BDA0002216415380000053
the noise matrix for sub-array i.
Considering the direction matrix of the developed co-prime matrix for the whole process, a ═ a (θ)1)，a(θ2)，…，a(θk)]，
Figure BDA0002216415380000054
S is the source matrix, J represents the fast beat number, N belongs to C(M+N-1)×JA noise matrix of a co-prime matrix is expanded. The received signal may be denoted AS X ═ AS + N
S2: solving a covariance matrix from the established mathematical model
Figure BDA0002216415380000055
And to
Figure BDA0002216415380000056
And (3) carrying out characteristic value decomposition to obtain a signal subspace and a noise subspace:
obtaining covariance matrixes obtained by J snapshots
Figure BDA0002216415380000057
Is estimated as
Figure BDA0002216415380000058
The eigenvalue decomposition is performed on the signal covariance matrix and can be expressed as
Figure BDA0002216415380000059
Wherein, USRepresenting a signal subspace, UNRepresenting a noise subspace, DSAnd DNEach represents a diagonal matrix.
S3: constructing an extraction matrix according to the relationship between the developed co-prime linear arrays and the one-dimensional uniform linear arrays, and constructing a root-seeking polynomial according to the relationship between the extraction matrix and the noise subspace:
in a one-dimensional uniform linear array, a polynomial is firstly defined based on a ROOT-MUSIC algorithm:
Figure BDA00022164153800000510
ulis the l-th eigenvector of the matrix R, and p1(z)＝[1，z，…，zM-1]T；
Considering now the case of the developed co-prime array, referring to fig. 2, the developed co-prime array has array element defaults in comparison to the one-dimensional uniform linear arrays of the same start and end points. Setting the array element position of the one-dimensional uniform linear array as Ls1＝{(-(M-1)d，…，0，…，(N-1)d)}
Defining A as the decimation matrix, Ls1The following relationship exists between Ls and
Ls＝ALs1
the developed co-prime matrix is considered to be the result of the extraction from a one-dimensional uniform linear array, the above relationship being reflected in p (z)
p(z)＝Ap1(z)
To extract information from all noise feature vectors simultaneously, the following root polynomial is obtained:
Figure BDA0002216415380000061
s4: and (3) carrying out root finding through the root finding polynomial, and determining a root corresponding to the information source angle:
f (z) is not yet a polynomial for z because of the power of z terms present. Since only the z value on the unit circle is of interest, p can be usedT(z-1) In place of pH(z) this gives the root-finding MUSIC polynomial, i.e.
Figure BDA0002216415380000062
As a result of this, it is possible to,f (z) is a polynomial of degree 2(2MN- (M + N)) whose roots are mirror pairs with respect to the unit circle. Wherein the K roots with the maximum amplitude
Figure BDA0002216415380000063
Gives an estimate of the direction of arrival.
S5: completing one-dimensional DOA estimation:
Figure BDA0002216415380000064
in this embodiment, to verify the effect of the present invention, the operation complexity of the algorithm of the present invention is analyzed, which is specifically as follows: because the array element number of the sub-array 1 in the expanded co-prime linear array is M, the array element number of the sub-array 2 is N, the information source number is K, and the fast beat number is L, the main complexity of the algorithm comprises the following steps: the covariance matrix of the received signal needs O { (M + N-1)2L, the eigenvalue decomposition requires O { (M + N-1)3The high-order polynomial root-finding requires O { (2(2MN- (M + N))3So the overall complexity of the ROOT-MUSIC algorithm is O { (M + N-1)2J+(M+N-1)3+(2(2MN-(M+N))3And because the algorithm does not need to search the angle of the information source, the complexity of the algorithm is far lower than that of the fuzzy MUSIC algorithm under the same array structure.
In order to further verify the effect of the algorithm of the present invention, the embodiment performs a simulation test, and the specific result is as follows:
as shown in fig. 4, the number of snapshots increases, i.e., the sample data increases. It can be seen that the angular estimation performance of the algorithm becomes better as the number of snapshots increases. Wherein the angle parameter theta of the incident signal is theta1＝10°，θ2＝30°，θ3The size of the developed linear array is M3 and N4, which are 50 deg..
As shown in fig. 5, the number of array elements increases, i.e., the diversity gain increases. It can be concluded that the angle estimation performance of the algorithm becomes better as the number of array elements increases. Wherein the angle parameter theta of the incident signal is theta1＝10°，θ2＝30°，θ350 degrees, the size of the unfolded linear array isM is 3, N is 4. The fast beat number L is 100.
Comparing the expanded co-prime array of this embodiment with the ULA and conventional co-prime array simulations, the result is shown in fig. 6, when the number of array elements is the same, the co-prime array has better angle estimation performance than the uniform linear array due to the increased array aperture, and the expanded co-prime array has better angle estimation performance than the conventional co-prime array due to the reduced redundancy of the differential array. Wherein the angle parameter theta of the incident signal is theta1＝10°，θ2＝30°，θ3The size of the developed linear array is M3 and N4, which are 50 deg.. The fast beat number L is 100.
The ROOT-MUSIC algorithm, the ESPRIT algorithm based on the deblurring, the MUSIC algorithm and the full-array MUSIC algorithm are compared in a simulation mode, the result is shown in figure 7, the ROOT-MUSIC algorithm can fully utilize the property of expanding a co-prime matrix, and a certain spatial degree of freedom can be lost in the deblurring method, so that the angle estimation performance of the ROOT-MUSIC algorithm is superior to that of the two methods for deblurring. Wherein the angle parameter theta of the incident signal is theta1＝10°，θ2＝30°，θ3And when the angle is 50 degrees, the size of the developed linear array is M-5, and N-4. The fast beat number L is 100.

Claims (8)

1. An expanded co-prime matrix direction finding estimation method based on polynomial root solving is characterized by comprising the following steps: the method comprises the following steps:
s1: establishing a mathematical model of an array signal of an expanded co-prime linear array;
s2: solving a covariance matrix according to the established mathematical model, and performing eigenvalue decomposition on the covariance matrix to obtain a signal subspace and a noise subspace;
s3: constructing an extraction matrix according to the relationship between the expanded co-prime linear arrays and the one-dimensional uniform linear arrays, and constructing a root-seeking polynomial according to the relationship between the extraction matrix and the noise subspace;
s4: performing root finding through the root finding polynomial obtained in the step S3, and determining a root corresponding to the information source angle;
s5: and completing one-dimensional DOA estimation.
2. The method according to claim 1, wherein the method comprises: the establishment of the mathematical model in the step S1 specifically includes:
the expanded co-prime linear array is composed of two sub-arrays, namely a first array and a second array on the left side and the right side of an original point, the number of array elements of the first array and the second array is M and N respectively, the array element spacing is N lambda/2 and M lambda/2 respectively, M and N are prime numbers and lambda is a wavelength, the expanded co-prime arrays are overlapped at the original point, except the array elements at the original point, the positions of other array elements are not overlapped, and the array element spacing d is recorded1＝Nλ/2，d2Where M λ/2, the position of each array element on the extended reciprocal linear array can be expressed as
Ls＝{(-m1d1，0)，|m1＝0，1，2…M-1}∪{(0，m2d2)，|m2＝0，1，2…N-1}
The array elements at the original point are shared by the first array and the second array, and the total number of the array elements of the mutually prime linear array is M + N-1;
setting K independent information sources in the space domain, and recording the incidence angle of the K signal as thetakAnd K is 1, 2, …, K, the received signal of the ith second array is represented as
Xi＝AiS+Ni
Ai＝[ai(θ1)，ai(θ2)，…，ai(θk)]，
Figure FDA0002216415370000011
Subarray 1 differs from subarray 2 only in that:
Figure FDA0002216415370000012
Aidirection matrix representing sub-array i, S ═ S1，s2，…，sK]TIs a source matrix, sk＝[sk(1)，sk(2)，…，sk(J)](K is 1, 2, …, K), J represents fast beat number,
Figure FDA0002216415370000013
a noise matrix which is a subarray i;
considering the direction matrix of the developed co-prime matrix for the whole process, a ═ a (θ)1)，a(θ2)，…，a(θk)]，
Figure FDA0002216415370000014
S is a source matrix, J represents a fast beat number, N belongs to C(M+N-1)×JTo expand the noise matrix of the co-prime matrix, the received signal is represented as
X＝AS+N。
3. The method according to claim 2, wherein the method comprises: the covariance matrix in step S2The specific solution of (a) is as follows:
obtaining covariance matrixes obtained by J snapshots
Figure FDA0002216415370000022
Estimation of (2):
Figure FDA0002216415370000023
4. the method according to claim 3, wherein the method comprises: the expression of the eigenvalue decomposition of the covariance matrix in step S2 is as follows:
Figure FDA0002216415370000024
wherein, USRepresenting a signal subspace, UNRepresenting a noise subspace, DSAnd DNEach represents a diagonal matrix.
5. The method according to claim 1, wherein the method comprises: the construction of the extraction matrix in the step S3 is specifically: in the one-dimensional uniform linear array, a polynomial is defined based on a ROOT-MUSIC algorithm:
Figure FDA0002216415370000025
wherein u islIs the l-th eigenvector of the matrix R, and p1(z)＝[1，z，…，zM-1]T；
The position of the array element of the one-dimensional uniform linear array is expressed as
Ls1＝{(-(M-1)d，…，0，…，(N-1)d)}
Defining A as the decimation matrix, Ls1The following relationship exists between Ls and
Ls＝ALs1
considering the developed co-prime array as the result of the extraction from a one-dimensional uniform linear array, the above relationship is reflected in p (z):
p(z)＝Ap1(z)。
6. the method according to claim 5, wherein the method comprises: the expression of the root polynomial in step S3 is as follows:
Figure FDA0002216415370000026
7. the method according to claim 6, wherein the method comprises: in said step S4, p is usedT(z-1) In place of pH(z) obtaining a root-seeking MUSIC polynomial, i.e.
Figure FDA0002216415370000027
(z) is a 2(2MN- (M + N)) degree polynomial whose roots are mirror pairs with respect to the unit circle, where the K roots with the largest magnitude
Figure FDA0002216415370000028
Gives an estimate of the direction of arrival.
8. The method according to claim 7, wherein the method comprises: the one-dimensional DOA estimation expression in step S5 is as follows:
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