CN110888106A - Angle and frequency joint estimation augmented DOA matrix method - Google Patents

Angle and frequency joint estimation augmented DOA matrix method Download PDF

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CN110888106A
CN110888106A CN201911147303.5A CN201911147303A CN110888106A CN 110888106 A CN110888106 A CN 110888106A CN 201911147303 A CN201911147303 A CN 201911147303A CN 110888106 A CN110888106 A CN 110888106A
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戴祥瑞
张小飞
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses an angle and frequency joint estimation augmented DOA matrix method, which discusses the problem of angle and frequency joint estimation in a single time delay array signal receiving system by taking array signal processing as the background. The method comprises the steps of constructing an augmented DOA matrix by utilizing an autocorrelation matrix and a cross-correlation matrix of received data of the signal system, directly obtaining a signal direction vector and a signal direction element to be estimated through characteristic decomposition of the DOA matrix, and obtaining DOA angle and frequency estimation of the signal to be estimated. Compared with the traditional DOA matrix method, the method completely utilizes the autocorrelation matrix and the cross-correlation matrix of the received data of the signal receiving system, so that the method has better angle and frequency estimation performance. The method does not need space spectrum search, has lower algorithm complexity, and can realize automatic pairing of the obtained DOA estimation angle and frequency estimation.

Description

Angle and frequency joint estimation augmented DOA matrix method
Technical Field
The invention relates to a signal source positioning method under a sensor array, in particular to an angle and frequency joint estimation DOA matrix augmentation method, and belongs to the technical field of array signal processing.
Background
The traditional DOA matrix method constructs a DOA matrix according to the properties of the covariance matrix. By means of characteristic decomposition of the DOA matrix, a signal direction vector and a signal direction element to be estimated can be directly obtained, and signal parameters can be estimated accordingly, so that polynomial search is completely avoided, the operation amount is small, but the DOA angle and frequency joint estimation performance is low because the autocorrelation information and the cross correlation information of the array received signals are not completely utilized.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the DOA matrix augmentation method based on angle and frequency joint estimation is provided, the problem of two-dimensional DOA estimation under a single-delay sensor array receiving system is solved, and the DOA matrix augmentation method based on angle and frequency joint estimation has high estimation performance.
The invention adopts the following technical scheme for solving the technical problems:
an angle and frequency joint estimation augmented DOA matrix method comprises the following steps:
step 1, a linear array is arranged in space, when K uncorrelated narrow-band co-carrier signals enter the linear array, estimation of an autocorrelation matrix of the linear array receiving signals is solved
Figure BDA0002282561210000011
Adding a delay output tau to a received signal of a linear array, and solving an estimate of an autocorrelation matrix of the received signal after adding the delay output
Figure BDA0002282561210000012
Estimation of cross-correlation matrix for solving linear array received signal and delayed output received signal
Figure BDA0002282561210000013
And estimating the cross-correlation matrix of the delayed output received signal and the linear array received signal
Figure BDA0002282561210000014
Step 2, estimating the autocorrelation matrix of the linear array received signal
Figure BDA0002282561210000021
Decomposing the characteristic value and removing the noise influence to obtain a matrix
Figure BDA0002282561210000022
Similarly, estimation of the autocorrelation matrix of the received signal to which the delay output is added
Figure BDA0002282561210000023
Decomposing the characteristic value and removing the noise influence to obtain a matrix
Figure BDA0002282561210000024
Step 3, estimating according to the cross correlation matrix
Figure BDA0002282561210000025
And
Figure BDA0002282561210000026
and a matrix
Figure BDA0002282561210000027
And
Figure BDA0002282561210000028
definition matrix R1And R2And constructing an extended DOA matrix
Figure BDA0002282561210000029
And 4, performing characteristic decomposition on the expanded DOA matrix R' to obtain a characteristic value and a characteristic vector, obtaining frequency estimation according to the characteristic value, and obtaining DOA angle estimation according to the characteristic vector.
As a preferred scheme of the invention, the estimation of the autocorrelation matrix in the step 1
Figure BDA00022825612100000210
And
Figure BDA00022825612100000211
and estimation of cross-correlation matrix
Figure BDA00022825612100000212
And
Figure BDA00022825612100000213
the following formula is obtained:
Figure BDA00022825612100000214
Figure BDA00022825612100000215
Figure BDA00022825612100000216
Figure BDA00022825612100000217
wherein N is fast beat number, x (t) represents the receiving signal of the linear array at time t, y (t) represents the receiving signal of the linear array after delay output is added at time t ·)HRepresenting a matrix conjugate transpose.
As a preferred embodiment of the present invention, the matrix of step 2
Figure BDA00022825612100000218
And
Figure BDA00022825612100000219
the formula of (1) is as follows:
Figure BDA00022825612100000220
Figure BDA00022825612100000221
wherein the content of the first and second substances,
Figure BDA00022825612100000222
an estimate of an autocorrelation matrix representing a linear array of received signals,
Figure BDA00022825612100000223
representing an estimate of the autocorrelation matrix of the linear array received signal after addition of the delayed output,
Figure BDA00022825612100000224
represents an estimate of the variance of additive white gaussian noise and I represents the identity matrix.
As a preferred embodiment of the present invention, the formula of the extended DOA matrix R' in step 3 is as follows:
Figure BDA00022825612100000225
wherein R is1And R2Each of which represents a matrix of the image data,
Figure BDA0002282561210000031
an estimate of a cross-correlation matrix representing the linear array received signal and the delayed output added received signal,
Figure BDA0002282561210000032
representing an estimate of a cross-correlation matrix of the delayed output received signal with the linear array received signal,
Figure BDA0002282561210000033
and
Figure BDA0002282561210000034
each of which represents a matrix of the image data,
Figure BDA0002282561210000035
representing a matrix conjugate transpose.
As a preferred embodiment of the present invention, the specific process of step 4 is:
performing characteristic decomposition on the expanded DOA matrix R' to obtain a matrix AEAnd a gamma-ray that is different from the gamma-ray,
Figure BDA0002282561210000036
A＝[a(f1,θ1),a(f2,θ2),…,a(fK,θK)]represents a direction matrix and has
Figure BDA0002282561210000037
Γ＝diag{exp(-j2πf1τ),exp(-j2πf2τ),…,exp(-j2πfKτ)}
Wherein f iskRepresenting the carrier frequency, thetakRepresenting the included angle between the kth narrowband same carrier signal and the linear array, c representing the light speed, tau representing delay output, K being the number of the narrowband same carrier signals, and j representing an imaginary unit;
obtaining the frequency f according to the characteristic value in the gammakEstimation of (2):
Figure BDA0002282561210000038
wherein λ iskRepresenting the kth characteristic value;
according to AEThe definition of (1) is divided into A and A Γ-1Two parts, after feature decomposition, the estimation of the two parts is respectively
Figure BDA0002282561210000039
And
Figure BDA00022825612100000310
for matrix
Figure BDA00022825612100000311
Normalizing a certain column to make the first term of the column be 1, taking a phase angle of the normalized column, estimating the phase difference between the narrow-band common-carrier signal and the linear array according to the phase angle, and finally obtaining the estimation of the DOA angle by using a least square method
Figure BDA00022825612100000312
Compared with the prior art, the invention adopting the technical scheme has the following technical effects:
1. the DOA matrix augmentation method provided by the invention keeps the advantages that the traditional DOA matrix method can completely avoid polynomial search and has small calculation amount, and simultaneously, the method completely utilizes the autocorrelation information and the cross-correlation information of array receiving signals to construct an augmented DOA matrix and improve the joint estimation performance of DOA angles and frequencies.
2. The DOA angle and the frequency estimated by the method can realize automatic pairing.
3. The method has lower complexity.
Drawings
FIG. 1 is a topological diagram of an array structure of the present invention.
Fig. 2 is a diagram of a signal receiving system for the method of the present invention.
FIG. 3 is a scatter plot for the method of the present invention.
FIG. 4 is a graph comparing the angular RMSE performance of the inventive method and the conventional DOA matrix method under different SNR conditions.
FIG. 5 is a graph comparing the frequency RMSE performance of the inventive method and the conventional DOA matrix method under different SNR conditions.
FIG. 6 is a graph comparing the angle RMSE performance of the method of the present invention and the conventional DOA matrix method under different snapshot conditions.
FIG. 7 is a graph comparing the frequency RMSE performance of the method of the present invention and the conventional DOA matrix method under different snapshot conditions.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
The symbols represent: used in the invention (·)TRepresentation matrix transposition, (.)HRepresenting the conjugate transpose of the matrix (.)*Representing the conjugate of the matrix, the capital letter X representing the matrix, the lower case letter X (·) representing the vector, I representing the identity matrix, diag (v) representing the diagonal matrix made up of the elements in v, E [ ·]Indicating the expectation of the matrix, and angle (·) indicates the phase angle operation.
Data model
The signal receiving array consists of a non-uniform linear array of one of the sensors shown in FIG. 1, the array having M sensors with the M-th sensor spaced d from the first sensorm(M ═ 1, …, M), where d10. Suppose that there are K uncorrelated narrowband co-carrier signals s in spacek(t) (K is 1. ltoreq. K. ltoreq.K) is incident on the array at an angle theta to the arraykCarrier frequency of fk. So the received signal of the mth sensor is:
Figure BDA0002282561210000051
wherein n isk(t) is zero mean, variance σ2C is the speed of light. To estimate the frequency of the signal, a delay output τ is added to the received signal of the sensor, as shown in FIG. 2, and it is assumed that 0 < 2 τ < 1/max (f)k). The output signal after adding the delay τ is therefore:
Figure BDA0002282561210000052
writing the output signal in vector/matrix form, i.e.
x(t)＝As(t)+n(t)
y(t)＝As(t-τ)+n(t-τ)＝AΓs(t)+n(t-τ)
Wherein x (t) ═ x1(t),x2(t),…,xM(t)]T，y(t)＝[y1(t),y2(t),…,yM(t)]T，s(t)＝[s1(t),s2(t),…,sK(t)]T，n(t)＝[n1(t),n2(t),…,nM(t)]T。A＝[a(f1,θ1),a(f2,θ2),…,a(fK,θK)]Represents a direction matrix and has
Figure BDA0002282561210000053
Γ＝diag{exp(-j2πf1τ),exp(-j2πf2τ),…,exp(-j2πfKτ)}
Second, method derivation
The received data x (t) has an autocorrelation matrix RxxThe expression is
Rxx＝E[x(t)xH(t)]＝AΨAH+σ2I
Where Ψ ═ E [ s (t) sH(t)]Is a covariance matrix, σ, of the signal source2Is the variance of additive white gaussian noise.
The autocorrelation matrix of the received data y (t) is RyyThe expression is
Ryy＝E[y(t)yH(t)]＝AΓΨΓHAH+σ2I
＝AΨΓΓHAH+σ2I
＝AΨAH+σ2I
Considering the independence of noise itself and independent of signal, let the cross-correlation matrix of y (t) and x (t) be RyxThen, then
Ryx＝E[y(t)xH(t)]＝AΓΨAH
Similarly, the cross-correlation matrix of x (t) and y (t) is
Rxy＝E[x(t)yH(t)]＝AΓHΨA＝AΓ-1ΨA
To RxxPerforming eigenvalue decomposition (EVD) to let ε1,…,εKIs a matrix RxxUnder the assumption of white noise, the noise variance σ can be obtained by averaging the M-K small eigenvalues2Is estimated. Then, by removing the influence of noise, it is possible to obtain
Figure BDA0002282561210000061
The same can be obtained
Cyy＝AΨAH＝Ryy-σ2I
Definition of
Figure BDA0002282561210000062
Figure BDA0002282561210000063
Figure BDA0002282561210000064
We define
Figure BDA0002282561210000065
Figure BDA0002282561210000066
Thus, it is possible to provide
R1＝AEΨAH
R2＝AEΓΨAH
According to the idea of DOA matrix method, the following DOA matrix can be defined
Figure BDA0002282561210000067
Figure BDA0002282561210000068
Wherein
Figure BDA0002282561210000069
If A and Ψ full rank, Γ, do not have identical diagonal elements, then the K non-zero eigenvectors of the DOA matrix R' are equal to the K diagonal elements in Γ, and the eigenvectors for these values are equal to the corresponding signal direction vectors, i.e., the
R′AE＝AEΓ
The matrix A can be obtained by performing characteristic decomposition on the DOA matrix REAnd Γ. From the eigenvalues in Γ, the frequency f can be derivedkEstimation of (2):
Figure BDA0002282561210000071
according to AEBy definition of (A) and (A) Γ we divide it into-1Two parts, after feature decomposition, the estimation of the two parts is respectively
Figure BDA0002282561210000072
And
Figure BDA0002282561210000073
estimate out
Figure BDA0002282561210000074
And
Figure BDA0002282561210000075
then is provided with
Figure BDA0002282561210000076
And (c) in a certain column a, performing DOA angle estimation on the direction matrix by using the Vandermonde characteristics of the direction matrix. The direction vector a is firstly normalized, so that the initial term is 1. And (b) taking angle (a), estimating the phase difference between the arrays, and finally estimating the DOA angle by using a least square method. Because of the fact that
Figure BDA0002282561210000077
Can therefore obtain
uk＝-angle(a(θk))
＝[0,2πd2fksinθk/c,…,2πdMfksinθk/c]T
Least squares fit to
Figure BDA0002282561210000078
Wherein
Figure BDA0002282561210000079
Wherein
Figure BDA00022825612100000710
For estimation of frequency, e1＝sinθk. Thus e can be estimated by least squaresk
Figure BDA00022825612100000711
So that the angle is estimated as
Figure BDA00022825612100000712
In the same way, we can get from
Figure BDA00022825612100000713
To obtain
Figure BDA00022825612100000714
Thus DOA angle thetakIs estimated by
Figure BDA00022825612100000715
The method comprises the following steps:
[1]estimation of autocorrelation and cross-correlation matrices for received data x (t) and y (t)
Figure BDA00022825612100000716
And
Figure BDA0002282561210000081
[2]removing noise influence on the autocorrelation matrix to obtain
Figure BDA0002282561210000082
And
Figure BDA0002282561210000083
[3]definition of R1And R2And constructing an augmented DOA matrix
Figure BDA0002282561210000084
[4] And decomposing the characteristic value of R', and respectively obtaining the frequency and the DOA angle estimation according to the characteristic value and the characteristic vector.
Third, method analysis and simulation
The DOA angle estimation method of the invention is subjected to complexity analysis to obtain the complexity O {4M } of the autocorrelation and cross-correlation matrix2N, wherein N represents the fast beat number of the received signal; computing
Figure BDA0002282561210000085
Has a complexity of O {5M }3}; computing
Figure BDA0002282561210000086
Has a complexity of O {4M }3}; the complexity of characteristic decomposition of R' is O {8M3}. The total complexity of the calculation algorithm is O {4M2N+17M3}。
The method of the invention completely utilizes the autocorrelation information and the cross-correlation information of the array received data to construct an augmented DOA matrix, and the traditional DOA matrix method does not completely utilize the autocorrelation information and the cross-correlation information of the array received data, so that the method of the invention has higher DOA angle and frequency estimation performance than the traditional DOA matrix method.
And (3) simulation results:
three narrow-band signals (theta) in a far field of space are assumed1,f1)＝(10°,6MHz)，(θ2,f2) ═ 20 °,8MHz) and (θ3,f3) Incident on the array of figure 1 at (30 °,10MHz) the signals are uncorrelated. The DOA angle and frequency estimation performance is evaluated using 1000 monte carlo simulations, defining the Root Mean Square Error (RMSE) expression as follows:
Figure BDA0002282561210000087
Figure BDA0002282561210000088
wherein
Figure BDA0002282561210000089
And
Figure BDA00022825612100000810
represents the parameter estimation result theta of the kth information source in the ith Monte Carlo simulationkAnd fkRepresenting the true value of the parameter for the kth source.
Fig. 3 shows the scatter distribution diagram of the algorithm of the present invention, with the simulation parameters M-12, N-500, K-3 and SNR-10 dB. The DOA angle (theta) versus frequency (frequency) of the source is evident from the figure.
Fig. 4 and 5 show graphs of angle versus frequency estimation performance and comparison to CRB performance for conventional DOA matrix algorithms and augmented DOA matrix algorithms as a function of signal-to-noise ratio (SNR) under the same conditions. The simulation parameters are set as array element number M of the array being 12 and snapshot number N being 500.
As can be seen from fig. 4 and 5, the augmented DOA matrix method has high angle and frequency estimation performance.
Fig. 6 and 7 show performance graphs of angle and frequency estimation performance of the conventional DOA matrix algorithm and the augmented DOA matrix algorithm as a function of a snapshot under the same SNR, which is set to 10 dB.
As can be seen from fig. 6 and 7, as the number of fast beats increases, the performance of both algorithms is improved, and the angle and frequency estimation performance of the augmented DOA matrix method is significantly better than that of the conventional DOA matrix method.
The above embodiments are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modifications made on the basis of the technical scheme according to the technical idea of the present invention fall within the protection scope of the present invention.

Claims (5)

1. An angle and frequency joint estimation augmented DOA matrix method is characterized by comprising the following steps:
step 1, a linear array is arranged in space, when K uncorrelated narrow-band co-carrier signals enter the linear array, estimation of an autocorrelation matrix of the linear array receiving signals is solved
Figure FDA0002282561200000011
Adding a delay output tau to a received signal of a linear array, and solving an estimate of an autocorrelation matrix of the received signal after adding the delay output
Figure FDA0002282561200000012
Estimation of cross-correlation matrix for solving linear array received signal and delayed output received signal
Figure FDA0002282561200000013
And estimating the cross-correlation matrix of the delayed output received signal and the linear array received signal
Figure FDA0002282561200000014
Step 2, estimating the autocorrelation matrix of the linear array received signal
Figure FDA0002282561200000015
Decomposing the characteristic value and removing the noise influence to obtain a matrix
Figure FDA0002282561200000016
Similarly, estimation of the autocorrelation matrix of the received signal to which the delay output is added
Figure FDA0002282561200000017
Decomposing the characteristic value and removing the noise influence to obtain a matrix
Figure FDA0002282561200000018
Step 3, estimating according to the cross correlation matrix
Figure FDA0002282561200000019
And
Figure FDA00022825612000000110
and a matrix
Figure FDA00022825612000000111
And
Figure FDA00022825612000000112
definition matrix R1And R2And constructing an extended DOA matrix
Figure FDA00022825612000000113
And 4, performing characteristic decomposition on the expanded DOA matrix R' to obtain a characteristic value and a characteristic vector, obtaining frequency estimation according to the characteristic value, and obtaining DOA angle estimation according to the characteristic vector.
2. The DOA matrix method for angle and frequency joint estimation according to claim 1, wherein the estimation of the autocorrelation matrix in step 1
Figure FDA00022825612000000114
And
Figure FDA00022825612000000115
and estimation of cross-correlation matrix
Figure FDA00022825612000000116
And
Figure FDA00022825612000000117
the following formula is obtained:
Figure FDA00022825612000000118
Figure FDA00022825612000000119
Figure FDA00022825612000000120
Figure FDA00022825612000000121
wherein N is fast beat number, x (t) represents the receiving signal of the linear array at time t, y (t) represents the receiving signal of the linear array after delay output is added at time t ·)HRepresenting a matrix conjugate transpose.
3. The DOA matrix method for angle and frequency joint estimation according to claim 1, wherein the matrix of step 2 is the DOA matrix
Figure FDA0002282561200000021
And
Figure FDA0002282561200000022
the formula of (1) is as follows:
Figure FDA0002282561200000023
Figure FDA0002282561200000024
wherein the content of the first and second substances,
Figure FDA0002282561200000025
an estimate of an autocorrelation matrix representing a linear array of received signals,
Figure FDA0002282561200000026
representing an estimate of the autocorrelation matrix of the linear array received signal after addition of the delayed output,
Figure FDA0002282561200000027
represents an estimate of the variance of additive white gaussian noise and I represents the identity matrix.
4. The method for amplifying the DOA matrix for joint angle and frequency estimation according to claim 1, wherein the formula of the extended DOA matrix R' in step 3 is as follows:
Figure FDA0002282561200000028
wherein R is1And R2Each of which represents a matrix of the image data,
Figure FDA0002282561200000029
an estimate of a cross-correlation matrix representing the linear array received signal and the delayed output added received signal,
Figure FDA00022825612000000210
representing an estimate of a cross-correlation matrix of the delayed output received signal with the linear array received signal,
Figure FDA00022825612000000211
and
Figure FDA00022825612000000212
each of which represents a matrix of the image data,
Figure FDA00022825612000000213
(·)Hrepresenting a matrix conjugate transpose.
5. The method for amplifying the DOA matrix of the joint estimation of the angle and the frequency according to the claim 1, wherein the specific process of the step 4 is as follows:
performing characteristic decomposition on the expanded DOA matrix R' to obtain a matrix AEAnd a gamma-ray that is different from the gamma-ray,
Figure FDA00022825612000000214
A＝[a(f1,θ1),a(f2,θ2),…,a(fK,θK)]represents a direction matrix and has
Figure FDA00022825612000000215
Γ＝diag{exp(-j2πf1τ),exp(-j2πf2τ),…,exp(-j2πfKτ)}
Wherein f iskRepresenting the carrier frequency, thetakRepresenting the included angle between the kth narrowband same carrier signal and the linear array, c representing the light speed, tau representing delay output, K being the number of the narrowband same carrier signals, and j representing an imaginary unit;
obtaining the frequency f according to the characteristic value in the gammakEstimation of (2):
Figure FDA0002282561200000031
wherein λ iskRepresenting the kth characteristic value;
according to AEThe definition of (1) is divided into A and A Γ-1Two parts, after feature decomposition, the estimation of the two parts is respectively
Figure FDA0002282561200000032
And
Figure FDA0002282561200000033
for matrix
Figure FDA0002282561200000034
Normalizing a certain column to make its first term 1, taking phase angle of the normalized column, estimating phase difference between narrow-band same-carrier signal and linear array according to the phase angle, and finally utilizingEstimation of DOA angle by least squares method
Figure FDA0002282561200000035
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CN113253193A (en) * 2021-04-15 2021-08-13 南京航空航天大学 Two-dimensional DOA estimation method of single snapshot data

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