CN111366891A - Pseudo covariance matrix-based uniform circular array single snapshot direction finding method - Google Patents

Pseudo covariance matrix-based uniform circular array single snapshot direction finding method Download PDF

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CN111366891A
CN111366891A CN202010208559.9A CN202010208559A CN111366891A CN 111366891 A CN111366891 A CN 111366891A CN 202010208559 A CN202010208559 A CN 202010208559A CN 111366891 A CN111366891 A CN 111366891A
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circular array
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covariance matrix
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CN111366891B (en
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熊钊
彭晓燕
魏逸凡
李万春
魏平
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NORTH AUTOMATIC CONTROL TECHNOLOGY INSTITUTE
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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

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Abstract

The invention belongs to the technical field of electronic countermeasure, and particularly relates to a uniform circular array single snapshot direction finding method based on a pseudo covariance matrix. For the array receiving data of the uniform circular array, the invention firstly converts the data into the receiving data of the virtual linear array through mode space transformation, thereby leading the array flow pattern to have a Vandemonde form. And then constructing the pseudo covariance matrix to enable the pseudo covariance matrix for subsequent processing to reach a full rank, so that subsequent eigen decomposition can correctly divide the signal subspace and the noise subspace. Finally, the MUSIC algorithm is used for the pseudo-covariance matrix, namely, the covariance matrix is subjected to characteristic decomposition, and then the orthogonality of the guide vector and the noise subspace is searched, so that the incident azimuth angle of the signal is obtained. The method has the advantages that the single snapshot direction finding can be carried out by using the uniform circular array, the method is simple, and the effect is good.

Description

Pseudo covariance matrix-based uniform circular array single snapshot direction finding method
Technical Field
The invention belongs to the technical field of electronic countermeasure, and relates to a uniform circular array single snapshot direction finding method based on a pseudo covariance matrix.
Background
Angle of arrival (DOA) estimation is one of the main problems to be studied in array signal processing, and has wide application in the fields of radio communication, electronic reconnaissance and the like. And single snapshot DOA estimation, namely, only the data of a single snapshot is processed, so that the estimation of the direction of arrival of the input signal is realized. At present, with the continuous improvement of DOA estimation on the real-time requirement, the algorithm operation amount is reduced by reducing the sampling fast beat number, the system real-time is improved, and the DOA estimation method becomes a hot spot of research in the DOA estimation field in recent years.
Among the single-snapshot direction finding algorithms, the most widely applied is the study based on the pseudo covariance matrix and spatial smoothing. The spatial smoothing method divides an array under a certain rule into a plurality of sub-arrays, each sub-array has the same rule, and an array covariance matrix corresponding to each sub-array is obtained. The new covariance matrix is constructed by processing the data to replace the whole matrix which is not divided into sub-matrices. The other method is a pseudo covariance matrix method, which converts a covariance matrix into a pseudo covariance matrix by processing received data under the condition of single snapshot, thereby increasing useful information quantity, enabling the rank of the matrix to reach the signal source number, and further applying the classical direction finding algorithm to the condition of single snapshot with the limit of fast snapshot number. However, both methods can only process matrices having the vandermode form, and when uniform circular arrays are used, single snapshot direction finding cannot be directly performed.
Disclosure of Invention
Aiming at the problems, the invention provides a uniform circular array single snapshot direction finding method based on a pseudo covariance matrix.
The technical scheme adopted by the invention is as follows:
and for the received single snapshot data, converting the array flow pattern of the circular array into a virtual linear array by using mode space conversion, wherein the processed data is equivalent to the array flow pattern of the virtual linear array multiplied by a signal. The array flow pattern now has the vandenonde form. And constructing a pseudo covariance matrix by taking the processed data. And (3) carrying out eigenvalue decomposition on the matrix, and utilizing the fact that the guide vector of the signal subspace is orthogonal to the noise subspace under the ideal condition, the product is very small under the actual condition, carrying out spectrum peak search on the azimuth angle from 0 degree to 360 degrees, wherein the angle corresponding to the maximum value point is the signal incidence direction.
Consider a uniform circular array of discrete fourier transforms that receive data. Suppose N far-field signal sources (θ)1,…,θN) The signal of (A) is incident on a uniform circular array of M array elements, the circular array elements are isotropic, and a received data matrix is
x＝As+n (1)
Where n is the noise matrix.
If each signal only receives a single snapshot, the snapshot data on the array element l is:
Figure BDA0002422026520000021
wherein the content of the first and second substances,
Figure BDA0002422026520000022
θiis the azimuth angle, s, of the i-th signal incidenceiIs the ith signal.
The steering vector of the circular array is
Figure BDA0002422026520000023
Spatial DFT of array snapshot data
Figure BDA0002422026520000024
Figure BDA0002422026520000026
Wherein the Bessel function:
Figure BDA0002422026520000025
when the number of the array elements is uniform
Figure BDA0002422026520000027
When there is JkM-q(- β) ≈ 0, so that the above formula can be simplified to
Figure BDA0002422026520000031
Wherein the content of the first and second substances,
Figure BDA0002422026520000037
if u orderedq＝v-qThen the above equation can be written in the form of a matrix as follows:
Figure BDA0002422026520000032
writing the above equation in matrix form:
Figure BDA0002422026520000033
namely:
Figure BDA0002422026520000034
Figure BDA0002422026520000035
the re-exploration space DFT itself has
Figure BDA0002422026520000036
The above formula is abbreviated as
u＝FHx (12)
Namely, it is
Figure BDA0002422026520000041
Is provided with
FHF＝MI (14)
F is an orthogonal matrix.
By
Figure BDA0002422026520000042
Figure BDA0002422026520000043
Obtaining a pre-processing matrix
Figure BDA0002422026520000044
Carrying out mode space transformation on single snapshot data x obtained by us from a uniform circular array through a matrix:
Figure BDA0002422026520000045
after transformation, the uniform circular array becomes a virtual linear array, and the number of array elements is M' ═ 2K +1,
Figure BDA0002422026520000046
taking the data from the K +1 th line to the 2K +1 th line of y, namely the data from the K +1 th array element to the 2K +1 th array element of the virtual linear array, including
Figure BDA0002422026520000051
Wherein y iskIs the data on the k-th row of the matrix y, i.e. the k-th array element of the virtual line array.
Figure BDA0002422026520000052
nzLines K +1 to 2K +1 of Tn. It can be seen that the component of the kth array element of the steering vector corresponding to signal i is:
bk(θi)＝exp(j(k-1)θi) (19)
constructing the pseudo covariance matrix with z as follows:
Figure BDA0002422026520000053
wherein z iskIs the kth data of matrix z. Is provided with
Figure BDA0002422026520000054
p,k＝1,2,…K+1(p+k-1≤K+1)
In the formula nkIs a noise matrix nzThe kth element of (1).
R can be written as
R＝BDBH+Nr(22)
Wherein the content of the first and second substances,
Figure BDA0002422026520000061
Figure BDA0002422026520000062
it is clear that the rank of D is the number of incident signals N, and since B is a Van der Monde matrix, B and BDBHThe rank of R is N, so that the rank of R is N, and the subsequent eigen decomposition can separate N large eigenvalues and M-N small eigenvalues, namely, the signal subspace and the noise subspace can be decomposed correctly.
Performing characteristic decomposition on the pseudo covariance matrix R obtained above:
Figure BDA0002422026520000063
obtaining a signal subspace USSum noise subspace UN. Steering vector b of signal subspace after mode space conversion under ideal conditionsH(theta) and noise subspace UNOrthogonal:
bH(θ)UN＝0 (26)
and in practice bH(theta) and UNAnd are not perfectly orthogonal. The azimuth angle θ can be obtained from a 1 degree to 360 degree search by a minimum optimization search:
Figure BDA0002422026520000064
the spectral estimation formula is:
Figure BDA0002422026520000065
drawings
FIG. 1 is a schematic diagram of a circular array received signal model, discussed herein as pitch angle
Figure BDA0002422026520000071
In the case of 90 deg..
FIG. 2 is a flow chart of a pseudo covariance matrix-based uniform circular array single snapshot array direction finding algorithm.
FIG. 3 is a plot of the spectral estimation of direction finding using the present algorithm.
Fig. 4 shows the positioning error for different signal-to-noise ratios.
Detailed Description
The present invention will be described in detail with reference to examples below:
examples
In this embodiment, matlab is used to verify the single snapshot array direction finding algorithm scheme based on the uniform circular array, and for simplification, the following assumptions are made for the algorithm model:
1. all engineering errors are superposed into equivalent noise;
2. assuming the target is stationary;
step 1, 4 targets are arranged at azimuth angles (100 degrees, 150 degrees, 200 degrees and 250 degrees). The target is measured using a uniform circular array having 24 array elements. The radius r of the circular array is 1.5 lambda. Assuming that the noise of the observation station follows Gaussian distribution with the mean value of zero;
step 2, obtaining received array signal data, wherein each array element only receives one snapshot;
step 3, β is calculated according to the radius-wavelength ratio of the circular array, and K is obtained;
step 4, constructing matrixes F and J according to K and the above formula (13) and formula (9), and obtaining a pretreatment matrix T according to formula (12);
step 5, preprocessing the received signal x, wherein y is Tx;
step 6, obtaining a matrix z from the K +1 th row to the 2K +1 th row of y;
step 7, constructing a matrix R by the formula (20);
step 8, performing characteristic decomposition on the R to obtain a signal subspace USSum noise subspace UN；
And 9, according to the formula (28), performing minimum optimization search on the azimuth angle theta from 1 degree to 360 degrees to obtain an accurate positioning result of the target.
The pseudo covariance matrix-based uniform circular array single snapshot direction finding algorithm has the effects that:
as shown in fig. 3, a spectrum peak diagram of the target appearing in the observation region can be seen, and the position of the spectrum peak is the positioning result of the target. Fig. 4 shows the trend of the positioning error with the signal-to-noise ratio.

Claims (1)

1. A pseudo covariance matrix-based uniform circular array single snapshot direction finding method is characterized by comprising the following steps:
s1, receiving data by using a uniform circular array, N far-field signal sources (theta)1,…,θN) The signal is incident on a uniform circular array of M array elements, the circular array elements are isotropic, and single snapshot data received by the array are
x＝As+n
Where A is the array pattern of the uniform circular array, and s is the signal vector formed by the N signals, i.e. s ═ s1,s2,…sN]TAnd n is a noise data,
Figure FDA0002422026510000011
if each signal only receives a single snapshot, the snapshot data on the array element l is:
Figure FDA0002422026510000012
wherein the content of the first and second substances,
Figure FDA0002422026510000013
λ is the wavelength, r is the radius of the circular array, nlIs the i-th element, θ, of the noise data niIs the azimuth angle, s, of the i-th signal incidenceiIs the ith signal;
the steering vector of the circular array is
Figure FDA0002422026510000014
S2, making mode space transformation to the received data matrix, and constructing the mode space transformation matrix
Figure FDA0002422026510000015
Wherein
Figure FDA0002422026510000016
Figure FDA0002422026510000021
In J, Jk(- β) represents the K-th order Bessel function of the first kind, K ∈ -K, -K +1, … K-1, K, K being the maximum number of phase modes that this uniform circular array can excite:
Figure FDA0002422026510000022
r is the radius of the circular array;
s3, preprocessing single snapshot data x obtained from the uniform circular array:
Figure FDA0002422026510000023
wherein
Figure FDA0002422026510000024
For spatial transformation of modesThen, the array flow pattern of the virtual linear array:
Figure FDA0002422026510000025
after transformation, the uniform circular array becomes a virtual linear array, and the number of array elements is M' ═ 2K +1,
Figure FDA0002422026510000026
s4, taking the data from the K +1 th line to the 2K +1 th line of y, namely the data from the K +1 th array element to the 2K +1 th array element of the virtual linear array, including
Figure FDA0002422026510000027
Wherein y iskIs the data on the k-th row of the matrix y, i.e. the k-th array element of the virtual line array,
Figure FDA0002422026510000031
b is an array flow pattern formed by the K array element to the 2K +1 array element of the virtual linear array, nzTaking lines K +1 to 2K +1 for Tn, the component of the kth array element of the steering vector corresponding to signal i is:
bk(θi)＝exp(j(k-1)θi)
s5, constructing a pseudo covariance matrix by using z as follows:
Figure FDA0002422026510000032
wherein z iskIs the kth data of matrix z; is provided with
Figure FDA0002422026510000033
In the formula nkIs a noise matrix nzThe kth element of (1);
r can be written as
R＝BDBH+Nr
Wherein the content of the first and second substances,
Figure FDA0002422026510000034
Figure FDA0002422026510000041
the rank of D is the number of incident signals N, and B is a Van der Monde matrix, so B and BDBHIs N, so the rank of R is N;
s6, performing characteristic decomposition on the pseudo covariance matrix R obtained in the step:
Figure FDA0002422026510000042
obtaining a signal subspace USSum noise subspace UN；
S7, obtaining an azimuth angle theta from 1-360-degree search through minimum optimization search:
Figure FDA0002422026510000043
wherein B (theta) is a guide vector corresponding to the array flow pattern B, namely:
Figure FDA0002422026510000044
the spectral estimation formula is:
Figure FDA0002422026510000045
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