CN114397619A - Two-dimensional positioning algorithm based on non-uniform sparse array - Google Patents

Two-dimensional positioning algorithm based on non-uniform sparse array Download PDF

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CN114397619A
CN114397619A CN202210002270.0A CN202210002270A CN114397619A CN 114397619 A CN114397619 A CN 114397619A CN 202210002270 A CN202210002270 A CN 202210002270A CN 114397619 A CN114397619 A CN 114397619A
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matrix
elements
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virtual
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王兰美
王乐
王桂宝
廖桂生
张志伟
孙长征
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Xidian University
Shaanxi University of Technology
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Shaanxi University of Technology
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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
    • 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
    • G01S3/04—Details
    • G01S3/043—Receivers

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  • General Physics & Mathematics (AREA)
  • Radar, Positioning & Navigation (AREA)
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  • Measurement Of Velocity Or Position Using Acoustic Or Ultrasonic Waves (AREA)
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Abstract

The invention constructs a non-uniform sparse array to carry out two-dimensional DOA estimation, calculates the difference and sum correlation matrix of the received data of two non-uniform sub-arrays, vectorizes the two correlation matrices to obtain the received data of the difference and sum virtual array element single snapshot, combines the received data of the two virtual arrays, removes redundancy and rearranges the combined received data to obtain the received data of the sum-difference virtual uniform array
Figure DDA0003455248950000011
Receiving data from a virtual uniform array
Figure DDA0003455248950000012
Performing spatial smoothing to construct a covariance matrix, and performing unitary transformation on the covariance matrix to obtain a real matrix; performing characteristic decomposition on a real data matrix of unitary transformation to obtain a signal subspace, and solving a signal and an x-axis square by using an ESPRIT algorithm according to a characteristic value of a rotation invariant relation matrixAnd substituting the estimated value of the included angle in the x-axis direction into the single snapshot virtual difference array receiving data to obtain the estimated value of the included angle between the signal and the y-axis direction.

Description

Two-dimensional positioning algorithm based on non-uniform sparse array
Technical Field
The invention belongs to the field of array signal processing, and relates to a two-dimensional positioning algorithm of a non-uniform sparse array.
Background
Direction of arrival (DOA) estimation is an important direction in the field of array signal processing, and is fully applied and developed in military and civil fields such as radar, sonar, electronic countermeasure, radio astronomy, mobile communication and the like. The traditional uniform linear array needs to meet the Nyquist sampling theorem, and the distance between adjacent array elements needs to be larger than the half wavelength of an estimation signal, so that the estimation signal is ensured not to generate signal ambiguity. Compared with a full array, the non-uniform array has the advantages that the required array elements are fewer under the condition that the same array aperture can be met, and mutual coupling and cost power consumption among the array elements are reduced. The parameters of the two-dimensional DOA estimation are a pitch angle and an azimuth angle, the incoming wave direction can be accurately positioned, and the engineering application is wide. Commonly used array arrangements include L-shaped arrays, planar arrays, parallel line arrays and circular arrays. In the non-uniform array, the array element spacing distribution is non-uniform and is an integral multiple of the wavelength, which easily causes the fuzzy of the estimated signal and the occurrence of false peaks. In order to solve this problem, the arrangement of the positions of the elements is studied. The array elements are reasonably arranged, and the received data is subjected to matrix transformation, so that a uniform continuous linear array can be generated, and the data is processed. And vectorizing and removing redundancy processing are carried out on the covariance matrix of the data to enable the covariance matrix to become the received data of a single snapshot of a uniform linear array. And performing rank recovery on the data covariance matrix, wherein the method comprises a space smoothing algorithm and constructing a Topritz matrix. The traditional uniform array is limited by the number of physical array elements in the DOA estimation problem, and the estimated number of information sources cannot exceed the number of the physical array elements. The scholars usually adopt the non-uniform array to solve the problem, the array freedom degree can be greatly improved by applying different array arrangement modes and algorithms, the array aperture is increased, the estimated number of information sources can exceed the actual number of physical array elements, common non-uniform linear arrays have the minimum redundant array, the co-prime array and the nested array, and the problem that the number of target sources is far more than the number of the array elements can be solved.
The invention provides a two-dimensional arrival angle estimation method based on a non-uniform sparse array, which is characterized in that two sub-arrays are placed in parallel, a uniform linear array is constructed by utilizing a cross covariance matrix of the two sub-arrays, a two-dimensional DOA problem is converted into a solution of two one-dimensional DOA problems according to a special structure of the array, a complex matrix is converted into a real matrix by utilizing unitary transformation of the matrix, and the solution is carried out by utilizing a traditional rotation invariant subspace method (ESPRIT). Compared with the sum-difference array algorithm, the invention greatly improves the array aperture, improves the estimation precision and has good performance under low signal-to-noise ratio. Two parameters of the pitch angle and the azimuth angle can be accurately estimated.
Disclosure of Invention
The invention aims to provide a two-dimensional DOA estimation algorithm of a non-uniform sparse array.
The two-dimensional DOA estimation algorithm of the non-uniform sparse array comprises the following steps:
step one, constructing a non-uniform sparse array;
the method comprises the following steps that a sparse non-uniform array is formed by two sparse uniform sub-arrays, the two sparse sub-arrays are placed on an x-y plane, a sub-array 1 is placed on an x axis, a sub-array 2 is parallel to the sub-array 1, and the distance between the sub-arrays is d; subarray 1 has 2M1Each array element having an array element interval of M2d, subarray 2 has M2Each array element having an array element interval of M1d，M1And M2Are two numbers that are relatively prime; the position of the element of the subarray 1 can be expressed as (x)10), wherein
Figure BDA0003455248930000021
Figure BDA0003455248930000022
Is aThe position set of the array elements of the array 1,
Figure BDA0003455248930000023
the position of the element of the sub-array 2 can be expressed as (x)2D) wherein
Figure BDA0003455248930000024
Figure BDA0003455248930000025
Is the array element position set of the sub-array 2,
Figure BDA0003455248930000026
step two, solving the position difference set, the collection and the sum difference set of the subarrays 1 and 2;
position difference set of array elements of subarray 1 and subarray 2
Figure BDA0003455248930000027
Position set of array elements of subarray 1 and subarray 2
Figure BDA0003455248930000028
The array element positions and difference sets of the subarrays 1 and 2 are
Figure BDA0003455248930000029
Figure BDA00034552489300000210
Wherein, U represents the union of the sets;
step three, the nonuniform sparse array constructed in the step one is used as a receiving array to receive K far-field, narrow-band and incoherent signals, and a difference correlation matrix of the subarray 1 and the subarray 2 is solved
Figure BDA00034552489300000211
And correlation matrix
Figure BDA00034552489300000212
The k-th incident signal has an arrival direction of (α)k,βk) Wherein α isk∈[0,π]Denotes the angle, beta, of the k-th incident signal with respect to the x-axisk∈[0,π]Representing the angle between the kth incident signal and the positive direction of the y axis; the received signal of the sub-array 1 is
Figure BDA0003455248930000031
Wherein M is1＝[m1(α1),…,m1(αk),…,m1(αK)]，M1Is an array flow pattern matrix of the sub-array 1,
Figure BDA0003455248930000032
is the space-domain steering vector of the kth signal of sub-array 1,
Figure BDA0003455248930000033
is additive white gaussian noise for sub-array 1; the received signal of the sub-array 2 is
Figure BDA0003455248930000034
Wherein the content of the first and second substances,
Figure BDA0003455248930000035
λ is the wavelength of the incident signal, M2＝[m2(α1),…,m2(αk),…,m2(αK)]，
Figure BDA0003455248930000036
Is the space-domain steering vector of the kth signal of sub-array 2,
Figure BDA0003455248930000037
is additive white gaussian noise for sub-array 2; corresponding rotation matrix M between subarrays 1 and 22Is an array flow pattern matrix of the subarray 2, S (t) ═ s1(t),...,sk(t),...,sK(t)]TIs an incident signal, where RY1And RY2The elements of the data correlation matrix may be represented as:
Figure BDA0003455248930000038
Figure BDA0003455248930000039
step four, the sum correlation matrix RY1D, d correlation matrix RY2Vectorization is carried out to obtain the received data C of the single snapshot of the virtual array element1＝vec(RY1) And C2＝vec(RY2) Mixing C with1And C2Are combined to obtain
Figure BDA00034552489300000310
Only one repeating element in the C is reserved and is arranged according to the sequence from small to large of the continuous virtual array elements, and the received data of the sum-difference virtual uniform array is obtained
Figure BDA00034552489300000311
Will be provided with
Figure BDA00034552489300000312
Performing spatial smoothing, recovering the rank of covariance matrix, dividing the virtual array into L identical uniform sub-arrays, and calculating the covariance matrix of the array after spatial smoothing of the virtual array
Figure BDA00034552489300000313
Wherein the content of the first and second substances,
Figure BDA00034552489300000314
indicating the first sub-array received signal,
Figure BDA00034552489300000315
Figure BDA00034552489300000316
Figure BDA0003455248930000041
vec (-) denotes vectorizing the matrix, i.e.The matrix elements are arranged in columns into a column vector,
Figure BDA0003455248930000042
represents the Kronecker product, ()*It is indicated that the conjugate is taken,
Figure BDA0003455248930000043
is a vectorized single snapshot virtual array guide vector,
Figure BDA0003455248930000044
vectorized single snapshot and virtual array steering vectors,
Figure BDA0003455248930000045
represents the signal power, and then C1And C2Are combined into
Figure BDA0003455248930000046
Obtaining single snapshot data of the virtual array, reserving only one repeated element according to the position of the virtual array element, arranging the repeated elements according to the sequence of the continuous virtual array elements from small to large for redundancy elimination rearrangement, and carrying out the next step of arranging the repeated elements according to the sequence of the continuous virtual array elements from small to large
Figure BDA0003455248930000047
And
Figure BDA0003455248930000048
after combination, the repeated elements are removed, and
Figure BDA0003455248930000049
and
Figure BDA00034552489300000410
only one array element with the same middle position is reserved, and rearrangement is performed according to the position size to obtain the array element
Figure BDA00034552489300000411
Step five, receiving data by the sum-difference virtual uniform array
Figure BDA00034552489300000412
Performing spatial smoothing to obtain a covariance matrix
Figure BDA00034552489300000413
Will be provided with
Figure BDA00034552489300000414
Performing unitary transformation to obtain matrix
Figure BDA00034552489300000415
To REExtracting signal subspace P by eigenvalue decompositionSCalculating Ψu＝(B1PS)#B2PSTo ΨuDecomposing the eigenvalue to obtain the eigenvalue { xi1,…,ξk,…,ξKTherein of
Figure BDA00034552489300000416
Xi is composed ofkObtaining angle estimation information
Figure BDA00034552489300000417
Will be provided with
Figure BDA00034552489300000418
Substituting single snapshot difference virtual array guide vector to obtain
Figure BDA00034552489300000419
By
Figure BDA00034552489300000420
Obtaining angle estimation information
Figure BDA00034552489300000421
Figure BDA00034552489300000422
A pseudo-inverse matrix representing a matrix;
unitary matrix UnIs an n x n dimensional square matrix, and when n is an even number, the unitary matrix is
Figure BDA00034552489300000423
When n is an odd number, the unitary matrix is
Figure BDA00034552489300000424
0 is a matrix with all zero p × 1 dimensional elements; b is1And B2Is a selection matrix of unitary transformation expressed as
Figure BDA00034552489300000425
Figure BDA00034552489300000426
Figure BDA00034552489300000427
Is an n multiplied by n square matrix with all diagonal elements being 1 and all other elements being 0;
Figure BDA0003455248930000051
is an n multiplied by n dimensional square matrix with the anti-angle elements all being 1 and the other elements all being 0,
Figure BDA0003455248930000052
is a p multiplied by p square matrix with all diagonal elements being 1 and all other elements being 0;
Figure BDA0003455248930000053
is a p x p dimensional square matrix with the inverse angle elements all being 1 and the other elements all being 0,
Figure BDA0003455248930000054
is a (n-1) x (n-1) square matrix with all diagonal elements being 1 and all other elements being 0;
Figure BDA0003455248930000055
is a (n-1) x (n-1) dimensional square matrix with all anti-angle elements being 1 and all other elements being 0;
Figure BDA0003455248930000056
the k-th element in (1) is denoted by
Figure BDA0003455248930000057
n represents the number of matrices in the spatial smoothing process
Figure BDA0003455248930000058
When n is an even number, n is 2p, and when n is an odd number, n is 2p + 1.
In the foregoing step, K denotes the number of signal sources, where K denotes the reference number of the signal source, and L denotes the number of spatially smoothed subarrays.
1. The method increases the virtual array elements by using the sum and difference set method, increases the array aperture compared with the difference co-prime double parallel array, and has better calculation precision of a positioning algorithm.
2. The invention adopts unitary-ESPRIT algorithm, has fast operation speed compared with the traditional algorithm, combines the correlation matrixes of summation and difference in the step of processing the data correlation matrix, effectively expands the virtual position of the array, has high calculation precision and relatively higher success probability of estimating signals.
3. The algorithm of the invention can reduce the cost, greatly improves the aperture and the degree of freedom of the array compared with the common traditional array, and can estimate the number of signals with large number by using few array elements.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic diagram of an array structure according to the present invention;
FIG. 2 is a flow chart of the present invention;
FIG. 3 is a scatter plot of the estimated value and the true value of the algorithm of the present invention and the conventional algorithm under the same conditions;
FIG. 4 is a diagram of the root mean square error of the angle estimation value of the present invention and the conventional algorithm under different SNR;
FIG. 5 is a graph of the success probability of angle estimates for different SNR's of the present invention versus the conventional method;
Detailed Description
In order to make the aforementioned and other objects, features and advantages of the present invention more apparent, embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
The invention aims to provide a two-dimensional positioning algorithm of a non-uniform sparse parallel array.
In order to achieve the purpose, the invention adopts the following technical solutions:
step one, constructing a non-uniform sparse array;
the method comprises the following steps that a sparse non-uniform array is formed by two sparse uniform sub-arrays, the two sparse sub-arrays are placed on an x-y plane, a sub-array 1 is placed on an x axis, a sub-array 2 is parallel to the sub-array 1, and the distance between the sub-arrays is d; subarray 1 has 2M1Each array element having an array element interval of M2d, subarray 2 has M2Each array element having an array element interval of M1d，M1And M2Are two numbers that are relatively prime; the position of the element of the subarray 1 can be expressed as (x)10), wherein
Figure BDA0003455248930000061
Figure BDA0003455248930000062
Is the array element position set of the sub-array 1,
Figure BDA0003455248930000063
the position of the element of the sub-array 2 can be expressed as (x)2D) wherein
Figure BDA0003455248930000064
Figure BDA0003455248930000065
Is the array element position set of the sub-array 2,
Figure BDA0003455248930000066
with M1＝3,M2For example, 4, the array element position set of the sub-array 1 is
Figure BDA0003455248930000067
The array element position of the subarray 2 is set as
Figure BDA0003455248930000068
Step two, solving the position difference set, the collection and the sum difference set of the subarrays 1 and 2;
position difference set of array elements of subarray 1 and subarray 2
Figure BDA0003455248930000069
Position set of array elements of subarray 1 and subarray 2
Figure BDA00034552489300000610
The array element positions and difference sets of the subarrays 1 and 2 are
Figure BDA00034552489300000611
Figure BDA00034552489300000612
Wherein, U represents the union of the sets; with M1＝3,M2For example, the number of array elements 6 in sub-array 1 and the number of array elements in sub-array 2 are 4, and the sum of the position difference of the array elements and the position of the array elements is respectively
Figure BDA00034552489300000613
Figure BDA00034552489300000614
Step three, the nonuniform sparse array constructed in the step one is used as a receiving array to receive K far-field, narrow-band and incoherent signals, and a difference correlation matrix of the subarray 1 and the subarray 2 is solved
Figure BDA0003455248930000071
And correlation matrix
Figure BDA0003455248930000072
The k-th incident signal has an arrival direction of (α)k,βk) Wherein α isk∈[0,π]Denotes the angle, beta, of the k-th incident signal with respect to the x-axisk∈[0,π]Representing the angle between the kth incident signal and the positive direction of the y axis; the received signal of the sub-array 1 is
Figure BDA0003455248930000073
Wherein M is1＝[m1(α1),…,m1(αk),…,m1(αK)]，M1Is an array flow pattern matrix of the sub-array 1,
Figure BDA0003455248930000074
is the space-domain steering vector of the kth signal of sub-array 1,
Figure BDA0003455248930000075
is additive white gaussian noise for sub-array 1; the received signal of the sub-array 2 is
Figure BDA0003455248930000076
Wherein
Figure BDA0003455248930000077
λ is the wavelength of the incident signal, M2＝[m2(α1),…,m2(αk),…,m2(αK)]，
Figure BDA0003455248930000078
Is the space-domain steering vector of the kth signal of sub-array 2,
Figure BDA0003455248930000079
is additive white gaussian noise for sub-array 2; corresponding rotation matrix M between subarrays 1 and 22Is an array flow pattern matrix of the subarray 2, S (t) ═ s1(t),...,sk(t),...,sK(t)]TIs an incident signal, where RY1And RY2The elements of the data correlation matrix may be represented as:
Figure BDA00034552489300000710
Figure BDA00034552489300000711
with M1＝3,M2As an example of the case of 4,
Figure BDA00034552489300000712
Figure BDA00034552489300000713
Figure BDA00034552489300000714
and
Figure BDA00034552489300000715
in the middle element of RY1And RY2The sum of differences corresponding to positions in the data correlation matrix is P1And P2；
Figure BDA00034552489300000716
Step four, the sum correlation matrix RY1D, d correlation matrix RY2Vectorization is carried out to obtain the received data C of the single snapshot of the virtual array element1＝vec(RY1) And C2＝vec(RY2) Mixing C with1And C2Are combined to obtain
Figure BDA0003455248930000081
Only one repeating element in the C is reserved and is arranged according to the sequence from small to large of the continuous virtual array elements, and the received data of the sum-difference virtual uniform array is obtained
Figure BDA0003455248930000082
Will be provided with
Figure BDA0003455248930000083
Performing spatial smoothing, recovering the rank of covariance matrix, dividing the virtual array into L identical uniform sub-arrays, and calculating the covariance matrix of the array after spatial smoothing of the virtual array
Figure BDA0003455248930000084
Wherein the content of the first and second substances,
Figure BDA0003455248930000085
indicating the first sub-array received signal,
Figure BDA0003455248930000086
Figure BDA0003455248930000087
Figure BDA0003455248930000088
vec (-) denotes vectorizing the matrix, i.e. arranging the matrix elements column by column into a column vector,
Figure BDA0003455248930000089
represents the Kronecker product, ()*It is indicated that the conjugate is taken,
Figure BDA00034552489300000810
is a vectorized single snapshot virtual array guide vector,
Figure BDA00034552489300000811
vectorized single snapshot and virtual array steering vectors,
Figure BDA00034552489300000812
represents the signal power, and then C1And C2Are combined into
Figure BDA00034552489300000813
Obtaining single snapshot data of the virtual array and then according to the virtualOnly one repeated element of the pseudo-array element position is reserved, and the repeated elements are arranged according to the sequence from small to large of the continuous virtual array elements to carry out redundancy elimination rearrangement, so that
Figure BDA00034552489300000814
And
Figure BDA00034552489300000815
after combination, the repeated elements are removed, and
Figure BDA00034552489300000816
and
Figure BDA00034552489300000817
only one array element with the same middle position is reserved, and rearrangement is performed according to the position size to obtain the array element
Figure BDA00034552489300000818
When in use
Figure BDA00034552489300000819
Figure BDA00034552489300000820
Step five, smoothing the sum-difference virtual uniform array received data space to obtain a covariance matrix
Figure BDA00034552489300000821
Will be provided with
Figure BDA00034552489300000822
Performing unitary transformation to obtain matrix
Figure BDA00034552489300000823
To REExtracting signal subspace P by eigenvalue decompositionSCalculating Ψu＝(B1PS)#B2PSTo ΨuDecomposing the eigenvalue to obtain the eigenvalue { xi1,…,ξk,…,ξKTherein of
Figure BDA0003455248930000091
Xi is composed ofkObtaining angle estimation information
Figure BDA0003455248930000092
Will be provided with
Figure BDA0003455248930000093
Substituting single snapshot difference virtual array guide vector to obtain
Figure BDA0003455248930000094
By
Figure BDA0003455248930000095
Obtaining angle estimation information
Figure BDA0003455248930000096
Figure BDA0003455248930000097
A pseudo-inverse matrix representing a matrix;
unitary matrix UnIs an n x n dimensional square matrix, and when n is an even number, the unitary matrix is
Figure BDA0003455248930000098
When n is an odd number, the unitary matrix is
Figure BDA0003455248930000099
0 is a matrix with all zero p × 1 dimensional elements; b is1And B2Is a selection matrix of unitary transformation expressed as
Figure BDA00034552489300000910
Figure BDA00034552489300000911
Figure BDA00034552489300000912
Is an n multiplied by n square matrix with all diagonal elements being 1 and all other elements being 0;
Figure BDA00034552489300000913
is an n multiplied by n dimensional square matrix with the anti-angle elements all being 1 and the other elements all being 0,
Figure BDA00034552489300000914
is a p multiplied by p square matrix with all diagonal elements being 1 and all other elements being 0;
Figure BDA00034552489300000915
is a p x p dimensional square matrix with the inverse angle elements all being 1 and the other elements all being 0,
Figure BDA00034552489300000916
is a (n-1) x (n-1) square matrix with all diagonal elements being 1 and all other elements being 0;
Figure BDA00034552489300000917
is a (n-1) x (n-1) dimensional square matrix with all anti-angle elements being 1 and all other elements being 0;
Figure BDA00034552489300000918
the k-th element in (1) is denoted by
Figure BDA00034552489300000919
n represents the number of matrices in the spatial smoothing process
Figure BDA00034552489300000920
When n is an even number, n is 2p, and when n is an odd number, n is 2p + 1.
In the foregoing step, K denotes the number of signal sources, where K denotes the reference number of the signal source, and L denotes the number of spatially smoothed subarrays.
The invention adopts a new virtual array element expansion method, obtains a sum-difference virtual uniform array by utilizing a difference sum correlation matrix of two sub-arrays, enlarges the array aperture and improves the parameter estimation precision, realizes two-dimensional parameter separation by combining a y-axis phase item and signal power, respectively estimates alpha and beta angle information by utilizing a twiddle factor matrix and an array guide vector, converts a two-dimensional estimation problem into two one-dimensional estimation problems, converts a vectorized data correlation matrix into a real matrix by utilizing unitary transformation of the matrix, solves an x-axis direction included angle according to an ESPRIT algorithm, and substitutes an x-axis direction included angle estimation value into virtual received data to obtain y-axis direction included angle estimation.
The effect of the present invention can be further illustrated by the following simulation results:
in simulation, the existing Root-MUSIC, ESPRIT and SS-MUSIC (spatial smoothing-MUSIC) algorithms are compared with the algorithm of the invention.
The simulation conditions were as follows:
taking M as parameter of non-uniform sparse array1＝3,M2The number of array elements 6 of sub-array 1 and the number of array elements of sub-array 2 are 4. The directions of arrival of the four signals are: the simulation experiment was performed under the conditions of (20 °,40 °), (35 °,36 °), (55 °,60 °), (65 °,75 °), and fast beat number T of 1000.
Simulation 1: fig. 3(a), 3(b), 3(c), and 3(d) are scatter diagrams of the algorithm, ESPRIT, SS-MUISC, and ROOT-MUSIC of the present invention at a signal-to-noise ratio of 0dB, respectively, and it can be seen from fig. 3 that the error between the estimated value and the true value of the algorithm of the present invention is the smallest and the estimation accuracy is the highest because the algorithm of the present invention jointly utilizes the difference and the virtual array, increasing the number of virtual array elements and increasing the aperture of the virtual array, so the estimation accuracy is improved.
Simulation 2: the signal-to-noise ratio is [ -15dB,15dB ], 50 Monte Carlo simulation experiments are carried out every 3dB, and the curves of the root mean square error of the arrival angle estimation and the success probability along with the change of the signal-to-noise ratio are shown in FIG. 4 and FIG. 5. It can be seen from fig. 4 that the performance of the algorithm of the present invention is obviously superior to that of the other three methods when the signal-to-noise ratio is low, the root mean square error is minimum, the sum-difference virtual array is used in combination, the aperture of the virtual array is enlarged, and the algorithm precision is improved.
As can be seen from FIG. 5, the success probability reaches over 90% when the signal-to-noise ratio is-15 dB, the success probability of the other three methods is lower than 60%, and the simulation result verifies the superiority of the method of the invention.
Although the present invention has been described with reference to a preferred embodiment, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. The two-dimensional positioning algorithm based on the non-uniform sparse array is characterized by comprising the following steps of: k far-field, narrow-band, incoherent signals are simultaneously incident on a sparse non-uniform array, which is a non-uniform array consisting of two sparse uniform sub-arrays,
step one, constructing a non-uniform sparse array;
the method comprises the following steps that a sparse non-uniform array is formed by two sparse uniform sub-arrays, the two sparse sub-arrays are placed on an x-y plane, a sub-array 1 is placed on an x axis, a sub-array 2 is parallel to the sub-array 1, and the distance between the sub-arrays is d; subarray 1 has 2M1Each array element having an array element interval of M2d, subarray 2 has M2Each array element having an array element interval of M1d，M1And M2Are two numbers that are relatively prime; the position of the element of the subarray 1 can be expressed as (x)10), wherein
Figure FDA0003455248920000011
Figure FDA0003455248920000012
Is the array element position set of the sub-array 1,
Figure FDA0003455248920000013
the position of the element of the sub-array 2 can be expressed as (x)2D) wherein
Figure FDA0003455248920000014
Figure FDA0003455248920000015
Is the array element position set of the sub-array 2,
Figure FDA0003455248920000016
step two, solving the position difference set, the collection and the sum difference set of the subarrays 1 and 2;
position difference set of array elements of subarray 1 and subarray 2
Figure FDA0003455248920000017
Position set of array elements of subarray 1 and subarray 2
Figure FDA0003455248920000018
The array element positions and difference sets of the subarrays 1 and 2 are
Figure FDA0003455248920000019
Figure FDA00034552489200000110
Wherein, U represents the union of the sets;
step three, the nonuniform sparse array constructed in the step one is used as a receiving array to receive K far-field, narrow-band and incoherent signals, and a difference correlation matrix of the subarray 1 and the subarray 2 is solved
Figure FDA00034552489200000111
And correlation matrix
Figure FDA00034552489200000112
The k-th incident signal has an arrival direction of (α)k,βk) Wherein α isk∈[0,π]Denotes the angle, beta, of the k-th incident signal with respect to the x-axisk∈[0,π]Representing the angle between the kth incident signal and the positive direction of the y axis; the received signal of the sub-array 1 is
Figure FDA00034552489200000114
Wherein M is1＝[m1(α1),…,m1(αk),…,m1(αK)]，M1Is an array flow pattern matrix of the sub-array 1,
Figure FDA00034552489200000113
is the space-domain steering vector of the kth signal of sub-array 1,
Figure FDA0003455248920000021
is additive white gaussian noise for sub-array 1; the received signal of the sub-array 2 is
Figure FDA0003455248920000022
Wherein
Figure FDA0003455248920000023
λ is the wavelength of the incident signal, M2＝[m2(α1),…,m2(αk),…,m2(αK)]，
Figure FDA0003455248920000024
Is the space-domain steering vector of the kth signal of sub-array 2,
Figure FDA0003455248920000025
is additive white gaussian noise for sub-array 2; corresponding rotation matrix M between subarrays 1 and 22Is an array flow pattern matrix of the subarray 2, S (t) ═ s1(t),...,sk(t),...,sK(t)]TIs an incident signal, where RY1And RY2The elements of the data correlation matrix may be represented as:
Figure FDA0003455248920000026
Figure FDA0003455248920000027
step four, the sum correlation matrix RY1D, d correlation matrix RY2Vectorization is carried out to obtain the received data C of the single snapshot of the virtual array element1＝vec(RY1) And C2＝vec(RY2) Mixing C with1And C2Are combined to obtain
Figure FDA0003455248920000028
Only one repeating element in the C is reserved and is arranged according to the sequence from small to large of the continuous virtual array elements, and the received data of the sum-difference virtual uniform array is obtained
Figure FDA0003455248920000029
Will be provided with
Figure FDA00034552489200000210
Performing spatial smoothing, recovering the rank of covariance matrix, dividing the virtual array into L identical uniform sub-arrays, and calculating the covariance matrix of the array after spatial smoothing of the virtual array
Figure FDA00034552489200000211
Wherein the content of the first and second substances,
Figure FDA00034552489200000212
representing the first sub-array received signal;
Figure FDA00034552489200000213
Figure FDA00034552489200000214
Figure FDA00034552489200000215
vec (-) denotes vectorizing the matrix, i.e. arranging the matrix elements column by column into a column vector,
Figure FDA00034552489200000216
represents the Kronecker product, ()*It is indicated that the conjugate is taken,
Figure FDA00034552489200000217
is a vectorized single snapshot virtual array guide vector,
Figure FDA00034552489200000218
vectorized single snapshot and virtual array steering vectors,
Figure FDA00034552489200000219
represents the signal power, and then C1And C2Are combined into
Figure FDA0003455248920000031
Obtaining single snapshot data of the virtual array, reserving only one repeated element according to the position of the virtual array element, arranging the repeated elements according to the sequence of the continuous virtual array elements from small to large for redundancy elimination rearrangement, and carrying out the next step of arranging the repeated elements according to the sequence of the continuous virtual array elements from small to large
Figure FDA0003455248920000032
And
Figure FDA0003455248920000033
after combination, the repeated elements are removed, and
Figure FDA0003455248920000034
and
Figure FDA0003455248920000035
only one array element with the same middle position is reserved, and rearrangement is performed according to the position size to obtain the array element
Figure FDA0003455248920000036
Step five, virtually homogenizing the sum and the differenceObtaining covariance matrix by array data space smoothing
Figure FDA0003455248920000037
Will be provided with
Figure FDA0003455248920000038
Performing unitary transformation to obtain matrix
Figure FDA0003455248920000039
To REExtracting signal subspace P by eigenvalue decompositionSCalculating Ψu＝(B1PS)#B2PSTo ΨuDecomposing the eigenvalue to obtain the eigenvalue { xi1,…,ξk,…,ξKTherein of
Figure FDA00034552489200000310
Xi is composed ofkObtaining angle estimation information
Figure FDA00034552489200000311
Will be provided with
Figure FDA00034552489200000312
Substituting single snapshot difference virtual array guide vector to obtain
Figure FDA00034552489200000313
By
Figure FDA00034552489200000314
Obtaining angle estimation information
Figure FDA00034552489200000315
Figure FDA00034552489200000316
A pseudo-inverse matrix representing a matrix;
unitary matrix UnIs an n x n dimensional square matrix when n is evenSeveral times, unitary matrix is
Figure FDA00034552489200000317
When n is an odd number, the unitary matrix is
Figure FDA00034552489200000318
0 is a matrix with all zero p × 1 dimensional elements; b is1And B2Is a selection matrix of unitary transformation expressed as
Figure FDA00034552489200000319
Figure FDA00034552489200000320
Figure FDA00034552489200000321
Is an n multiplied by n square matrix with all diagonal elements being 1 and all other elements being 0;
Figure FDA00034552489200000322
is an n multiplied by n dimensional square matrix with the anti-angle elements all being 1 and the other elements all being 0,
Figure FDA00034552489200000323
is a p multiplied by p square matrix with all diagonal elements being 1 and all other elements being 0;
Figure FDA00034552489200000324
is a p x p dimensional square matrix with the inverse angle elements all being 1 and the other elements all being 0,
Figure FDA00034552489200000325
is a (n-1) x (n-1) square matrix with all diagonal elements being 1 and all other elements being 0;
Figure FDA0003455248920000041
is a (n-1) x (n-1) dimensional square matrix with all anti-angle elements being 1 and all other elements being 0;
Figure FDA0003455248920000042
the k-th element in (1) is denoted by
Figure FDA0003455248920000043
n represents the number of matrices in the spatial smoothing process
Figure FDA0003455248920000044
When n is an even number, n is 2p, and when n is an odd number, n is 2p + 1;
in the foregoing step, K denotes the number of signal sources, where K denotes the reference number of the signal source, and L denotes the number of spatially smoothed subarrays.
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