CN109655799B - IAA-based covariance matrix vectorization non-uniform sparse array direction finding method - Google Patents

IAA-based covariance matrix vectorization non-uniform sparse array direction finding method Download PDF

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CN109655799B
CN109655799B CN201811596703.XA CN201811596703A CN109655799B CN 109655799 B CN109655799 B CN 109655799B CN 201811596703 A CN201811596703 A CN 201811596703A CN 109655799 B CN109655799 B CN 109655799B
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CN109655799A (en
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干鹏
陈卓
朱晓丹
侯庆禹
汤永浩
赵洪冰
李贵显
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8511 Research Institute of CASIC
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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
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
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    • G01S7/4004—Means for monitoring or calibrating of parts of a radar system
    • 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
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
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    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
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Abstract

The invention discloses an IAA-based covariance matrix vectorization non-uniform sparse array direction-finding method, which adopts an IAA algorithm to carry out power estimation and covariance matrix rank recovery on the basis of obtaining virtual array equivalent single snapshot received data by non-uniform sparse array received data covariance matrix vectorization, redundancy removal and reordering, and further obtains a direction-finding result. The method is applicable to the condition of known information source number and unknown information source number, the calculation amount is moderate, the advantages of improving the degree of freedom, the resolution and the direction finding precision of the non-uniform sparse array are reserved, and the underdetermined DOA estimation problem that the information source number is more than the array element number can be processed.

Description

IAA-based covariance matrix vectorization non-uniform sparse array direction finding method
Technical Field
The invention belongs to the field of array signal processing, and particularly relates to an IAA-based covariance matrix vectorization non-uniform sparse array direction finding method.
Background
Direction-of-Arrival (DOA) estimation is one of the core research contents in the field of radar reconnaissance and countermeasure, and plays a crucial role in acquiring battlefield initiative and competing for information right. In a modern complex electromagnetic environment, information sources are dense, complex and changeable, accurate direction finding is often needed for multiple information sources, and a mission is difficult to complete by a traditional direction finding system. The super-resolution array direction-finding technology can accurately estimate the directions of a plurality of signals simultaneously, and overcomes the defects of the traditional direction-finding system. The music (multiple Signal classification) algorithm is a classical super-resolution algorithm, which has been improved by many scholars since the proposal, but most of the optimization algorithms are designed for Uniform Linear Arrays (ULA) with Array element spacing smaller than half wavelength. The uniform linear array has the defects of small array aperture, low direction finding precision, poor resolution ratio and the like, and is difficult to adapt to modern complex electromagnetic environment. The sparse array is an array system with the array element spacing larger than half wavelength, has larger array aperture, higher resolution and higher degree of freedom compared with a uniform linear array with the same array element number, and can process the underdetermined DOA estimation problem that the information source number is more than the array element number. But when the array element spacing is longer than half wavelength, the problem of direction finding ambiguity can occur. Through the reasonable design of the array structure, the occurrence of direction finding ambiguity can be avoided.
In recent years, many researchers have introduced sparse arrays into the field of DOA estimation, with more heterogeneous sparse arrays being applied, namely Nested arrays (Nested arrays) and Coprime arrays (Coprime arrays). The array is virtually expanded through the vectorization covariance matrix, signal processing is carried out under the virtual array, fuzzy direction finding without the condition that the number of array elements is less than the number of information sources is achieved, the array aperture is enlarged under the condition that the number of the array elements is the same, the degree of freedom is improved, and the resolution and the angle measuring precision are improved. The existing processing method for the non-uniform sparse array under the virtual array mainly comprises three types: restoring the rank of the covariance matrix of the equivalent received signals of the virtual array by adopting a space smoothing method under the virtual array, and then carrying out DOA estimation by adopting an MUSIC algorithm; secondly, constructing a Toeplitz matrix by using the virtual array equivalent single snapshot received data, and performing DOA estimation on the matrix by adopting an MUSIC algorithm; and thirdly, constructing a compressed sensing model by adopting a compressed sensing method under the virtual array, and solving an optimization problem by adopting an LASSO method to obtain a DOA estimation result.
Disclosure of Invention
The invention aims to provide an IAA-based covariance matrix vectorization non-uniform sparse array direction finding method, which can accurately find directions no matter whether the number of information sources is known, and keeps the characteristics of sparse array expansion array aperture, improvement of freedom degree and improvement of resolution ratio.
The technical solution for realizing the purpose of the invention is as follows: an IAA-based covariance matrix vectorization non-uniform sparse array direction finding method comprises the following steps:
step 1: the receiving end antennas are arranged according to the non-uniform array structure to obtain a non-uniform array structure receiving signal model:
for a non-uniform sparse array of M array elements, K represents the number of incident incoherent signals, i.e. the number of sources, and then the received signal model x (n) of the non-uniform array is represented as:
Figure BDA0001921487730000021
wherein
Figure BDA0001921487730000022
For fast beat, v (n) is an independent identically distributed additive white Gaussian noise vector, a (θ)k) For the steering vector of the kth signal, the signal vector s (n) and the direction matrix a are respectively defined as:
s(n)＝[s1(n),s2(n),…,sK(n)]T∈CK×1 (4)
A＝[a(θ1),a(θ2),…,a(θK)]∈CM×K (5)
then the multi-snapshot received data of the array signal model is written in a matrix form X as follows:
X＝AS+V (6)
wherein
Figure BDA0001921487730000023
A∈CM×K，
Figure BDA0001921487730000024
To represent
Figure BDA0001921487730000025
A complex matrix of dimensions;
step 2: calculating a covariance matrix of the non-uniform array:
the covariance matrix of the non-uniform array is calculated according to equation (3) as follows:
Figure BDA0001921487730000026
wherein E [. C]Represents a statistical average, (.)HDenotes the conjugate transpose, RsIs an autocorrelation matrix of the incident signal, which is a diagonal matrix because the incident signal is an incoherent signal,
Figure BDA0001921487730000027
is the noise power, I is the identity matrix,
Figure BDA0001921487730000028
is the incident signal power; time-averaged estimation of covariance matrix R using finite number of samples
Figure BDA0001921487730000029
Covariance matrix of ready-to-use data
Figure BDA00019214877300000210
To replace the theoretical covariance matrix R; data covariance matrix
Figure BDA00019214877300000211
Can be calculated from the following formula:
Figure BDA00019214877300000212
and 3, step 3: vectorizing, removing redundancy and reordering the obtained covariance matrix to obtain equivalent single snapshot received data z under the virtual array;
the vectorization processing for equation (7) includes:
Figure BDA0001921487730000031
wherein
Figure BDA0001921487730000032
The result of vectorization of the identity matrix for the incident signal power vector
Figure BDA0001921487730000033
Figure BDA0001921487730000034
Represents M2Real column vector of dimension, sign (·)TDenotes transposition, eiIs a column vector of 0 except the ith position as 1, and a direction matrix of the virtual array
Figure BDA0001921487730000035
Symbol (·)*Indicating a conjugate, the symbol |, indicates a KR product,
Figure BDA0001921487730000036
represents the Kronecker product;
due to the Kronecker product operation,
Figure BDA0001921487730000037
and z there are many repeated rows that will
Figure BDA0001921487730000038
And different rows in the Z are extracted and sequenced in sequence, and a new received signal model under the virtual array is obtained after redundancy is removed
Figure BDA0001921487730000039
Comprises the following steps:
Figure BDA00019214877300000310
wherein
Figure BDA00019214877300000311
Is a direction matrix corresponding to the virtual array,
Figure BDA00019214877300000312
for the new noise vector, the covariance matrix under the virtual array is obtained according to the formula (10)
Figure BDA00019214877300000313
Figure BDA00019214877300000314
And 4, step 4: processing by adopting an IAA algorithm;
assuming that a large number of potential signals are uniformly distributed at a spatial domain L (L > M) point, D (θ) ═ D is defined1(θ),…,dL(θ)]，dlA steering vector representing the l-th potential signal, the vector of the corresponding potential signal being denoted as s (n) ═ s1(n),…,sL(n)]TInitialization is as follows:
Figure BDA00019214877300000315
Figure BDA00019214877300000316
carrying out iterative processing:
Figure BDA00019214877300000317
Figure BDA00019214877300000318
performing iterative calculation until convergence;
and 5: calculating a direction of arrival estimation result:
in step 4, the power estimation result obtained after the iteration is finished
Figure BDA00019214877300000319
The position is the estimation result of the direction of arrival of the incident signal after iteration
Figure BDA00019214877300000320
The covariance matrix of the virtual array with the rank recovered through iterative processing can also be matched under the condition of knowing the information source number
Figure BDA0001921487730000041
And performing characteristic decomposition, dividing the characteristic decomposition into a signal subspace and a noise subspace, and estimating the direction of arrival by adopting an MUSIC algorithm.
Compared with the prior art, the invention has the remarkable advantages that: (1) the method is applicable to the condition of known information source number and unknown information source number, and the calculation amount is moderate.
(2) The method has the advantages that the non-uniform sparse array improves the degree of freedom, the resolution and the direction finding precision, and can solve the underdetermined DOA estimation problem that the number of information sources is more than that of array elements.
Drawings
FIG. 1 is a block diagram of the overall process of the method of the present invention.
FIG. 2 is a schematic diagram of a nested array configuration of the present invention.
FIG. 3 is a schematic diagram of a relatively prime array structure according to the present invention.
FIG. 4 is a DOA estimation result graph obtained according to power estimation under a nested array by the method of the present invention.
FIG. 5 is a DOA estimation result diagram obtained by decomposition according to the matrix characteristics with the restored rank under the nested array.
FIG. 6 is a diagram of DOA estimation results obtained by the method of the present invention based on power estimation under co-prime matrix.
FIG. 7 is a DOA estimation result diagram obtained by decomposition according to the matrix characteristics with the recovered rank under the co-prime matrix.
FIG. 8 is a graph comparing the resolution performance of the method of the present invention with that of the spatial smoothing method.
FIG. 9 is a graph comparing the resolution performance of the method of the present invention with that of the uniform linear array direction finding method.
Detailed Description
The present invention is described in further detail below with reference to the attached drawing figures.
Referring to fig. 1, the embodiment of the present invention is as follows:
step 1, receiving end array antenna pressAnd according to the structural arrangement of the non-uniform sparse array, obtaining a non-uniform array structure receiving signal model. The nested array is composed of two uniform linear arrays, as shown in FIG. 2, the number of the array elements of the inner uniform linear array is M1Array element spacing of d1(half wavelength is taken); the number of the array elements of the outer uniform linear array is M2Array element spacing of d2＝(M1+1)d1The array element positions of the two sub-arrays are respectively as follows:
Figure BDA0001921487730000042
the co-prime array is composed of two sparse sub-arrays, a pair of co-prime integers M, N is selected, and M is less than N, as shown in fig. 3, a pair of sparse uniform linear sub-arrays is constructed, wherein the first sub-array comprises 2M antenna array elements with the array element interval Nd, the second sub-array comprises N antenna array elements with the array element interval Md, then the two sub-arrays are combined together according to the overlapping mode of the first array element, a co-prime array structure is obtained, the co-prime array structure actually comprises 2M + N-1 antenna array elements, and the array element positions are respectively:
Figure BDA0001921487730000051
and receiving signals by adopting a non-uniform sparse array and modeling. Suppose there are K from θ1,θ2,…,θKThe directional far-field narrow-band incoherent signal is incident to the non-uniform array, the signal is received by adopting a non-uniform array structure, and the modeling can be carried out as follows:
Figure BDA0001921487730000052
wherein
Figure BDA0001921487730000053
For snap-shot number, sk(n) is a signal waveform, v (n) is noise independent of each signal source, and a (theta)k) Is thetakA signal steering vector of direction, s (n) being the signalThe vector, A is a direction matrix, defined as follows:
s(n)＝[s1(n),s2(n),…,sK(n)]T∈CK×1 (4)
A＝[a(θ1),a(θ2),…,a(θK)]∈CM×K (5)
the multi-snapshot received data of the array signal model is written in a matrix form X as follows:
X＝AS+V (6)
wherein
Figure BDA0001921487730000054
A∈CM×K，
Figure BDA0001921487730000055
To represent
Figure BDA0001921487730000056
A complex matrix of dimensions;
and 2, calculating a covariance matrix of the non-uniform array received signals. The covariance matrix of the non-uniform array received signal is:
Figure BDA0001921487730000057
wherein E [. C]Represents a statistical average, (.)HDenotes the conjugate transpose, RsIs an autocorrelation matrix of the incident signal, which is a diagonal matrix because the incident signal is an incoherent signal,
Figure BDA0001921487730000058
is the noise power, I is the identity matrix,
Figure BDA0001921487730000059
is the incident signal power.
Since the computation of the covariance matrix R of equation (7) requires infinite samples, which cannot be realized in practical engineering, in practical situations, the covariance matrix R is often computed using finite number of samplesTime-averaged estimation of variance matrix R
Figure BDA00019214877300000510
Covariance matrix of ready-to-use data
Figure BDA00019214877300000511
Instead of the theoretical covariance matrix R. Use of
Figure BDA00019214877300000512
Sampling data of each snapshot to obtain a data covariance matrix
Figure BDA00019214877300000513
Figure BDA0001921487730000061
And 3, vectorizing, removing redundancy and reordering the obtained covariance matrix to obtain equivalent single snapshot received data z under the virtual array. For covariance matrix
Figure BDA0001921487730000062
The vectorization processing is carried out as follows:
Figure BDA0001921487730000063
wherein
Figure BDA0001921487730000064
As a vector of the power of the incident signal,
Figure BDA0001921487730000065
represents M2Real column vector of dimension, sign (·)TDenotes transposition, eiIs a column vector whose position is 0 except the ith position is 1,
Figure BDA0001921487730000066
symbol (·)*Indicating a conjugate, the symbol |, indicates a KR product,
Figure BDA0001921487730000067
representing the Kronecker product. Due to the Kronecker product operation,
Figure BDA0001921487730000068
and z there are many rows that are duplicated, requiring de-redundancy and reordering operations.
Will be provided with
Figure BDA0001921487730000069
And different rows in the z are extracted and sequenced in sequence to obtain equivalent receiving signals of the virtual array corresponding to the non-uniform sparse array. The new received signal model under the virtual array obtained after removing the redundancy is:
Figure BDA00019214877300000610
wherein
Figure BDA00019214877300000611
Is a direction matrix corresponding to the virtual array,
Figure BDA00019214877300000612
is a new noise vector. The covariance matrix under the virtual array is:
Figure BDA00019214877300000613
and 4, processing by adopting an IAA algorithm under the virtual array. The IAA algorithm is an algorithm based on an iterative idea. From the equation (9), it can be seen that the covariance matrix of the non-uniform array received signal is vectorized to obtain the equivalent received data of the virtual array, which is equivalent to the coherent signal received by the virtual array, so that the covariance matrix of the equivalent received signal of the virtual array
Figure BDA00019214877300000614
Are rank-deficient, the different methods are mainly to recover the rank of the matrix for DOA estimation. In the IAA algorithm, the coherent or correlated signal is decorrelated due to its initialization process and the power estimation method in an iterative process. After iteration, an accurate spatial power spectrum can be obtained while recovering the covariance matrix
Figure BDA00019214877300000615
Is determined by the rank of (c). The processing flow of the IAA algorithm is as follows:
initialization:
Figure BDA00019214877300000616
Figure BDA00019214877300000617
and (3) iterative processing:
Figure BDA00019214877300000618
for L ═ 1,2, …, L:
Figure BDA0001921487730000071
until convergence, convergence is reached by typically iterating 20 times.
And 5, calculating the estimation result of the direction of arrival. Through the iterative processing of the IAA algorithm in the step 4, the real power of the signal can be estimated,
Figure BDA0001921487730000072
the position of the spectral peak is the angle of the incident signal, i.e. the estimation result of the direction of arrival. In addition, the method can be used for producing a composite material
Figure BDA0001921487730000073
Is a matrix with the rank recovered after iterative processing, and can also be used for the matrix under the condition of known information source number
Figure BDA0001921487730000074
And (5) carrying out characteristic decomposition and carrying out DOA estimation by adopting a MUSIC algorithm.
Simulation test 1: the method comprises the following steps of 6 array element nested arrays, arranging antenna array elements according to the nested array structure, enabling 10 signals to be incident, enabling incident angles to be uniformly distributed between minus 60 degrees and 60 degrees, enabling signal-to-noise ratio (SNR) to be 10dB, enabling snapshot number to be 500, carrying out direction-of-arrival estimation through the method, and enabling simulation results to be shown in fig. 4 and 5.
FIG. 4 is a graph of a result of power estimation
Figure BDA0001921487730000075
The resulting direction of arrival estimation results, FIG. 5 is a matrix based on the recovered rank
Figure BDA0001921487730000076
The characteristic decomposition obtains the estimation result of the direction of arrival, and the simulation result shows that the effectiveness of the method provided by the invention can determine the signal incidence direction directly according to the power estimation result, and the direction of arrival can be well estimated under the condition of not knowing the number of signal sources. In addition, the characteristic that the non-uniform array improves the degree of freedom is reserved, and the condition that the number of information sources is more than that of array elements can be processed.
Simulation test 2: the 9 array element co-prime array is arranged according to the above co-prime array structure, M is 3, N is 4, 13 signals are incident, the incident angle is uniformly distributed between-60 degrees and 60 degrees, the signal-to-noise ratio SNR is 10dB, the snapshot number is 500, the direction of arrival estimation is performed by the method of the present invention, and the simulation result is shown in fig. 6 and 7.
FIG. 6 is a graph of a result of power estimation
Figure BDA0001921487730000077
The resulting direction of arrival estimation results, FIG. 7 is a matrix based on the recovered rank
Figure BDA0001921487730000078
The characteristic decomposition obtains the estimation result of the direction of arrival, and the simulation result shows that the method provided by the invention is effective under the co-prime matrix.
Simulation test 3: 6 array element nested arrays, 6 array element uniform linear arrays and 2 incident signals, wherein the incident angle is 0 degree and 2 degrees, the signal-to-noise ratio SNR is 10dB, the snapshot number is 500, the uniform linear array direction finding, the direction finding of the method and the space smoothing mode direction finding are respectively adopted, and the simulation results are shown in figures 8 and 9.
As can be seen from fig. 8 and 9, under the condition that the incident angles of the incident signals are 2 ° apart, the uniform linear arrays with the same array element number cannot be correctly direction-finding, but the direction of arrival can be well estimated by adopting the nested array structure, and the spectral peak obtained by the method provided by the invention is sharper than that obtained by the spatial smoothing method, has higher resolution performance, and embodies the superiority of the method provided by the invention.

Claims (1)

1. An IAA-based covariance matrix vectorization non-uniform sparse array direction finding method is characterized by comprising the following steps:
step 1: the receiving end antennas are arranged according to the non-uniform array structure to obtain a non-uniform array structure receiving signal model:
for a non-uniform sparse array of M array elements, K represents the number of incident incoherent signals, i.e. the number of sources, and then the received signal model x (n) of the non-uniform array is represented as:
Figure FDA0001921487720000011
wherein
Figure FDA0001921487720000012
For fast beat, v (n) is an independent identically distributed additive white Gaussian noise vector, a (θ)k) For the steering vector of the kth signal, the signal vector s (n) and the direction matrix a are respectively defined as:
s(n)＝[s1(n),s2(n),…,sK(n)]T∈CK×1 (4)
A＝[a(θ1),a(θ2),…,a(θK)]∈CM×K (5)
then the multi-snapshot received data of the array signal model is written in a matrix form X as follows:
X＝AS+V (6)
wherein
Figure FDA0001921487720000013
A∈CM×K，
Figure FDA0001921487720000014
Figure FDA0001921487720000015
To represent
Figure FDA0001921487720000016
A complex matrix of dimensions;
step 2: calculating a covariance matrix of the non-uniform array:
the covariance matrix of the non-uniform array is calculated according to equation (3) as follows:
Figure FDA0001921487720000017
wherein E [. C]Represents a statistical average, (.)HDenotes the conjugate transpose, RsIs an autocorrelation matrix of the incident signal, which is a diagonal matrix because the incident signal is an incoherent signal,
Figure FDA0001921487720000018
is the noise power, I is the identity matrix,
Figure FDA0001921487720000019
is the incident signal power; computing covariance matrices using finite order samplesTime-averaged estimation of R
Figure FDA00019214877200000110
Covariance matrix of ready-to-use data
Figure FDA00019214877200000111
To replace the theoretical covariance matrix R; data covariance matrix
Figure FDA00019214877200000112
Can be calculated from the following formula:
Figure FDA00019214877200000113
and step 3: vectorizing, removing redundancy and reordering the obtained covariance matrix to obtain equivalent single snapshot received data z under the virtual array;
the vectorization processing for equation (7) includes:
Figure FDA00019214877200000114
wherein
Figure FDA0001921487720000021
The result of vectorization of the identity matrix for the incident signal power vector
Figure FDA0001921487720000022
Figure FDA0001921487720000023
Represents M2Real column vector of dimension, sign (·)TDenotes transposition, eiIs a column vector of 0 except the ith position as 1, and a direction matrix of the virtual array
Figure FDA0001921487720000024
Symbol (·)*Indicating a conjugate, the symbol |, indicates a KR product,
Figure FDA0001921487720000025
represents the Kronecker product;
due to the Kronecker product operation,
Figure FDA0001921487720000026
and z there are many repeated rows, will
Figure FDA0001921487720000027
And different rows in the Z are extracted and sequenced in sequence, and a new received signal model under the virtual array is obtained after redundancy is removed
Figure FDA0001921487720000028
Comprises the following steps:
Figure FDA0001921487720000029
wherein
Figure FDA00019214877200000210
Is a direction matrix corresponding to the virtual array,
Figure FDA00019214877200000211
for the new noise vector, the covariance matrix under the virtual array is obtained according to the equation (10)
Figure FDA00019214877200000212
Figure FDA00019214877200000213
And 4, step 4: processing by adopting an IAA algorithm;
assuming that a large number of potential signals are uniformly distributed at a spatial domain L (L > M) point, D (θ) ═ D is defined1(θ),…,dL(θ)]，dlA steering vector representing the l-th potential signal, the vector of the corresponding potential signal being denoted as s (n) ═ s1(n),…,sL(n)]TInitialization is as follows:
Figure FDA00019214877200000214
Figure FDA00019214877200000215
carrying out iterative processing:
Figure FDA00019214877200000216
Figure FDA00019214877200000217
performing iterative calculation until convergence;
and 5: calculating a direction of arrival estimation result:
in step 4, the power estimation result obtained after the iteration is finished
Figure FDA00019214877200000218
The position is the estimation result of the direction of arrival of the incident signal, after iteration
Figure FDA00019214877200000219
The covariance matrix of the virtual array with the rank recovered through iterative processing can also be matched under the condition of knowing the information source number
Figure FDA00019214877200000220
Go on speciallyAnd (4) sign decomposition, namely dividing the sign decomposition into a signal subspace and a noise subspace, and estimating the direction of arrival by adopting an MUSIC algorithm.
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