CN111707985A - Off-grid DOA estimation method based on covariance matrix reconstruction - Google Patents

Off-grid DOA estimation method based on covariance matrix reconstruction Download PDF

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CN111707985A
CN111707985A CN202010539779.XA CN202010539779A CN111707985A CN 111707985 A CN111707985 A CN 111707985A CN 202010539779 A CN202010539779 A CN 202010539779A CN 111707985 A CN111707985 A CN 111707985A
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matrix
doa
covariance matrix
estimation
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王洪雁
于若男
薛喜扬
汪祖民
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Zhejiang University of Technology ZJUT
Zhejiang Sci Tech University ZSTU
Zhejiang University of Science and Technology ZUST
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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
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Abstract

An off-grid DOA estimation method based on covariance matrix reconstruction belongs to the field of signal processing, and aims to solve the problem that large estimation errors exist in DOA estimation caused by grid mismatch in a sparse representation model, and the key points are as follows: firstly, including the offset between DOA and grid points into a constructed receiving data space domain discrete sparse representation model; then, establishing a sparse representation convex optimization problem about DOA estimation based on the reconstructed signal covariance matrix; then, a sampling covariance matrix estimation error convex model is constructed, and the convex set is explicitly contained in a sparse representation model to improve the sparse signal reconstruction performance; and finally, solving the obtained joint optimization problem by adopting an alternating iteration method to obtain grid offset parameters and off-grid DOA estimation, wherein the effect is better angle resolution and higher DOA estimation precision.

Description

Off-grid DOA estimation method based on covariance matrix reconstruction
Technical Field
The invention belongs to the field of signal processing, and particularly relates to an off-grid DOA estimation method based on covariance matrix reconstruction.
Background
As one Of the research hotspots in the field Of array signal processing, Direction Of Arrival (DOA) estimation techniques are widely used in the fields Of wireless communication, target tracking, voice processing, radar, radio astronomy, and the like. With the continuous and deep research of DOA estimation theory, various DOA estimation methods are proposed successively. Classical subspace-like methods can achieve super-resolution direction finding, such as MUltiple SIgnal Classification (MUSIC), rotation invariant subspace (ESPRIT), etc., but their Estimation performance will be significantly degraded under low snr or insufficient snapshot. Aiming at the problem, Pesavento M and the like provide a Maximum Likelihood (ML) DOA estimation algorithm, which solves the log-likelihood function of signals and noise through gradual iteration to realize DOA estimation, and the performance of the DOA estimation algorithm is obviously superior to that of the MUSIC algorithm under the condition of low signal-to-noise ratio, but the performance of the DOA estimation algorithm depends on the selection of an initial value and the complexity of the algorithm is higher, so that the practical application of the DOA estimation algorithm is limited. Therefore, how to improve the performance of the DOA estimation algorithm while reducing the computational complexity is one of the research hotspots in the field of current array signal processing.
In recent years, sparse signal representation and compressed sensing theory have gradually become powerful tools in the field of parameter estimation. With the continuous and deep research of sparse reconstruction algorithm, researchers successively put forward a plurality of DOA estimation methods based on the sparse characteristics of signal space domain, wherein the most representative l1SVD algorithm, which makes use of l1The norm builds a sparse model, and then the computation complexity is reduced through singular value decomposition, however, under the condition that the information source number is unknown, the algorithm cannot effectively distinguish the adjacent information source angles. To address this problem, Wer et al propose a method based on weighting l1Norm sparse reconstruction DOA estimation algorithm which utilizes signal sparsity and is based on cepstrum function design weight of improved Capon algorithm and constructs weighting l1And 4, carrying out norm convex optimization to realize DOA estimation under the scene of unknown information source number. YANG Jie et al propose a Sparse Bayesian Inference (SBI) -based mutual-prime array DOA estimation algorithm that eliminates noise variance from covariance vectors using linear transformation, and then combines parameter dictionary learning and Sparse recovery iterative update to achieve DOA estimation. Wang et al propose a robust DOA estimation method based on low rank recovery, which utilizes the sparseness of a sampling covariance matrix and the low rank characteristic to construct a convex problem about a signal and noise covariance matrix, and then utilizes an MVDR (mean square deviation) squareThe method achieves DOA estimation. It should be noted that the above DOA estimation algorithms all assume that the source DOA is exactly located at the predetermined discrete grid point, however, the actual angle of arrival of the signal may be offset from the predetermined discrete grid, and therefore, the obtained estimation has a certain error. The most intuitive solution to this problem is to reduce the inter-grid distance, i.e., to cover the probe space with a smaller step size, i.e., a denser search grid, to reduce the estimation error, however, this approach significantly increases the computational complexity and increases the coherence between the overcomplete dictionary atoms, thereby violating the limited Isometry Property (RIP), which leads to an increase in the estimation error. Aiming at the problems, WU Xiaohuan et al propose an Off-Grid Bayesian Learning (OGSBL) DOA estimation algorithm based on Sparse Bayesian Learning, and solve the Grid division problem by introducing an offset parameter to a DOA Sparse representation model and solving based on SBL. DAI Jisheng et al propose a Root sparse Off-Grid Bayesian inference (Root Off-Grid SBI, ROGSBI) method, which combines a coarse-divided Grid and an iterative subdivided Grid to optimize the Grid to achieve high-precision DOA estimation. It should be noted that, although the above algorithm reduces the off-grid effect and reduces the amount of calculation based on SBI, it does not consider the influence of parameter estimation caused by finite snapshots. It is known that finite snapshots can cause estimation errors in covariance matrices of sampled signals, thereby resulting in a finite improvement in DOA estimation accuracy obtained by the above algorithm.
Disclosure of Invention
Aiming at the problem that a large estimation error exists in direction of arrival (DOA) estimation caused by grid mismatch in a sparse representation model, the invention provides an off-grid (DOA) estimation method based on covariance matrix reconstruction.
The technical scheme of the invention is as follows: an off-grid DOA estimation method based on covariance matrix reconstruction comprises
A1. And (3) including the offset between the DOA and the grid points into a sparse representation model of the received data, and establishing a sparse representation model related to DOA estimation based on a covariance matrix of the reconstructed signal.
A2. Constructing a sampling covariance matrix estimation error convex model, deriving an estimation error upper bound based on the statistical characteristic that the sampling covariance matrix estimation error obeys progressive normal distribution, then explicitly including the convex set into a sparse representation model, and solving the obtained joint optimization problem by adopting an alternative iteration method to obtain sparse DOA and grid offset parameter estimation.
Further, the specific steps of a1 include:
step 1: establishing a receiving data space domain discrete sparse representation model
Assume K far-field narrowband signals
Figure BDA0002538498410000021
Incident on a uniform linear array with M array elements, the received signal model at time t can be expressed as
Figure BDA0002538498410000022
Wherein, x (t) is received data,
Figure BDA0002538498410000023
and sk(t) is respectively the guide vector and signal amplitude of the kth signal source, d and lambda are respectively the array element spacing and the carrier wavelength, d is not more than lambda/2, theta1θ2…θKFor K sources DOA,
Figure BDA0002538498410000024
a matrix of vectors is directed to the array,
Figure BDA0002538498410000025
is a waveform vector, n (t) ═ n1(t) n2(t) … nM(t)]TAre white gaussian noise that are not correlated.
For L snapshots, (1) the received signal model is denoted X ═ a (θ) S + N (2)
Where X ═ X (1) X (2) … X (l) is a received signal matrix, S ═ S (1) S (2) … S (l)) is a signal amplitude matrix, and N ═ N (1) N (2) … N (l)) is a noise matrix.
Assuming that the signal and noise are uncorrelated and the sources are independent of each other, the received signal covariance is expressed as
Figure BDA0002538498410000031
Wherein R issIs a noise-free signal covariance matrix, I is an identity matrix, and P ═ E { SS }HIs the signal power covariance matrix, σ2Is the noise power.
In practical application, the receiving covariance matrix R is obtained based on L finite sampling snapshots, i.e.
Figure BDA0002538498410000032
Step 2: off-grid DOA estimation model
Suppose the source azimuth is discretely divided into N grids covering all possible target incident directions, i.e.
Figure BDA0002538498410000033
And the number N > M > K of the grids, for the grid model, the information source is supposed to be incident on the preset grid point without bias, namely
Figure BDA0002538498410000034
Then (1) the received signal model is restated as
Figure BDA0002538498410000035
Wherein the content of the first and second substances,
Figure BDA0002538498410000036
in order to be an ultra-complete dictionary,
Figure BDA0002538498410000037
the matrix is a row sparse signal matrix obtained after expansion.
Based on an overcomplete matrix
Figure BDA0002538498410000038
In the representation (2), the covariance matrix R is sparseIs shown as
Figure BDA0002538498410000039
Wherein R ═ vec (R-sigma)2I)，
Figure BDA00025384984100000310
Is the product of the Kronecker reaction,
Figure BDA00025384984100000311
the signal power vector obtained after the expansion is obtained.
And step 3: introducing an offset into an Off-grid sparse representation model
Introducing a set of offset parameters rho ═ rho1ρ2… ρN-1]TCorrecting grid offset:
Figure BDA00025384984100000312
where ρ isiIs an offset parameter and is greater than 0,
Figure BDA00025384984100000313
and
Figure BDA00025384984100000314
respectively true DOA thetaiLeft and right adjacent grid points.
Array steering vector matrix representation as
Figure BDA00025384984100000315
Wherein A (u: v) represents the u-th to v-th columns of the acquisition matrix A, I-＝[IN-1,0(N-1)×1]T,I-＝[0(N-1)×1,IN-1]T. Let Δ be diag (ρ), the array steering vector matrix is re-expressed as
Figure BDA0002538498410000041
Wherein the content of the first and second substances,
Figure BDA0002538498410000042
from equations (5) and (9), the off-grid DOA estimation model is expressed as
Figure BDA0002538498410000043
The covariance matrix R is sparsely represented as
Figure BDA0002538498410000044
Wherein the content of the first and second substances,
Figure BDA0002538498410000045
actual received signal covariance matrix
Figure BDA0002538498410000046
Is shown as
Figure BDA0002538498410000047
Wherein the content of the first and second substances,
Figure BDA0002538498410000048
further, the specific steps of a2 include:
step 1: estimation error model by introducing sampling covariance matrix
Constructing an error model for R:
Figure BDA0002538498410000049
wherein | · | purple sweetFIs the matrix Frobenius norm and is the error parameter factor.
Sparse signal reconstruction problem base based on error convex setIn l1Solving by a norm constraint optimization algorithm:
Figure BDA00025384984100000410
vectorized covariance matrix error
Figure BDA00025384984100000411
Obeying a progressive normal distribution
Figure BDA00025384984100000412
Wherein vec (-) is a vectorization operator, AsN (mu, sigma) represents a progressive normal distribution with a mean value mu and a variance sigma,
Figure BDA00025384984100000413
through matrix operations, the vectorized covariance matrix error is re-represented as
Figure BDA00025384984100000414
Wherein the content of the first and second substances,
Figure BDA00025384984100000415
represents M2An order unit matrix.
From a normal distribution characteristic
Figure BDA0002538498410000051
Wherein, χ2(. represents chi-square distribution, As-2(M2) Represents a degree of freedom of M2Progressive chi-square distribution.
Since the vector R is a vectorized form of the covariance matrix R, the vector R is obtained
Figure BDA0002538498410000052
General formula (11)
Figure BDA0002538498410000053
The sparse signal reconstruction problem is re-expressed as
Figure BDA0002538498410000054
Wherein eta is an error parameter factor, and is obtained by the following formula
Pr{χ2(M2)≤η}＝p，η＝χp 2(M2) (21)
Wherein Pr {. cndot } represents a probability distribution, p is a probability value, χp 2(M2) Representing a probability value p and a degree of freedom M2The lower chi fang is distributed.
Step 2: implementing offset solution and sparse signal reconstruction
Is represented by the formula (4) and
Figure BDA0002538498410000055
(20) the optimization problem is equivalent to
Figure BDA0002538498410000056
The convex optimization problem formula (22) is solved based on an alternate iteration method, in the (l + 1) th iteration,
(1) fixed rho, update
Figure BDA0002538498410000057
Figure BDA0002538498410000058
(2) Fixing
Figure BDA0002538498410000059
Updating rho:
Figure BDA00025384984100000510
dimension reduction processing by using singular value decomposition method
Figure BDA00025384984100000511
Wherein the content of the first and second substances,
Figure BDA0002538498410000061
the azimuth of the wave crest is taken as the azimuth of the wave crest,
Figure BDA0002538498410000062
by
Figure BDA0002538498410000063
K maximum peaks.
After matrix operation, neglecting the items irrelevant to the optimization variables:
Figure BDA0002538498410000064
wherein R iseThe representation is taken in the real part,
Figure BDA00025384984100000611
representing the Hadamard product of the matrix.
Let the derivative of equation (26) with respect to ρ be zero, get ρKIs updated to
Figure BDA0002538498410000065
Wherein the content of the first and second substances,
Figure BDA0002538498410000066
the pseudo-inverse is represented by a pseudo-inverse,
Figure BDA0002538498410000067
and step 3: implementing DOA estimation
Based on the results
Figure BDA0002538498410000068
And the offset rho, DOA is expressed as
Figure BDA0002538498410000069
Wherein the intervals are divided for the grid.
Iterative calculation until the DOA estimated values of two adjacent times have no obvious change, i.e.
Figure BDA00025384984100000610
An effective DOA estimate is obtained.
Has the advantages that: the algorithm provided by the invention has higher angle resolution and DOA estimation precision under the condition of grid mismatch. And has better DOA estimation performance and can replace the improvement of the DOA estimation performance with less time. In addition, the proposed algorithm has a narrower main lobe and lower side lobes because the proposed OGCMR algorithm incorporates the offsets between the DOA and the grid points into the received data sparse representation model, constructs a convex model of the sampling covariance matrix estimation error, and explicitly incorporates this convex set into the convex sparse representation model to improve the sparse signal reconstruction performance, thereby achieving efficient DOA estimation.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention.
FIG. 2 is a comparison graph of spatial spectrum of incoherent signal under different SNR and snapshot conditions.
Fig. 3 is a spatial spectrum of an incoherent signal.
FIG. 4 is a graph of the change in RMSE with SNR for DOA estimation.
FIG. 5 is a graph of estimated RMSE versus snapshot number for DOA.
FIG. 6 is a graph showing the variation of the arithmetic operation time with the number of snapshots.
Detailed Description
The implementation steps of the present invention are further described in detail below with reference to fig. 1:
the invention provides an Off-Grid based on Covariance Matrix Reconstruction (OGCMR) discrete Grid DOA estimation method under a Grid mismatch condition. Firstly, including offsets between DOA and grid points into a received data sparse representation model; then, establishing a sparse representation model related to DOA estimation based on the reconstructed signal covariance matrix; then, constructing a sampling covariance matrix estimation error convex model, deriving an estimation error upper bound based on the statistical characteristic that the sampling covariance matrix estimation error obeys progressive normal distribution, and then explicitly including the convex set into a sparse representation model to improve the sparse signal reconstruction performance so as to improve the DOA estimation precision; and finally, solving the obtained joint optimization problem by adopting an alternating iteration method to obtain sparse DOA and grid offset parameter estimation.
Numerical simulation shows that1Compared with estimation algorithms such as SVD and robust MVDR (SLRD-RMVDR) based on sparse and low rank recovery, the algorithm has better angular resolution and higher DOA estimation precision. The basic idea for realizing the method is that firstly, a receiving data space domain discrete sparse representation model is established; secondly, constructing an Off-grid DOA estimation model; then, based on the Off-grid reconstruction sparse signal, introducing an offset to an Off-grid sparse representation model and a sampling covariance matrix estimation error model so as to realize offset solution and sparse signal reconstruction; and finally, DOA estimation is realized.
The method comprises the following specific steps:
1, establishing a space domain discrete sparse representation model of received data
Assume K far-field narrowband signals
Figure BDA0002538498410000071
Incident on a uniform linear array with M array elements, the received signal model at time t can be expressed as
Figure BDA0002538498410000072
Wherein, x (t) is received data,
Figure BDA0002538498410000073
and sk(t) is the steering vector and signal amplitude of the kth signal source, d and λ are the array element spacing and carrier wavelength, d is usually ≦ λ/2, { θ ≦ λ/21θ2… θKK source DOAs (the invention only considers azimuth, the obtained conclusion can be extended to two-dimensional DOA estimation),
Figure BDA0002538498410000074
a matrix of vectors is directed to the array,
Figure BDA0002538498410000075
is a waveform vector, n (t) ═ n1(t) n2(t) … nM(t)]TAre white gaussian noise that are not correlated. The input signal of the received signal model is the baseband signal of the receiver in the radar system.
For L snapshots, the received signal model of equation (1) may be further expressed as X ═ a (θ) S + N (30)
Where X ═ X (1) X (2) … X (l) is a received signal matrix, S ═ S (1) S (2) … S (l)) is a signal amplitude matrix, and N ═ N (1) N (2) … N (l)) is a noise matrix.
Assuming that the signal and noise are uncorrelated and the sources are independent of each other, the received signal covariance can be expressed as
Figure BDA0002538498410000081
Wherein R issFor noise-free signal covariance matrix, P ═ E { SSHIs the signal power covariance matrix, σ2Is the noise power.
In practical application, the receiving covariance matrix R is obtained based on L finite sampling snapshots, i.e.
Figure BDA0002538498410000082
2off-grid DOA estimation model
Assuming a discrete division of the source azimuth into N grids covering all possible target incident directions,namely, it is
Figure BDA0002538498410000083
And the number of grids N > M > K. For the gridded model, it is assumed that the source is incident unbiased on a preset grid point, i.e. the source is incident on the preset grid point
Figure BDA0002538498410000084
Then the formula (2) can be restated as
Figure BDA0002538498410000085
Wherein the content of the first and second substances,
Figure BDA0002538498410000086
in order to be an ultra-complete dictionary,
Figure BDA0002538498410000087
the matrix is a row sparse signal matrix obtained after expansion.
Based on an overcomplete matrix
Figure BDA0002538498410000088
With respect to the covariance matrix R, the covariance matrix R can be sparsely expressed as
Figure BDA0002538498410000089
Wherein R ═ vec (R-sigma)2I)，
Figure BDA00025384984100000810
Is the product of the Kronecker reaction,
Figure BDA00025384984100000811
the signal power vector obtained after the expansion is obtained.
3 reconstructing sparse signals based on off-grid
(1) Introducing an offset into an Off-grid sparse representation model
In practice, no matter the grid
Figure BDA00025384984100000812
Dividing into multiple densities, each true information source azimuth angle thetaiIs unlikely to be located exactly on the divided grid, i.e.
Figure BDA00025384984100000813
Furthermore, as the lattice density increases, the amount of computation increases, and the correlation between atoms in the dictionary is enhanced, resulting in deterioration of DOA estimation performance. Aiming at the problem that grid mismatch causes the performance degradation of DOA estimation, a set of offset parameters rho ═ rho is introduced1ρ2… ρN-1]TTo correct the grid offset to reduce the DOA estimation error, thereby obtaining
Figure BDA00025384984100000814
Where ρ isiIs an offset parameter and is greater than 0,
Figure BDA00025384984100000815
and
Figure BDA00025384984100000816
respectively true DOA thetaiLeft and right adjacent grid points.
Based on equation (7), the array steering vector matrix can be expressed as
Figure BDA0002538498410000091
Wherein A (u: v) represents the u-th to v-th columns of the acquisition matrix A, I-＝[IN-1,0(N-1)×1]T,I-＝[0(N-1)×1,IN-1]T. Let Δ be diag (ρ), equation (8) can be re-expressed as
Figure BDA0002538498410000092
Wherein the content of the first and second substances,
Figure BDA0002538498410000093
from equations (5) and (9), the off-grid DOA estimation model can be expressed as
Figure BDA0002538498410000094
Based on the above discussion, the covariance matrix R can be sparsely represented as
Figure BDA0002538498410000095
Wherein the content of the first and second substances,
Figure BDA0002538498410000096
as previously mentioned, the actual received signal covariance matrix
Figure BDA0002538498410000097
Can be obtained from the formula (4), from which
Figure BDA0002538498410000098
Wherein the content of the first and second substances,
Figure BDA0002538498410000099
(2) estimation error model by introducing sampling covariance matrix
However, in practical applications, because the sampling times are limited, the signal covariance matrix has estimation errors, and thus the DOA estimation performance is degraded due to the estimation errors also existing on the basis of the sparse representation DOA. To address this problem, an error model for R is constructed as follows:
Figure BDA00025384984100000910
wherein | · | purple sweetFIs the matrix Frobenius norm and is the error parameter factor.
Sparse signal reconstruction problem ground based on error convex setIn l1Norm-constrained optimization algorithm solution, i.e.
Figure BDA00025384984100000911
It should be noted that solving the above optimization problem requires a well-defined upper error bound. However, this value is difficult to ascertain in practice, and is often determined empirically. Based on the above, the method for determining the upper bound of the error is derived based on the statistical characteristic that the covariance estimation error follows the progressive normal distribution. Vectorized covariance matrix error
Figure BDA00025384984100000912
Obeying an asymptotic Normal (AsN) distribution, i.e.
Figure BDA00025384984100000913
Where vec (-) is the vectorization operator, AsN (. mu., ∑) represents the progressive normal distribution with mean μ and variance ∑,
Figure BDA0002538498410000101
through simple matrix operation, equation (15) can be re-expressed as
Figure BDA0002538498410000102
From the normal distribution characteristic
Figure BDA0002538498410000103
Wherein, χ2(. represents chi-square distribution, As-2(M2) Represents a degree of freedom of M2Progressive chi-square distribution.
Since the vector R is a vectorized form of the covariance matrix R, it is thus available
Figure BDA0002538498410000104
Similar to the formula (17), it can be seen that
Figure BDA0002538498410000105
In summary, the sparse signal reconstruction problem can be re-expressed as
Figure BDA0002538498410000106
Wherein eta is an error parameter factor, and can be obtained by the following formula
Pr{χ2(M2)≤η}＝p，η＝χp 2(M2) (49)
Wherein Pr {. cndot } represents a probability distribution, and p is a probability value.
(3) Implementing offset solution and sparse signal reconstruction
In practical application, the covariance matrix of the received signal
Figure BDA0002538498410000107
Obtainable from formula (4), accordingly
Figure BDA0002538498410000108
Thus, the optimization problem of equation (20) can be equated with
Figure BDA0002538498410000109
The convex optimization problem equation (22) can be solved based on an alternating iterative method, i.e., in the (l + 1) th iteration,
(1) fixed rho, update
Figure BDA00025384984100001010
Figure BDA00025384984100001011
(2) Fixing
Figure BDA0002538498410000111
Updating rho:
Figure BDA0002538498410000112
considering the large amount of calculation, the singular value decomposition method is used for dimensionality reduction of the above formula, namely
Figure BDA0002538498410000113
Wherein the content of the first and second substances,
Figure BDA0002538498410000114
the azimuth of the wave crest is taken as the azimuth of the wave crest,
Figure BDA0002538498410000115
by
Figure BDA0002538498410000116
K maximum peaks.
Equation (25) can be further expressed by a simple matrix operation, ignoring the optimization variable independent terms
Figure BDA0002538498410000117
Wherein R iseThe representation is taken in the real part,
Figure BDA00025384984100001115
representing the Hadamard product of the matrix.
Let the derivative of equation (26) with respect to ρ be zero, and thus ρ is obtainedKIs updated to
Figure BDA0002538498410000118
Wherein the content of the first and second substances,
Figure BDA0002538498410000119
the pseudo-inverse is represented by a pseudo-inverse,
Figure BDA00025384984100001110
4-implementation of DOA estimation
Based on the above results
Figure BDA00025384984100001111
And an offset ρ, DOA, which can be expressed as
Figure BDA00025384984100001112
Wherein the intervals are divided for the grid.
In summary, the OGCMR algorithm provided by the present invention can be briefly expressed as
(1) Inputting an array received signal matrix X;
(2) dividing a grid and introducing an offset rho to obtain an array flow pattern matrix A (rho);
(3) solving the equation (21) to obtain an optimal estimation value of the error parameter eta;
(4) solving the formula (23) update
Figure BDA00025384984100001113
(5) Solving formula (27) to update rho;
(6) solving equation (28) to obtain a DOA estimate;
(7) repeating the steps (4) to (6) until the DOA estimated values of two adjacent times have no obvious change, namely:
Figure BDA00025384984100001114
the effects of the present invention can be further illustrated by the following simulations:
simulation conditions are as follows: the simulation software is MATLAB R2014a, and the hardware environment is as follows: the processor is Intel Core (TM) i7-7700, the main frequency is 4GHz, and the memory is 8 GB. The simulation conditions were set as follows: the number of array elements M is 8, the number of snapshots L is 200, and the distance d between array elements is lambda/2. Wherein, the measurement standard of DOA estimation accuracy can adopt Root-Mean-Square Error (RMSE), which is defined as
Figure BDA0002538498410000121
Wherein K is the Monte Carlo test frequency, N is the target number,
Figure BDA0002538498410000122
estimate the ith DOA for the kth experiment, θiIs the ith real DOA.
Simulation content:
simulation 1: and (4) a non-coherent signal spatial spectrum contrast diagram under different signal-to-noise ratios and snapshot conditions. Considering two incoherent signals with incidence angles of [ -10.6 degrees and 5.3 degrees ], fig. 2 shows spatial power spectrum comparison of MUSIC, -SVD, SLRD-RMVDR and OGCMR algorithm proposed by the present invention under four conditions.
As can be seen from fig. 2(a), under the conditions of SNR 0dB and snapshot number L40, MUSIC can approximately distinguish the target signal angle but its peak is low at-10.6 °, L1Neither SVD nor SLRD-RMVDR algorithms are able to effectively resolve the target signal, whereas the OGCMR algorithm proposed by the present invention is able to effectively discriminate two target signal angles and has a narrower main lobe. As can be seen from FIG. 2(b), as the SNR increases, the angle resolution of the SLRD-RMVDR algorithm increases, and l1The SVD algorithm can only approximately resolve the target signal at-10.6 degrees, and the OGCMR algorithm provided by the invention still has good angular resolution. Fig. 2(c) is a comparison graph of spatial spectrum estimation of four algorithms under the conditions of SNR 0dB and L200 dB. As can be seen from FIG. 2(c), under low SNR, the spectrum peak of MUSIC algorithm is lower at 5.3 deg. position, the spectrum peak of SLRD-RMVDR algorithm is lower at-10.6 deg., and l1The SVD algorithm cannot effectively distinguish the target signal, and the OGCMR algorithm provided by the invention has better estimation performance. Fig. 2(d) is a comparison graph of spatial spectrum estimation of four algorithms under the conditions of SNR 10dB and L200 dB. As can be seen from fig. 2(d), as the SNR and the number of snapshots increase, all four algorithms have sharper spectral peaks, and the proposed algorithm has a narrower main lobe, which is closer to the true DOA, and thus has better DOA estimation performance.
Simulation 2: and (3) a non-coherent signal spatial spectrum. Considering four incoherent signals with incidence angles of [ -35.3 °, -10.6 °, 5.3 °, 26.5 ° ] respectively, SNR is 10dB and snapshot number L is 200. FIG. 3 is a comparison diagram of spatial spectrum estimation of four algorithms.
As can be seen from FIG. 3, given simulation conditions, MUSIC, l1The angles estimated by the SVD algorithm and the SLRD-RMVDR algorithm have larger deviation with the actual target signal angle, but the OGCMR algorithm provided by the invention can effectively distinguish four target angles and has a narrower main lobe and a lower side lobe compared with three comparison algorithms, because the OGCMR algorithm includes the offset between DOA and grid points into a received data sparse representation model, constructs a convex model of sampling covariance matrix estimation error, and explicitly includes the convex set into a convex sparse representation model to improve sparse signal reconstruction performance, thereby realizing effective DOA estimation.
Simulation 3: DOA estimate RMSE versus SNR plot. 200 Monte Carlo independent replicates were performed taking into account the incoherent signals with incident angles of-10.6 and 5.3, respectively. Fig. 4 shows the curves of the RMSE versus SNR for the four DOA algorithms, where SNR [ -6:2:14], and snapshot number L [ -200.
As can be seen from fig. 4, the RMSE for the four algorithm DOA estimates decreases gradually as the SNR increases. It should be noted that conventional MUSIC and l1The DOA estimation RMSE of the SVD algorithm is relatively high, the RMSE of the SLRD-RMVDR and the OGCMR algorithm is low, and the estimation performance of the proposed algorithm is obviously better than that of the SLRD-RMVDR, because the proposed algorithm considers the offset between the DOA and the grid points and includes the DOA and the grid points into the sparse representation model of the received data, and particularly under the condition of low SNR, the advantages of the proposed algorithm are more prominent, thereby showing that the proposed OGCMR algorithm has better DOA estimation performance.
And (4) simulation: DOA estimated RMSE is plotted as a function of snapshot number. Considering incoherent signals with incidence angles of-10.6 ° and 5.3 °, SNR 10dB, and snapshot number L [50:50:500], respectively, 200 monte carlo independent repeat experiments were performed, and fig. 5 is a plot of RMSE estimated by four algorithms DOA as a function of snapshot number.
As can be seen from FIG. 5, as the number of snapshots increases, MUSIC, l1-SVD, SLRD-RMVDR and RMSE of OGCMR algorithm proposed by the present inventionAnd gradually decreases. In addition, it should be noted that the OGCMR algorithm has better DOA estimation performance than the other three algorithms under the same fast-beat condition because the OGCMR algorithm considers that the DOA and the grid point have deviation and the sampling covariance matrix has error, and constructs a correlation joint optimization problem to improve the sparse signal reconstruction performance.
And (5) simulation: the algorithm operation time is changed along with the snapshot number. In order to evaluate the complexity of the algorithm, the algorithm operation time comparison under different fast beat number conditions is analyzed. Considering two incoherent signals with incidence angles of-10.6 ° and 5.3 °, respectively, SNR is 10dB, fig. 6 is a plot of operating time versus snapshot count for the four algorithms, and L is [50:50:500 ].
As can be seen from FIG. 6, the four algorithms have an increasing operation time as the number of snapshots increases. Proposed OGCMR and l1The SVD algorithm sparsely reconstructs signals by utilizing an overcomplete dictionary matrix, the SLRD-RMVDR reconstructs a noise-free covariance matrix based on a low-rank recovery theory, the operation time of the three algorithms is slightly longer than that of the traditional MUSIC algorithm, and l1The SVD algorithm takes a long time because it employs a convex optimization method to reconstruct the signal on the non-dimensionality-reduced data. However, it should be noted that although the running time of the proposed OGCMR algorithm is slightly higher than that of the MUSIC algorithm, the DOA estimation accuracy and the angular resolution are significantly better than those of the MUSIC algorithm, i.e., the proposed algorithm can trade off the improvement of DOA estimation performance with a small time.
And (6) simulation: in order to more fully evaluate the performance of the proposed OGCMR algorithm, the influence of the error parameter eta on the reconstruction performance of the proposed algorithm is analyzed. Considering incoherent signals with incident angles of-10.6 ° and 5.3 °, respectively, the SNR is 10dB and the snapshot count L is 200. As can be seen from table 1, when the error parameter is 0.1, the peak power of the reconstructed signal of the OGCMR algorithm is large, and the peak power of the reconstructed signal of the OGCMR algorithm gradually decreases with the increase of the error parameter, which indicates that the value of the error parameter η has a large influence on the reconstruction performance of the algorithm.
TABLE 1 Effect of error parameters on Algorithm reconstruction Performance
Figure BDA0002538498410000141
In summary, the present invention provides an off-grid DOA estimation (OGCMR) method based on a reconstructed covariance matrix. According to the algorithm, firstly, the offset between the DOA and the grid points is included in a received data sparse representation model, then a sampling covariance matrix estimation error convex model is constructed, the convex set is explicitly included in the sparse representation model, and finally the obtained joint optimization problem is solved by adopting an alternative iteration method to obtain sparse DOA and grid offset parameter estimation. And multiple Signal Classification (MUSIC), l1Compared with DOA estimation algorithms such as SVD (sparse representation and low rank recovery) -based robust MVDR (SLRD-RMVDR), simulation results show that the algorithm has higher angle resolution and DOA estimation accuracy under the grid mismatch condition. Therefore, the algorithm provided by the invention can provide a solid theory and implementation basis for the research of DOA estimation performance in the field of array signal processing in engineering application.
The above description is only for the purpose of creating a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can substitute or change the technical solution and the inventive concept of the present invention within the technical scope of the present invention.

Claims (3)

1. An off-grid DOA estimation method based on covariance matrix reconstruction is characterized in that:
A1. the offset between the DOA and the grid points is included in a received data sparse representation model, and a sparse representation model related to DOA estimation is established based on a reconstructed signal covariance matrix;
A2. constructing a sampling covariance matrix estimation error convex model, deriving an estimation error upper bound based on the statistical characteristic that the sampling covariance matrix estimation error obeys progressive normal distribution, then explicitly including the convex set into a sparse representation model, and solving the obtained joint optimization problem by adopting an alternative iteration method to obtain sparse DOA and grid offset parameter estimation.
2. The covariance matrix reconstruction-based off-grid DOA estimation method of claim 1, wherein: the specific steps of A1 include:
step 1: establishing a receiving data space domain discrete sparse representation model
Assume K far-field narrowband signals
Figure FDA0002538498400000011
Incident on a uniform linear array with M array elements, the received signal model at time t can be expressed as
Figure FDA0002538498400000012
Wherein, x (t) is received data,
Figure FDA0002538498400000013
and sk(t) is respectively the guide vector and signal amplitude of the kth signal source, d and lambda are respectively the array element spacing and the carrier wavelength, d is not more than lambda/2, theta1θ2…θKFor K sources DOA,
Figure FDA0002538498400000014
a matrix of vectors is directed to the array,
Figure FDA0002538498400000015
is a waveform vector, n (t) ═ n1(t) n2(t)…nM(t)]TIs uncorrelated white Gaussian noise;
for L snapshots, (1) the received signal model is expressed as
X＝A(θ)S+N (2)
Wherein X ═ X (1) X (2) … X (l) is a received signal matrix, S ═ S (1) S (2) … S (l)) is a signal amplitude matrix, and N ═ N (1) N (2) … N (l)) is a noise matrix;
assuming that the signal and noise are uncorrelated and the sources are independent of each other, the received signal covariance is expressed as
Figure FDA0002538498400000016
Wherein R issIs a noise-free signal covariance matrix, I is an identity matrix, and P ═ E { SS }HIs the signal power covariance matrix, σ2Is the noise power.
In practical application, the receiving covariance matrix R is obtained based on L finite sampling snapshots, i.e.
Figure FDA0002538498400000021
Step 2: off-grid DOA estimation model
Suppose the source azimuth is discretely divided into N grids covering all possible target incident directions, i.e.
Figure FDA0002538498400000022
And the number N > M > K of the grids, for the grid model, the information source is supposed to be incident on the preset grid point without bias, namely
Figure FDA0002538498400000023
Then (1) the received signal model is restated as
Figure FDA0002538498400000024
Wherein the content of the first and second substances,
Figure FDA0002538498400000025
in order to be an ultra-complete dictionary,
Figure FDA0002538498400000026
obtaining a row sparse signal matrix after expansion;
based on an overcomplete matrix
Figure FDA0002538498400000027
Under the expression of (2), the covariance matrix R is expressed as sparse
Figure FDA0002538498400000028
Wherein R ═ vec (R-sigma)2I)，
Figure FDA0002538498400000029
Is the product of the Kronecker reaction,
Figure FDA00025384984000000210
the signal power vector obtained after the expansion is obtained;
and step 3: introducing an offset into an Off-grid sparse representation model
Introducing a set of offset parameters rho ═ rho1ρ2…ρN-1]TCorrecting grid offset:
Figure FDA00025384984000000211
where ρ isiIs an offset parameter and is greater than 0,
Figure FDA00025384984000000212
and
Figure FDA00025384984000000213
respectively true DOA thetaiLeft and right adjacent grid points;
array steering vector matrix representation as
Figure FDA00025384984000000214
Wherein A (u: v) represents the u-th to v-th columns of the acquisition matrix A, I-＝[IN-1,0(N-1)×1]T,I-＝[0(N-1)×1,IN-1]T. Let Δ be diag (ρ), the array steering vector matrix is re-expressed as
Figure FDA00025384984000000215
Wherein the content of the first and second substances,
Figure FDA00025384984000000216
from equations (5) and (9), the off-grid DOA estimation model is expressed as
Figure FDA00025384984000000217
The covariance matrix R is sparsely represented as
Figure FDA00025384984000000218
Wherein the content of the first and second substances,
Figure FDA00025384984000000219
actual received signal covariance matrix
Figure FDA0002538498400000031
Is shown as
Figure FDA0002538498400000032
Wherein the content of the first and second substances,
Figure FDA0002538498400000033
3. the covariance matrix reconstruction-based off-grid DOA estimation method of claim 2, wherein: the specific steps of A2 include:
step 1: estimation error model by introducing sampling covariance matrix
Constructing an error model for R:
Figure FDA0002538498400000034
wherein | · | purple sweetFIs a matrix Frobenius norm and is an error parameter factor;
based on the error convex set, the sparse signal reconstruction problem is based on l1Solving by a norm constraint optimization algorithm:
Figure FDA0002538498400000035
vectorized covariance matrix error
Figure FDA0002538498400000036
Obeying a progressive normal distribution
Figure FDA0002538498400000037
Wherein vec (-) is a vectorization operator, AsN (mu, ∑) represents an asymptotic normal distribution with mean mu and variance ∑,
Figure FDA0002538498400000038
through matrix operations, the vectorized covariance matrix error is re-represented as
Figure FDA0002538498400000039
Wherein the content of the first and second substances,
Figure FDA00025384984000000310
represents M2An order unit matrix.
From a normal distribution characteristic
Figure FDA00025384984000000311
Wherein, χ2(. represents chi-square distribution, As-2(M2) Represents a degree of freedom of M2The progressive chi-square distribution of (2);
since the vector R is a vectorized form of the covariance matrix R, the vector R is obtained
Figure FDA00025384984000000312
General formula (11)
Figure FDA00025384984000000313
The sparse signal reconstruction problem is re-expressed as
Figure FDA0002538498400000041
Wherein eta is an error parameter factor, and is obtained by the following formula
Figure FDA0002538498400000042
Wherein Pr {. cndot } represents a probability distribution, p is a probability value, χp 2(M2) Representing a probability value p and a degree of freedom M2The lower chi fang is distributed;
step 2: implementing offset solution and sparse signal reconstruction
Is represented by the formula (4) and
Figure FDA0002538498400000043
(20) the optimization problem is equivalent to
Figure FDA0002538498400000044
The convex optimization problem formula (22) is solved based on an alternate iteration method, in the (l + 1) th iteration,
(1) fixed rho, update
Figure FDA0002538498400000045
Figure FDA0002538498400000046
(2) Fixing
Figure FDA0002538498400000047
Updating rho:
Figure FDA0002538498400000048
dimension reduction processing by using singular value decomposition method
Figure FDA0002538498400000049
Wherein the content of the first and second substances,
Figure FDA00025384984000000410
Figure FDA00025384984000000411
the azimuth of the wave crest is taken as the azimuth of the wave crest,
Figure FDA00025384984000000412
by
Figure FDA00025384984000000413
K maximum peak values of (a);
after matrix operation, neglecting the items irrelevant to the optimization variables:
Figure FDA00025384984000000414
wherein R iseThe representation is taken in the real part,
Figure FDA00025384984000000415
representing a Hadamard product of the matrix;
order type(26) The derivative with respect to p is zero, resulting in pKIs updated to
Figure FDA0002538498400000051
Wherein the content of the first and second substances,
Figure FDA0002538498400000052
Figure FDA0002538498400000053
the pseudo-inverse is represented by a pseudo-inverse,
Figure FDA0002538498400000054
and step 3: implementing DOA estimation
Based on the results
Figure FDA0002538498400000055
And the offset rho, DOA is expressed as
Figure FDA0002538498400000056
Wherein, the grid is divided into intervals;
iterative calculation until the DOA estimated values of two adjacent times have no obvious change, i.e.
Figure FDA0002538498400000057
An effective DOA estimate is obtained.
CN202010539779.XA 2020-06-15 2020-06-15 Off-grid DOA estimation method based on covariance matrix reconstruction Pending CN111707985A (en)

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