CN107544051A - Wave arrival direction estimating method of the nested array based on K R subspaces - Google Patents
Wave arrival direction estimating method of the nested array based on K R subspaces Download PDFInfo
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Abstract
Wave arrival direction estimating method of the nested array based on K R subspaces, the present invention relates to Wave arrival direction estimating method.The present invention is in order to solve the problems, such as that existing method of estimation error is larger and aerial array can only estimate that the ripple fewer than array element quantity reaches the arrival bearing of quantity.The present invention includes：One：According to linear array array signal model at equal intervals, nested array model is established；Two：The nested array model for being established step 1 according to K R product transformations principles carries out LS-SVM sparseness, obtains sparse matrix Φ；Three：The sparse matrix Φ obtained to step 2 optimizes reconstruct, obtains arrival bearing's estimate.The present invention can estimate to break through more incident wave source numbers of Rayleigh limit；When signal to noise ratio is 20dB, root-mean-square error drops to 0.06952 by 0.15；When fast umber of beats is 500, root-mean-square error drops to 0.1949 by 0.4031.The present invention is used for smart antenna and Mutual coupling field.
Description
Technical Field
The invention relates to the field of intelligent antenna and direction of arrival estimation, in particular to a direction of arrival estimation method.
Background
Array signals arrange a set of sensors at spatially distinct locations in a manner to form an array of sensors. And (3) receiving the space signals by using a sensor array, namely sampling the field signals distributed in space to obtain space discrete observation data of the signal source. The array signal contains signal space domain characteristics besides signal time domain, frequency domain or time-frequency domain characteristics. The array signal processing fully excavates and utilizes the spatial characteristics of the array signals, so that the array output signals and the characteristic parameters thereof are extracted, and meanwhile, interference and noise are inhibited. Compared with the traditional single directional sensor, the sensor array has the characteristics of flexible beam control, high signal gain, very strong interference suppression capability, high spatial resolution and the like, is an important branch of array signal processing when being estimated, and is widely applied to the military and civil fields of wireless communication, radar, sonar, radio astronomy, medical images and the like. The destination angle of arrival of the information is one of the important characteristic parameters of the array signal.
The earliest array signal DOA estimation method is derived from classical Fourier transform and is a simple expansion of traditional time domain Fourier spectrum estimation in a space domain, wherein a representative algorithm is a beam forming algorithm, but the angular resolution of the algorithm is limited by Rayleigh limit. Then, the time domain nonlinear spectrum estimation technology is also popularized to a space domain, a high-resolution spectrum estimation technology is developed, and the Rayleigh limit is broken through, wherein a typical algorithm comprises a maximum entropy method and a minimum variance method. However, since the high-resolution spectrum estimation technique cannot effectively utilize the inherent structure of the array signal correlation matrix, the high-resolution spectrum estimation technique is still limited by the characteristics of the array, such as structure and aperture, and is not fully super-resolution. Schmidt et al then introduced subspace decomposition into the spatial spectrum estimation and proposed multiple signal classification (MUSIC) to achieve true full-scale super-resolution. The advent of the MUSIC algorithm opened a new door to DOA estimation, and many subspace-like algorithms were developed, among which the more typical algorithms are the MUSIC algorithm, root-finding MUSIC (Root-MUSIC) algorithm, minimum modulus algorithm and projection matrix algorithm, and least squares rotation invariant subspace technique (LS-ESPRIT) algorithm, total least squares ESPRIT (TLS-ESPRIT) algorithm, matrix beam (ESPRIT) algorithm, and real-valued space ESPRIT algorithm. The super-resolution performance of the subspace-based algorithm is based on signal independence. If the signals are correlated, the signal subspace and the noise subspace cannot be completely orthogonal, which results in serious deterioration of estimation performance, and even very poor estimation effect when the signals are coherent. In order to improve the problem, a probability statistical model of the array signals is introduced into the DOA estimation, a Maximum Likelihood (ML) algorithm appears, compared with a subspace decomposition algorithm, the ML algorithm is suitable for coherent sources, the DOA estimation angle value is more accurate, but the solving process has the problems of overlarge calculated amount, convergence to a local optimal value and the like. Therefore, efforts are made to simplify the above solving process and a large number of computationally efficient algorithms are proposed, representative of which are genetic algorithms, iterative quadratic ML algorithms, gaussian-newton algorithms and alternative projection algorithms. In addition, a subspace fitting algorithm is another method capable of solving the above problem, and is typically represented by a signal subspace fitting algorithm and a noise subspace fitting algorithm. Moreover, most of the estimation algorithms require that the array satisfies the space nyquist sampling theorem (the array satisfying the theorem is called a full array), that is, the distance between adjacent array elements is not more than half wavelength of an incident signal, so as to ensure the uniqueness of the DOA estimation. However, this kind of method is based on the prior information such as the known number of sources, and the incorrect estimation of the number of sources will possibly result in the degradation of the algorithm effect, and the effective array aperture is also related to the number of sources that can be processed simultaneously, so there is a certain limitation in the application of the electronic reconnaissance system.
Disclosure of Invention
The invention aims to solve the problems that the existing estimation method has larger error and an antenna array can only estimate the incoming wave direction of the arrival number less than the array element number, and provides a K-R subspace-based arrival direction estimation method of a nested array.
The method for estimating the arrival direction of the nested array based on the K-R subspace comprises the following steps:
the method comprises the following steps: establishing a nested array model according to the equidistant linear array signal model;
step two: performing sparsification treatment on the nested array model established in the step one according to a K-R product transformation principle to obtain a sparse matrix phi;
step three: and D, performing optimization reconstruction on the sparse matrix phi obtained in the step two to obtain an incoming wave direction estimation value.
The invention has the beneficial effects that:
due to the particularity of array arrangement of the nested array, when the nested array is compared with a traditional uniform linear array under a classical MUSIC algorithm, the estimation result of the direction of arrival of the nested array can have smaller error under the same condition, namely higher estimation accuracy. Secondly, the resolution of the nested array is analyzed by the MUSIC algorithm, and the nested array has higher resolution under the same condition, so that the nested array has practical significance of being better applied to practical production. Finally, in the direction finding of the nested array, the method of the invention can break through the Rayleigh limit which can not be broken through by the traditional method. The method can estimate the direction of arrival of the number of incident waves larger than the array element number of the array. The invention also carries out error statistics on the estimation result of the direction of arrival of the method, and can conclude that the estimation error of the method can be allowed to be within the range of actual life, so that the method can embody more definite significance in the actual application of production and life.
According to the invention, DOA estimation of the nested array with two different arrangement modes based on K-R subspace is firstly carried out, and based on the algorithm, the nested array can estimate the incident wave source number exceeding the number of the array elements per se. The main reason is that the nested array can generate virtual array elements when being under the K-R subspace algorithm, and the array aperture of the uniform nested array is larger than the aperture of the uniform linear array with the same array element number, so that the DOA estimation result is well influenced. Subsequently, the performance of four arrays in the DOA estimation method based on the K-R subspace is compared, results under different signal-to-noise ratios and different snapshot numbers are simulated respectively, and error statistics is carried out on the results, so that the error of the algorithm is generally small, the performance is good, and the algorithm can be considered to be used in practice. The performance of the nested array is still better than that of a uniform linear array with the same array element number in the DOA estimation method.
The invention can estimate more incident wave source numbers for breaking through Rayleigh limit; when the signal-to-noise ratio is 20dB, the root mean square error is reduced from 0.15 to 0.06952, about 53.65%; when the fast beat number is 500, the root mean square error is reduced from 0.4031 to 0.1949, which is about 51.65%.
Drawings
FIG. 1 is a diagram of a two-dimensional nested array arrangement with six array elements;
FIG. 2 is a diagram of a novel nested array arrangement with six array elements;
FIG. 3 is a diagram of a virtual array element array layout generated by a nested array with six array elements;
FIG. 4 is a diagram of a mathematical model for compressed sensing;
FIG. 5 is a DOA estimation space spectrogram of a two-dimensional nested array under a K-R subspace algorithm;
FIG. 6 is a DOA estimation space spectrogram of the novel nested array under the K-R subspace algorithm; the novel nesting in the figure is the method of the invention;
FIG. 7 is a performance analysis graph of DOA estimation of two nested arrays with signal-to-noise ratio variation based on K-R subspace algorithm; in the graph, SNR is a signal-to-noise ratio, and RMSE is a root mean square error;
FIG. 8 is a graph of performance analysis of four arrays as a function of signal to noise ratio under a K-R subspace-based algorithm;
FIG. 9 is a graph of performance analysis of DOA estimation of two nested arrays as a function of snapshot number under a K-R subspace algorithm;
FIG. 10 is a graph of performance analysis of four arrays as a function of snapshot number under a K-R subspace-based algorithm.
Detailed Description
The first embodiment is as follows: the method for estimating the direction of arrival (DOA) of the nested array based on the K-R subspace comprises the following steps:
the method comprises the following steps: establishing a nested array model according to the equidistant linear array signal model;
step two: performing sparsification treatment on the nested array model established in the step one according to a K-R product transformation principle to obtain a sparse matrix phi;
step three: and (4) performing optimization reconstruction on the sparse matrix phi obtained in the second step to obtain an incoming wave direction estimation value.
The second embodiment is as follows: the first difference between the present embodiment and the specific embodiment is: the first medium-interval linear array signal model specifically comprises:
X(t)＝A(Θ)S(t)+N(t)
wherein X (t) is a received signal, S (t) is a transmitted signal, N (t) represents system noise, t is time, and A (theta) is a steering vector matrix.
Any column vector a (theta) in matrix A (theta) i ) Is the direction of the array in the space source signal as theta i And is an M × 1-dimensional column vector, having:
other steps and parameters are the same as those in the first embodiment.
The third concrete implementation mode: the present embodiment differs from the first or second embodiment in that: the establishing of the nested array model in the first step specifically comprises the following steps:
if D signals are incident on the nested array, the input data vector received by the nested array of M array elements is represented as a linear combination of incident waveforms of the D incident signals and noise, that is:
where x (t) is the received signal vector, a (φ) i ) Is the steering vector, s, of the array of directions of arrival of the ith signal i (t) is the vector of the i-th incident signal, and n (t) is the vector of the system noise.
Since the array element position spacing must be redundant for array element numbers greater than 4, the researchers are working on finding the optimal array for less redundancy and freedom, thereby creating the concept of minimal redundancy. The arrangement mode of the nested array is a special minimum redundant array, and the resolution is improved while the aperture of the array is enlarged. According to the property of K-R product, the virtual uniform linear array generated by the second-order nested array containing N array elements hasAnd (4) array elements. The array flow pattern researched by the invention consists of 2 adjacent uniform linear arrays in size, and totally comprises N = N 1 +N 2 N is an even number 1 ＝N/2-1,N 2 = N/2+1. In the present invention, the case where N is even number is mainly discussed, and the case where N is odd number can be obtained by analogy. The smaller uniform linear array is a first-order sub-array containing N 1 Array elements with an array element interval of d 1 (ii) a The larger uniform linear array is a second-order sub-array containing N 2 Individual array elements with an array element interval of d 2 And d is 2 ＝(N 1 +1)d 1 . If the position of the array element is represented by the set S, the array element position S of the 1 st array can be obtained 1 ＝{md 1 ，m＝1，2，…，N 1 Array element position S of the 2 nd array 2 ＝{n(N 1 +1)d 1 ，n＝1，2，…，N 2 }. Table 1 compares the array arrangement of the second-order nested array and the novel nested array, and fig. 1 and fig. 2 specifically show two nested array arrangements when the number of array elements is six. FIG. 3 shows an array elementAnd the number of the virtual array elements generated by the six-time nested array is arranged.
Table 1 shows a comparison table between a second-order nested array and a novel nested array
Other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode: the difference between this embodiment and one of the first to third embodiments is: in the second step, the nested array model established in the first step is subjected to sparsification according to a K-R product transformation principle, and a specific process for obtaining a sparse matrix phi is as follows:
discretizing the received signal vector X (t) to obtain a received signal matrix X (n) (omitting time variables), and defining a covariance matrix R of the received signal matrix X (n) xx Comprises the following steps:
R xx ＝E{X(n)X H (n)}
wherein E represents desire, X H X (n) is conjugate transpose;
r is to be xx Vectorization processing is carried out:
where z is the result of the vectorization process, Λ = [ a (φ) 1 ) a(φ 2 )...a(φ D )]For vectors consisting of guide vector elements, Λ * Is a adjoint matrix of a,representing the Khatri-Rao product, p is a diagonal matrix of the signal correlation matrix, n =1,2, …, D, which is the nth diagonal element of the diagonal matrix of the signal correlation matrix; vec (-) denotes matrix vectorization, I N Is a unit diagonal matrix of unit NxN, N being R xx Dimension of (A), R ss Is a signal correlation matrix, R, obtained by singular value decomposition NN A covariance matrix that is noise;
will be provided withEquivalent to one signal sampled by the virtual array at one snapshot,representing the real number domain, from the signal model, we obtain:
in the formulaIs a directional matrix of the sampled signal, and:
whereinRepresents the Kronecker product, a * (φ 1 )、a * (φ 2 )...a * (φ D ) Are guide vector elements;
for is toWhen blocking by row, N 2 Repeated items exist in each row vector, the repeated items are subjected to arithmetic mean, each row vector is rearranged according to the position of the virtual array element, and the rearranged row vector is obtainedThe resulting sparse matrix is denoted as Φ.
Definition of sparse signals as stated by doruoho: if a signal x ∈ R N Can be expressed as α = ψ in an orthogonal matrix ψ T x, wherein α = [ α = 1 ，α 2 ，…，α N ] T Coefficients are sparsely represented, containing only a limited number of non-zero values, and the p-norm of α satisfies:
wherein, K is more than 0 and p is more than or equal to 0 and less than 2, the signal x can be expressed as a sparse signal under the orthogonal matrix psi. And if K =0, x can be recorded as a K-sparse signal.
Therefore, signals are compressed through the inner product of the transformation matrix and the original signals, a few sparse representation coefficients of the signals are obtained, namely, the projection vector alpha is obtained, whether the projection vector is sparse or not is closely related to the selected transformation matrix psi.
The most common transformation matrix in normal use is an orthogonal matrix, such as the well-known fourier transform, which represents a set of orthogonal functions for all the basis functions of a signal. When the signal characteristics are the same as those of the basis functions in the transformation matrix, the signal can be represented with high accuracy by a small number of coefficients. However, these fixed orthogonal matrix representation signals have two problems, one is that it is difficult to effectively represent sound and image signals of unknown regularity in nature. For example, the basis functions of the discrete cosine transform matrix lack spatial and temporal resolution and do not represent well signals featuring temporal localization. Secondly, a single orthogonal matrix can only represent signals of the same structure type, and mixed signals in nature are difficult to represent. A mixed signal, such as a pulse signal, cannot be represented by a single pulse basis, but also by a single sine basis. To solve these problems, an adaptive orthogonal matrix and a cascaded orthogonal matrix are proposed. Peler indicates that the non-stationary wavelet transform coefficients of a one-dimensional signal have an arbitrary C d Bandelet coefficients of regular image signals are sufficiently sparse to form an adaptationThe basis functions of the method can self-adapt signals under the condition that the structural characteristics of the signals are not known in advance, and the optimal sparse representation can be obtained. However, this method often requires a large-scale transformation matrix to be constructed, and is computationally expensive. Two irrelevant orthogonal matrixes can be cascaded to obtain a cascaded matrix, so that the defect of the representation of the basis function of each orthogonal matrix can be made up, and natural signals with various structures can be more effectively represented.
Unlike the method of representing signals by orthogonal matrix, the signal representation method based on redundant atom library has been developed rapidly in recent years, and its basic idea is: the orthogonal basis is replaced by the signal of a redundant atom library, the element in which is a redundancy function because it does not satisfy orthogonality, called atom: the constitution of the atom library is not limited at all, and the atom library can conform to the structure of an original signal as well as possible: few atoms from the atom pool can be selected to approximate the signal. Its mathematical description is:
atom library D = { g k K =1,2, …, K }, atom g i ∈R N For any signal x ∈ R N The m best atoms from the atom pool can be selected to approach:
in the formula I m Is a selected atom g γ Is given by the subscript set of (1) and has dimension m. K > N denotes the redundancy of the atom library, m<<N
Sparseness of the signal is reflected. The process of sparse representation thereof involves two main directions: the first is to select the best or best atom from a given atom pool; the second is to select m combinations from the good atoms selected to approximate the signal. The signal representation method of the redundant atom library solves the problem that a single orthogonal matrix represents signals, and a large number of effective algorithms appear in the aspect of signal reconstruction, so that the method is an important direction for researching signal representation at present.
Other steps and parameters are the same as those in one of the first to third embodiments.
The fifth concrete implementation mode: and in the third step, the sparse matrix phi obtained in the second step is subjected to optimization reconstruction, and the specific process of obtaining the incoming wave direction estimation value is as follows:
initialization residual r 0 = y, index setm =1,m is the number of iterations;
defining: m-dimensional observation signal y, signal sparsity K, M × N sparse matrix phi, K-sparse approximation of received signal X (t)
The iterative process is as follows:
step three, firstly: combining the residual r with the sparse matrix array vectorPerforming inner product, taking the maximum value corresponding to the subscript lambda, namely:
step three: update index set Λ m ＝Λ m-1 ∪{λ m And the column vector set in the sparse matrix phi corresponding to the index set
Step three: obtaining an approximation signal by a least square method:
wherein x m For the m-th estimated signal, the signal is,transposing the sparse matrix phi in the mth iteration;
step three and four: and (3) residual error updating:
r m ＝y-Φ m x m ，m＝m+1
step three and five: judging whether the condition m is larger than K, and if so, stopping the iteration process; if not, returning to the first step to continue the circulation.
The reconstruction of the signal is to solve the following equation:
wherein | a | purple o The number of elements in a, which are not 0, is referred to as the lo norm, and the sparsity of a. The original signal is then recovered from a small number of observation vectors. The reconstruction algorithm mainly aims at accuracy, rapidness and stability. The main reconstruction algorithms that exist today can be divided into four categories: convex optimization algorithm, greedy algorithm, combined algorithm and statistical optimization algorithm.
The invention mainly applies the OMP algorithm in the greedy algorithm for simulation, and the greedy algorithm directly solves the problem of l o Norm problem, allowing for some reconstruction error, the basic idea is: the observation signal is regarded as a linear combination of certain atoms in a redundant atom library, an optimal solution or a local optimal solution is sought through a certain number of iteration processes, so that the residual error between the estimation signal and the original signal is smaller and smaller, and the approximation to the original signal is finally realized. Their most obvious advantage is the fast operation speed, which is very meaningful for compressed sensing theory, because the compressed sensing theory is sampling althoughThe rate is much lower than that of the traditional method, but because only a small amount of observed signals are obtained, and complex calculation is generally carried out when the signals are recovered, reducing the calculation amount of the sparse recovery algorithm is a very important research content in the compressed sensing theory.
Many signals can be represented by dimension sparse signals by selecting a proper sparse representation matrix, but the selection of the sparse representation matrix is a difficulty, but the space signals processed by the array signals generally meet the sparsity of the space and can be directly represented as sparse signals, because many target signals do not appear simultaneously in the space, the target signals are sparse compared with the whole space.
In the conventional array angle estimation model, each column a of the array manifold j (w 0 ) Corresponds to a real target signal in space, but under the compressed sensing framework we extend it to the whole space, as shown in fig. 4.
Suppose that in the range of space-90 deg., there are 2N +1 signals, and the incident angle is represented as [ theta ] 1 ，θ 2 ，…，θ 2N+1 ]Each angle corresponds to a steering vector, and the corresponding signal is denoted as S = [ S ] 1 s 2 …s 2N+1 ] T This is a 2N × 1 dimensional vector. When there is actually a signal in the assumed signal of 2n +1, the corresponding element and steering vector in S are not zero. When there is actually a signal in this hypothetical 2N +1 signal, the element and steering vector in the corresponding S is zero. If K (2N K) signals are actually present in space, then vector S [ S ] 1 s 2 …s 2N+1 ] T Becomes a K sparse signal and can be analyzed by the compressed sensing theory.
According to the basic knowledge of array signal processing, the equal-interval linear array signal model can be rewritten into:
Y＝AS+N
wherein Y = [ Y = 1 y 2 …y M ] T Indicating the array received signal at a certain time, A ∈ R M×2N+1 Represents an array flow pattern matrix, S = [ S ] 1 s 2 …s 2N+1 ] T Representing a spatially sparse signal containing only K actual signals, and N represents the noise received by the array. From the above discussion, it can be seen that DOA estimation in the compressed sensing framework is equivalent to reconstructing a spatially sparse signal S = [ S ] from the resulting array received signal Y 1 s 2 …s 2N+1 ] T According to { theta 1 ，θ 2 ，…，θ 2N+1 And S = [ S ] 1 s 2 …s 2N+1 ] T The one-to-one correspondence of (a) can determine the angular information of the spatial target. Therefore, the array DOA estimation mathematical model based on compressed sensing can be written as:
where σ is a possible noise level estimate.
In the array DOA estimation model based on compressed sensing, because the spatial signal S is sparse originally, a sparse representation matrix does not need to be considered, the array manifold matrix A is equivalent to a measurement matrix phi in the compressed sensing theory, the measurement matrix in the compressed sensing theory needs to meet a series of conditions, and the array popular matrix A also needs to meet corresponding conditions. Unlike a freely designed measurement matrix, the array manifold is determined in a basic form and only relates to a plurality of factors such as an array structure, a spatial signal incidence angle, a signal wavelength and the like, so that the change condition of the factors can influence the sparse reconstruction condition met by the array manifold.
Other steps and parameters are the same as in one of the first to fourth embodiments.
The first embodiment is as follows:
by using the method, the number of incident waves larger than the array element number of the nested array signal can be estimated. The invention carries out DOA estimation on two nested arrays by using a K-R subspace method and carries out analysis simulation on the performances of the two nested arrays.
DOA estimation of two-dimensional nested array under K-R subspace algorithm
The simulation results of the two-dimensional nested array using DOA estimation based on K-R subspace are shown in fig. 5 when the SNR is =10dB, the number of snapshots n =50, and the number of incident waves is 8, respectively, 30 degrees, 40 degrees, 60 degrees, 90 degrees, 100 degrees, 120 degrees, 140 degrees, and 160 degrees.
The simulation results of the new nested array using DOA estimation based on K-R subspace are shown in fig. 6 when the SNR of the signal to noise ratio =10dB, the number of snapshots n =50, and the number of incident waves is 8, respectively, 30 degrees, 40 degrees, 60 degrees, 90 degrees, 100 degrees, 120 degrees, 140 degrees, and 160 degrees.
According to the two simulation results, the number of incident waves exceeding the number of array elements per se can be estimated by the nested arrays with the two arrangement modes under the K-R subspace-based algorithm. The simulation result is better.
According to the simulation result, the DOA estimation algorithm based on the K-R subspace can be well applied to the nested array, and incident wave sources of redundant array elements are estimated, so that the algorithm can break through the Rayleigh limit which cannot be broken through by the traditional algorithm. The reliability of this algorithm is simulated in the following text, i.e. the error (RMSE) of this algorithm is simulated under different conditions and the DOA estimation performance of four different arrays is compared under different signal-to-noise ratios and different fast beat numbers.
Performance comparison of nested arrays at different signal-to-noise ratios
And carrying out DOA estimation based on K-R subspace on the two-dimensional nested array of six array elements and the novel nested array of six array elements. And comparing the DOA estimation result with the angle of the incident wave source, and finally carrying out error statistics. In the simulation test, the number of snapshots n is 200, the incident angles are 40 degrees and 120 degrees, so that the signal-to-noise ratio SNR is changed from-10 dB to 20dB, and the simulation result is shown in FIG. 7.
According to the simulation results, the root-mean-square RMSE of DOA estimation errors of the nested arrays with two different arrangement modes under the K-R subspace-based algorithm is reduced along with the increase of the SNR of the signal to noise ratio, namely the larger the SNR is, the more accurate the estimation result is.
Performance comparison of nested array and uniform linear array under conditions of different signal-to-noise ratios
And carrying out DOA estimation based on K-R subspace on four different arrays, namely a six-array-element uniform linear array, a twenty-three-array-element uniform linear array, a six-array-element two-dimensional nested array and a six-array-element novel nested array. And comparing the DOA estimation result with the angle of the incident wave source, and finally carrying out error statistics. In the simulation test, the number of snapshots n is 200, the incident angles are 40 degrees and 120 degrees, so that the signal-to-noise ratio SNR is changed from-10 dB to 20dB, and the simulation result is shown in FIG. 8.
The simulation results show that under the same conditions, generally speaking, the DOA estimation errors of the nested arrays in the two arrangement modes can be smaller than the uniform linear arrays with the same array element number, that is, the performance of the nested arrays is superior to the uniform linear arrays with the same array element number in most cases. However, nested arrays do not always perform as well as uniform lines of their virtual array elements.
And carrying out DOA estimation based on the K-R subspace on the two-dimensional nested array and the novel nested array. And comparing the DOA estimation result with the angle of the incident wave source, and finally carrying out error statistics. In the simulation test, when the signal-to-noise ratio SNR =10dB and the incidence angles are 40 degrees and 120 degrees, and the fast beat count is changed from 10, 20, 50, 80, 100, 150, 200, and 500, the simulation result is shown in fig. 9.
According to the simulation results, the root-mean-square RMSE of DOA estimation errors of the nested arrays with two different arrangement modes under the K-R subspace-based algorithm is reduced along with the increase of the fast beat number, namely, the larger the fast beat number is, the more accurate the estimation result is.
Performance comparison of nested array and uniform linear array under different fast-beat conditions
And carrying out DOA estimation based on K-R subspace by using four different arrays, namely a uniform linear array with six array elements, a uniform linear array with twenty-three array elements, a two-dimensional nested array and a novel nested array. And comparing the DOA estimation result with the angle of the incident wave source, and finally carrying out error statistics. In the simulation test, the simulation results are shown in fig. 10 when the signal-to-noise ratio SNR is 10dB, the incident angles are 40 degrees and 120 degrees, and the fast beat count is changed from 10, 20, 50, 80, 100, 150, 200, and 500.
The following simulation results show that under the same conditions, generally, the DOA estimation error of the nested arrays in the two arrangement modes can be smaller than that of the uniform linear arrays with the same array element number, that is, the performance of the nested arrays is superior to that of the uniform linear arrays with the same array element number in most cases. However, nested arrays do not always perform as well as uniform lines of their virtual array elements. The estimation errors of the four arrays gradually decrease as the number of fast beats increases.
Through the performance analysis of the four simulations, the performance of the nested array is basically superior to that of a uniform linear array with the same array element number under a DOA estimation calculation method based on K-R subspace.
The present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof, and it is therefore intended that all such changes and modifications be considered as within the spirit and scope of the appended claims.
Claims (5)
1. The method for estimating the direction of arrival of the nested array based on the K-R subspace is characterized by comprising the following steps: the direction of arrival estimation method comprises the following steps:
the method comprises the following steps: establishing a nested array model according to the equidistant linear array signal model;
step two: performing sparsification treatment on the nested array model established in the step one according to a K-R product transformation principle to obtain a sparse matrix phi;
step three: and D, performing optimization reconstruction on the sparse matrix phi obtained in the step two to obtain an incoming wave direction estimation value.
2. The nested array K-R subspace-based direction of arrival estimation method of claim 1, wherein: the first medium-interval linear array signal model specifically comprises:
X(t)＝A(Θ)S(t)+N(t)
wherein X (t) is a received signal, S (t) is a transmitted signal, N (t) represents system noise, t is time, and A (theta) is a steering vector matrix.
3. The nested array K-R subspace-based direction of arrival estimation method of claim 2, wherein: the establishing of the nested array model in the first step specifically comprises the following steps:
if D signals are incident on the nested array, the input data vector received by the nested array of M array elements is represented as a linear combination of incident waveforms of the D incident signals and noise, that is:
where x (t) is the received signal vector, a (φ) i ) Is the steering vector, s, of the array of directions of arrival of the ith signal i (t) is the vector of the i-th incident signal, and n (t) is the vector of the system noise.
4. The nested array K-R subspace-based direction of arrival estimation method of claim 3, wherein: in the second step, the nested array model established in the first step is subjected to sparsification according to a K-R product transformation principle, and a specific process for obtaining a sparse matrix phi is as follows:
discretizing the received signal vector X (t) to obtain a received signal matrix X (n), and defining a covariance matrix R of the received signal matrix X (n) xx Comprises the following steps:
R xx ＝E{X(n)X H (n)}
wherein E represents desire, X H X (n) is conjugate transpose;
r is to be xx Vectorization processing is carried out:
where z is the result of the vectorization process, Λ = [ a (φ) 1 ) a(φ 2 ) ... a(φ D )]For vectors consisting of guide vector elements, Λ * Is a co-matrix of a and,representing the Khatri-Rao product, p is a diagonal matrix of the signal correlation matrix,n =1,2, …, D, which is the nth diagonal element of the diagonal matrix of the signal correlation matrix; vec (-) denotes matrix vectorization, I N Is a unit diagonal matrix of unit NxN, N being R xx Dimension of (A), R ss Is a signal correlation matrix, R, obtained by singular value decomposition NN A covariance matrix that is noise;
will be provided withEquivalent to one signal sampled by the virtual array at one snapshot,representing the real number domain yields:
in the formulaIs a directional matrix of the sampled signal, and:
whereinRepresents the Kronecker product;
to pairWhen blocking by row, N 2 Repeated items exist in each row vector, the repeated items are subjected to arithmetic mean, each row vector is rearranged according to the position of the virtual array element, and the sparse matrix obtained after rearrangement is represented as phi.
5. The nested array K-R subspace-based direction of arrival estimation method of claim 4, wherein: and in the third step, the sparse matrix phi obtained in the second step is subjected to optimization reconstruction, and the specific process of obtaining the incoming wave direction estimation value is as follows:
initializing the residual r 0 = y, index setm =1,m for iteration number;
defining: m-dimensional observation signal y, signal sparsity K, M × N sparse matrix phi, K-sparse approximation of received signal X (t)
The iterative process is as follows:
step three, firstly: combining the residual r with the sparse matrix array vectorPerforming inner product, taking the maximum value corresponding to the subscript lambda, namely:
step three: updating index set Λ m ＝Λ m-1 ∪{λ m And the column vector set in the sparse matrix phi corresponding to the index set
Step three: obtaining an approximation signal by a least square method:
wherein x is m For the m-th estimated signal, the signal is,transposing the sparse matrix phi in the mth iteration;
step three and four: and (3) residual error updating:
r m ＝y-Φ m x m ，m＝m+1
step three and five: judging whether the condition m is larger than K, and if so, stopping the iteration process; if not, returning to the first step to continue the circulation.
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