CN106526529A - Sparse representation-based direction-of-arrival estimation method in mismatched condition of steering vectors - Google Patents

Abstract

The invention belongs to the array signal processing field and provides a method for estimating the direction-of-arrival of a signal source based on sparse representation in the steering vector mismatched condition of a sensor. According to the technical scheme of the sparse representation-based direction-of-arrival estimation method in the mismatched condition of steering vectors, firstly, a receiving signal model of a sensor array is constructed. Secondly, a heavy-tailed actual ambient noise signal is simulated through the synthesized circle symmetrical type generalized Gaussian distribution. Meanwhile, the receiving signal model is processed based on the fraction low-order moment method and the unknown gain value of the processed receiving signal model, namely the gain value of uncertainty parameters caused by the steering vector mismatched condition, is optimized. Finally, the direction-of-arrival (DOA) of a signal source is estimated based on the sparse representation manner. In this way, the optimal estimation value of the direction-of-arrival (DOA) of the signal source is obtained. The method is mainly applied in the signal processing field of array signals.

Description

Wave arrival direction estimating method in the case of steering vector mismatch based on rarefaction representation

Technical field

The invention belongs to Array Signal Processing field, and in particular to a kind of in non-ideal Gaussian noise environment to proposing In, under sensor steering vector mating situation, the algorithm of signal source Mutual coupling is carried out using sparse representation method.

Background technology

Signal source direction of arrival (DOA) is estimated to be always one of important research direction of target positioning and tracking field, The aspects such as radar, sonar, the theory of calamity, communication suffer from greatly application.In the last few years as people are in space flight, ocean, disaster The research in the fields such as prediction is increasingly deep, and target DOA estimation receives more extensive concern, and is actually applied in extraterrestrial target The fields such as tracking, underwater sound communication, pipeline protection and earthquake prediction.

Very long development course experienced to the research that DOA estimates, a series of algorithm for estimating of classics is generated, such as Capon, MUSIC and ESPRIT etc., but these algorithms signal to noise ratio is relatively low, fast umber of beats it is less in the case of signal resolution ratio and Direction finding precision declines serious.In the last few years, based on signal spatial distribution openness feature, researcher is by sparse reconstruct Thought is applied in terms of DOA estimations, and proposes some DOA algorithm for estimating based on rarefaction representation.Early stage sparse representation method It is by building only related to signal source DOA cumulant domain data, using weighting l₁Norm minimum method carries out DOA and estimates Meter, and rarefaction is carried out on this basis to DOA estimates for estimated distance parameter, the method can be effectively prevented from signal Source physical distance is close to estimate the impact for producing to DOA.Researcher proposes to propose one kind using focus method thought recently Based on wideband signal source DOA algorithm for estimating (FSP) of rarefaction representation, replace frequency and angle joint with the basic matrix of single frequency The basic matrix of structure, reduces the columns of basic matrix, solves in traditional sparse representation method as basic matrix dimension is excessive So that amount of storage is big and calculates the difficult problems such as complexity.Existing researcher is verified using sparse representation method in actual environment The performance that DOA estimates, derives thinned array signal transacting model according to the type structure of two kinds of arrays of nested battle array and mutual primitive matrix, DOA estimation is carried out with the sparse two kinds of models of grade sine space using spatial domain is angularly sparse, this method can be with less battle array Unit reaches the larger free degree, so as to improve the precision of target resolution, but does not consider destabilizing factor docking in actual environment The impact that the collection of letters number is produced.

In some actual environments, as the heavy-tailed phenomenon of noise component(s) is than more serious, it is therefore desirable to select one suitably Noise model generally uses impact noise or S α S noises is simulated, and the present invention is adopted for noise signal process The symmetrical generalized Gaussian distribution of synthesis circle carrys out analogue noise environment, and its rate of decay is slower, more truly can describe Actual environment noise profile.Simultaneously in actual applications, due to being affected by itself and outside environmental elements, sensor Steering vector may produce fluctuation, generate a undesired gain, be allowed to the presence of error between actual value, and this is estimated to DOA The accuracy of meter can have a huge impact.

The content of the invention

To overcome the deficiencies in the prior art, it is contemplated that realizing in the case of sensor steering vector mismatch using dilute Thin method for expressing is estimated to signal source direction of arrival.The technical solution used in the present invention is, in the case of steering vector mismatch Based on the Wave arrival direction estimating method of rarefaction representation, the receipt signal model of sensor array is built first；Then using synthesis The symmetrical generalized Gaussian distribution of circle is simulated to heavy-tailed serious actual environment noise signal, is docked using fractional lower-order Moment Methods Receive signal model to be processed, and to unknown yield value present in the signal model after process be steering vector mismatch and generate Uncertain parameter yield value be optimized；Finally, estimated using carrying out signal source direction of arrival DOA based on sparse representation method Meter, obtains the signal source direction of arrival DOA estimates of optimum.

The receipt signal model of sensor array is built first, it is assumed that the Q narrowband random signal under the environment of far field Source transmitting signal wave reach by P sensor group into uniform linear array ULA, it is separate between wherein Q signal source, And noise is orthogonal with signal；The angle of arrival of q-th signal source is designated as θ_q, the reception at t, p-th sensor Signal isIn formula, a (θ_q) for q-th signal source s_qT () is in its direction of arrival θ_qUpper p-th Steering vector at sensor, expression formula isD is the distance between sensor, and λ is signal wavelength, g_p For the steering vector yield value at p-th sensor in actual environment, s_qThe narrow band signal of (t) for random distribution in space, n_p T () is spatial noise signal；

The receipt signal model of sensor array is tried to achieve according to the reception signal at each sensor：

X (t)=GAs (t)+n (t)

Wherein

X (t)=[x₁(t),x₂(t),...,x_P(t)]^T

G=diag [g₁,g₂,...,g_P]

A=[a (θ₁),a(θ₂),...,a(θ_Q)]

S (t)=[s₁(t),s₂(t),...,s_q(t)]^T

N (t)=[n₁(t),n₂(t,...,n_p(t))]^T

In formula, x (t) is sensor array receipt signal model, []^TArray transposition is represented, G is at each sensor The diagonal matrix of yield value composition, A is the array manifold of all steering vectors composition, a (θ_q) it is each sensor at signal source q Steering vector, s (t) is signal source matrix, s_qT () is the transmission signal of q-th signal source, n (t) for all the sensors at Noise signal, n_pT () is the noise signal at p-th sensor.

Actual noise environment is simulated using the symmetrical generalized Gaussian distribution of synthesis circle, noise signal is with regard to stochastic variable w Probability density function is expressed as：

In formula, Γ () represents gamma function, and α is Stable distritation coefficient, and σ is function variance, and when α=2, noise Gaussian Profile is presented.

Using being processed to which based on Fractional Lower Order Moments matrix method FLOM, comprise the following steps that：Homogenous linear is sensed Reception signal x (t) the Fractional Lower Order Moments matrix of device array is the matrix of a P × P, and its (i, k) individual component is

In formula, E () represents desired value, x^*T () represents its conjugate function, orthogonal between each signal in x (t), ε It is the index parameters less than α；

Signal Model in Time Domain is updated in the Fractional Lower Order Moments matrix expression that sensor array receives signal x (t)

By R_ikIt is decomposed into two parts C_ikAnd D_ik, wherein

In formula,For kronecker delta, the subscript of i, k for Fractional Lower Order Moments matrix component, θ_q,θ_rRespectively For the angle of arrival of q-th and r-th signal source, s_q(t),s_rThe transmission signal of (t) respectively q and r-th signal source, and In above formula

Therefore, the Fractional Lower Order Moments matrix of sensor array is expressed as

R=GA Γ A^H+γI

In formula,Unit matrixs of the I for P × P ranks, A is battle array Row manifold matrix, H represent conjugate transposition.

Due to the yield value G produced containing sensor displacement and external environmental interference in above formula, using weighting most Little least square method carries out optimum estimation to which, obtains its optimal solution：The sensor array for obtaining is received into signal x (t) first K sampling is carried out according to time interval T and obtains sampled signal x (n), then obtain the expression formula of its covariance matrix

According to the sampled data for getting, gain function is obtained using weighted least-squares method, then by asking for gain letter Several minimum of a values is obtaining the optimal solution of omnidirectional gain

In formula, W is weight coefficient, logical which to be initialized first with W=Ι here, Zhi HouyongIt is right Each sampled data is modified, then the gain function in above formula is rewritten as

Due to sensor array be by P sensor group into, then the gain function at p-th sensor is designated as：

In formula,w_n=ζ (W_n) and z=ζ_p(GAΓA^H)/g_p, wherein operator ζ_p() is represented from a matrix Middle its pth of selection is arranged as column vector, and removes p-th element in vector.

So optimal gain values Solve problems translate into P Linear least squares minimization problem of solution, then to gain function Minimize to try to achieve final result, introduce formula z here_w=z ⊙ w_n, then the optimal gain values obtained are

Finally, DOA estimations are carried out based on sparse representation method, is comprised the concrete steps that：First to the signal after yield value optimization Model carries out vectorized process；After the Fractional Lower Order Moments matrix for obtaining vector quantization, LS-SVM sparseness is carried out to DOA angles, The angle set of construction redundancy, and the method using singular value decomposition SVD obtains signal subspace, angle is solved and is converted into two Rank bores planning problem.

Vector quantization is carried out to the Fractional Lower Order Moments matrix after yield value optimization

In formula, vec () represents vectored calculations,The Kronecker products and Khatri-Rao of representing matrix are distinguished with ⊙ Product.

Then obtain after the K all of Fractional Lower Order Moments matrix vector of frame in of sampling

In formula, y_kFor each sampling frame in vector quantization Fractional Lower Order Moments matrix, DefinitionDue to response matrixMatrix Y Dimension be changed into P², detectable number of sources Q and array number P need to meet Q≤2 (P-1).

After the Fractional Lower Order Moments matrix for obtaining vector quantization, redundant is carried out to DOA angles using sparse representation method Process, the main thought of rarefaction representation DOA estimation method is to use angle setCover all possible target Signal incident direction, and meet N ＞ ＞ Q, N are the numbers of the space angle for dividing, based on this may make up one it is excessively complete superfluous Remaining dictionaryIts all possible positional information comprising source, takes some methods from these positional informations In extract needed for angle information, the vector quantization Fractional Lower Order Moments matrix of K frame ins is entered using the method for singular value decomposition SVD Row is processed

In formula,WithRespectively left singular matrix and right singular matrix,For diagonal matrix；

Obtain signal subspace

In formula,Unit matrixs of the I for Q × Q ranks,

Therefore, signal source DOA Solve problems are converted into

In formula, | | | |₁With | | | |₂L is represented respectively₁Norm and l₂Norm, it is dilute that regularisation parameter β is used for balanced signal The impact of thin property and noise；

Can only obtain according to the limited fast umber of beats of array output dataEstimated resultAnd their approximately equalsThe optimization problem of above formula is converted into

In formula, parameter lambda is used for balancing l₁Norm and l₂Norm, the previous item reflection mismatch of object function, latter are anti- Openness requirement is reflected, above formula is a Second-order cone programming problem, is solved using interior point method, and then according to the position of nonzero element Put the DOA estimates for trying to achieve signal source.

The characteristics of of the invention and beneficial effect are：

1. when in present invention consideration actual environment, noise is heavy-tailed serious mould is carried out using synthesis circle symmetrical generalized Gaussian distribution Intend, and signal model is processed using Fractional Lower Order Moments matrix, overcome under this noise circumstance, noise second order above square is not In the presence of traditional algorithm can not carry out the shortcoming of accurate estimation；

2. the present invention is processed to the nonideal situation of sensor steering vector in actual environment, using a weighting most young waiter in a wineshop or an inn Method is taken advantage of to be optimized estimation to steering vector yield value；

3. instant invention overcomes can not carry out accurately when number of sensors is less than source number in tradition DOA algorithm for estimating The problem that DOA estimates；

4. the present invention carries out the estimation of signal source DOA using improved rarefaction representation algorithm, improves estimated accuracy, and adopts Singular value decomposition reduces operand.

Description of the drawings：

Accompanying drawing 1 is the direction flow chart of inventive algorithm；

Accompanying drawing 2 is the DOA estimated results of inventive algorithm and other classic algorithms；

Accompanying drawing 3 is the relation curve of the mean square error with signal to noise ratio of signal DOA estimation；

Accompanying drawing 4 is the relation curve of the mean square error with fast umber of beats of signal DOA estimation；

Accompanying drawing 5 is the relation curve of the detection probability with signal to noise ratio of signal DOA estimation.

Specific embodiment：

For problem of the prior art, the present invention proposes a kind of new DOA algorithm for estimating based on rarefaction representation.The calculation Method simulates actual environment noise from the symmetrical generalized Gaussian distribution of synthesis circle, but as its second order above square is not present, tradition The algorithm decomposed based on covariance matrix be no longer suitable for, therefore inventive algorithm is using the method pair based on fractional lower-order matrix Noise signal is processed.Then the yield value for generating to steering vector mismatch, is optimized using weighted least-squares method Estimate.Complete redundant dictionary was built using sparse representation method finally, and dimension was reduced using singular value decomposition, by its turn Turn to Second-order cone programming problem to be solved, the DOA estimates of optimum are obtained using interior point method.

Present invention mainly solves problem be in the case of sensor steering vector mismatch utilize sparse representation method pair Signal source direction of arrival is estimated.The present invention's realizes that process is as follows：

Step one：Build signal model.

Assume the signal wave arrival of Q narrowband random signal source transmitting under the environment of far field by P sensor group into Uniform linear array (ULA), it is wherein separate between this Q signal source, and noise is orthogonal with signal.Q-th The angle of arrival of signal source is designated as θ_q, the reception signal at t, p-th sensor is In formula, a (θ_q) for signal s_qT () is in its direction of arrival θ_qSteering vector at upper p-th sensor, expression formula isD is the distance between sensor, and λ is signal wavelength.g_pAt p-th sensor in actual environment Steering vector yield value, s_qThe narrow band signal of (t) for random distribution in space, n_pT () is spatial noise signal.

Can be in the hope of the receipt signal model of sensor array according to the reception signal at each sensor

X (t)=GAs (t)+n (t)

Wherein

X (t)=[x₁(t),x₂(t),...,x_P(t)]^T

G=diag [g₁,g₂,...,g_P]

A=[a (θ₁),a(θ₂),...,a(θ_Q)]

S (t)=[s₁(t),s₂(t),...,s_Q(t)]^T

N (t)=[n₁(t),n₂(t,...,n_P(t))]^T

In formula, x (t) is sensor array receipt signal model, and G is the diagonal of the yield value composition at each sensor Battle array, A is the array manifold of all steering vectors composition, a (θ_q) it is steering vector of each sensor at signal source q, s (t) For signal source matrix, s_qT () is the transmission signal of q-th signal source, n (t) is the noise signal at all the sensors, n_pT () is Noise signal at p-th sensor.

Step 2：Signal model based on Fractional Lower Order Moments is processed and yield value optimum estimation.

Noise signal in actual environment not always preferable white Gaussian noise, this can estimate to produce certain to angle on target Impact, especially the heavy-tailed phenomenon of noise than it is more serious when it is this affect it is especially prominent.For this situation, we take following Step is solved：Heavy-tailed serious noise signal is simulated initially with the symmetrical generalized Gaussian distribution of synthesis circle, and profit Receipt signal model is processed with fractional lower-order Moment Methods, next unknown yield value present in signal model is optimized.

The heavy-tailed phenomenon of noise component(s) in some actual environments is serious in the extreme, needs to select to close when noise processed is carried out Suitable distribution pattern is simulated to which, generally uses impact noise or S α S noise profile types, and the present invention is adopted The symmetrical generalized Gaussian distribution of synthesis circle carrys out analogue noise environment, and its rate of decay is slower, more truly can describe Actual environment noise profile.Noise signal is expressed as with regard to the probability density function of stochastic variable w

In formula, Γ () represents gamma function, and α is Stable distritation coefficient, and σ is function variance, and when α=2, noise Gaussian Profile is presented.

During symmetrical Generalized Gaussian Noise round for synthesis due to noise signal, the second order above square of signal covariance matrix is received Do not exist, therefore traditional MUSIC, ESPRIT etc. is no longer suitable for based on the DOA algorithm for estimating of covariance matrix feature decomposition. Here we are processed to noise component(s) using Fractional Lower Order Moments matrix (FLOM) method.

Reception signal x (t) the Fractional Lower Order Moments matrix of homogenous linear sensor array is the matrix of a P × P, its (i, k) individual component isIn formula, E () represents desired value, x^*T () represents its conjugation Function, it is orthogonal between each signal in x (t).

Due to there is the uncertain parameter yield value g generated because of steering vector mismatch in FLOM matrixes, this is to follow-up number According to process cause greatly puzzlement, therefore we need the g values for taking rational method to include which to estimate, obtain which Optimal solution.We obtain optimal gain values using weighted least-squares method iteration in the present invention.By asking for gain function Minimum of a value obtaining the optimal solution of omnidirectional gainIn formula, W is weight coefficient, generally In the case of be unknown, but due in actual environment, impact of the ambient noise to signal transacting can be than larger, therefore here first Which is initialized using W=Ι, Zhi HouyongEach sampled data is modified.

Step 3：DOA estimations are carried out based on sparse representation method.

We are frequently encountered number of sensors less than signal number purpose situation in actual applications, at this moment using tradition DOA estimation method be no longer able to carry out accurate measurement, take vectorization method to be processed herein for such case, increase The dimension of FLOM matrixes, and then increase detectable source number.

After the Fractional Lower Order Moments matrix for obtaining vector quantization, redundant is carried out to DOA angles using sparse representation method Process.The main thought of rarefaction representation DOA estimation method is into a mistake by the manifold matrix-expand comprising steering vector information Complete redundant dictionary, its all possible positional information comprising source, and then needed for extracting from the positional information of redundancy Angle information.With the increase of dictionary redundancy, amount of calculation is also sharply increased, in order to reduce amount of calculation, herein using singular value The method for decomposing (SVD) is processed.

The present invention is further described with reference to the accompanying drawings and detailed description.

The present invention proposes the signal source DOA algorithm for estimating based on rarefaction representation in the case of a kind of steering vector mismatch, Specific implementation process can be divided into three parts.

Structure of the Part I for signal model.

Assume the signal wave arrival of Q narrowband random signal source transmitting under the environment of far field by P sensor group into Uniform linear array (ULA), it is wherein separate between this Q signal source, and noise is orthogonal with signal.Q-th The angle of arrival of signal source is designated as θ_q, the reception signal at t, p-th sensor is In formula, a (θ_q) for q-th signal source s_qT () is in its direction of arrival θ_qSteering vector at upper p-th sensor, expression formula isD is the distance between sensor, and λ is signal wavelength.g_pAt p-th sensor in actual environment Steering vector yield value, s_qThe narrow band signal of (t) for random distribution in space, n_pT () is spatial noise signal.

Part II is that the signal model based on Fractional Lower Order Moments is processed and yield value optimum estimation.

The present invention is simulated to actual noise environment using the symmetrical Generalized Gaussian Noise of synthesis circle, relatively common noise Be distributed its rate of decay slower, can more true simulation actual environment noise profile, its probability density function (probability density function, PDF) is expressed as

In formula, Γ () represents gamma function, and α is Stable distritation coefficient, and σ is function variance, and when α=2, noise Gaussian Profile is presented.

We are processed to this noise using the method based on Fractional Lower Order Moments (FLOM) in the present invention.Uniform line Reception signal x (t) the Fractional Lower Order Moments matrix of property sensor array is the matrix of a P × P, and its (i, k) individual component is

In formula, E () represents desired value, x^*T () represents its conjugate function, orthogonal between each signal in x (t), ε It is the index parameters less than α.

Signal Model in Time Domain is updated in the Fractional Lower Order Moments matrix expression that sensor array receives signal x (t)

By R_ikIt is decomposed into two parts C_ikAnd D_ik, wherein

In formula,For kronecker delta, the subscript of i, k for Fractional Lower Order Moments matrix component, θ_q,θ_rRespectively For the angle of arrival of q-th and r-th signal source, s_q(t),s_rThe transmission signal of (t) respectively q and r-th signal source, and In above formula

Therefore, the Fractional Lower Order Moments matrix of sensor array is represented by

R=GA Γ A^H+γI

In formula,Unit matrixs of the I for P × P ranks, A is battle array Row manifold matrix, H represent conjugate transposition.

The sensor array for obtaining reception signal x (t) is carried out into K sampling according to time interval T and obtains sampled signal x N (), then can obtain the expression formula of its covariance matrix

According to the sampled data for getting, we can obtain gain function using weighted least-squares method, then by asking Take the minimum of a value of gain function to obtain the optimal solution of omnidirectional gain

In formula, W is weight coefficient, is unknown under normal circumstances, but as, in neritic environment, ambient noise is to signal The impact of process than larger, therefore can be initialized to which first with W=Ι, Zhi Houyong hereTo each Sampled data modify, then the gain function in above formula is rewritable is

Due to sensor array be by P sensor group into, then the gain function at each sensor can be designated as

In formula,w_n=ζ (W_n) and z=ζ_p(GAΓA^H)/g_p, wherein operator ζ_p() is represented from a matrix Middle its pth of selection is arranged as column vector, and removes p-th element in vector.

So optimal gain values Solve problems translate into P Linear least squares minimization problem of solution, then to gain function Minimize to try to achieve final result, we introduce formula z here_w=z ⊙ w_n, then the optimal gain values obtained are

Part III is that the DOA based on rarefaction representation estimates.

In order to increase the detectable number of sources of sensor array, the Fractional Lower Order Moments matrix after yield value optimization is entered Row vector quantization

In formula, vec () represents vectored calculations,The Kronecker products and Khatri-Rao of representing matrix are distinguished with ⊙ Product.

Then can obtain after the K all of Fractional Lower Order Moments matrix vector of frame in of sampling

In formula, y_kFor each sampling frame in vector quantization Fractional Lower Order Moments matrix, DefinitionIt may be seen that due to response matrix The dimension of matrix Y is changed into P², more than the physical dimensions P (P for generally being adopted>When 1), it is few therefore, it is possible to use Which is measured in the sensor of target source number, but detectable number of sources Q and array number P need to meet Q≤2 (P- 1)。

DOA estimations are carried out using the method based on rarefaction representation below with the vector quantization FLOM matrixes for obtaining.Obtaining After the Fractional Lower Order Moments matrix of vector quantization, redundant process is carried out to DOA angles using sparse representation method.Rarefaction representation The main thought of DOA estimation method is to use angle setAll possible echo signal incident direction is covered, And meeting N ＞ ＞ Q, N is the number of the space angle for dividing, and may make up an excessively complete redundant dictionary based on thisIts all possible positional information comprising source, and then take positional information of some methods from redundancy In extract needed for angle information.With the increase of dictionary redundancy, amount of calculation is also sharply increased, in order to reduce amount of calculation, The vector quantization Fractional Lower Order Moments matrix of K frame ins is processed using the method for singular value decomposition (SVD) herein.

In formula,WithRespectively left singular matrix and right singular matrix,For diagonal matrix.

Signal subspace can be obtained

In formula,Unit matrixs of the I for Q × Q ranks,

Therefore, it can be converted into signal source DOA Solve problems

In formula, | | | |₁With | | | |₂L is represented respectively₁Norm and l₂Norm, it is dilute that regularisation parameter β is used for balanced signal The impact of thin property and noise.

According to the limited fast umber of beats of array output data, we can only obtainEstimated resultAnd their approximate phases DengTherefore we are converted into the optimization problem of above formula

In formula, parameter lambda is used for balancing l₁Norm and l₂Norm, the previous item reflection mismatch of object function, latter are anti- Reflect openness requirement.

Above formula is a Second-order cone programming problem, and we can be solved using interior point method, and then according to nonzero element Position try to achieve the DOA estimates of signal source, as a result as shown in Figure 2.

Claims (6)

1. the Wave arrival direction estimating method in the case of a kind of steering vector mismatch based on rarefaction representation, is characterized in that, build first The receipt signal model of sensor array；Then heavy-tailed serious actual environment is made an uproar using the symmetrical generalized Gaussian distribution of synthesis circle Acoustical signal is simulated, and receives signal model using the docking of fractional lower-order Moment Methods and is processed, and to the signal model after process Present in unknown yield value be steering vector mismatch and the uncertain parameter yield value that generates is optimized；Finally, using base Signal source direction of arrival DOA estimations are carried out in sparse representation method, the signal source direction of arrival DOA estimates of optimum are obtained.

2. the Wave arrival direction estimating method in the case of steering vector mismatch as claimed in claim 1 based on rarefaction representation, which is special Levying is, builds the receipt signal model of sensor array first, it is assumed that the Q narrowband random signal source under the environment of far field is sent out The signal wave penetrated reach by P sensor group into uniform linear array ULA, it is separate between wherein Q signal source, and Noise is orthogonal with signal；The angle of arrival of q-th signal source is designated as θ_q, the reception signal at t, p-th sensor ForIn formula, a (θ_q) for q-th signal source s_qT () is in its direction of arrival θ_qUpper p-th sensing Steering vector at device, expression formula isD is the distance between sensor, and λ is signal wavelength, g_pFor reality Steering vector yield value in the environment of border at p-th sensor, s_qThe narrow band signal of (t) for random distribution in space, n_pT () is Spatial noise signal；

The receipt signal model of sensor array is tried to achieve according to the reception signal at each sensor：

X (t)=GAs (t)+n (t)

Wherein

X (t)=[x₁(t),x₂(t),...,x_P(t)]^T

G=diag [g₁,g₂,...,g_P]

A=[a (θ₁),a(θ₂),...,a(θ_Q)]

$a\left({\mathrm{\θ}}_q\right)={\[1,e^{-\frac{j2\mathrm{\π}d}{\mathrm{\λ}}sin\left({\mathrm{\θ}}_q\right)},...,e^{-\frac{j2\mathrm{\π}d(P-1)}{\mathrm{\λ}}sin\left({\mathrm{\θ}}_q\right)}\]}^T$

S (t)=[s₁(t),s₂(t),...,s_q(t)]^T

N (t)=[n₁(t),n₂(t,...,n_p(t))]^T

In formula, x (t) is sensor array receipt signal model, []^TArray transposition is represented, G is the gain at each sensor The diagonal matrix of value composition, A is the array manifold of all steering vectors composition, a (θ_q) it is each sensor leading at signal source q To vector, s (t) is signal source matrix, s_qT () is the transmission signal of q-th signal source, n (t) is the noise at all the sensors Signal, n_pT () is the noise signal at p-th sensor.

3. the Wave arrival direction estimating method in the case of steering vector mismatch as claimed in claim 1 based on rarefaction representation, which is special Levying is, simulates actual noise environment, probability of the noise signal with regard to stochastic variable w using the symmetrical generalized Gaussian distribution of synthesis circle Density function is expressed as：

$f\left(w\right)=\left(\frac{\mathrm{\&alpha;}\mathrm{\&Gamma;}(4/\mathrm{\&alpha;})}{2{\mathrm{\&pi;\&sigma;}}^2{\&lsqb;\mathrm{\&Gamma;}(2/\mathrm{\&alpha;})\&rsqb;}^2}\right)\exp (-\frac{1}{{\mathrm{\&sigma;}}^{\mathrm{\&alpha;}}}{\&lsqb;\frac{\mathrm{\&Gamma;}(4/\mathrm{\&alpha;})}{\mathrm{\&Gamma;}(2/\mathrm{\&alpha;})}\&rsqb;}^{\mathrm{\&alpha;}/2}|w{|}^{\mathrm{\&alpha;}}),0<\mathrm{\&alpha;}\&le;2$

In formula, Γ () represents gamma function, and α is Stable distritation coefficient, and σ is function variance, and when α=2, noise is presented Gaussian Profile.

4. the Wave arrival direction estimating method in the case of steering vector mismatch as claimed in claim 1 based on rarefaction representation, which is special Levying is, using being processed to which based on Fractional Lower Order Moments matrix method FLOM, comprises the following steps that：Homogenous linear sensor array Reception signal x (t) the Fractional Lower Order Moments matrix of row is the matrix of a P × P, and its (i, k) individual component is：

$R_{ik}=E\&lsqb;x_i\left(t\right)|x_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{x}_k^{*}\left(t\right)\&rsqb;,1<\mathrm{\&epsiv;}<\mathrm{\&alpha;}\&le;2$

In formula, E () represents desired value, x^*T () represents its conjugate function, orthogonal between each signal in x (t), and ε is little In the index parameters of α；

Signal Model in Time Domain is updated in the Fractional Lower Order Moments matrix expression that sensor array receives signal x (t)

$\begin{array}{c}{R}_{ik}=g_i\underset{q=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_i\left({\mathrm{\&theta;}}_q\right)E\left\{s_q\left(t\right)|g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\right\}+\\ E\left\{n_i\left(t\right)|g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\right\}\end{array}$

By R_ikIt is decomposed into two parts C_ikAnd D_ik, wherein

$\begin{array}{c}{C}_{ik}=g_i\underset{q=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_i\left({\mathrm{\&theta;}}_q\right)E\left\{s_q\left(t\right)|g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\right\}\\ =g_i\underset{q=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_i\left({\mathrm{\&theta;}}_q\right)E\left\{s_q\left(t\right)|g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{s}_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\mathrm{\&Sigma;}}}{s}_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\right\}{a}_k^{*}\left({\mathrm{\&theta;}}_q\right)\\ =g_i\underset{q=1}{\overset{Q}{\mathrm{\&Sigma;}}}{a}_i\left({\mathrm{\&theta;}}_q\right){\mathrm{\&Gamma;}}^q{a}_k^{*}\left({\mathrm{\&theta;}}_q\right)\end{array}$

$D_{ik}=E\left\{n_i\left(t\right)\right|g_k\underset{r=1}{\overset{Q}{\&Sigma;}}{a}_k\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\&Sigma;}}a\left({\mathrm{\&theta;}}_r\right)s_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\}={\mathrm{\&gamma;\&delta;}}_{ik}$

In formula,For kronecker delta, the subscript of i, k for Fractional Lower Order Moments matrix component, θ_q,θ_rRespectively q The angle of arrival of individual and r-th signal source, s_q(t),s_rT () is respectively the transmission signal of q and r-th signal source, and in above formula

${\mathrm{\&Gamma;}}^q=E\left\{s_q\left(t\right)\right|g_k\underset{r=1}{\overset{Q}{\&Sigma;}}{s}_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\&Sigma;}}{s}_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\}$

$\mathrm{\&gamma;}=E\left\{n_i\left(t\right)\right|g_k\underset{r=1}{\overset{Q}{\&Sigma;}}{s}_r\left(t\right)+n_k\left(t\right){|}^{\mathrm{\&epsiv;}-2}{\&lsqb;g_k\underset{r=1}{\overset{Q}{\&Sigma;}}{s}_r\left(t\right)+n_k\left(t\right)\&rsqb;}^{*}\}$

Therefore, the Fractional Lower Order Moments matrix of sensor array is expressed as

R=GA Γ A^H+γI

In formula,Unit matrixs of the I for P × P ranks, A are array stream Shape matrix, H represent conjugate transposition；Due to the yield value produced containing sensor displacement and external environmental interference in above formula G, carries out optimum estimation to which using weighted least-squares method, obtains its optimal solution：First by the sensor array for obtaining Reception signal x (t) carries out K sampling according to time interval T and obtains sampled signal x (n), then obtain the expression of its covariance matrix Formula：

$\hat{R}=\frac{1}{K}\underset{n=1}{\overset{K}{\&Sigma;}}x\left(n\right)x^H\left(n\right)$

According to the sampled data for getting, gain function is obtained using weighted least-squares method, then by asking for gain function Minimum of a value is obtaining the optimal solution of omnidirectional gain：

$\mathrm{\&eta;}\left(g\right)=\mathrm{argmin}\left|\right|W^H(R-\hat{R})W|{|}_F^2$

In formula, W is weight coefficient, logical which to be initialized first with W=Ι here, Zhi HouyongTo each Sampled data modify, then the gain function in above formula is rewritten as

${\mathrm{\&chi;}}_n\left(g\right)=\left|\right|W_n^H\left(R-\hat{R}\right)W_n|{|}_F^2$

Due to sensor array be by P sensor group into, then the gain function at p-th sensor is designated as：

${\mathrm{\&chi;}}_n^p\left(g_p\right)=\left|\right|w_n^H\left({\hat{r}}_p-{\mathrm{zg}}_p\right)w_n|{|}_2^2$

In formula,w_n=ζ (W_n) and z=ζ_p(GAΓA^H)/g_p, wherein operator ζ_p() represents and selects from a matrix Select its pth to arrange as column vector, and remove p-th element in vector；

So optimal gain values Solve problems translate into P Linear least squares minimization problem of solution, and then gain function is asked most Little value introduces formula z trying to achieve final result, here_w=z ⊙ w_n, then the optimal gain values obtained are

${\hat{g}}_p=\frac{z_w^H{\hat{r}}_p}{z_w^{H}z}.$

5. the Wave arrival direction estimating method in the case of steering vector mismatch as claimed in claim 1 based on rarefaction representation, which is special Levying is, carries out DOA estimations based on sparse representation method, comprises the concrete steps that：First the signal model after yield value optimization is carried out Vectorized process；After the Fractional Lower Order Moments matrix for obtaining vector quantization, LS-SVM sparseness is carried out to DOA angles, construct redundancy Angle set, and the method using singular value decomposition SVD obtains signal subspace, angle is solved and is converted into Second-order cone programming Problem.

6. the Wave arrival direction estimating method in the case of steering vector mismatch as claimed in claim 5 based on rarefaction representation, which is special Levying is, carries out vector quantization to the Fractional Lower Order Moments matrix after yield value optimization：

In formula, vec () represents vectored calculations,The Kronecker products and Khatri-Rao products of representing matrix are distinguished with ⊙, then K After the individual all of Fractional Lower Order Moments matrix vector of frame in of sampling

In formula, y_kFor each sampling frame in vector quantization Fractional Lower Order Moments matrix, DefinitionDue to response matrix：

The dimension of matrix Y is changed into P², the number of sources Q that measures and array number P need to meet Q≤2 (P-1)；Obtaining vector quantization After Fractional Lower Order Moments matrix, redundant process, rarefaction representation DOA estimation sides are carried out using sparse representation method to DOA angles The main thought of method is to use angle setAll possible echo signal incident direction is covered, and meets N>> Q, N are the numbers of the space angle for dividing, and may make up an excessively complete redundant dictionary based on thisIt All possible positional information comprising source, the angle information needed for taking some methods to extract from these positional informations, The vector quantization Fractional Lower Order Moments matrix of K frame ins is processed using the method for singular value decomposition SVD：

$\tilde{Y}={\mathrm{U\&Sigma;V}}^H$

In formula,WithRespectively left singular matrix and right singular matrix,For diagonal matrix；

Obtain signal subspace

In formula,Unit matrixs of the I for Q × Q ranks,

Therefore, signal source DOA Solve problems are converted into：

In formula, | | | |₁With | | | |₂L is represented respectively₁Norm and l₂Norm, regularisation parameter β be used for balanced signal it is openness and The impact of noise；

Can only obtain according to the limited fast umber of beats of array output dataEstimated resultAnd their approximately equals The optimization problem of above formula is converted into：

In formula, parameter lambda is used for balancing l₁Norm and l₂Norm, the previous item reflection mismatch of object function, latter reflection are dilute Thin property requires that above formula is a Second-order cone programming problem, is solved using interior point method, and then is asked according to the position of nonzero element Obtain the DOA estimates of signal source.