CN105223541A - Mutual coupling existing between elements error calibration method in the direction finding of broadband signal super-resolution - Google Patents

Abstract

Mutual coupling existing between elements error calibration method in the direction finding of broadband signal super-resolution, relates to the bearing calibration of the array error existed in the direction finding of broadband signal super-resolution.The present invention is in order to solve existing electromagnetic measurement or carry out by method of moment the problem that correction that the not high problem of the precision of mutual coupling effect correction and parametrization bearing calibration carry out mutual coupling effect is not suitable for broadband signal.The present invention utilizes the signal on each frequency to build corresponding majorized function, utilize the spatial domain of signal openness afterwards, respectively iteration optimization process is carried out to the function on each frequency by management loading method, finally fusion is carried out to the information on all frequencies and estimate direction of arrival of signal.The array error that the method can effectively realize when mutual coupling existing between elements error exists corrects, and utilizes multi-disc digital signal processor effectively to improve the travelling speed of algorithm.The present invention is applicable to the correction field of the array error existed in the direction finding of broadband signal super-resolution.

Description

Mutual coupling existing between elements error calibration method in the direction finding of broadband signal super-resolution

Technical field

The present invention relates to the bearing calibration of the array error existed in the direction finding of broadband signal super-resolution.

Background technology

Super-resolution direction finding is an important research content in Array Signal Processing, has apply more widely in fields such as radio monitoring, Internet of Things and electronic countermeasures.The direction-finding method of current majority is all to grasp premised on array manifold accurately.And in the middle of the direction-finding system of reality, often there is higher-order of oscillation device, often along with mutual coupling existing between elements error when causing direction finding to be estimated, this directly results in the penalty of a lot of super-resolution direction-finding methods, even loses efficacy, so be necessary to carry out correction process to it.

For the situation that only there is mutual coupling effect, way comparatively is early carried out electromagnetic measurement to mutual coupling or carries out electromagnetism calculating by method of moment to the mutual coupling coefficient, then the mutual coupling matrix obtained carried out to the compensation correction of mutual coupling effect.The correction accuracy of these methods often can not meet the requirement of Practical Project, and especially for high-frequency ground wave radar, electromagnetic environment is around complicated, is to measure or the precision that electromagnetism calculates all cannot meet the requirement of array calibration.Along with the development of correcting algorithm, use more method to be the special construction utilizing array mutual coupling matrix now, by the mutual coupling coefficient parametrization, the method that then operation parameter is estimated realizes the estimation of mutual coupling matrix and pair array corrects.Parameterized bearing calibration not only has very high correction accuracy, but also the change of the electromagnetic parameter along with environment and array element that can be real-time carrys out pair array mutual coupling corrects.Wherein more typically king's cloth is grand carries out fast algorithm that correct, that only need linear search with Wang Yongliang in the pair array mutual coupling effect that 2004 propose, this algorithm does not need loaded down with trivial details iterative process, but it is only applicable to narrow band signal, for the alignment technique of mutual coupling existing between elements error during the direction finding of broadband signal super-resolution, the document published is actually rare.

Summary of the invention

The present invention is in order to solve existing electromagnetic measurement or carry out by method of moment the problem that correction that the not high problem of the precision of mutual coupling effect correction and parametrization bearing calibration carry out mutual coupling effect is not suitable for broadband signal.

Mutual coupling existing between elements error calibration method in the direction finding of broadband signal super-resolution, comprises the steps:

Step 1: set up the array signal model containing mutual coupling existing between elements error:

When there is mutual coupling existing between elements error in the middle of array, array exports and can be expressed as

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(12)

Wherein, S (f _i) be signal s _k(t) signal phasor matrix after Fourier transform; N (f _i) be noise n _mt () noise vector matrix after Fourier transform, average is 0, and variance is μ ²(f _i);

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]，i＝1,2,…,J(13)

For there is mutual coupling existing between elements error time frequency point f _ion array manifold matrix; A'(f _i, α _k) for there is mutual coupling existing between elements error time frequency point f _ithe array steering vector of a upper kth signal; Then have

R'(f _i)＝E{X'(f _i)(X'(f _i)) ^H}，i＝1,2,…,J(14)

R'(f _i) for there is mutual coupling existing between elements error time frequency point f _ion Received signal strength covariance matrix;

A (f _i, α) and=[a (f _i, α ₁) ..., a (f _i, α _k) ..., a (f _i, α _k)] be ideally frequency f _ion array manifold matrix, a (f _i, α _k) be ideally frequency f _ithe array steering vector of a upper kth signal;

If W is (f _i) be frequency f _ion array perturbation matrix, represent the mutual coupling degree between array element, then when there is mutual coupling existing between elements error time frequency point f _ion array steering vector be

a'(f _i,α _k)＝W(f _i)a(f _i,α _k)(15)

Corresponding array manifold matrix can be expressed as

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]

(16)

＝W(f _i)A(f _i,α)

Wherein

Wherein, c _q(f _i) expression spacing is q, signal frequency is f _itime array element between the mutual coupling coefficient, q=1,2 ..., Q;

Then have

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)

＝A(f _i,α)S(f _i)+(W(f _i)-I _M)A(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(18)

＝A(f _i,α)S(f _i)+Λ(f _i)w(f _i)+N(f _i)

Wherein, Λ (f _i) be only relevant with an original signal parameter, have nothing to do with error; W (f _i)=[c ₁(f _i) ..., c _q(f _i)] ^trepresent frequency f _ion mutual coupling existing between elements perturbing vector; I _mit is the unit matrix of M × M dimension;

Step 2: the array signal parameter containing mutual coupling existing between elements error is estimated:

First be some discrete angle grids by search space partition l represents L the direction that signal may arrive; Thus frequency f can be drawn _ithe rarefaction representation of upper array manifold matrix

$A(f_i,\mathrm{\Ω})=\[a(f_i,{\stackrel{\‾}{\mathrm{\α}}}_1),...,a(f,{\stackrel{\‾}{\mathrm{\α}}}_l),...,a(f_i,{\stackrel{\‾}{\mathrm{\α}}}_L)\]$

Wherein, $a(f_i,{\stackrel{\‾}{\mathrm{\α}}}_l)={\[1,...,\exp (-jm\frac{2{\mathrm{\πf}}_{i}d}{c}sin{\stackrel{\‾}{\mathrm{\α}}}_l),...,\exp \left(-j\left(M-1\right)\frac{2{\mathrm{\πf}}_{i}d}{c}sin{\stackrel{\‾}{\mathrm{\α}}}_l\right)\]}^T$ For frequency f _ithe array steering vector of upper l sparse signal, can obtain accordingly and there is mutual coupling existing between elements error time frequency point f _ithe rarefaction representation of upper array manifold matrix

$A^{\′}(f_i,\mathrm{\Ω})=\[a^{\′}(f_i,{\stackrel{\‾}{\mathrm{\α}}}_1),...,a^{\′}(f_i,{\stackrel{\‾}{\mathrm{\α}}}_l),...,a^{\′}(f_i,{\stackrel{\‾}{\mathrm{\α}}}_L)\]=W\left(f_i\right)A(f_i,\mathrm{\Ω})$

Wherein, for there is mutual coupling existing between elements error time frequency point f _ithe array steering vector of upper l sparse signal, draws to there is mutual coupling existing between elements error time frequency point f _ithe rarefaction representation of upper array output signal

$\begin{array}{c}{\stackrel{\‾}{X}}^{\′}\left(f_i\right)A^{\′}\left(f_i,\mathrm{\Ω}\right)\stackrel{\‾}{S}\left(f_i\right)+N\left(f_i\right)\\ =A\left(f_i,\mathrm{\Ω}\right)\stackrel{\‾}{S}\left(f_i\right)+\stackrel{\‾}{\mathrm{\Λ}}\left(f_i\right)w\left(f_i\right)+N\left(f_i\right)\end{array},i=1,2,\mathrm{...},J---\left(19\right)$

Wherein, Λ (f _i) be only relevant with an original signal parameter, have nothing to do with error; for Λ (f _i) rarefaction representation;

covariance matrix be

${\stackrel{\‾}{R}}^{\′}\left(f_i\right)=E\left\{{\stackrel{\‾}{X}}^{\′}\left(f_i\right){\left({\stackrel{\‾}{X}}^{\′}\right(f_i\left)\right)}^H\right\},i=1,2,...,J---\left(20\right)$

In formula (19) $\stackrel{\‾}{S}\left(f_i\right)=\[\stackrel{\‾}{S}(f_i,1),...,\stackrel{\‾}{S}(f_i,kp),...,\stackrel{\‾}{S}(f_i,KP)\]$ For S (f _i) rarefaction representation,

Wherein, $\stackrel{\‾}{S}(f_i,kp)={\[{\stackrel{\‾}{S}}_1(f_i,kp),...{\stackrel{\‾}{S}}_l(f_i,kp),...,{\stackrel{\‾}{S}}_L(f_i,kp)\]}^T$ For S (f _i, kp) rarefaction representation, in only comprise K nonzero element, for in l element, and if only if time in element be entirely not zero and have l=1,2 ..., L, k=1,2 ..., K; So s (f can be regarded as _i) in add many 0 elements after the matrix that obtains;

If δ is (f _i)=[δ ₁(f _i) ..., δ _l(f _i) ..., δ _l(f _i)] ^tfor the variance of middle element, reflects the energy of signal, namely has

$\stackrel{\‾}{S}\left(f_i\right)~N(0,\mathrm{\Σ}(f_i))---\left(21\right)$

Wherein, Σ (f _i)=diag (δ (f _i)), namely obeying average is 0, and variance is δ (f _i) Gaussian distribution;

Due to s (f can be regarded as _i) in add many 0 elements after the vector that obtains, so δ (f _i) contain K nonzero element, and have K<<L, according to δ (f _i), in conjunction with w (f _i) and noise variance μ ²(f _i) estimate thus reconstruct original signal, error is corrected simultaneously;

Known according to formula (19), there is mutual coupling existing between elements error time frequency point f _ion the probability density of array output signal be

$\begin{array}{c}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)|\stackrel{\&OverBar;}{S}\left(f_i\right);w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)\\ ={\left|{\mathrm{\&pi;\&mu;}}^2\left(f_i\right)I_M\right|}^{-KP}\exp \left\{-{\mathrm{\&mu;}}^2\left(f_i\right)\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2\right\}\\ ={\left|{\mathrm{\&pi;\&mu;}}^2\left(f_i\right)I_M\right|}^{-KP}\exp \left\{-{\mathrm{\&mu;}}^2\left(f_i\right)\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)-\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)w\left(f_i\right)|{|}_2^2\right\}\end{array}---\left(22\right)$

Convolution (19), (21) and (22) can obtain

$\begin{array}{c}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right),w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)\\ =\&Integral;P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)|\stackrel{\&OverBar;}{S}\left(f_i\right);w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)P\left(\stackrel{\&OverBar;}{S}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right)\right)d\stackrel{\&OverBar;}{S}\left(f_i\right)\\ ={\left|\mathrm{\&pi;}\left({\mathrm{\&mu;}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\mathrm{\&Sigma;}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)\right|}^{-KP}\&times;\\ \exp \left\{-KP\&times;tr\left({\left({\mathrm{\&mu;}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\mathrm{\&Sigma;}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\right\}\end{array}---\left(23\right)$

Expectation maximization (ExpectationMaximization, EM) method is adopted to come w (f _i), μ ²(f _i) and δ _l(f _i) carry out iterative estimate, draw estimated value with corresponding obtains $\widehat{\mathrm{\&delta;}}\left(f_i\right)={\&lsqb;{\widehat{\mathrm{\&delta;}}}_1\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_l\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_L\left(f_i\right)\&rsqb;}^T$ And $\widehat{\mathrm{\&Sigma;}}\left(f_i\right)=diag\left(\widehat{\mathrm{\&delta;}}\right(f_i));$

Step 3: utilize with pair array error carries out correcting and solving direction of arrival of signal:

Make X be all frequency signals that in one section of observation time, array received arrives with form vector, because the signal of each frequency has statistical independence, therefore the joint probability density of each frequency Received signal strength is

$\begin{array}{c}P\left(X\right)=\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right);\widehat{\mathrm{\&delta;}}\left(f_i\right),\hat{w}\left(f_i\right),{\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)\right)\\ {\left|\mathrm{\&pi;}\right|}^{-J\&times;KP}\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}{\left|\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)\right|}^{-KP}\&times;\\ \exp \left\{-KP\&times;\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}tr\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\right\}\end{array}---\left(36\right)$

Taken the logarithm in formula (36) two ends

$\begin{array}{c}In\left(P\left(X\right)\right)\\ =-J\&times;KP\&times;In\mathrm{\&pi;}-KP\&times;\left(\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}In\left|{\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right|\right)-\\ KP\&times;\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}tr\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\end{array}---\left(37\right)$

Therefore formula (37) is made to maximize the estimated value can trying to achieve direction of arrival of signal k=1,2 ..., K, namely can pass through formula (38) and try to achieve;

$\frac{\&part;In\left(P\right(X))}{\&part;\mathrm{\&alpha;}}=0---\left(38\right)$

Have through derivation

${\widehat{\mathrm{\&alpha;}}}_k=\arg \underset{{\mathrm{\&alpha;}}_k}{\max }{\left|\mathrm{Re}\left\{\begin{array}{c}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H{\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right){\widehat{\mathrm{\&Sigma;}}}_{-k}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}\right)\&times;\\ \left[\begin{array}{c}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right){\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H{\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\\ -\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right){\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}{a}^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right){\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H\right)\end{array}\right]\&times;\\ \underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}\&times;\frac{\&part;a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)}{\&part;{\mathrm{\&alpha;}}_k}\right)\end{array}\right\}\right|}^{-1}---\left(39\right)$

Wherein, Re{} is for asking the real part of { }; Ω _-k, represent respectively from Ω and in remove a kth element wherein; K=1,2 ..., K;

According to expression formula try to achieve c ₁(f _i) ..., c _q(f _i), then try to achieve W (f according to formula (17) _i), utilize it to carry out array calibration and try to achieve a'(f _i, α _k) and A'(f _i, Ω _-k), then according to above parameter and formula (39), the estimated value of the direction of arrival of signal after array calibration can be obtained

The present invention has following beneficial effect:

The present invention proposes a kind of broadband signal super-resolution angle measurement error bearing calibration when there is mutual coupling existing between elements error, the signal on each frequency is utilized to build corresponding majorized function, utilize the spatial domain of signal openness afterwards, respectively iteration optimization process is carried out to the function on each frequency by management loading method, finally fusion is carried out to the information on all frequencies and estimate direction of arrival of signal.The present invention effectively can realize array error when mutual coupling existing between elements error exists and correct, and when signal to noise ratio (S/N ratio) is 10dB, each frequency samples fast umber of beats when being 40, and precision can reach 0.5 °/σ.

And method of the present invention can process with multi-disc digital signal processor, the travelling speed of algorithm effectively can be improved.

Accompanying drawing explanation

Fig. 1 is broadband signal super-resolution direction finding array signal model schematic;

Fig. 2 is broadband signal detection system installation drawing;

Fig. 3 is the broadband signal super-resolution direction-finding device figure of embodiment five;

Fig. 4 is the broadband signal super-resolution direction-finding device figure of embodiment six;

Fig. 5 is the broadband signal super-resolution direction-finding device figure of embodiment seven.

Embodiment

Embodiment one:

Mutual coupling existing between elements error calibration method in the direction finding of broadband signal super-resolution, comprises the steps:

Step 1: set up the array signal model containing mutual coupling existing between elements error:

When there is mutual coupling existing between elements error in the middle of array, array exports and can be expressed as

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(12)

Wherein, S (f _i) be signal s _k(t) signal phasor matrix after Fourier transform; N (f _i) be noise n _mt () noise vector matrix after Fourier transform, average is 0, and variance is μ ²(f _i);

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]，i＝1,2,…,J(13)

For there is mutual coupling existing between elements error time frequency point f _ion array manifold matrix; A'(f _i, α _k) for there is mutual coupling existing between elements error time frequency point f _ithe array steering vector of a upper kth signal; Then have

R'(f _i)＝E{X'(f _i)(X'(f _i)) ^H}，i＝1,2,…,J(14)

R'(f _i) for there is mutual coupling existing between elements error time frequency point f _ion Received signal strength covariance matrix;

A (f _i, α) and=[a (f _i, α ₁) ..., a (f _i, α _k) ..., a (f _i, α _k)] be ideally frequency f _ion array manifold matrix, a (f _i, α _k) be ideally frequency f _ithe array steering vector of a upper kth signal;

If W is (f _i) be frequency f _ion array perturbation matrix, represent the mutual coupling degree between array element, then when there is mutual coupling existing between elements error time frequency point f _ion array steering vector be

a'(f _i,α _k)＝W(f _i)a(f _i,α _k)(15)

Corresponding array manifold matrix can be expressed as

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]

(16)

＝W(f _i)A(f _i,α)

Wherein

Wherein, c _q(f _i) expression spacing is q, signal frequency is f _itime array element between the mutual coupling coefficient, q=1,2 ..., Q;

Then have

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)

＝A(f _i,α)S(f _i)+(W(f _i)-I _M)A(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(18)

＝A(f _i,α)S(f _i)+Λ(f _i)w(f _i)+N(f _i)

Wherein, Λ (f _i) be only relevant with an original signal parameter, have nothing to do with error; W (f _i)=[c ₁(f _i) ..., c _q(f _i)] ^trepresent frequency f _ion mutual coupling existing between elements perturbing vector; I _mit is the unit matrix of M × M dimension;

Step 2: the array signal parameter containing mutual coupling existing between elements error is estimated:

First be some discrete angle grids by search space partition l represents L the direction that signal may arrive; Thus frequency f can be drawn _ithe rarefaction representation of upper array manifold matrix

$A(f_i,\mathrm{\&Omega;})=\&lsqb;a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_1),...,a(f,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l),...,a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_L)\&rsqb;$

Wherein, $a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l)={\&lsqb;1,...,\exp (-jm\frac{2{\mathrm{\&pi;f}}_{i}d}{c}sin{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l),...,\exp \left(-j\left(M-1\right)\frac{2{\mathrm{\&pi;f}}_{i}d}{c}sin{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l\right)\&rsqb;}^T$ For frequency f _ithe array steering vector of upper l sparse signal, can obtain accordingly and there is mutual coupling existing between elements error time frequency point f _ithe rarefaction representation of upper array manifold matrix

$A^{\&prime;}(f_i,\mathrm{\&Omega;})=\&lsqb;a^{\&prime;}(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_1),...,a^{\&prime;}(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l),...,a^{\&prime;}(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_L)\&rsqb;=W\left(f_i\right)A(f_i,\mathrm{\&Omega;})$

Wherein, for there is mutual coupling existing between elements error time frequency point f _ithe array steering vector of upper l sparse signal, draws to there is mutual coupling existing between elements error time frequency point f _ithe rarefaction representation of upper array output signal

$\begin{array}{c}{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)+N\left(f_i\right)\\ =A\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)+\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)w\left(f_i\right)+N\left(f_i\right)\end{array},i=1,2,\mathrm{...},J---\left(19\right)$

Wherein, Λ (f _i) be only relevant with an original signal parameter, have nothing to do with error; for Λ (f _i) rarefaction representation;

covariance matrix be

${\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)=E\left\{{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right){\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i\left)\right)}^H\right\},i=1,2,...,J---\left(20\right)$

In formula (19) $\stackrel{\&OverBar;}{S}\left(f_i\right)=\&lsqb;\stackrel{\&OverBar;}{S}(f_i,1),...,\stackrel{\&OverBar;}{S}(f_i,kp),...,\stackrel{\&OverBar;}{S}(f_i,KP)\&rsqb;$ For S (f _i) rarefaction representation,

Wherein, $\stackrel{\&OverBar;}{S}(f_i,kp)={\&lsqb;{\stackrel{\&OverBar;}{S}}_1(f_i,kp),...{\stackrel{\&OverBar;}{S}}_l(f_i,kp),...,{\stackrel{\&OverBar;}{S}}_L(f_i,kp)\&rsqb;}^T$ For S (f _i, kp) rarefaction representation, in only comprise K nonzero element, for in l element, and if only if time in element be entirely not zero and have l=1,2 ..., L, k=1,2 ..., K; So s (f can be regarded as _i) in add many 0 elements after the matrix that obtains;

If δ is (f _i)=[δ ₁(f _i) ..., δ _l(f _i) ..., δ _l(f _i)] ^tfor the variance of middle element, reflects the energy of signal, namely has

$\stackrel{\&OverBar;}{S}\left(f_i\right)~N(0,\mathrm{\&Sigma;}(f_i))---\left(21\right)$

Wherein, Σ (f _i)=diag (δ (f _i)), namely obeying average is 0, and variance is δ (f _i) Gaussian distribution;

Due to s (f can be regarded as _i) in add many 0 elements after the vector that obtains, so δ (f _i) contain K nonzero element, and have K<<L, according to δ (f _i), in conjunction with w (f _i) and noise variance μ ²(f _i) estimate thus reconstruct original signal, error is corrected simultaneously;

Known according to formula (19), there is mutual coupling existing between elements error time frequency point f _ion the probability density of array output signal be

$\begin{array}{c}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)|\stackrel{\&OverBar;}{S}\left(f_i\right);w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)\\ ={\left|{\mathrm{\&pi;\&mu;}}^2\left(f_i\right)I_M\right|}^{-KP}\exp \left\{-{\mathrm{\&mu;}}^2\left(f_i\right)\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2\right\}\\ ={\left|{\mathrm{\&pi;\&mu;}}^2\left(f_i\right)I_M\right|}^{-KP}\exp \left\{-{\mathrm{\&mu;}}^2\left(f_i\right)\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)-\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)w\left(f_i\right)|{|}_2^2\right\}\end{array}---\left(22\right)$

Convolution (19), (21) and (22) can obtain

$\begin{array}{c}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right),w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)\\ =\&Integral;P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)|\stackrel{\&OverBar;}{S}\left(f_i\right);w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)P\left(\stackrel{\&OverBar;}{S}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right)\right)d\stackrel{\&OverBar;}{S}\left(f_i\right)\\ ={\left|\mathrm{\&pi;}\left({\mathrm{\&mu;}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\mathrm{\&Sigma;}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)\right|}^{-KP}\&times;\\ \exp \left\{-KP\&times;tr\left({\left({\mathrm{\&mu;}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\mathrm{\&Sigma;}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\right\}\end{array}---\left(23\right)$

Expectation maximization (ExpectationMaximization, EM) method is adopted to come w (f _i), μ ²(f _i) and δ _l(f _i) carry out iterative estimate, draw estimated value with corresponding obtains $\widehat{\mathrm{\&delta;}}\left(f_i\right)={\&lsqb;{\widehat{\mathrm{\&delta;}}}_1\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_l\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_L\left(f_i\right)\&rsqb;}^T$ And $\widehat{\mathrm{\&Sigma;}}\left(f_i\right)=diag\left(\widehat{\mathrm{\&delta;}}\right(f_i));$

Step 3: utilize with pair array error carries out correcting and solving direction of arrival of signal:

Make X be all frequency signals that in one section of observation time, array received arrives with form vector, because the signal of each frequency has statistical independence, therefore the joint probability density of each frequency Received signal strength is

$\begin{array}{c}P\left(X\right)=\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right);\widehat{\mathrm{\&delta;}}\left(f_i\right),\hat{w}\left(f_i\right),{\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)\right)\\ {\left|\mathrm{\&pi;}\right|}^{-J\&times;KP}\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}{\left|\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)\right|}^{-KP}\&times;\\ \exp \left\{-KP\&times;\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}tr\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\right\}\end{array}---\left(36\right)$

Taken the logarithm in formula (36) two ends

$\begin{array}{c}In\left(P\left(X\right)\right)\\ =-J\&times;KP\&times;In\mathrm{\&pi;}-KP\&times;\left(\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}In\left|{\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right|\right)-\\ KP\&times;\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}tr\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\end{array}---\left(37\right)$

Therefore formula (37) is made to maximize the estimated value can trying to achieve direction of arrival of signal k=1,2 ..., K, namely can pass through formula (38) and try to achieve;

$\frac{\&part;In\left(P\right(X))}{\&part;\mathrm{\&alpha;}}=0---\left(38\right)$

Have through derivation

${\widehat{\mathrm{\&alpha;}}}_k=\arg \underset{{\mathrm{\&alpha;}}_k}{\max }{\left|\mathrm{Re}\left\{\begin{array}{c}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H{\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right){\widehat{\mathrm{\&Sigma;}}}_{-k}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}\right)\&times;\\ \left[\begin{array}{c}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right){\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H{\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\\ -\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right){\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}{a}^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right){\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H\right)\end{array}\right]\&times;\\ \underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}\&times;\frac{\&part;a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)}{\&part;{\mathrm{\&alpha;}}_k}\right)\end{array}\right\}\right|}^{-1}---\left(39\right)$

Wherein, Re{} is for asking the real part of { }; Ω _-k, represent respectively from Ω and in remove a kth element wherein; K=1,2 ..., K;

According to expression formula try to achieve c ₁(f _i) ..., c _q(f _i), then try to achieve W (f according to formula (17) _i), utilize it to carry out array calibration and try to achieve a'(f _i, α _k) and A'(f _i, Ω _-k), then according to above parameter and formula (39), the estimated value of the direction of arrival of signal after array calibration can be obtained

Embodiment two:

The concrete steps setting up the array signal model containing mutual coupling existing between elements error described in present embodiment step 1 are as follows:

Step 1.1: set up ideal array signal model:

As shown in Figure 1, K far field broadband signal s is provided with _k(t), k=1,2 ..., K, incide on the broadband uniform linear array of M omnidirectional's array element composition, arrival direction is α=[α ₁..., α _k..., α _k], array element distance is d; Far field broadband signal s _kt (), is called for short broadband signal s _k(t);

Using the 1st array element as phase reference point, in the ideal case, the output of m array element is expressed as

$x_m\left(t\right)=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}{s}_k(t-{\mathrm{\&tau;}}_m({\mathrm{\&alpha;}}_k))+n_m\left(t\right),m=1,2,...,M---\left(1\right)$

Wherein, represent a kth broadband signal s _kt () arrives m array element arrives phase reference point time delay relative to it, c is electromagnetic wave velocity of propagation in a vacuum, n _mt () is the white Gaussian noise that m array element receives;

Suppose that the frequency range of broadband signal is for [f _low, f _high], utilize discrete Fourier transformation broadband signal to be divided into J frequency, they separated through narrow band filter group, then i-th group of filter array output signal is expressed as

X(f _i)＝A(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(2)

Wherein, f _low≤ f _i≤ f _high, i=1,2 ..., J;

Suppose at each frequency f _ion carried out KP time sampling, X (f _i) matrix representation be

X(f _i)＝[X(f _i,1),…,X(f _i,kp),…,X(f _i,KP)]，i＝1,2,…,J(3)

Wherein, X (f _i, kp) and be X (f _i) kth p secondary data sampling matrix,

X(f _i,kp)＝[X ₁(f _i,kp),…,X _m(f _i,kp),…,X _M(f _i,kp)] ^T，i＝1,2,…,J，(4)

X _m(f _i, kp) and be that m array element is at frequency f _ion the kth p secondary data sampled value that obtains;

A (f _i, α) and be ideally frequency f _ion array manifold matrix,

A(f _i,α)＝[a(f _i,α ₁),…,a(f _i,α _k),…,a(f _i,α _K)]，i＝1,2,…,J，(5)

A (f _i, α _k) be ideally frequency f _ithe array steering vector of a upper kth signal,

Wherein, it is the phase place of a kth signal; J is plural number mark;

S(f _i)＝[S(f _i,1),…,S(f _i,kp),…,S(f _i,KP)]，i＝1,2,…,J，(8)

For signal s _k(t) signal phasor matrix after Fourier transform, k=1,2 ..., K;

Wherein, S (f _i, kp) and be S (f _i) kth p signal sampling matrix,

S(f _i,kp)＝[S ₁(f _i,kp),…S _k(f _i,kp),…,S _K(f _i,kp)] ^Ti＝1,2,…,J(9)

S _k(f _i, kp) and for a kth signal is at frequency f _ion kth p signal sampling value of obtaining;

N(f _i)＝[N(f _i,1),…,N(f _i,kp),…,N(f _i,KP)]i＝1,2,…,J(10)

For noise n _mt () noise vector matrix after Fourier transform, average is 0, and variance is μ ²(f _i); M=1,2 ..., M;

Wherein, N (f _i, kp) and be N (f _i) kth p noise samples matrix,

N(f _i,kp)＝[N ₁(f _i,kp),…,N _m(f _i,kp),…,N _M(f _i,kp)] ^Ti＝1,2,…,J(11)

N _m(f _i, kp) and be that m array element is at frequency f _ion kth p noise samples value of obtaining;

Step 1.2: set up the array signal model containing mutual coupling existing between elements error on desirable array signal model basis:

When there is mutual coupling existing between elements error in the middle of array, frequency f _ion array export can be expressed as

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(12)

Wherein

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]，i＝1,2,…,J(13)

For there is mutual coupling existing between elements error time frequency point f _ion array manifold matrix, a'(f _i, α _k) be corresponding array steering vector;

Then have

R'(f _i)＝E{X'(f _i)(X'(f _i)) ^H}，i＝1,2,…,J(14)

R'(f _i) for there is mutual coupling existing between elements error time frequency point f _ion Received signal strength covariance matrix;

If W is (f _i) be array perturbation matrix, represent frequency f _imutual coupling degree between upper array element, then the array steering vector when there is mutual coupling existing between elements error needs to be modified to

a'(f _i,α _k)＝W(f _i)a(f _i,α _k)(15)

Corresponding array manifold matrix can be expressed as

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]

(16)

＝W(f _i)A(f _i,α)

Wherein, c _q(f _i) expression spacing is q, signal frequency is f _itime array element between the mutual coupling coefficient, q=1,2 ..., Q,

Then have

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)

＝A(f _i,α)S(f _i)+(W(f _i)-I _M)A(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(18)

＝A(f _i,α)S(f _i)+Λ(f _i)w(f _i)+N(f _i)

Wherein, Λ (f _i) be only relevant with an original signal parameter, have nothing to do with error; W (f _i)=[c ₁(f _i) ..., c _q(f _i)] ^trepresent frequency f _ion mutual coupling existing between elements perturbing vector; I _mit is the unit matrix of M × M dimension.

Other step is identical with embodiment one with parameter.

Embodiment three:

Perturbation matrix W (f described in present embodiment step 1.2 _i) to solve concrete steps as follows:

U, v is made to be respectively the index value of the row and column of perturbation matrix, then perturbation matrix W (f _i) in element can use c _uv(f _i) represent, represent frequency f _ithe mutual coupling coefficient between upper u array element and v array element,

The array element mutual coupling of even linear array has following character:

(1), when the distance between array element is greater than mutual coupling degree of freedom Q, between array element, not can think to there is mutual coupling;

The mutual coupling coefficient between (2) u array element and v array element is equal with the mutual coupling coefficient between v array element and u array element;

(3) the mutual coupling coefficient corresponding to the array element that spacing is equal is equal;

According to the array element mutual coupling character of even linear array, adopt following expression formula to describe perturbation matrix W (f when there is mutual coupling existing between elements error _i):

Other step is identical with embodiment two with parameter.

Embodiment four:

Employing expectation maximization method described in present embodiment step 2 is come w (f _i), μ ²(f _i) and δ _l(f _i) carry out iterative estimate concrete steps as follows:

In E-step step in expectation maximization method, first right distribution function calculate

Wherein operational symbol <> represents that solving condition is expected;

In M-step step in expectation maximization method, ask for distribution function respectively to the derivative of each unknown parameter, namely right get extreme value to solve each unknown parameter;

$\begin{array}{c}\frac{\&part;F\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right),\stackrel{\&OverBar;}{S}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right),w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)}{\&part;w\left(f_i\right)}\\ =-2{\mathrm{\&mu;}}^{-2}\left(f_i\right)\&lsqb;<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)>w\left(f_i\right)-<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)\right)>\&rsqb;\end{array}---\left(25\right)$

$\begin{array}{c}\frac{\&part;F\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right),\stackrel{\&OverBar;}{S}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right),w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)}{\&part;{\mathrm{\&mu;}}^2\left(f_i\right)}\\ =-\frac{M\&times;KP}{{\mathrm{\&mu;}}^2\left(f_i\right)}+\frac{1}{{\left({\mathrm{\&mu;}}^2\left(f_i\right)\right)}^2}<\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2>\end{array}---\left(26\right)$

$\begin{array}{c}\frac{\&part;F\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right),\stackrel{\&OverBar;}{S}\left(f_i\right);\mathrm{\&delta;}\left(f_i\right),w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right)\right)}{\&part;{\mathrm{\&delta;}}_l\left(f_i\right)}\\ =-\frac{KP}{{\mathrm{\&delta;}}_l\left(f_i\right)}+\frac{1}{{\mathrm{\&delta;}}_l^2\left(f_i\right)}<\underset{kp=1}{\overset{KP}{\mathrm{\&Sigma;}}}{\left|{\stackrel{\&OverBar;}{S}}_l\left(f_i,kp\right)\right|}^2>\end{array}---\left(27\right)$

Make above derivative be 0 respectively, the estimated value of each unknown parameter during the p time iteration can be tried to achieve

$w^{\left(p\right)}\left(f_i\right)=<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right){>}^{-1}<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)\right)>---\left(28\right)$

${\left({\mathrm{\&mu;}}^2\left(f_i\right)\right)}^{\left(p\right)}=\frac{1}{M\&times;KP}<\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-{\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2>---\left(29\right)$

${\mathrm{\&delta;}}_l^{\left(p\right)}\left(f_i\right)=\frac{1}{KP}<\underset{kp=1}{\overset{KP}{\mathrm{\&Sigma;}}}{\left|{\stackrel{\&OverBar;}{S}}_l\left(f_i,kp\right)\right|}^2>---\left(30\right)$

Wherein (p) represents iterations, in formula (28)

$\begin{array}{c}<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right){>}_{r1,r2}\\ =tr\&lsqb;{\mathrm{\&Psi;}}_{r1}^H\left(f_i\right){\mathrm{\&Psi;}}_{r2}\left(f_i\right)A\left(f_i,\mathrm{\&Omega;}\right)\left(O\left(f_i\right)O^H\left(f_i\right)+KP\&times;\mathrm{\&Xi;}\left(f_i\right)\right)A^H\left(f_i,\mathrm{\&Omega;}\right)\&rsqb;\end{array}---\left(31\right)$

For matrix r1 is capable, the element of r2 row, and wherein mark computing is asked in tr [] expression;

In formula (31)

O(f _i)＝Σ(f _i)(A'(f _i,Ω)) ^H(μ ²(f _i)I _M+A'(f _i,Ω)Σ(f _i)(A'(f _i,Ω)) ^H) ^-1X(f _i)(32)

For intermediate variable;

Ξ(f _i)

(33)

＝Σ(f _i)-Σ(f _i)(A'(f _i,Ω)) ^H(μ ²(f _i)I _M+A'(f _i,Ω)Σ(f _i)(A'(f _i,Ω)) ^H) ^-1A'(f _i,Ω)Σ(f _i)

For intermediate variable;

In formula (28)

$\begin{array}{c}<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A\left(f_i,\mathrm{\&Omega;}\right)\stackrel{\&OverBar;}{S}\left(f_i\right)\right)>\\ =tr\&lsqb;{\mathrm{\&Psi;}}_r^H\left(f_i\right)X\left(f_i\right)O^H\left(f_i\right)A^H\left(f_i,\mathrm{\&Omega;}\right)\&rsqb;-\\ tr\&lsqb;{\mathrm{\&Psi;}}_r^H\left(f_i\right)A\left(f_i,\mathrm{\&Omega;}\right)\left(O\left(f_i\right)O^H\left(f_i\right)+KP\&times;\mathrm{\&Xi;}\left(f_i\right)\right)A^H\left(f_i,\mathrm{\&Omega;}\right)\&rsqb;\end{array}---\left(34\right)$

In formula (34), Ψ _r(f _i) be intermediate variable, be the matrix of M × M dimension, the element only on the ± r diagonal line is 1 entirely, and all the other elements are 0 entirely;

In formula (29)

$\begin{array}{c}<\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-{\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2>\\ =\left|\right|X\left(f_i\right)-{\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}O\left(f_i\right)|{|}_F^2+KP\&times;tr\left({\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}\mathrm{\&Xi;}\left(f_i\right){\left({\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}\right)}^H\right)\end{array}---\left(35\right)$

W (f is calculated owing to directly utilizing formula (28) ~ (29) _i) and μ ²(f _i) more complicated, therefore formula (31) ~ (35) can be substituted into peer-to-peer in formula (28) ~ (29) and carry out abbreviation and to w (f _i) and μ ²(f _i) solve;

When after the some steps of iteration, w (f _i), μ ²(f _i) and δ _l(f _i) change of three amount estimated values is tending towards 0, now can think that they are restrained, then can draw last estimated value with correspondence obtains $\widehat{\mathrm{\&delta;}}\left(f_i\right)={\&lsqb;{\widehat{\mathrm{\&delta;}}}_1\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_l\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_L\left(f_i\right)\&rsqb;}^T$ And $\widehat{\mathrm{\&Sigma;}}\left(f_i\right)=diag\left(\widehat{\mathrm{\&delta;}}\right(f_i)).$

Other step is identical with embodiment three with parameter.

Embodiment five: illustrate present embodiment with reference to Fig. 2 and Fig. 3,

The method that present embodiment detects for the broadband signal detection system and realization realizing method described in embodiment one to four,

As shown in Figure 2, broadband signal detection system comprises: broadband uniform linear array 1, hyperchannel wideband digital receiver 2 and broadband signal super-resolution direction-finding device 3;

As shown in Figure 3, broadband signal super-resolution direction-finding device 3 comprises 6 digital signal processors, i.e. DSP, adopts Fast Sequential input/output port, i.e. SRIO mouth, and composition multicomputer system realizes parallel processing.Wherein, DSP3-1 is main DSP, DSP3-2 ~ DSP3-6 is from DSP; Broadband signal super-resolution direction-finding device 3 also comprises CPLD3-7, PROM3-8, FLASH3-9, SRAM3-10, JTAG3-11, power supply, crystal oscillator and reset.

Digital signal processor adopts the TMS320C6678 of TexasInstruments (TI) company, adopt 6 processor parallel processings, 6 DSP are connected by SRIO mouth, after powering on, first program loads to CPLD3-7 by PROM3-8, program also loads to these 6 pieces of DSP (3-1 ~ 3-6) by FLASH3-9, main DSP3-1 starts the observation data receiving J the frequency that hyperchannel wideband digital receiver 2 transmits afterwards, they are divided into W group, suppose J=30, W=6, then every sheet DSP can process the observation data of U=30/6=5 frequency, other observation data being responsible for process from DSP (3-2 ~ 3-6) is passed to them by SRIO mouth by main DSP3-1, each DSP (3-1 ~ 3-6) solves according to the step of above theory deduction afterwards, main DSP3-1 is given respective error estimate by SRIO oral instructions for 5 afterwards from DSP (3-2 ~ 3-6), main DSP3-1 recycles these results, convolution (39) draws direction of arrival degree.Wherein SRAM3-10 is responsible for storing data, and JTAG3-11 is responsible for debugging DSP (3-1 ~ 3-6), and power supply is responsible for bulk supply, and crystal oscillator is responsible for providing clock, resets and is responsible for providing reset signal.

Embodiment six: illustrate present embodiment with reference to Fig. 2 and Fig. 4,

The method that present embodiment detects for the broadband signal detection system and realization realizing method described in embodiment one to four,

As shown in Figure 2, broadband signal detection system comprises: broadband uniform linear array 1, hyperchannel wideband digital receiver 2 and broadband signal super-resolution direction-finding device 3;

As shown in Figure 4, broadband signal super-resolution direction-finding device 3 comprises 6 digital signal processors, i.e. DSP, adopts shared bus close coupled system composition multicomputer system to realize parallel processing.Wherein, DSP3-1 is main DSP, DSP3-2 ~ DSP3-6 is from DSP; Broadband signal super-resolution direction-finding device 3 also comprises CPLD3-7, PROM3-8, FLASH3-9, SRAM3-10, JTAG3-11, power supply, crystal oscillator and reset.

Digital signal processor adopts the ADSP-TS201S of AnalogDeviceInstruments (ADI) company, adopt 6 DSP parallel processings, 6 DSP are connected by shared bus close coupled system, after powering on, first program loads and is configured DSP (3-1 ~ 3-6) to CPLD3-7 by PROM3-8, program loads to these 6 pieces of DSP (3-1 ~ 3-6) by FLASH3-9 afterwards, main DSP3-1 starts the observation data receiving J the frequency that hyperchannel wideband digital receiver 2 transmits, they are divided into W group, suppose J=30, W=6, then every sheet DSP can process the observation data of U=30/6=5 frequency, other observation data being responsible for process from DSP (3-2 ~ 3-6) is passed to them by bus by main DSP3-1, each DSP (3-1 ~ 3-6) solves according to the step of above theory deduction afterwards, from DSP (3-2 ~ 3-6), respective error estimate is passed to main DSP3-1 by bus for 5 afterwards, main DSP3-1 recycles these results, convolution (39) draws direction of arrival degree.Wherein SRAM3-10 is responsible for storing data, and JTAG3-11 is responsible for debugging DSP (3-1 ~ 3-6), and power supply is responsible for bulk supply, and crystal oscillator is responsible for providing clock, resets and is responsible for providing reset signal.

Embodiment seven: illustrate present embodiment with reference to Fig. 2 and Fig. 5,

The method that present embodiment detects for the broadband signal detection system and realization realizing method described in embodiment one to four,

As shown in Figure 2, broadband signal detection system comprises: broadband uniform linear array 1, hyperchannel wideband digital receiver 2 and broadband signal super-resolution direction-finding device 3;

As shown in Figure 5, broadband signal super-resolution direction-finding device 3 comprises 6 digital signal processors, i.e. DSP, adopts link port cascade loose coupling mode to form multicomputer system and realizes parallel processing.Wherein, DSP3-1 is main DSP, DSP3-2 ~ DSP3-6 is from DSP; Broadband signal super-resolution direction-finding device 3 also comprises CPLD3-7, PROM3-8, FLASH3-9, SRAM3-10, JTAG3-11, power supply, crystal oscillator and reset.

Digital signal processor adopts the ADSP-TS201S of AnalogDeviceInstruments (ADI) company, adopt 6 processor parallel processings, 6 DSP are connected by link port cascade loose coupling mode, after powering on, first program loads to CPLD3-7 by PROM3-8, the program of these 6 DSP loads to main DSP3-1 by FLASH3-9, other is passed to them from the program of DSP (3-2 ~ 3-6) by link port one-level one-level by main DSP3-1 more successively, main DSP3-1 starts the observation data receiving J the frequency that hyperchannel wideband digital receiver 2 transmits afterwards, they are divided into W group, suppose J=30, W=6, then every sheet DSP can process the observation data of U=30/6=5 frequency, the observation data one-level one-level that other DSP (3-2 ~ 3-6) is responsible for process is successively passed to them by link port by main DSP3-1 again, each DSP (3-1 ~ 3-6) solves according to the step of above theory deduction afterwards, respective error estimate is successively uploaded to main DSP3-1 by link port one-level one-level from DSP (3-2 ~ 3-6) by 5 afterwards, main DSP3-1 recycles these results, convolution (39) draws direction of arrival degree.Wherein SRAM3-10 is responsible for storing data, and JTAG3-11 is responsible for debugging DSP (3-1 ~ 3-6), and power supply is responsible for bulk supply, and crystal oscillator is responsible for providing clock, resets and is responsible for providing reset signal.

Claims (4)

1. the mutual coupling existing between elements error calibration method in the direction finding of broadband signal super-resolution, is characterized in that comprising the steps:

Step 1: set up the array signal model containing mutual coupling existing between elements error:

When there is mutual coupling existing between elements error in the middle of array, array exports and is expressed as

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(12)

Wherein, S (f _i) be signal s _k(t) signal phasor matrix after Fourier transform; N (f _i) be noise n _mt () noise vector matrix after Fourier transform, average is 0, and variance is μ ²(f _i);

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]，i＝1,2,…,J(13)

For there is mutual coupling existing between elements error time frequency point f _ion array manifold matrix; A'(f _i, α _k) for there is mutual coupling existing between elements error time frequency point f _ithe array steering vector of a upper kth signal; Then have

R'(f _i)＝E{X'(f _i)(X'(f _i)) ^H}，i＝1,2,…,J(14)

R'(f _i) for there is mutual coupling existing between elements error time frequency point f _ion Received signal strength covariance matrix;

A (f _i, α) and=[a (f _i, α ₁) ..., a (f _i, α _k) ..., a (f _i, α _k)] be ideally frequency f _ion array manifold matrix, a (f _i, α _k) be ideally frequency f _ithe array steering vector of a upper kth signal;

If W is (f _i) be frequency f _ion array perturbation matrix, represent the mutual coupling degree between array element, then when there is mutual coupling existing between elements error time frequency point f _ion array steering vector be

a'(f _i,α _k)＝W(f _i)a(f _i,α _k)(15)

Corresponding array manifold matrix representation is

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]

(16)

＝W(f _i)A(f _i,α)

Wherein

Wherein, c _q(f _i) expression spacing is q, signal frequency is f _itime array element between the mutual coupling coefficient, q=1,2 ..., Q;

Then have

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)

，i＝1,2,…,J(18)

＝A(f _i,α)S(f _i)+Λ(f _i)w(f _i)+N(f _i)

Wherein, Λ (f _i) be only relevant with an original signal parameter; W (f _i)=[c ₁(f _i) ..., c _q(f _i)] ^trepresent frequency f _ion mutual coupling existing between elements perturbing vector; I _mit is the unit matrix of M × M dimension;

Step 2: the array signal parameter containing mutual coupling existing between elements error is estimated:

First be some discrete angle grids by search space partition l represents L the direction that signal may arrive; Thus draw frequency f _ithe rarefaction representation of upper array manifold matrix

$A(f_i,\mathrm{\&Omega;})=\&lsqb;a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_1),...,a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l),...,a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_L)\&rsqb;$

Wherein, $a(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l)={\&lsqb;1,...,\exp (-jm\frac{2{\mathrm{\&pi;f}}_{i}d}{c}\sin {\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l),...,\exp (-j(\mathrm{M-1}\left)\frac{2{\mathrm{\&pi;f}}_{i}d}{c}\sin {\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l\right)\&rsqb;}^T$ For frequency f _ithe array steering vector of upper l sparse signal, there is mutual coupling existing between elements error time frequency point f in corresponding acquisition _ithe rarefaction representation of upper array manifold matrix

$A^{\&prime;}(f_i,\mathrm{\&Omega;})=\&lsqb;a^{\&prime;}(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_1),...,a^{\&prime;}(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_l),...,a^{\&prime;}(f_i,{\stackrel{\&OverBar;}{\mathrm{\&alpha;}}}_L)\&rsqb;=W\left(f_i\right)A(f_i,\mathrm{\&Omega;})$

Wherein, for there is mutual coupling existing between elements error time frequency point f _ithe array steering vector of upper l sparse signal, draws to there is mutual coupling existing between elements error time frequency point f _ithe rarefaction representation of upper array output signal

$\begin{array}{c}{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)=A^{\&prime;}(f_i,\mathrm{\&Omega;})\stackrel{\&OverBar;}{S}\left(f_i\right)+N\left(f_i\right)\\ =A(f_i,\mathrm{\&Omega;})\stackrel{\&OverBar;}{S}\left(f_i\right)+\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)w\left(f_i\right)+N\left(f_i\right)\end{array},i=1,2,...,J---\left(19\right)$

Wherein, Λ (f _i) be only relevant with an original signal parameter; for Λ (f _i) rarefaction representation;

covariance matrix be

${\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)=E\left\{{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right){\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i\left)\right)}^H\right\},i=1,2,...,J---\left(20\right)$

In formula (19) $\stackrel{\&OverBar;}{S}\left(f_i\right)=\&lsqb;\stackrel{\&OverBar;}{S}(f_i,1),...,\stackrel{\&OverBar;}{S}(f_i,kp),...,\stackrel{\&OverBar;}{S}(f_i,KP)\&rsqb;$ For S (f _i) rarefaction representation,

Wherein, $\stackrel{\&OverBar;}{S}(f_i,kp)={\&lsqb;{\stackrel{\&OverBar;}{S}}_1(f_i,kp),...{\stackrel{\&OverBar;}{S}}_l(f_i,kp),...,{\stackrel{\&OverBar;}{S}}_L(f_i,kp)\&rsqb;}^T$ For S (f _i, kp) rarefaction representation, in only comprise K nonzero element, for in l element, and if only if time in element be entirely not zero and have l=1,2 ..., L, k=1,2 ..., K; So s (f _i) in add many 0 elements after the matrix that obtains;

If δ is (f _i)=[δ ₁(f _i) ..., δ _l(f _i) ..., δ _l(f _i)] ^tfor the variance of middle element, reflects the energy of signal, namely has

$\stackrel{\&OverBar;}{S}\left(f_i\right)~N(0,\mathrm{\&Sigma;}(f_i\left)\right)---\left(21\right)$

Wherein, Σ (f _i)=diag (δ (f _i)), namely obeying average is 0, and variance is δ (f _i) Gaussian distribution;

According to formula (19), there is mutual coupling existing between elements error time frequency point f _ion the probability density of array output signal be

$\begin{array}{c}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i\left)\right|\stackrel{\&OverBar;}{S}\left(f_i\right);w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right))\\ =\left|{\mathrm{\&pi;\&mu;}}^2\left(f_i\right)I_M{|}^{-KP}\exp \right\{-{\mathrm{\&mu;}}^2\left(f_i\right)\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A(f_i,\mathrm{\&Omega;})\stackrel{\&OverBar;}{S}\left(f_i\right)-\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)w\left(f_i\right)\left|{|}_2^2\right\}\end{array}---\left(22\right)$

Convolution (19), (21) and (22) obtain

$\begin{array}{c}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i);|\mathrm{\&delta;}\left(f_i\right);w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right))\\ =|\mathrm{\&pi;}\left({\mathrm{\&mu;}}^2\right(f_i)I_M+A^{\&prime;}(f_i,\mathrm{\&Omega;}\left)\mathrm{\&Sigma;}\right(f_i\left){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right){|}^{-KP}\&times;\\ \exp \{-KP\&times;tr\left({\left({\mathrm{\&mu;}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\mathrm{\&Sigma;}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\right(f_i\left)\right)\}\end{array}---\left(23\right)$

Expectation maximization method is adopted to come w (f _i), μ ²(f _i) and δ _l(f _i) carry out iterative estimate, draw estimated value with corresponding obtains $\widehat{\mathrm{\&delta;}}\left(f_i\right)={\&lsqb;{\widehat{\mathrm{\&delta;}}}_1\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_l\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_L\left(f_i\right)\&rsqb;}^T$ And $\widehat{\mathrm{\&Sigma;}}\left(f_i\right)=diag\left(\widehat{\mathrm{\&delta;}}\right(f_i\left)\right);$

Step 3: utilize with pair array error carries out correcting and solving direction of arrival of signal:

Make X be all frequency signals that in one section of observation time, array received arrives with form vector, because the signal of each frequency has statistical independence, therefore the joint probability density of each frequency Received signal strength is

$\begin{array}{c}P\left(X\right)=\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}P\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i);\widehat{\mathrm{\&delta;}}(f_i),\hat{w}(f_i),{\widehat{\mathrm{\&mu;}}}^2(f_i\left)\right)\\ =\left|\mathrm{\&pi;}{|}^{-J\&times;KP}\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}\right|\left({\widehat{\mathrm{\&mu;}}}^2\right(f_i)I_M+A^{\&prime;}(f_i,\mathrm{\&Omega;}\left)\widehat{\mathrm{\&Sigma;}}\right(f_i\left){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right){|}^{-KP}\&times;\\ \exp \{-KP\&times;\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}tr\left({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right)\right)\}\end{array}---\left(36\right)$

Taken the logarithm in formula (36) two ends

$\begin{array}{c}In\left(P\right(X\left)\right)\\ =-J\&times;JP\&times;In\mathrm{\&pi;}-KP\&times;\left(\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}In\right|{\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\left|\right)-\\ KP\&times;\underset{i=1}{\overset{J}{\mathrm{\&Pi;}}}tr{\left({\widehat{\mathrm{\&mu;}}}^2\right(f_i)I_M+A^{\&prime;}(f_i,\mathrm{\&Omega;}\left)\widehat{\mathrm{\&Sigma;}}\right(f_i\left){\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right))\end{array}---\left(37\right)$

Formula (37) is made to maximize the estimated value of namely trying to achieve direction of arrival of signal k=1,2 ..., K, is namely tried to achieve by formula (38);

$\frac{\&part;In\left(P\right(X))}{\&part;\mathrm{\&alpha;}}=0---\left(38\right)$

Have through derivation

${\widehat{\mathrm{\&alpha;}}}_k=\arg \underset{{\mathrm{\&alpha;}}_k}{\min }|\mathrm{Re}\left\{\begin{array}{c}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H{\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right){\widehat{\mathrm{\&Sigma;}}}_{-k}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}\right)\&times;\\ \left[\begin{array}{c}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left(a^{\&prime;}\right(f_i,{\mathrm{\&alpha;}}_k\left){\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H{\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}{\stackrel{\&OverBar;}{R}}^{\&prime;}\right(f_i\left)\right)\\ -\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left({\stackrel{\&OverBar;}{R}}^{\&prime;}\left(f_i\right){\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}{a}^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right){\left(a^{\&prime;}\left(f_i,{\mathrm{\&alpha;}}_k\right)\right)}^H\right)\end{array}\right]\&times;\\ \underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}({\left({\widehat{\mathrm{\&mu;}}}^2\left(f_i\right)I_M+A\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\widehat{\mathrm{\&Sigma;}}\left(f_i\right){\left(A^{\&prime;}\left(f_i,{\mathrm{\&Omega;}}_{-k}\right)\right)}^H\right)}^{-1}\&times;\frac{\&part;a^{\&prime;}(f_i,{\mathrm{\&alpha;}}_k)}{\&part;{\mathrm{\&alpha;}}_k})\end{array}\right\}{|}^{-1}---\left(39\right)$

Wherein, Re{} is for asking the real part of { }; Ω _-k, represent respectively from Ω and in remove a kth element wherein; K=1,2 ..., K;

According to expression formula try to achieve c ₁(f _i) ..., c _q(f _i), then try to achieve W (f according to formula (17) _i), utilize it to carry out array calibration and try to achieve a'(f _i, α _k) and A'(f _i, Ω _-k), then according to above parameter and formula (39), the estimated value of the direction of arrival of signal after array calibration can be obtained

2. the mutual coupling existing between elements error calibration method in broadband signal super-resolution according to claim 1 direction finding, is characterized in that the concrete steps setting up the array signal model containing mutual coupling existing between elements error described in step 1 are as follows:

Step 1.1: set up ideal array signal model:

Be provided with K far field broadband signal s _k(t), k=1,2 ..., K, incide on the broadband uniform linear array of M omnidirectional's array element composition, arrival direction is α=[α ₁..., α _k..., α _k], array element distance is d; Far field broadband signal s _kt (), is called for short broadband signal s _k(t);

Using the 1st array element as phase reference point, in the ideal case, the output of m array element is expressed as

$x_m\left(t\right)=\underset{k=1}{\overset{K}{\&Sigma;}}{s}_k(t-{\mathrm{\&tau;}}_m({\mathrm{\&alpha;}}_k\left)\right)+n_m\left(t\right),m=1,2,...,M---\left(1\right)$

Wherein, represent a kth broadband signal s _kt () arrives m array element arrives phase reference point time delay relative to it, c is electromagnetic wave velocity of propagation in a vacuum, n _mt () is the white Gaussian noise that m array element receives;

Suppose that the frequency range of broadband signal is for [f _low, f _high], utilize discrete Fourier transformation broadband signal to be divided into J frequency, they separated through narrow band filter group, then i-th group of filter array output signal is expressed as

X(f _i)＝A(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(2)

Wherein, f _low≤ f _i≤ f _high, i=1,2 ..., J;

Suppose at each frequency f _ion carried out KP time sampling, X (f _i) matrix representation be

X(f _i)＝[X(f _i,1),…,X(f _i,kp),…,X(f _i,KP)]，i＝1,2,…,J(3)

Wherein, X (f _i, kp) and be X (f _i) kth p secondary data sampling matrix,

X(f _i,kp)＝[X ₁(f _i,kp),…,X _m(f _i,kp),…,X _M(f _i,kp)] ^T，i＝1,2,…,J，(4)

X _m(f _i, kp) and be that m array element is at frequency f _ion the kth p secondary data sampled value that obtains;

A (f _i, α) and be ideally frequency f _ion array manifold matrix,

A(f _i,α)＝[a(f _i,α ₁),…,a(f _i,α _k),…,a(f _i,α _K)]，i＝1,2,…,J，(5)

A (f _i, α _k) be ideally frequency f _ithe array steering vector of a upper kth signal,

Wherein, it is the phase place of a kth signal; J is plural number mark;

S(f _i)＝[S(f _i,1),…,S(f _i,kp),…,S(f _i,KP)]，i＝1,2,…,J，(8)

For signal s _k(t) signal phasor matrix after Fourier transform, k=1,2 ..., K;

Wherein, S (f _i, kp) and be S (f _i) kth p signal sampling matrix,

S(f _i,kp)＝[S ₁(f _i,kp),…S _k(f _i,kp),…,S _K(f _i,kp)] ^Ti＝1,2,…,J(9)

S _k(f _i, kp) and for a kth signal is at frequency f _ion kth p signal sampling value of obtaining;

N(f _i)＝[N(f _i,1),…,N(f _i,kp),…,N(f _i,KP)]i＝1,2,…,J(10)

For noise n _mt () noise vector matrix after Fourier transform, average is 0, and variance is μ ²(f _i); M=1,2 ..., M;

Wherein, N (f _i, kp) and be N (f _i) kth p noise samples matrix,

N(f _i,kp)＝[N ₁(f _i,kp),…,N _m(f _i,kp),…,N _M(f _i,kp)] ^Ti＝1,2,…,J(11)

N _m(f _i, kp) and be that m array element is at frequency f _ion kth p noise samples value of obtaining;

Step 1.2: set up the array signal model containing mutual coupling existing between elements error on desirable array signal model basis:

When there is mutual coupling existing between elements error in the middle of array, frequency f _ion array export be expressed as

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)，i＝1,2,…,J(12)

Wherein

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]，i＝1,2,…,J(13)

For there is mutual coupling existing between elements error time frequency point f _ion array manifold matrix, a'(f _i, α _k) be corresponding array steering vector;

Then have

R'(f _i)＝E{X'(f _i)(X'(f _i)) ^H}，i＝1,2,…,J(14)

R'(f _i) for there is mutual coupling existing between elements error time frequency point f _ion Received signal strength covariance matrix;

If W is (f _i) be array perturbation matrix, represent frequency f _imutual coupling degree between upper array element, then the array steering vector when there is mutual coupling existing between elements error needs to be modified to

a'(f _i,α _k)＝W(f _i)a(f _i,α _k)(15)

Corresponding array manifold matrix representation is

A'(f _i,α)＝[a'(f _i,α ₁),…,a'(f _i,α _k),…,a'(f _i,α _K)]

(16)

＝W(f _i)A(f _i,α)

Wherein, c _q(f _i) expression spacing is q, signal frequency is f _itime array element between the mutual coupling coefficient, q=1,2 ..., Q,

Then have

X'(f _i)＝A'(f _i,α)S(f _i)+N(f _i)

，i＝1,2,…,J(18)

＝A(f _i,α)S(f _i)+Λ(f _i)w(f _i)+N(f _i)

Wherein, Λ (f _i) be only relevant with an original signal parameter; W (f _i)=[c ₁(f _i) ..., c _q(f _i)] ^trepresent frequency f _ion mutual coupling existing between elements perturbing vector; I _mit is the unit matrix of M × M dimension.

3. the mutual coupling existing between elements error calibration method in broadband signal super-resolution according to claim 2 direction finding, is characterized in that the perturbation matrix W (f described in step 1.2 _i) to solve concrete steps as follows:

U, v is made to be respectively the index value of the row and column of perturbation matrix, then perturbation matrix W (f _i) in element c _uv(f _i) represent, represent frequency f _ithe mutual coupling coefficient between upper u array element and v array element,

The array element mutual coupling of even linear array has following character:

(1), when the distance between array element is greater than mutual coupling degree of freedom Q, between array element, not think to there is mutual coupling;

The mutual coupling coefficient between (2) u array element and v array element is equal with the mutual coupling coefficient between v array element and u array element;

(3) the mutual coupling coefficient corresponding to the array element that spacing is equal is equal;

According to the array element mutual coupling character of even linear array, adopt following expression formula to describe perturbation matrix W (f when there is mutual coupling existing between elements error _i):

4. the mutual coupling existing between elements error calibration method in broadband signal super-resolution according to claim 3 direction finding, is characterized in that the employing expectation maximization method described in step 2 is come w (f _i), μ ²(f _i) and δ _l(f _i) carry out iterative estimate concrete steps as follows:

In E-step step in expectation maximization method, first right distribution function calculate

Wherein operational symbol <> represents that solving condition is expected;

In M-step step in expectation maximization method, ask for distribution function respectively to the derivative of each unknown parameter, namely right get extreme value to solve each unknown parameter;

$\begin{array}{c}\frac{\&part;F\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i),\stackrel{\&OverBar;}{S}(f_i);\mathrm{\&delta;}(f_i),w(f_i),{\mathrm{\&mu;}}^2(f_i\left)\right)}{\&part;w\left(f_i\right)}\\ =-2{\mathrm{\&mu;}}^2\left(f_i\right)\&lsqb;<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right)>w\left(f_i\right)-<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i)-A(f_i,\mathrm{\&Omega;}\left)\stackrel{\&OverBar;}{S}\right(f_i\left)\right)>\&rsqb;\end{array}---\left(25\right)$

$\begin{array}{c}\frac{\&part;F\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i),\stackrel{\&OverBar;}{S}(f_i);\mathrm{\&delta;}(f_i),w(f_i),{\mathrm{\&mu;}}^2(f_i\left)\right)}{\&part;{\mathrm{\&mu;}}^2\left(f_i\right)}\\ =-\frac{M\&times;KP}{{\mathrm{\&mu;}}^2\left(f_i\right)}\frac{1}{{\left({\mathrm{\&mu;}}^2\right(f_i\left)\right)}^2}<\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-A^{\&prime;}(f_i,\mathrm{\&Omega;})\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2>\end{array}---\left(26\right)$

$\begin{array}{c}\frac{\&part;F\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i),\stackrel{\&OverBar;}{S}(f_i);\mathrm{\&delta;}\left(f_i\right),w\left(f_i\right),{\mathrm{\&mu;}}^2\left(f_i\right))}{\&part;{\mathrm{\&delta;}}_l\left(f_i\right)}\\ =-\frac{KP}{{\mathrm{\&delta;}}_l\left(f_i\right)}+\frac{1}{{\mathrm{\&delta;}}_l^2\left(f_i\right)}<\underset{kp=1}{\overset{KP}{\mathrm{\&Sigma;}}}|{\stackrel{\&OverBar;}{S}}_l(f_i,kp){|}^2>\end{array}---\left(27\right)$

Make above derivative be 0 respectively, namely try to achieve the estimated value of each unknown parameter during the p time iteration

$w^{\left(p\right)}\left(f_i\right)=<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right){>}^{-1}<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i)-A(f_i,\mathrm{\&Omega;}\left)\stackrel{\&OverBar;}{S}\right(f_i\left)\right)>---\left(28\right)$

${\left({\mathrm{\&mu;}}^2\right(f_i\left)\right)}^{\left(p\right)}=\frac{1}{M\&times;KP}<\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-{\left(A^{\&prime;}\right(f_i,\mathrm{\&Omega;}\left)\right)}^{\left(p\right)}\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2>---\left(29\right)$

${\mathrm{\&delta;}}_l^{\left(p\right)}\left(f_i\right)=\frac{1}{KP}<\underset{kp=1}{\overset{KP}{\mathrm{\&Sigma;}}}|{\stackrel{\&OverBar;}{S}}_l(f_i,kp){|}^2>---\left(30\right)$

Wherein (p) represents iterations, in formula (28)

$\begin{array}{c}<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}\left(f_i\right){>}_{r1,r2}\\ =tr\&lsqb;{\mathrm{\&Psi;}}_{r1}^H\left(f_i\right){\mathrm{\&Psi;}}_{r2}\left(f_i\right)A(f_i,\mathrm{\&Omega;})\left(O\right(f_i\left)O^H\right(f_i)+KP\&times;\mathrm{\&Xi;}(f_i\left)\right)A^H(f_i,\mathrm{\&Omega;})\&rsqb;\end{array}---\left(31\right)$

For matrix r1 is capable, the element of r2 row, and wherein mark computing is asked in tr [] expression;

In formula (31)

O(f _i)＝Σ(f _i)(A'(f _i,Ω)) ^H(μ ²(f _i)I _M+A'(f _i,Ω)Σ(f _i)(A'(f _i,Ω)) ^H) ^-1X(f _i)(32)

For intermediate variable;

Ξ(f _i)

(33)

＝Σ(f _i)-Σ(f _i)(A'(f _i,Ω)) ^H(μ ²(f _i)I _M+A'(f _i,Ω)Σ(f _i)(A'(f _i,Ω)) ^H) ^-1A'(f _i,Ω)Σ(f _i)

For intermediate variable;

In formula (28)

$\begin{array}{c}<{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}^H\left(f_i\right)\left({\stackrel{\&OverBar;}{X}}^{\&prime;}\right(f_i)-A(f_i,\mathrm{\&Omega;})\stackrel{\&OverBar;}{S}\left(f_i\right))>\\ =tr\&lsqb;{\mathrm{\&Psi;}}_r^H\left(f_i\right)X\left(f_i\right)O^H\left(f_i\right)A^H(f_i,\mathrm{\&Omega;})\&rsqb;-\\ tr\&lsqb;{\mathrm{\&Psi;}}_r^H\left(f_i\right)A(f_i,\mathrm{\&Omega;})\left(O\right(f_i\left)O^H\right(f_i)+KP\&times;\mathrm{\&Xi;}(f_i\left)\right)A^H(f_i,\mathrm{\&Omega;})\&rsqb;\end{array}---\left(34\right)$

In formula (34), Ψ _r(f _i) be intermediate variable, be the matrix of M × M dimension, the element only on the ± r diagonal line is 1 entirely, and all the other elements are 0 entirely;

In formula (29)

$\begin{array}{c}<\left|\right|{\stackrel{\&OverBar;}{X}}^{\&prime;}\left(f_i\right)-{\left(A^{\&prime;}\right(f_i,\mathrm{\&Omega;}\left)\right)}^{\left(p\right)}\stackrel{\&OverBar;}{S}\left(f_i\right)|{|}_2^2>\\ =\left|\right|X\left(f_i\right)-{\left(A^{\&prime;}\right(f_i,\mathrm{\&Omega;}\left)\right)}^{\left(p\right)}O\left(f_i\right)|{|}_F^2+KP\&times;tr({\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}\mathrm{\&Xi;}\left(f_i\right){\left({\left(A^{\&prime;}\left(f_i,\mathrm{\&Omega;}\right)\right)}^{\left(p\right)}\right)}^H)\end{array}---\left(35\right)$

Formula (31) ~ (35) are substituted into peer-to-peer in formula (28) ~ (29) and carry out abbreviation and to w (f _i) and μ ²(f _i) solve;

When after the some steps of iteration, w (f _i), μ ²(f _i) and δ _l(f _i) change of three amount estimated values is tending towards 0, then draw last estimated value with correspondence obtains $\widehat{\mathrm{\&delta;}}\left(f_i\right)={\&lsqb;{\widehat{\mathrm{\&delta;}}}_1\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_l\left(f_i\right),...,{\widehat{\mathrm{\&delta;}}}_L\left(f_i\right)\&rsqb;}^T$ And $\hat{E}\left(f_i\right)=diag\left(\widehat{\mathrm{\&delta;}}\right(f_i\left)\right).$