CN103245956B - A kind of GPS anti-multipath method based on robust ada-ptive beamformer algorithm - Google Patents

Abstract

The invention discloses a kind of GPS anti-multipath method based on robust ada-ptive beamformer algorithm, comprise step one: adopt even linear array, modeling is carried out to GPS multipath channel; Step 2: utilize the collection of letters number of front-rear space smooth interface differential technique to carry out decorrelation LMS process, obtain the best initial weights of new data covariance matrix and Capon Beam-former; Step 3: on the basis of step 2, sets up the cost function of the poorest best performance robust ada-ptive beamformer algorithm, solves the best initial weights under steering vector mismatch.Step 4: give other array response mismatchs uncertain modular constraint on the basis of step 3, obtain a kind of model of the poorest best performance robust ada-ptive beamformer algorithm of improvement; Step 5: the model determined according to step 4 calculates the optimum weight vector of Capon Beam-former, utilizes Newton iteration method to determine heap(ed) capacity.

Description

A kind of GPS anti-multipath method based on robust ada-ptive beamformer algorithm

Technical field

The present invention relates to a kind of GPS anti-multipath method based on robust ada-ptive beamformer algorithm, belong to the technical field utilizing Adaptive beamformer technology to process coherent interference signal.

Background technology

Adaptive beamformer is widely used in the various fields such as radar, communication, sonar and medical science.Ideally, conventional Capon (optimal beam forming device) Beamforming Method has higher resolving power and stronger interference rejection capability.And the interference environment of reality often more complicated, GPS receiving system is vulnerable to multipath signal interference, the direct Capon Adaptive beamformer method of routine that uses will cause wanted signal to disappear mutually, Beam-former performance degradation, this is because the difference between the array response of the putative signal angle of arrival and the actual signal angle of arrival or supposition and real array response causes.In this case, Capon Beam-former is used as useful signal as AF panel and is fallen.So the people such as Evans propose a kind of decorrelation LMS method based on space smoothing (spatial-smoothing), a main deficiency of space smoothing can only be used in uniform linear array exactly, this structure is easy to be destroyed because the coupling, array response mismatch etc. be subject between array element affects, so just cause the hydraulic performance decline of spatial smoothing method, therefore, robustness just becomes necessarily requiring of adaptive array processing.

For improving the output performance of Capon Beam-former under error condition, emerge many outstanding robustness methods.As the robust method in the feature based space for mismatch improving, covariance matrix disappear cone method, conventional diagonal angle loading technique, the people such as S.A.Vorobyov are also had to propose a kind of sane Adaptive beamformer method based on the poorest best performance of steering vector mismatch, utilize uncertain collection to describe the uncertainty of steering vector, data covariance matrix is changed with this, ensure that Beam-former still can keep good performance when there is steering vector error, these class methods belong to the one of diagonal angle loading method.But these methods when array accept to exist between data covariance matrix and corresponding actual value mismatch, useful signal and undesired signal relevant, the performance of Capon Beam-former will sharply decline.

Summary of the invention

The object of the invention is when GPS is subject to multi-path jamming, steering vector mismatch and other array response mismatchs, the situation that the performance of Capon Beam-former will sharply decline, propose a kind of GPS anti-multipath method based on robust ada-ptive beamformer algorithm.The present invention effectively can suppress multi-path jamming, and when signal exists various mismatch, the output performance of Capon Beam-former is well improved.

Based on a GPS anti-multipath method for robust ada-ptive beamformer algorithm, comprise following step:

Step one: adopt even linear array, modeling is carried out to GPS multipath channel;

Step 2: utilize the collection of letters number of front-rear space smooth interface differential technique to carry out decorrelation LMS process, obtain the best initial weights of new data covariance matrix and Capon Beam-former;

Step 3: on the basis of step 2, sets up the cost function of the poorest best performance robust ada-ptive beamformer method, solves the best initial weights under steering vector mismatch.

Step 4: give other array response mismatchs uncertain modular constraint on the basis of step 3, obtain a kind of model of the poorest best performance robust ada-ptive beamformer method of improvement;

Step 5: the model determined according to step 4 calculates the optimum weight vector of Capon Beam-former, utilizes Newton iteration method to determine heap(ed) capacity.

The present invention is directed to the situation that useful signal and undesired signal are relevant, a kind of improvement robust adaptive beamforming method based on space smoothing of proposition.When the method is mainly for steering vector and other array response mismatchs, Capon adaptive beam former output performance degradation.Search Space Smoothing can suppress coherent interference effectively, carries out decorrelation LMS process, and the poorest Robust Performance Beamforming Method of improvement can improve the situation of various mismatch effectively.The maximum feature of the method is that its load factor and steering vector error and error of covariance have and directly contacts, namely suitable heap(ed) capacity can be determined according to different error, can better analyze its output performance like this, because the method considers all possible mismatch condition, so have general practicality.

The invention has the advantages that:

(1) the present invention adopts front-rear space smooth technology to carry out decorrelation LMS process to GPS Received signal strength, reduces the number of sacrificing array element, reduces the correlativity of useful signal and coherent signal;

(2) the present invention has carried out uncertain constraint to steering vector error and other array response mismatchs, the method can process various mismatch, the problem that the performance effectively solving Beam-former sharply declines, the Signal to Interference plus Noise Ratio of output is greatly improved, reduce main peak skew simultaneously, and there is lower sidelobe level.

Accompanying drawing explanation

Fig. 1 is the schematic diagram that GPS multipath signal produces;

Fig. 2 is method flow diagram of the present invention;

Fig. 3 is forward direction space smoothing principle schematic.

Embodiment

Below in conjunction with accompanying drawing and implementation step, the present invention is described in further detail.

The key problem of decorrelation LMS is exactly: how by a series of process or conversion, the order of signal covariance matrix to be effectively restored.As shown in Figure 1: multipath signal is exactly the summation of direct signal and corresponding reflected signal thereof, owing to being subject to the reflection of the reverberation such as buildings, ground, the signal that receiver is received is not the direct signal of satellite launch, but the superposed signal of direct signal and reflected signal.Two coherent signals enter identical array simultaneously will make the vector of array covariance matrix signal subspace few, if but two coherent signals enter different arrays simultaneously, so the vector of the signal subspace of these two array covariance matrix sums does not just likely reduce, the design philosophy of Search Space Smoothing that Here it is.No matter be forward direction smoothing processing or backward smoothing processing, although can obtain the object of decorrelation LMS, effectively array element sacrifice is too many, in order to reduce the number of sacrificing array element as much as possible, front and back can be combined to smoothing technique.

The present invention is a kind of GPS anti-multipath method of robust ada-ptive beamformer algorithm, and flow process as shown in Figure 2, comprises following step:

Step one: adopt even linear array, modeling is carried out to GPS multipath channel;

Suppose adopt even linear array, whole array has M array element, adjacent array element be spaced apart λ/2, λ is the wavelength of GPS, is 19cm, and the deflection of wanted signal is α=0 °, the deflection of reflected signal is β=30 °, then can be expressed as at the snap of the multipath signal of certain sampling instant n:

$x\[n\]=s_0\[n\]+\underset{i=1}{\overset{L-1}{\Σ}}{s}_i\[n\]+n\[n\],i=1,...,L---\left(1\right)$

Wherein: x [n] is for receiver is at the Received signal strength of sampling instant n, and L is the number of coherent interference, s ₀[n] represents the vector representation form of direct signal in even linear array, s _i[n] is the vector representation form of i-th coherent interference signal in even linear array, and n [n] is separate zero mean Gaussian white noise, and uncorrelated with multipath signal.S again ₀[n] and s _it () represents s respectively ₀[n] and s _ithe expression-form of the element in [n], s ₀(t)=Ad (t) c (t) cos (ω _ct), wherein A represents the carrier amplitude of direct signal; It is 1 that d (t) represents navigation message; C (t) represents the pseudo-code of GPS; w _cbe IF-FRE (supposing not consider the impact of Doppler shift) be 1/1.5e6Hz; a _irepresent signal amplitude attenuation coefficient, a _i∈ (0,1]; C (t-τ _i) be the pseudo-code of different delayed time, τ _iit is the time delay of the relative direct signal of the i-th road signal pseudo-code; θ _iit is the carrier phase of the i-th road signal.

Step 2: utilize the collection of letters number of front-rear space smooth interface differential technique to carry out decorrelation LMS process, obtain the best initial weights of new data covariance matrix and Capon Beam-former;

Because useful signal and undesired signal are concerned with, so utilize space smoothing first to carry out decorrelation LMS process.As shown in Figure 3: whole array is divided into K subarray, the array number of each subarray is P, P>L, each subarray progressively moves to right, then have array number P=M-K+1, getting first submatrix is R-matrix, then the Received signal strength of each submatrix that forward direction is level and smooth is x _i[n]=[x _i[n] ..., x _i+p-1[n]], i=1 ..., K, wherein x _i[n] equation left side represents the vector form of acknowledge(ment) signal, x _i[n] represents the output of i-th array element at sampling instant n.All subarray covariance matrixes can be added the original covariance matrix of rear average replacement, that is:

${\tilde{R}}_{ss}=\frac{1}{NK}\underset{i=1}{\overset{K}{\Σ}}\underset{n=1}{\overset{N}{\Σ}}{x}_i\[n\]x_i{\[n\]}^H=\frac{1}{NK}\underset{i=1}{\overset{K}{\Σ}}{\hat{x}}_i{{\hat{x}}_i}^H---\left(2\right)$

Wherein: represent the covariance matrix of forward direction space smoothing, N is the fast umber of beats of sampling. the Received signal strength of i-th array element, no matter be forward direction smoothing processing or backward smoothing processing, although can obtain the object of decorrelation LMS, effectively array element sacrifice is too many, in order to reduce the number of sacrificing array element as much as possible, front and back can be combined to smoothing technique, can obtain thus:

${\hat{R}}_{ss}=\frac{{\tilde{R}}_{ss}+J\left({\tilde{R}}_{ss}\right)*J}{2}---\left(3\right)$

Wherein:

be the covariance matrix of front-rear space smooth, when practical application, utilize and estimate steering vector and covariance replace actual steering vector a ₀with covariance R _ss, so based on the best initial weights of the Capon Beam-former of front-rear space smooth be:

${\hat{w}}_{c,fo}=\frac{{{\hat{R}}_{ss}}^{-1}{\hat{a}}_0}{{{\hat{a}}_0}^H{{\hat{R}}_{ss}}^{-1}{\hat{a}}_0}---\left(4\right)$

Wherein: what represent is the best initial weights of the Capon Beam-former of front-rear space smooth, ${\hat{a}}_0=\exp (j\mathrm{\π}\×{\[0:M-1\]}^{\′}\×sin(\mathrm{\α})),$ And j ²=-1.

Step 3: on the basis of step 2, sets up the cost function of most best performance robust ada-ptive beamformer method, obtains the Capon best initial weights under steering vector mismatch.

But due to the impact of the factors such as error in pointing, sensor position uncertainties and each array element characteristic are inconsistent, the limited bat of sampled data, Search Space Smoothing error, DOAs estimation, steering vector is made to there is certain error, have a strong impact on the performance of Adaptive beamformer method, therefore the poorest Robust Performance Beamforming Method becomes an emphasis of Recent study.

Because steering vector is impossible accurately obtain in actual applications, therefore suppose signal steering vector mismatch sLM Signal Label Mismatch amount 2 ⁰, then the cost function constructing robust adaptive beamforming device is:

$\left\{\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H\left({\hat{R}}_{ss}\right)\hat{w}\\ s.t.& \left|{\hat{w}}^H({\hat{a}}_0+\mathrm{\Δ}{\hat{a}}_0)\right|\≥1\\ & \left|\right|\mathrm{\Δ}{\hat{a}}_0\left|\right|\≤\mathrm{\ϵ}\end{array}\right.---\left(5\right)$

Wherein, optimum weight vector, || || be Frobenius norm, ε is a positive number, represents the amount of restraint of steering vector mismatch.Application triangle inequality, Cauchy-schwarz inequality, above formula can change into the minimization problem of single non-linear constrain, and formula (5) can be converted into following reduced form again:

$\left\{\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H\left({\hat{R}}_{ss}\right)\hat{w}\\ s.t.& \left|{\hat{w}}^H{\hat{a}}_0\right|-\mathrm{\ϵ}\left|\right|\hat{w}\left|\right|\≥1\\ & \mathrm{Im}\left\{{\hat{w}}^H{\hat{a}}_0\right\}=0\end{array}\right.---\left(6\right)$

Wherein, Im{}=0 represents the imaginary part of formula, when when carrying out Arbitrary Rotation, cost function does not change, like this can when not affecting objective function, right carry out rotating thus making for real number, namely real part is greater than 1, and imaginary part equals 0, inequality can be written to the ratio of the signal not affecting output and interference and noise and SINR.So formula (6) can be expressed as following form:

$\left\{\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H{\hat{R}}_{ss}\hat{w}\\ s.t& |{\hat{w}}^H{\hat{a}}_0-1{|}^2={\mathrm{\ϵ}}^2{\hat{w}}^H\hat{w}\end{array}\right.---\left(7\right)$

The method of Lagrange's multiplier is utilized to solve above formula, its optimum weight vector solution can obtain by minimizing following function:

$Q({\hat{w}}^H,{\mathrm{\λ}}_1)={\hat{w}}^H{\hat{R}}_{ss}\hat{w}-{\mathrm{\λ}}_1(|{\hat{w}}^H{\hat{a}}_0-1{|}^2-{\mathrm{\ϵ}}^2{\hat{w}}^H\hat{w})---\left(8\right)$

Wherein, be about and λ ₁lagrangian function, λ ₁it is Lagrange factor.Ask formula (8) relative gradient, and make it equal zero, then finally solve optimum weight vector solution be:

$\begin{array}{c}\hat{w}=-{\mathrm{\λ}}_1{({\hat{R}}_{ss}+{\mathrm{\λ}}_1{\mathrm{\ϵ}}^2-{\mathrm{\λ}}_1{\hat{a}}_0{{\hat{a}}_0}^H)}^{\mathrm{-1}}{\hat{a}}_0\\ ={({\hat{R}}_{ss}+{\mathrm{\λ}}_1{\mathrm{\ϵ}}^2+{\hat{a}}_0(-{\mathrm{\λ}}_{1}I\left){{\hat{a}}_0}^H\right)}^{-1}{\hat{a}}_0(-{\mathrm{\λ}}_{1}I)\end{array}---\left(9\right)$

Utilize the law of reciprovity of matrix: (A+BCD) ^-1bC=A ^-1b (C ^-1+ DA ^-1b) ^-1, final required optimum weight vector can be obtained

${\hat{w}}_{fb}=\frac{{\mathrm{\λ}}_1{({\hat{R}}_{ss}+{\mathrm{\λ}}_1{\mathrm{\ϵ}}^{2}I)}^{-1}{\hat{a}}_0}{{\mathrm{\λ}}_1{{\hat{a}}_0}^H{({\hat{R}}_{ss}+{\mathrm{\λ}}_1{\mathrm{\ϵ}}^{2}I)}^{-1}{\hat{a}}_0-1}---\left(10\right)$

Wherein what represent is the optimum weight vector of the poorest best performance robust ada-ptive beamformer algorithm.

Step 4: give the constraint of other array response mismatchs one uncertainty on the basis of step 3, obtain a kind of model of the poorest best performance robust ada-ptive beamformer method of improvement;

Due in the Wave beam forming application of reality, can there is certain error in signal covariance matrix formula (5) can be written to:

$\left\{\begin{array}{ccc}\underset{\hat{w}}{min}& \underset{\left|\right|\mathrm{\Δ}{\hat{R}}_{ss}\left|\right|\≤\mathrm{\γ}}{\max }& {\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\Δ}{\hat{R}}_{ss})\hat{w}\\ s.t.& \left|{\hat{w}}^H({\hat{a}}_0+\mathrm{\Δ}{\hat{a}}_0)\right|\≥1& \left|\right|\mathrm{\Δ}{\hat{a}}_0\left|\right|\≤\mathrm{\ϵ}\end{array}\right.---\left(11\right)$

Wherein: γ is arbitrary positive number, the amount of restraint of signal covariance matrix error is represented.Because be Hermitian (Hermite) error matrix an of the unknown, so any given maximal value exist border obtain, γ represents any given positive number, and formula (11) is rewritten into following form:

$\left\{\begin{array}{cc}\underset{\mathrm{\Δ}{\hat{R}}_{ss}}{min}& -{\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\Δ}{\hat{R}}_{ss})\hat{w}\\ s.t.& \left|\right|\mathrm{\Δ}{\hat{R}}_{ss}\left|\right|=\mathrm{\γ}\end{array}\right.---\left(12\right)$

The solution of formula (12) above utilizes Lagrange multiplier method to solve, its solution can obtain by minimizing following function λ ₂for Lagrange multiplier, ask the gradient of (12) and make it be zero, $\mathrm{\Δ}{\hat{R}}_{ss}=\hat{w}{\hat{w}}^H/2{\mathrm{\λ}}_2,$ When $\left|\right|\mathrm{\Δ}{\hat{R}}_{ss}|{|}^2={\mathrm{\γ}}^2,$ Obtain further:

$\mathrm{\Δ}{\hat{R}}_{ss}=\mathrm{\γ}\frac{\hat{w}{\hat{w}}^H}{{\hat{w}}^H\hat{w}}---\left(13\right)$

(13) are updated to (10), and (13) can wait and be converted into following form:

$\left\{\begin{array}{ccc}\underset{\hat{w}}{min}& {\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\γ}I)\hat{w}& \\ s.t.& \left|{\hat{w}}^H({\hat{a}}_0+\mathrm{\Δ}{\hat{a}}_0)\right|\≥1& \left|\right|\mathrm{\Δ}{\hat{a}}_0\left|\right|\≤\mathrm{\ϵ}\end{array}\right.---\left(14\right)$

Equally abbreviation is carried out to this problem, following reduced form can be obtained:

$\left\{\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\γ}I)\hat{w}\\ s.t& |{\hat{w}}^H{\hat{a}}_0-1|={\mathrm{\ϵ}}^2{\hat{w}}^H\hat{w}\end{array}\right.---\left(15\right)$

Step 5: the model determined according to step 4, determines each operational factor in model, finally calculates the optimum weight vector of Capon Beam-former, wherein utilizes Newton iteration method to determine heap(ed) capacity.

The method solved with step 3 is similar, can obtain final required optimum weight vector

${\hat{w}}_{rfo}=\frac{{\mathrm{\λ}}_3{({\hat{R}}_{ss}+\mathrm{\γ}I+{\mathrm{\λ}}_3{\mathrm{\ϵ}}^{2}I)}^{-1}{\hat{a}}_0}{{\mathrm{\λ}}_3{{\hat{a}}_0}^H{({\hat{R}}_{ss}+\mathrm{\γ}I+{\mathrm{\λ}}_3{\mathrm{\ϵ}}^{2}I)}^{-1}{\hat{a}}_0-1}---\left(16\right)$

Wherein, λ ₃for Lagrange's multiplier, the poorest Robust Performance Wave beam forming that representative improves, conveniently solves this equation, given γ=2, ε=2.First to covariance matrix carry out feature decomposition u is eigenvectors matrix, Λ=diag [δ ₁, δ ₂..., δ _p], δ in formula _ibe i-th eigenwert.Equation (16) can be simplified to further:

${\hat{w}}_{rfo}=\frac{{\mathrm{\λ}}_3{U}^H{(\mathrm{\Λ}+\mathrm{\γ}I+{\mathrm{\λ}}_3{\mathrm{\ϵ}}^{2}I)}^{-1}U{\hat{a}}_0}{{\mathrm{\λ}}_3{U}^H{(\mathrm{\Λ}+\mathrm{\γ}I+{\mathrm{\λ}}_3{\mathrm{\ϵ}}^{2}I)}^{-1}U{\hat{a}}_0-1}---\left(17\right)$

Formula (17) is updated to available equation in (15) order z _ibe i-th element of vectorial z, so equation can be write as following form again:

$f\left({\mathrm{\λ}}_3\right)={\mathrm{\λ}}_3{\mathrm{\ϵ}}^2\underset{m=1}{\overset{p}{\Σ}}\frac{|z_m{|}^2}{{\mathrm{\δ}}_m+\mathrm{\γ}+{\mathrm{\λ}}_3{\mathrm{\ϵ}}^2}=1---\left(18\right)$

Due to with f (0)=0 < 1, so the solution of formula (18) is unique.Newton iteration method is utilized to obtain λ ₃, λ ₃span:

$\frac{{\mathrm{\&delta;}}_{min}+\mathrm{\&gamma;}}{\left|\right|{\hat{a}}_0\left|\right|-1}\&CenterDot;\frac{1}{{\mathrm{\&epsiv;}}^2}\&le;{\mathrm{\&lambda;}}_3\&le;\frac{{\mathrm{\&delta;}}_{max}+\mathrm{\&gamma;}}{\left|\right|{\hat{a}}_0\left|\right|-1}\&CenterDot;\frac{1}{{\mathrm{\&epsiv;}}^2}---\left(19\right)$

Finally obtain the optimum weight vector of the improvement the poorest Robust Performance wave beam based on space smoothing so exported utilize array antenna 1, array antenna 2 and array antenna M etc. to receive gps signal, afterwards radio-frequency front-end the radio-frequency input signals on each road is amplified, filtering and be down-converted to a fixing intermediate frequency.A/D completes the acquisition function of data, the output of intermediate frequency AGC amplifier adjustment signal makes it meet the requirement of A/D dynamic range, then Adaptive Signal Processing module is given by signal, first Search Space Smoothing is adopted to carry out pre-service to multipath signal signal, adopt the various mismatch of robust ada-ptive beamformer algorithm process of improvement again, reduce main peak skew, sense forms wave beam, form zero adaptively at interference radiating way to fall into, thus reach the object improving Signal to Interference plus Noise Ratio, suppress multi-path jamming.So export the signal after y [n] is through multipaths restraint, so just achieve multipaths restraint.

Claims (1)

1. a GPS anti-multipath method for robust ada-ptive beamformer algorithm, comprises following step:

Step one: adopt even linear array, modeling is carried out to GPS multipath channel;

Suppose adopt even linear array, whole array has M array element, adjacent array element be spaced apart λ/2, λ is the wavelength of GPS, be then expressed as at the snap of the multipath signal of certain sampling instant n:

$x\&lsqb;n\&rsqb;=s_0\&lsqb;n\&rsqb;+\underset{i=1}{\overset{L-1}{\&Sigma;}}{s}_i\&lsqb;n\&rsqb;+n\&lsqb;n\&rsqb;---\left(1\right)$

Wherein: x [n] for receiver is at the Received signal strength of sampling instant n, i=1 ..., L, L are the numbers of coherent interference, s ₀[n] represents the vector representation form of direct signal in even linear array, s _i[n] is the vector representation form of i-th coherent interference signal in even linear array, and n [n] is separate zero mean Gaussian white noise, and uncorrelated with multipath signal, s ₀(n) and s _it () represents s respectively ₀[n] and s _ielement in [n], s ₀(t)=Ad (t) c (t) cos (ω _ct), wherein A represents the carrier amplitude of direct signal; It is the pseudo-code that 1, c (t) represents GPS that d (t) represents navigation message, ω _cbe IF-FRE be 1/1.5e6Hz;

a _irepresent signal amplitude attenuation coefficient, a _i∈ (0,1]; C (t-τ _i) be the pseudo-code of different delayed time, τ _iit is the time delay of the relative direct signal of the i-th road signal pseudo-code; θ _iit is the carrier phase of the i-th road signal;

Step 2: utilize the collection of letters number of front-rear space smooth interface differential technique to carry out decorrelation LMS process, obtain the best initial weights of new data covariance matrix and Capon Beam-former;

Whole array is divided into K subarray, and the array number of each subarray is P, P>L, and each subarray progressively moves to right, then have array number P=M-K+1, and getting first submatrix is R-matrix, then the Received signal strength of each submatrix that forward direction is level and smooth is x _m[n]=[x _m[n] ..., x _m+p-1[n]], m=1 ..., K, wherein x _m[n] equation left side represents the vector form of Received signal strength, x _m[n] represents the output of m array element at sampling instant n; The covariance matrix that after all subarray covariance matrixes are added, average replacement is original, that is:

${\tilde{R}}_{ss}=\frac{1}{NK}\underset{m=1}{\overset{K}{\&Sigma;}}\underset{n=1}{\overset{N}{\&Sigma;}}{x}_m\&lsqb;n\&rsqb;x_m{\&lsqb;n\&rsqb;}^H=\frac{1}{NK}\underset{i=1}{\overset{K}{\&Sigma;}}{\hat{x}}_i{{\hat{x}}_i}^H---\left(2\right)$

Wherein: represent the covariance matrix of forward direction space smoothing, N is the fast umber of beats of sampling; the Received signal strength of i-th array element, front and back are combined to smoothing technique, can obtain thus:

${\hat{R}}_{ss}=\frac{{\tilde{R}}_{ss}+J\left({\tilde{R}}_{ss}\right)*J}{2}---\left(3\right)$

Wherein:

be the covariance matrix of front-rear space smooth, when practical application, use symbol with represent the steering vector and covariance estimating to obtain, so based on the best initial weights of the Capon Beam-former of front-rear space smooth be:

${\hat{w}}_{c,fo}=\frac{{{\hat{R}}_{ss}}^{-1}{\hat{a}}_0}{{{\hat{a}}_0}^H{{\hat{R}}_{ss}}^{-1}{\hat{a}}_0}---\left(4\right)$

Wherein: what represent is the best initial weights of the Capon Beam-former of front-rear space smooth, ${\hat{a}}_0=\exp (j\mathrm{\&pi;}\&times;{\&lsqb;0:M-1\&rsqb;}^{\&prime;}\&times;sin(\mathrm{\&alpha;}\left)\right),$ And j ²=-1;

Step 3: on the basis of step 2, sets up the cost function of the poorest best performance robust ada-ptive beamformer method, obtains the Capon best initial weights under steering vector mismatch;

Suppose signal steering vector mismatch sLM Signal Label Mismatch amount 2 °, then the cost function constructing robust adaptive beamforming device is:

$\left\{\begin{array}{c}\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H\left({\hat{R}}_{ss}\right)\hat{w}\end{array}\\ \begin{array}{cc}s.t.& \left|{\hat{w}}^H({\hat{a}}_0+\mathrm{\&Delta;}{\hat{a}}_0)\right|\&GreaterEqual;1\end{array}\\ \left|\right|\mathrm{\&Delta;}{\hat{a}}_0\left|\right|\&le;\mathrm{\&epsiv;}\end{array}\right.---\left(5\right)$

Wherein, optimum weight vector, || .|| is Frobenius norm, and ε is a positive number, represents the amount of restraint of steering vector mismatch; Application triangle inequality, Cauchy-schwarz inequality, formula (5) changes into the minimization problem of single non-linear constrain, and formula (5) is converted into again following reduced form:

$\left\{\begin{array}{c}\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H\left({\hat{R}}_{ss}\right)\hat{w}\end{array}\\ \begin{array}{cc}s.t.& \left|{\hat{w}}^H{\hat{a}}_0\right|-\mathrm{\&epsiv;}\mathrm{||}\hat{w}\mathrm{||}\&GreaterEqual;1\end{array}\\ \mathrm{Im}\left\{{\hat{w}}^H{\hat{a}}_0\right\}=0\end{array}\right.---\left(6\right)$

Wherein, Im{.}=0 represents the imaginary part of formula, when when carrying out Arbitrary Rotation, cost function does not change, like this when not affecting objective function, right carry out rotating thus making for real number, namely real part is greater than 1, and imaginary part equals 0, inequality be written to the ratio of the signal not affecting output and interference and noise and SINR; So formula (6) is expressed as following form:

$\left\{\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H{\hat{R}}_{ss}\hat{w}\\ s.t& |{\hat{w}}^H{\hat{a}}_0-1{|}^2={\mathrm{\&epsiv;}}^2{\hat{w}}^H\hat{w}\end{array}\right.---\left(7\right)$

The method of Lagrange's multiplier is utilized to solve formula (7), its optimum weight vector solution obtain by minimizing following function:

$Q({\hat{w}}^H,{\mathrm{\&lambda;}}_1)={\hat{w}}^H{\hat{R}}_{ss}\hat{w}-{\mathrm{\&lambda;}}_1(|{\hat{w}}^H{\hat{a}}_0-1{|}^2-{\mathrm{\&epsiv;}}^2{\hat{w}}^H\hat{w})---\left(8\right)$

Wherein, be about and λ ₁lagrangian function, λ ₁it is Lagrange factor; Ask formula (8) relative gradient, and make gradient function equal zero, then finally solve optimum weight vector solution be:

$\begin{array}{c}\hat{w}=-{\mathrm{\&lambda;}}_1{({\hat{R}}_{ss}+{\mathrm{\&lambda;}}_1{\mathrm{\&epsiv;}}^2-{\mathrm{\&lambda;}}_1{\hat{a}}_0{{\hat{a}}_0}^H)}^{-1}{\hat{a}}_0\\ ={({\hat{R}}_{ss}+{\mathrm{\&lambda;}}_1{\mathrm{\&epsiv;}}^2+{\hat{a}}_0(-{\mathrm{\&lambda;}}_{1}I\left){{\hat{a}}_0}^H\right)}^{-1}{\hat{a}}_0(-{\mathrm{\&lambda;}}_{1}I)\end{array}---\left(9\right)$

Utilize the law of reciprovity of matrix: (A+BCD) ^-1bC=A ^-1b (C ^-1+ DA ^-1b) ^-1, obtain final required optimum weight vector

${\hat{w}}_{fb}=\frac{{\mathrm{\&lambda;}}_1{({\hat{R}}_{ss}+{\mathrm{\&lambda;}}_1{\mathrm{\&epsiv;}}^{2}I)}^{-1}{\hat{a}}_0}{{\mathrm{\&lambda;}}_1{{\hat{a}}_0}^H{({\hat{R}}_{ss}+{\mathrm{\&lambda;}}_1{\mathrm{\&epsiv;}}^{2}I)}^{-1}{\hat{a}}_0-1}---\left(10\right)$

Wherein what represent is the optimum weight vector of the poorest best performance robust ada-ptive beamformer algorithm;

Step 4: increase the constraint of a signal covariance uncertainty on the basis of step 3 obtain a kind of model of the poorest best performance robust ada-ptive beamformer method of improvement;

Due in the Wave beam forming application of reality, can there is certain error in signal covariance matrix formula (5) is written to:

$\left\{\begin{array}{ccc}\underset{\hat{w}}{min}& \underset{\mathrm{||}\mathrm{\&Delta;}{\hat{R}}_{ss}\left|\right|\&le;\mathrm{\&gamma;}}{max}& {\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\&Delta;}{\hat{R}}_{ss})\hat{w}\\ s.t.& \left|{\hat{w}}^H({\hat{a}}_0+\mathrm{\&Delta;}{\hat{a}}_0)\right|\&GreaterEqual;1& \left|\right|\mathrm{\&Delta;}{\hat{a}}_0\left|\right|\&le;\mathrm{\&epsiv;}\end{array}\right.---(11$

Wherein: γ is arbitrary positive number, the amount of restraint of signal covariance matrix error is represented; Because be the Hermite error matrix an of the unknown, so any given maximal value exist border obtain, γ represents any given positive number, and formula (11) is rewritten into following form:

$\left\{\begin{array}{cc}\underset{\mathrm{\&Delta;}{\hat{R}}_{ss}}{min}& -{\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\&Delta;}{\hat{R}}_{ss})\hat{w}\\ s.t& \left|\right|\mathrm{\&Delta;}{\hat{R}}_{ss}\left|\right|=\mathrm{\&gamma;}\end{array}\right.---\left(12\right)$

The solution of formula (12) above utilizes Lagrange multiplier method to solve, its solution obtain by minimizing following function $Q(\hat{w},{\mathrm{\&lambda;}}_2)=-{\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\&Delta;}{\hat{R}}_{ss})\hat{w}+{\mathrm{\&lambda;}}_2\left(\right|\left|\mathrm{\&Delta;}{\hat{R}}_{ss}\right|{|}^2+{\mathrm{\&gamma;}}^2),$ λ ₂for Lagrange factor, ask (12) about gradient and make gradient function be zero, when obtain further:

$\mathrm{\&Delta;}{\hat{R}}_{ss}=\mathrm{\&gamma;}\frac{\hat{w}{\hat{w}}^H}{{\hat{w}}^H\hat{w}}---\left(13\right)$

Following form is obtained after (13) are updated to (11):

$\left\{\begin{array}{ccc}\underset{\hat{w}}{min}& {\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\&gamma;}I)\hat{w}& \\ s.t.& \left|{\hat{w}}^H({\hat{a}}_0+\mathrm{\&Delta;}{\hat{a}}_0)\right|\&GreaterEqual;1& \left|\right|\mathrm{\&Delta;}{\hat{a}}_0\left|\right|\&le;\mathrm{\&epsiv;}\end{array}\right.---\left(14\right)$

According to the step of (5) ~ (7), abbreviation is carried out to formula (14):

$\left\{\begin{array}{cc}\underset{\hat{w}}{min}& {\hat{w}}^H({\hat{R}}_{ss}+\mathrm{\&gamma;}I)\hat{w}\\ s.t& |{\hat{w}}^H{\hat{a}}_0-1|={\mathrm{\&epsiv;}}^2{\hat{w}}^H\hat{w}\end{array}\right.---\left(15\right)$

Step 5: the model determined according to step 4, determines each operational factor in model, finally calculates the optimum weight vector of Capon Beam-former, wherein utilizes Newton iteration method to determine heap(ed) capacity;

The optimum weight vector of the final required improvement the poorest Robust Performance wave beam based on space smoothing

${\hat{w}}_{rfo}=\frac{{\mathrm{\&lambda;}}_3{({\hat{R}}_{ss}+\mathrm{\&gamma;}I+{\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^{2}I)}^{-1}{\hat{a}}_0}{{\mathrm{\&lambda;}}_3{{\hat{a}}_0}^H{({\hat{R}}_{ss}+\mathrm{\&gamma;}I+{\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^{2}I)}^{-1}{\hat{a}}_0-1}---\left(16\right)$

Wherein, λ ₃for Lagrange factor, represent the optimum weight vector based on the improvement the poorest Robust Performance wave beam of space smoothing, first to covariance matrix carry out feature decomposition u is eigenvectors matrix, Λ=diag [δ ₁, δ ₂..., δ _p], δ in formula _ibe i-th eigenwert, equation (16) is simplified to further:

${\hat{w}}_{rfo}=\frac{{\mathrm{\&lambda;}}_3{U}^H{(\mathrm{\&Lambda;}+\mathrm{\&gamma;}I+{\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^{2}I)}^{-1}U{\hat{a}}_0}{{\mathrm{\&lambda;}}_3{U}^H{(\mathrm{\&Lambda;}+\mathrm{\&gamma;}I+{\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^{2}I)}^{-1}U{\hat{a}}_0-1}---\left(17\right)$

The equality condition that formula (17) is updated in formula (15) is obtained order z _ii-th element of vectorial z, so write as following form:

$f\left({\mathrm{\&lambda;}}_3\right)={\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^2\underset{m=1}{\overset{p}{\&Sigma;}}\frac{|z_m{|}^2}{{\mathrm{\&delta;}}_m+\mathrm{\&gamma;}+{\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^2}=1---\left(18\right)$

Due to so the solution of formula (18) is unique, Newton iteration method is utilized to obtain λ ₃, λ ₃span:

$\frac{{\mathrm{\&delta;}}_{min}+\mathrm{\&gamma;}}{\left|\right|{\hat{a}}_0\left|\right|-1}.\frac{1}{{\mathrm{\&epsiv;}}^2}\&le;{\mathrm{\&lambda;}}_3\&le;\frac{{\mathrm{\&delta;}}_{max}+\mathrm{\&gamma;}}{\left|\right|{\hat{a}}_0\left|\right|-1}.\frac{1}{{\mathrm{\&epsiv;}}^2}---\left(19\right)$

Finally obtain the optimum weight vector of the improvement the poorest Robust Performance wave beam based on space smoothing exported $y\&lsqb;n\&rsqb;={{\hat{w}}_{rfo}}^{H}x\&lsqb;n\&rsqb;,$ Realize multipaths restraint.