CN103245956A - GPS (global positioning system) multipath mitigation method based on robust beam forming algorithm - Google Patents

Abstract

The invention discloses a GPS (global positioning system) multi-path mitigation method based on a robust beam forming algorithm, which comprises the following steps: step I, modeling a GPS multipath channel by adopting a uniform linear array; step II, performing decorrelation on received signals by adopting the forward-backward spatial-smoothing technology, so as to obtain a new data covariance matrix and the optimal weight of a Capon beam former; step III, on the basis of the step II, establishing a cost function of the worst-case performance optimization robust beam forming algorithm, so as to solve the optimal weight under steering vector mismatch; step IV, on the basis of the step III, exerting uncertainty constraint on other array response errors, so as to obtain an improved model for the worst-case performance optimization robust beam forming algorithm; and step V, according to the model determined in the step IV, calculating to obtain the optimal weight vector of the Capon beam former, and determining the loading capacity by utilizing the Newton iteration method.

Description

GPS anti-multipath method based on robust beam forming algorithm

Technical Field

The invention relates to a GPS anti-multipath method based on a steady beam forming algorithm, belonging to the technical field of processing coherent interference signals by utilizing a self-adaptive beam forming technology.

Background

Adaptive beamforming is widely used in radar, communication, sonar, medicine and other fields. Ideally, the conventional Capon (optimal beamformer) beamforming method has high resolution and strong interference suppression capability. While the actual interference environment is often complex, the GPS receiving system is susceptible to multi-path signal interference, and directly using the conventional Capon adaptive beamforming method will result in cancellation of the desired signal, and the performance of the beamformer will be severely degraded due to the difference between the assumed signal arrival angle and the true signal arrival angle or the assumed array response and the true array response. In this case, the Capon beamformer rejects the desired signal as interference. Evans et al therefore propose a decorrelation method based on spatial-smoothing, one of the main disadvantages of spatial smoothing is that it can only be used in uniform linear arrays, and the structure is easily damaged due to the influence of coupling between array elements, array response errors, etc., which results in the performance degradation of the spatial smoothing method, and therefore, robustness becomes an essential requirement for adaptive array processing.

To improve Capon beamformer output performance in error situations, a number of excellent robust methods emerge. For example, a robust method based on a feature space for improving general mismatch, a covariance matrix cone elimination method, a common diagonal loading technique, and the worst performance optimization based on steering vector mismatch of s.a. vorobyov et al propose a robust adaptive beam forming method, which uses an uncertainty set to describe uncertainty of a steering vector so as to change a data covariance matrix and ensure that a beam forming device can still maintain good performance when a steering vector error exists, and the method belongs to one of diagonal loading methods. However, in these methods, the performance of the Capon beamformer is drastically degraded in the case where the array accepts a mismatch between the data covariance matrix and the corresponding true values, and the desired signal and the interfering signal are coherent.

Disclosure of Invention

The invention aims to provide a GPS anti-multipath method based on a robust beam forming algorithm under the condition that the performance of a Capon beam former is sharply reduced when a GPS is subjected to multipath interference, steering vector mismatch and other array response mismatches. The invention can effectively inhibit multipath interference and can well improve the output performance of the Capon beam former under the condition that various mismatching exists in signals.

A GPS anti-multipath method based on a robust beam forming algorithm comprises the following steps:

the method comprises the following steps: modeling a GPS multipath channel by adopting a uniform linear array;

step two: performing decorrelation processing on the received signals by utilizing a forward-backward space smoothing technology to obtain a new data covariance matrix and an optimal weight of a Capon beam former;

step three: and on the basis of the second step, establishing a cost function of the worst performance optimal steady beam forming method, and solving the optimal weight under the mismatching of the steering vectors.

Step four: giving an uncertainty mode constraint to other array response errors on the basis of the third step to obtain an improved model of the worst performance optimal steady beam forming method;

step five: and C, calculating the optimal weight vector of the Capon beam former according to the model determined in the step four, and determining the loading capacity by utilizing a Newton iteration method.

The invention provides an improved robust adaptive beam forming method based on spatial smoothing, aiming at the condition that a useful signal and an interference signal are coherent. The method mainly aims at the problem that when the steering vector and other array responses are mismatched, the output performance of the Capon self-adaptive beam former is seriously reduced. The spatial smoothing technology can effectively inhibit coherent interference and carry out coherent de-processing, and the improved worst-performance robust beam forming method can effectively improve various mismatch conditions. The method is mainly characterized in that the loading factor of the method is directly connected with the error of the guide vector and the error of the covariance, namely, the proper loading amount can be determined according to different errors, so that the output performance of the method can be better analyzed, and the method has universal practicability because all possible mismatch conditions are considered.

The invention has the advantages that:

(1) the invention adopts the forward and backward space smoothing technology to carry out the decorrelation processing on the GPS receiving signal, thereby reducing the number of sacrificial array elements and reducing the correlation between useful signals and coherent signals;

(2) the invention carries out uncertainty constraint on the steering vector error and other array response errors, can process various mismatches, effectively solves the problem of rapid performance reduction of a beam former, greatly improves the output signal-to-interference-and-noise ratio, reduces the main peak offset and has lower sidelobe level.

Drawings

FIG. 1 is a schematic diagram of GPS multipath signal generation;

FIG. 2 is a flow chart of a method of the present invention;

fig. 3 is a schematic diagram of the forward spatial smoothing principle.

Detailed Description

The invention will be described in further detail below with reference to the accompanying drawings and implementation steps.

The core problem of decoherence is: how to effectively recover the rank of the signal covariance matrix through a series of processes or transformations. As shown in fig. 1: the multipath signal is the sum of the direct signal and its corresponding reflected signal, and due to the reflection of the reflectors such as buildings and the ground, the signal received by the receiver is not the direct signal transmitted by the satellite, but the superimposed signal of the direct signal and the reflected signal. The simultaneous entry of two coherent signals into the same array will result in fewer vectors in the signal subspace of the array covariance matrices, but if two coherent signals enter different arrays simultaneously, the vector in the signal subspace of the sum of the two array covariance matrices may not be reduced, which is the design idea of spatial smoothing techniques. Both forward smoothing and backward smoothing can achieve the purpose of decoherence, but the effective array elements are sacrificed too much, and forward and backward smoothing techniques can be combined in order to reduce the number of the sacrificed array elements as much as possible.

The invention relates to a GPS anti-multipath method of a robust beam forming algorithm, the flow is shown as figure 2, and the method comprises the following steps:

the method comprises the following steps: modeling a GPS multipath channel by adopting a uniform linear array;

assuming that a uniform linear array is used, the entire array has M array elements, the interval between adjacent array elements is λ/2, λ is the wavelength of GPS and is 19cm, the direction angle α of the desired signal is 0 °, the direction angle β of the reflected signal is 30 °, and the snapshot of the multipath signal at a certain sampling time n can be expressed as:

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

wherein: x [ n ]]For the received signal at the sampling instant n of the receiver, L is the number of coherent interferers, s₀[n]Representing the vector representation, s, of the direct signal in a uniform linear array_i[n]N [ n ] is the vector representation form of the ith coherent interference signal in the uniform linear array]Is independent zero mean white gaussian noise and is uncorrelated with multipath signals. And s₀[n]And s_i(t) each represents s₀[n]And s_i[n]The expression form of the element in (1), s₀(t)＝Ad(t)c(t)cos(ω_ct), where a represents the carrier amplitude of the direct signal; d (t) represents a navigation message of 1; c (t) pseudo code representing GPS; w is a_cIs an intermediate frequency (assuming the effect of doppler shift is not taken into account) of 11.5e6 Hz;

a_irepresenting the attenuation coefficient of the signal amplitude, a_i∈(0,1]；c(t-τ_i) For differently delayed pseudo-codes, tau_iThe delay time of the ith signal pseudo code relative to the direct signal is obtained; theta_iIs the carrier phase of the ith signal.

Step two: performing decorrelation processing on the received signals by utilizing a forward-backward space smoothing technology to obtain a new data covariance matrix and an optimal weight of a Capon beam former;

since the desired signal and the interfering signal are coherent, the decorrelation process is first performed using spatial smoothing. As shown in fig. 3: dividing the whole array into K sub-arrays, wherein the number of array elements of each sub-array is P>And L, if P is M-K +1, each subarray is gradually shifted to the right, the first subarray is taken as a reference matrix, and the received signal of each forward smooth subarray is x_i[n]＝[x_i[n],...,x_i+p-1[n]]1, K, where x is_i[n]The left side of the equation represents the vector form of the received signal, x_i[n]Representing the output of the ith array element at sample time n. All subarray covariance matrices can be added and then averagely substituted for the original covariance matrix, that is:

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

wherein:

representing the forward spatially smoothed covariance matrix, N being the fast beat number of the sample.The received signal of the ith array element,although the objective of decoherence can be obtained by both forward smoothing and backward smoothing, the effective array elements are sacrificed too much, and in order to reduce the number of the sacrificed array elements as much as possible, the forward and backward smoothing techniques can be combined, so that the method can be obtained:

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

wherein:

is a covariance matrix with smooth front and back space, and utilizes the estimated guide vector in practical applicationSum covariance

Instead of the actual steering vector a₀Sum covariance R_ssTherefore, the optimal weight of Capon beamformer based on forward and backward spatial smoothing is:

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

wherein:

the optimal weights for the Capon beamformer are shown for forward and backward spatial smoothing, ${\hat{a}}_0=\exp (\mathrm{j\π}\×{[0:M-1]}^{\′}\×\sin \left(\mathrm{\α}\right)),$ and j is²＝-1。

Step three: and on the basis of the second step, establishing a cost function of the most-performance optimal robust beam forming method to obtain the Capon optimal weight under the mismatching of the steering vectors.

However, due to the influence of pointing errors, array element position errors, inconsistent characteristics of each array element, limited beat of sampled data, technical errors of spatial smoothing, DOAs estimation and other factors, a certain error exists in a steering vector, and the performance of the adaptive beam forming method is seriously affected, so that the beam forming method with the worst performance and the steady performance becomes a key point of research in recent years.

Since the steering vector cannot be accurately obtained in practical applications, the deviation of the signal direction vector is assumed

Amount of signal mismatch 2⁰Then constructing a cost function of the robust adaptive beamformer as：

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

Wherein,

is the optimal weight vector, | | | | is the Frobenius norm, and epsilon is a positive number, representing the constrained quantity of the mismatching of the guide vector. Applying the trigonometric inequality, the cauchy-schwarz inequality, the above equation can be converted into a minimization problem of a single nonlinear constraint, and equation (5) can be converted into a simplified form as follows:

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

wherein Im { } 0 represents the imaginary part of the equation, whenWhen any angle rotation is carried out, the cost function is not changed, so that the objective function can be adjusted under the condition of not influencing the objective function

Is rotated to make

Is real, i.e. real part is greater than 1, imaginary part is equal to 0, inequality

Can be written asThe ratio of the output signal to the interference and noise, i.e. the SINR, is not affected. Equation (6) can be expressed in the form:

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

the method of Lagrange multiplier is used for solving the above formula, and the optimal weight vector of the formula is obtained

Can be obtained by minimizing the following function:

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

wherein,

is about

And λ₁Of the Lagrangian function, λ₁Is the lagrange factor. Relative equation (8)

And is made equal to zero, the optimal weight vector is finally solved

The solution of (a) is:

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

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

using the reversible theorem of matrices: (A + BCD)^-1BC＝A^-1B(C^-1+DA^-1B)^-1The optimal weight vector required finally can be obtained

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

Wherein

The optimal weight vector for the worst performance optimal robust beamforming algorithm is shown.

Step four: giving an uncertainty constraint to other array response errors on the basis of the third step to obtain an improved model of the worst performance optimal steady beam forming method;

because in practical beam forming application, the signal covariance matrix has certain errorEquation (5) can be written as:

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

wherein: γ is an arbitrary positive number and represents a constraint amount of the signal covariance matrix error. Because of the fact that

Is an unknown Hermitian error matrix, and is therefore arbitrarily given

Has a maximum value of

γ represents an arbitrary given positive number, and equation (11) is rewritten into the following form:

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

the solution of equation (12) above is solved using the Lagrange multiplier method, which

Can be solved byMiniaturize function acquisition as followsλ₂For the Lagrange multiplier, the gradient of (12) is calculated and made zero, $\mathrm{\Δ}{\hat{R}}_{\mathrm{ss}}=\hat{w}{\hat{w}}^H/2{\mathrm{\λ}}_2,$ when in use ${\left|\right|\mathrm{\Δ}{\hat{R}}_{\mathrm{ss}}\left|\right|}^2={\mathrm{\γ}}^2,$ Further obtaining:

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

substitution of (13) into (10), (13) can be converted into the following forms:

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

the problem is also simplified and the following simplification can be obtained:

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

step five: and determining each operation parameter in the model according to the model determined in the step four, and finally calculating to obtain the optimal weight vector of the Capon beam former, wherein the load capacity is determined by utilizing a Newton iteration method.

Similar to the method of solving in step three, the optimal weight vector required finally can be obtained

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

Wherein λ is₃In order to be a lagrange multiplier, the lagrange multiplier,

representing improved worst-performance robust beamforming, γ -2 and e-2 are given for the convenience of solving the equation. First, to covariance matrix

Performing feature decomposition

U is a feature vector matrix, and Λ is diag [ delta ]₁,δ₂,...,δ_p]In the formula, wherein delta_iIs that

The ith characteristic value of (1). Equation (16) can be further simplified to:

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

equation obtained by substituting equation (17) into equation (16)Order to

z_iIs the ith element of the vector z, then the equation can again be written as follows:

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

due to the fact that

And f (0) < 1, so the solution of equation (18) is unique. By Newton's iteration method lambda can be found₃，λ₃The value range of (A):

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

finally, the optimal weight vector of the improved worst performance robust beam based on the spatial smoothing is obtained

So as to obtain an output

The array antenna 1, the array antenna 2, the array antenna M and the like are used for receiving GPS signals, and then the radio frequency front end amplifies, filters and down-converts the radio frequency input signals of each path to a fixed intermediate frequency. The A/D finishes the data acquisition function, the intermediate frequency AGC amplifier adjusts the output of the signal to make the signal meet the requirement of the A/D dynamic range, then the signal is sent to the self-adaptive signal processing module, firstly the space smoothing technology is adopted to preprocess the multipath signal, then the improved steady beam forming algorithm is adopted to process various mismatching, the main peak deviation is reduced, the beam is formed in the signal direction, the null notch is formed in the interference direction in a self-adaptive way, and thus the purposes of improving the signal-to-interference-noise ratio and restraining the multipath interference are achieved. So output y [ n ]]Is the signal after multipath mitigation, thusMultipath mitigation is achieved.

Claims (1)

1. A GPS anti-multipath method of a robust beam forming algorithm comprises the following steps:

the method comprises the following steps: modeling a GPS multipath channel by adopting a uniform linear array;

assuming that a uniform linear array is adopted, the whole array has M array elements, the interval between adjacent array elements is λ/2, and λ is the wavelength of the GPS, the snapshot of the multipath signal at a certain sampling time n is expressed as:

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

wherein: x [ n ]]For the received signal of the receiver at sampling instant n, i 1₀[n]Representing the vector representation, s, of the direct signal in a uniform linear array_i[n]N [ n ] is the vector representation form of the ith coherent interference signal in the uniform linear array]Is independent zero mean white Gaussian noise and is uncorrelated with multipath signals, and s₀[n]And s_i(t) each represents s₀[n]And s_i[n]The expression form of the element in (1), s₀(t)＝Ad(t)c(t)cos(ω_ct), where a represents the carrier amplitude of the direct signal; d (t) represents navigation message 1, c (t) represents pseudo code of GPS, w_cIs an intermediate frequency of 11.5e6 Hz;

a_irepresenting the attenuation coefficient of the signal amplitude, a_i∈(0,1]；c(t-τ_i) For differently delayed pseudo-codes, tau_iThe delay time of the ith signal pseudo code relative to the direct signal is obtained; theta_iIs the carrier phase of the ith signal;

step two: performing decorrelation processing on the received signals by utilizing a forward-backward space smoothing technology to obtain a new data covariance matrix and an optimal weight of a Capon beam former;

dividing the whole array into K sub-arrays, wherein the number of array elements of each sub-array is P, P>And L, if P is M-K +1, each subarray is gradually shifted to the right, the first subarray is taken as a reference matrix, and the received signal of each forward smooth subarray is x_i[n]＝[x_i[n],...,x_i+p-1[n]]1, K, where x is_i[n]The left side of the equation represents the vector form of the received signal, x_i[n]The output of the ith array element at the sampling time n is shown; after all the subarray covariance matrixes are added, the original covariance matrix is averagely replaced, namely:

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

wherein:

representing a forward spatially smoothed covariance matrix, N being the number of fast beats sampled;

the received signal of the ith array element,

by combining the forward and backward smoothing techniques, it is possible to obtain:

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

wherein:

is a covariance matrix with smooth front and back space, and utilizes the estimated guide vector in practical application

Sum covariance

Instead of the actual steering vector a₀Sum covariance R_ssTherefore, the optimal weight of Capon beamformer based on forward and backward spatial smoothing is:

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

wherein:

the optimal weights for the Capon beamformer are shown for forward and backward spatial smoothing, ${\hat{a}}_0=\exp (\mathrm{j\&pi;}\&times;{[0:M-1]}^{\&prime;}\&times;\sin \left(\mathrm{\&alpha;}\right)),$ and j is²＝-1；

Step three: on the basis of the second step, establishing a cost function of the most-performance optimal robust beam forming method to obtain a Capon optimal weight under the mismatching of the steering vectors;

assuming signal directional vector deviation

Amount of signal mismatch 2⁰Then the cost function for constructing the robust adaptive beamformer is:

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

wherein,

the method is characterized in that the method is an optimal weight vector, | | | is a Frobenius norm, and epsilon is a positive number and represents the constrained quantity of mismatching of a guide vector; applying the trigonometric inequality, the cauchy-schwarz inequality, the above equation translates to a minimization problem of a single nonlinear constraint, and equation (5) translates to a simplified form as follows:

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

wherein Im { } 0 represents the imaginary part of the equation, when

When any angle rotation is carried out, the cost function is not changed, so that the objective function is not influenced

Is rotated to make

Is real, i.e. real part is greater than 1, imaginary part is equal to 0, inequality

Is written as

Does not affectThe ratio of the output signal to the interference and noise, namely SINR; formula (6) is therefore expressed in the following form:

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

the method of Lagrange multiplier is used for solving the above formula, and the optimal weight vector of the formula is obtained

Is obtained by minimizing the following function:

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

wherein,

is about

And λ₁Of the Lagrangian function, λ₁Is the lagrange factor; relative equation (8)

And is made equal to zero, the optimal weight vector is finally solved

The solution of (a) is:

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

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

using the reversible theorem of matrices: (A + BCD)^-1BC＝A^-1B(C^-1+DA^-1B)^-1To obtain the final required optimumWeight vector

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

WhereinThe optimal weight vector of the worst performance optimal robust beam forming algorithm is represented;

step four: giving an uncertainty constraint to other array response errors on the basis of the third step to obtain an improved model of the worst performance optimal steady beam forming method;

because in practical beam forming application, the signal covariance matrix has certain error

Equation (5) is written as:

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

wherein: gamma is an arbitrary positive number and represents the constrained quantity of the signal covariance matrix error; because of the fact that

Is an unknown Hermite error matrix, so any given

Has a maximum value of

γ represents an arbitrary given positive number, and equation (11) is rewritten into the following form:

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

the solution of equation (12) above is solved using the Lagrange multiplier method, whichIs obtained by minimizing the function

λ₂For the Lagrange multiplier, the gradient of (12) is calculated and made zero, $\mathrm{\&Delta;}{\hat{R}}_{\mathrm{ss}}=\hat{w}{\hat{w}}^H/2{\mathrm{\&lambda;}}_2,$ when in use ${\left|\right|\mathrm{\&Delta;}{\hat{R}}_{\mathrm{ss}}\left|\right|}^2={\mathrm{\&gamma;}}^2,$ Further obtaining:

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

substitution of (13) to (10), (13), etc. converts to the following form:

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

the problem is also simplified and the following simplification can be obtained:

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

step five: determining each operation parameter in the model according to the model determined in the step four, and finally calculating to obtain the optimal weight vector of the Capon beam former, wherein the load capacity is determined by utilizing a Newton iteration method;

the final required optimal weight vector

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

Wherein λ is₃In order to be a lagrange multiplier, the lagrange multiplier,

representing improved worst-performance robust beamforming, the covariance matrix is first aligned

Performing feature decomposition

U is a feature vector matrix, and Λ is diag [ delta ]₁,δ₂,...,δ_p]In the formula, wherein delta_iIs that

For the ith eigenvalue, equation (16) is further simplified to:

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

equation obtained by substituting equation (17) into equation (16) ${\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^2\left|{{\hat{a}}_0}^H{U}^H{(\mathrm{\&Lambda;}+\mathrm{\&gamma;I}+{\mathrm{\&lambda;}}_3{\mathrm{\&epsiv;}}^{2}I)}^{-1}U{\hat{a}}_0\right|=1,$ Order to $z=U^H{\hat{a}}_0,$ z_iIs the ith element of the vector z, then the equation is written as follows:

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

due to the fact that

Since f (0) < 1, the solution of equation (18) is unique, and λ is determined by newton's iteration₃，λ₃The value range of (A):

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

finally, the optimal weight vector of the improved worst performance robust beam based on the spatial smoothing is obtained

Obtain an output

Multipath mitigation is achieved.

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