1. Introduction

2. Array Model

3. Methods

3.1. MVDR

As one of the core tasks of array signal processing technology, beam forming can realize functions such as beam steering, interference and noise suppression, target identification and positioning, etc. The MVDR beamforming proposed by Capon is a commonly used algorithm [23], which weights and sums the output of the array and focuses the gain in one direction. The appropriate filter coefficients are chosen such that the average power at the output of the array is minimized under the constraint of the desired signal without distortion.

Figure 2. Beam forming basic structure.

Figure 2. Beam forming basic structure.

Assuming that the desired signal is undistorted, appropriate filter coefficients must be selected to minimise the average power of the array output. The MVDR weight optimisation problem can be expressed as:

$$\underset{w}{min}{w}^H{R}_{i+n}w\mathrm{s}.\mathrm{t}.w^H{a}_0=1$$

(12)

The above MVDR optimisation problem can be solved using the Lagrange multiplier method as follows:

$$w_{opt}=\frac{R_{i+n}^{-1}{a}_0}{a_0^H{R}_{i+n}^{-1}{a}_0}$$

(13)

Where $w={(w_1,\mathrm{...}{w}_M)}^T$ is the complex weighting vector of the beamformer, $R_{i+n}^{-1}$ and $a_0$ represent the ideal interference plus noise covariance matrix and the precise desired signal steering vector. In practice they are usually unavailable, so the sampled covariance matrix ${\tilde{R}}^{-1}$and the nominal guiding vector ${\bar{a}}_0$ based on the array geometry computed are commonly used instead [50], and in this case the MVDR beamformer is also referred to as the sampled covariance matrix inversion (SMI) beamformer [9]:

$$w_{SMI}=a{\tilde{R}}^{-1}{\bar{a}}_0=\frac{{\tilde{R}}^{-1}{\bar{a}}_0}{{\bar{a}}_0^H{\tilde{R}}^{-1}{\bar{a}}_0}$$

(14)

$a=\frac{1}{{\bar{a}}_0^H}$ The output sequence of the beamformer is

$$y\left(k\right)=w_{SMI}^{H}x\left(k\right)$$

(15)

The output SINR of the beamformer is:

$$\begin{array}{c}SINR\triangleq \frac{E\left[w^H{x}_s\left(k\right)x_s^H\left(k\right)w\right]}{E\left[w^H{x}_{i+n}\left(k\right)x_{i+n}^H\left(k\right)w\right]}\\ =\frac{w^H{R}_{s}w}{w^H{R}_{i+n}w}=\frac{{\sigma }_0^2{\left|w^H{a}_0\right|}^2}{w^H{R}_{i+n}w},\end{array}$$

(16)

${\sigma }_0^2=E\left[s_i\left(k\right)s_i(k{)}^H\right]$ is the expected signal power. The array pattern of the Capon beamformer represents the response relationship between the complex weighted vector and the incident signal in a given direction.

Also known as the beam response, it can be expressed as

$$B\left(\theta \right)=w^{H}a\left(\theta \right)$$

(17)

The overall execution process steps of the MVDR algorithm are as follows:

(1)

Conduct L snapshot observations of the signal source at time t, and use formula $R_k=\frac{1}{L}\sum ^{\underset{L}{i=1}}{G}_{ki}{G}_{ki}^H$ to construct a covariance matrix from the 2M − 1 signal data received by the array.

(2)

Calculate the inverse matrix or pseudo-inverse matrix of the covariance matrix to represent the relationship between signals.

(3)

Calculate the weight vector, which is the inverse matrix (or pseudo-inverse matrix) of the covariance matrix and the received signal.

(4)

Sort according to the size of the eigenroots, take the eigenvectors corresponding to the first K larger eigenvalues to form the signal subspace, and the remaining eigenvectors are the noise subspace;

(5)

Change $(\theta ,\varphi )$, and calculate the spectral function according to the formula to find the position of the maximum value, and then determine the azimuth angle and elevation angle of the source.

3.3. MVDR+PF Nonlinear Dynamical System Modeling

In order to predict the state of the system, the particle filter algorithm needs to build a model of the system. When the system model is difficult to obtain, a real model can be constructed by constructing an approximate model. This paper mainly implements the system model construction of moving targets in angle estimation navigation in the form of state equations and measurement equations.

1. Establishment of state equations

The motion of the target object generally includes uniform velocity motion and variable velocity motion, in order to better describe the operation of the object, the state of the target is described by the angular position quantity, angular velocity quantity and angular acceleration quantity in the study. Assuming that the target object moves along a straight line in a short period of time, the sampling time is $\Delta T$，the true azimuth angle of the target at the sampling moment $k\Delta T$ is $\theta \left(k\right)$, the target’s movement speed is $\dot{\theta }\left(k\right)$ and the acceleration is $\ddot{\theta }\left(k\right)$. The formula for the change of the target’s angular state can be expressed as:

$$\begin{gathered} \theta \left(k+1\right)=\theta \left(k\right)+\dot{\theta }\left(k\right)\Delta T+\frac{1}{2\ddot{\theta }\left(k\right)\Delta T^2}+\frac{1}{6\nu \Delta T^3} \\ \dot{\theta }\left(k+1\right)=\dot{\theta }\left(k\right)+\ddot{\theta }\left(k\right)\Delta T+\frac{1}{2v\Delta T^2} \\ \ddot{\theta }(k+1)=\ddot{\theta }\left(k\right)+\Delta T\mathcal{V} \end{gathered}$$

(19)

Among them, $\mathcal{V}$ represents the process noise of the system, which is independent of the observation noise and satisfies the Gaussian distribution of $\mathcal{V}\sim N(0,Q)$. Noise causes fluctuations in angle, angular velocity, and angular acceleration. The state space model of the target motion is obtained by combining the expressions, and the expression is as follows:

$$\left[\begin{array}{c}\theta (k+1)\\ \dot{\theta }(k+1)\\ \ddot{\theta }(k+1)\end{array}\right]=\left[\begin{array}{ccc}1& \Delta T& 1/2\Delta T^2\\ 0& 1& \Delta T\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\theta \left(k\right)\\ \dot{\theta }\left(k\right)\\ \ddot{\theta }\left(k\right)\end{array}\right]+\left[\begin{array}{c}1/6\Delta T^3\\ 1/2\Delta T^2\\ \Delta T\end{array}\right]\mathcal{V}$$

(20)

Expand the signal source into three-dimensional space, the target direction angle includes azimuth angle and elevation angle, The state x is expressed as

$$x=[\theta ,\dot{\theta },\ddot{\theta },\phi ,\dot{\phi },\ddot{\phi }{]}^T$$

(21)

The state vector at time $k$ is expressed as

$$x_k=A{x}_{k-1}+B\nu$$

(22)

Among them, the state transition matrix:

$$A=\left[\begin{array}{ccc}1& \Delta T& 1/2\Delta T^2\\ 0& 1& \Delta T\\ 0& 0& 1\\ 1& \Delta T& 1/2\Delta T^2\\ 0& 1& \Delta T\\ 0& 0& 1\end{array}\right],\hspace{1em}B=\left[\begin{array}{c}1/6\Delta T^3\\ 1/2\Delta T^2\\ \Delta T\\ 1/6\Delta T^3\\ 1/2\Delta T^2\\ \Delta T\end{array}\right]$$

(23)

2. Establishment of observation model

The observation model is another important model in the particle filter framework, which is expressed in the form of observation equations. The observation equation is mainly to solve the matching degree between the predicted state and the observed state of the moving target, and then update the state prediction estimate of the target. Most observation models are obtained in the form of data collected by sensors. The DOA navigation and tracking method for moving targets collects signals containing status information and noise information of moving targets through array sensors, and builds a system observation model through the collected data. The corresponding representation is as follows:

$$G_k=f\left(x_k,\omega \right)$$

(24)

Among them, $x_k$ represents the status information of the target at the time $k$ of the system, and $\omega$ represents the observation noise, which is independent of the process noise. The establishment of the observation model of moving objects is based on the signal data received by the array. In this paper, the L-shaped array model is used as the signal model, and the original data of signal observation obtained is

$$G=AS\left(t\right)+N\left(t\right)$$

(25)

where $S\left(t\right)$ is the source signal, $A$ is the steering matrix of the array, and $N\left(t\right)$ is the observation noise of the array elements in the array. In actual situations, in order to determine the effectiveness of the collected signal and the accuracy of the algorithm, the signal is often collected in multiple snapshots. Assuming that the signal is observed in snapshots at time $k$, the observation output matrix of the array is:

$$G_k=\left[G\right(kL+1),\hspace{1em}G(kL+2),\hspace{1em}\cdots ,\hspace{1em}G(kL+L)$$

(26)

The covariance matrix of the original data can be obtained from the above formula, expressed as follows:

$$R_k=\frac{1}{L}{G}_k{G}_k^H$$

(27)

The above equation represents the observation equation of the system. To further explain the quantities in the moving target DOA navigation and tracking algorithm based on the particle filter algorithm, the entire algorithm design mainly includes the relationship between three quantities, namely the observed value, the predicted value and the real value. The true value represents the true direction position of the target source, the predicted value represents the result value obtained by the particle filter algorithm, and the observed value is the measured value of the observation equation that reflects the size of the true value. The relationship between the three is shown in Figure 3.

In summary, the equations of the system model in the moving target DOA navigation and tracking algorithm based on the particle filter algorithm represent the prior probability and conditional probability expressions in the particle filter algorithm. Among them, $p\left(x_k\mid x_{k-1}\right),p\left(y_k\mid x_k\right)$ essentially represents the state equation and observation equation in this section. The robust angle estimation fusion algorithm proposed in this article uses the basic particle filter algorithm process as the basic implementation framework. The algorithm design is carried out through the setting of state conditions and parameters, which can be carried out in the following steps.

Figure 4. Flowchart of the algorithm for MVDR+PF.

Figure 4. Flowchart of the algorithm for MVDR+PF.

1. Initialization

Assume there are particles $N$, the number of snapshots is $L$ , and the total number of iterations is $Y$;

For $i=1:N$, initialize the particle state, azimuth angle ${\theta }_0\left(i\right)\sim U[0,\pi ]$;

The pitch angle ${\phi }_0\left(i\right)\sim U[0,\pi /2],{\dot{\phi }}_0\left(i\right),\dot{{\theta }_0}\left(i\right)\sim N\left({\mu }_0,{\epsilon }_0^2\right),i=1,\dots ,N,U\left[e_1,e_2\right]$ is evenly distributed, $N\left(\mu ,{\epsilon }^2\right)$normal distribution, ${\ddot{\theta }}_0\left(i\right)$ and ${\ddot{\phi }}_0\left(i\right)$ are initialized to 0, and the weight value is 1/N;

2. Iteration: $\mathrm{for}\mathrm{i}=1:Y$

(a) Obtain observation data

$Fori=1:L;$ Obtain the observation data of the system through the array at time K, Obtain the data covariance matrix according to $R_k=\frac{1}{L}{G}_k{G}_k^H$;

(b) Importance sampling stage

For transfer equation according to status $q\left(x_k\left(i\right)\mid x_{0:k-1}\left(i\right),Y_{1:k}\right)=p\left(x_k\left(i\right)\mid x_{k-1}\left(i\right)\right)$ carry out sampling $\left({\theta }_k\left(i\right),{\phi }_k\left(i\right)\right)$,

For $i=1:N$; Update the weight of the particle based on the weight calculation formula $W_k\left(x_k^{\left(i\right)}\right)=W_{k-1}\left(x_{k-1}^{\left(i\right)}\right)p\left(y_k\mid x_k\right)$

Normalize the weights of particles according to ${\tilde{W}}_k\left(x_k^{\left(i\right)}\right)=\frac{W_k\left(x_k^{\left(i\right)}\right)}{\sum ^{\underset{N}{i=1}}{W}_k\left(x_k^{\left(i\right)}\right)}$

(c) Algorithm resampling stage

Resample the particle set $\left\{x_k^{\left(i\right)},w_k\left(x_k^{\left(i\right)}\right)\right\}$ and reinitialize the weight values.

Get new particle set $\left\{x_k^{\left(i\right)},w_k\left(x_k^{\left(i\right)}\right)\right\}$

(d) Result output

Output the prediction results according to $x_k=\sum ^{\underset{N}{i=1}}{w}_k\left(x_k^{\left(i\right)}\right)x_k^{\left(i\right)}$

(e) Reinitialize weights

For $i=1:N$, reset the weight of the particles $w_k\left(x_k^{\left(i\right)}\right)=\frac{1}{N}$

3. End

4. Simulation Experiments and Analyses

5. Conclusions

6. Patents

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