## 3 Realization of Adaptive Square Root Cubature Kalman Filter Algorithm in Vector Receiver  
### Cubature Kalman Filter  
First, the volumetric Kalman filter is a filtering method based on the spherical-radial criterion. Suppose there is the following vector function integral in the Cartesian coordinate system [18]:  
\[\mathbf{I}(g)=\int_{\mathbf{R}^{n}}g(\mathbf{x})e^{-\mathbf{x}^{T}\mathbf{x}}d\mathbf{x} \tag{10}\]  
among them, if \(\mathbf{x}=r\mathbf{y}\), \(\mathbf{y}^{T}\mathbf{y}=1\), the integral in the Cartesian coordinate system is converted into the following spherical-radial coordinate integral:  
\[\mathbf{I}(g)=\int_{0}^{\infty}r^{n-1}e^{-\mathbf{x}^{2}}dr\int_{U_{n}}g(\mathbf{r}y)d\delta (\mathbf{y}) \tag{11}\]  
Assuming that the spherical integral \(\int_{U_{n}}g(\mathbf{r}\mathbf{y})\ d\mathbf{\langle}\mathbf{y}\ \text{is numerically calculated by the}\ L_{s}\) point spherical rule, and the radial integral \(\int_{0}^{\infty}S(r)r^{n-1}e^{-\mathbf{x}^{2}}dr\) is numerically calculated by the \(L_{r}\) point Gaussian integral rule, Formula (12) can be approximated by the spherical-radial rule as [18]  
\[\mathbf{I}(g)\approx\int_{0}^{\infty}r^{n-1}e^{-\mathbf{x}^{2}}\sum_{i=1}^{L_{s}}w_{s,i}g(ry_{i})d\pi\approx\sum_{j=1}^{L_{r}}\sum_{i=1}^{L_{s}}w_{r,i}w_{s,i}g(r_{ j}y_{i})=\sum_{i=1}^{L}w_{i}g(\xi_{i}) \tag{12}\]  
When CKF processes nonlinear equations, it generates a point set based on the mean value and covariance of the prior probability density distribution of the system state and performs nonlinear propagation of the sampling points to approximate the Gaussian integral in nonlinear Gaussian filtering [19]. The cubature Kalman filter is similar to the Kalman filter, and it is also divided into two processes: prediction and correction. The implementation steps are as follows [11]:  
1. Prediction 1. Initialization size quantity, error covariance, process noise, and measurement noise; 2. Calculate and propagate volume points; 3. Calculate the predicted value of states and error covariance.
2. Amendment 1. Calculate and propagate volume point; 2. Calculate the predicted value of the measurement; 3. Calculate measurement error covariance and cross-covariance; 4. Gain update, states, and error covariance update.  
This process does not include linearization errors, and there is no need to calculate the Jacobian matrix; however, to ensure the stability of the filtering process value, the state error covariance matrix needs to have two properties of symmetry and positive definiteness. The iterative process of the CKF algorithm may destroy the positive definiteness and symmetry of the covariance and cause numerical instability. For this reason, this article is based on the square root filtering idea of the Kalman filter to propose the square root cubature Kalman filter algorithm, the algorithm directly uses the process noise, measures the square root factor of the noise and error covariance matrix, and generates a set of volume points to spread throughout the nonlinear system model. In each iteration, the square root value of the error covariance will be updated and propagated through the even number of cubature points generated.