## 3 Realization of Adaptive Square Root Cubature Kalman Filter Algorithm in Vector Receiver
### Adaptive Square Root Cubature Kalman Filter Algorithm  
In practical applications, due to continuous changes in environmental factors, the actual noise items will change accordingly. At this time, if the measurement noise covariance \(R_{k}\) is always a fixed value, it may not reflect the actual situation of the received signal, resulting in key parameter values, such as position and speed, that cannot be accurately estimated. Therefore, it is necessary to realize real-time adjustment of the measurement noise covariance \(R_{k}\) in the SRCKF algorithm and real-time estimation and correction of noise statistical characteristics in the filtering process to reduce state estimation errors. Based on the innovation sequence [14], this paper proposes an adaptive square root cubature Kalman filter algorithm to realize the adaptive adjustment of \(R_{k}\) and compares the tracking results with the SRCKF VLL and ASRCKF SLL. The results prove that the algorithm proposed has a more accurate estimation performance and is more stable than the other two.  
Since the observation is the output of the code phase discriminator and the carrier frequency discriminator, \(R_{k}\) contains the code phase and carrier frequency errors. Normally, \(R_{k}\) is set by calculating the variance of the discriminator output for a certain length of time. The longer this length of time, the more accurate the obtained variance, but if the length is set too long, the state value cannot be estimated in real time [20].  
In Formula (32), \(\Big{(}z_{k}-\hat{z}_{k|k-1}\Big{)}\) is defined as innovation. Therefore, the innovation is a linear combination of the observation vector and its predicted value. In an ideal state, the innovation is zero-mean Gaussian white noise, and the innovation at different moments should be orthogonal. However, in the target-tracking process, especially when the carrier is in the high dynamic state of motion, the observation noise will increase so that the innovation is no longer orthogonal. In order to solve the above problems, the adaptability of the algorithm must be improved [21].  
The so-called adaptive filtering matches innovation with its theoretical covariance. Therefore, this paper applies the idea of covariance matching based on the innovation sequence to the square root cubature Kalman filter algorithm and realizes the adaptive adjustment of the measurement noise covariance matrix \(R_{k}\) in the algorithm [16].  
According to the literature [22; 23], the measurement noise error covariance matrix \(R_{k}\) can be estimated as  
\[\hat{R}_{k}=E\Big{[}z_{k}-\hat{z}_{k|k-1}z_{k}-\hat{z}_{k|k-1}^{T}\Big{]}+ \frac{1}{2n}\sum_{j=0}^{2n}z_{j,k|k}z_{j,k|k}^{T}-\hat{z}_{k|k}\hat{z}_{k|k}^{T} \tag{27}\]where \(z_{j,k|k}=h\Big{(}X_{j,k|k}\Big{)}\) is the volume point propagated through the function relationship of the measurement equation, \(X_{j,k|k}=S_{k|k}\xi_{i}+\hat{x}_{k|k}\) is the measurement update cubature point, and \(\hat{z}_{k|k}=\frac{1}{m}\sum_{i=1}^{2n}Z_{i,\ k|k}\) is the measurement prediction.  
The covariance matrix of innovation is  
\[P_{ino}=E[(z_{k}-\hat{z}_{k|k-1})(z_{k}-\hat{z}_{k|k-1})^{T}] \tag{28}\]  
If a piece of sample data of innovation is used to approximate the covariance matrix of innovation [10], then there is  
\[P_{ino}\approx\frac{1}{N}\sum_{j=0}^{N-1}(z_{k-j}-\hat{z}_{k-j|k-j-1})(z_{k-j} -\hat{z}_{k-j|k-j-1})^{T} \tag{29}\]  
Therefore, we can get  
\[S_{R,k}=\sqrt{R_{k}}=\sqrt{\frac{1}{N}\sum_{j=0}^{N-1}(z_{k-j}-\hat{z}_{k-j|k- j-1})(z_{k-j}-\hat{z}_{k-j|k-j-1})^{T}}+\frac{1}{2n}\sum_{j=j_{0}}^{2n}z_{j,k|k} z_{j,k|k}^{T}-\hat{z}_{k|k}\hat{z}_{k|k}^{T} \tag{30}\]