# Underwater Acoustic Target Tracking: A Review
## 4 Theory of Target Tracking
### State Filtering and Estimation
 \tag{10}\]  
Assuming the estimated state of the target at time \(k\) is \(\hat{x}(k|k)\), the linearization of the state function at time \(k+1\) can be represented as Equation (10) shows. The higher order is omitted here, where \(e_{i}\) is the ith standard vector of unit matrix. The summation formula superscripts n represents the dimension of state vector. \(f^{\prime}_{x}(k)\) and \(f^{\prime}_{xx}(k)\) represents the Jacobian matrix and Hessian matrix of the state transfer vector \(f\) at the nearest estimation value of the state respectively. The linearization of the measurement equation at time \(k+1\) is similar, represented as Equation (11) shows. The higher order is also omitted here. \[ \begin{array}{l}z(k+1)\approx h(k+1,\hat{x}(k|k))+h^{\prime}_{x}(k+1)[x(k+1) -\hat{x}(k+1|k)]+p(k+1)\\ \qquad+\frac{1}{2}\sum\limits_{i=1}^{n}e_{i}[x(k+1)-\hat{x}(k+1|k)]^{T}h^{ \prime}_{xx}(k+1)[x(k+1)-\hat{x}(k+1|k)]\end{array}  \tag{11}\]  
The second and higher order items are omitted in Equations (10) and (11) when the first order EKF is adopted, while the second order items should be kept in the second order EKF. Then the state estimation can be completed according to the basic KF. However, if the nonlinearity of the system is serious, tracking algorithms adopting EKF will result in large errors due to the loss of numerous information in the linearization process using Taylor series. _UKF:_ Based on the unscented transform (UT), UKF approximates the nonlinear distribution using determined sample particles. Different from approximately linearizing the state, UT performs an approximation on the posterior PDF of the state by some determined symmetrical sampling points [46]. The mean and variance of Gaussian PDF can be fully reflected by these sampling points. Therefore, UKF is widely used for solving nonlinear problems with noise in Gaussian distributions. The state prediction and estimation can be carried out by computing the mean and variance of the sampling points undergoing the nonlinear system. The precision of UKF can be compared to the second order EKF because it is not necessary to linearize the nonlinear system which needs complex computation. _PF:_ It is a sub-optimal algorithm of the Bayesian filter based on random sampling [47]. Because it is hard to calculate the posterior PDF in nonlinear or non-Gaussian problems, PF was developed to approximate the posterior PDF based on a set of discrete and random sampling particles. Every particle is equipped with a weight which indicates its quality, and the weighted sum of all particles is the final estimate result. Like the Bayesian filter, PF can be divided into two phases: prediction and update. The prediction phase estimates the state of particles at the next time step according to the settled target dynamic motion model. The update phase recalculates the weights of particles using newly obtained measurements at the current time. As these particles propagate over time, weights of some particles will become increasingly smaller, which is called particle degeneration problem. If the problem is ignored, numerous computations will be wasted on the particles that make few contributions to the estimated result. To address the problem, the re-sampling step, which means updating the set of particles by copying these particles with greater weights and eliminate those with smaller weights, is introduced into PF. The performance of EKF and PF is compared in [48], which shows that PF has superiority in real scenarios. EKF can effectively solve nonlinear problems on the condition that the noise is Gaussian. However, the PDF of the target state cannot be accurately described when the noise is non-Gaussian. PF approximates PDF directly by a set of random sampling points, therefore, it has more precise estimate results compared with the EKF. The results in [44] demonstrate that no other algorithms can perform better than KF when the assumption (with a linear model and Gaussian noise) holds because KF is the optimal solution for that case.