## 2 Theory
### Kalman Filter  
Kalman filter algorithm is an optimal linear recursive filtering algorithm based on the minimum mean square error (MMSE) and it can predict the next state according to the current state of the target. The proposed Kalman filter has three basic assumptions: (1) the modeled system is linear, (2) the noise affecting measurement is white noise, and (3) the noise follows a Gaussian distribution. The first hypothesis is that the state of the system is linear, as shown in the state prediction Equation (7) below. The latter two assumptions are that noise is white noise and follow Gaussian distribution, namely white Gaussian noise (WGN), which means that noise is random and not related to time, and it can be modeled accurately by the mean and covariance. Moreover, in the application of the Kalman filter, three kinds of motion are generally considered: dynamic motion, control motion, and random motion. Dynamic motion refers to the direct result of the system state in the previous measurement, such as uniform motion. Controlled motion is a mode of motion that is imposed on the system by some known external factors for some reason, such as accelerated motion. Random motion means random irregular movement, which can be modeled using a Gaussian model. Unlike simulation where noise can be artificially designed, the real noise is unknowable in an actual situation. Thus, in order to conform to the actual situation to the greatest extent, the motion model chooses a random model and the noise is assumed to be white noise and follows Gaussian distribution as WGN is the simplest optimalestimation of the actual noise. Then, based on the assumptions above, the state prediction equation and systematic observation equation are:  
\[X(k)=A\times X(k-1)+B\times U(k-1)+W(k-1) \tag{7}\]  
\[Z(k)=H\times X(k)+V(k) \tag{8}\]  
where \(X(k)/X(k-1)\) is the state value at the \(k/k-1\) moment; \(Z(k)\) is the observation value at the \(k\) moment; and \(A\), \(B\), and \(H\) represents state transition matrix, control matrix, and measurement matrix, respectively. \(U(k-1)\) is the external control vector of moment \(k-1\). \(W(k)\) and \(V(k)\) are the process noise and measurement noise, which are assumed to be white noise, independent of each other, and follow a Gaussian distribution. The corresponding covariances are \(Q\) and \(R\), respectively. In practice, a transition matrix \(A\) and measurement matrix \(H\) might change with each time step, but they are assumed to be constant in the proposed system to simplify the model. A control matrix \(B\) is not considered here for there is no external control vector \(U\) in the process. Moreover, the system and observation covariances \(Q\) and \(R\) are assumed to be constant. Furthermore, in the proposed system, the state matrix \(X\) is a four-dimensional vector, including the coordinates of the center position of the target signal source and velocity component in the direction of \(x\) and \(y\). Thus, \(X\), \(Z\), \(W\), \(V\), \(A\), \(H\), \(P\), \(Q\), and \(R\) are designed to be \(4\times 1\), \(2\times 1\), \(4\times 1\), \(2\times 1\), \(4\times 4\), \(2\times 4\), \(4\times 4\), \(4\times 4\), and \(2\times 2\) matrices, respectively. A Kalman filter algorithm has two major parts: predict and update. It predicts the target's current state based on the previous state of the target and obtains the predicted value of the target's current state. Furthermore, the Kalman filter is updated with the measured value of the target of the current state. The process of predicting and updating is shown in Figure 5.  
Predict (time update): Suppose that at the \(k-1\) moment, the state vector is \(X(k-1|k-1)\) and the posterior covariance matrix is \(P(k-1|k-1)\). Then the predicted state vector \(X(k|k-1)\) and prior covariance \(P(k|k-1)\) can be forecasted based on the Kalman filter model.