## 3 Application for Target Tracking Problem  
As an application of the proposed unified algorithm of FMS filtering and smoothing, a target tracking problem is considered through extensive computer simulations. In these days, a target tracking problem in indoor positioning systems for wireless network environments has been an interesting research problem and thus been applied successfully for locating and tracking people within a building and products stored within a warehouse using wireless sensors and mobile devices. Due to a variety of indoor wireless channel characteristics, an accurate tracking estimation of a moving target is required for indoor positioning systems. However, a couple of noises such as system noise and measurement noise can contaminate the estimated position and thus can cause deterioration of target tracking performance. These noises can be caused by various factors such as environmental interference, turbulence affecting the target's movement, inaccuracies of sensor measurement data, and human's navigation error in a perfect straight \begin{table}
\begin{tabular}{c c c} \hline \hline  & **FMS Filter and Smoother** & **Fixed-Lag Kalman Smoothing Filter** \\ \hline On-line computation (flops) & \(n\times M\) (respectively) & \(4n^{3}+8n^{2}+6n+1\) \\ \hline Memory shift & \(M-1\) & _None_ \\ \hline Memory size & \(M\) & \(d\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of on-line computation and memory size. line. Hence, these noises must be filtered for tracking a true path of a moving target. To provide the most accurate position and velocity of a moving target on the real-time indoor positioning in wireless network environments, some research adopted both IMS and FMS filters to reduce the effect of noises [20]. This paper considers only the one dimensional moving target for the \(X\) direction for the simplicity. The state estimation filter for the target tracking is typically designed with the state-space model called the constant velocity motion model. The constant velocity motion model assumes that targets move with constant velocity, that is, zero-acceleration within a short sampling time. However, in real situations, moving targets maneuver and change velocity and thus move temporarily with nonzero acceleration. Thus, the moving target's successive locations for the \(X\) direction can be represented by the 3 rd-order discrete-time state-space model with sampling time \(T\) and acceleration term:  
\[x_{i+1} = Ax_{i}+Gw_{i},\] \[z_{i} = Cx_{i}+v_{i} \tag{15}\]  
where  
\[x_{i}\stackrel{{\triangle}}{{=}}\left[ \begin{array}{c}x_{p}: \text{position}\\ x_{v}:\text{velocity}\\ x_{a}:\text{acceleration}\end{array} \right],\ A=\left[ \begin{array}{ccc}1&T&T^{2 }/2\\ 0&1&T\\ 0&0&1\end{array} \right],\ G=\left[ \begin{array}{cc}T/2&0\\ 1&0\\ 0&1\end{array} \right],\ C=[1\ \ 0\ 0]. \tag{16}\]  
The state vector \(x_{i}\) at time \(i\) consists of three state variables, \(x_{p}\) to represent the moving target's random _position_, \(x_{v}\) to represent the corresponding _velocity_, \(x_{a}\) to represent the corresponding _accelerations_. Typically, since the acceleration occurs over a short time interval, this can be considered as a temporary uncertainty, that is, an unknown input and thus considered as an unknown state term. Hence, since the target tracking problem in this paper does not need the control input term, the proposed unified algorithm (13) is applied with assuming that the control input matrix is zero, that is, \(B=0\). Because of random disturbances by fading and shadowing, the state vector \(x_{i}\) cannot be measured directly. Thus, the measurement vector \(z_{i}\) is modeled by only the position to take these effects into account filtering or smoothing estimates. As shown in [23], target tracking applications using the constant velocity motion model often adopt the state-space model where the state equation is linear and only the measurement equation is nonlinear.