# Optimal Steady-State Range Prediction Filter for Tracking with LFM Waveforms
## 1 Introduction  
Maneuvering-target tracking using laser, radar, and/or sonar is essential for monitoring systems in robots, aircrafts, and intelligent vehicles. For some applications in such technical fields, linear frequency-modulated (LFM) waveforms are used to improve the tracking accuracy in the range direction [1; 2; 3; 4; 5]. Known as range-Doppler coupling, the important consideration when using LFM waveforms is the error in the range measurement in relation to the Doppler effect [5]. Hence, Kalman [6; 7] and \(\alpha\)-\(\beta\) filters [8; 9; 10] based on the state equations that take range-Doppler coupling into account are used for the tracking of moving objects using LFM waveforms to reduce this type of error. In particular, the analysis of the \(\alpha\)-\(\beta\) filter using LFM waveforms (LFM-\(\alpha\)-\(\beta\) filter) is important because it theoretically clarifies the mathematical formulation of the steady-state performance indices of tracking systems and establishes design criteria of the filter [8; 9; 10; 11].  
The steady-state filter analysis that accounts for the effects of range-Doppler coupling is important in designing the tracking filter with LFM waveforms. Ref. [8] presents a theoretical analysis of the fundamental performance of the LFM-\(\alpha\)-\(\beta\) filter and reveals the relationship between the coefficient of range-Doppler coupling and the steady-state tracking performance. Although Ref. [8] only considers random process noise associated with acceleration in modeling the motion, this investigation is extended to general process noise, which considers random velocities and random jerks, among others, as given in Ref. [9]. However, in these studies, the design strategy of the tracking filter is not clearly presented and an empirical design is required. Ref. [10] proposes a gain design method for the LFM \(\alpha\)-\(\beta\) filter based on theoretical performance indices. In this design method, a design parameter called the deterministic tracking index is introduced. However, the paper optimizes the estimation error of range or range rate but the prediction performance is not optimized. Moreover, a steady-state Kalmanfilter is assumed in conventional studies. Therefore, the relationship between filter gains is limited to Kalman filter equations. This assumption does not lead to minimum prediction errors because the Kalman filter optimizes not the range prediction but the estimation of the state vector. For these reasons, the range prediction using the LFM \(\alpha\)-\(\beta\) filter is not optimized.  
In contrast, with respect to a general tracking filter (without LFM waveforms), our previous work [12; 13] introduced an efficiency performance index called a root-mean-square (RMS)-index, which expresses the steady-state root-mean-square error in position (range) predictions. Using this index, we achieved accurate tracking compared with conventional gains determined based on the Kalman filter equations. It is believed that the application of the RMS-index to the LFM \(\alpha\)-\(\beta\) filter resolves the above-mentioned problems.  
In this communication, a gain design method used to compose an LFM \(\alpha\)-\(\beta\) filter that minimizes the steady-state range prediction errors is proposed. The RMS-index of the LFM \(\alpha\)-\(\beta\) filter is derived and the proposed design method is introduced using the results obtained. A theoretical analysis proves that, using the proposed method, a smaller steady-state tracking error is achieved compared with the conventional design method based Kalman filter. Moreover, an application of the proposed method to an LFM radar simulation is demonstrated to show its effectiveness.