## 3 OTFS-Based Radar Sensing
### Fractional Parameter Estimation  
Upon examining the peaks within the DD plane denoted as \(\mathbf{\Gamma}\), we are limited to discerning only the integer components of the DD indices [63; 64; 65; 66]. This constraint implies that the fractional segments of these respective indices remain undisclosed, thereby diminishing the precision of the sensing outcomes, as elaborated upon in references [67; 68; 69; 70]. To rectify these concerns, this subsection will present an uncomplicated technique for estimating the fractional parameters.  
Upon a closer examination of the 2D MF matrix illustrated in Figure 2, it becomes evident that targets with fractional indices exhibit a phenomenon known as power leakage (PL) into neighboring DD bins. A thorough exploration of the origins of this PL resulting from fractional indices is available in [53; 71]. Building upon this insight, we introduce a distinctive method for estimating these fractional indices differentially, as detailed below.  
To calculate the fractional component of the Doppler index of the \(i\)-th target, we first assume that the delay or Doppler shift is different for each target. Following the 2D MF operation, the row indices associated with the highest and second highest magnitudes within the \(l_{i}\)-th column of the DD matrix, denoted as \(k_{1}^{\prime}\) and \(k_{2}^{\prime}\), are:  
\[\begin{split} k_{1}^{\prime}&=\operatorname*{arg \,max}_{k\in\{[-N_{T}/2],\ldots,[N_{T}/2]-1\}}|V[k,l_{i}]|,\\ k_{2}^{\prime}&=\operatorname*{arg\,max}_{k\in\{[-N _{T}/2],\ldots,[N_{T}/2]-1\}\setminus\{k_{\tau_{1}}^{\prime}\}}|V[k,l_{i}]|. \end{split} \tag{24}\]  
Upon establishing the values of \(k_{1}^{\prime}\) and \(k_{2}^{\prime}\), a significant observation comes to light. In noise-free scenarios, the genuine Doppler index, represented as \(k_{i}+\kappa_{\
u_{i}}\) and linked to the \(i\)-th target, logically falls within the interval delineated by \(k_{\
u_{1}}^{\prime}\) and \(k_{\
u_{2}}^{\prime}\). It is worth emphasizing that \(k_{i}\) coincides with \(k_{\
u_{1}}^{\prime}\), and the discrepancy between \(k_{\
u_{2}}^{\prime}\) and \(k_{\
u_{1}}^{\prime}\) precisely equals 1. Moreover, the proportion between the magnitudes of the MF coefficients associated with Doppler indices \(k_{\
u_{1}}^{\prime}\) and \(k_{\
u_{2}}^{\prime}\) at the same delay index is reasonably estimated in the following manner:  
\[\frac{|V[k_{\
u_{1}}^{\prime},l_{i}]|}{|V[k_{\
u_{2}}^{\prime},l_{i}]|}\approx \frac{|k_{\
u_{2}}^{\prime}-k_{\
u_{1}}^{\prime}-\kappa_{\
u_{i}}|}{|-\kappa_ {\
u_{1}}|}. \tag{25}\]  
The error of the approximation above is \(\mathcal{O}\left(\frac{1}{M_{5}N_{T}}\right)\).  
As a result, we can calculate the fractional Doppler value as:  
\[\kappa_{\
u_{i}}=\frac{(k_{\
u_{2}}^{\prime}-k_{\
u_{1}}^{\prime})|V[k_{\
u_{ 2}}^{\prime},l_{i}]|}{|V[k_{\
u_{1}}^{\prime},l_{i}]|+|V[k_{\
u_{2}}^{\prime},l_{i}]|}. \tag{26}\]  
Likewise, employing the same approach described above for fractional delay taps, we obtain:  
Figure 2: An illustration of the corrected matrix under \(P=3\) targets.  
\[\iota_{\tau_{i}}=\frac{(l^{\prime}_{\tau_{1}}-l^{\prime}_{\tau_{1}})|V[k_{i},l^{ \prime}_{\tau_{2}}]|}{|V[k_{i},l^{\prime}_{\tau_{1}}]|+|V[k_{i},l^{\prime}_{\tau _{2}}]|}. \tag{27}\]