# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 4 The System Model
### CIR Estimation  
Analyzing the instantaneous CIR estimation processes in the case of OFDM and OTFS, there are some common points. The main peculiarity is that in the case of OFDM, the data and the pilots are both inside an OFDM symbol. In contrast, in the case of the RP-OTFS, the data are located separately between OFDM-based pilots. After obtaining the estimation of the instantaneous CIR for each delay in the range of \([0\cdots L]\) samples, the sample in position \(n=L+l+iM\), \(i\in\{0,\cdots,N-1\}\), \(l\in\{0\cdots L\}\) contains instantaneous CIR information at the \(l\) samples delay. In the RP-OTFS, we obtain the CIR estimate for the delay \(l\) once per \(M\) samples. The complete CIR can be recovered by interpolating the instantaneous CIR estimations for each \(l\) separately (in OFDM, the CFR estimate is interpolated). The choice of the interpolation method is based on the expected behavior of the CIR function, mainly on its derivatives. We will base our interpolation analysis on (6), which takes into account the effect of fractional reflections. But during analysis of the CIR function, we do not consider the particular DD grid; therefore, we can assume that no fractional reflection occurs. Then we can also assume that \((\kappa-k)\) and \((l-\ell)\in\mathbf{Z}\), hence \(\zeta_{N}(\kappa-k)=\sqrt{N},\mathrm{sinc}(l-\ell)=0\) and  
\[\
u_{m,l}(k)=\left\{ \begin{array}{ll}\
u_{\ell}(\kappa)e^{2\pi j\kappa(m-l )/MN},&\mbox{if $l=\ell$ and $k=[\kappa]_{N}$,}\\ 0,&\mbox{otherwise.}\end{array} \right. \tag{15}\]  
By calculating the derivatives of (15) over \(m\), one obtains the following recurrent formula:  
\[\frac{\partial^{p}\
u_{m,l}(k)}{\partial m^{p}}=(2\pi j\kappa/MN)^{p}e^{2\pi j \kappa(m-l)/MN}. \tag{16}\]  
As \(\lim_{p\to\infty}(2\pi j\kappa/MN)^{p}=0\) for \(MN>>\kappa\), the magnitude of the derivatives will decrease with increasing \(p\). The rate of decrease in derivatives depends on the size of the grid \(MN\) and on the Doppler frequency shift \(\kappa\). This means that for the same grid size, the optimal  
Figure 7: Corresponding DD matrix obtained from estimated exemplary \(\mathbf{H}\) matrix. interpolation method depends on the environment. For the cases where the Doppler effect is not very high, low-order interpolating polynomials can be used, for example, simple linear interpolation. At the same time, when the value of \(\kappa\) increases, more advanced interpolation methods, such as cubic spline interpolation, should be exploited. Generalizing to the more general case: when the Doppler frequency shift is not constant during the RP-OTFS frame, for example, when a target is accelerating, analysis of Equations (15) and (16) is more complicated. In this case \(\kappa\) becomes dependent on \(m\) and additional degrees of freedom of an interpolation function are required to fit it. After obtaining CSI estimation in the time domain (i.e., CIR \(\
u_{m,l}(k)\)), Equations (6)-(9) are used to transform CSI to the DD domain and model data transmission directly in the DD domain with (10). Based on the system model described above the block diagram of the receiver processing is shown in Figure 8. On the receiver side, it is assumed that the signal is perfectly synchronized with the strongest path of the multipath environment. It is also assumed that the signal is already transformed to the DD domain, so the processing is conducted mainly in the DD domain, except for the channel estimation part, because of the nature of OFDM-based pilots. The subsequent processing is divided into two branches: radar system processing and communication system processing. In the radar subsystem, only OFDM pilots are processed; the rest of the DD grid is ignored. First, the pilots are transformed to the TF domain where the instantaneous CFRs are estimated for each OFDM pilot separately using the Least Squares method. The CFR estimates are then transformed to the TD to obtain an instantaneous CIR. The CIR estimates are used in the communication processing branch.