# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 4 The System Model
### The Radar System Model  
The radar channel model is based on the communication model (10). Suppose that the radar channel has \(\tilde{\mathcal{L}}=\tilde{\ell}_{i}\) delays and the set of all Doppler frequency shifts \(\tilde{K}_{\ell}=\{\tilde{\kappa}_{i}\mid\tilde{\ell}=\tilde{\ell}_{i}\}\). Then similarly to (6), we can rewrite  
\[\tilde{v}_{m,l}(k)=\frac{1}{\sqrt{N}}\sum_{\tilde{\ell}\in\mathcal{L}}\left( \sum_{\tilde{k}\in\tilde{K}_{\ell}}\tilde{v}_{\tilde{\ell}}(\tilde{\kappa})e^ {l2\pi k(m-l)}\tilde{\varsigma}_{N}(\tilde{\kappa}-k)\right)\text{sinc}(l- \tilde{\ell}), \tag{12}\]  
where \(\tilde{v}_{\tilde{\ell}}(\tilde{\kappa})\) is the RCS of the target with delay \(\tilde{\ell}\) and \(\tilde{\kappa}\) Doppler shift.  
By applying transformations analogous to those used in the communication model, the final radar channel model is derived:  
\[\tilde{\mathbf{y}}=\tilde{\mathbf{H}}\cdot\mathbf{x}+\mathbf{w}. \tag{13}\]  
In this article, we focus only on the CSI-based approach, explained in [23; 26], as it has a more common processing part with the communication subsystem than CAF. In the CSI-based approach, the results of the OFDM-like pilot-based CIR estimation are used to detect the targets. This reduces the complexity of radar processing significantly lower than in the case of CAF-based processing. Another not-so-straightforward disadvantage of the CAF is that the limited freedom of choice of waveform parameters can be critical for the shape of the ambiguity function. This limitation comes from the fact that the waveform structure we choose should allow the decoding of the transmitted data in a receiver. For this reason, some additional communication components should be inserted into the transmitted signal that are treated as unwanted \"artifacts\" by the radar subsystem, e.g., repeating reference pilot signals and repeating fragments of OFDM waveforms called a cyclic prefix. The repeatable parts of the signal generate undesirable peaks in CAF, so-called ghosts [25].  
In the CSI-based approach, the processing begins with the \(\tilde{\mathbf{H}}\) matrix estimation. The channel matrix estimation step is common for both subsystems: communication and radar. The process of the CSI estimation in case of RP-OTFS is described in [26].  
Each subdiagonal of the \(\tilde{\mathbf{H}}\) contains information about specific \(\tilde{\ell}\), \(\tilde{\kappa}\) and \(\tilde{v}_{\tilde{\ell}}(\tilde{\kappa})\). This way of presenting information is not suitable for standard radar processing algorithms. Therefore, before target detection, an additional step is needed to transform \(\tilde{\mathbf{H}}\) to the DD grid (different delays in rows, different Doppler frequency shifts in columns). The transformation should be performed as follows:  
\[\tilde{\mathbf{H}}_{DD}[m,k]=\tilde{\mathbf{H}}[m\cdot N+k,l_{0}\cdot N], \tag{14}\]  
where \(l_{0}\) is the time shift of the CSI estimation relative to the beginning of the OTFS frame. In most practical cases \(l_{0}=0\), the range of \(l_{0}\) is \(0\ldots L-1\).  
In Figure 7 an exemplary result of the matrix \(\tilde{\mathbf{H}}_{\mathbf{DD}}\) conversion is shown. Reflections from two static objects (so-called clutter) and from one moving object/target are visible in it.  
The obtained DD matrix should be processed by any standard radar detection algorithm, such as the constant false alarm rate (CFAR) [19]. The detection of targets performed by the radar subsystem is beyond the scope of this paper. However, in order to compare the performance of the RP-OTFS in simulations with other wave the processing gain for different modulation schemes. This gain was defined as a ratio of the output SNR (defined as the power of the DD cell with the target and noise, after radar processing, divided by the power of the DD cells with noise only) to the input SNR (signal-to-noise ratio at the input of the receiver) and was calculated for different input SNR values.