## 2 System Model
### Introduction of OTFS Modulation  
We utilize the symbols \(N_{T}\in\mathbb{N}\) and \(M_{S}\in\mathbb{N}\) to signify the quantity of time slots and subcarriers allocated to each OTFS frame, respectively, with \(N_{T}\) belonging to the set of positive integers. The discrete DD plane can be succinctly depicted as follows:  
\[\boldsymbol{\Gamma}=\bigg{\{}\bigg{(}\frac{k}{N_{T}T},\frac{l}{M_{S}\Delta f} |k\in[0,N_{T}-1],l\in[0,M_{S}-1]\bigg{)}\bigg{\}}. \tag{1}\]The duration of a single time slot can be represented by \(T\), and the separation between subcarriers is denoted as \(\Delta f\). Consequently, the entire duration of an OTFS frame is given by \(N_{T}T\), while the occupied bandwidth of the system is \(M_{S}\Delta f\). In terms of resolution, we attain a delay resolution of \(\frac{1}{M_{S}\Delta f}\) and a Doppler shift resolution of \(\frac{1}{N_{T}T}\).  
The cross-ambiguity function (CAF) between the pulse-shaping filters (PSFs) at the transmitter \(g_{tx}(t)\) and the receiver \(g_{rx}\) can be expressed as follows:  
\[A_{g_{rx}g_{tx}}(t,f)\overset{\Delta}{=}\int_{t}g_{tx}(t^{\prime})g_{rx}^{*}(t ^{\prime}-t)e^{\beta\pi\
u(t^{\prime}-t)}dt^{\prime}. \tag{2}\]  
To ensure clarity in our discourse, we focus our analysis on situations in which both the transmitter and receiver operating within the OTFS framework are endowed with ideal PSFs. As elucidated in [53], these filters exhibit bi-orthogonal characteristics, which is delineated as follows:  
\[\begin{split}& A_{g_{rx}g_{tx}}(t,f)|_{t=(n-n^{\prime})T-\frac{ l_{rx}}{M_{S}^{*}\Delta f}(m-m^{\prime})\Delta f-\frac{l_{rx}}{N_{T}T}}\\ &\quad=\delta[n]\delta[m]q_{\tau_{max}}(t-nT)q_{\tau_{max}}(f-m \Delta f),\end{split} \tag{3}\]  
where  
\[q_{a}(t)=\left\{\begin{array}{cc}1&-a<x<a\\ 0&otherwise.\end{array}\right.\]  
While the OTFS framework presents promising capabilities for addressing challenges in wireless communication and sensing systems, there are several practical challenges that need to be considered during implementation in real-world scenarios, including:  
1. Computational Complexity: The OTFS framework involves complex mathematical transformations, including the conversion from the time-delay domain to the DD domain. These operations can be computationally intensive, especially in real-time applications or resource-constrained environments.
2. Channel Estimation and Tracking: Accurate channel estimation and tracking are crucial for the success of the OTFS framework. In dynamic environments with rapidly changing channel conditions, developing robust algorithms for real-time channel estimation and tracking becomes a challenge.
3. Synchronization: Achieving precise synchronization is essential for accurate transformation between domains in the OTFS framework. Synchronization challenges may arise in scenarios with high mobility, multipath propagation, and varying signal delays.
4. Adaptability to Non-Ideal Conditions: Real-world environments often introduce non-ideal conditions such as frequency-selective fading, non-line-of-sight (NLOS) scenarios, and interference. The OTFS framework needs to be adaptable to such conditions, requiring advanced signal processing techniques and algorithms.  
While the OTFS framework shows promise, its successful implementation in real-world wireless communication and sensing systems requires careful consideration of these practical challenges.