## 2 System Model
### DD Domain Input-Output Relationship in OTFS Modulation  
We will now delve into a comprehensive elucidation of the OTFS transmission process. To begin with, a sequence of information bits is subjected to a mapping procedure, resulting in a symbol set situated within the DD domain, as described in previous references [54; 55; 56; 57; 58]. This symbol set is formally denoted as \(\{X_{\text{DD}}[k,l]|k=0,\dots,N_{T}-1,\)\(l=0,\dots,M_{S}-1\}\). It is essential to underscore that both the indices k and l correspond to the DD references, respectively. Furthermore, it is worth noting that \(X_{\text{DD}}\) belongs to the modulation alphabet set \(\mathbb{A}\), defined as \(\mathbb{A}=\{a_{i}\in\mathbb{C}|i=1,2,\dots,|A|\}\), encompassing a range of modulation schemes, including quadrature amplitude modulation (QAM). Subsequently, the OTFS symbols un dergo a transformation using the inverse symplectic (IS) FFT. This transformation process will be further elaborated upon in the subsequent discussion. \[X_{\text{TF}}[n,m]=\sum_{k=0}^{N_{T}-1}\sum_{l=0}^{M_{S}-1}X_{\text{DD}}[k,l]e^{ 2\pi\left(\frac{nR}{N_{T}}-\frac{mI}{M_{S}}\right)}, \tag{4}\]  
for \(n=0,\dots,N_{T}-1\) and \(m=0,\dots,M_{S}-1\). In the next step, the modulator, which functions in the TF domain, processes the symbols \(X_{\text{TF}}[n,m]\), transforming them into a continuous-time waveform represented as \(s(t)\). This conversion is realized by harnessing the transmitter's pulse-shaping function (PSF) \(g_{tx}(t)\), making use of the Heisenberg transform technique as detailed in reference [59]. \[s(t)=\sum_{n=0}^{N_{T}-1}\sum_{m=0}^{M_{S}-1}X_{\text{TF}}[n,m]g(t-nT)e^{j2\pi m \Delta f(t-nT)}. \tag{5}\]  
Following its propagation through a time-varying channel, as elucidated in reference [60], the signal, initially denoted as \(s(t)\), reaches the receiver. So, we can obtain the received signal, designated as \(r(t)\):  
\[r(t)=\int\int h(\tau,\
u)s(t-\tau)e^{j2\pi\
u(t-\tau)}d\tau d\
u+w(t). \tag{6}\]  
Within this equation, the term \(w(t)\) signifies the existence of additive white Gaussian noise (AWGN), represented as \(\mathcal{N}_{0}\). On the other hand, \(h(\tau,\
u)\in\mathbb{C}\) denotes a complex baseband channel impulse response, where \(\tau\) and \(\
u\) represent the channel's reaction to an impulse with DD shift (\(\tau\) and \(\
u\), i.e.). The equation is  
\[h(\tau,\
u)=\sum_{i=1}^{P}h_{i}\delta(\tau-\tau_{i})\delta(\
u-\
u_{i}). \tag{7}\]  
In this context, our notation includes \(P\in\mathbb{Z}\) to represent the number of targets present. We further employ \(h_{i}\), \(\tau_{i}\), and \(\
u_{i}\) to signify the attributes of the ith target, specifically the reflection DD shift and channel coefficient, respectively. Additionally, the velocity and range of the \(i\)th target are denoted by \(R_{i}\) and \(\
u_{i}\). Within this framework, we can express the corresponding Doppler frequency \(\
u_{i}\in\mathbb{R}\) and delay \(\tau_{i}\in\mathbb{R}\) in a mathematical formulation. \[\tau_{i}=\frac{2R_{i}}{c}=\frac{l_{\tau_{i}}}{M_{S}\Delta f^{\prime}},\
u_{i} =\frac{2f_{c}v_{i}}{c}=\frac{k_{\
u_{i}}}{N_{T}T}. \tag{8}\]  
Subsequently, we establish a notation where \(f_{c}\) denotes the carrier frequency and \(c\) represents the speed of light. Furthermore, we introduce the utilization of \(l_{\tau_{i}}\in\mathbb{R}\) and \(k_{\
u_{i}}\in\mathbb{R}\) as the indices characterizing Doppler and delay properties for the \(i\)th target. In the context of this study, we expand our analysis to encompass fractional elements of Doppler and delay. These indices, specifically \(l_{\tau}=l_{i}+i_{i}\) and \(k_{\
u_{i}}=k_{i}+\kappa_{i}\), incorporate the following elements: \(l_{i}\in\mathbb{Z}\) and \(k_{i}\in\mathbb{Z}\), while \(i_{i}\in\mathbb{R}\) and \(\kappa_{i}\in\mathbb{R}\) signify the real numbers representing the fractional aspects. Upon reception at the receiver, the incoming signal, denoted as \(r(t)\), is subjected to a transformation to the TF domain.