# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 4 The System Model
### The Communication System Model
It is shown in [16], that in the general case of fractional delay and fractional Doppler, the CSI can be expressed in the DD domain by the following formula:  
\[\mathbf{v}_{m,l}(k)=\frac{1}{\sqrt{N}}\sum_{\ell\in\mathcal{L}}\left(\sum_{\kappa \in\mathcal{K}_{\ell}}\
u_{\ell}(\kappa)\underbrace{e^{j2\pi k(m-l)}\zeta_{N} (\kappa-k)}_{Doppler\ component}\right)\underbrace{\text{sinc}(l-\ell)}_{Delay\ component}, \tag{6}\]  
where \(\zeta_{N}(x)=\frac{1}{\sqrt{N}}\frac{\sin(\pi x)}{\sin(\frac{\pi x}{N})}e^{ \frac{i\pi(N-1)}{N}}\) is a periodic sinc function, \(m\) and \(k\) are the sample coordinates on the DD grid in the delay and Doppler directions, respectively, \(m\in 0\cdots M-1,k\in 0\cdots N-1\). \(l\) is a time shift of the CSI. We assume that CSI changes in time due to the Doppler effect: \(l\in 0\cdots(M-m)N\), \(m,k,l\in\mathbb{Z}\). The resulting signal, after passing the Doppler multipath wireless channel, is equal to the following summation of circular convolutions (for each delay) in the DD domain [16]:  
\[\mathbf{y}_{m}=\sum_{l\in\ell}\mathbf{v}_{m,l}\mathbin{\text{\text{\textcircled{ \textcircled{\textcircled{\textcircled{\textcircled{\textThe arrangement of the submatrices \(\mathbf{K}_{m,l}\) (8) inside the matrix \(\mathbf{H}\) is as follows:  
\[\mathbf{H}[ \begin{array}{l}mN+1\ldots(m+1)N,\ (m-l)N+1\ldots(m-l+1)N \end{array} ]=\] \[\quad=\left\{ \begin{array}{ll}\mathbf{K}_{m,l},&m\in 0\cdots M-1, \ l\in 0\cdots(M-m)N,\\ 0,&\text{otherwise}.\end{array} \right. \tag{9}\]  
The visualization of the placement of the submatrices \(\mathbf{K}_{m,l}\) within the matrix \(\mathbf{H}\) is given in Figure 6.  
Here we want to draw the Readers' attention to two important details about \(\mathbf{H}\): the matrix \(\mathbf{H}\) describes the CSI directly in the DD domain and the format of the matrix \(\mathbf{H}\) is uncommon. The change in the format of the matrix \(\mathbf{H}\) is due to the fact that the OTFS symbol has two dimensions (2D), unlike OFDM (1D). Based on the above description, the signal that passes through the multipath channel with Doppler frequency shifts can be written as [16]:  
\[\mathbf{y}=\mathbf{H}\cdot\mathbf{x}+\mathbf{w}, \tag{10}\]  
where \(\mathbf{w}\) is a noise vector in the DD domain. The vector \(\mathbf{w}\) is defined as \(\mathbf{w}_{m}=[W(m,0),W(m,1),\ldots,W(m,N-1)]^{T}\), \(\mathbf{w}=[\,\mathbf{w}_{0},\,\mathbf{w}_{1},\ \ldots,\mathbf{w}_{M-1}]^{T}\). The \(\mathbf{W}\) matrix is the result of the Zak transform performed on the time domain noise vector \(w\sim\mathcal{N}(0,\sigma^{2})\), where \(\sigma^{2}\) is the variance of the noise (here we assume that the rectangle windowing function was used in the Zak transform). After the Zak transform of \(w\), one obtains \(\mathbf{w}\sim\mathcal{N}(0,\sigma^{2}/N)\)[42]. Based on the above description, the signal that passes through the multipath channel with Doppler frequency shift can be modeled directly in the DD domain using the Bayesian General Linear Model [43]. This allows us to directly use the well-known estimation and bit detection methods designed/developed for OTFS signals, for example, the linear minimum mean square error (LMMSE). The LMMSE estimate of the transmitted data \(d_{k}\) is defined as [15]:  
\[\widehat{\mathbf{d}}=[\mathbf{F}_{N}\otimes\mathbf{I}_{M}]^{\dagger}\mathbf{H }^{\dagger}\Bigg{[}\mathbf{H}\mathbf{H}^{\dagger}+\frac{\sigma_{w}^{2}}{ \sigma_{d}^{2}}\mathbf{I}\Bigg{]}^{-1}\mathbf{y}, \tag{11}\]  
where \(\mathbf{F}_{N}\) is the discrete Fourier transform matrix of size \(N\), \(\mathbf{I}_{M}\) is the identity matrix of size \(M\), \(\otimes\) is the Kronecker product, \(\dagger\) denotes complex conjugation and transposition, and  
Figure 6: Input-output relation for OTFS modulation with zero padding in the DD domain [16]. finally, \(\sigma_{w}^{2}\) and \(\sigma_{d}^{2}\) are noise and data variances (see Figure 6). The fast algorithm for solving (11) is proposed in [44]. The whole channel estimation and equalization process can be performed in the DD domain.