# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 4 The System Model  
### The Communication System Model  
The considerations presented in this section are based on [15; 16]. The OTFS waveform uses the DD domain to modulate the IQ data. It is important to note that below, the exchange of meaning between variables \(m\) and \(k\) is made compared with Section 2, where \(k\) denotes the delay and \(m\) the Doppler frequency shift. Currently, the notation is reversed. As a consequence, Doppler-delay matrices are now considered as delay-Doppler, as in [15]. This is a result of the inconsistency in the historically established names of the domains: time-frequency and Doppler delay. The time dimension in TF (or TD) results in the Doppler dimension in the Doppler-delay representation. However, the term \"delay-Doppler\" is commonly used in modern literature. Figure 5: The RP-OTFS and ZP-OTFS waveforms in DD, TD and time domains. Let us assume that on the DD grid on the transmitter side, we have \(\mathbf{X}\in\mathbb{C}^{M\times N}\), where \(M\) is the number of carriers in the delay direction and \(N\) is the number of carriers in the Doppler directions. Let the signal sampling frequency be \(f_{s}\). Now we can calculate basic signal and resolution parameters, as well as the maximum delay and maximum Doppler of the DD-grid. The length of the OTFS frame is \(T=M\cdot N/f_{s}\). We assume that the OTFS signal is critically sampled and, therefore, the bandwidth of the signal \(B=f_{s}\). The delay resolution, Doppler resolution, maximum delay, and maximum Doppler are equal to \(1/f_{s},f_{s}/M,\,M/f_{s}\), and \(N\cdot f_{s}/M\), respectively. Transformation from DD to TD domain is performed using the inverse DFT over DD matrix rows (operation known as the inverse Zak transform), while transformation from TD to DD domain--using the DFT over TD matrix rows (operation known as the Zak transform) [16]. Let the received signal be denoted on the DD grid as \(\mathbf{Y}\in\mathbb{C}^{M\times N}\). The vertical vectors \(\mathbf{x}_{m}=[X(m,0),\,X(m,1),\ldots,X(m,N-1)]^{T}\) and \(\mathbf{y}_{m}=[Y(m,0),\,Y(m,1),\ldots,Y(m,N-1)]^{T}\) are the \(m\)-th rows of \(\mathbf{X}\) and \(\mathbf{Y}\), respectively: \(\mathbf{x}_{m}\), \(\mathbf{y}_{m}\in\mathbb{C}^{N\times 1}\). By combining vectors \(\mathbf{x}_{m}\) we define the vertical vector \(\mathbf{x}\in\mathbb{C}^{M\times 1}\), \(\mathbf{x}=[\mathbf{x}_{0},\mathbf{x}_{1},...,\mathbf{x}_{M-1}]^{T}\), similarly to \(\mathbf{y}\in\mathbb{C}^{M\times 1}\), \(\mathbf{y}=[\mathbf{y}_{0},\mathbf{y}_{1},\ldots,\mathbf{y}_{M-1}]^{T}\). For simplicity, we will normalize the delay and Doppler to the DD grid resolution in the following way: \(\ell_{i}=\tau_{i}\cdot\frac{T}{MN}\) is a normalized delay, and \(\kappa_{i}=\frac{v_{i}}{NT}\) is a normalized Doppler. Here, it is important to note that \(\ell_{i}\) and \(\kappa_{i}\) are not necessarily integers; they could be fractional values, as mentioned in the Introduction. We will denote the set of all time delays by \(\mathcal{L}=\{\ell_{i}\}\), and the set of all Doppler frequency shifts by \(K_{\ell}=\{\kappa_{i}\mid\ell=\ell_{i}\}\). The complex gain/attenuation, corresponding to the delay \(\ell_{i}\) and the Doppler offset \(\kappa_{i}\), is denoted as \(\
u_{\ell}(\kappa)\). At this stage, we assume that the maximum delay of reflections is not very large (a few samples). We also assume that the [PERSON] sampling theorem is also met in the Doppler direction, that is, \(-N/2\leq\kappa_{i}\leq N/2\).