# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 2 The OTFS Waveform  
In this section, the algorithmic basics of an OTFS waveform generation are presented only. Our goal is to provide the necessary background for the next part of the article. A detailed description of the OTFS modulation, with mathematical equations of definitions, features and relations, can be found in [14; 15; 21]. Here, as in our research, the application of the rectangular windowing is assumed only. For a better understanding of the OTFS, we will also describe the OFDM waveform generation here since in some of our simulations the OFDM performance is used as a baseline. In Figure 1 the time domain samples \(s(n)\) of a signal/waveform used in OTFS or OFDM are shown. The OTFS waveform is generated using the IQ data of the DD grid cells, while the OFDM waveform--the TF grid cells. Using a simple rearrangement, the signal/waveform \(s(n)\) can be represented in the time-delay (TD) domain as a complex-value matrix \(\mathbf{S}\) with \(M\) rows (\(k=0,1,\ldots,M-1\)) and \(N\) columns (\(n=0,1,\ldots,N-1\)) (in some sources this matrix is called a long time-short time or slow time-fast time). The DD and TD domains are connected with a pair of direct and inverse Zak transforms (ZT / IZT), more properties of ZT can be found in [15]. As shown in [16], the ZT (or IZT) of \(s(n)\) could be computed by applying the discrete Fourier transform (DFT) (or its inverse (IDFT)) to the rows of \(\mathbf{S}\). Applying DFT (IDFT) in the columns of \(\mathbf{S}\) switches the signal between the TD and TF domains. Concluding from Figure 1 and assuming the following relations: \(t\leftrightarrow n\) (time), \(\tau\leftrightarrow k\) (delay), \(v\leftrightarrow m\) (Doppler) and \(f\leftrightarrow l\) (frequency), the generated and transmitted OTFS waveform \(s_{TD}(n,k)\) can be computed in two ways from \(s_{DD}(m,k)\) grid cells (i.e., from \(s_{DD}(v,\tau)\)):  
1. directly via IZT, i.e., IDFT of \(s_{DD}(m,k)\) rows: \[\text{DD}\rightarrow\text{TD},\text{ IDFT[rows]}:\ \ s_{TD}(n,k)=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}s_{DD}(m,k)e^{j\frac{2 }{N}mn};\] (1)around via 2D inverse SFT of \(s_{DD}(m,k)\) followed by IDFTs of columns of obtained \(s_{TF}(m,k)\):  
\[\text{DD}\rightarrow\text{DF},\ \text{DFT[cols]}: s_{DF}(m,l)=\frac{1}{\sqrt{M}}\sum_{l=0}^{M-1}s_{DD}(m,k)e^{-j \frac{2\pi l}{M}l}, \tag{2}\] \[\text{DF}\rightarrow\text{TF},\ \text{IDFT[rows]}: s_{TF}(n,l)=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}s_{DF}(m,l)e^{j \frac{2\pi}{N}mn},\] (3) \[\text{TF}\rightarrow\text{TD},\ \text{IDFT[cols]}: s_{TD}(n,k)=\frac{1}{\sqrt{M}}\sum_{l=0}^{M-1}s_{TF}(n,l)e^{j \frac{2\pi}{M}l}. \tag{4}\]  
It can be observed that the last step (4) in the second method is equivalent to the generation of the OFDM waveform \(s_{TD}(n,k)\) from the given \(s_{TF}(n,l)\) IQ values. Therefore, OTFS modulation can be interpreted as OFDM modulation with specific precoding. Consequently, the existing OFDM hardware can be used in joint OFDM and OTFS-based ISAC systems. Here, it is important to note and remember that the transition from one domain to another can be accomplished by (I)DFTs in various ways (see the inner diamond in the lower part of Figure 1). When the original matrix (in the TD domain) contains the time-varying impulse response of the transmission channel in its columns, i.e., \(h(t,\tau)\) replaces/exchanges \(s(t,\tau)\), the DFTs of its rows give a Doppler-delay map of the transmission system (i.e., velocity-range map after appropriate scaling), while the DFTs of its columns result in a time-varying channel frequency response (CFR) of the system. The idea of OTFS modulation is explained separately with more details in the upper part of Figure 2, while in its lower part, different types of OTFS (different pilot vs. data arrangements) are compared. \(s_{DD}(m,k)\), here and in the remainder of the paper, is noted as \(x(m,k)\)--transmitted IQ carriers (the received ones will be marked as \(y(m,k)\)). The complex value QAM (Quadrature Amplitude Modulation) symbols are entries in the DD matrix \(\mathbf{X}\).