# OTFS: A Potential Waveform for Space-Air-Ground Integrated Networks in 6G and Beyond
## 3 System Model
### OTFS Modulation and Demodulation  
At the transmitter, information symbols \(X[k,l]\) are initially mapped onto an \(M\times N\) two-dimensional DD transmit data grid \(x_{dd}[k,l]\), where \(k\) and \(l\) are the discrete delay and Doppler axis indices, \(k=0,1,\ldots,M-1\) and \(l=0,1,\ldots,N-1\), respectively. The grid represents one OTFS symbol, and each column of the grid is a sub-symbol [24]. Hence, \(k\) and \(l\) can be referred to as the normalized delay and Doppler shifts, while the actual discretized delay (\(\tau_{k}\)) and Doppler shift (\(\
u_{l}\)) are given by  
\[\tau_{k}=\frac{k}{M\Delta f},\;k=0,1,\ldots,M-1 \tag{14}\]  
\[\
u_{l}=\frac{l}{NT},\;l=0,1,\ldots,N-1 \tag{15}\]  
where \(\Delta f\) is the subcarrier spacing in Hertz and \(T\) is the OTFS sub-symbol duration in seconds (plus the cyclic prefix, if any). These symbols are then transformed into the TF domain using the ISFFT, facilitating transmission over the wireless channel. The received signal is converted back to the DD domain at the receiver using the SFFT, enabling effective demodulation and decoding of the transmitted information. The discrete channel response \(H[m,n]\) can be derived from the continuous channel \(h(\tau,\
u)\) using the relationship \(t=nT\) and \(f=m\Delta f\) as shown:  
\[H[m,n]=\iint h(\tau,\
u)e^{i2\pi(vnT-m\Delta f\tau)}d\tau dv \tag{16}\]  
For a discrete channel with \(P\) paths, the DD domain channel is given by  
\[h[k,l]=\sum_{i=1}^{P}g_{i}\delta(k-k_{i})\delta(l-l_{i}) \tag{17}\]and the discrete ISFT transforms this DD domain channel \(h[k,l]\) into the discrete TF domain channel \(H[m,n]\) as follows:  
\[H[m,n]=\frac{1}{\sqrt{MN}}\sum_{k=0}^{M-1}\sum_{l=0}^{N-1}h[k,l]e^{(j2\pi(\frac{ n}{N}-\frac{mk}{M}))} \tag{18}\]  
The process of OTFS modulation and demodulation can be performed on an existing OFDM system, where \(M\) corresponds to the number of subcarriers (in an OFDM symbol) and \(N\) the number of OFDM symbols contained in a single frame. Figure 4 shows the relationship between OTFS and OFDM systems in the DD and TF domains, respectively. Hence, to convert modulated symbols from the delay-Doppler domain \(x_{dd}[k,l]\) to the time-frequency domain \(X_{tf}[m,n]\), the ISFFT is employed as shown [13]:  
\[X_{tf}[m,n]=\frac{1}{\sqrt{MN}}\sum_{k=0}^{M-1}\sum_{l=0}^{N-1}x_{dd}[k,l]e^{(j 2\pi(\frac{n}{N}-\frac{mk}{M}))} \tag{19}\]  
where \(m\) and \(n\) denote the frequency and time indices, respectively. The ISFFT can be decomposed into two steps. First, an inverse Discrete Fourier Transform (IDFT) is applied to the transpose of \(x_{dd}(x_{dd}^{T})\), which operates along the Doppler axis and transforms \(x_{dd}\) from the DD domain to the delay-time domain \(X_{dt}\), and then the result is transposed back to the proper array dimension as shown:  
\[X_{dt}[k,n]=\frac{1}{\sqrt{N}}\sum_{l=0}^{N-1}x_{dd}[k,l]e^{j\frac{2\pi nl}{N}} =IFFT\Big{(}x_{dd}^{T}\Big{)}^{T} \tag{20}\]  
Then, a Discrete Fourier Transform (DFT) is applied along the delay axis of \(X_{dt}\), to obtain the TF domain representation as shown:  
\[X_{tf}[m,n]=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}X_{dt}[k,n]e^{-j\frac{2\pi nl}{M} }=FFT(X_{dt}) \tag{21}\]  
Combining (20) and (21), the ISFFT can be simplified and implemented as  
\[X_{tf}[m,n]=FFT\Big{(}IFFT\Big{(}x_{dd}^{T}\Big{)}^{T}\Big{)} \tag{22}\]  
Figure 4: OTFS modulation and demodulation block diagram. The time-frequency domain signal \(X_{tf}[m,n]\), can now be converted to a time-domain signal \(x(t)\) using the Heisenberg transform (equivalent to an OFDM modulator) as shown:  
\[\
u_{l}=\frac{l}{NT},\;l=0,1,\ldots,N-1 \tag{15}\]0  
where \(T\) is the duration of the OFDM symbol (plus cyclic prefix if any) and \(g(t)\) is the pulse-shaping filter. The pulse-shaping filter aims to mitigate the channel spreading caused by fractional Doppler shifts (when the Doppler frequency does not fall on a multiple of \(1/(NT)\)).