# OTFS: A Potential Waveform for Space-Air-Ground Integrated Networks in 6G and Beyond
## 3 System Model
### Received Signal and Channel Model  
Consider a static multipath wireless channel shown in Figure 3a. In this scenario, there is no Doppler effect on the transmitted signal because there is no mobility in the system. However, due to the difference in the propagation delay between the line of sight (LoS) path and reflected paths, multiple copies of the signal s(t) arrive at the receiver at different times. Hence, the received signal \(\mathrm{y}(t)\) can be given by [6]:  
\[\mathrm{y}(t)=g_{1}\mathrm{s}(t-\tau_{1})+g_{2}\mathrm{s}(t-\tau_{2}) \tag{1}\]  
where \(r\) is the distance, \(\tau_{1}=\tau_{1}/c\) is the LoS path delay, \(\tau_{2}=(\tau_{2}+\tau_{3})/c\) is the reflected path delay, and \(c\) is the speed of light (\(3\times 10^{8}\mathrm{m/s}\)). The difference between the maximum and minimum propagation delays (\(\tau_{\mathrm{max}}-\tau_{\mathrm{min}}\)) gives the delay spread. Consider another scenario where there is mobility between the transmitter, receiver, or reflectors in the environment, with a velocity \(v\), as shown in Figure 3b. The Doppler shift due to a relative velocity \(v\) can be expressed as \(\frac{v}{c}f_{c}\). Hence, the received signal is given  
Figure 3: Wireless channel scenarios. (**a**) Static multipath channel with different propagation delay. (**b**) Multipath channel with Doppler effects due to mobility and different angles of arrival [6]. as the sum of the different copies of the transmitted signal, which is delayed and Doppler shifted as shown [6]:  
\[y(t)=\underbrace{g_{1}\mathrm{e}^{2\pi\
u_{1}(t-\tau_{1})}}_{g(\tau_{1},t)}s(t- \tau_{1})+\underbrace{g_{2}\mathrm{e}^{\beta\tau\
u_{2}(t-\tau_{2})}}_{g(\tau_{2 },t)}s(t-\tau_{2}) \tag{2}\]  
where \(g(\tau_{i},t)=g_{i}\mathrm{e}^{2\pi\
u_{i}(t-\tau_{i})},i=1,2\), is the time-varying attenuation due to channel effects, \(\
u_{1}=\frac{\pi}{c}f_{c}\), and \(\
u_{2}=\frac{\mathrm{t}\mathrm{t}\mathrm{t}\mathrm{t}\mathrm{t}\mathrm{t} \mathrm{t}}{c}f_{c}\) represents the Doppler shift of the direct LoS path and reflected path, respectively, and \(|\
u_{2}-\
u_{1}|\) is the Doppler spread. The received signal can, therefore, be represented by  
\[y(t)=\int_{0}^{\infty}g(\tau,t)s(t-\tau)d\tau \tag{3}\]  
where \(g(\tau,t)\) is the channel impulse response, which is a function of time \(t\) and delay \(\tau\). Taking the Fourier transform of the delay-time channel impulse response \(g(\tau,t)\) along the delay domain gives the time-frequency channel impulse response at the time \(t\) as shown:  
\[H(f,t)=\int\tau g(\tau,t)\mathrm{e}^{-j2\pi\tau}d\tau \tag{4}\]  
If the channel has multiple propagation paths P, and the time-varying attenuation is given by \(g(\tau_{i},t)=g_{i}\mathrm{e}^{j2\pi\
u_{i}(t-\tau_{i})},i=1,2\), then the frequency response of the channel can be given as  
\[H(f,t)=\sum_{i=1}^{P}g_{i}\mathrm{e}^{-\beta\tau\
u_{i}}\mathrm{e}^{-j2\pi(f \tau_{i}-\
u_{i}t)} \tag{5}\]  
where \(H(f,t)\) is assumed to be slowly time-varying. Hence, when the channel is static, and \(\
u_{i}=0\), then the channel becomes time-independent \(H(f)\). Therefore, due to the limited number of channel paths \(P\), the sparse representation of the channel in the DD domain is given by  
\[h(\tau,\
u)=\sum_{i=1}^{P}g_{i}\mathrm{e}^{-\beta\tau\
u_{i}\tau_{i}}\delta( \tau-\tau_{i})\delta(\
u-\
u_{i}) \tag{6}\]  
If the constant phase shift \(\mathrm{e}^{-j2\pi\
u_{i}\tau_{i}}\) is absorbed into the channel coefficient \(g_{i}\), then the wireless channel in the DD domain can be completely represented by the parameters \((g_{i},\tau_{i},\
u_{i})\), for \(i=1,\ldots,P\) as shown:  
\[h(\tau,\
u)=\sum_{i=1}^{P}g_{i}\delta(\tau-\tau_{i})\delta(\
u-\
u_{i}) \tag{7}\]  
Hence, the received signal given in (2) can therefore be expressed as  
\[y(t)=\sum_{i=1}^{P}g_{i}\mathrm{e}^{j2\pi\
u_{i}(t-\tau_{i})}s(t-\tau_{i}) \tag{8}\]  
which is the summation of the different copies of the transmitted signal, which are delayed and Doppler-shifted.