# A Comprehensive Experimental Emulation For OTFS Waveform RF-Impairments
## 3 Channel Effects Furthermore, RF-Impairments
### Carrier Frequency Offset
#### 3.1.1 CFO Due to Doppler Shift  
Firstly, in our setup, to create the Doppler effect, a metallic fan (stirrer) with 40 cm \(\times\) 40 cm \(\times\) 20 cm dimensions and a paddle radius of 0.15 m is used to create Doppler shifts for high-speed. In order to convert these shifts into Doppler spread, a reverberation chamber with the dimensions of 120 cm \(\times\) 68 cm \(\times\) 55 cm is used to create more multipath and provide enough Doppler spread for the experiment. One method for generating Doppler dispersion within a reverberation chamber is to use a stirrer that moves continuously while keeping a steady receiver/transmitter setup. as shown in Figure 3[48]. The amount of the shift in the carrier frequency \(f_{D}\) is evaluated as follows  
\[f_{D}=\frac{V}{\lambda}\cos\theta=\frac{\omega_{p}\cdot r}{\lambda}\cos\theta, \tag{12}\]  
where \(V\), \(\omega_{p}\), \(r\), and \(\theta\) represent stirring paddles' effective linear speed, paddles angular speed, the size of the stirrer, and the effect of the angle of arrival (AOA), respectively. The \(\lambda\) is the wavelength of the signal in free space and is equal to \(\lambda=C/f_{c}\), where \(C\) is the speed of the light and \(f_{c}\) is the carrier frequency. In order to have a sufficient impact on the shapes of the Doppler power spectrum, the RF absorbers are arranged in such a manner that they are positioned around the stirrer. Due to the fact that the stirrer's blades move in opposite directions at the same effective linear speed [48].  
Figure 4: Laboratory equipment setup.  
Doppler shift is created through experimentation by utilizing a stirrer with dimensions of 40 cm \(\times\) 40 cm \(\times\) 20 cm and a paddle with a radius of \(r=0.15\) m meters that rotates at an angler velocity of \(\omega_{p}=1414.71\) rpm with AoA in the same rotation angle \(\theta=0\), where the linear velocity is equal to \(V\approx 80\) Km/h). According to Equation (12), this results in a Doppler shift of about \(f_{D}\approx 370\) Hz at a carrier frequency of \(fc=5\) GHz as it shown in Figure 5. Before turning on the stirrer, it is seen that the received signal has a constant bandwidth over transmission time, however, as soon as the stirrer is turned on the spectrum enlarges taking up more bandwidth due to the spreading. When the Doppler shift is generated, the BER performance of both the OTFS and OFDM waveforms degrades, as seen in Figure 6. This degradation takes place as a result of the Doppler shift, which leads to the loss of orthogonality of the sub-carrier and eventually results in inter-carrier interference (ICI). However, as compared to the OFDM waveform, the OTFS waveform provides better BER performance. This is because the multipath components and Doppler shifts are resolvable in the delay-Doppler domain.  
Figure 5: _Cont._  
Assuming the resolved delays as \(\tau_{i}\) and the Doppler frequency \(\
u_{i}\), the received signal is given by the following weighted summation  
\[y(t)=\sum_{i=1}^{L_{tap}}h_{i}x(t-\tau_{i})e^{|2\pi\
u_{i}t}=\sum_{i=1}^{L_{tap} }h_{i}(\mathbb{M}_{\
u_{i}}\mathbb{D}_{\tau_{i}}x)(t), \tag{13}\]  
where \(h_{i}\) is the \(i\)-th channel complex gain and \(L_{tap}\) is the total number of resolvable paths.  
Note that when using the OTFS waveform, the multipath components and Doppler shifts are resolvable in the delay-Doppler domain, as seen in Figure 2c where each bin represents a tap with a specific delay and Doppler. This feature made the OTFS more suitable for rich scattering and high mobility environments than OFDM.