# A Comprehensive Experimental Emulation For OTFS Waveform RF-Impairments
## 3 Channel Effects Furthermore, RF-Impairments
### 1/Q imbalance  
In communication systems, direct-conversion receivers were used to convert RF signals to base-band signals immediately. In contrast to heterodyne receivers, direct-conversion receivers did not need the signal to be down-converted to an intermediate frequency (IF). Unfortunately, imperfections in the local oscillator (LO) might cause significant RF-impairments, such as I/Q imbalance and DC offset [51]. A local oscillator (LO) is a device that generates sinusoidal signals, which are used to represent in-phase and quadrature signals during modulation and demodulation processes. To make a cosine signal, the LO produces a sine signal and shifts it \(90^{\circ}\) degrees to produce a cosine signal. Unfortunately, the signal gain created and the phase \(90^{\circ}\) are not in sync in terms of their practical implementation. [47]. I/Q imbalance impairment is the combination of two effects; the first is the amplitude or gain imbalance (I) and the quadrature (Q) paths known as \(\epsilon\), and the second  
Figure 8: CCDF comparison of OTFS waveform and OFDM waveform at Carrier frequency 2.4 GHz. is the phase mismatch given by \(\Delta\phi\). The following model is used to add the I/Q imbalance on the transmitted signal \(x(t)=I+J\)\(Q\)[53; 52], as follows  
\[ \begin{split} y(t)&=(1+\epsilon)\cos\Delta\phi\Re\{x (t)\}-J(1-\epsilon)\sin\Delta\phi\Re\{x(t)\}\\ &+J(1-\epsilon)\cos\Delta\phi\Im\{x(t)\}-(1+\epsilon)\sin\Delta \phi\Im\{x(t)\},\end{split}  \tag{16}\]  
where \(\Re(\cdot)\) and \(\Im(\cdot)\) symbolize the real and the imaginary part, respectively. For more simplicity, (16) can be written as  
\[y(t)=\alpha\cdot x(t)+\beta\cdot x(t)^{*}, \tag{17}\]  
where \((\cdot)^{*}\) denote complex conjugate, \(\alpha=\cos\Delta\phi+\jmath\epsilon\sin\Delta\phi\), and \(\beta=\epsilon\cos\Delta\phi-\jmath\sin\Delta\phi\). As it is observed in (17) that the I/Q imbalance does not exist if \(\alpha=1\) and \(\beta=0\). In the N5172B-VSG, there is a specification for an internal I/Q baseband generator that may be adjusted either internally or externally, depending on the application. In this experiment, we adjust the internal I/Q baseband of the N5172B-VSG to evaluate the effect of the I/Q imbalance on the performance of the OTFS system. Figure 9 shows the effect of I/Q imbalance on the average BER performance for both OTFS and OFDM systems. In general, I/Q imbalance degrades the system's performance for both waveforms as \(\epsilon\) and/or \(\Delta\phi\) increase, whereas the I/Q imbalances are introduced in the system, the performance directly converges to a constant error floor at certain SNR value. Beyond this value, even increasing SNR does not help in improving the BER performance as given in [26]. Additionally, it can be seen that changing the value of gain \(\epsilon\) causes a bad influence that is greater than that of \(\Delta\phi\). This is because changing \(\epsilon\) leads to narrowing the received symbols in constellations, which in turn leads to narrowing the threshold regions that the demodulation should use to distinguish the received symbols. In other words, changing \(\epsilon\) causes a more negative effect than changing \(\Delta\phi\). Since the points in the constellation are now quite close to each other, while changing \(\Delta\phi\) causes symbols to shift without changing the distance between neighboring points as shown in Figure 10b-d.  
Figure 9: BER performance comparison between OTFS waveform with \(M=32,N=32\) and OFDM waveform with \(N=1024\) considering different gains imbalance (\(\epsilon\)) and phase mismatch (\(\Delta\phi\)) I/Q imbalances at 2.4 GHz carrier frequency. Figure 10: Constellation diagram shows the effect of the different I/Q and DC offset on the \(M=32,N=32\) OTFS received information symbols of \(\mathfrak{g}[k,l]\) for 4-QAM at 2.4 GHz carrier frequency.