# Multiple-Mode Orthogonal Time Frequency Space with Index Modulation
## 4 Simulation Results and Discussion  
In this section, we present the simulation results for the proposed MM-OTFS-IM scheme over time-varying channels with different configurations compared with the classical OTFS, OTFS-IM and OTFS-DM-IM. A channel with \(p=5\) paths was used in the simulation, where the Doppler profile was given by \(v_{i}=v_{max}cos(\theta_{i})\) with \(v_{max}\) as the maximum Doppler shift and \(\theta_{i}\sim U[-\pi,\pi]\), and the delay profile is defined in Table 1. The maximum Doppler shift was determined by the user equipment (UE) speed, when the UE speed was 120 km/h, 300 km/h, and 500 km/h with carrier frequency 4 GHz, the corresponding maximum Doppler shift was \(v_{max}=444\) Hz, 300 Hz, and 1850 Hz respectively. The other simulation parameters used are given in Tables 2 and 3. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Path Index** & & & & & \\
**(i)** & 1 & 2 & 3 & 4 & 5 \\ \hline
**Delay** & & & & & \\
**(\(\tau_{1}\)), μs** & 0 & 2.08 & 4.164 & 6.246 & 8.328 \\ \hline \end{tabular}
\end{table} 
Table 1: Delay profile of channel (\(p=5\)) for simulations. \begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & & **Values** \\ \hline Carrier frequency (GHz) & & 4 \\ Subcarrier spacing (kHz) & & 15 \\ Size of an MM-OTFS-IM frame (M,N) & & (32, 32) \\ UE speed (Kmph) & & 120, 300, 500 \\ \(v_{max}\) (Hz) & & 444, 1111, 1850 \\ Channel state information & & Ideal \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. \begin{table}
\begin{tabular}{c c c c} \hline
**SE** & **OTFS** & **OTFS-IM** & **OTFS-DM-IM** & **MM-OTFS-IM** \\ \hline
**1.5** & NA & (\(m\), \(n\), \(k\)) = (2, 2, 2) & NA & (\(m\), \(n\)) = (1, 2) \\  & QPSK & & BPSK \\
**2.5** & NA & (\(m\), \(n\), \(k\)) = (2, 2, 2) & NA & (\(m\), \(n\)) = (1, 2) \\  & 16 PSK & & QPSK \\
**3** & 8 PSK & (\(m\), \(n\), \(k\)) = (4, 2, 7) & (\(m\), \(n\), \(k\)) = (2, 2, 2) & (\(m\), \(n\)) = (2, 2) \\  & 8 PSK/8 QAM & \(M_{modA}\):QPSK 1, 0, 0/QAM & QPSK \\  & & \(M_{modB}\): 8 PSK 1, 0, 0/8 QAM \\
**4** & 16 PSK & (\(m\), \(n\), \(k\)) = (2, 2, 2) & (\(m\), \(n\)) = (2, 2) \\  & 16 PSK/16 QAM & \(M_{modA}\): 8 PSK/8 QAM & 8 PSK \\  & & \(M_{modB}\): 16 PSK/16 QAM & \(M_{modB}\): 16 PSK/16 QAM & \\
**5** & 32 PSK & (\(m\), \(n\), \(k\)) = (2, 2, 3) & (\(m\), \(n\), \(k\)) = (2, 2, 2) & (\(m\), \(n\)) = (2, 2) \\  & 64 PSK/64 QAM & \(M_{modA}\): 16 PSK/16 QAM & 16 PSK \\  & & \(M_{modB}\): 32 PSK/32 QAM & \\ \hline \end{tabular}
\end{table} 
Table 3: System parameters of the compared systems. In Figure 5, we compare the BER performance of the proposed MM-OTFS-IM system with the OTFS-IM system under different SEs. It can be seen from Figure 5 that when the SE was 1.5 bps/Hz and 2.5 bps/Hz, the BER performance of the MM-OTFS-IM system was better than that of the OTFS-IM system. For example, when the SE was 2.5 bps/Hz, the MM-OTFS-IM system achieved a performance gain of 2 dB compared to the OTFS-IM system at the BER level of \(10^{-3}\). When the SE was 1.5 bps/Hz, the MM-OTFS-IM system achieved a 5 dB gain over the OTFS-IM system at the BER level of \(10^{-5}\). Under the same SE, the MM-OTFS-IM system used a lower order constellation to modulate the transmission symbols, which resulted in a larger Euclidean distance between the transmission symbols; thus, the BER performance of the proposed system was improved. The BER performance comparisons between the MM-OTFS-IM, the OTFS, the OTFS-IM, and the OTFS-DM-IM system with SEs of 3, 4, and 5 bps/Hz are shown in Figures 6-8, respectively. Both the OTFS-IM and the OTFS-DM-IM system used \(M\)-PSK and \(M\)-QAM modulation modes, and the OTFS and the MM-OTFS-IM system used \(M\)-PSK modulation. The MM-OTFS-IM system obtained the best BER performance compared with the traditional OTFS, OTFS-IM, and OTFS-DM-IM systems with the same spectral efficiency at a high SNR region when using \(M\)-PSK modulation and the OTFS-IM with \(M\)-QAM.