# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## 4 Simulation Results and Discussion
### Number of multipaths is two (\(L=2\))  
In the first part of the simulation, \(L\) is assumed to be two and maximum-likelihood (ML) detection is employed at the receiver. QPSK space-time block coding (STBC) is considered and two streams (\(M=2\)) of Alamouti code are multiplexed based on our design. The block size \(K\) is therefore equal to 4. For random rotation, phases are randomly generated from \([0,2\pi)\). For deterministic rotation, according to Proposition 3, \(\theta=2\pi/11\) is chosen because \(\cos(\theta)\) has a degree of 5. Figure 5 depicts the simulation results assuming \(M_{R}=1\). It also contains the performance result of the original Alamouti code in such channel. It is clear from the slope of the performance curves that the original Alamouti code exhibits inferior performance whereas our approach is able to utilize the original Alamouti code in the new design so as to exploit the frequency-diversity gain in addition to the spatial-diversity gain. In addition, random rotation and deterministic rotation show comparable performance as they both attain the maximum diversity gain.  
Next, the number of transmit antenna increases to four and two streams of QPSK-modulated quasi-orthogonal space-time block code (QOSTBC) [32] designed for maximum space diversity in a flat fading channel are utilized. Figure 6 depicts the simulation results assuming \(M_{T}=4,M_{R}=1,L=2\). As a comparison, the performance of original QOSTBC in the flat fading channel \((L=1)\) with four transmit antennas is also shown. It is clear that the proposed scheme achieves better diversity gain, with an order of \(4\times 2=8\), whereas the QOSTBC only achieves a diversity order of 4. We remark that the ML detection complexity of the proposed scheme is larger than that of QOSTBC, as the latter can exploit the property  
Figure 5: Performance of the proposed scheme based on QPSK\(-\)modulated STBC, \(M_{T}=2,M_{R}=1\), \(L=M=2\).  
of QOSTBC to divide the detection task into two for separate groups of decodable symbols, which however can be performed only in a flat fading channel.  
BPSK space-time trellis code (STTC) is then considered. The block size, \(K\), is 130. The constituent STTC used in our scheme is the optimal 8-state BPSK STTC code designed for the flat fading \(M_{T}=2\) channel (obtained by computer search in [33]), and has a generator matrix given by \(G_{8}=[1110;0101]^{T}\). Again, we consider multiplexing two streams of such STC, each one has a block size of 65 (\(K=65\cdot 2=130\)). For convenience, we name our scheme as Proposed-\(G_{8}\) scheme. For deterministic rotation, \(\theta=2\pi/5\) is chosen according to Proposition 2. Figures 7 and 8 depict the simulation results, for \(M_{R}=1\) and \(M_{R}=2\), respectively. Comparing with the original 8-state \(G_{8}\)-STTC, the Proposed-\(G_{8}\) scheme is superior since it is able to achieve full space-time-multipath diversity gain. We also compare our design with the best 8-state STTC (\(G_{8}\): \(=[1111;1001]^{T}\)) previously found for the frequency-selective channel [23]. For both random and deterministic rotation, the Proposed-\(G_{8}\) scheme outperforms the \(G_{8}\)-STTC by around 1 dB at FER\(=0.01\) for \(M_{R}=1\). In terms of complexity, the Proposed-\(G_{8}\) scheme however requires 128 ML states whereas the \(G_{8}\)- STTC requires only 16 ML states. Nonetheless, the simulations have validated our design that it attains the maximum diversity order by using simple space-time code designed for a flat fading channel.  
Figure 6: Performance of the proposed scheme based on QPSK\(-\)modulated STBC, \(M_{T}=4,M_{R}=1\), \(L=M=2\).  
Figure 8: Performance of the proposed scheme based on BPSK\(-\)modulated STTC, \(M_{T}=2\), \(M_{R}=2\), \(L=M=2\).  
Figure 7: Performance of the proposed scheme based on BPSK\(-\)modulated STTC, \(M_{T}=2\), \(M_{R}=1\), \(L=M=2\).