# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## 3 Rotation-Interleaved Multi-Stream Space-Time Code
### Frequency-Domain Iterative Receiver  
Single-carrier space-time trellis coding schemes operating in multipath-rich channels are known to have prohibitive receiver complexity if maximum-likelihood sequence detection is employed. For instance, the complexity of the optimal receiver for the STTC developed in [21] is \(\mathcal{O}(|\Omega|^{(\
u+(L-1))})\), where \(\
u\) is the memory length of the trellis code and \(|\Omega|\) is the size of the constellation. The complexity renders such coding schemes impractical for large \(L\) and \(\
u\). For the proposed scheme, the complexity of the optimal receiver is \(\mathcal{O}(|\Omega|^{(\
u+1)L-1)})\), which is also prohibitively high due to the joint multi-stream detection as well as the presence of multipath. Alternatively, in view of the unique multi-stream time-interleaved structure of the proposed scheme, we propose to use a simpler and effective receiver based on iterative block-based MMSE frequency domain equalization (FDE) and decoding formulated in [24; 29]. Figure 4 depicts the corresponding turbo receiver structure that, after subtracting the inter-symbol and inter-stream interference, jointly detects the block of signals from multiple streams through the linear MMSE FDE followed by stream-based BCJR decoding. The time-interleaved multiplexer in the proposed scheme plays the role of interleaver in the turbo processing and allows the equalizer to approximate the inter-symbols as independent signals. The detail of the MMSE-FDE and the iterative processing are described in Appendix B.  
The MMSE-FDE process has a complexity of \(\mathcal{O}(KM^{3})\) which is a linear function of the block size \(K\), while the stream-based decoding (e.g., BCJR algorithm) has a complexity depending on the constituent STC and the number of trellis states which is given by \(\mathcal{O}(|\Omega|^{\
u})\). For large \(L\) and \(\
u\), our approach clearly has a reasonable complexity unmatched by the optimal receiver of the traditional space-time trellis codes [21]. Besides the complexity, the memory requirement and the decoding delay also need to be addressed for the proposed iterative receiver. For a binary linear block code of blocksize \(P\), the BCJR algorithm requires storage up to \(K_{i}(2^{P-K_{i}})\) real numbers, where \(K_{i}\) is the number of information-carrying bits in the code. Modified BCJR algorithms have been previously proposed to reduce the complexity and memory requirement by a fixed factor \(K_{i}\)[30]. With regard to the decoding delay, it is well known that the decoding delay due to iteration adversely affects the throughput performance. As an illustration, it is found that in an iterative-based turbo decoder implemented with a parallel structure, the throughput is proportionally to \(P_{m}F/p\), where \(P_{m}\) is the number of parallel MAP-decoders, \(F\) is the operating clock frequency and \(p\) is the number of iterations [31]. Existing works such as [31] studied improved parallel architectures and sliding-window-based logarithmic-BCJR algorithm to increase the throughput and managed to match the LTE-advanced standards throughput requirements. In our iterative receiver, it is expected that the multi-stream-based structure can also be implemented with a similar parallel architecture consisting of logarithmic BCJR-based decoders to reduce the throughput degradation.  
Figure 4: Iterative receiver based on linear MMSE-FDE and turbo processing.