# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## 3 Rotation-Interleaved Multi-Stream Space-Time Code  
We now outline the new method, namely the rotation-interleaved multi-stream space-time code, which may obtain the maximum diversity gain order (\(M_{T}M_{R}L\)) in a frequency-selective MIMO channel. The basic idea is to employ an STC scheme which is optimized for a flat fading (\(L=1\)) MIMO channel, and multiplex independent streams of such STCs in a time-interleaved fashion followed by symbol-wise phase rotations. This new method is in theory applicable for arbitrary \(M_{T},M_{R}\) and \(L\) provided that the STC employed in each stream is able to achieve full diversity in the corresponding flat fading \(M_{T}\)-by-\(M_{R}\) MIMO channel. Furthermore, our design is bandwidth-efficient as it neither alters the transmission rate of the original STC nor increases any redundancy.  
Figure 1 depicts the diagram of our design. The steps to construct the multi-stream-based space-time codewords are outlined as follows:  
1. The input data stream \(\mathbf{d}\) is demultiplexed into \(M\geq L\) parallel sub-streams \(\mathbf{d}^{(m)}\), \(m=1,\cdots,M\).
2. Each sub-stream is encoded by the same STC encoder. Denote the output codewords from the STC encoder corresponding to different sub-streams as \(\mathbf{C}^{(m)},m=1,\cdots M\), which are \(M_{T}\times P\) matrices, where \(P=K/M\).
3. Each codeword \(\mathbf{C}^{(m)},m=1,\cdots,M\) is multiplied by a phasor \(e^{i\theta_{m}}\) to give \(e^{i\theta_{m}}\mathbf{C}^{(m)}\).
4. The codewords from the rotated sub-streams, \(e^{i\theta_{m}}\mathbf{C}^{(m)},m=1,\cdots,M\) enter a multiplexer, which does the following. Let \(e^{i\theta_{m}}\mathbf{C}^{(m)}=[e^{i\theta_{m}}\mathbf{c}^{(m)}_{0}\,e^{i \theta_{m}}\mathbf{c}^{(m)}_{1}\cdots\,e^{i\theta_{m}}\mathbf{c}^{(m)}_{P-1}]\), where \(\mathbf{c}^{(m)}_{t}\) denotes the \((t+1)\)-th column vector in \(\mathbf{C}^{(m)}\), i.e., the code symbols transmitted at time \(t\). By multiplexing these column vectors from different rotated STC sub-streams in a time-division fashion (see Figure 2), the super-codeword from the outputs of the multiplexer can now be expressed as \[\mathbf{C}(0) = [e^{i\theta_{1}}\mathbf{c}^{(1)}_{0}\,e^{i\theta_{2}}\mathbf{c}^ {(2)}_{0}\,\cdots\,e^{i\theta_{M}}\mathbf{c}^{(M)}_{0}e^{i\theta_{1}}\mathbf{ c}^{(1)}_{1}\,\cdots\] (9) \[e^{i\theta_{M}}\mathbf{c}^{(M)}_{1}\cdots\,e^{i\theta_{M}} \mathbf{c}^{(M)}_{P-1}].\]  
The resulting \(M_{T}\times K\) codeword \(\mathbf{C}(0)\) is called a super-codeword because it consists of multiple codewords from the phase-rotated sub-streams. Note that, due to the interleaving, the adjacent columns in any \(\mathbf{C}^{(m)}\) are now separated by \(M\) columns in \(\mathbf{C}(0)\), and it is clear that no bandwidth efficiency reduction occurs as the separation gap is filled with columns from different sub-codewords. By stacking all the delayed versions of this super-codeword, the final codeword \(\mathbf{C}\) is in the form of \((5)\).  
Note that the input data stream \(\mathbf{d}\) can also be replaced by a coded data stream. As such, our scheme becomes the inner code whereas the incoming coded stream is from an outer code. Due to limited space, such concatenated scheme will be a subject of future investigation.  
Figure 1: Proposed method.