# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## 3 Rotation-Interleaved Multi-Stream Space-Time Code
### Symbol-Wise Random Phase Rotation  
As the multiple streams in the proposed scheme are independently generated using the same STC encoder, we need to ensure that the resulting super-codewords fulfill the full-rank criterion. Hence, for each stream, phase rotation is uniformly applied to rotate each symbol while different streams are subject to different angles of rotation. The rationale is that rotation, being either random or deterministic, can be viewed as a means to separate the multiple streams in the algebraic space. The following is the first proposition concerning the property of our design when the symbol-wise rotation is assumed to be random. **Proposition 1**. : _Assume that the phase-rotation angles \(\theta_{m},m=1,\cdots,M\) are chosen to be \(\theta_{m}=m\cdot\theta\), where \(\theta\) is randomly chosen from \([0,2\pi)\). Then, the time-interleaved multi-stream space-time code achieves the maximum diversity gain order (\(M_{T}M_{R}L\)) with probability one, provided that \(M\geq L\) and the constituent space-time code used in the construction achieves the maximum diversity gain order in the corresponding flat fading MIMO (\(M_{T}M_{R}\)) channel._  
Proof of Proposition 1.: For simplicity, it is assumed that \(M=L\). The proof can be easily extended to \(M>L\) and thus omitted here. First, for the super-codeword \(\mathbf{C}\) (see (5)), we permute the columns as follows:  
\[[\mathbf{c}_{0}\mathbf{c}_{1}\cdots\mathbf{c}_{MP-1}] \rightarrow [\mathbf{c}_{0}\mathbf{c}_{P}\mathbf{c}_{2P}\cdots\mathbf{c}_{(M- 1)P}\mathbf{c}_{1}\mathbf{c}_{P+1}\cdots \tag{10}\] \[\mathbf{c}_{(M-1)P+1}\cdots\mathbf{c}_{MP-1}]\]  
where \(\mathbf{c}_{j}\) denotes the \((j+1)\)-th column of \(\mathbf{C}\). Since column-wise permutation would not affect the rank of the matrix \(\mathbf{A}=\mathbf{B}\mathbf{B}^{H}\), our analysis from now on is based on the permuted super-codewords. Due to the permutation, the super-codeword can be expressed as a block-Toeplitz matrix in which the blocks are the individual codewords from the STC streams. Now, consider the pairwise error probability and a pair of super-codewords (\(\mathbf{B}=\mathbf{C}-\mathbf{\hat{C}}\)). It is clear that \(\mathbf{B}\) is also a block-Toepliz matrix in which the blocks are given by \(\mathbf{B}^{(m)}=\mathbf{C}^{(m)}-\mathbf{\hat{C}}^{(m)},m=1,\cdots,M\) (or its column-permutated version) and  
\[\mathbf{B}=\left[ \begin{array}{ccccc}e^{\beta_{1}}\mathbf{B}^{(1)}&e^{ \beta_{2}}\mathbf{B}^{(2)}&\cdots&\cdots&e^{\beta_{M}}\mathbf{B}^{(M)}\\ e^{\beta_{M}}\mathbf{B}^{(M)}&e^{\beta_{1}}\mathbf{B}^{(1)}&e^{\beta_{2}} \mathbf{B}^{(2)}&\ddots&\ddots\\ \ddots&\ddots&\ddots&\ddots&\ddots\\ e^{\beta_{2}}\mathbf{B}^{(2)}&\cdots&\cdots&\cdots&e^{\beta_{1}}\mathbf{B}^{ (1)}\end{array} \right]. \tag{11}\]  
Figure 2: The time-interleaved multiplexer of rotated streams (\(L=3\), \(M=3\), \(M_{T}=2\)). Due to the assumption that the STC achieves full diversity gain in the flat fading channel, the rows in each block \(\mathbf{B}^{(m)},m=1,\cdots,M\) are linearly independent. Note that \(\mathbf{B}\) must contain at least one non-zero block. To begin, let us assume that \(\mathbf{B}^{(1)}\) is a non-zero block. In addition, \(\mathbf{B}\) clearly has full rank when other blocks are zero because \(\mathbf{B}^{(1)}\) occupies the main diagonal. The more difficult task is to prove that \(\mathbf{B}\) has full rank when there are more than one block among \(\mathbf{B}^{(m)},m=1,\cdots,M\) being non-zero. To proceed, we apply Gaussian elimination (a sequence of elementary row operations) on the first \(M_{T}\) rows so that the block \(\mathbf{B}^{(1)}\) is reduced to a full-rank upper triangular matrix \(\mathbf{T}^{(1)}\). The same set of elementary row operations are applied to every block of \(M_{T}\) rows in \(\mathbf{B}\) and we obtain the following \(M_{T}M\times K\) matrix  
\[\mathbf{B}\rightarrow\left[