# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## 3 Rotation-Interleaved Multi-Stream Space-Time Code
### Symbol-Wise Deterministic Phase Rotation  
To justify the applicability of the proposed scheme in practice, deterministic rotation as opposed to random rotation is considered in the following. To understand the role of rotation, it is observed that the proposed interleaved multi-steam structure has led to a layered space-time structure in which each stream occupies a _thread_ in the resulting space-time matrix **B** (11). This space-time matrix which contains the extra spatial dimensions introduced by the delayed taps in the multipath channel resembles the space-time threading framework proposed in [26]. Specifically, a thread in the proposed scheme is shown in Figure 3 and is defined as follows: \(b^{(l)}_{ij};i=(i^{\prime}-1)\cdot M_{T}+q_{i},j=(j^{\prime}-1)\cdot M_{T}+q_{ j},j^{\prime}=[i^{\prime}+l]_{M},q_{i}\in\{1,\cdots,M_{T}\}\), \(q_{j}\in\{1,\cdots,M_{T}\}\), \(i^{\prime}\in\{1,\cdots,M\},l\in\{1,\cdots,M\}\), where \(l\) denotes the \(l\)-th thread and \(i^{\prime}\), \(j^{\prime}\) denote the row-block index and column-block index, respectively. With the above definition, each thread is now block-based (with \(M_{T}\) rows) instead of row-based as in [26]. We emphasize that the block-based thread structure is naturally formed due to the delayed taps as the STC codewords are shifted successively in time over different spatial dimensions. As such, the analysis in [26] and the algebraic codeword constructions can also be employed here in the proposed structure after some modification. Figure 3: The block-based space-time threading structure (\(M=L=4\)). 1. BPSK codes  
Firstly, we consider BPSK modulation in the proposed scheme where the constituent STCs are BPSK codes and the angles of rotation are chosen based on algebraic number theory. For more background on algebraic number theory, interested readers may refer to [27]. A few important definitions relevant to our development are highlighted here. Let \(\mathbb{Q}\) be the set of rational numbers. Let \(\alpha\in\mathbb{C}\), then \(\alpha\) is called an algebraic number over \(\mathbb{Q}\) if there exists a non-zero irreducible monic polynomial \(f\in Q[X]\) such that \(f(\alpha)=0\). The degree of \(\alpha\) is defined to be the degree of \(f\), which is called the minimal polynomial of \(\alpha\). If \(\alpha\) is complex, one can also define its minimal polynomial over \(\mathbb{Q}[iX]\). Based on the algebraic code construction proposed in [26], the following proposition is obtained concerning our proposed scheme using BPSK codes:  
**Proposition 2**. : _Assume that the phase-rotation angles \(\theta_{m},m=1,\cdots,M\) are chosen to be \(\theta_{m}=\theta\cdot(m-1)/M\). If \(\{1,e^{i\theta},e^{i2\theta},\cdots,e^{i(M-1)M_{T}\theta}\}\) are algebraically independent over \(\mathbb{Q}\), then the time-interleaved multi-stream space-time code achieves the maximum diversity gain order (\(M_{T}M_{R}L\)) provided that \(M\geq L\) and the constituent BPSK space-time code used in the construction achieves the maximum diversity gain order in the corresponding flat fading MIMO (\(M_{T}M_{R}\)) channel. In order to achieve the maximum diversity, there are two ways to choose \(\theta\):_  
1. \(\theta\) _is chosen such that_ \(e^{i\theta}\) _is an algebraic number with degree of_ \((M-1)M_{T}+1\) _or greater._
2. \(\theta\
eq 0\) _is an algebraic number (i.e.,_ \(e^{i\theta}\) _is transcendental)._  
Proof of Proposition 2.: It is assumed that \(M=L\) and \(K=MM_{T}\). The extension to \(M>L\) and \(K>MM_{T}\) is straightforward and omitted here. The first part of the proof is as follows. As in the Proof of Proposition 1, the permuted super-codewords are considered, and for any pair of distinct super-codewords, \(\mathbf{B}\) is a \(K\times K\) block-Toepliz matrix as shown in (11). To begin, let us assume that \(\mathbf{B}^{(1)}\), the main diagonal of \(\mathbf{B}\), is non-zero.