# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## Appendix A Lemmas for Propositions 2 and 3  
The following lemmas are required in the proofs of Propositions 2 and 3. The determinant of \(\mathbf{B}\) can be expressed as  
\[\det(\mathbf{B})=c_{0}+c_{1}e^{i\theta}+c_{2}e^{i2\theta}+\cdots c_{(M-1)M_{T }}e^{i(M-1)M_{T}\theta}. \tag{1}\]  
Proof. : The proof is an extension of the one used in Lemma 2 from [26]. In [26], the space-time thread traverses one row only for each column it visits whereas the thread in the proposed structure traverses a block of \(M_{T}\) rows for each block of \(M_{T}\) columns it visits (see Figure 3). Let \(\sigma\) be a permutation of the set \((1,2,\cdots,MM_{T})\). We decompose each permutation into \(M_{T}\) partitions in such a way that the determinant (14) can be expressed as  
\[\det(\mathbf{B})=\sum_{\sigma\in S_{K}}\lambda(\sigma)\prod_{q=1}^{M_{T}} \prod_{p=1}^{M}b_{(p-1)\cdot M_{T}+q\sigma_{(p-1)\cdot M_{T}+q}}. \tag{2}\]  
That is, the \(q\)-th partition contains the rows \((p-1)\cdot M_{T}+q\) where \(p=1,\cdots,M\). Let \(\phi=e^{i\theta}\) and denote \(\exp_{\phi}(b_{i,j})\) as the operator which returns the exponent of \(\phi\) in \(b_{i,j}\) (e.g., if \(b_{i,j}=e^{i2\theta}\), then \(\exp_{\phi}(b_{i,j})=2\)). Due to the thread structure of \(\mathbf{B}\), we have  
\[\exp_{\phi}(b_{\lfloor k-j\rfloor_{M}\cdot M_{T}+q_{1},(k-1)\cdot M _{T}+q_{2}})=\tfrac{j-1}{M} \tag{3}\] \[\forall q_{1},q_{2}\in\{1,\cdots,M_{T}\}\]where \(j,k\in\{1,\cdots,M\}\). It can then be shown that  
\[ \begin{split}\lfloor M\exp_{\phi}(b_{(j_{1}-1)\cdot M_{T}+q_{1},(k_{ 0}-1)\cdot M_{T}+q_{2}})-\\ M\exp_{\phi}(b_{(j_{2}-1)\cdot M_{T}+q_{3},(k_{0}-1)\cdot M_{T}+q_{4}}) \rfloor_{M}=j_{2}-j_{1}\\ \forall q_{1},q_{2},q_{3},q_{4}\in\{1,\cdots,M_{T}\}\end{split}  \tag{10}\]  
where \(j_{2}>j_{1},j_{1},j_{2}\in\{1,\cdots,M\}\) and \(k_{0}\in\{1,\cdots,M\}\). Note that the above relationship holds despite the values of \(q_{1},q_{2},q_{3},q_{4}\). Furthermore, the sum of the exponents over any row or any combinations of rows within the \(r\)-th block of rows can be expressed as  
\[\sum_{q=1}^{M_{T}}\sum_{p=1}^{M}\exp_{\phi}(b_{f_{r}(p,q),\sigma_{(p-1)M_{T}+q} })=\frac{M_{T}(M-1)}{2} \tag{11}\]  
where \(f_{r}(p,q)\) is any function of \(p,q\) such that \((r-1)M_{T}+1\leq f(p,q)\leq rM_{T}\). Next, for one permutation \(\sigma\), the sum of exponents can be expressed as  
\[ \begin{split}\lfloor M&\exp_{\phi}(\sigma)\rfloor_{M }\\ &=\lfloor M\exp_{\phi}\biggl{(}\prod_{q=1}^{M_{T}}\prod_{p=1}^{M}b_ {(p-1)M_{T}+q,\sigma_{(p-1)M_{T}+q}}\biggr{)}\rfloor_{M}\\ &=\lfloor M\sum_{q=1}^{M_{T}}\sum_{p=1}^{M}\exp_{\phi}(b_{(p-1)M_ {T}+q,\sigma_{(p-1)M_{T}+q}})\rfloor_{M}\\ &=\lfloor M\sum_{q=1}^{M_{T}}\biggl{\{}\sum_{p=1}^{M-1}\exp_{ \phi}(b_{(p-1)M_{T}+q,\sigma_{(p-1)M_{T}+q}})\biggr{.}\biggr{.}\\ &\qquad\qquad=\exp_{\phi}(b_{(M-1)M_{T}+q,\sigma_{(M-1)M_{T}+q}} )\biggr{\}}\rfloor_{M}.\end{split}  \tag{12}\]  
The last term in the above equation can be expressed as the sum of the exponents of the whole row subtracting the remaining entries in the same row and gives  
\[ \begin{split}&\exp_{\phi}(b_{(M-1)M_{T}+q,\sigma_{(M-1)M_{T}+q} })\\ &=\frac{M_{T}(M-1)}{2}-\sum_{p=1}^{M-1}\exp_{\phi}(b_{(M-1)M_{T}+ q,\sigma_{(p-1)M_{T}+q}})\\ &\qquad-\sum_{p^{\prime}=1}^{M}\sum_{q^{\prime}\
eq q}\exp_{ \phi}(b_{(M-1)M_{T}+q^{\prime},\sigma_{(p^{\prime}-1)M_{T}+q^{\prime}}}).\end{split}  \tag{13}\]  
The above equation is substituted into (12) and then one of the summations in (12) gives  
\[ \begin{split}\sum_{q=1}^{M_{T}}&\sum_{p^{\prime}=1}^{ M}\sum_{q^{\prime}\
eq q}\exp_{\phi}(b_{(M-1)M_{T}+q^{\prime},\sigma_{(p^{ \prime}-1)M_{T}+q^{\prime}}})\\ &=\sum_{q=1}^{M_{T}}\sum_{p^{\prime}=1}^{M}\left\{\sum_{q^{ \prime}=1}^{M_{T}}\exp_{\phi}(b_{(M-1)M_{T}+q^{\prime},\sigma_{(p^{\prime}-1)M _{T}+q^{\prime}}})\right.\\ &\qquad\qquad\left.-\exp_{\phi}(b_{(M-1)M_{T}+q,\sigma_{(p^{\prime }-1)M_{T}+q}})\right\}\\ &=\sum_{q=1}^{M_{T}}\frac{M_{T}(M-1)}{2}-\frac{M_{T}(M-1)}{2}\\ &=\frac{M_{T}(M_{T}-1)(M-1)}{2}\end{split}  \tag{14}\]where we have used the relationship of (A5).