# 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
The determinant can be expressed as follows,  
\[\det(\mathbf{B})=\sum_{\sigma\in S_{K}}\lambda(\sigma)\prod_{i=1}^{K}b_{i, \sigma_{i}} \tag{14}\]  
The above sum is computed over all the permutations \(\sigma\) of the set \(S_{K}=\{1,\cdots,K\}\). In addition, \(\lambda(\sigma)\in\{1,-1\}\) denotes the signature of \(\sigma\), and \(b_{i,\sigma_{i}}\) denotes the \((i,\sigma_{i})\) entry in the matrix. Thus, each permutation is traversing the matrix from top to bottom through different columns and the accessed entries are multiplied to yield the product in (14). As each entry contains a phasor, the products of some permutations in (14) will contain common phasor terms and they can be factored out in (14). It is proven in Lemma A1 (see Appendix A) that the determinant can be expressed as  
\[\det(\mathbf{B})=c_{0}+c_{1}e^{i\theta}+c_{2}e^{i2\theta}\cdots c_{(M-1)M_{T}} e^{i(M-1)M_{T}\theta} \tag{15}\]  
where \(c_{n}\in\mathbb{Q}\) and it can be expressed as  
\[c_{n}=\sum_{\sigma\in T_{n}}\prod_{i=1}^{K}b_{i,\sigma_{i}} \tag{16}\]  
where \(T_{n}\) denotes the set of all those permutations for which the product in (16) contains a common phasor term \(e^{j\mu\theta}\). Note that \(c_{0}\) must be due to those permutations traversing only the main block diagonal of \(\mathbf{B}\) as they yield the smallest sum of angles which is equal to zero. Likewise, \(c_{(M-1)M_{T}}\) is due to the permutations traversing only the columns from \(\mathbf{B}^{(M)}\) because they are the only permutations yielding the maximum angle equal to \((M-1)M_{T}\). Next, we observe that by definition, \(c_{0}\) can be expressed as the determinant of matrix \(\mathbf{B}\) when all the blocks are zero except the main block diagonal which contains \(\mathbf{B}^{(1)}\). Such block diagonal matrix is full-rank since \(\mathbf{B}^{(1)}\) is a full-rank matrix, implying that \(c_{0}\) is non-zero. The assumption that \(\{1,e^{i\theta},e^{j2\theta},\cdots,e^{j(M-1)M_{T}\theta}\}\) are algebraically independent over \(\mathbb{Q}\) leads to the following conclusion: that the determinant of \(\mathbf{B}\) in (11) is equal to zero if and only if all the coefficients \(c_{0},c_{1},\cdots,c_{(M-1)M_{T}}\) are equal to zero. Since \(c_{0}\) is not equal to zero, the determinant is not zero and \(\mathbf{B}\) is therefore a full-rank matrix. Next, as done in Proposition 1, we may repeat the above procedure considering other super-codewords with \(\mathbf{B}^{(1)}\) strictly being zero and \(\mathbf{B}^{(2)}\) being non-zero. The matrix \(\mathbf{B}\) can be permuted such that \(\mathbf{B}^{(2)}\) lies in the diagonal. As such, the coefficient \(c_{M_{T}}\) is now due to the permutations traversing the main block diagonal containing \(\mathbf{B}^{(2)}\) because the sum of angles is equal to \(M_{T}\). It can be shown to be non-zero using the same argument above, leading to the conclusion that \(\mathbf{B}\) is a full-rank matrix. Likewise, the procedure then repeats for all other remaining super-codewords with \(\mathbf{B}^{(m)},m=2,\cdots,M-1\), respectively being zero. The second part of the proof is to show that the construction methods for the phase rotations are valid. In the first method, having an algebraic number, \(e^{i\theta}\), with degree greater than \((M-1)M_{T}\) guarantees that the polynomial (15) is never equal to zero since \(c_{n}\in\mathbb{Q}\) and the highest degree in (15) is \((M-1)M_{T}\). Similarly, in the second method, the Link proposition (see Appendix A) asserts that the transcendental number, \(e^{i\theta}\), guarantees a non-zero polynomial (15). For the first construction method, the cyclotomic numbers, which are defined as the \(n\)th roots of unity \(e^{j2\pi/n}\), can be used. Their degrees over \(\mathbb{Q}\) are equal to \(\phi(n)\) (the Euler \(\phi\)-function). Since it was shown in [26] that the algebraic numbers yield better performance than transcendental numbers, the first method is preferred over the second method and our simulation studies will only focus on using cyclotomic numbers in the proposed scheme.