# 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
\begin{array}{ccccc}\beta_{1}\mathbf{T}^{(1)}& \beta_{1}^{2}\mathbf{S}^{(2)}&\beta_{1}^{3}\mathbf{S}^{(3)}&\cdots&\beta_{1}^{ M}\mathbf{S}^{(M)}\\ \beta_{1}^{M}\mathbf{S}^{(M)}&\beta_{1}\mathbf{T}^{(1)}&\beta_{1}^{2}\mathbf{ S}^{(2)}&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \beta_{1}^{2}\mathbf{S}^{(2)}&\cdots&\cdots&\cdots&\beta_{1}\mathbf{T}^{(1)} \end{array} \right] \tag{12}\]  
where \(\mathbf{S}^{m},m\in\{2,\cdots,M\}\) is the resulting matrix after applying the row operations (\(\mathbf{B}^{(1)}\rightarrow\mathbf{T}^{(1)}\)) on \(\mathbf{B}^{(m)}\). We also let \(\beta_{1}=e^{i\beta_{1}}\) and express all other phasors as a power of \(\beta_{1}\), i.e., \(e^{i\beta_{m}}=\beta_{1}^{m}\) in (12). Next, we want to convert the above matrix (12) to upper triangular form by elementary row operations. Denote \(g_{n}\) as the leading entry in the \(n\)-th row of the resulting upper-triangular matrix, and it can be shown that each leading entry can be reduced to a non-zero polynomial with indeterminate \(\beta_{1}\). Then,  
\[g_{n}=\bar{a}_{n}\beta_{1}^{p}+\sum_{p<k}a_{k}\beta_{1}^{k} \tag{13}\]  
where \(\bar{a}_{n}\) and \(a_{k}\) denote the coefficients, \(p\) is some integer. To see this, let us start the Gaussian elimination procedure at the (\(M_{T}+1\))-th row (since the first \(M_{T}\) rows are already triangularized). In order to clear an entry to the left of the diagonal entry, we need to first multiply the current row by a scalar (i.e., the leading entry with \(\beta_{1}\) from the corresponding row above), whereas the corresponding row above is to multiply the current entry that needs to be cleared. Such cross multiplication is to be followed by row-wise subtraction and there are \(M_{T}\) such operations to clear \(M_{T}\) entries. After each multiplication, the power of \(\beta_{1}\) in the polynomial at each entry will increase by one. When completed, the leading entry at the (\(M_{T}+1\),\(M_{T}+1\))-th position takes the form of (13) with \(\bar{a}_{n}\
eq 0\) and \(p=M_{T}+1\) because the (\(M_{T}+1\),\(M_{T}+1\))-th entry starts as a term containing \(\beta_{1}\) and during the row operations, it is always multiplied by the diagonal entries which contain \(\beta_{1}\) and are always non-zero. As such, it contains a term with the lowest power of \(\beta_{1}\) compared with all other entries in the current row. The other terms in (13) are due to cross-multiplication of different powers of \(\beta_{1}\) and the product can be expressed as \(\beta_{1}^{k}\), where \(k\) is greater than \(p\). This is because there are at least \(M_{T}\) such multiplications and some of them always involve \(\beta\) of power greater than one. Likewise, the above elimination procedure is then applied to the subsequent rows. The only difference is that the rows involved may contain leading entries already in the form of (13), and each multiplication will increase the respective powers of \(\beta_{1}\) in the polynomial accordingly. After completing the elimination, the leading entry takes the form of (13) for some \(p\) again because the terms with the smallest powers of \(\beta_{1}\) are always non-zero and occupy the main diagonal after each cross-multiplication and subtraction (\(\bar{a}_{n}\beta_{1}^{p}\) is in fact the product of the diagonal entries before the \(n\)-th row). The other terms in (13) are due to cross multiplications involving some \(\beta\)s of power greater than one. Next, as we know from the fundamental theorem of algebra, the number of roots of a polynomial (13) is finite. From measure theory, it can be shown that the zero set of a polynomial has measure zero (i.e., a polynomial function on \(C^{n}\) to \(\mathcal{C}\), is either identically \(0\), or non-zero everywhere). Since the leading coefficient, \(\bar{a}_{n}\) in (13), is non-zero, the event that \(g_{n}\
eq 0\) occurs with probability one. As the event of \(g_{n}\
eq 0\), \(n=1,\cdots,MM_{T}\) for all these super-codeword pairs is the intersection of all such individual events, the rank of **B** is \(M_{T}M\) with probability one.