## 7 RDFT Algorithm for \(N=7\)  
For \(N=7\), the Equation (17) will take the form  
\[\mathbf{y}_{7}=\mathbf{R}_{7}\mathbf{x}_{7} \tag{54}\]  
where  
\[\mathbf{R}_{7}= \begin{bmatrix}1&1&1&1&1&1\\ 1&\cos\phi_{7}&\cos 2\phi_{7}&\cos 3\phi_{7}&\cos 4\phi_{7}&\cos 5\phi_{7}&\cos 6 \phi_{7}\\ 1&\cos 2\phi_{7}&\cos 4\phi_{7}&\cos 6\phi_{7}&\cos 8\phi_{7}&\cos 10\phi_{7}&\cos 1 2\phi_{7}\\ 1&\cos 3\phi_{7}&\cos 6\phi_{7}&\cos 9\phi_{7}&\cos 12\phi_{7}&\cos 15\phi_{7}&\cos 1 8\phi_{7}\\ 0&-\sin 4\phi_{7}&-\sin 8\phi_{7}&-\sin 12\phi_{7}&-\sin 16\phi_{7}&-\sin 20\phi_{7}&- \sin 24\phi_{7}\\ 0&-\sin 5\phi_{7}&-\sin 10\phi_{7}&-\sin 15\phi_{7}&-\sin 20\phi_{7}&-\sin 25\phi_{7}&- \sin 30\phi_{7}\\ 0&-\sin 6\phi_{7}&-\sin 12\phi_{7}&-\sin 18\phi_{7}&-\sin 24\phi_{7}&-\sin 30\phi_{7}&- \sin 36\phi_{7}\end{bmatrix}  \tag{55}\]  
Since \(\phi_{7}=2\pi/7\) then \(\cos 4\phi_{7}=\cos 3\phi_{7},\cos 5\phi_{7}=\cos 2\phi_{7},\cos 6\phi_{7}=\cos \phi_{7},\cos 8\phi_{7}=\cos \phi_{7},\cos 9\phi_{7}=\cos 2\phi_{7},\cos 10\phi_{7}=\cos 3\phi_{7},\cos 12\phi_{7}= \cos 2\phi_{7},\cos 15\phi_{7}=\cos \phi_{7},\cos 18\phi_{7}=\cos 3\phi_{7}\) and \(\sin 4\phi_{7}=-\sin 3\phi_{7},\sin 5\phi_{7}=-\sin 2\phi_{7},\sin 6\phi_{7}=-\sin \phi_{7},\)\(\sin 8\phi_{7}=\sin\phi_{7},\)\(\sin 10\  
\(\sin 2\phi_{7}\), \(\sin 18\phi_{7}=-\sin 3\phi_{7}\), \(\sin 20\phi_{7}=-\sin\phi_{7}\), \(\sin 24\phi_{7}=\sin 3\phi_{7}\), \(\sin 25\phi_{7}=-\sin 3\phi_{7}\), \(\sin 30\phi_{7}=\sin 2\phi_{7}\), \(\sin 36\phi_{7}=\sin\phi_{7}\), so the \(\mathbf{R}_{7}\) matrix will take the form  
\[\mathbf{R}_{7}= \begin{bmatrix}1&1&1&1&1&1&1\\ 1&\cos\phi_{7}&\cos 2\phi_{7}&\cos 3\phi_{7}&\cos 3\phi_{7}&\cos 2\phi_{7}&\cos \phi_{7}\\ 1&\cos 2\phi_{7}&\cos 3\phi_{7}&\cos\phi_{7}&\cos\phi_{7}&\cos 3\phi_{7}&\cos 2 \phi_{7}\\ 1&\cos 3\phi_{7}&\cos\phi_{7}&\cos 2\phi_{7}&\cos 2\phi_{7}&\cos\phi_{7}&\cos 3 \phi_{7}\\ 0&\sin 3\phi_{7}&-\sin\phi_{7}&\sin 2\phi_{7}&-\sin 2\phi_{7}&\sin\phi_{7}&-\sin 3 \phi_{7}\\ 0&\sin 2\phi_{7}&-\sin 3\phi_{7}&-\sin\phi_{7}&\sin\phi_{7}&\sin 3\phi_{7}&-\sin 2 \phi_{7}\\ 0&\sin\phi_{7}&\sin 2\phi_{7}&\sin 3\phi_{7}&-\sin 3\phi_{7}&-\sin 2\phi_{7}&-\sin \phi_{7}\end{bmatrix}  \tag{56}\]  
When we calculate the product of this matrix by the input vector \(\mathbf{x}_{7}\) we obtain  
\[ \begin{bmatrix}y_{0}\\ y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\end{bmatrix} = \begin{bmatrix}x_{0}+\left[(x_{1}+x_{6})+(x_{2}+x_{5})+(x_{3 }+x_{4})\right]\\ x_{0}+\cos\phi_{7}(x_{1}+x_{6})+\cos 2\phi_{7}(x_{2}+x_{5})+\cos 3\phi_{7}(x_{3}+x_{4})\\ x_{0}+\cos 2\phi_{7}(x_{1}+x_{6})+\cos\phi_{7}(x_{2}+x_{5})+\cos\phi_{7}(x_{3}+ x_{4})\\ x_{0}+\cos 3\phi_{7}(x_{1}+x_{6})+\cos\phi_{7}(x_{2}+x_{5})+\cos 2\phi_{7}(x_{3}+ x_{4})\\ \sin 3\phi_{7}(x_{1}-x_{6})-\sin\phi_{7}(x_{2}-x_{5})+\sin 2\phi_{7}(x_{3}- x_{4})\\ \sin 2\phi_{7}(x_{1}-x_{6})-\sin 3\phi_{7}(x_{2}-x_{5})-\sin\phi_{7}(x_{3}- x_{4})\\ \sin\phi_{7}(x_{1}-x_{6})+\sin 2\phi_{7}(x_{2}-x_{5})+\sin 3\phi_{7}(x_{3}- x_{4})\end{bmatrix}  \tag{57}\]  
To better understand the construction of the RDFT algorithm for \(N=7\), we will introduce the notations \(t_{1}=x_{1}+x_{6}\), \(t_{2}=x_{2}+x_{5}\), \(t_{3}=x_{3}+x_{4}\), \(t_{4}=x_{3}-x_{4}\), \(t_{5}=x_{2}-x_{5}\), \(t_{6}=x_{1}-x_{6}\), \(t_{0}=x_{0}+t_{1}+t_{2}+t_{3}\) and consider the sub-blocks of the output vector \(\mathbf{y}_{7}\). The first sub-block is  
\[ \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix} = \begin{bmatrix}x_{0}\\ x_{0}\\ x_{0}\\ x_{0}\end{bmatrix} + \begin{bmatrix}\cos\phi_{7}&\cos 2\phi_{7}&\cos 3\phi_{7}\\ \cos 2\phi_{7}&\cos 3\phi_{7}&\cos\phi_{7}\\ \cos 3\phi_{7}&\cos\phi_{7}&\cos 2\phi_{7}\end{bmatrix}\begin{bmatrix}t_{1}\\ t_{2}\\ t_{3}\end{bmatrix} =\]  
\[= \begin{bmatrix}x_{0}\\ x_{0}\\ x_{0}\\ x_{0}\end{bmatrix} + \begin{bmatrix}1&1&1&0\\ 1&-1&0&1\\ 1&0&-1&-1\end{bmatrix}\begin{bmatrix}d_{1}&0&0&0\\ 0&d_{2}&0&0\\ 0&0&d_{3}&0\\ 0&0&0&d_{4}\end{bmatrix}\begin{bmatrix}1&1&1\\ 1&0&-1\\ -1&1&0\end{bmatrix}\begin{bmatrix}t_{1}\\ t_{2}\\ t_{3}\end{bmatrix}