## 8 RDFT Algorithm for \(N=8\)
 \tag{70}\]  
The algorithm of the RDFT for \(N=8\), presented in Figure 6, can be described by the following matrix-vector procedure, in which the matrix \(\mathbf{R}_{8}\) has been factorized:  
\[\mathbf{y}_{8}=\mathbf{C}_{8}\mathbf{D}_{8}\mathbf{\tilde{A}}_{8}\mathbf{ \overline{A}}_{8}\mathbf{\tilde{A}}_{8}\mathbf{x}_{8} \tag{71}\]  
where  
\[\mathbf{\hat{A}}_{8}\mathbf{=} \begin{bmatrix}1&0&0&0&1&0&0\\ 0&1&0&0&0&0&0&1\\ 0&0&1&0&0&0&1&0\\ 0&0&0&1&0&1&0&0\\ 1&0&0&0&-1&0&0&0\\ 0&0&0&1&0&-1&0&0\\ 0&0&1&0&0&0&-1&0\\ 0&1&0&0&0&0&-1\end{bmatrix} \quad\mathbf{\overline{A}}_{8}\mathbf{=} \begin{bmatrix} 1&0&1&0&0&0&0&0\\ 0&1&0&1&0&0&0&0\\ 1&0&-1&0&0&0&0&0\\ 0&1&0&-1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&1\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&-1&0&1\end{bmatrix}  \tag{72}\]  
Figure 6: Data flow diagram of the RDFT algorithm for \(N=8\). \(\mathbf{C}_{4}\) by reversing the order of its columns. When we calculate the product of the matrix \(\mathbf{R}_{9}\) by the input vector \(\mathbf{x}_{9}\) we obtain  
\[ \begin{bmatrix}y_{0}\\ y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\\ y_{7}\\ y_{8}\\ \end{bmatrix} = \begin{bmatrix}[x_{0}+(x_{3}\!+\!x_{6})]+[(x_{1}\!+\!x_{8})+(x_{2} \!+\!x_{7})+(x_{4}\!+\!x_{5})]\\ [x_{0}\!-\!0.5(x_{3}\!+\!x_{6})]+\cos\phi_{9}[(x_{1}\!+\!x_{8})+\cos 2\phi_{9}[(x_{ 2}\!+\!x_{7})+\cos 4\phi_{9}[(x_{4}\!+\!x_{5})]\\ [x_{0}\!-\!0.5(x_{3}\!+\!x_{6})]+\cos 2\phi_{9}[(x_{1}\!+\!x_{8})+\cos 4\phi_{9}[(x_{ 2}\!+\!x_{7})+\cos\phi_{9}[(x_{ 4}\!+\!x_{5})]\\ [x_{0}\!+(x_{3}\!+\!x_{6})]-0.5[(x_{1}\!+\!x_{8})+(x_{2}\!+\!x_{7})+(x_{4}\!+ \!x_{5})]\\ [x_{0}\!-\!0.5(x_{3}\!+\!x_{6})]+\cos 4\phi_{9}[(x_{1}\!+\!x_{8})+\cos\phi_{9}[(x_{ 2}\!+\!x_{7})+\cos 2\phi_{9}[(x_{ 4}\!+\!x_{5})\\ \sin 3\phi_{9}(x_{3}\!-\!x_{6})]-\sin 2\phi_{9}[(x_{ 4}-x_{5})-\sin\phi_{9}[(x_{ 2}-x_{7})+\sin 4\phi_{9}[(x_{ 1}-x_{8})\\ \sin 3\phi_{9}[(x_{ 4}\!-\!x_{5})-(x_{2}\!-\!x_{7})+(x_{1}\!-\!x_{8})]\\ -\sin 3\phi_{9}(x_{ 3}\!-\!x_{6})]-\sin\phi_{9}[(x_{ 4}\!-\!x_{5})+\sin 4\phi_{9}[(x_{ 2}\!-\!x_{7})+\sin 2\phi_{9}[(x_{ 1}\!-\!x_{8})\\ \sin 3\phi_{9}(x_{ 3}\!-\!x_{6})]+\sin 4\phi_{9}[(x_{ 4}\!-\!x_{5})+\sin 2\phi_{9}[(x_{ 2}\!-\!x_{7})+\sin\phi_{9}[(x_{ 1}\!-\!x_{8})\end{bmatrix}  \tag{81}\]  
To better understand the construction of the RDFT algorithm for \(N=9\), we will introduce the notations \(t_{1}=x_{1}+x_{8}\), \(t_{2}=x_{2}+x_{7}\), \(t_{3}=x_{3}+x_{6}\), \(t_{4}=x_{4}+x_{5}\), \(t_{5}=x_{4}-x_{5}\), \(t_{6}=x_{3}-x_{6}\), \(t_{7}=x_{2}-x_{7}\), \(t_{8}=x_{1}-x_{8}\) and consider the sub-blocks of the output vector \(\mathbf{y}_{8}\). The first sub-block is  
\[ \begin{bmatrix}y_{1}\\ y_{2}\\ y_{4}\end{bmatrix} = \begin{bmatrix}x_{0}-0.5t_{3}\\ x_{0}-0.5t_{3}\\ x_{0}-0.5t_{3}\end{bmatrix} + \begin{bmatrix}\cos\phi_{9}&\cos 2\phi_{9}&\cos 4 \phi_{9}\\ \cos 2\phi_{9}&\cos 4\phi_{9}&\cos\phi_{9}\\ \cos 4\phi_{9}&\cos\phi_{9}&\cos 2\phi_{9}\end{bmatrix}\begin{bmatrix}t_{1}\\ t_{2}\\ t_{4}\end{bmatrix} =\]  
\[= \begin{bmatrix}x_{0}-0.5t_{3}\\ x_{0}-0.5t_{3}\\ x_{0}-0.5t_{3}\end{bmatrix} + \begin{bmatrix}1&1&1&0\\ 1&-1&0&1\\ 1&0&-1&-1\end{bmatrix}\begin{bmatrix}d_{5}&0&0&0\\ 0&d_{6}&0&0\\ 0&0&d_{7}&0\\ 0&0&0&d_{8}\end{bmatrix}\begin{bmatrix}1&1&1\\ 1&0&-1\\ 0&1&-1\\ -1&1&0\end{bmatrix}\begin{bmatrix}t_{1}\\ t_{2}\\ t_{4}\end{bmatrix}  \tag{82}\]  
where \(d_{5}=\frac{1}{3}(\cos\phi_{9}+\cos 2\phi_{9}+\cos 4\phi_{9})\), \(d_{6}=\frac{1}{3}(2\cos\phi_{9}-\cos 2\phi_{9}-\cos 4\phi_{9})\),  
\[d_{7}=\frac{1}{3}(-\cos\phi_{9}+2\cos 2\phi_{9}-\cos 4\phi_{9})\), \(d_{8}=\frac{1}{3}(-\cos\phi_{9}-\cos 2\phi_{9}+2\cos 3\phi_{9})\). The second sub-block is  
\[ \begin{bmatrix}y_{5}\\ y_{7}\\ y_{8}\end{bmatrix} = \begin{bmatrix}t_{4}\sin 3\phi_{9}\\ -t_{4}\sin 3\phi_{9}\\ t_{4}\sin 3\phi_{9}\end{bmatrix} + \begin{bmatrix}-\sin 2\phi_{9}&-\sin\phi_{9}&\sin 4 \phi_{9}\\ -\sin\phi_{9}&\sin 4\phi_{9}&\sin 2\phi_{9}\\ \sin 4\phi_{9}&\sin 2\phi_{9}&\sin\phi_{9}\end{bmatrix}\begin{bmatrix}t_{5}\\ t_{7}\\ t_{8}\end{bmatrix} =\]  
\[= \begin{bmatrix}t_{4}\sin 3\phi_{9}\\ -t_{4}\sin 3\phi_{9}\\ t_{4}\sin 3\phi_{9}\end{bmatrix} +