## 7 RDFT Algorithm for \(N=7\)
 \tag{58}\]  
where \(\hat{d}_{1}=\frac{1}{3}(\cos\phi_{7}+\cos 2\phi_{7}+\cos 3\phi_{7})\), \(d_{2}=\frac{1}{3}(2\cos\phi_{7}-\cos 2\phi_{7}-\cos 3\phi_{7})\), \(d_{3}=\frac{1}{3}(-\cos\phi_{7}+2\cos 2\phi_{7}-\cos 3\phi_{7})\), \(d_{4}=\frac{1}{3}(-\cos\phi_{7}-\cos 2\phi_{7}+2\cos 3\phi_{7})\). It can also be written as  
\[ \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix} = \begin{bmatrix}t_{0}\\ t_{0}\\ t_{0}\end{bmatrix} + \begin{bmatrix}1&1&1&0\\ 1&-1&0&1\\ 1&0&-1&-1\end{bmatrix}\begin{bmatrix}d_{1}&0&0&0\\ 0&d_{2}&0&0\\ 0&0&d_{3}&0\\ 0&0&0&d_{4}\end{bmatrix}\begin{bmatrix}1&1&1\\ 1&0&-1\\ -1&1&0\end{bmatrix}\begin{bmatrix}t_{1}\\ t_{2}\\ t_{3}\end{bmatrix}  \tag{59}\]  
where \(d_{1}=\frac{1}{3}(\cos\phi_{7}+\cos 2\phi_{7}+\cos 3\phi_{7})-1\). The second sub-block is  
\[ \begin{bmatrix}y_{4}\\ y_{5}\\ y_{6}\end{bmatrix} = \begin{bmatrix}\sin 2\phi_{7}&-\sin\phi_{7}&\sin 3\phi_{7}\\ -\sin\phi_{7}&-\sin 3\phi_{7}&\sin 2\phi_{7}\\ \sin 3\phi_{7}&\sin 2\phi_{7}&\sin\phi_{7}\end{bmatrix}\begin{bmatrix}t_{4}\\ t_{5}\\ t_{6}\end{bmatrix} =\]  
\[= \begin{bmatrix}-1&0&1&-1\\ 1&-1&0&-1\\ 1&1&1&0\end{bmatrix}\begin{bmatrix}d_{5}&0&0&0\\ 0&d_{6}&0&0\\ 0&0&d_{7}&0\\ 0&0&0&d_{8}\end{bmatrix}\begin{bmatrix}-1&1&1\\ 1&0&1\\ -1&-1&0\\ 0&1&-1\end{bmatrix}\begin{bmatrix}t_{4}\\ t_{5}\\ t_{6}\end{bmatrix}  \tag{60}\]\[\text{where }d_{5}=\frac{1}{3}(\sin\phi_{7}+\sin 2\phi_{7}-\sin 3 \phi_{7}),\,d_{6}=\frac{1}{3}(2\sin\phi_{7}-\sin 2\phi_{7}+\sin 3\phi_{7}),\] \[d_{7}=\frac{1}{3}(\sin\phi_{7}-2\sin 2\phi_{7}-\sin 3\phi_{7}),\,d_{8 }=\frac{1}{3}(\sin\phi_{7}+\sin 2\phi_{7}+2\sin 3\phi_{7}).\]  
Figure 5 shows a data-flow diagram corresponding to the calculation of the output vector \(\mathbf{y}\gamma\). When the vector \(\mathbf{z}_{7}=\mathbf{a}_{7}+i\mathbf{b}_{7}\) of complex coefficients of DFT for the real input vector \(\mathbf{x}_{7}\) is needed, it can be easily obtained from the output vector \(\mathbf{y}_{7}\) of RDFT, according to (16)  
\[\mathbf{a}_{7}= \begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ a_{5}\\ a_{6}\end{bmatrix} = \begin{bmatrix}y_{0}\\ y_{1}\\ y_{2}\\ y_{3}\\ y_{3}\\ y_{2}\\ y_{1}\end{bmatrix} \mathbf{b}_{7}= \begin{bmatrix}b_{0}\\ b_{1}\\ b_{2}\\ b_{3}\\ b_{4}\\ b_{5}\\ b_{6}\end{bmatrix} = \begin{bmatrix}0\\ -y_{6}\\ -y_{5}\\ -y_{4}\\ y_{4}\\ y_{5}\\ y_{6}\end{bmatrix}  \tag{61}\]  
The algorithm of the RDFT for \(N=7\), presented in Figure 5, can be described by the following matrix-vector procedure, in which the matrix \(\mathbf{R}_{7}\) has been factorized:  
\[\mathbf{y}_{7}=\mathbf{C}_{7\times 9}\mathbf{\tilde{C}}_{9}\mathbf{D}_{9} \mathbf{\tilde{A}}_{9}\mathbf{\overline{A}}_{9\times 7}\mathbf{\tilde{A}}_{7} \mathbf{x}_{7} \tag{62}\]  
where  
\[\mathbf{\tilde{A}}_{7}= \begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0&1\\ 0&0&1&0&0&1&0\\ 0&0&0&1&1&0&0\\ 0&0&0&1&-1&0&0\\ 0&0&1&0&0&-1&0\\ 0&1&0&0&0&0&-1\end{bmatrix} \quad\mathbf{\overline{A}}_{9\times 7}= \begin{bmatrix}1&0&0&0&0&0&0\\ 0&1&1&1&0&0&0\\ 0&1&0&-1&0&0&0\\ 0&0&1&-1&0&0&0\\ 0&-1&1&0&0&0&0\\ 0&0&0&0&-1&1&1\\ 0&0&0&0&1&0&1\\ 0&0&0&0&-1&-1&0\\ 0&0&0&0&0&1&-1\end{bmatrix}  \tag{63}\]  
Figure 5: Data-flow diagram of the RDFT algorithm for \(N=7\). \[\tilde{\bf A}_{9}= \begin{bmatrix}1&1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\end{bmatrix} \quad\bf D_{9}= \begin{bmatrix}1&0&0&0&0&0&0&0&0\\ 0&d_{1}&0&0&0&0&0&0&0\\ 0&0&d_{2}&0&0&0&0&0&0\\ 0&0&0&d_{3}&0&0&0&0&0\\ 0&0&0&0&d_{4}&0&0&0&0\\ 0&0&0&0&0&d_{5}&0&0&0\\ 0&0&0&0&0&d_{6}&0&0\\ 0&0&0&0&0&0&0&d_{7}&0\\ 0&0&0&0&0&0&0&d_{8}\end{bmatrix}  \tag{64}\]  
\[\tilde{\bf C}_{9}= \begin{bmatrix}1&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\end{bmatrix} \quad\bf C_{7\gamma 9}= \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&1&0&0&0&0&0\\ 0&1&-1&0&1&0&0&0&0\\ 0&1&0&-1&-1&0&0&0&0\\ 0&0&0&0&0&-1&0&1&-1\\ 0&0&0&0&0&1&-1&0&-1\\ 0&0&0&0&0&1&1&1&0\end{bmatrix}