# The Fast Discrete Tchebichef Transform Algorithms for Short-Length Input Sequences
## 3 Reduced Complexity DTT Algorithms for Short-Length Input Sequences
### Algorithm for 7-Point DTT
\tag{37}\]  
Next, matrix \(\mathbf{C}_{7}^{(a)}\) is decomposed into two components:  
\[\mathbf{C}_{7}^{(a)}=\mathbf{C}_{7}^{(b)}+\mathbf{C}_{7}^{(c)}, \tag{38}\]  
where\[\mathbf{C}_{7}^{(b)}= \begin{bmatrix}&&&\\ \mathbf{a}_{7}&\mathbf{a}_{7}&\mathbf{a}_{7}&\mathbf{a}_{7}&\mathbf{a}_{7}&\mathbf{a}_{7}\\ &&-\mathbf{a}_{7}&&&\\ &&&f_{7}&&&\\ &&-k_{7}&&\end{bmatrix} ,\mathbf{C}_{7}^{(c)}= \begin{bmatrix}-c_{7}&c_{7}&-b_{7}&b_ {7}&-a_{7}&a_{7}\\ -a_{7}&a_{7}&a_{7}&-a_{7}&a_{7}&-a_{7}\\ -a_{7}&a_{7}&d_{7}&-d_{7}&-e_{7}&e_{7}\\ &&&&\\ \mathbf{e}_{7}&\mathbf{e}_{7}&&&-c_{7}&-c_{7}\\ \mathbf{c}_{7}&\mathbf{c}_{7}&-\mathbf{g}_{7}&-\mathbf{g}_{7}&\mathbf{a}_{7}&\mathbf{a}_{7}\\ \mathbf{a}_{7}&\mathbf{a}_{7}&-\mathbf{f}_{7}&-\mathbf{f}_{7}&\mathbf{h}_{7}&\mathbf{h}_{7}\end{bmatrix} . \tag{39}\]  
Matrix \(\mathbf{C}_{7}^{(b)}\) has identical entries in its fourth row. This allows us to reduce the number of operations without further transformations. Next, we eliminate the row and column containing only zero entries in matrix \(\mathbf{C}_{7}^{(c)}\) and obtain matrix \(\mathbf{C}_{6}^{(d)}\):  
\[\mathbf{C}_{6}^{(d)}= \begin{bmatrix}-c_{7}&c_{7}&-b_{7}&\mathbf{b}_{7}&-a_{7}&a_{7 }\\ -a_{7}&a_{7}&a_{7}&-a_{7}&a_{7}&-a_{7}\\ -a_{7}&a_{7}&d_{7}&-d_{7}&-e_{7}&e_{7}\\ \mathbf{e}_{7}&\mathbf{e}_{7}&\mathbf{0}&\mathbf{0}&-c_{7}&-c_{7}\\ \mathbf{c}_{7}&\mathbf{c}_{7}&-\mathbf{g}_{7}&-\mathbf{g}_{7}&a_{7}&a_{7}\\ a_{7}&a_{7}&-\mathbf{f}_{7}&-\mathbf{f}_{7}&h_{7}&h_{7}\end{bmatrix} . \tag{40}\]  
Matrix \(\mathbf{C}_{6}^{(d)}\) contains repeated elements, enabling the design of the data flow graph for the 7-point 1D DTT calculation. Figure 5 shows the obtained data flow graph of the developed 7-point 1D DTT algorithm. Figure 5: The data flow graph of the 7-point 1D DTT algorithm. Further, we create the factorization of the matrix of 7-point 1D DTT using the matrices for calculations at each layer of the graph. As a result, we obtain matrix \(\textbf{W}_{10\times 7}\) at the first layer of calculations:  
\[\textbf{W}_{10\times 7}= \begin{bmatrix}\textbf{1}&&&&\\ &\textbf{1}&&&&\\ &\textbf{1}&&&&\\ &\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}\\ &&&\textbf{1}&\\ &&&\textbf{1}\\ &&\textbf{1}&&&\\ &\textbf{1}&&&\\ &\textbf{1}&&&\\ &\textbf{1}&&&\\ &\textbf{1}&&&\\ \end{bmatrix} . \tag{41}\]  
The second layer of calculations is described by matrix \(\textbf{W}_{10}\) which can be expressed as follows: \(\textbf{W}_{10}=\textbf{W}_{2}\oplus\textbf{W}_{3}\oplus\textbf{W}_{2}\oplus \textbf{I}_{3}\), where \(\textbf{W}_{3}= \begin{bmatrix}-1&1\\ &1\\ &1&1\\ \end{bmatrix} \). Matrices \(\textbf{W}_{19\times 10}\) and \(\textbf{W}_{19}\) provide calculations at the third and fifth layers of the data flow graph in Figure 5:  
\[\textbf{W}_{19\times 10}=\textbf{1}_{2\times 1}\oplus\textbf{1}_{3\times 1} \oplus\textbf{1}_{3\times 1}\oplus\textbf{1}_{2\times 1}\oplus\textbf{1}_{2 \times 1}\oplus\textbf{1}_{3\times 1}\oplus\textbf{I}_{3}, \tag{42}\]  
\[\textbf{W}_{19}= \begin{bmatrix}1&1\\ &1\\ &1\\ \end{bmatrix} \oplus\textbf{H}_{2}\oplus 1\oplus \begin{bmatrix}1&-1\\ &1\\ \end{bmatrix} \oplus\mathbf{I}_{2}\oplus \begin{bmatrix}1&-1\\ &1\\ &1\\ \end{bmatrix} \oplus \begin{bmatrix}1&1\\ &1\\ &-1&1\\ \end{bmatrix} \oplus \begin{bmatrix}1&1\\ &1&1\\ 1&1&1\\ \end{bmatrix} . \tag{43}\]  
We have diagonal matrix  
\[\textbf{D}_{19}=diag(2,\,1,\,4,\,1,\,1,\,2,\,2,\,1,\,1,\,8,\,1,\,4,\,2,\,1,\,1, \,16,\,4,\,2,\,16) \tag{44}\]  
at the fourth layer of calculation, and matrix \(\textbf{W}_{7\times 19}\) at the sixth layer of calculation as follows:  
\[= \begin{bmatrix}1&&&&&&&\textbf{W}_{7\times 19}=&&&&\\ &1&&&&1&&&&\\ &1&&&-1&&&-1&&&\\ &1&&&-1&&&1&&&\\ &1&&&-1&&&1&&&\\ &&1&&&1&&&-1&&&1&-1\\ \end{bmatrix} .