# Fast Type-II Hartley Transform Algorithms for Short-Length Input Sequences
## 6 Conclusions  
As a result of research, the problem of designing the fast algorithms of type-II DHT was solved for short-length input signals based on a structural approach. The algorithms for short-length input signals are of special interest because they are subsequently used as building blocks of algorithms for long-length sequences. After a literature review, it was decided to use the structural properties of the type-II DHT coefficients matrix to develop fast algorithms for this transform. The proposed algorithms differ from existing solutions by applying pre-decomposition of the original \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{**Reference, Year of**} & \multicolumn{2}{c}{\(N=4\)} & \multicolumn{2}{c}{\(N=8\)} \\ \cline{3-6}  & & **Mulplications** & **Additions** & **Multiplications** & **Additions** \\ \hline [PERSON]’s radix-2 DIT algorithm & [24], 2007 & 2 (0\%) & 18 (\(-67\%\)) & 12 (\(-33\%\)) & 48 (\(-46\%\)) \\ Prime factor algorithm & [23], 2004 & 2 (0\%) & 10 (\(-40\%\)) & 8 (0\%) & 28 (\(-7\%\)) \\ Radix-2 algorithm & [24], 2007 & 4 (\(-50\%\)) & 18 (\(-67\%\)) & 14 (\(-43\%\)) & 42 (\(-38\%\)) \\ Radix-2 algorithm & [25], 2011 & – & – & – & 10 (\(-20\%\)) & 28 (\(-7\%\)) \\ Radix-2 DIT algorithm & [26], 2016 & 2 (0\%) & 6 (0\%) & 12 (\(-33\%\)) & 24 (+8\%) \\ Split-radix algorithm & [27], 2020 & 2 (0\%) & 6 (0\%) & 12 (\(-33\%\)) & 24 (+8\%) \\ Proposed algorithm & – & 2 & 6 & 8 & 26 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The number of operations for the algorithms known from the literature. matrices into submatrices, as well as rearranging the rows and/or columns of the resulting matrices to find the factorizations that would reduce the computational complexity. The relations between the values of the matrix entries were also considered. In an experiment, the computational complexity of the proposed algorithms was compared with the direct calculation of matrix-vector products. The resulting DHT-II matrix factorizations reduced the number of additions by 5% on average and the number of multiplications by 73% for DHT-II lengths in the range of 3 to 8. As a result, the amount of resources used on the signal processor was significantly reduced while allowing for easier operation in real-time. A significant contribution to the speedup of signal processing came from the reduction in multiplications since multiplications are more labor-intensive and time-expensive to use than additions. To achieve computational efficiency, the structure of the transform matrix and not the parameters of DHT-II were optimized. A comparison of the proposed algorithms based on the structural approach with algorithms known from the literature showed that the efficiency of the proposed algorithms significantly exceeded the efficiency of radix or split-radix algorithms and was close to the efficiency of prime factor algorithms. As for prospects for further research, we note the following. In [38], as an example, the application of the Hartley transform in wavelet-like decomposition was shown. The application of known orthogonal transformations in multiscale decomposition and reconstruction significantly reduced computational costs and gave the same effect as wavelet decomposition and reconstruction [39; 40]. Thus, we assume that the wavelet-like decomposition can be applied to increase the performance of texture image segmentation for skin disease diagnosis [41]. Conceptualization, A.C.; methodology, A.C., M.P. ; software, M.P. ; validation, A.C., M.P. ; formal analysis, A.C., M.P. ; investigation, M.P. ; writing--original draft preparation, M.P. ; writing--review and editing, A.C.; supervision, A.C. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors. The authors declare no conflicts of interest.