Аннотация:
Fast algorithms for type-five discrete cosine transform (DCT-V) for sequences of input data
of short length in the range of two to eight are elaborated in the paper. A matrix–vector product
representation of the DCT-V is the starting point for designing the algorithms. In each specific
case, the DCT-V matrices have remarkable structural properties that follow from the localization of
identical entries within the matrices. Each matrix of the DCT-V has only a few distinct entries that are
repeated at different positions in its structure. Using simple transformations such as permutations
of the rows and/or columns of this matrix or its favorable decomposition into two or more matrix
components, it is possible to obtain efficient matrix structures that lead to useful factorization schemes.
Based on the suitable factorization schemes we obtained, we developed fast algorithms that reduce
the number of arithmetic operations when calculating the DCT-V. The correctness of the obtained
algorithmic solutions was justified theoretically using a strict mathematical background of each of
them. The developed algorithms were then further tested using MATLAB R2023b software to finally
confirm their correctness. Finally, an evaluation of the computational complexity for each obtained
solution is presented. The evaluation results were compared with the computational complexity of
the direct calculation of matrix–vector products. The resulting factorizations of the matrices of the
DCT-V reduce the average number of multiplications by 57% but increase the number of additions
by 29%.
