The Design of Fast Type-V Discrete Cosine Transform Algorithms for Short-Length Input Sequences

Abstract

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%.