The Development of Fast DST-II Algorithms for Short-Length Input Sequences

Abstract

The subject of this work is the development of fast algorithms for the discrete sinusoidal transformation of the second type (DST-II) for sequences of input data of small length N = 2, 3, 4, 5, 6, 7, 8. The starting point for the development of algorithms is the well-known possibility of representing any discrete transformation in the form of a matrix–vector product. Due to the remarkable structural properties of the matrices of the DST-II transformation base, these matrices can be successfully factorized, which should lead to a reduction in the computational complexity of the procedure as a whole. You can factorize matrices in different ways. The art of designing fast algorithms is to find the factorization that produces the maximum effect. We justified the correctness of the obtained algorithmic solutions theoretically, using strict mathematical derivations of each of them. The developed algorithms were then further tested using MATLAB R2023b software to finally confirm their performance. Finally, we presented estimates of the computational complexity for each solution obtained and compared them with direct computational methods that rely on the direct calculation of matrix–vector products.