Signal flow graph approach to efficient and forward stable DST algorithms

Abstract

In this paper, signal flow graphs are mainly addressed for the efficient and forward stable Discrete Sine Transform (DST) algorithms having sparse, scaled orthogonal, rotational, rotational-reflection, and butterfly matrices. In electrical engineering, theoretical computer science, control theory, system engineering, etc., we often use signal flow graphs as a modeling tool which interconnects the system components, or represents the realization of a system as electronic devices. The objective in this paper is to establish the connection between algebraic operations used in sparse and scaled orthogonal factorizations of DST I–IV matrices, with the signal flow graph building blocks. This paper elaborates signal flow graphs for the foundational frameworks with n=8 and n=16, and use these together with the digital structures to describe the DST algorithms having a (n−1)-point signal flow graph for DST I and n-point signal flow graphs for DST II–IV. The presented DST algorithms are completely recursive, and solely based on corresponding matrices DST I–IV. These DST algorithms have low arithmetic complexity, especially the low number of multiplications, and significant speed improvement factor as opposed to most existing algorithms. Finally, this paper establishes that the presented algorithms are forward stable DST algorithms.