Roundoff error analysis of fast DCT algorithms in fixed point arithmetic

Abstract

Discrete cosine transforms (DCT) are essential tools in numerical analysis and digital signal processing. Processors in digital signal processing often use fixed point arithmetic. In this paper, we consider the numerical stability of fast DCT algorithms in fixed point arithmetic. The fast DCT algorithms are based on known factorizations of the corresponding cosine matrices into products of sparse, orthogonal matrices of simple structure. These algorithms are completely recursive, are easy to implement and use only permutations, scaling, butterfly operations, and plane rotations/rotation-reflections. In comparison with other fast DCT algorithms, these algorithms have low arithmetic costs. Using von Neumann–Goldstine’s model of fixed point arithmetic, we present a detailed roundoff error analysis for fast DCT algorithms in fixed point arithmetic. Numerical tests demonstrate the performance of our results.