Chapter 1 - Discrete Cosine and Sine Transforms

Abstract

This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loève transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.