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-Loeve 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.

CHAPTER 1 – Discrete Cosine and Sine Transforms

Fast and numerically stable algorithms for discrete cosine transforms

An in-place algorithm for the fast, direct computation of the forward and inverse discrete cosine transform is presented and evaluated. The transform length may be an arbitrary power of two.

Algorithm 749: fast discrete cosine transform

A new fast algorithm for computing prime-length DCT through cyclic convolutions

This paper presents an efficient serial–parallel multiplier algorithm that realizes the input data bit-serially, for implementation of the discrete cosine transform (DCT), which realizes the input data bit-serially First, the DCT equation is split into a few groups of equations by using some mathematical techniques, and index tables are constructed to facilitate efficient data permutations A new formulation of the DCT is then derived Second, we represent the cosine coefficient in binary form and realize multiplications using a new serial–parallel multiplier architecture that results in a simple structure for VLSI realization

Efficient multiplier structure for realization of the discrete cosine transform

An extension of an existing fast algorithm for the computation of the discrete cosine transform is presented. The proposed algorithm has a structure similar to the split-radix algorithm used for the computation of the FFT, the same computational complexity as the direct fast cosine algorithms and a high regularity which facilitates its implementation in VLSI technology. A comparison on different computer architectures shows that the proposed algorithm is superior to existing algorithms in terms of execution time.

A. N. SltODRAS

Papers

Split-radix fast cosine transform algorithm