Abstract — The general method for efficient computation of discrete cosine transform (DCT) using cyclic convolutions is considered. Forming hashing arrays on the basis of simplified arguments on the basis of discrete cosine transform are analyzed. The examples of four types of discrete cosine transform using proposed method are analyzed. Keywords — algorithm, cyclic convolution, discrete cosine transforms, hashing array, synthesis. I. INTRODUCTION Since the middle of 1960s fast computation of discrete Fourier transform (DFT) has been intensively developed. In many studies attention was paid to the prospects of applying only real computing. In 1974 a discrete cosine transform (DCT) [1], which recreates the real of basis function dependence similar to the DFT was proposed. There are 8 types of discrete cosine transform, which are in general examined in [2]. Cosine, sine transforms and Fourier transforms are interrelated with strict mathematical relations that allow us to find an effective technique to compute one transform via another [3]. DCT is widely applied for several reasons. Firstly, basis DCT functions are well approximated to Karhunen-Loyeva transfer function for a big number of stationary stochastic processes which allows presenting the signal of a given accuracy with a minimal number of components. Secondly, DCT is included as part of some efficient DFT algorithms such algorithm [4]. Thirdly, DCT contains a number of special properties, because the conversion is concentrated in the lower indices and more intense and zeroing the remaining output values does not lead to a significant loss of signal energy that prevents edge effects at the block encoded images [5]. DCT is used in many applications, especially in processing digital signals audio and video. Further intensive development of information technology sets higher requirements before DCT and algorithmic, software and hardware with the performance and development of functional and specific opportunities transforms. Efficient computing of one or two dimensional DCT, called fast discrete cosine transform (FDCT), has been investigated for more than four decades. A significant numbers of publications devoted of efficient computation of DCT [6] has Ihor Prots’ko with the CAD Department, Lviv Polytechnic University, S. Bandery Str.,12, Lviv, 79046, Ukraine, (e-mail: ioprotsko@gmail.com). been written. Multivariate effective computing algorithms can be divided into: size of radix two, split radix, mixed radix, odd size, and prime factors composite transform size. For the synthesis of efficient algorithms of DCT the following approaches are used: 1) direct matrix factorization of DCT; 2) indirect calculation by FFT or through discrete Hartley transform; 3) algorithms based on the theory of complexity. It is proved that two types of DCT are polynomial transforms, and in one case it was used to obtain a fast algorithm. It is shown that four types of DCT with group symmetry (properties pertaining to the theory of groups and their representations) and for each of them fast algorithm is derived purely algebraically. Works of fast DCT are summarized and systematized and final step in this direction is the theory of the creation of fast algorithms [7]. One approach of efficient algorithms is the possibility to compute DCT through the cyclic convolutions. Papers [8] use the following index mappings on base primitive root g of cyclic group. A lot of the papers [9]-[11] appeared about efficient hardware architectures connected with the computation through cyclic convolution. The paper is structured as follows. Section 2 describes four types of discrete cosine transform (subsection A) and simplified arguments of basis DCT I-IV (subsection B) for general algorithm on the basis of hashing arrays for DCT using cyclic convolutions (subsection C). In Section 3 the performance of the proposed general algorithm on examples of four DCT I-IV types for concrete sizes are analyzed (subsection A,B,C,D) and Section 4 conclusions are presented. II. EFFICIENT COMPUTATION OF DCT I-IV USING CYCLIC CONVOLUTIONS Computation types DCT and IDCT (direct and inverse) are one of the most long-term procedures in information technology, such as for example the compression frame images. This procedure requires the greatest degree of improvements that will speed up the work of software and hardware. Efficient computation of DCT using cyclic convolutions is important and needs development. Using the method of computation for each type of DCT based on cyclic convolutions has different specifics and is analyzed in this paper. Algorithm of efficient computation of DCT I-IV using cyclic convolutions Ihor Prots’ko INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Issue 1, Volume 7, 2013 1