Transactions Briefs A Cyclic Correlated Structure for the Realization of Discrete Cosine Transform

In this paper we propose using the correlated cosine structure (CCS) for the computation of the discrete cosine transform (DCT). This structure has circulant property and is most suitable for the hardware realization. We will show that there exists a close relationship between the CCS and the DCT. In such a case, a 2"' length DCT can be decomposed recursively into shorter length CCS and DCT. This new approach results in very simple and straightforward structure and gives the minimum number of multiplications for its realization. I. . Many algorithms for the computation of the discrete cosine transform (DCT) have been proposed since the introduction of the DCT in 1974 by Ahmed et al. [l]. These algorithms can broadly be classified into two groups: 1) indirect computation through the fast discrete Fourier transform and the Walsh Hadamard transform [2], [3], and 2) direct computation of DCT through matrix decomposition or recursive computation [4]-[8]. Among them, Lee's [4], Hou's [5], and Vetterli's [7] algorithms meet the minimum known number of multiplications to implement a length 2"' DCT algorithm. The convolution structure plays an important role in digital signal processing due to its regularity and simplicity during its hardware implementation [9]. Since a correlation of two sequences can always be converted into a convolution by inverting one of the input sequences, the correlation is also widely used in digital signal processing. In fact, there is a number of DFT algorithms developed depending upon these two structures [ 101, [ 111. In this paper, we first introduce a structure called the correlated cosine structure (CCS), and then present a new algorithm to compute an N = 2" length DCT by decomposing it into shorter length CCS and DCT recursively. Then we will elaborate the correlation property of the CCS. Finally, we will show that the proposed algorithm enables us to realize the DCT with the least number of multiplications compared with conventional approaches, and it also results in extremely regular structure such that its realization is very simple. E. ALGORITHM DERIVATION The DCT [l] of a real data sequence {x(i): i = 0,l; . a , N 1) is defined as X ( k ) = x ( i ) c o s [ 2 ~ ( 2 i + l ) k / 4 N ] , N 1 i = O for k = 0, l ; . . , N 1. (1) Let us split X ( k ) into odd and even sequences. Manuscript received November 20, 1989; revised April 19, 1991 and September 13, 1991. This paper was recommended by Associate Editor E. J. Coyle. The authors are with the Computer Engineering Section, Department of Electronic Engineering, Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong. IEEE Log Number 9105218. X ( 2 k )