New 2n discrete cosine transform algorithm using recursive filter structure

It is always desirable to look for more efficient algorithms for the realization of the discrete cosine transform (DCT). In this paper, we generalize a formulation for converting a lcngth-2" DCT into n groups of equations, then apply a novel technique for its implementation. The sizes of the groups are 2", for m = 11-1, .._., 0. While their structures are extremely regular. Tlie realization can then be converted into the simplest recursive filter form, which is of particularly simple for practical implementation. Introduction Tlie discrete cosine transform [I] is widely used in digital signal processing, particularly for digital image processing. Because of the complicated computational complexity, many eflicient algorithms were proposed to improve the computing speed and hardware complexity. These algorithms can broadly be classified into the following categories: 1) indirect computation through the discrete Fourier transform or the Walsh-Hadamard transform [2-3 1, 2) direct factorization [4-71, and 3) recursive computation [S-121. Among them, the ones using indirect computation method often involve extra opcrations. Tlie direct factorization decomposes the DCT directly, so that the total number of operations can be rcduccd. By implementing the recursive structure in an effective way, a regular and parallel VLSI structure can possibly be used and the computational complexity is greatly reduced. In this papcr, we present a formulation for converting a lcngtli-2" DCT into n groups of equations and the sizes of the groups are 2"-', 2"-2,....,20 respectively. The resultant formulation is extremely regular, which is suitable for the implementation using a recursive filter structure. Furthermore, the beauty of the formulation is enhanced by expanding the multiple angle cosine function into a series of high order cosine functions to effect the realization of the recursive filter structure. Derivation of algorithm The DCT of a data sequence (xo(i):i = 0, l,.... ,N 1) can be written as N-1 Y(k) = 1 x,(i)cos i=O for k = 0, 1, ..., N-1, where N = 2", n is an integer and the index "0" of x gives the stage of data representation (see below). For the convenience of realization, let us introduce a formulation, such that Y(k) is split into n groups, namely Y(2r + I), Y(2(2r + I)),......, Y(2"-'(2r + 1)) and Y(0) for I= 0, I ,... ., 2n-(m 'I) I Let us rewrite Y(k) in the form of Y(2"(2r+1)) and make some simplifications, for m = 0, 1, ..., n-I, where m is the group number starting from zero N-1 = C x~i)cos((2i + 1)2m0,) i=O (2r + 1)x 2N for 8, = We may rearrange the order of computation of the lower half of the left hand side, hence, f 4-1 1 2 (Cxo(N-l-i)cos((2N-2i-l)2~€Ir) i=O 1169 0-7803-2431 -5/95 $4.00