A New Convolution Structure for the Realisation of the Discrete Cosine Transform

In this p%er, we present a new formulation for converting a length-N( = 2 ) Discrete Cosine Transform into two length-N/2 correlations. This formulation enables us to realise the Discrete Cosine Transform with a reduced number of operations compared to conventional approaches and it also results in extremely regular structure which is most witable for the realisationusing distributed arithmetic. Introduction The discrete cosine transform (DCT) has been widely used as a tool for digital signal processing applications, such as image coding. Many algorithms for the computation of the 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 fast Discrete Fourier Transform and Walsh Hadamard transform[2-51 and 2) direct computation of DCT through matrix decomposition or recursive computation[6-151. Among them, Lee[6]'s Algorithm and Vetterli[ 151's algorithm meet the minimum known number of multiplications to implement a length 2* DCT. Lee[6]'s algorithm decomposes an N-point DCT into the sum of two N/2 point DCT's repeatedly to achieve the number of real multiplications as (N/2)logfl for an N-point DCT, with N =2'". Vetterli[lS]'s algorithm is special as it is a recursive algorithm which uses the property that DCT, DFT, sin-DFT and cos-DFT can be decomposed into two half-length of the individual transforms. Hence, it is difficult to classify it critically. Narasimha[S]'s algorithm is a typical example of group one. It uses an N-point discrete Fourier transform (DFT) algorithm to evaluate a DCT by a simple rearrangement of the input data, which requires Nlogfl-N + 2 real multiplications. In this pger , we present a new formulation for converting a length-N( = 2 ) Discrete Cosine Transform into two length-N/2 correlations. This formulation enables us to realise the Discrete Cosine Transform with a reduced number of operations compared to conventional approaches and it also results in extremely regular structure which is most suitable for the realisation using distributed arithmetic. The Algorithm Derivation The DCT[l] of a real data sequence {x(i):i=O,l,,..N-l}, where N = 2'" and m is an integer, is defined by N1 X(k) = 2 x(i) cos [2n(2i + l)k/4N] i=O for k = 0,1, ... N-1 (1) Let y(i) = x(2i) y(N-i-1) = x(2i + 1) for i = O,l, ... N/2-1 (2) N1 then, X(k) = y(i) cos [ h ( 4 + l)k/4N] for k = 0,l ... N-1 (3) Now let us split X(k) into odd and even sequences, say X(2k+ 1) and X(2k), and look for a formulation of the form cos[(4i + 1)(4k + 1)2~/4N] for cosine terms. The reason for such an arrangement will be clear at a latter stage. i=O For odd terms of X@): N1 X(2k + 1) = y(i) cos [h(4i + 1)(2k + 1)/4N] for k = O,l, ... N/2-1 i=O N1 We define X(k) = 2 y(i) cos [%(4i + 1)(4k + 1)/4N] for k = 0,1, ... N-1 for k = O,l, ... N/4-1 for k = N/4,N/4 + 1, ... N/2-1 i=O then it can be shown that X(4k + 1) = X'(k) X(2N-4k-1) = X(k) For even terms ofX@): X(2k) = N1 y(i) cos [h(4i + 1)(2k)/4N] for k = O,l, ... N/2-1 i=O If we define X*(k) = N1 z(i) cos[h(4i + 1)(2k+ 1)/4N] for k = 0,1, ... N-1 i=O where z(i) =2y(i)cos[(h/4N)(4i + l)] then w,e have X*(N/2-1) =X(N-2) and X (k) = X(2k) + X(2k + 2) for i = 0,1, ... N-1 (10) for k=0,1, ... N/2-2 (11) In order to have the required form, let us define again F(k) = . z(i) cos[h(4i + 1)(4k + 1)/4N] for k=O,1, ... N-1 (12) We can obtainX*(k) through the realisation of eqn. 12 since it is read;y shown that X (2k) = F(k) for k = OJ, ..., N/4-1 (13) X*(N-2k-1) = -F(k) for k = N/4,N/4 + 1, ... N/2-l(l4) N1