Novel Formulation and Realisation of Discrete Cosine Transform using Distributed Arithmetic
چکیده
In this paper, a new algorithm is introduced such that we can convert an odd prime length, say N, DCT into two (N-1)/2 length cyclic correlations. This formulation enables us to realise the DCT using the distributed arithmetic and it also results in extremely regular structure which is most suitable for VLSI realisation. An example is given to show the feasibility and the structural reBllarity of the algorithm. 1. INTRODUCTIQN The discrete cosine transform (DCT) is an important tool for digital signal processing as it performs much like the optimal Karhunen-Loeve transform ( U T ) under a variety of criteria. Many algorithms for the computation of the DCT have been proposed since the introduction of the DCT in 1974 by Ahmed, Natarajan and Rao[l]. Among these algorithms, three algorithms[2-41 meet the minimum known number of multiplications for their realisation. However, all of these algorithms still rely heavily on the use of multipliers which are expensive to be implemented using VLSI or other techniques. The distributed arithmetic[S-9] is essentially a technique which is extremely regular and structural in nature such that it is most suitable for VLSI realisation. The advantages of this type of structure include: (1) no actual multiplication involved as multipliers are replaced by memory look-up tables, (2) high accuracy as it suffers fewer rounding/truncation errors than the cases for other structures, (3) possible for modular circuit design as the structure is extremely regular and (4) simple structure which leads to a saving of gate count and makes routing easy. These features allow a high speed circuit design composed of memories, adders and registers only. Besides, due to the nature of the structure of the distributed arithmetic, it is most suitable for the realisation of convolutions/correlations. It can reduce the computation of an N-length cyclic convolution from requiring N2 multiplications and N(N-1) additions to N(M-1) additions only, where M is the wordlength of the input sequence {x(n):n= O,l..N-l}. Hence, it is also true that the success of using distributed arithmetic for the realisation of a certain transform relies heavily on whether we could be able to convert the transform into a convolutiodcorrelation form. Successful results[7-91 for the realisation ofdiscrete Fourier transforms have been reported. However, there are less significant results reported in the literature for the realisation of DCT using the distributed arithmetic. Recently, we proposed a new approach to realise a 2m-length DCT by correlations[lO]. In such case, one can make use of the distributed arithmetic technique to achieve an easy and fast chip implementation. It is found that most of fast discrete cosine transform algorithms were proposed for the computation of 2m length DCT. In this paper, a new algorithm is proposed such that we can convert an odd prime length, say N, DCTinto two (N-1)/2 length cyclic correlations. This formulation enables us to realise the DCT using the distributed arithmetic with the minimum number of operations and it also results in an extremely regular structure which is most suitable for VLSI realisation. An example will also be given to show the feasibility and the structural regularity of the present algorithm. 2. ALGORITHM DERIVATION: i = O,l, ... N-1} is defined as T h e DCT[ l ] of a real data sequence {y(i): N-1 Y(k) = y(i) cos [2n(2i + l)k/4N] i = O for k=O,1, ... N-1 (1) We firstly define another sequence {x(i):i = 0,l ... N-1) as x(N-1) = y(N-1) x(i) = y(i) x(i + 1) Y(k) = { 2T(k) + x(0) } cos[kn/2N] where T(k) is defined as T(k) = 2 x(i)cos[niklN] for i = 0,l ... N-2 (2) Then we have for k = 0,1, ... N-1 (3)