DFT computation with prime Ramanujan numbers

Ramanujan numbers were introduced by Bhatnagar (see Signal Processing, vol.43, p.93-101, 1995) to implement the discrete Fourier transform (DFT) without using any multiplication operation. Ramanujan numbers are related to /spl pi/ and integers which are powers of 2. If the transform size N, is a Ramanujan number, then the computational complexity of the algorithms used for computing DFT is O(N/sup 2/) addition and shift operations, and no multiplications. In these algorithms, the transform can be computed sequentially with a single adder in O(N/sup 2/) addition times. Parallel implementation of the algorithm can be executed in O(N) addition times, with O(N) number of adders. We analytically obtain the upper bound on the degree of approximation in the computation of the DFT if N is a prime Ramanujan number. In this case, the degree of approximation is shown to be equal to O(N/sup -1/).