On computation of certain discrete Fourier transforms using binary calculus

Abstract

Completely multiplyless algorithms are proposed to approximately compute Discrete Fourier Transform (DFT). The computational complexity of the proposed algorithms for a transform of size N, is O( N 2 ) addition and shift operations. In these algorithms, the DFT can be computed sequentially, with a single adder in O( N 2 ) addition times. Parallel versions of the algorithms can be executed in O( N) addition times, with O( N) number of adders. The accuracy of the trigonometric approximations used, is bounded by O( N −1). For more elaborate algorithms, the accuracy of the trigonometric approximations is bounded by O( N −2). The degree of approximation in the transformation can be characterized by the ratio of the average magnitude squared error in the transformed sequence, and the average magnitude square of the transformed sequence. This ratio is very low for higher values of the transform length, N. For smaller values of the transform length, this value is generally low. A stipulation is made, that the size of the transform be a Ramanujan number. These Ramanujan numbers are related to Π and integers which are powers of 2. It is assumed in this paper that it is more expensive to perform a multiplication than a shift operation.