Discrete Fourier transform using summation by parts

Abstract

An algorithm for evaluating the Discrete Fourier Transform (DFT) at particular output frequency is derived using a technique called summation by parts (SBP). This technique is shown to reduce the number of multiplications and the number of bits per multiplicative coefficient needed to implement the DFT. For many transform lengths, only two one-bit multiplications or simple memory shifts are needed to implement the DFT. When the DFT length is prime, a SBP algorithm designed for a fixed output frequency index can be used to evaluate the DFT at any other non-zero output frequency index simply by appropriately changing the order of the input sequence.