Fast Algorithms

Abstract

Direct computation of an N-point DFT requires nearly \( O({N^2}) \) complex arithmetic operations. An arithmetic operation implies a multiplication and an addition. However, this complexity can be significantly reduced by developing efficient algorithms. The key to this reduction in computational complexity is that in an \( (N \times N) \) DFT matrix (see (2.10) and (2.12)) of the \( {N^2} \) elements, only \( N \) elements are distinct. These algorithms are denoted as FFT (fast Fourier transform) algorithms []. Several techniques are developed for the FFT. We will initially develop the decimation-in-time (DIT) and decimation-in-frequency (DIF) FFT algorithms. The detailed development will be based on radix-2. This will then be extended to other radices such as radix-3, radix-4, etc. The reader can then foresee that innumerable combinations of fast algorithms exist for the FFT, i.e., mixed-radix, split-radix, DIT, DIF, DIT/DIF, vector radix, vector-split-radix, etc.