Pre-FFT computation of the DFT

Introduction

In this chapter we will study the problem of computing the components

of the discrete Fourier transform of given complex numbers f[0], f[1], …, f[N-1]. We write these relations in the compact form

F = Ff,

where

are complex N-component column vectors and where the N × N DFT matrix

is expressed in terms of powers of

ω ≔ e-2π/N = cos(2π/N) - i sin(2π/N).

We will use indices 0, 1, …, N - 1 (rather than 1, 2, …, N) for the rows of vectors and for the rows and columns of matrices. When it is necessary, we will use a subscript to specify the size of a matrix, e.g., I8, F16 will denote the 8 × 8 identity matrix and the 16 × 16 DFT matrix, respectively.

Given an N × N matrix

and an N-vector

we can evaluate the components of

by using the algorithm

The cost of this computation is approximately N2 operations when we define an operation to be the work we do as we execute the statement

S ≔ S + akn · bn

from the inner loop. [More specifically, we fetch akn, bn, and the “old” value of S from storage; we form the product akn · bn and the sum S + (akn · bn); and we store this result as the “new” value of S.] Of course, complex arithmetic requires more effort than real arithmetic, and by using the real-imaginary decomposition

we verify that

1 complex operation = 4 real operations.