   Next: Output of Fourier Transforms Up: Storage: What Goes Where Previous: Storage: What Goes Where

### Fourier Transform Definitions and Conventions

The Fourier transform of a function is defined as (1) (2)

A discrete Fourier transform takes a function known only at discrete points and gives back a function known only at the discrete points  (3) (4)

Note that the relationship between the discrete Fourier transform and the continuous one is (5)

where is the spacing between points and the spacing between frequencies is given by (6)

(Note that I am following the conventions of Numerical Recipes using to denote frequency rather than angular frequency. The conversion is simply .) Formally the discrete Fourier transform is periodic with period so can take any set of consecutive values. Practically speaking, though, if the points represent a typical region of the function then the results give the frequency components in the range (where by periodicity). The next section describes how these complex points are arranged in the output of the routines fftc1 and fftcn. The following section describes the additional information needed to interpret the results of the real Fourier transform routines fftr1 and fftrn.   Next: Output of Fourier Transforms Up: Storage: What Goes Where Previous: Storage: What Goes Where