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 GNU Scientific Library Reference Manual - Third Edition (v1.12) by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. RossiPaperback (6"x9"), 592 pages, 60 figuresISBN 0954612078RRP £24.95 (\$39.95)

## 15.2 Overview of complex data FFTs

The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, the following definition of a packed array of length 6,

```double x[3*2];
gsl_complex_packed_array data = x;
```

can be used to hold an array of three complex numbers, `z[3]`, in the following way,

```data[0] = Re(z[0])
data[1] = Im(z[0])
data[2] = Re(z[1])
data[3] = Im(z[1])
data[4] = Re(z[2])
data[5] = Im(z[2])
```

The array indices for the data have the same ordering as those in the definition of the DFT--i.e. there are no index transformations or permutations of the data.

A stride parameter allows the user to perform transforms on the elements `z[stride*i]` instead of `z[i]`. A stride greater than 1 can be used to take an in-place FFT of the column of a matrix. A stride of 1 accesses the array without any additional spacing between elements.

To perform an FFT on a vector argument, such as ```gsl_vector_complex * v```, use the following definitions (or their equivalents) when calling the functions described in this chapter:

```gsl_complex_packed_array data = v->data;
size_t stride = v->stride;
size_t n = v->size;
```

For physical applications it is important to remember that the index appearing in the DFT does not correspond directly to a physical frequency. If the time-step of the DFT is \Delta then the frequency-domain includes both positive and negative frequencies, ranging from -1/(2\Delta) through 0 to +1/(2\Delta). The positive frequencies are stored from the beginning of the array up to the middle, and the negative frequencies are stored backwards from the end of the array.

Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, and the frequency-domain data x.

```index    z               x = FFT(z)

0        z(t = 0)        x(f = 0)
1        z(t = 1)        x(f = 1/(N Delta))
2        z(t = 2)        x(f = 2/(N Delta))
.        ........        ..................
N/2      z(t = N/2)      x(f = +1/(2 Delta),
-1/(2 Delta))
.        ........        ..................
N-3      z(t = N-3)      x(f = -3/(N Delta))
N-2      z(t = N-2)      x(f = -2/(N Delta))
N-1      z(t = N-1)      x(f = -1/(N Delta))
```

When N is even the location N/2 contains the most positive and negative frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent. If N is odd then general structure of the table above still applies, but N/2 does not appear.

 ISBN 0954612078 GNU Scientific Library Reference Manual - Third Edition (v1.12) See the print edition