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## 2.1 Complex One-Dimensional DFTs

Plan: To bother about the best method of accomplishing an accidental result. [Ambrose Bierce, The Enlarged Devil’s Dictionary.]

The basic usage of FFTW to compute a one-dimensional DFT of size
`N`

is simple, and it typically looks something like this code:

```
#include <fftw3.h>
...
{
fftw_complex *in, *out;
fftw_plan p;
...
in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
...
fftw_execute(p); /* repeat as needed */
...
fftw_destroy_plan(p);
fftw_free(in); fftw_free(out);
}
```

You must link this code with the `fftw3`

library. On Unix systems,
link with `-lfftw3 -lm`

.

The example code first allocates the input and output arrays. You can
allocate them in any way that you like, but we recommend using
`fftw_malloc`

, which behaves like
`malloc`

except that it properly aligns the array when SIMD
instructions (such as SSE and Altivec) are available (see section SIMD alignment and fftw_malloc). [Alternatively, we provide a convenient wrapper function `fftw_alloc_complex(N)`

which has the same effect.]

The data is an array of type `fftw_complex`

, which is by default a
`double[2]`

composed of the real (`in[i][0]`

) and imaginary
(`in[i][1]`

) parts of a complex number.

The next step is to create a *plan*, which is an object
that contains all the data that FFTW needs to compute the FFT.
This function creates the plan:

fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out, int sign, unsigned flags);

The first argument, `n`

, is the size of the transform you are
trying to compute. The size `n`

can be any positive integer, but
sizes that are products of small factors are transformed most
efficiently (although prime sizes still use an *O*(*n* log *n*) algorithm).

The next two arguments are pointers to the input and output arrays of
the transform. These pointers can be equal, indicating an
*in-place* transform.

The fourth argument, `sign`

, can be either `FFTW_FORWARD`

(`-1`

) or `FFTW_BACKWARD`

(`+1`

),
and indicates the direction of the transform you are interested in;
technically, it is the sign of the exponent in the transform.

The `flags`

argument is usually either `FFTW_MEASURE`

or
`FFTW_ESTIMATE`

. `FFTW_MEASURE`

instructs FFTW to run
and measure the execution time of several FFTs in order to find the
best way to compute the transform of size `n`

. This process takes
some time (usually a few seconds), depending on your machine and on
the size of the transform. `FFTW_ESTIMATE`

, on the contrary,
does not run any computation and just builds a
reasonable plan that is probably sub-optimal. In short, if your
program performs many transforms of the same size and initialization
time is not important, use `FFTW_MEASURE`

; otherwise use the
estimate.

*You must create the plan before initializing the input*, because
`FFTW_MEASURE`

overwrites the `in`

/`out`

arrays.
(Technically, `FFTW_ESTIMATE`

does not touch your arrays, but you
should always create plans first just to be sure.)

Once the plan has been created, you can use it as many times as you
like for transforms on the specified `in`

/`out`

arrays,
computing the actual transforms via `fftw_execute(plan)`

:

void fftw_execute(const fftw_plan plan);

The DFT results are stored in-order in the array `out`

, with the
zero-frequency (DC) component in `out[0]`

.
If `in != out`

, the transform is *out-of-place* and the input
array `in`

is not modified. Otherwise, the input array is
overwritten with the transform.

If you want to transform a *different* array of the same size, you
can create a new plan with `fftw_plan_dft_1d`

and FFTW
automatically reuses the information from the previous plan, if
possible. Alternatively, with the “guru” interface you can apply a
given plan to a different array, if you are careful.
See section FFTW Reference.

When you are done with the plan, you deallocate it by calling
`fftw_destroy_plan(plan)`

:

void fftw_destroy_plan(fftw_plan plan);

If you allocate an array with `fftw_malloc()`

you must deallocate
it with `fftw_free()`

. Do not use `free()`

or, heaven
forbid, `delete`

.

FFTW computes an *unnormalized* DFT. Thus, computing a forward
followed by a backward transform (or vice versa) results in the original
array scaled by `n`

. For the definition of the DFT, see What FFTW Really Computes.

If you have a C compiler, such as `gcc`

, that supports the
C99 standard, and you `#include <complex.h>`

*before*
`<fftw3.h>`

, then `fftw_complex`

is the native
double-precision complex type and you can manipulate it with ordinary
arithmetic. Otherwise, FFTW defines its own complex type, which is
bit-compatible with the C99 complex type. See section Complex numbers.
(The C++ `<complex>`

template class may also be usable via a
typecast.)

To use single or long-double precision versions of FFTW, replace the
`fftw_`

prefix by `fftwf_`

or `fftwl_`

and link with
`-lfftw3f`

or `-lfftw3l`

, but use the *same*
`<fftw3.h>`

header file.

Many more flags exist besides `FFTW_MEASURE`

and
`FFTW_ESTIMATE`

. For example, use `FFTW_PATIENT`

if you’re
willing to wait even longer for a possibly even faster plan (see section FFTW Reference).
You can also save plans for future use, as described by Words of Wisdom—Saving Plans.

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