FFTW 3.3.5: Complex One-Dimensional DFTs

The basic usage of FFTW to compute a one-dimensional DFT of size Chris@42: N is simple, and it typically looks something like this code: Chris@42:

Chris@42:

#include <fftw3.h>
Chris@42: ...
Chris@42: {
Chris@42:     fftw_complex *in, *out;
Chris@42:     fftw_plan p;
Chris@42:     ...
Chris@42:     in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
Chris@42:     out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
Chris@42:     p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
Chris@42:     ...
Chris@42:     fftw_execute(p); /* repeat as needed */
Chris@42:     ...
Chris@42:     fftw_destroy_plan(p);
Chris@42:     fftw_free(in); fftw_free(out);
Chris@42: }
Chris@42:

You must link this code with the fftw3 library. On Unix systems, Chris@42: link with -lfftw3 -lm. Chris@42:

The example code first allocates the input and output arrays. You can Chris@42: allocate them in any way that you like, but we recommend using Chris@42: fftw_malloc, which behaves like Chris@42: Chris@42: malloc except that it properly aligns the array when SIMD Chris@42: instructions (such as SSE and Altivec) are available (see SIMD alignment and fftw_malloc). [Alternatively, we provide a convenient wrapper function fftw_alloc_complex(N) which has the same effect.] Chris@42: Chris@42: Chris@42:

The data is an array of type fftw_complex, which is by default a Chris@42: double[2] composed of the real (in[i][0]) and imaginary Chris@42: (in[i][1]) parts of a complex number. Chris@42: Chris@42:

The next step is to create a plan, which is an object Chris@42: Chris@42: that contains all the data that FFTW needs to compute the FFT. Chris@42: This function creates the plan: Chris@42:

Chris@42:

fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out,
Chris@42:                            int sign, unsigned flags);
Chris@42:

The first argument, n, is the size of the transform you are Chris@42: trying to compute. The size n can be any positive integer, but Chris@42: sizes that are products of small factors are transformed most Chris@42: efficiently (although prime sizes still use an O(n log n) algorithm). Chris@42:

The next two arguments are pointers to the input and output arrays of Chris@42: the transform. These pointers can be equal, indicating an Chris@42: in-place transform. Chris@42: Chris@42:

The fourth argument, sign, can be either FFTW_FORWARD Chris@42: (-1) or FFTW_BACKWARD (+1), Chris@42: Chris@42: Chris@42: and indicates the direction of the transform you are interested in; Chris@42: technically, it is the sign of the exponent in the transform. Chris@42:

The flags argument is usually either FFTW_MEASURE or Chris@42: Chris@42: FFTW_ESTIMATE. FFTW_MEASURE instructs FFTW to run Chris@42: Chris@42: and measure the execution time of several FFTs in order to find the Chris@42: best way to compute the transform of size n. This process takes Chris@42: some time (usually a few seconds), depending on your machine and on Chris@42: the size of the transform. FFTW_ESTIMATE, on the contrary, Chris@42: does not run any computation and just builds a Chris@42: Chris@42: reasonable plan that is probably sub-optimal. In short, if your Chris@42: program performs many transforms of the same size and initialization Chris@42: time is not important, use FFTW_MEASURE; otherwise use the Chris@42: estimate. Chris@42:

You must create the plan before initializing the input, because Chris@42: FFTW_MEASURE overwrites the in/out arrays. Chris@42: (Technically, FFTW_ESTIMATE does not touch your arrays, but you Chris@42: should always create plans first just to be sure.) Chris@42:

Once the plan has been created, you can use it as many times as you Chris@42: like for transforms on the specified in/out arrays, Chris@42: computing the actual transforms via fftw_execute(plan): Chris@42:

Chris@42:

void fftw_execute(const fftw_plan plan);
Chris@42:

The DFT results are stored in-order in the array out, with the Chris@42: zero-frequency (DC) component in out[0]. Chris@42: Chris@42: If in != out, the transform is out-of-place and the input Chris@42: array in is not modified. Otherwise, the input array is Chris@42: overwritten with the transform. Chris@42:

If you want to transform a different array of the same size, you Chris@42: can create a new plan with fftw_plan_dft_1d and FFTW Chris@42: automatically reuses the information from the previous plan, if Chris@42: possible. Alternatively, with the “guru” interface you can apply a Chris@42: given plan to a different array, if you are careful. Chris@42: See FFTW Reference. Chris@42:

When you are done with the plan, you deallocate it by calling Chris@42: fftw_destroy_plan(plan): Chris@42:

Chris@42:

void fftw_destroy_plan(fftw_plan plan);
Chris@42:

If you allocate an array with fftw_malloc() you must deallocate Chris@42: it with fftw_free(). Do not use free() or, heaven Chris@42: forbid, delete. Chris@42: Chris@42:

FFTW computes an unnormalized DFT. Thus, computing a forward Chris@42: followed by a backward transform (or vice versa) results in the original Chris@42: array scaled by n. For the definition of the DFT, see What FFTW Really Computes. Chris@42: Chris@42: Chris@42:

If you have a C compiler, such as gcc, that supports the Chris@42: C99 standard, and you #include <complex.h> before Chris@42: <fftw3.h>, then fftw_complex is the native Chris@42: double-precision complex type and you can manipulate it with ordinary Chris@42: arithmetic. Otherwise, FFTW defines its own complex type, which is Chris@42: bit-compatible with the C99 complex type. See Complex numbers. Chris@42: (The C++ <complex> template class may also be usable via a Chris@42: typecast.) Chris@42: Chris@42:

To use single or long-double precision versions of FFTW, replace the Chris@42: fftw_ prefix by fftwf_ or fftwl_ and link with Chris@42: -lfftw3f or -lfftw3l, but use the same Chris@42: <fftw3.h> header file. Chris@42: Chris@42:

Many more flags exist besides FFTW_MEASURE and Chris@42: FFTW_ESTIMATE. For example, use FFTW_PATIENT if you’re Chris@42: willing to wait even longer for a possibly even faster plan (see FFTW Reference). Chris@42: Chris@42: You can also save plans for future use, as described by Words of Wisdom-Saving Plans. Chris@42:

2.1 Complex One-Dimensional DFTs