FFTW 3.3.8: One-Dimensional DFTs of Real Data

In many practical applications, the input data in[i] are purely Chris@82: real numbers, in which case the DFT output satisfies the “Hermitian” Chris@82: Chris@82: redundancy: out[i] is the conjugate of out[n-i]. It is Chris@82: possible to take advantage of these circumstances in order to achieve Chris@82: roughly a factor of two improvement in both speed and memory usage. Chris@82:

In exchange for these speed and space advantages, the user sacrifices Chris@82: some of the simplicity of FFTW’s complex transforms. First of all, the Chris@82: input and output arrays are of different sizes and types: the Chris@82: input is n real numbers, while the output is n/2+1 Chris@82: complex numbers (the non-redundant outputs); this also requires slight Chris@82: “padding” of the input array for Chris@82: Chris@82: in-place transforms. Second, the inverse transform (complex to real) Chris@82: has the side-effect of overwriting its input array, by default. Chris@82: Neither of these inconveniences should pose a serious problem for Chris@82: users, but it is important to be aware of them. Chris@82:

The routines to perform real-data transforms are almost the same as Chris@82: those for complex transforms: you allocate arrays of double Chris@82: and/or fftw_complex (preferably using fftw_malloc or Chris@82: fftw_alloc_complex), create an fftw_plan, execute it as Chris@82: many times as you want with fftw_execute(plan), and clean up Chris@82: with fftw_destroy_plan(plan) (and fftw_free). The only Chris@82: differences are that the input (or output) is of type double Chris@82: and there are new routines to create the plan. In one dimension: Chris@82:

Chris@82:

fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
Chris@82:                                unsigned flags);
Chris@82: fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
Chris@82:                                unsigned flags);
Chris@82:

for the real input to complex-Hermitian output (r2c) and Chris@82: complex-Hermitian input to real output (c2r) transforms. Chris@82: Chris@82: Chris@82: Unlike the complex DFT planner, there is no sign argument. Chris@82: Instead, r2c DFTs are always FFTW_FORWARD and c2r DFTs are Chris@82: always FFTW_BACKWARD. Chris@82: Chris@82: Chris@82: (For single/long-double precision Chris@82: fftwf and fftwl, double should be replaced by Chris@82: float and long double, respectively.) Chris@82: Chris@82:

Here, n is the “logical” size of the DFT, not necessarily the Chris@82: physical size of the array. In particular, the real (double) Chris@82: array has n elements, while the complex (fftw_complex) Chris@82: array has n/2+1 elements (where the division is rounded down). Chris@82: For an in-place transform, Chris@82: Chris@82: in and out are aliased to the same array, which must be Chris@82: big enough to hold both; so, the real array would actually have Chris@82: 2*(n/2+1) elements, where the elements beyond the first Chris@82: n are unused padding. (Note that this is very different from Chris@82: the concept of “zero-padding” a transform to a larger length, which Chris@82: changes the logical size of the DFT by actually adding new input Chris@82: data.) The kth element of the complex array is exactly the Chris@82: same as the kth element of the corresponding complex DFT. All Chris@82: positive n are supported; products of small factors are most Chris@82: efficient, but an O(n log n) Chris@82: algorithm is used even for prime sizes. Chris@82:

As noted above, the c2r transform destroys its input array even for Chris@82: out-of-place transforms. This can be prevented, if necessary, by Chris@82: including FFTW_PRESERVE_INPUT in the flags, with Chris@82: unfortunately some sacrifice in performance. Chris@82: Chris@82: Chris@82: This flag is also not currently supported for multi-dimensional real Chris@82: DFTs (next section). Chris@82:

Readers familiar with DFTs of real data will recall that the 0th (the Chris@82: “DC”) and n/2-th (the “Nyquist” frequency, when n is Chris@82: even) elements of the complex output are purely real. Some Chris@82: implementations therefore store the Nyquist element where the DC Chris@82: imaginary part would go, in order to make the input and output arrays Chris@82: the same size. Such packing, however, does not generalize well to Chris@82: multi-dimensional transforms, and the space savings are miniscule in Chris@82: any case; FFTW does not support it. Chris@82:

An alternative interface for one-dimensional r2c and c2r DFTs can be Chris@82: found in the ‘r2r’ interface (see The Halfcomplex-format DFT), with “halfcomplex”-format output that is the same size Chris@82: (and type) as the input array. Chris@82: Chris@82: That interface, although it is not very useful for multi-dimensional Chris@82: transforms, may sometimes yield better performance. Chris@82:

2.3 One-Dimensional DFTs of Real Data