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2.3 One-Dimensional DFTs of Real Data

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

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

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

```fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
unsigned flags);
fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
unsigned flags);
```

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

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

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

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

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

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