Chapter 2: Tutorial 11
such a separable product is somewhat problematic in more than one dimension, however,
as is described below.)
In the current version of FFTW, all r2r transforms except for the halfcomplex type are com-
puted via pre- or post-processing of halfcomplex transforms, and they are therefore not as
fast as they could be. Since most other general DCT/DST codes employ a similar algorithm,
however, FFTW’s implementation should provide at least competitive performance.
2.5.1 The Halfcomplex-format DFT
An r2r kind of FFTW_R2HC (r2hc) corresponds to an r2c DFT (see
Section 2.3
[One-Dimensional DFTs of Real Data], page 6) but with “halfcomplex” format output,
and may sometimes be faster and/or more convenient than the latter. The inverse hc2r
transform is of kind FFTW_HC2R. This consists of the non-redundant half of the complex
output for a 1d real-input DFT of size n, stored as a sequence of n real numbers (double)
in the format:
r
0
, r
1
, r
2
, . . . , r
n/2
, i
(n+1)/2−1
, . . . , i
2
, i
1
Here, r
k
is the real part of the kth output, and i
k
is the imaginary part. (Division by 2
is rounded down.) For a halfcomplex array hc[n], the kth component thus has its real
part in hc[k] and its imaginary part in hc[n-k], with the exception of k == 0 or n/2 (the
latter only if n is even)—in these two cases, the imaginary part is zero due to symmetries
of the real-input DFT, and is not stored. Thus, the r2hc transform of n real values is a
halfcomplex array of length n, and vice versa for hc2r.
Aside from the differing format, the output of FFTW_R2HC/FFTW_HC2R is otherwise exactly
the same as for the corresponding 1d r2c/c2r transform (i.e. FFTW_FORWARD/FFTW_BACKWARD
transforms, respectively). Recall that these transforms are unnormalized, so r2hc followed
by hc2r will result in the original data multiplied by n. Furthermore, like the c2r transform,
an out-of-place hc2r transform will destroy its input array.
Although these halfcomplex transforms can be used with the multi-dimensional r2r interface,
the interpretation of such a separable product of transforms along each dimension is prob-
lematic. For example, consider a two-dimensional n0 by n1, r2hc by r2hc transform planned
by fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC, FFTW_R2HC, FFTW_MEASURE). Con-
ceptually, FFTW first transforms the rows (of size n1) to produce halfcomplex rows, and
then transforms the columns (of size n0). Half of these column transforms, however, are
of imaginary parts, and should therefore be multiplied by i and combined with the r2hc
transforms of the real columns to produce the 2d DFT amplitudes; FFTW’s r2r transform
does not perform this combination for you. Thus, if a multi-dimensional real-input/output
DFT is required, we recommend using the ordinary r2c/c2r interface (see Section 2.4 [Multi-
Dimensional DFTs of Real Data], page 7).
2.5.2 Real even/odd DFTs (cosine/sine transforms)
The Fourier transform of a real-even function f(−x) = f(x) is real-even, and i times
the Fourier transform of a real-odd function f(−x) = −f(x) is real-odd. Similar results
hold for a discrete Fourier transform, and thus for these symmetries the need for complex
inputs/outputs is entirely eliminated. Moreover, one gains a factor of two in speed/space
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