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2.5.1 The Halfcomplex-format DFT

An r2r kind of FFTW_R2HC (r2hc) corresponds to an r2c DFT (see One-Dimensional DFTs of Real Data) 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:

r₀, r₁, r₂, ..., r_n/2, i_(n+1)/2-1, ..., i₂, i₁

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 problematic. 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). Conceptually, 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 Multi-Dimensional DFTs of Real Data).