#### 4.8.2 The 1d Real-data DFT

The real-input (r2c) DFT in FFTW computes the *forward* transform
*Y* of the size `n`

real array *X*, exactly as defined
above, i.e.

.
This output array *Y* can easily be shown to possess the
“Hermitian” symmetry
*Y*_{k} = Y_{n-k}^{*},
where we take *Y* to be periodic so that
*Y*_{n} = Y_{0}.
As a result of this symmetry, half of the output *Y* is redundant
(being the complex conjugate of the other half), and so the 1d r2c
transforms only output elements *0*…*n/2* of *Y*
(*n/2+1* complex numbers), where the division by *2* is
rounded down.

Moreover, the Hermitian symmetry implies that
*Y*_{0}
and, if *n* is even, the
*Y*_{n/2}
element, are purely real. So, for the `R2HC`

r2r transform, the
halfcomplex format does not store the imaginary parts of these elements.

The c2r and `H2RC`

r2r transforms compute the backward DFT of the
*complex* array *X* with Hermitian symmetry, stored in the
r2c/`R2HC`

output formats, respectively, where the backward
transform is defined exactly as for the complex case:

.
The outputs `Y`

of this transform can easily be seen to be purely
real, and are stored as an array of real numbers.
Like FFTW’s complex DFT, these transforms are unnormalized. In other
words, applying the real-to-complex (forward) and then the
complex-to-real (backward) transform will multiply the input by
*n*.