The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized forward (and backward) DFTs as defined above, where the input array X of length N is purely real and is also even symmetry. In this case, the output array is likewise real and even symmetry.
For the case of
REDFT00, this even symmetry means that
Xj = XN-j,
where we take X to be periodic so that
XN = X0.
Because of this redundancy, only the first n real numbers are
actually stored, where N = 2(n-1).
The proper definition of even symmetry for
REDFT11 transforms is somewhat more intricate
because of the shifts by 1/2 of the input and/or output, although
the corresponding boundary conditions are given in Real even/odd DFTs (cosine/sine transforms). Because of the even symmetry, however,
the sine terms in the DFT all cancel and the remaining cosine terms are
written explicitly below. This formulation often leads people to call
such a transform a discrete cosine transform (DCT), although it is
really just a special case of the DFT.
In each of the definitions below, we transform a real array X of length n to a real array Y of length n:
REDFT00 transform (type-I DCT) in FFTW is defined by:
REDFT10 transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by:
REDFT01 transform (type-III DCT) in FFTW is defined by:
REDFT10(“the” DCT), and so the
REDFT01(DCT-III) is sometimes called the “IDCT”.
REDFT11 transform (type-IV DCT) in FFTW is defined by:
These definitions correspond directly to the unnormalized DFTs used
elsewhere in FFTW (hence the factors of 2 in front of the
summations). The unnormalized inverse of
REDFT01 and vice versa, and
REDFT11. Each unnormalized inverse results
in the original array multiplied by N, where N is the
logical DFT size. For
REDFT00, N=2(n-1) (note that
n=1 is not defined); otherwise, N=2n.
In defining the discrete cosine transform, some authors also include additional factors of √2 (or its inverse) multiplying selected inputs and/or outputs. This is a mostly cosmetic change that makes the transform orthogonal, but sacrifices the direct equivalence to a symmetric DFT.