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4.8.3 1d Real-even DFTs (DCTs)

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 X_j = X_N-j, where we take X to be periodic so that X_N = 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 REDFT10, REDFT01, and 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 (DCT-I)

An REDFT00 transform (type-I DCT) in FFTW is defined by:

. Note that this transform is not defined for n=1. For n=2, the summation term above is dropped as you might expect.

REDFT10 (DCT-II)

An REDFT10 transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by:

.

REDFT01 (DCT-III)

An REDFT01 transform (type-III DCT) in FFTW is defined by:

. In the case of n=1, this reduces to Y₀ = X₀. Up to a scale factor (see below), this is the inverse of REDFT10 (“the” DCT), and so the REDFT01 (DCT-III) is sometimes called the “IDCT”.

REDFT11 (DCT-IV)

An REDFT11 transform (type-IV DCT) in FFTW is defined by:

.

Inverses and Normalization

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 REDFT00 is REDFT00, of REDFT10 is REDFT01 and vice versa, and of REDFT11 is 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.