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4.8.3 1d Real-even DFTs (DCTs)

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The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized cannam@127: forward (and backward) DFTs as defined above, where the input array cannam@127: X of length N is purely real and is also even symmetry. In cannam@127: this case, the output array is likewise real and even symmetry. cannam@127: cannam@127: cannam@127:

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For the case of REDFT00, this even symmetry means that cannam@127: Xj = XN-j,where we take X to be periodic so that cannam@127: XN = X0.Because of this redundancy, only the first n real numbers are cannam@127: actually stored, where N = 2(n-1). cannam@127:

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The proper definition of even symmetry for REDFT10, cannam@127: REDFT01, and REDFT11 transforms is somewhat more intricate cannam@127: because of the shifts by 1/2 of the input and/or output, although cannam@127: the corresponding boundary conditions are given in Real even/odd DFTs (cosine/sine transforms). Because of the even symmetry, however, cannam@127: the sine terms in the DFT all cancel and the remaining cosine terms are cannam@127: written explicitly below. This formulation often leads people to call cannam@127: such a transform a discrete cosine transform (DCT), although it is cannam@127: really just a special case of the DFT. cannam@127: cannam@127: cannam@127:

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In each of the definitions below, we transform a real array X of cannam@127: length n to a real array Y of length n: cannam@127:

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REDFT00 (DCT-I)

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An REDFT00 transform (type-I DCT) in FFTW is defined by: cannam@127:

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Note that this transform is not defined for n=1. For n=2, cannam@127: the summation term above is dropped as you might expect. cannam@127:

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REDFT10 (DCT-II)

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An REDFT10 transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by: cannam@127:

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REDFT01 (DCT-III)

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An REDFT01 transform (type-III DCT) in FFTW is defined by: cannam@127:

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In the case of n=1, this reduces to cannam@127: 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”. cannam@127: cannam@127:

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REDFT11 (DCT-IV)

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An REDFT11 transform (type-IV DCT) in FFTW is defined by: cannam@127:

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Inverses and Normalization

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These definitions correspond directly to the unnormalized DFTs used cannam@127: elsewhere in FFTW (hence the factors of 2 in front of the cannam@127: summations). The unnormalized inverse of REDFT00 is cannam@127: REDFT00, of REDFT10 is REDFT01 and vice versa, and cannam@127: of REDFT11 is REDFT11. Each unnormalized inverse results cannam@127: in the original array multiplied by N, where N is the cannam@127: logical DFT size. For REDFT00, N=2(n-1) (note that cannam@127: n=1 is not defined); otherwise, N=2n. cannam@127: cannam@127:

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In defining the discrete cosine transform, some authors also include cannam@127: additional factors of cannam@127: √2(or its inverse) multiplying selected inputs and/or outputs. This is a cannam@127: mostly cosmetic change that makes the transform orthogonal, but cannam@127: sacrifices the direct equivalence to a symmetric DFT. cannam@127:

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