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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 Chris@42: forward (and backward) DFTs as defined above, where the input array Chris@42: X of length N is purely real and is also even symmetry. In Chris@42: this case, the output array is likewise real and even symmetry. Chris@42: Chris@42: Chris@42:

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For the case of REDFT00, this even symmetry means that Chris@42: Xj = XN-j,where we take X to be periodic so that Chris@42: XN = X0.Because of this redundancy, only the first n real numbers are Chris@42: actually stored, where N = 2(n-1). Chris@42:

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The proper definition of even symmetry for REDFT10, Chris@42: REDFT01, and REDFT11 transforms is somewhat more intricate Chris@42: because of the shifts by 1/2 of the input and/or output, although Chris@42: the corresponding boundary conditions are given in Real even/odd DFTs (cosine/sine transforms). Because of the even symmetry, however, Chris@42: the sine terms in the DFT all cancel and the remaining cosine terms are Chris@42: written explicitly below. This formulation often leads people to call Chris@42: such a transform a discrete cosine transform (DCT), although it is Chris@42: really just a special case of the DFT. Chris@42: Chris@42: Chris@42:

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In each of the definitions below, we transform a real array X of Chris@42: length n to a real array Y of length n: Chris@42:

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REDFT00 (DCT-I)

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An REDFT00 transform (type-I DCT) in FFTW is defined by: Chris@42:

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Note that this transform is not defined for n=1. For n=2, Chris@42: the summation term above is dropped as you might expect. Chris@42:

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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: Chris@42:

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REDFT01 (DCT-III)

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An REDFT01 transform (type-III DCT) in FFTW is defined by: Chris@42:

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In the case of n=1, this reduces to Chris@42: 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”. Chris@42: Chris@42:

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REDFT11 (DCT-IV)

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An REDFT11 transform (type-IV DCT) in FFTW is defined by: Chris@42:

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Inverses and Normalization

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These definitions correspond directly to the unnormalized DFTs used Chris@42: elsewhere in FFTW (hence the factors of 2 in front of the Chris@42: summations). The unnormalized inverse of REDFT00 is Chris@42: REDFT00, of REDFT10 is REDFT01 and vice versa, and Chris@42: of REDFT11 is REDFT11. Each unnormalized inverse results Chris@42: in the original array multiplied by N, where N is the Chris@42: logical DFT size. For REDFT00, N=2(n-1) (note that Chris@42: n=1 is not defined); otherwise, N=2n. Chris@42: Chris@42:

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In defining the discrete cosine transform, some authors also include Chris@42: additional factors of Chris@42: √2(or its inverse) multiplying selected inputs and/or outputs. This is a Chris@42: mostly cosmetic change that makes the transform orthogonal, but Chris@42: sacrifices the direct equivalence to a symmetric DFT. Chris@42:

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