cannam@95: cannam@95: cannam@95: 1d Real-odd DFTs (DSTs) - FFTW 3.3.3 cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95: cannam@95:
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4.8.4 1d Real-odd DFTs (DSTs)

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The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized cannam@95: forward (and backward) DFTs as defined above, where the input array cannam@95: X of length N is purely real and is also odd symmetry. In cannam@95: this case, the output is odd symmetry and purely imaginary. cannam@95: cannam@95: cannam@95:

For the case of RODFT00, this odd symmetry means that cannam@95: Xj = -XN-j,where we take X to be periodic so that cannam@95: XN = X0. Because of this redundancy, only the first n real numbers cannam@95: starting at j=1 are actually stored (the j=0 element is cannam@95: zero), where N = 2(n+1). cannam@95: cannam@95:

The proper definition of odd symmetry for RODFT10, cannam@95: RODFT01, and RODFT11 transforms is somewhat more intricate cannam@95: because of the shifts by 1/2 of the input and/or output, although cannam@95: the corresponding boundary conditions are given in Real even/odd DFTs (cosine/sine transforms). Because of the odd symmetry, however, cannam@95: the cosine terms in the DFT all cancel and the remaining sine terms are cannam@95: written explicitly below. This formulation often leads people to call cannam@95: such a transform a discrete sine transform (DST), although it is cannam@95: really just a special case of the DFT. cannam@95: cannam@95: cannam@95:

In each of the definitions below, we transform a real array X of cannam@95: length n to a real array Y of length n: cannam@95: cannam@95:

RODFT00 (DST-I)
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An RODFT00 transform (type-I DST) in FFTW is defined by: cannam@95:

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RODFT10 (DST-II)
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An RODFT10 transform (type-II DST) in FFTW is defined by: cannam@95:

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RODFT01 (DST-III)
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An RODFT01 transform (type-III DST) in FFTW is defined by: cannam@95:

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In the case of n=1, this reduces to cannam@95: Y0 = X0. cannam@95: cannam@95:
RODFT11 (DST-IV)
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An RODFT11 transform (type-IV DST) in FFTW is defined by: cannam@95:

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Inverses and Normalization
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These definitions correspond directly to the unnormalized DFTs used cannam@95: elsewhere in FFTW (hence the factors of 2 in front of the cannam@95: summations). The unnormalized inverse of RODFT00 is cannam@95: RODFT00, of RODFT10 is RODFT01 and vice versa, and cannam@95: of RODFT11 is RODFT11. Each unnormalized inverse results cannam@95: in the original array multiplied by N, where N is the cannam@95: logical DFT size. For RODFT00, N=2(n+1); cannam@95: otherwise, N=2n. cannam@95: cannam@95: cannam@95:

In defining the discrete sine transform, some authors also include cannam@95: additional factors of cannam@95: √2(or its inverse) multiplying selected inputs and/or outputs. This is a cannam@95: mostly cosmetic change that makes the transform orthogonal, but cannam@95: sacrifices the direct equivalence to an antisymmetric DFT. cannam@95: cannam@95: cannam@95: cannam@95: