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The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
d@0: forward (and backward) DFTs as defined above, where the input array
d@0: X of length N is purely real and is also odd symmetry. In
d@0: this case, the output is odd symmetry and purely imaginary.
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d@0: For the case of RODFT00, this odd symmetry means that
d@0: Xj = -XN-j,where we take X to be periodic so that
d@0: XN = X0. Because of this redundancy, only the first n real numbers
d@0: starting at j=1 are actually stored (the j=0 element is
d@0: zero), where N = 2(n+1).
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The proper definition of odd symmetry for RODFT10,
d@0: RODFT01, and RODFT11 transforms is somewhat more intricate
d@0: because of the shifts by 1/2 of the input and/or output, although
d@0: the corresponding boundary conditions are given in Real even/odd DFTs (cosine/sine transforms). Because of the odd symmetry, however,
d@0: the cosine terms in the DFT all cancel and the remaining sine terms are
d@0: written explicitly below. This formulation often leads people to call
d@0: such a transform a discrete sine transform (DST), although it is
d@0: really just a special case of the DFT.
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d@0: In each of the definitions below, we transform a real array X of
d@0: length n to a real array Y of length n:
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An RODFT00 transform (type-I DST) in FFTW is defined by:
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.An RODFT10 transform (type-II DST) in FFTW is defined by:
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.An RODFT01 transform (type-III DST) in FFTW is defined by:
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.An RODFT11 transform (type-IV DST) in FFTW is defined by:
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.These definitions correspond directly to the unnormalized DFTs used
d@0: elsewhere in FFTW (hence the factors of 2 in front of the
d@0: summations). The unnormalized inverse of RODFT00 is
d@0: RODFT00, of RODFT10 is RODFT01 and vice versa, and
d@0: of RODFT11 is RODFT11. Each unnormalized inverse results
d@0: in the original array multiplied by N, where N is the
d@0: logical DFT size. For RODFT00, N=2(n+1);
d@0: otherwise, N=2n.
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d@0: In defining the discrete sine transform, some authors also include
d@0: additional factors of
d@0: √2(or its inverse) multiplying selected inputs and/or outputs. This is a
d@0: mostly cosmetic change that makes the transform orthogonal, but
d@0: sacrifices the direct equivalence to an antisymmetric DFT.
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