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The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized Chris@10: forward (and backward) DFTs as defined above, where the input array Chris@10: X of length N is purely real and is also odd symmetry. In Chris@10: this case, the output is odd symmetry and purely imaginary. Chris@10: Chris@10: Chris@10:
For the case of RODFT00, this odd symmetry means that
Chris@10: Xj = -XN-j,where we take X to be periodic so that
Chris@10: XN = X0. Because of this redundancy, only the first n real numbers
Chris@10: starting at j=1 are actually stored (the j=0 element is
Chris@10: zero), where N = 2(n+1).
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The proper definition of odd symmetry for RODFT10,
Chris@10: RODFT01, and RODFT11 transforms is somewhat more intricate
Chris@10: because of the shifts by 1/2 of the input and/or output, although
Chris@10: the corresponding boundary conditions are given in Real even/odd DFTs (cosine/sine transforms). Because of the odd symmetry, however,
Chris@10: the cosine terms in the DFT all cancel and the remaining sine terms are
Chris@10: written explicitly below. This formulation often leads people to call
Chris@10: such a transform a discrete sine transform (DST), although it is
Chris@10: really just a special case of the DFT.
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In each of the definitions below, we transform a real array X of Chris@10: length n to a real array Y of length n: Chris@10: Chris@10:
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:
Chris@10:
.These definitions correspond directly to the unnormalized DFTs used
Chris@10: elsewhere in FFTW (hence the factors of 2 in front of the
Chris@10: summations). The unnormalized inverse of RODFT00 is
Chris@10: RODFT00, of RODFT10 is RODFT01 and vice versa, and
Chris@10: of RODFT11 is RODFT11. Each unnormalized inverse results
Chris@10: in the original array multiplied by N, where N is the
Chris@10: logical DFT size. For RODFT00, N=2(n+1);
Chris@10: otherwise, N=2n.
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In defining the discrete sine transform, some authors also include Chris@10: additional factors of Chris@10: √2(or its inverse) multiplying selected inputs and/or outputs. This is a Chris@10: mostly cosmetic change that makes the transform orthogonal, but Chris@10: sacrifices the direct equivalence to an antisymmetric DFT. Chris@10: Chris@10: Chris@10: Chris@10: