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cannam@127:The Real-odd 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 odd symmetry. In cannam@127: this case, the output is odd symmetry and purely imaginary. cannam@127: cannam@127: cannam@127:
cannam@127: cannam@127: cannam@127:For the case of RODFT00, this odd 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
cannam@127: starting at j=1 are actually stored (the j=0 element is
cannam@127: zero), where N = 2(n+1).
cannam@127:
The proper definition of odd symmetry for RODFT10,
cannam@127: RODFT01, and RODFT11 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 odd symmetry, however,
cannam@127: the cosine terms in the DFT all cancel and the remaining sine terms are
cannam@127: written explicitly below. This formulation often leads people to call
cannam@127: such a transform a discrete sine transform (DST), although it is
cannam@127: really just a special case of the DFT.
cannam@127:
cannam@127:
cannam@127:
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:
cannam@127: cannam@127:An RODFT00 transform (type-I DST) in FFTW is defined by:
cannam@127:
.An RODFT10 transform (type-II DST) in FFTW is defined by:
cannam@127:
.An RODFT01 transform (type-III DST) in FFTW is defined by:
cannam@127:
.An RODFT11 transform (type-IV DST) in FFTW is defined by:
cannam@127:
.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 RODFT00 is
cannam@127: RODFT00, of RODFT10 is RODFT01 and vice versa, and
cannam@127: of RODFT11 is RODFT11. Each unnormalized inverse results
cannam@127: in the original array multiplied by N, where N is the
cannam@127: logical DFT size. For RODFT00, N=2(n+1);
cannam@127: otherwise, N=2n.
cannam@127:
cannam@127:
In defining the discrete sine 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 an antisymmetric DFT. cannam@127:
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cannam@127: