fftw3: 1d Real-odd DFTs (DSTs)
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 forward (and backward) DFTs as defined above, where the
input array X of length N is purely real and is also "odd" symmetry. In
this case, the output is odd symmetry and purely imaginary.
For the case of 'RODFT00', this odd symmetry means that X[j] =
-X[N-j], where we take X to be periodic so that X[N] = X[0]. Because of
this redundancy, only the first n real numbers starting at j=1 are
actually stored (the j=0 element is zero), where N = 2(n+1).
The proper definition of odd symmetry for 'RODFT10', 'RODFT01', and
'RODFT11' transforms is somewhat more intricate because of the shifts by
1/2 of the input and/or output, although the corresponding boundary
conditions are given in 
Real even/odd DFTs (cosine/sine
transforms). Because of the odd symmetry, however, the cosine terms
in the DFT all cancel and the remaining sine terms are written
explicitly below. This formulation often leads people to call such a
transform a "discrete sine transform" (DST), although it is really just
a special case of the DFT.
In each of the definitions below, we transform a real array X of
length n to a real array Y of length n:
RODFT00 (DST-I)
...............
An 'RODFT00' transform (type-I DST) in FFTW is defined by: Y[k] = 2 (sum
for j = 0 to n-1 of X[j] sin(pi (j+1)(k+1) / (n+1))).
RODFT10 (DST-II)
................
An 'RODFT10' transform (type-II DST) in FFTW is defined by: Y[k] = 2
(sum for j = 0 to n-1 of X[j] sin(pi (j+1/2) (k+1) / n)).
RODFT01 (DST-III)
.................
An 'RODFT01' transform (type-III DST) in FFTW is defined by: Y[k] =
(-1)^k X[n-1] + 2 (sum for j = 0 to n-2 of X[j] sin(pi (j+1) (k+1/2) /
n)). In the case of n=1, this reduces to Y[0] = X[0].
RODFT11 (DST-IV)
................
An 'RODFT11' transform (type-IV DST) in FFTW is defined by: Y[k] = 2
(sum for j = 0 to n-1 of X[j] sin(pi (j+1/2) (k+1/2) / n)).
Inverses and Normalization
..........................
These definitions correspond directly to the unnormalized DFTs used
elsewhere in FFTW (hence the factors of 2 in front of the summations).
The unnormalized inverse of 'RODFT00' is 'RODFT00', of 'RODFT10' is
'RODFT01' and vice versa, and of 'RODFT11' is 'RODFT11'. Each
unnormalized inverse results in the original array multiplied by N,
where N is the _logical_ DFT size. For 'RODFT00', N=2(n+1); otherwise,
N=2n.
In defining the discrete sine transform, some authors also include
additional factors of sqrt(2) (or its inverse) multiplying selected
inputs and/or outputs. This is a mostly cosmetic change that makes the
transform orthogonal, but sacrifices the direct equivalence to an
antisymmetric DFT.
Info Catalog
fftw3: 1d Real-even DFTs (DCTs)
fftw3: What FFTW Really Computes
fftw3: 1d Discrete Hartley Transforms (DHTs)