FFTW 3.3.5: Real even/odd DFTs (cosine/sine transforms)

The Fourier transform of a real-even function f(-x) = f(x) is Chris@42: real-even, and i times the Fourier transform of a real-odd Chris@42: function f(-x) = -f(x) is real-odd. Similar results hold for a Chris@42: discrete Fourier transform, and thus for these symmetries the need for Chris@42: complex inputs/outputs is entirely eliminated. Moreover, one gains a Chris@42: factor of two in speed/space from the fact that the data are real, and Chris@42: an additional factor of two from the even/odd symmetry: only the Chris@42: non-redundant (first) half of the array need be stored. The result is Chris@42: the real-even DFT (REDFT) and the real-odd DFT (RODFT), also Chris@42: known as the discrete cosine and sine transforms (DCT and Chris@42: DST), respectively. Chris@42: Chris@42: Chris@42: Chris@42: Chris@42: Chris@42: Chris@42: Chris@42: Chris@42:

(In this section, we describe the 1d transforms; multi-dimensional Chris@42: transforms are just a separable product of these transforms operating Chris@42: along each dimension.) Chris@42:

Because of the discrete sampling, one has an additional choice: is the Chris@42: data even/odd around a sampling point, or around the point halfway Chris@42: between two samples? The latter corresponds to shifting the Chris@42: samples by half an interval, and gives rise to several transform Chris@42: variants denoted by REDFTab and RODFTab: a and Chris@42: b are 0 or 1, and indicate whether the input Chris@42: (a) and/or output (b) are shifted by half a sample Chris@42: (1 means it is shifted). These are also known as types I-IV of Chris@42: the DCT and DST, and all four types are supported by FFTW’s r2r Chris@42: interface.³ Chris@42:

The r2r kinds for the various REDFT and RODFT types supported by FFTW, Chris@42: along with the boundary conditions at both ends of the input Chris@42: array (n real numbers in[j=0..n-1]), are: Chris@42:

Note that these symmetries apply to the “logical” array being Chris@42: transformed; there are no constraints on your physical input Chris@42: data. So, for example, if you specify a size-5 REDFT00 (DCT-I) of the Chris@42: data abcde, it corresponds to the DFT of the logical even array Chris@42: abcdedcb of size 8. A size-4 REDFT10 (DCT-II) of the data Chris@42: abcd corresponds to the size-8 logical DFT of the even array Chris@42: abcddcba, shifted by half a sample. Chris@42:

All of these transforms are invertible. The inverse of R*DFT00 is Chris@42: R*DFT00; of R*DFT10 is R*DFT01 and vice versa (these are often called Chris@42: simply “the” DCT and IDCT, respectively); and of R*DFT11 is R*DFT11. Chris@42: However, the transforms computed by FFTW are unnormalized, exactly Chris@42: like the corresponding real and complex DFTs, so computing a transform Chris@42: followed by its inverse yields the original array scaled by N, Chris@42: where N is the logical DFT size. For REDFT00, Chris@42: N=2(n-1); for RODFT00, N=2(n+1); otherwise, N=2n. Chris@42: Chris@42: Chris@42:

Note that the boundary conditions of the transform output array are Chris@42: given by the input boundary conditions of the inverse transform. Chris@42: Thus, the above transforms are all inequivalent in terms of Chris@42: input/output boundary conditions, even neglecting the 0.5 shift Chris@42: difference. Chris@42:

FFTW is most efficient when N is a product of small factors; note Chris@42: that this differs from the factorization of the physical size Chris@42: n for REDFT00 and RODFT00! There is another oddity: n=1 Chris@42: REDFT00 transforms correspond to N=0, and so are not Chris@42: defined (the planner will return NULL). Otherwise, any positive Chris@42: n is supported. Chris@42:

For the precise mathematical definitions of these transforms as used by Chris@42: FFTW, see What FFTW Really Computes. (For people accustomed to Chris@42: the DCT/DST, FFTW’s definitions have a coefficient of 2 in front Chris@42: of the cos/sin functions so that they correspond precisely to an Chris@42: even/odd DFT of size N. Some authors also include additional Chris@42: multiplicative factors of Chris@42: √2for selected inputs and outputs; this makes Chris@42: the transform orthogonal, but sacrifices the direct equivalence to a Chris@42: symmetric DFT.) Chris@42:

Which type do you need?

Since the required flavor of even/odd DFT depends upon your problem, Chris@42: you are the best judge of this choice, but we can make a few comments Chris@42: on relative efficiency to help you in your selection. In particular, Chris@42: R*DFT01 and R*DFT10 tend to be slightly faster than R*DFT11 Chris@42: (especially for odd sizes), while the R*DFT00 transforms are sometimes Chris@42: significantly slower (especially for even sizes).⁴ Chris@42:

Thus, if only the boundary conditions on the transform inputs are Chris@42: specified, we generally recommend R*DFT10 over R*DFT00 and R*DFT01 over Chris@42: R*DFT11 (unless the half-sample shift or the self-inverse property is Chris@42: significant for your problem). Chris@42:

If performance is important to you and you are using only small sizes Chris@42: (say n<200), e.g. for multi-dimensional transforms, then you Chris@42: might consider generating hard-coded transforms of those sizes and types Chris@42: that you are interested in (see Generating your own code). Chris@42:

We are interested in hearing what types of symmetric transforms you find Chris@42: most useful. Chris@42:

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Footnotes

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There are also type V-VIII transforms, which Chris@42: correspond to a logical DFT of odd size N, independent of Chris@42: whether the physical size n is odd, but we do not support these Chris@42: variants.

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R*DFT00 is Chris@42: sometimes slower in FFTW because we discovered that the standard Chris@42: algorithm for computing this by a pre/post-processed real DFT—the Chris@42: algorithm used in FFTPACK, Numerical Recipes, and other sources for Chris@42: decades now—has serious numerical problems: it already loses several Chris@42: decimal places of accuracy for 16k sizes. There seem to be only two Chris@42: alternatives in the literature that do not suffer similarly: a Chris@42: recursive decomposition into smaller DCTs, which would require a large Chris@42: set of codelets for efficiency and generality, or sacrificing a factor of Chris@42: 2 Chris@42: in speed to use a real DFT of twice the size. We currently Chris@42: employ the latter technique for general n, as well as a limited Chris@42: form of the former method: a split-radix decomposition when n Chris@42: is odd (N a multiple of 4). For N containing many Chris@42: factors of 2, the split-radix method seems to recover most of the Chris@42: speed of the standard algorithm without the accuracy tradeoff.

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2.5.2 Real even/odd DFTs (cosine/sine transforms)

Which type do you need?

Footnotes

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