Real even/odd DFTs (cosine/sine transforms)

The Fourier transform of a real-even function f(-x) = f(x) is cannam@95: real-even, and i times the Fourier transform of a real-odd cannam@95: function f(-x) = -f(x) is real-odd. Similar results hold for a cannam@95: discrete Fourier transform, and thus for these symmetries the need for cannam@95: complex inputs/outputs is entirely eliminated. Moreover, one gains a cannam@95: factor of two in speed/space from the fact that the data are real, and cannam@95: an additional factor of two from the even/odd symmetry: only the cannam@95: non-redundant (first) half of the array need be stored. The result is cannam@95: the real-even DFT (REDFT) and the real-odd DFT (RODFT), also cannam@95: known as the discrete cosine and sine transforms (DCT and cannam@95: DST), respectively. cannam@95: cannam@95: cannam@95:

(In this section, we describe the 1d transforms; multi-dimensional cannam@95: transforms are just a separable product of these transforms operating cannam@95: along each dimension.) cannam@95: cannam@95:

Because of the discrete sampling, one has an additional choice: is the cannam@95: data even/odd around a sampling point, or around the point halfway cannam@95: between two samples? The latter corresponds to shifting the cannam@95: samples by half an interval, and gives rise to several transform cannam@95: variants denoted by REDFTab and RODFTab: a and cannam@95: b are 0 or 1, and indicate whether the input cannam@95: (a) and/or output (b) are shifted by half a sample cannam@95: (1 means it is shifted). These are also known as types I-IV of cannam@95: the DCT and DST, and all four types are supported by FFTW's r2r cannam@95: interface.¹ cannam@95: cannam@95:

The r2r kinds for the various REDFT and RODFT types supported by FFTW, cannam@95: along with the boundary conditions at both ends of the input cannam@95: array (n real numbers in[j=0..n-1]), are: cannam@95: cannam@95:

Note that these symmetries apply to the “logical” array being cannam@95: transformed; there are no constraints on your physical input cannam@95: data. So, for example, if you specify a size-5 REDFT00 (DCT-I) of the cannam@95: data abcde, it corresponds to the DFT of the logical even array cannam@95: abcdedcb of size 8. A size-4 REDFT10 (DCT-II) of the data cannam@95: abcd corresponds to the size-8 logical DFT of the even array cannam@95: abcddcba, shifted by half a sample. cannam@95: cannam@95:

All of these transforms are invertible. The inverse of R*DFT00 is cannam@95: R*DFT00; of R*DFT10 is R*DFT01 and vice versa (these are often called cannam@95: simply “the” DCT and IDCT, respectively); and of R*DFT11 is R*DFT11. cannam@95: However, the transforms computed by FFTW are unnormalized, exactly cannam@95: like the corresponding real and complex DFTs, so computing a transform cannam@95: followed by its inverse yields the original array scaled by N, cannam@95: where N is the logical DFT size. For REDFT00, cannam@95: N=2(n-1); for RODFT00, N=2(n+1); otherwise, N=2n. cannam@95: cannam@95: cannam@95:

Note that the boundary conditions of the transform output array are cannam@95: given by the input boundary conditions of the inverse transform. cannam@95: Thus, the above transforms are all inequivalent in terms of cannam@95: input/output boundary conditions, even neglecting the 0.5 shift cannam@95: difference. cannam@95: cannam@95:

FFTW is most efficient when N is a product of small factors; note cannam@95: that this differs from the factorization of the physical size cannam@95: n for REDFT00 and RODFT00! There is another oddity: n=1 cannam@95: REDFT00 transforms correspond to N=0, and so are not cannam@95: defined (the planner will return NULL). Otherwise, any positive cannam@95: n is supported. cannam@95: cannam@95:

For the precise mathematical definitions of these transforms as used by cannam@95: FFTW, see What FFTW Really Computes. (For people accustomed to cannam@95: the DCT/DST, FFTW's definitions have a coefficient of 2 in front cannam@95: of the cos/sin functions so that they correspond precisely to an cannam@95: even/odd DFT of size N. Some authors also include additional cannam@95: multiplicative factors of cannam@95: √2for selected inputs and outputs; this makes cannam@95: the transform orthogonal, but sacrifices the direct equivalence to a cannam@95: symmetric DFT.) cannam@95: cannam@95:

Which type do you need?

Since the required flavor of even/odd DFT depends upon your problem, cannam@95: you are the best judge of this choice, but we can make a few comments cannam@95: on relative efficiency to help you in your selection. In particular, cannam@95: R*DFT01 and R*DFT10 tend to be slightly faster than R*DFT11 cannam@95: (especially for odd sizes), while the R*DFT00 transforms are sometimes cannam@95: significantly slower (especially for even sizes).² cannam@95: cannam@95:

Thus, if only the boundary conditions on the transform inputs are cannam@95: specified, we generally recommend R*DFT10 over R*DFT00 and R*DFT01 over cannam@95: R*DFT11 (unless the half-sample shift or the self-inverse property is cannam@95: significant for your problem). cannam@95: cannam@95:

If performance is important to you and you are using only small sizes cannam@95: (say n<200), e.g. for multi-dimensional transforms, then you cannam@95: might consider generating hard-coded transforms of those sizes and types cannam@95: that you are interested in (see Generating your own code). cannam@95: cannam@95:

We are interested in hearing what types of symmetric transforms you find cannam@95: most useful. cannam@95: cannam@95: cannam@95:

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Footnotes

[1] There are also type V-VIII transforms, which cannam@95: correspond to a logical DFT of odd size N, independent of cannam@95: whether the physical size n is odd, but we do not support these cannam@95: variants.

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[2] R*DFT00 is cannam@95: sometimes slower in FFTW because we discovered that the standard cannam@95: algorithm for computing this by a pre/post-processed real DFT—the cannam@95: algorithm used in FFTPACK, Numerical Recipes, and other sources for cannam@95: decades now—has serious numerical problems: it already loses several cannam@95: decimal places of accuracy for 16k sizes. There seem to be only two cannam@95: alternatives in the literature that do not suffer similarly: a cannam@95: recursive decomposition into smaller DCTs, which would require a large cannam@95: set of codelets for efficiency and generality, or sacrificing a factor of cannam@95: 2 cannam@95: in speed to use a real DFT of twice the size. We currently cannam@95: employ the latter technique for general n, as well as a limited cannam@95: form of the former method: a split-radix decomposition when n cannam@95: is odd (N a multiple of 4). For N containing many cannam@95: factors of 2, the split-radix method seems to recover most of the cannam@95: 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