Real even/odd DFTs (cosine/sine transforms)

The Fourier transform of a real-even function f(-x) = f(x) is Chris@10: real-even, and i times the Fourier transform of a real-odd Chris@10: function f(-x) = -f(x) is real-odd. Similar results hold for a Chris@10: discrete Fourier transform, and thus for these symmetries the need for Chris@10: complex inputs/outputs is entirely eliminated. Moreover, one gains a Chris@10: factor of two in speed/space from the fact that the data are real, and Chris@10: an additional factor of two from the even/odd symmetry: only the Chris@10: non-redundant (first) half of the array need be stored. The result is Chris@10: the real-even DFT (REDFT) and the real-odd DFT (RODFT), also Chris@10: known as the discrete cosine and sine transforms (DCT and Chris@10: DST), respectively. Chris@10: Chris@10: Chris@10:

(In this section, we describe the 1d transforms; multi-dimensional Chris@10: transforms are just a separable product of these transforms operating Chris@10: along each dimension.) Chris@10: Chris@10:

Because of the discrete sampling, one has an additional choice: is the Chris@10: data even/odd around a sampling point, or around the point halfway Chris@10: between two samples? The latter corresponds to shifting the Chris@10: samples by half an interval, and gives rise to several transform Chris@10: variants denoted by REDFTab and RODFTab: a and Chris@10: b are 0 or 1, and indicate whether the input Chris@10: (a) and/or output (b) are shifted by half a sample Chris@10: (1 means it is shifted). These are also known as types I-IV of Chris@10: the DCT and DST, and all four types are supported by FFTW's r2r Chris@10: interface.¹ Chris@10: Chris@10:

The r2r kinds for the various REDFT and RODFT types supported by FFTW, Chris@10: along with the boundary conditions at both ends of the input Chris@10: array (n real numbers in[j=0..n-1]), are: Chris@10: Chris@10:

Note that these symmetries apply to the “logical” array being Chris@10: transformed; there are no constraints on your physical input Chris@10: data. So, for example, if you specify a size-5 REDFT00 (DCT-I) of the Chris@10: data abcde, it corresponds to the DFT of the logical even array Chris@10: abcdedcb of size 8. A size-4 REDFT10 (DCT-II) of the data Chris@10: abcd corresponds to the size-8 logical DFT of the even array Chris@10: abcddcba, shifted by half a sample. Chris@10: Chris@10:

All of these transforms are invertible. The inverse of R*DFT00 is Chris@10: R*DFT00; of R*DFT10 is R*DFT01 and vice versa (these are often called Chris@10: simply “the” DCT and IDCT, respectively); and of R*DFT11 is R*DFT11. Chris@10: However, the transforms computed by FFTW are unnormalized, exactly Chris@10: like the corresponding real and complex DFTs, so computing a transform Chris@10: followed by its inverse yields the original array scaled by N, Chris@10: where N is the logical DFT size. For REDFT00, Chris@10: N=2(n-1); for RODFT00, N=2(n+1); otherwise, N=2n. Chris@10: Chris@10: Chris@10:

Note that the boundary conditions of the transform output array are Chris@10: given by the input boundary conditions of the inverse transform. Chris@10: Thus, the above transforms are all inequivalent in terms of Chris@10: input/output boundary conditions, even neglecting the 0.5 shift Chris@10: difference. Chris@10: Chris@10:

FFTW is most efficient when N is a product of small factors; note Chris@10: that this differs from the factorization of the physical size Chris@10: n for REDFT00 and RODFT00! There is another oddity: n=1 Chris@10: REDFT00 transforms correspond to N=0, and so are not Chris@10: defined (the planner will return NULL). Otherwise, any positive Chris@10: n is supported. Chris@10: Chris@10:

For the precise mathematical definitions of these transforms as used by Chris@10: FFTW, see What FFTW Really Computes. (For people accustomed to Chris@10: the DCT/DST, FFTW's definitions have a coefficient of 2 in front Chris@10: of the cos/sin functions so that they correspond precisely to an Chris@10: even/odd DFT of size N. Some authors also include additional Chris@10: multiplicative factors of Chris@10: √2for selected inputs and outputs; this makes Chris@10: the transform orthogonal, but sacrifices the direct equivalence to a Chris@10: symmetric DFT.) Chris@10: Chris@10:

Which type do you need?

Since the required flavor of even/odd DFT depends upon your problem, Chris@10: you are the best judge of this choice, but we can make a few comments Chris@10: on relative efficiency to help you in your selection. In particular, Chris@10: R*DFT01 and R*DFT10 tend to be slightly faster than R*DFT11 Chris@10: (especially for odd sizes), while the R*DFT00 transforms are sometimes Chris@10: significantly slower (especially for even sizes).² Chris@10: Chris@10:

Thus, if only the boundary conditions on the transform inputs are Chris@10: specified, we generally recommend R*DFT10 over R*DFT00 and R*DFT01 over Chris@10: R*DFT11 (unless the half-sample shift or the self-inverse property is Chris@10: significant for your problem). Chris@10: Chris@10:

If performance is important to you and you are using only small sizes Chris@10: (say n<200), e.g. for multi-dimensional transforms, then you Chris@10: might consider generating hard-coded transforms of those sizes and types Chris@10: that you are interested in (see Generating your own code). Chris@10: Chris@10:

We are interested in hearing what types of symmetric transforms you find Chris@10: most useful. Chris@10: Chris@10: Chris@10:

Chris@10:

Footnotes

[1] There are also type V-VIII transforms, which Chris@10: correspond to a logical DFT of odd size N, independent of Chris@10: whether the physical size n is odd, but we do not support these Chris@10: variants.

Chris@10: Chris@10:

[2] R*DFT00 is Chris@10: sometimes slower in FFTW because we discovered that the standard Chris@10: algorithm for computing this by a pre/post-processed real DFT—the Chris@10: algorithm used in FFTPACK, Numerical Recipes, and other sources for Chris@10: decades now—has serious numerical problems: it already loses several Chris@10: decimal places of accuracy for 16k sizes. There seem to be only two Chris@10: alternatives in the literature that do not suffer similarly: a Chris@10: recursive decomposition into smaller DCTs, which would require a large Chris@10: set of codelets for efficiency and generality, or sacrificing a factor of Chris@10: 2 Chris@10: in speed to use a real DFT of twice the size. We currently Chris@10: employ the latter technique for general n, as well as a limited Chris@10: form of the former method: a split-radix decomposition when n Chris@10: is odd (N a multiple of 4). For N containing many Chris@10: factors of 2, the split-radix method seems to recover most of the Chris@10: speed of the standard algorithm without the accuracy tradeoff.

Chris@10: Chris@10:

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

Which type do you need?

Footnotes