0.10 Implementing ffts in practice  (Page 13/21)

We call $\mathbf{N}$ the size of the problem. The rank of a problem is defined to be the rank of its size (i.e., the dimensionality of the DFT). Similarly, we call $\mathbf{V}$ the vector size of the problem, and the vector rank of a problem is correspondingly defined to be the rank of its vector size.Intuitively, the vector size can be interpreted as a set of “loops” wrapped around a single DFT, and we therefore refer to a singleI/O dimension of $\mathbf{V}$ as a vector loop . (Alternatively, one can view the problem as describing a DFT over a $\left|\mathbf{V}\right|$ -dimensional vector space.) The problem does not specify the order of execution of these loops, however, and thereforeFFTW is free to choose the fastest or most convenient order.

Dft problem examples

A more detailed discussion of the space of problems in FFTW can be found in [link] , but a simple understanding can be gained by examining a few examples demonstrating that the I/O tensorrepresentation is sufficiently general to cover many situations that arise in practice, including some that are not usually considered tobe instances of the DFT.

A single one-dimensional DFT of length $n$ , with stride-1 input $\mathbf{X}$ and output $\mathbf{Y}$ , as in [link] , is denoted by the problem $dft(\left\{(,n,,,1,,,1,)\right\},\left\{\right\},\mathbf{X},\mathbf{Y})$ (no loops: vector-rank zero).

As a more complicated example, suppose we have an $n_1\times n_2$ matrix $\mathbf{X}$ stored as $n_1$ consecutive blocks of contiguous length- $n_2$ rows (this is called row-major format). The in-place DFT of all the rows of this matrix would be denoted by the problem $dft(\left\{(,n_2,,,1,,,1,)\right\},\left\{(,n_1,,,n_2,,,n_2,)\right\},\mathbf{X},\mathbf{X})$ : a length- $n_1$ loop of size- $n_2$ contiguous DFTs, where each iteration of the loop offsets its input/output data by a stride $n_2$ . Conversely, the in-place DFT of all the columns of this matrix would be denoted by $dft(\left\{(,n_1,,,n_2,,,n_2,)\right\},\left\{(,n_2,,,1,,,1,)\right\},\mathbf{X},\mathbf{X})$ : compared to the previous example, $\mathbf{N}$ and $\mathbf{V}$ are swapped. In the latter case, each DFT operates on discontiguous data, andFFTW might well choose to interchange the loops: instead of performing a loop of DFTs computed individually, the subtransformsthemselves could act on $n_2$ -component vectors, as described in "The space of plans in FFTW" .

A size-1 DFT is simply a copy $Y\left[0\right]=X\left[0\right]$ , and here this can also be denoted by $\mathbf{N}=\left\{\right\}$ (rank zero, a “zero-dimensional” DFT). This allows FFTW's problems to represent many kinds of copies andpermutations of the data within the same problem framework, which is convenient because these sorts of operations arise frequently inFFT algorithms. For example, to copy $n$ consecutive numbers from $\mathbf{I}$ to $\mathbf{O}$ , one would use the rank-zero problem $dft(\left\{\right\},\left\{(,n,,,1,,,1,)\right\},\mathbf{I},\mathbf{O})$ . More interestingly, the in-place transpose of an $n_1\times n_2$ matrix $\mathbf{X}$ stored in row-major format, as described above, is denoted by $dft(\left\{\right\},\left\{(n_1,n_2,1),,,(n_2,1,n_1)\right\},\mathbf{X},\mathbf{X})$ (rank zero, vector-rank two).

The space of plans in fftw

Here, we describe a subset of the possible plans considered by FFTW; while not exhaustive [link] , this subset is enough to illustrate the basic structure of FFTW and the necessity ofincluding the vector loop(s) in the problem definition to enable several interesting algorithms. The plans that we now describeusually perform some simple “atomic” operation, and it may not be apparent how these operations fit together to actually computeDFTs, or why certain operations are useful at all. We shall discuss those matters in "Discussion" .