Chris@19: @node Tutorial, Other Important Topics, Introduction, Top Chris@19: @chapter Tutorial Chris@19: @menu Chris@19: * Complex One-Dimensional DFTs:: Chris@19: * Complex Multi-Dimensional DFTs:: Chris@19: * One-Dimensional DFTs of Real Data:: Chris@19: * Multi-Dimensional DFTs of Real Data:: Chris@19: * More DFTs of Real Data:: Chris@19: @end menu Chris@19: Chris@19: This chapter describes the basic usage of FFTW, i.e., how to compute Chris@19: @cindex basic interface Chris@19: the Fourier transform of a single array. This chapter tells the Chris@19: truth, but not the @emph{whole} truth. Specifically, FFTW implements Chris@19: additional routines and flags that are not documented here, although Chris@19: in many cases we try to indicate where added capabilities exist. For Chris@19: more complete information, see @ref{FFTW Reference}. (Note that you Chris@19: need to compile and install FFTW before you can use it in a program. Chris@19: For the details of the installation, see @ref{Installation and Chris@19: Customization}.) Chris@19: Chris@19: We recommend that you read this tutorial in order.@footnote{You can Chris@19: read the tutorial in bit-reversed order after computing your first Chris@19: transform.} At the least, read the first section (@pxref{Complex Chris@19: One-Dimensional DFTs}) before reading any of the others, even if your Chris@19: main interest lies in one of the other transform types. Chris@19: Chris@19: Users of FFTW version 2 and earlier may also want to read @ref{Upgrading Chris@19: from FFTW version 2}. Chris@19: Chris@19: @c ------------------------------------------------------------ Chris@19: @node Complex One-Dimensional DFTs, Complex Multi-Dimensional DFTs, Tutorial, Tutorial Chris@19: @section Complex One-Dimensional DFTs Chris@19: Chris@19: @quotation Chris@19: Plan: To bother about the best method of accomplishing an accidental result. Chris@19: [Ambrose Bierce, @cite{The Enlarged Devil's Dictionary}.] Chris@19: @cindex Devil Chris@19: @end quotation Chris@19: Chris@19: @iftex Chris@19: @medskip Chris@19: @end iftex Chris@19: Chris@19: The basic usage of FFTW to compute a one-dimensional DFT of size Chris@19: @code{N} is simple, and it typically looks something like this code: Chris@19: Chris@19: @example Chris@19: #include Chris@19: ... Chris@19: @{ Chris@19: fftw_complex *in, *out; Chris@19: fftw_plan p; Chris@19: ... Chris@19: in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N); Chris@19: out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N); Chris@19: p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE); Chris@19: ... Chris@19: fftw_execute(p); /* @r{repeat as needed} */ Chris@19: ... Chris@19: fftw_destroy_plan(p); Chris@19: fftw_free(in); fftw_free(out); Chris@19: @} Chris@19: @end example Chris@19: Chris@19: You must link this code with the @code{fftw3} library. On Unix systems, Chris@19: link with @code{-lfftw3 -lm}. Chris@19: Chris@19: The example code first allocates the input and output arrays. You can Chris@19: allocate them in any way that you like, but we recommend using Chris@19: @code{fftw_malloc}, which behaves like Chris@19: @findex fftw_malloc Chris@19: @code{malloc} except that it properly aligns the array when SIMD Chris@19: instructions (such as SSE and Altivec) are available (@pxref{SIMD Chris@19: alignment and fftw_malloc}). [Alternatively, we provide a convenient wrapper function @code{fftw_alloc_complex(N)} which has the same effect.] Chris@19: @findex fftw_alloc_complex Chris@19: @cindex SIMD Chris@19: Chris@19: Chris@19: The data is an array of type @code{fftw_complex}, which is by default a Chris@19: @code{double[2]} composed of the real (@code{in[i][0]}) and imaginary Chris@19: (@code{in[i][1]}) parts of a complex number. Chris@19: @tindex fftw_complex Chris@19: Chris@19: The next step is to create a @dfn{plan}, which is an object Chris@19: @cindex plan Chris@19: that contains all the data that FFTW needs to compute the FFT. Chris@19: This function creates the plan: Chris@19: Chris@19: @example Chris@19: fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out, Chris@19: int sign, unsigned flags); Chris@19: @end example Chris@19: @findex fftw_plan_dft_1d Chris@19: @tindex fftw_plan Chris@19: Chris@19: The first argument, @code{n}, is the size of the transform you are Chris@19: trying to compute. The size @code{n} can be any positive integer, but Chris@19: sizes that are products of small factors are transformed most Chris@19: efficiently (although prime sizes still use an @Onlogn{} algorithm). Chris@19: Chris@19: The next two arguments are pointers to the input and output arrays of Chris@19: the transform. These pointers can be equal, indicating an Chris@19: @dfn{in-place} transform. Chris@19: @cindex in-place Chris@19: Chris@19: Chris@19: The fourth argument, @code{sign}, can be either @code{FFTW_FORWARD} Chris@19: (@code{-1}) or @code{FFTW_BACKWARD} (@code{+1}), Chris@19: @ctindex FFTW_FORWARD Chris@19: @ctindex FFTW_BACKWARD Chris@19: and indicates the direction of the transform you are interested in; Chris@19: technically, it is the sign of the exponent in the transform. Chris@19: Chris@19: The @code{flags} argument is usually either @code{FFTW_MEASURE} or Chris@19: @cindex flags Chris@19: @code{FFTW_ESTIMATE}. @code{FFTW_MEASURE} instructs FFTW to run Chris@19: @ctindex FFTW_MEASURE Chris@19: and measure the execution time of several FFTs in order to find the Chris@19: best way to compute the transform of size @code{n}. This process takes Chris@19: some time (usually a few seconds), depending on your machine and on Chris@19: the size of the transform. @code{FFTW_ESTIMATE}, on the contrary, Chris@19: does not run any computation and just builds a Chris@19: @ctindex FFTW_ESTIMATE Chris@19: reasonable plan that is probably sub-optimal. In short, if your Chris@19: program performs many transforms of the same size and initialization Chris@19: time is not important, use @code{FFTW_MEASURE}; otherwise use the Chris@19: estimate. Chris@19: Chris@19: @emph{You must create the plan before initializing the input}, because Chris@19: @code{FFTW_MEASURE} overwrites the @code{in}/@code{out} arrays. Chris@19: (Technically, @code{FFTW_ESTIMATE} does not touch your arrays, but you Chris@19: should always create plans first just to be sure.) Chris@19: Chris@19: Once the plan has been created, you can use it as many times as you Chris@19: like for transforms on the specified @code{in}/@code{out} arrays, Chris@19: computing the actual transforms via @code{fftw_execute(plan)}: Chris@19: @example Chris@19: void fftw_execute(const fftw_plan plan); Chris@19: @end example Chris@19: @findex fftw_execute Chris@19: Chris@19: The DFT results are stored in-order in the array @code{out}, with the Chris@19: zero-frequency (DC) component in @code{out[0]}. Chris@19: @cindex frequency Chris@19: If @code{in != out}, the transform is @dfn{out-of-place} and the input Chris@19: array @code{in} is not modified. Otherwise, the input array is Chris@19: overwritten with the transform. Chris@19: Chris@19: @cindex execute Chris@19: If you want to transform a @emph{different} array of the same size, you Chris@19: can create a new plan with @code{fftw_plan_dft_1d} and FFTW Chris@19: automatically reuses the information from the previous plan, if Chris@19: possible. Alternatively, with the ``guru'' interface you can apply a Chris@19: given plan to a different array, if you are careful. Chris@19: @xref{FFTW Reference}. Chris@19: Chris@19: When you are done with the plan, you deallocate it by calling Chris@19: @code{fftw_destroy_plan(plan)}: Chris@19: @example Chris@19: void fftw_destroy_plan(fftw_plan plan); Chris@19: @end example Chris@19: @findex fftw_destroy_plan Chris@19: If you allocate an array with @code{fftw_malloc()} you must deallocate Chris@19: it with @code{fftw_free()}. Do not use @code{free()} or, heaven Chris@19: forbid, @code{delete}. Chris@19: @findex fftw_free Chris@19: Chris@19: FFTW computes an @emph{unnormalized} DFT. Thus, computing a forward Chris@19: followed by a backward transform (or vice versa) results in the original Chris@19: array scaled by @code{n}. For the definition of the DFT, see @ref{What Chris@19: FFTW Really Computes}. Chris@19: @cindex DFT Chris@19: @cindex normalization Chris@19: Chris@19: Chris@19: If you have a C compiler, such as @code{gcc}, that supports the Chris@19: C99 standard, and you @code{#include } @emph{before} Chris@19: @code{}, then @code{fftw_complex} is the native Chris@19: double-precision complex type and you can manipulate it with ordinary Chris@19: arithmetic. Otherwise, FFTW defines its own complex type, which is Chris@19: bit-compatible with the C99 complex type. @xref{Complex numbers}. Chris@19: (The C++ @code{} template class may also be usable via a Chris@19: typecast.) Chris@19: @cindex C++ Chris@19: Chris@19: To use single or long-double precision versions of FFTW, replace the Chris@19: @code{fftw_} prefix by @code{fftwf_} or @code{fftwl_} and link with Chris@19: @code{-lfftw3f} or @code{-lfftw3l}, but use the @emph{same} Chris@19: @code{} header file. Chris@19: @cindex precision Chris@19: Chris@19: Chris@19: Many more flags exist besides @code{FFTW_MEASURE} and Chris@19: @code{FFTW_ESTIMATE}. For example, use @code{FFTW_PATIENT} if you're Chris@19: willing to wait even longer for a possibly even faster plan (@pxref{FFTW Chris@19: Reference}). Chris@19: @ctindex FFTW_PATIENT Chris@19: You can also save plans for future use, as described by @ref{Words of Chris@19: Wisdom-Saving Plans}. Chris@19: Chris@19: @c ------------------------------------------------------------ Chris@19: @node Complex Multi-Dimensional DFTs, One-Dimensional DFTs of Real Data, Complex One-Dimensional DFTs, Tutorial Chris@19: @section Complex Multi-Dimensional DFTs Chris@19: Chris@19: Multi-dimensional transforms work much the same way as one-dimensional Chris@19: transforms: you allocate arrays of @code{fftw_complex} (preferably Chris@19: using @code{fftw_malloc}), create an @code{fftw_plan}, execute it as Chris@19: many times as you want with @code{fftw_execute(plan)}, and clean up Chris@19: with @code{fftw_destroy_plan(plan)} (and @code{fftw_free}). Chris@19: Chris@19: FFTW provides two routines for creating plans for 2d and 3d transforms, Chris@19: and one routine for creating plans of arbitrary dimensionality. Chris@19: The 2d and 3d routines have the following signature: Chris@19: @example Chris@19: fftw_plan fftw_plan_dft_2d(int n0, int n1, Chris@19: fftw_complex *in, fftw_complex *out, Chris@19: int sign, unsigned flags); Chris@19: fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2, Chris@19: fftw_complex *in, fftw_complex *out, Chris@19: int sign, unsigned flags); Chris@19: @end example Chris@19: @findex fftw_plan_dft_2d Chris@19: @findex fftw_plan_dft_3d Chris@19: Chris@19: These routines create plans for @code{n0} by @code{n1} two-dimensional Chris@19: (2d) transforms and @code{n0} by @code{n1} by @code{n2} 3d transforms, Chris@19: respectively. All of these transforms operate on contiguous arrays in Chris@19: the C-standard @dfn{row-major} order, so that the last dimension has the Chris@19: fastest-varying index in the array. This layout is described further in Chris@19: @ref{Multi-dimensional Array Format}. Chris@19: Chris@19: FFTW can also compute transforms of higher dimensionality. In order to Chris@19: avoid confusion between the various meanings of the the word Chris@19: ``dimension'', we use the term @emph{rank} Chris@19: @cindex rank Chris@19: to denote the number of independent indices in an array.@footnote{The Chris@19: term ``rank'' is commonly used in the APL, FORTRAN, and Common Lisp Chris@19: traditions, although it is not so common in the C@tie{}world.} For Chris@19: example, we say that a 2d transform has rank@tie{}2, a 3d transform has Chris@19: rank@tie{}3, and so on. You can plan transforms of arbitrary rank by Chris@19: means of the following function: Chris@19: Chris@19: @example Chris@19: fftw_plan fftw_plan_dft(int rank, const int *n, Chris@19: fftw_complex *in, fftw_complex *out, Chris@19: int sign, unsigned flags); Chris@19: @end example Chris@19: @findex fftw_plan_dft Chris@19: Chris@19: Here, @code{n} is a pointer to an array @code{n[rank]} denoting an Chris@19: @code{n[0]} by @code{n[1]} by @dots{} by @code{n[rank-1]} transform. Chris@19: Thus, for example, the call Chris@19: @example Chris@19: fftw_plan_dft_2d(n0, n1, in, out, sign, flags); Chris@19: @end example Chris@19: is equivalent to the following code fragment: Chris@19: @example Chris@19: int n[2]; Chris@19: n[0] = n0; Chris@19: n[1] = n1; Chris@19: fftw_plan_dft(2, n, in, out, sign, flags); Chris@19: @end example Chris@19: @code{fftw_plan_dft} is not restricted to 2d and 3d transforms, Chris@19: however, but it can plan transforms of arbitrary rank. Chris@19: Chris@19: You may have noticed that all the planner routines described so far Chris@19: have overlapping functionality. For example, you can plan a 1d or 2d Chris@19: transform by using @code{fftw_plan_dft} with a @code{rank} of @code{1} Chris@19: or @code{2}, or even by calling @code{fftw_plan_dft_3d} with @code{n0} Chris@19: and/or @code{n1} equal to @code{1} (with no loss in efficiency). This Chris@19: pattern continues, and FFTW's planning routines in general form a Chris@19: ``partial order,'' sequences of Chris@19: @cindex partial order Chris@19: interfaces with strictly increasing generality but correspondingly Chris@19: greater complexity. Chris@19: Chris@19: @code{fftw_plan_dft} is the most general complex-DFT routine that we Chris@19: describe in this tutorial, but there are also the advanced and guru interfaces, Chris@19: @cindex advanced interface Chris@19: @cindex guru interface Chris@19: which allow one to efficiently combine multiple/strided transforms Chris@19: into a single FFTW plan, transform a subset of a larger Chris@19: multi-dimensional array, and/or to handle more general complex-number Chris@19: formats. For more information, see @ref{FFTW Reference}. Chris@19: Chris@19: @c ------------------------------------------------------------ Chris@19: @node One-Dimensional DFTs of Real Data, Multi-Dimensional DFTs of Real Data, Complex Multi-Dimensional DFTs, Tutorial Chris@19: @section One-Dimensional DFTs of Real Data Chris@19: Chris@19: In many practical applications, the input data @code{in[i]} are purely Chris@19: real numbers, in which case the DFT output satisfies the ``Hermitian'' Chris@19: @cindex Hermitian Chris@19: redundancy: @code{out[i]} is the conjugate of @code{out[n-i]}. It is Chris@19: possible to take advantage of these circumstances in order to achieve Chris@19: roughly a factor of two improvement in both speed and memory usage. Chris@19: Chris@19: In exchange for these speed and space advantages, the user sacrifices Chris@19: some of the simplicity of FFTW's complex transforms. First of all, the Chris@19: input and output arrays are of @emph{different sizes and types}: the Chris@19: input is @code{n} real numbers, while the output is @code{n/2+1} Chris@19: complex numbers (the non-redundant outputs); this also requires slight Chris@19: ``padding'' of the input array for Chris@19: @cindex padding Chris@19: in-place transforms. Second, the inverse transform (complex to real) Chris@19: has the side-effect of @emph{overwriting its input array}, by default. Chris@19: Neither of these inconveniences should pose a serious problem for Chris@19: users, but it is important to be aware of them. Chris@19: Chris@19: The routines to perform real-data transforms are almost the same as Chris@19: those for complex transforms: you allocate arrays of @code{double} Chris@19: and/or @code{fftw_complex} (preferably using @code{fftw_malloc} or Chris@19: @code{fftw_alloc_complex}), create an @code{fftw_plan}, execute it as Chris@19: many times as you want with @code{fftw_execute(plan)}, and clean up Chris@19: with @code{fftw_destroy_plan(plan)} (and @code{fftw_free}). The only Chris@19: differences are that the input (or output) is of type @code{double} Chris@19: and there are new routines to create the plan. In one dimension: Chris@19: Chris@19: @example Chris@19: fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out, Chris@19: unsigned flags); Chris@19: fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out, Chris@19: unsigned flags); Chris@19: @end example Chris@19: @findex fftw_plan_dft_r2c_1d Chris@19: @findex fftw_plan_dft_c2r_1d Chris@19: Chris@19: for the real input to complex-Hermitian output (@dfn{r2c}) and Chris@19: complex-Hermitian input to real output (@dfn{c2r}) transforms. Chris@19: @cindex r2c Chris@19: @cindex c2r Chris@19: Unlike the complex DFT planner, there is no @code{sign} argument. Chris@19: Instead, r2c DFTs are always @code{FFTW_FORWARD} and c2r DFTs are Chris@19: always @code{FFTW_BACKWARD}. Chris@19: @ctindex FFTW_FORWARD Chris@19: @ctindex FFTW_BACKWARD Chris@19: (For single/long-double precision Chris@19: @code{fftwf} and @code{fftwl}, @code{double} should be replaced by Chris@19: @code{float} and @code{long double}, respectively.) Chris@19: @cindex precision Chris@19: Chris@19: Chris@19: Here, @code{n} is the ``logical'' size of the DFT, not necessarily the Chris@19: physical size of the array. In particular, the real (@code{double}) Chris@19: array has @code{n} elements, while the complex (@code{fftw_complex}) Chris@19: array has @code{n/2+1} elements (where the division is rounded down). Chris@19: For an in-place transform, Chris@19: @cindex in-place Chris@19: @code{in} and @code{out} are aliased to the same array, which must be Chris@19: big enough to hold both; so, the real array would actually have Chris@19: @code{2*(n/2+1)} elements, where the elements beyond the first Chris@19: @code{n} are unused padding. (Note that this is very different from Chris@19: the concept of ``zero-padding'' a transform to a larger length, which Chris@19: changes the logical size of the DFT by actually adding new input Chris@19: data.) The @math{k}th element of the complex array is exactly the Chris@19: same as the @math{k}th element of the corresponding complex DFT. All Chris@19: positive @code{n} are supported; products of small factors are most Chris@19: efficient, but an @Onlogn algorithm is used even for prime sizes. Chris@19: Chris@19: As noted above, the c2r transform destroys its input array even for Chris@19: out-of-place transforms. This can be prevented, if necessary, by Chris@19: including @code{FFTW_PRESERVE_INPUT} in the @code{flags}, with Chris@19: unfortunately some sacrifice in performance. Chris@19: @cindex flags Chris@19: @ctindex FFTW_PRESERVE_INPUT Chris@19: This flag is also not currently supported for multi-dimensional real Chris@19: DFTs (next section). Chris@19: Chris@19: Readers familiar with DFTs of real data will recall that the 0th (the Chris@19: ``DC'') and @code{n/2}-th (the ``Nyquist'' frequency, when @code{n} is Chris@19: even) elements of the complex output are purely real. Some Chris@19: implementations therefore store the Nyquist element where the DC Chris@19: imaginary part would go, in order to make the input and output arrays Chris@19: the same size. Such packing, however, does not generalize well to Chris@19: multi-dimensional transforms, and the space savings are miniscule in Chris@19: any case; FFTW does not support it. Chris@19: Chris@19: An alternative interface for one-dimensional r2c and c2r DFTs can be Chris@19: found in the @samp{r2r} interface (@pxref{The Halfcomplex-format Chris@19: DFT}), with ``halfcomplex''-format output that @emph{is} the same size Chris@19: (and type) as the input array. Chris@19: @cindex halfcomplex format Chris@19: That interface, although it is not very useful for multi-dimensional Chris@19: transforms, may sometimes yield better performance. Chris@19: Chris@19: @c ------------------------------------------------------------ Chris@19: @node Multi-Dimensional DFTs of Real Data, More DFTs of Real Data, One-Dimensional DFTs of Real Data, Tutorial Chris@19: @section Multi-Dimensional DFTs of Real Data Chris@19: Chris@19: Multi-dimensional DFTs of real data use the following planner routines: Chris@19: Chris@19: @example Chris@19: fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1, Chris@19: double *in, fftw_complex *out, Chris@19: unsigned flags); Chris@19: fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2, Chris@19: double *in, fftw_complex *out, Chris@19: unsigned flags); Chris@19: fftw_plan fftw_plan_dft_r2c(int rank, const int *n, Chris@19: double *in, fftw_complex *out, Chris@19: unsigned flags); Chris@19: @end example Chris@19: @findex fftw_plan_dft_r2c_2d Chris@19: @findex fftw_plan_dft_r2c_3d Chris@19: @findex fftw_plan_dft_r2c Chris@19: Chris@19: as well as the corresponding @code{c2r} routines with the input/output Chris@19: types swapped. These routines work similarly to their complex Chris@19: analogues, except for the fact that here the complex output array is cut Chris@19: roughly in half and the real array requires padding for in-place Chris@19: transforms (as in 1d, above). Chris@19: Chris@19: As before, @code{n} is the logical size of the array, and the Chris@19: consequences of this on the the format of the complex arrays deserve Chris@19: careful attention. Chris@19: @cindex r2c/c2r multi-dimensional array format Chris@19: Suppose that the real data has dimensions @ndims (in row-major order). Chris@19: Then, after an r2c transform, the output is an @ndimshalf array of Chris@19: @code{fftw_complex} values in row-major order, corresponding to slightly Chris@19: over half of the output of the corresponding complex DFT. (The division Chris@19: is rounded down.) The ordering of the data is otherwise exactly the Chris@19: same as in the complex-DFT case. Chris@19: Chris@19: For out-of-place transforms, this is the end of the story: the real Chris@19: data is stored as a row-major array of size @ndims and the complex Chris@19: data is stored as a row-major array of size @ndimshalf{}. Chris@19: Chris@19: For in-place transforms, however, extra padding of the real-data array Chris@19: is necessary because the complex array is larger than the real array, Chris@19: and the two arrays share the same memory locations. Thus, for Chris@19: in-place transforms, the final dimension of the real-data array must Chris@19: be padded with extra values to accommodate the size of the complex Chris@19: data---two values if the last dimension is even and one if it is odd. Chris@19: @cindex padding Chris@19: That is, the last dimension of the real data must physically contain Chris@19: @tex Chris@19: $2 (n_{d-1}/2+1)$ Chris@19: @end tex Chris@19: @ifinfo Chris@19: 2 * (n[d-1]/2+1) Chris@19: @end ifinfo Chris@19: @html Chris@19: 2 * (nd-1/2+1) Chris@19: @end html Chris@19: @code{double} values (exactly enough to hold the complex data). Chris@19: This physical array size does not, however, change the @emph{logical} Chris@19: array size---only Chris@19: @tex Chris@19: $n_{d-1}$ Chris@19: @end tex Chris@19: @ifinfo Chris@19: n[d-1] Chris@19: @end ifinfo Chris@19: @html Chris@19: nd-1 Chris@19: @end html Chris@19: values are actually stored in the last dimension, and Chris@19: @tex Chris@19: $n_{d-1}$ Chris@19: @end tex Chris@19: @ifinfo Chris@19: n[d-1] Chris@19: @end ifinfo Chris@19: @html Chris@19: nd-1 Chris@19: @end html Chris@19: is the last dimension passed to the plan-creation routine. Chris@19: Chris@19: For example, consider the transform of a two-dimensional real array of Chris@19: size @code{n0} by @code{n1}. The output of the r2c transform is a Chris@19: two-dimensional complex array of size @code{n0} by @code{n1/2+1}, where Chris@19: the @code{y} dimension has been cut nearly in half because of Chris@19: redundancies in the output. Because @code{fftw_complex} is twice the Chris@19: size of @code{double}, the output array is slightly bigger than the Chris@19: input array. Thus, if we want to compute the transform in place, we Chris@19: must @emph{pad} the input array so that it is of size @code{n0} by Chris@19: @code{2*(n1/2+1)}. If @code{n1} is even, then there are two padding Chris@19: elements at the end of each row (which need not be initialized, as they Chris@19: are only used for output). Chris@19: Chris@19: @ifhtml Chris@19: The following illustration depicts the input and output arrays just Chris@19: described, for both the out-of-place and in-place transforms (with the Chris@19: arrows indicating consecutive memory locations): Chris@19: @image{rfftwnd-for-html} Chris@19: @end ifhtml Chris@19: @ifnotinfo Chris@19: @ifnothtml Chris@19: @float Figure,fig:rfftwnd Chris@19: @center @image{rfftwnd} Chris@19: @caption{Illustration of the data layout for a 2d @code{nx} by @code{ny} Chris@19: real-to-complex transform.} Chris@19: @end float Chris@19: @ref{fig:rfftwnd} depicts the input and output arrays just Chris@19: described, for both the out-of-place and in-place transforms (with the Chris@19: arrows indicating consecutive memory locations): Chris@19: @end ifnothtml Chris@19: @end ifnotinfo Chris@19: Chris@19: These transforms are unnormalized, so an r2c followed by a c2r Chris@19: transform (or vice versa) will result in the original data scaled by Chris@19: the number of real data elements---that is, the product of the Chris@19: (logical) dimensions of the real data. Chris@19: @cindex normalization Chris@19: Chris@19: Chris@19: (Because the last dimension is treated specially, if it is equal to Chris@19: @code{1} the transform is @emph{not} equivalent to a lower-dimensional Chris@19: r2c/c2r transform. In that case, the last complex dimension also has Chris@19: size @code{1} (@code{=1/2+1}), and no advantage is gained over the Chris@19: complex transforms.) Chris@19: Chris@19: @c ------------------------------------------------------------ Chris@19: @node More DFTs of Real Data, , Multi-Dimensional DFTs of Real Data, Tutorial Chris@19: @section More DFTs of Real Data Chris@19: @menu Chris@19: * The Halfcomplex-format DFT:: Chris@19: * Real even/odd DFTs (cosine/sine transforms):: Chris@19: * The Discrete Hartley Transform:: Chris@19: @end menu Chris@19: Chris@19: FFTW supports several other transform types via a unified @dfn{r2r} Chris@19: (real-to-real) interface, Chris@19: @cindex r2r Chris@19: so called because it takes a real (@code{double}) array and outputs a Chris@19: real array of the same size. These r2r transforms currently fall into Chris@19: three categories: DFTs of real input and complex-Hermitian output in Chris@19: halfcomplex format, DFTs of real input with even/odd symmetry Chris@19: (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete Chris@19: Hartley transforms (DHTs), all described in more detail by the Chris@19: following sections. Chris@19: Chris@19: The r2r transforms follow the by now familiar interface of creating an Chris@19: @code{fftw_plan}, executing it with @code{fftw_execute(plan)}, and Chris@19: destroying it with @code{fftw_destroy_plan(plan)}. Furthermore, all Chris@19: r2r transforms share the same planner interface: Chris@19: Chris@19: @example Chris@19: fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out, Chris@19: fftw_r2r_kind kind, unsigned flags); Chris@19: fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out, Chris@19: fftw_r2r_kind kind0, fftw_r2r_kind kind1, Chris@19: unsigned flags); Chris@19: fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2, Chris@19: double *in, double *out, Chris@19: fftw_r2r_kind kind0, Chris@19: fftw_r2r_kind kind1, Chris@19: fftw_r2r_kind kind2, Chris@19: unsigned flags); Chris@19: fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out, Chris@19: const fftw_r2r_kind *kind, unsigned flags); Chris@19: @end example Chris@19: @findex fftw_plan_r2r_1d Chris@19: @findex fftw_plan_r2r_2d Chris@19: @findex fftw_plan_r2r_3d Chris@19: @findex fftw_plan_r2r Chris@19: Chris@19: Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional Chris@19: transforms for contiguous arrays in row-major order, transforming (real) Chris@19: input to output of the same size, where @code{n} specifies the Chris@19: @emph{physical} dimensions of the arrays. All positive @code{n} are Chris@19: supported (with the exception of @code{n=1} for the @code{FFTW_REDFT00} Chris@19: kind, noted in the real-even subsection below); products of small Chris@19: factors are most efficient (factorizing @code{n-1} and @code{n+1} for Chris@19: @code{FFTW_REDFT00} and @code{FFTW_RODFT00} kinds, described below), but Chris@19: an @Onlogn algorithm is used even for prime sizes. Chris@19: Chris@19: Each dimension has a @dfn{kind} parameter, of type Chris@19: @code{fftw_r2r_kind}, specifying the kind of r2r transform to be used Chris@19: for that dimension. Chris@19: @cindex kind (r2r) Chris@19: @tindex fftw_r2r_kind Chris@19: (In the case of @code{fftw_plan_r2r}, this is an array @code{kind[rank]} Chris@19: where @code{kind[i]} is the transform kind for the dimension Chris@19: @code{n[i]}.) The kind can be one of a set of predefined constants, Chris@19: defined in the following subsections. Chris@19: Chris@19: In other words, FFTW computes the separable product of the specified Chris@19: r2r transforms over each dimension, which can be used e.g. for partial Chris@19: differential equations with mixed boundary conditions. (For some r2r Chris@19: kinds, notably the halfcomplex DFT and the DHT, such a separable Chris@19: product is somewhat problematic in more than one dimension, however, Chris@19: as is described below.) Chris@19: Chris@19: In the current version of FFTW, all r2r transforms except for the Chris@19: halfcomplex type are computed via pre- or post-processing of Chris@19: halfcomplex transforms, and they are therefore not as fast as they Chris@19: could be. Since most other general DCT/DST codes employ a similar Chris@19: algorithm, however, FFTW's implementation should provide at least Chris@19: competitive performance. Chris@19: Chris@19: @c =========> Chris@19: @node The Halfcomplex-format DFT, Real even/odd DFTs (cosine/sine transforms), More DFTs of Real Data, More DFTs of Real Data Chris@19: @subsection The Halfcomplex-format DFT Chris@19: Chris@19: An r2r kind of @code{FFTW_R2HC} (@dfn{r2hc}) corresponds to an r2c DFT Chris@19: @ctindex FFTW_R2HC Chris@19: @cindex r2c Chris@19: @cindex r2hc Chris@19: (@pxref{One-Dimensional DFTs of Real Data}) but with ``halfcomplex'' Chris@19: format output, and may sometimes be faster and/or more convenient than Chris@19: the latter. Chris@19: @cindex halfcomplex format Chris@19: The inverse @dfn{hc2r} transform is of kind @code{FFTW_HC2R}. Chris@19: @ctindex FFTW_HC2R Chris@19: @cindex hc2r Chris@19: This consists of the non-redundant half of the complex output for a 1d Chris@19: real-input DFT of size @code{n}, stored as a sequence of @code{n} real Chris@19: numbers (@code{double}) in the format: Chris@19: Chris@19: @tex Chris@19: $$ Chris@19: r_0, r_1, r_2, \ldots, r_{n/2}, i_{(n+1)/2-1}, \ldots, i_2, i_1 Chris@19: $$ Chris@19: @end tex Chris@19: @ifinfo Chris@19: r0, r1, r2, r(n/2), i((n+1)/2-1), ..., i2, i1 Chris@19: @end ifinfo Chris@19: @html Chris@19:

Chris@19: r0, r1, r2, ..., rn/2, i(n+1)/2-1, ..., i2, i1 Chris@19:

Chris@19: @end html Chris@19: Chris@19: Here, Chris@19: @ifinfo Chris@19: rk Chris@19: @end ifinfo Chris@19: @tex Chris@19: $r_k$ Chris@19: @end tex Chris@19: @html Chris@19: rk Chris@19: @end html Chris@19: is the real part of the @math{k}th output, and Chris@19: @ifinfo Chris@19: ik Chris@19: @end ifinfo Chris@19: @tex Chris@19: $i_k$ Chris@19: @end tex Chris@19: @html Chris@19: ik Chris@19: @end html Chris@19: is the imaginary part. (Division by 2 is rounded down.) For a Chris@19: halfcomplex array @code{hc[n]}, the @math{k}th component thus has its Chris@19: real part in @code{hc[k]} and its imaginary part in @code{hc[n-k]}, with Chris@19: the exception of @code{k} @code{==} @code{0} or @code{n/2} (the latter Chris@19: only if @code{n} is even)---in these two cases, the imaginary part is Chris@19: zero due to symmetries of the real-input DFT, and is not stored. Chris@19: Thus, the r2hc transform of @code{n} real values is a halfcomplex array of Chris@19: length @code{n}, and vice versa for hc2r. Chris@19: @cindex normalization Chris@19: Chris@19: Chris@19: Aside from the differing format, the output of Chris@19: @code{FFTW_R2HC}/@code{FFTW_HC2R} is otherwise exactly the same as for Chris@19: the corresponding 1d r2c/c2r transform Chris@19: (i.e. @code{FFTW_FORWARD}/@code{FFTW_BACKWARD} transforms, respectively). Chris@19: Recall that these transforms are unnormalized, so r2hc followed by hc2r Chris@19: will result in the original data multiplied by @code{n}. Furthermore, Chris@19: like the c2r transform, an out-of-place hc2r transform will Chris@19: @emph{destroy its input} array. Chris@19: Chris@19: Although these halfcomplex transforms can be used with the Chris@19: multi-dimensional r2r interface, the interpretation of such a separable Chris@19: product of transforms along each dimension is problematic. For example, Chris@19: consider a two-dimensional @code{n0} by @code{n1}, r2hc by r2hc Chris@19: transform planned by @code{fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC, Chris@19: FFTW_R2HC, FFTW_MEASURE)}. Conceptually, FFTW first transforms the rows Chris@19: (of size @code{n1}) to produce halfcomplex rows, and then transforms the Chris@19: columns (of size @code{n0}). Half of these column transforms, however, Chris@19: are of imaginary parts, and should therefore be multiplied by @math{i} Chris@19: and combined with the r2hc transforms of the real columns to produce the Chris@19: 2d DFT amplitudes; FFTW's r2r transform does @emph{not} perform this Chris@19: combination for you. Thus, if a multi-dimensional real-input/output DFT Chris@19: is required, we recommend using the ordinary r2c/c2r Chris@19: interface (@pxref{Multi-Dimensional DFTs of Real Data}). Chris@19: Chris@19: @c =========> Chris@19: @node Real even/odd DFTs (cosine/sine transforms), The Discrete Hartley Transform, The Halfcomplex-format DFT, More DFTs of Real Data Chris@19: @subsection Real even/odd DFTs (cosine/sine transforms) Chris@19: Chris@19: The Fourier transform of a real-even function @math{f(-x) = f(x)} is Chris@19: real-even, and @math{i} times the Fourier transform of a real-odd Chris@19: function @math{f(-x) = -f(x)} is real-odd. Similar results hold for a Chris@19: discrete Fourier transform, and thus for these symmetries the need for Chris@19: complex inputs/outputs is entirely eliminated. Moreover, one gains a Chris@19: factor of two in speed/space from the fact that the data are real, and Chris@19: an additional factor of two from the even/odd symmetry: only the Chris@19: non-redundant (first) half of the array need be stored. The result is Chris@19: the real-even DFT (@dfn{REDFT}) and the real-odd DFT (@dfn{RODFT}), also Chris@19: known as the discrete cosine and sine transforms (@dfn{DCT} and Chris@19: @dfn{DST}), respectively. Chris@19: @cindex real-even DFT Chris@19: @cindex REDFT Chris@19: @cindex real-odd DFT Chris@19: @cindex RODFT Chris@19: @cindex discrete cosine transform Chris@19: @cindex DCT Chris@19: @cindex discrete sine transform Chris@19: @cindex DST Chris@19: Chris@19: Chris@19: (In this section, we describe the 1d transforms; multi-dimensional Chris@19: transforms are just a separable product of these transforms operating Chris@19: along each dimension.) Chris@19: Chris@19: Because of the discrete sampling, one has an additional choice: is the Chris@19: data even/odd around a sampling point, or around the point halfway Chris@19: between two samples? The latter corresponds to @emph{shifting} the Chris@19: samples by @emph{half} an interval, and gives rise to several transform Chris@19: variants denoted by REDFT@math{ab} and RODFT@math{ab}: @math{a} and Chris@19: @math{b} are @math{0} or @math{1}, and indicate whether the input Chris@19: (@math{a}) and/or output (@math{b}) are shifted by half a sample Chris@19: (@math{1} means it is shifted). These are also known as types I-IV of Chris@19: the DCT and DST, and all four types are supported by FFTW's r2r Chris@19: interface.@footnote{There are also type V-VIII transforms, which Chris@19: correspond to a logical DFT of @emph{odd} size @math{N}, independent of Chris@19: whether the physical size @code{n} is odd, but we do not support these Chris@19: variants.} Chris@19: Chris@19: The r2r kinds for the various REDFT and RODFT types supported by FFTW, Chris@19: along with the boundary conditions at both ends of the @emph{input} Chris@19: array (@code{n} real numbers @code{in[j=0..n-1]}), are: Chris@19: Chris@19: @itemize @bullet Chris@19: Chris@19: @item Chris@19: @code{FFTW_REDFT00} (DCT-I): even around @math{j=0} and even around @math{j=n-1}. Chris@19: @ctindex FFTW_REDFT00 Chris@19: Chris@19: @item Chris@19: @code{FFTW_REDFT10} (DCT-II, ``the'' DCT): even around @math{j=-0.5} and even around @math{j=n-0.5}. Chris@19: @ctindex FFTW_REDFT10 Chris@19: Chris@19: @item Chris@19: @code{FFTW_REDFT01} (DCT-III, ``the'' IDCT): even around @math{j=0} and odd around @math{j=n}. Chris@19: @ctindex FFTW_REDFT01 Chris@19: @cindex IDCT Chris@19: Chris@19: @item Chris@19: @code{FFTW_REDFT11} (DCT-IV): even around @math{j=-0.5} and odd around @math{j=n-0.5}. Chris@19: @ctindex FFTW_REDFT11 Chris@19: Chris@19: @item Chris@19: @code{FFTW_RODFT00} (DST-I): odd around @math{j=-1} and odd around @math{j=n}. Chris@19: @ctindex FFTW_RODFT00 Chris@19: Chris@19: @item Chris@19: @code{FFTW_RODFT10} (DST-II): odd around @math{j=-0.5} and odd around @math{j=n-0.5}. Chris@19: @ctindex FFTW_RODFT10 Chris@19: Chris@19: @item Chris@19: @code{FFTW_RODFT01} (DST-III): odd around @math{j=-1} and even around @math{j=n-1}. Chris@19: @ctindex FFTW_RODFT01 Chris@19: Chris@19: @item Chris@19: @code{FFTW_RODFT11} (DST-IV): odd around @math{j=-0.5} and even around @math{j=n-0.5}. Chris@19: @ctindex FFTW_RODFT11 Chris@19: Chris@19: @end itemize Chris@19: Chris@19: Note that these symmetries apply to the ``logical'' array being Chris@19: transformed; @strong{there are no constraints on your physical input Chris@19: data}. So, for example, if you specify a size-5 REDFT00 (DCT-I) of the Chris@19: data @math{abcde}, it corresponds to the DFT of the logical even array Chris@19: @math{abcdedcb} of size 8. A size-4 REDFT10 (DCT-II) of the data Chris@19: @math{abcd} corresponds to the size-8 logical DFT of the even array Chris@19: @math{abcddcba}, shifted by half a sample. Chris@19: Chris@19: All of these transforms are invertible. The inverse of R*DFT00 is Chris@19: R*DFT00; of R*DFT10 is R*DFT01 and vice versa (these are often called Chris@19: simply ``the'' DCT and IDCT, respectively); and of R*DFT11 is R*DFT11. Chris@19: However, the transforms computed by FFTW are unnormalized, exactly Chris@19: like the corresponding real and complex DFTs, so computing a transform Chris@19: followed by its inverse yields the original array scaled by @math{N}, Chris@19: where @math{N} is the @emph{logical} DFT size. For REDFT00, Chris@19: @math{N=2(n-1)}; for RODFT00, @math{N=2(n+1)}; otherwise, @math{N=2n}. Chris@19: @cindex normalization Chris@19: @cindex IDCT Chris@19: Chris@19: Chris@19: Note that the boundary conditions of the transform output array are Chris@19: given by the input boundary conditions of the inverse transform. Chris@19: Thus, the above transforms are all inequivalent in terms of Chris@19: input/output boundary conditions, even neglecting the 0.5 shift Chris@19: difference. Chris@19: Chris@19: FFTW is most efficient when @math{N} is a product of small factors; note Chris@19: that this @emph{differs} from the factorization of the physical size Chris@19: @code{n} for REDFT00 and RODFT00! There is another oddity: @code{n=1} Chris@19: REDFT00 transforms correspond to @math{N=0}, and so are @emph{not Chris@19: defined} (the planner will return @code{NULL}). Otherwise, any positive Chris@19: @code{n} is supported. Chris@19: Chris@19: For the precise mathematical definitions of these transforms as used by Chris@19: FFTW, see @ref{What FFTW Really Computes}. (For people accustomed to Chris@19: the DCT/DST, FFTW's definitions have a coefficient of @math{2} in front Chris@19: of the cos/sin functions so that they correspond precisely to an Chris@19: even/odd DFT of size @math{N}. Some authors also include additional Chris@19: multiplicative factors of Chris@19: @ifinfo Chris@19: sqrt(2) Chris@19: @end ifinfo Chris@19: @html Chris@19: √2 Chris@19: @end html Chris@19: @tex Chris@19: $\sqrt{2}$ Chris@19: @end tex Chris@19: for selected inputs and outputs; this makes Chris@19: the transform orthogonal, but sacrifices the direct equivalence to a Chris@19: symmetric DFT.) Chris@19: Chris@19: @subsubheading Which type do you need? Chris@19: Chris@19: Since the required flavor of even/odd DFT depends upon your problem, Chris@19: you are the best judge of this choice, but we can make a few comments Chris@19: on relative efficiency to help you in your selection. In particular, Chris@19: R*DFT01 and R*DFT10 tend to be slightly faster than R*DFT11 Chris@19: (especially for odd sizes), while the R*DFT00 transforms are sometimes Chris@19: significantly slower (especially for even sizes).@footnote{R*DFT00 is Chris@19: sometimes slower in FFTW because we discovered that the standard Chris@19: algorithm for computing this by a pre/post-processed real DFT---the Chris@19: algorithm used in FFTPACK, Numerical Recipes, and other sources for Chris@19: decades now---has serious numerical problems: it already loses several Chris@19: decimal places of accuracy for 16k sizes. There seem to be only two Chris@19: alternatives in the literature that do not suffer similarly: a Chris@19: recursive decomposition into smaller DCTs, which would require a large Chris@19: set of codelets for efficiency and generality, or sacrificing a factor of Chris@19: @tex Chris@19: $\sim 2$ Chris@19: @end tex Chris@19: @ifnottex Chris@19: 2 Chris@19: @end ifnottex Chris@19: in speed to use a real DFT of twice the size. We currently Chris@19: employ the latter technique for general @math{n}, as well as a limited Chris@19: form of the former method: a split-radix decomposition when @math{n} Chris@19: is odd (@math{N} a multiple of 4). For @math{N} containing many Chris@19: factors of 2, the split-radix method seems to recover most of the Chris@19: speed of the standard algorithm without the accuracy tradeoff.} Chris@19: Chris@19: Thus, if only the boundary conditions on the transform inputs are Chris@19: specified, we generally recommend R*DFT10 over R*DFT00 and R*DFT01 over Chris@19: R*DFT11 (unless the half-sample shift or the self-inverse property is Chris@19: significant for your problem). Chris@19: Chris@19: If performance is important to you and you are using only small sizes Chris@19: (say @math{n<200}), e.g. for multi-dimensional transforms, then you Chris@19: might consider generating hard-coded transforms of those sizes and types Chris@19: that you are interested in (@pxref{Generating your own code}). Chris@19: Chris@19: We are interested in hearing what types of symmetric transforms you find Chris@19: most useful. Chris@19: Chris@19: @c =========> Chris@19: @node The Discrete Hartley Transform, , Real even/odd DFTs (cosine/sine transforms), More DFTs of Real Data Chris@19: @subsection The Discrete Hartley Transform Chris@19: Chris@19: If you are planning to use the DHT because you've heard that it is Chris@19: ``faster'' than the DFT (FFT), @strong{stop here}. The DHT is not Chris@19: faster than the DFT. That story is an old but enduring misconception Chris@19: that was debunked in 1987. Chris@19: Chris@19: The discrete Hartley transform (DHT) is an invertible linear transform Chris@19: closely related to the DFT. In the DFT, one multiplies each input by Chris@19: @math{cos - i * sin} (a complex exponential), whereas in the DHT each Chris@19: input is multiplied by simply @math{cos + sin}. Thus, the DHT Chris@19: transforms @code{n} real numbers to @code{n} real numbers, and has the Chris@19: convenient property of being its own inverse. In FFTW, a DHT (of any Chris@19: positive @code{n}) can be specified by an r2r kind of @code{FFTW_DHT}. Chris@19: @ctindex FFTW_DHT Chris@19: @cindex discrete Hartley transform Chris@19: @cindex DHT Chris@19: Chris@19: Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of Chris@19: size @code{n} followed by another DHT of the same size will result in Chris@19: the original array multiplied by @code{n}. Chris@19: @cindex normalization Chris@19: Chris@19: The DHT was originally proposed as a more efficient alternative to the Chris@19: DFT for real data, but it was subsequently shown that a specialized DFT Chris@19: (such as FFTW's r2hc or r2c transforms) could be just as fast. In FFTW, Chris@19: the DHT is actually computed by post-processing an r2hc transform, so Chris@19: there is ordinarily no reason to prefer it from a performance Chris@19: perspective.@footnote{We provide the DHT mainly as a byproduct of some Chris@19: internal algorithms. FFTW computes a real input/output DFT of Chris@19: @emph{prime} size by re-expressing it as a DHT plus post/pre-processing Chris@19: and then using Rader's prime-DFT algorithm adapted to the DHT.} Chris@19: However, we have heard rumors that the DHT might be the most appropriate Chris@19: transform in its own right for certain applications, and we would be Chris@19: very interested to hear from anyone who finds it useful. Chris@19: Chris@19: If @code{FFTW_DHT} is specified for multiple dimensions of a Chris@19: multi-dimensional transform, FFTW computes the separable product of 1d Chris@19: DHTs along each dimension. Unfortunately, this is not quite the same Chris@19: thing as a true multi-dimensional DHT; you can compute the latter, if Chris@19: necessary, with at most @code{rank-1} post-processing passes Chris@19: [see e.g. H. Hao and R. N. Bracewell, @i{Proc. IEEE} @b{75}, 264--266 (1987)]. Chris@19: Chris@19: For the precise mathematical definition of the DHT as used by FFTW, see Chris@19: @ref{What FFTW Really Computes}. Chris@19: