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2456 lines
90 KiB
Text
2456 lines
90 KiB
Text
@node FFTW Reference, Multi-threaded FFTW, Other Important Topics, Top
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@chapter FFTW Reference
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This chapter provides a complete reference for all sequential (i.e.,
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one-processor) FFTW functions. Parallel transforms are described in
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later chapters.
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@menu
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* Data Types and Files::
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* Using Plans::
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* Basic Interface::
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* Advanced Interface::
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* Guru Interface::
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* New-array Execute Functions::
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* Wisdom::
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* What FFTW Really Computes::
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@end menu
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@c ------------------------------------------------------------
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@node Data Types and Files, Using Plans, FFTW Reference, FFTW Reference
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@section Data Types and Files
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All programs using FFTW should include its header file:
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@example
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#include <fftw3.h>
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@end example
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You must also link to the FFTW library. On Unix, this
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means adding @code{-lfftw3 -lm} at the @emph{end} of the link command.
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@menu
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* Complex numbers::
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* Precision::
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* Memory Allocation::
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@end menu
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@c =========>
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@node Complex numbers, Precision, Data Types and Files, Data Types and Files
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@subsection Complex numbers
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The default FFTW interface uses @code{double} precision for all
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floating-point numbers, and defines a @code{fftw_complex} type to hold
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complex numbers as:
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@example
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typedef double fftw_complex[2];
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@end example
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@tindex fftw_complex
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Here, the @code{[0]} element holds the real part and the @code{[1]}
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element holds the imaginary part.
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Alternatively, if you have a C compiler (such as @code{gcc}) that
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supports the C99 revision of the ANSI C standard, you can use C's new
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native complex type (which is binary-compatible with the typedef above).
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In particular, if you @code{#include <complex.h>} @emph{before}
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@code{<fftw3.h>}, then @code{fftw_complex} is defined to be the native
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complex type and you can manipulate it with ordinary arithmetic
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(e.g. @code{x = y * (3+4*I)}, where @code{x} and @code{y} are
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@code{fftw_complex} and @code{I} is the standard symbol for the
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imaginary unit);
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@cindex C99
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C++ has its own @code{complex<T>} template class, defined in the
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standard @code{<complex>} header file. Reportedly, the C++ standards
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committee has recently agreed to mandate that the storage format used
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for this type be binary-compatible with the C99 type, i.e. an array
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@code{T[2]} with consecutive real @code{[0]} and imaginary @code{[1]}
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parts. (See report
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@uref{http://www.open-std.org/jtc1/sc22/WG21/docs/papers/2002/n1388.pdf
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WG21/N1388}.) Although not part of the official standard as of this
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writing, the proposal stated that: ``This solution has been tested with
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all current major implementations of the standard library and shown to
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be working.'' To the extent that this is true, if you have a variable
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@code{complex<double> *x}, you can pass it directly to FFTW via
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@code{reinterpret_cast<fftw_complex*>(x)}.
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@cindex C++
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@cindex portability
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@c =========>
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@node Precision, Memory Allocation, Complex numbers, Data Types and Files
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@subsection Precision
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@cindex precision
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You can install single and long-double precision versions of FFTW,
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which replace @code{double} with @code{float} and @code{long double},
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respectively (@pxref{Installation and Customization}). To use these
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interfaces, you:
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@itemize @bullet
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@item
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Link to the single/long-double libraries; on Unix, @code{-lfftw3f} or
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@code{-lfftw3l} instead of (or in addition to) @code{-lfftw3}. (You
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can link to the different-precision libraries simultaneously.)
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@item
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Include the @emph{same} @code{<fftw3.h>} header file.
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@item
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Replace all lowercase instances of @samp{fftw_} with @samp{fftwf_} or
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@samp{fftwl_} for single or long-double precision, respectively.
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(@code{fftw_complex} becomes @code{fftwf_complex}, @code{fftw_execute}
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becomes @code{fftwf_execute}, etcetera.)
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@item
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Uppercase names, i.e. names beginning with @samp{FFTW_}, remain the
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same.
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@item
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Replace @code{double} with @code{float} or @code{long double} for
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subroutine parameters.
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@end itemize
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Depending upon your compiler and/or hardware, @code{long double} may not
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be any more precise than @code{double} (or may not be supported at all,
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although it is standard in C99).
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@cindex C99
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We also support using the nonstandard @code{__float128}
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quadruple-precision type provided by recent versions of @code{gcc} on
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32- and 64-bit x86 hardware (@pxref{Installation and Customization}).
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To use this type, link with @code{-lfftw3q -lquadmath -lm} (the
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@code{libquadmath} library provided by @code{gcc} is needed for
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quadruple-precision trigonometric functions) and use @samp{fftwq_}
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identifiers.
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@c =========>
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@node Memory Allocation, , Precision, Data Types and Files
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@subsection Memory Allocation
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@example
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void *fftw_malloc(size_t n);
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void fftw_free(void *p);
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@end example
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@findex fftw_malloc
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@findex fftw_free
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These are functions that behave identically to @code{malloc} and
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@code{free}, except that they guarantee that the returned pointer obeys
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any special alignment restrictions imposed by any algorithm in FFTW
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(e.g. for SIMD acceleration). @xref{SIMD alignment and fftw_malloc}.
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@cindex alignment
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Data allocated by @code{fftw_malloc} @emph{must} be deallocated by
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@code{fftw_free} and not by the ordinary @code{free}.
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These routines simply call through to your operating system's
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@code{malloc} or, if necessary, its aligned equivalent
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(e.g. @code{memalign}), so you normally need not worry about any
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significant time or space overhead. You are @emph{not required} to use
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them to allocate your data, but we strongly recommend it.
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Note: in C++, just as with ordinary @code{malloc}, you must typecast
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the output of @code{fftw_malloc} to whatever pointer type you are
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allocating.
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@cindex C++
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We also provide the following two convenience functions to allocate
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real and complex arrays with @code{n} elements, which are equivalent
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to @code{(double *) fftw_malloc(sizeof(double) * n)} and
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@code{(fftw_complex *) fftw_malloc(sizeof(fftw_complex) * n)},
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respectively:
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@example
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double *fftw_alloc_real(size_t n);
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fftw_complex *fftw_alloc_complex(size_t n);
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@end example
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@findex fftw_alloc_real
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@findex fftw_alloc_complex
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The equivalent functions in other precisions allocate arrays of @code{n}
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elements in that precision. e.g. @code{fftwf_alloc_real(n)} is
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equivalent to @code{(float *) fftwf_malloc(sizeof(float) * n)}.
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@cindex precision
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@c ------------------------------------------------------------
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@node Using Plans, Basic Interface, Data Types and Files, FFTW Reference
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@section Using Plans
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Plans for all transform types in FFTW are stored as type
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@code{fftw_plan} (an opaque pointer type), and are created by one of the
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various planning routines described in the following sections.
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@tindex fftw_plan
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An @code{fftw_plan} contains all information necessary to compute the
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transform, including the pointers to the input and output arrays.
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@example
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void fftw_execute(const fftw_plan plan);
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@end example
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@findex fftw_execute
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This executes the @code{plan}, to compute the corresponding transform on
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the arrays for which it was planned (which must still exist). The plan
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is not modified, and @code{fftw_execute} can be called as many times as
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desired.
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To apply a given plan to a different array, you can use the new-array execute
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interface. @xref{New-array Execute Functions}.
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@code{fftw_execute} (and equivalents) is the only function in FFTW
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guaranteed to be thread-safe; see @ref{Thread safety}.
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This function:
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@example
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void fftw_destroy_plan(fftw_plan plan);
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@end example
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@findex fftw_destroy_plan
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deallocates the @code{plan} and all its associated data.
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FFTW's planner saves some other persistent data, such as the
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accumulated wisdom and a list of algorithms available in the current
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configuration. If you want to deallocate all of that and reset FFTW
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to the pristine state it was in when you started your program, you can
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call:
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@example
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void fftw_cleanup(void);
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@end example
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@findex fftw_cleanup
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After calling @code{fftw_cleanup}, all existing plans become undefined,
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and you should not attempt to execute them nor to destroy them. You can
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however create and execute/destroy new plans, in which case FFTW starts
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accumulating wisdom information again.
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@code{fftw_cleanup} does not deallocate your plans, however. To prevent
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memory leaks, you must still call @code{fftw_destroy_plan} before
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executing @code{fftw_cleanup}.
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Occasionally, it may useful to know FFTW's internal ``cost'' metric
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that it uses to compare plans to one another; this cost is
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proportional to an execution time of the plan, in undocumented units,
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if the plan was created with the @code{FFTW_MEASURE} or other
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timing-based options, or alternatively is a heuristic cost function
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for @code{FFTW_ESTIMATE} plans. (The cost values of measured and
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estimated plans are not comparable, being in different units. Also,
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costs from different FFTW versions or the same version compiled
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differently may not be in the same units. Plans created from wisdom
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have a cost of 0 since no timing measurement is performed for them.
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Finally, certain problems for which only one top-level algorithm was
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possible may have required no measurements of the cost of the whole
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plan, in which case @code{fftw_cost} will also return 0.) The cost
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metric for a given plan is returned by:
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@example
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double fftw_cost(const fftw_plan plan);
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@end example
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@findex fftw_cost
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The following two routines are provided purely for academic purposes
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(that is, for entertainment).
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@example
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void fftw_flops(const fftw_plan plan,
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double *add, double *mul, double *fma);
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@end example
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@findex fftw_flops
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Given a @code{plan}, set @code{add}, @code{mul}, and @code{fma} to an
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exact count of the number of floating-point additions, multiplications,
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and fused multiply-add operations involved in the plan's execution. The
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total number of floating-point operations (flops) is @code{add + mul +
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2*fma}, or @code{add + mul + fma} if the hardware supports fused
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multiply-add instructions (although the number of FMA operations is only
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approximate because of compiler voodoo). (The number of operations
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should be an integer, but we use @code{double} to avoid overflowing
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@code{int} for large transforms; the arguments are of type @code{double}
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even for single and long-double precision versions of FFTW.)
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@example
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void fftw_fprint_plan(const fftw_plan plan, FILE *output_file);
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void fftw_print_plan(const fftw_plan plan);
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char *fftw_sprint_plan(const fftw_plan plan);
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@end example
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@findex fftw_fprint_plan
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@findex fftw_print_plan
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This outputs a ``nerd-readable'' representation of the @code{plan} to
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the given file, to @code{stdout}, or two a newly allocated
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NUL-terminated string (which the caller is responsible for deallocating
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with @code{free}), respectively.
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@c ------------------------------------------------------------
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@node Basic Interface, Advanced Interface, Using Plans, FFTW Reference
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@section Basic Interface
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@cindex basic interface
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Recall that the FFTW API is divided into three parts@footnote{@i{Gallia est
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omnis divisa in partes tres} (Julius Caesar).}: the @dfn{basic interface}
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computes a single transform of contiguous data, the @dfn{advanced
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interface} computes transforms of multiple or strided arrays, and the
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@dfn{guru interface} supports the most general data layouts,
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multiplicities, and strides. This section describes the basic
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interface, which we expect to satisfy the needs of most users.
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@menu
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* Complex DFTs::
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* Planner Flags::
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* Real-data DFTs::
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* Real-data DFT Array Format::
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* Real-to-Real Transforms::
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* Real-to-Real Transform Kinds::
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@end menu
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@c =========>
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@node Complex DFTs, Planner Flags, Basic Interface, Basic Interface
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@subsection Complex DFTs
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@example
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fftw_plan fftw_plan_dft_1d(int n0,
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fftw_complex *in, fftw_complex *out,
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int sign, unsigned flags);
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fftw_plan fftw_plan_dft_2d(int n0, int n1,
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fftw_complex *in, fftw_complex *out,
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int sign, unsigned flags);
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fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
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fftw_complex *in, fftw_complex *out,
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int sign, unsigned flags);
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fftw_plan fftw_plan_dft(int rank, const int *n,
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fftw_complex *in, fftw_complex *out,
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int sign, unsigned flags);
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@end example
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@findex fftw_plan_dft_1d
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@findex fftw_plan_dft_2d
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@findex fftw_plan_dft_3d
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@findex fftw_plan_dft
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Plan a complex input/output discrete Fourier transform (DFT) in zero or
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more dimensions, returning an @code{fftw_plan} (@pxref{Using Plans}).
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Once you have created a plan for a certain transform type and
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parameters, then creating another plan of the same type and parameters,
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but for different arrays, is fast and shares constant data with the
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first plan (if it still exists).
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The planner returns @code{NULL} if the plan cannot be created. In the
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standard FFTW distribution, the basic interface is guaranteed to return
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a non-@code{NULL} plan. A plan may be @code{NULL}, however, if you are
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using a customized FFTW configuration supporting a restricted set of
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transforms.
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@subsubheading Arguments
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@itemize @bullet
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@item
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@code{rank} is the rank of the transform (it should be the size of the
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array @code{*n}), and can be any non-negative integer. (@xref{Complex
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Multi-Dimensional DFTs}, for the definition of ``rank''.) The
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@samp{_1d}, @samp{_2d}, and @samp{_3d} planners correspond to a
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@code{rank} of @code{1}, @code{2}, and @code{3}, respectively. The rank
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may be zero, which is equivalent to a rank-1 transform of size 1, i.e. a
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copy of one number from input to output.
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@item
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@code{n0}, @code{n1}, @code{n2}, or @code{n[0..rank-1]} (as appropriate
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for each routine) specify the size of the transform dimensions. They
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can be any positive integer.
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@itemize @minus
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@item
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@cindex row-major
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Multi-dimensional arrays are stored in row-major order with dimensions:
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@code{n0} x @code{n1}; or @code{n0} x @code{n1} x @code{n2}; or
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@code{n[0]} x @code{n[1]} x ... x @code{n[rank-1]}.
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@xref{Multi-dimensional Array Format}.
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@item
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FFTW is best at handling sizes of the form
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@ifinfo
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@math{2^a 3^b 5^c 7^d 11^e 13^f},
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@end ifinfo
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@tex
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$2^a 3^b 5^c 7^d 11^e 13^f$,
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@end tex
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@html
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2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup> 7<sup>d</sup>
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11<sup>e</sup> 13<sup>f</sup>,
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@end html
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where @math{e+f} is either @math{0} or @math{1}, and the other exponents
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are arbitrary. Other sizes are computed by means of a slow,
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general-purpose algorithm (which nevertheless retains @Onlogn{} performance even for prime sizes). It is possible to customize FFTW
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for different array sizes; see @ref{Installation and Customization}.
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Transforms whose sizes are powers of @math{2} are especially fast.
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@end itemize
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@item
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@code{in} and @code{out} point to the input and output arrays of the
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transform, which may be the same (yielding an in-place transform).
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@cindex in-place
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These arrays are overwritten during planning, unless
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@code{FFTW_ESTIMATE} is used in the flags. (The arrays need not be
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initialized, but they must be allocated.)
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If @code{in == out}, the transform is @dfn{in-place} and the input
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array is overwritten. If @code{in != out}, the two arrays must
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not overlap (but FFTW does not check for this condition).
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@item
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@ctindex FFTW_FORWARD
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@ctindex FFTW_BACKWARD
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@code{sign} is the sign of the exponent in the formula that defines the
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Fourier transform. It can be @math{-1} (= @code{FFTW_FORWARD}) or
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@math{+1} (= @code{FFTW_BACKWARD}).
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@item
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@cindex flags
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@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
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as defined in @ref{Planner Flags}.
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@end itemize
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FFTW computes an unnormalized transform: computing a forward followed by
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a backward transform (or vice versa) will result in the original data
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multiplied by the size of the transform (the product of the dimensions).
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@cindex normalization
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For more information, see @ref{What FFTW Really Computes}.
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@c =========>
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@node Planner Flags, Real-data DFTs, Complex DFTs, Basic Interface
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@subsection Planner Flags
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All of the planner routines in FFTW accept an integer @code{flags}
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argument, which is a bitwise OR (@samp{|}) of zero or more of the flag
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constants defined below. These flags control the rigor (and time) of
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the planning process, and can also impose (or lift) restrictions on the
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type of transform algorithm that is employed.
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@emph{Important:} the planner overwrites the input array during
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planning unless a saved plan (@pxref{Wisdom}) is available for that
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problem, so you should initialize your input data after creating the
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plan. The only exceptions to this are the @code{FFTW_ESTIMATE} and
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@code{FFTW_WISDOM_ONLY} flags, as mentioned below.
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In all cases, if wisdom is available for the given problem that was
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created with equal-or-greater planning rigor, then the more rigorous
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wisdom is used. For example, in @code{FFTW_ESTIMATE} mode any available
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wisdom is used, whereas in @code{FFTW_PATIENT} mode only wisdom created
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in patient or exhaustive mode can be used. @xref{Words of Wisdom-Saving
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Plans}.
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@subsubheading Planning-rigor flags
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@itemize @bullet
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@item
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@ctindex FFTW_ESTIMATE
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@code{FFTW_ESTIMATE} specifies that, instead of actual measurements of
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different algorithms, a simple heuristic is used to pick a (probably
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sub-optimal) plan quickly. With this flag, the input/output arrays are
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not overwritten during planning.
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@item
|
|
@ctindex FFTW_MEASURE
|
|
@code{FFTW_MEASURE} tells FFTW to find an optimized plan by actually
|
|
@emph{computing} several FFTs and measuring their execution time.
|
|
Depending on your machine, this can take some time (often a few
|
|
seconds). @code{FFTW_MEASURE} is the default planning option.
|
|
|
|
@item
|
|
@ctindex FFTW_PATIENT
|
|
@code{FFTW_PATIENT} is like @code{FFTW_MEASURE}, but considers a wider
|
|
range of algorithms and often produces a ``more optimal'' plan
|
|
(especially for large transforms), but at the expense of several times
|
|
longer planning time (especially for large transforms).
|
|
|
|
@item
|
|
@ctindex FFTW_EXHAUSTIVE
|
|
@code{FFTW_EXHAUSTIVE} is like @code{FFTW_PATIENT}, but considers an
|
|
even wider range of algorithms, including many that we think are
|
|
unlikely to be fast, to produce the most optimal plan but with a
|
|
substantially increased planning time.
|
|
|
|
@item
|
|
@ctindex FFTW_WISDOM_ONLY
|
|
@code{FFTW_WISDOM_ONLY} is a special planning mode in which the plan
|
|
is only created if wisdom is available for the given problem, and
|
|
otherwise a @code{NULL} plan is returned. This can be combined with
|
|
other flags, e.g. @samp{FFTW_WISDOM_ONLY | FFTW_PATIENT} creates a
|
|
plan only if wisdom is available that was created in
|
|
@code{FFTW_PATIENT} or @code{FFTW_EXHAUSTIVE} mode. The
|
|
@code{FFTW_WISDOM_ONLY} flag is intended for users who need to detect
|
|
whether wisdom is available; for example, if wisdom is not available
|
|
one may wish to allocate new arrays for planning so that user data is
|
|
not overwritten.
|
|
|
|
@end itemize
|
|
|
|
@subsubheading Algorithm-restriction flags
|
|
@itemize @bullet
|
|
|
|
@item
|
|
@ctindex FFTW_DESTROY_INPUT
|
|
@code{FFTW_DESTROY_INPUT} specifies that an out-of-place transform is
|
|
allowed to @emph{overwrite its input} array with arbitrary data; this
|
|
can sometimes allow more efficient algorithms to be employed.
|
|
@cindex out-of-place
|
|
|
|
@item
|
|
@ctindex FFTW_PRESERVE_INPUT
|
|
@code{FFTW_PRESERVE_INPUT} specifies that an out-of-place transform must
|
|
@emph{not change its input} array. This is ordinarily the
|
|
@emph{default}, except for c2r and hc2r (i.e. complex-to-real)
|
|
transforms for which @code{FFTW_DESTROY_INPUT} is the default. In the
|
|
latter cases, passing @code{FFTW_PRESERVE_INPUT} will attempt to use
|
|
algorithms that do not destroy the input, at the expense of worse
|
|
performance; for multi-dimensional c2r transforms, however, no
|
|
input-preserving algorithms are implemented and the planner will return
|
|
@code{NULL} if one is requested.
|
|
@cindex c2r
|
|
@cindex hc2r
|
|
|
|
@item
|
|
@ctindex FFTW_UNALIGNED
|
|
@cindex alignment
|
|
@findex fftw_malloc
|
|
@findex fftw_alignment_of
|
|
@code{FFTW_UNALIGNED} specifies that the algorithm may not impose any
|
|
unusual alignment requirements on the input/output arrays (i.e. no
|
|
SIMD may be used). This flag is normally @emph{not necessary}, since
|
|
the planner automatically detects misaligned arrays. The only use for
|
|
this flag is if you want to use the new-array execute interface to
|
|
execute a given plan on a different array that may not be aligned like
|
|
the original. (Using @code{fftw_malloc} makes this flag unnecessary
|
|
even then. You can also use @code{fftw_alignment_of} to detect
|
|
whether two arrays are equivalently aligned.)
|
|
|
|
@end itemize
|
|
|
|
@subsubheading Limiting planning time
|
|
|
|
@example
|
|
extern void fftw_set_timelimit(double seconds);
|
|
@end example
|
|
@findex fftw_set_timelimit
|
|
|
|
This function instructs FFTW to spend at most @code{seconds} seconds
|
|
(approximately) in the planner. If @code{seconds ==
|
|
FFTW_NO_TIMELIMIT} (the default value, which is negative), then
|
|
planning time is unbounded. Otherwise, FFTW plans with a
|
|
progressively wider range of algorithms until the given time limit
|
|
is reached or the given range of algorithms is explored, returning the
|
|
best available plan.
|
|
@ctindex FFTW_NO_TIMELIMIT
|
|
|
|
|
|
For example, specifying @code{FFTW_PATIENT} first plans in
|
|
@code{FFTW_ESTIMATE} mode, then in @code{FFTW_MEASURE} mode, then
|
|
finally (time permitting) in @code{FFTW_PATIENT}. If
|
|
@code{FFTW_EXHAUSTIVE} is specified instead, the planner will further
|
|
progress to @code{FFTW_EXHAUSTIVE} mode.
|
|
|
|
Note that the @code{seconds} argument specifies only a rough limit; in
|
|
practice, the planner may use somewhat more time if the time limit is
|
|
reached when the planner is in the middle of an operation that cannot
|
|
be interrupted. At the very least, the planner will complete planning
|
|
in @code{FFTW_ESTIMATE} mode (which is thus equivalent to a time limit
|
|
of 0).
|
|
|
|
|
|
@c =========>
|
|
@node Real-data DFTs, Real-data DFT Array Format, Planner Flags, Basic Interface
|
|
@subsection Real-data DFTs
|
|
|
|
@example
|
|
fftw_plan fftw_plan_dft_r2c_1d(int n0,
|
|
double *in, fftw_complex *out,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
|
|
double *in, fftw_complex *out,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
|
|
double *in, fftw_complex *out,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
|
|
double *in, fftw_complex *out,
|
|
unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_dft_r2c_1d
|
|
@findex fftw_plan_dft_r2c_2d
|
|
@findex fftw_plan_dft_r2c_3d
|
|
@findex fftw_plan_dft_r2c
|
|
@cindex r2c
|
|
|
|
Plan a real-input/complex-output discrete Fourier transform (DFT) in
|
|
zero or more dimensions, returning an @code{fftw_plan} (@pxref{Using
|
|
Plans}).
|
|
|
|
Once you have created a plan for a certain transform type and
|
|
parameters, then creating another plan of the same type and parameters,
|
|
but for different arrays, is fast and shares constant data with the
|
|
first plan (if it still exists).
|
|
|
|
The planner returns @code{NULL} if the plan cannot be created. A
|
|
non-@code{NULL} plan is always returned by the basic interface unless
|
|
you are using a customized FFTW configuration supporting a restricted
|
|
set of transforms, or if you use the @code{FFTW_PRESERVE_INPUT} flag
|
|
with a multi-dimensional out-of-place c2r transform (see below).
|
|
|
|
@subsubheading Arguments
|
|
@itemize @bullet
|
|
|
|
@item
|
|
@code{rank} is the rank of the transform (it should be the size of the
|
|
array @code{*n}), and can be any non-negative integer. (@xref{Complex
|
|
Multi-Dimensional DFTs}, for the definition of ``rank''.) The
|
|
@samp{_1d}, @samp{_2d}, and @samp{_3d} planners correspond to a
|
|
@code{rank} of @code{1}, @code{2}, and @code{3}, respectively. The rank
|
|
may be zero, which is equivalent to a rank-1 transform of size 1, i.e. a
|
|
copy of one real number (with zero imaginary part) from input to output.
|
|
|
|
@item
|
|
@code{n0}, @code{n1}, @code{n2}, or @code{n[0..rank-1]}, (as appropriate
|
|
for each routine) specify the size of the transform dimensions. They
|
|
can be any positive integer. This is different in general from the
|
|
@emph{physical} array dimensions, which are described in @ref{Real-data
|
|
DFT Array Format}.
|
|
|
|
@itemize @minus
|
|
@item
|
|
FFTW is best at handling sizes of the form
|
|
@ifinfo
|
|
@math{2^a 3^b 5^c 7^d 11^e 13^f},
|
|
@end ifinfo
|
|
@tex
|
|
$2^a 3^b 5^c 7^d 11^e 13^f$,
|
|
@end tex
|
|
@html
|
|
2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup> 7<sup>d</sup>
|
|
11<sup>e</sup> 13<sup>f</sup>,
|
|
@end html
|
|
where @math{e+f} is either @math{0} or @math{1}, and the other exponents
|
|
are arbitrary. Other sizes are computed by means of a slow,
|
|
general-purpose algorithm (which nevertheless retains @Onlogn{} performance even for prime sizes). (It is possible to customize FFTW
|
|
for different array sizes; see @ref{Installation and Customization}.)
|
|
Transforms whose sizes are powers of @math{2} are especially fast, and
|
|
it is generally beneficial for the @emph{last} dimension of an r2c/c2r
|
|
transform to be @emph{even}.
|
|
@end itemize
|
|
|
|
@item
|
|
@code{in} and @code{out} point to the input and output arrays of the
|
|
transform, which may be the same (yielding an in-place transform).
|
|
@cindex in-place
|
|
These arrays are overwritten during planning, unless
|
|
@code{FFTW_ESTIMATE} is used in the flags. (The arrays need not be
|
|
initialized, but they must be allocated.) For an in-place transform, it
|
|
is important to remember that the real array will require padding,
|
|
described in @ref{Real-data DFT Array Format}.
|
|
@cindex padding
|
|
|
|
@item
|
|
@cindex flags
|
|
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
|
|
as defined in @ref{Planner Flags}.
|
|
|
|
@end itemize
|
|
|
|
The inverse transforms, taking complex input (storing the non-redundant
|
|
half of a logically Hermitian array) to real output, are given by:
|
|
|
|
@example
|
|
fftw_plan fftw_plan_dft_c2r_1d(int n0,
|
|
fftw_complex *in, double *out,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_dft_c2r_2d(int n0, int n1,
|
|
fftw_complex *in, double *out,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_dft_c2r_3d(int n0, int n1, int n2,
|
|
fftw_complex *in, double *out,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_dft_c2r(int rank, const int *n,
|
|
fftw_complex *in, double *out,
|
|
unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_dft_c2r_1d
|
|
@findex fftw_plan_dft_c2r_2d
|
|
@findex fftw_plan_dft_c2r_3d
|
|
@findex fftw_plan_dft_c2r
|
|
@cindex c2r
|
|
|
|
The arguments are the same as for the r2c transforms, except that the
|
|
input and output data formats are reversed.
|
|
|
|
FFTW computes an unnormalized transform: computing an r2c followed by a
|
|
c2r transform (or vice versa) will result in the original data
|
|
multiplied by the size of the transform (the product of the logical
|
|
dimensions).
|
|
@cindex normalization
|
|
An r2c transform produces the same output as a @code{FFTW_FORWARD}
|
|
complex DFT of the same input, and a c2r transform is correspondingly
|
|
equivalent to @code{FFTW_BACKWARD}. For more information, see @ref{What
|
|
FFTW Really Computes}.
|
|
|
|
@c =========>
|
|
@node Real-data DFT Array Format, Real-to-Real Transforms, Real-data DFTs, Basic Interface
|
|
@subsection Real-data DFT Array Format
|
|
@cindex r2c/c2r multi-dimensional array format
|
|
|
|
The output of a DFT of real data (r2c) contains symmetries that, in
|
|
principle, make half of the outputs redundant (@pxref{What FFTW Really
|
|
Computes}). (Similarly for the input of an inverse c2r transform.) In
|
|
practice, it is not possible to entirely realize these savings in an
|
|
efficient and understandable format that generalizes to
|
|
multi-dimensional transforms. Instead, the output of the r2c
|
|
transforms is @emph{slightly} over half of the output of the
|
|
corresponding complex transform. We do not ``pack'' the data in any
|
|
way, but store it as an ordinary array of @code{fftw_complex} values.
|
|
In fact, this data is simply a subsection of what would be the array in
|
|
the corresponding complex transform.
|
|
|
|
Specifically, for a real transform of @math{d} (= @code{rank})
|
|
dimensions @ndims{}, the complex data is an @ndimshalf array of
|
|
@code{fftw_complex} values in row-major order (with the division rounded
|
|
down). That is, we only store the @emph{lower} half (non-negative
|
|
frequencies), plus one element, of the last dimension of the data from
|
|
the ordinary complex transform. (We could have instead taken half of
|
|
any other dimension, but implementation turns out to be simpler if the
|
|
last, contiguous, dimension is used.)
|
|
|
|
@cindex out-of-place
|
|
For an out-of-place transform, the real data is simply an array with
|
|
physical dimensions @ndims in row-major order.
|
|
|
|
@cindex in-place
|
|
@cindex padding
|
|
For an in-place transform, some complications arise since the complex data
|
|
is slightly larger than the real data. In this case, the final
|
|
dimension of the real data must be @emph{padded} with extra values to
|
|
accommodate the size of the complex data---two extra if the last
|
|
dimension is even and one if it is odd. That is, the last dimension of
|
|
the real data must physically contain
|
|
@tex
|
|
$2 (n_{d-1}/2+1)$
|
|
@end tex
|
|
@ifinfo
|
|
2 * (n[d-1]/2+1)
|
|
@end ifinfo
|
|
@html
|
|
2 * (n<sub>d-1</sub>/2+1)
|
|
@end html
|
|
@code{double} values (exactly enough to hold the complex data). This
|
|
physical array size does not, however, change the @emph{logical} array
|
|
size---only
|
|
@tex
|
|
$n_{d-1}$
|
|
@end tex
|
|
@ifinfo
|
|
n[d-1]
|
|
@end ifinfo
|
|
@html
|
|
n<sub>d-1</sub>
|
|
@end html
|
|
values are actually stored in the last dimension, and
|
|
@tex
|
|
$n_{d-1}$
|
|
@end tex
|
|
@ifinfo
|
|
n[d-1]
|
|
@end ifinfo
|
|
@html
|
|
n<sub>d-1</sub>
|
|
@end html
|
|
is the last dimension passed to the planner.
|
|
|
|
@c =========>
|
|
@node Real-to-Real Transforms, Real-to-Real Transform Kinds, Real-data DFT Array Format, Basic Interface
|
|
@subsection Real-to-Real Transforms
|
|
@cindex r2r
|
|
|
|
@example
|
|
fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
|
|
fftw_r2r_kind kind, unsigned flags);
|
|
fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
|
|
fftw_r2r_kind kind0, fftw_r2r_kind kind1,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
|
|
double *in, double *out,
|
|
fftw_r2r_kind kind0,
|
|
fftw_r2r_kind kind1,
|
|
fftw_r2r_kind kind2,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
|
|
const fftw_r2r_kind *kind, unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_r2r_1d
|
|
@findex fftw_plan_r2r_2d
|
|
@findex fftw_plan_r2r_3d
|
|
@findex fftw_plan_r2r
|
|
|
|
Plan a real input/output (r2r) transform of various kinds in zero or
|
|
more dimensions, returning an @code{fftw_plan} (@pxref{Using Plans}).
|
|
|
|
Once you have created a plan for a certain transform type and
|
|
parameters, then creating another plan of the same type and parameters,
|
|
but for different arrays, is fast and shares constant data with the
|
|
first plan (if it still exists).
|
|
|
|
The planner returns @code{NULL} if the plan cannot be created. A
|
|
non-@code{NULL} plan is always returned by the basic interface unless
|
|
you are using a customized FFTW configuration supporting a restricted
|
|
set of transforms, or for size-1 @code{FFTW_REDFT00} kinds (which are
|
|
not defined).
|
|
@ctindex FFTW_REDFT00
|
|
|
|
@subsubheading Arguments
|
|
@itemize @bullet
|
|
|
|
@item
|
|
@code{rank} is the dimensionality of the transform (it should be the
|
|
size of the arrays @code{*n} and @code{*kind}), and can be any
|
|
non-negative integer. The @samp{_1d}, @samp{_2d}, and @samp{_3d}
|
|
planners correspond to a @code{rank} of @code{1}, @code{2}, and
|
|
@code{3}, respectively. A @code{rank} of zero is equivalent to a copy
|
|
of one number from input to output.
|
|
|
|
@item
|
|
@code{n}, or @code{n0}/@code{n1}/@code{n2}, or @code{n[rank]},
|
|
respectively, gives the (physical) size of the transform dimensions.
|
|
They can be any positive integer.
|
|
|
|
@itemize @minus
|
|
@item
|
|
@cindex row-major
|
|
Multi-dimensional arrays are stored in row-major order with dimensions:
|
|
@code{n0} x @code{n1}; or @code{n0} x @code{n1} x @code{n2}; or
|
|
@code{n[0]} x @code{n[1]} x ... x @code{n[rank-1]}.
|
|
@xref{Multi-dimensional Array Format}.
|
|
@item
|
|
FFTW is generally best at handling sizes of the form
|
|
@ifinfo
|
|
@math{2^a 3^b 5^c 7^d 11^e 13^f},
|
|
@end ifinfo
|
|
@tex
|
|
$2^a 3^b 5^c 7^d 11^e 13^f$,
|
|
@end tex
|
|
@html
|
|
2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup> 7<sup>d</sup>
|
|
11<sup>e</sup> 13<sup>f</sup>,
|
|
@end html
|
|
where @math{e+f} is either @math{0} or @math{1}, and the other exponents
|
|
are arbitrary. Other sizes are computed by means of a slow,
|
|
general-purpose algorithm (which nevertheless retains @Onlogn{} performance even for prime sizes). (It is possible to customize FFTW
|
|
for different array sizes; see @ref{Installation and Customization}.)
|
|
Transforms whose sizes are powers of @math{2} are especially fast.
|
|
@item
|
|
For a @code{REDFT00} or @code{RODFT00} transform kind in a dimension of
|
|
size @math{n}, it is @math{n-1} or @math{n+1}, respectively, that
|
|
should be factorizable in the above form.
|
|
@end itemize
|
|
|
|
@item
|
|
@code{in} and @code{out} point to the input and output arrays of the
|
|
transform, which may be the same (yielding an in-place transform).
|
|
@cindex in-place
|
|
These arrays are overwritten during planning, unless
|
|
@code{FFTW_ESTIMATE} is used in the flags. (The arrays need not be
|
|
initialized, but they must be allocated.)
|
|
|
|
@item
|
|
@code{kind}, or @code{kind0}/@code{kind1}/@code{kind2}, or
|
|
@code{kind[rank]}, is the kind of r2r transform used for the
|
|
corresponding dimension. The valid kind constants are described in
|
|
@ref{Real-to-Real Transform Kinds}. In a multi-dimensional transform,
|
|
what is computed is the separable product formed by taking each
|
|
transform kind along the corresponding dimension, one dimension after
|
|
another.
|
|
|
|
@item
|
|
@cindex flags
|
|
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
|
|
as defined in @ref{Planner Flags}.
|
|
|
|
@end itemize
|
|
|
|
@c =========>
|
|
@node Real-to-Real Transform Kinds, , Real-to-Real Transforms, Basic Interface
|
|
@subsection Real-to-Real Transform Kinds
|
|
@cindex kind (r2r)
|
|
|
|
FFTW currently supports 11 different r2r transform kinds, specified by
|
|
one of the constants below. For the precise definitions of these
|
|
transforms, see @ref{What FFTW Really Computes}. For a more colloquial
|
|
introduction to these transform kinds, see @ref{More DFTs of Real Data}.
|
|
|
|
For dimension of size @code{n}, there is a corresponding ``logical''
|
|
dimension @code{N} that determines the normalization (and the optimal
|
|
factorization); the formula for @code{N} is given for each kind below.
|
|
Also, with each transform kind is listed its corrsponding inverse
|
|
transform. FFTW computes unnormalized transforms: a transform followed
|
|
by its inverse will result in the original data multiplied by @code{N}
|
|
(or the product of the @code{N}'s for each dimension, in
|
|
multi-dimensions).
|
|
@cindex normalization
|
|
|
|
@itemize @bullet
|
|
|
|
@item
|
|
@ctindex FFTW_R2HC
|
|
@code{FFTW_R2HC} computes a real-input DFT with output in
|
|
``halfcomplex'' format, i.e. real and imaginary parts for a transform of
|
|
size @code{n} stored as:
|
|
@tex
|
|
$$
|
|
r_0, r_1, r_2, \ldots, r_{n/2}, i_{(n+1)/2-1}, \ldots, i_2, i_1
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
r0, r1, r2, r(n/2), i((n+1)/2-1), ..., i2, i1
|
|
@end ifinfo
|
|
@html
|
|
<p align=center>
|
|
r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub>
|
|
</p>
|
|
@end html
|
|
(Logical @code{N=n}, inverse is @code{FFTW_HC2R}.)
|
|
|
|
@item
|
|
@ctindex FFTW_HC2R
|
|
@code{FFTW_HC2R} computes the reverse of @code{FFTW_R2HC}, above.
|
|
(Logical @code{N=n}, inverse is @code{FFTW_R2HC}.)
|
|
|
|
@item
|
|
@ctindex FFTW_DHT
|
|
@code{FFTW_DHT} computes a discrete Hartley transform.
|
|
(Logical @code{N=n}, inverse is @code{FFTW_DHT}.)
|
|
@cindex discrete Hartley transform
|
|
|
|
@item
|
|
@ctindex FFTW_REDFT00
|
|
@code{FFTW_REDFT00} computes an REDFT00 transform, i.e. a DCT-I.
|
|
(Logical @code{N=2*(n-1)}, inverse is @code{FFTW_REDFT00}.)
|
|
@cindex discrete cosine transform
|
|
@cindex DCT
|
|
|
|
@item
|
|
@ctindex FFTW_REDFT10
|
|
@code{FFTW_REDFT10} computes an REDFT10 transform, i.e. a DCT-II (sometimes called ``the'' DCT).
|
|
(Logical @code{N=2*n}, inverse is @code{FFTW_REDFT01}.)
|
|
|
|
@item
|
|
@ctindex FFTW_REDFT01
|
|
@code{FFTW_REDFT01} computes an REDFT01 transform, i.e. a DCT-III (sometimes called ``the'' IDCT, being the inverse of DCT-II).
|
|
(Logical @code{N=2*n}, inverse is @code{FFTW_REDFT=10}.)
|
|
@cindex IDCT
|
|
|
|
@item
|
|
@ctindex FFTW_REDFT11
|
|
@code{FFTW_REDFT11} computes an REDFT11 transform, i.e. a DCT-IV.
|
|
(Logical @code{N=2*n}, inverse is @code{FFTW_REDFT11}.)
|
|
|
|
@item
|
|
@ctindex FFTW_RODFT00
|
|
@code{FFTW_RODFT00} computes an RODFT00 transform, i.e. a DST-I.
|
|
(Logical @code{N=2*(n+1)}, inverse is @code{FFTW_RODFT00}.)
|
|
@cindex discrete sine transform
|
|
@cindex DST
|
|
|
|
@item
|
|
@ctindex FFTW_RODFT10
|
|
@code{FFTW_RODFT10} computes an RODFT10 transform, i.e. a DST-II.
|
|
(Logical @code{N=2*n}, inverse is @code{FFTW_RODFT01}.)
|
|
|
|
@item
|
|
@ctindex FFTW_RODFT01
|
|
@code{FFTW_RODFT01} computes an RODFT01 transform, i.e. a DST-III.
|
|
(Logical @code{N=2*n}, inverse is @code{FFTW_RODFT=10}.)
|
|
|
|
@item
|
|
@ctindex FFTW_RODFT11
|
|
@code{FFTW_RODFT11} computes an RODFT11 transform, i.e. a DST-IV.
|
|
(Logical @code{N=2*n}, inverse is @code{FFTW_RODFT11}.)
|
|
|
|
@end itemize
|
|
|
|
@c ------------------------------------------------------------
|
|
@node Advanced Interface, Guru Interface, Basic Interface, FFTW Reference
|
|
@section Advanced Interface
|
|
@cindex advanced interface
|
|
|
|
FFTW's ``advanced'' interface supplements the basic interface with four
|
|
new planner routines, providing a new level of flexibility: you can plan
|
|
a transform of multiple arrays simultaneously, operate on non-contiguous
|
|
(strided) data, and transform a subset of a larger multi-dimensional
|
|
array. Other than these additional features, the planner operates in
|
|
the same fashion as in the basic interface, and the resulting
|
|
@code{fftw_plan} is used in the same way (@pxref{Using Plans}).
|
|
|
|
@menu
|
|
* Advanced Complex DFTs::
|
|
* Advanced Real-data DFTs::
|
|
* Advanced Real-to-real Transforms::
|
|
@end menu
|
|
|
|
@c =========>
|
|
@node Advanced Complex DFTs, Advanced Real-data DFTs, Advanced Interface, Advanced Interface
|
|
@subsection Advanced Complex DFTs
|
|
|
|
@example
|
|
fftw_plan fftw_plan_many_dft(int rank, const int *n, int howmany,
|
|
fftw_complex *in, const int *inembed,
|
|
int istride, int idist,
|
|
fftw_complex *out, const int *onembed,
|
|
int ostride, int odist,
|
|
int sign, unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_many_dft
|
|
|
|
This routine plans multiple multidimensional complex DFTs, and it
|
|
extends the @code{fftw_plan_dft} routine (@pxref{Complex DFTs}) to
|
|
compute @code{howmany} transforms, each having rank @code{rank} and size
|
|
@code{n}. In addition, the transform data need not be contiguous, but
|
|
it may be laid out in memory with an arbitrary stride. To account for
|
|
these possibilities, @code{fftw_plan_many_dft} adds the new parameters
|
|
@code{howmany}, @{@code{i},@code{o}@}@code{nembed},
|
|
@{@code{i},@code{o}@}@code{stride}, and
|
|
@{@code{i},@code{o}@}@code{dist}. The FFTW basic interface
|
|
(@pxref{Complex DFTs}) provides routines specialized for ranks 1, 2,
|
|
and@tie{}3, but the advanced interface handles only the general-rank
|
|
case.
|
|
|
|
@code{howmany} is the (nonnegative) number of transforms to compute. The resulting
|
|
plan computes @code{howmany} transforms, where the input of the
|
|
@code{k}-th transform is at location @code{in+k*idist} (in C pointer
|
|
arithmetic), and its output is at location @code{out+k*odist}. Plans
|
|
obtained in this way can often be faster than calling FFTW multiple
|
|
times for the individual transforms. The basic @code{fftw_plan_dft}
|
|
interface corresponds to @code{howmany=1} (in which case the @code{dist}
|
|
parameters are ignored).
|
|
@cindex howmany parameter
|
|
@cindex dist
|
|
|
|
|
|
Each of the @code{howmany} transforms has rank @code{rank} and size
|
|
@code{n}, as in the basic interface. In addition, the advanced
|
|
interface allows the input and output arrays of each transform to be
|
|
row-major subarrays of larger rank-@code{rank} arrays, described by
|
|
@code{inembed} and @code{onembed} parameters, respectively.
|
|
@{@code{i},@code{o}@}@code{nembed} must be arrays of length @code{rank},
|
|
and @code{n} should be elementwise less than or equal to
|
|
@{@code{i},@code{o}@}@code{nembed}. Passing @code{NULL} for an
|
|
@code{nembed} parameter is equivalent to passing @code{n} (i.e. same
|
|
physical and logical dimensions, as in the basic interface.)
|
|
|
|
The @code{stride} parameters indicate that the @code{j}-th element of
|
|
the input or output arrays is located at @code{j*istride} or
|
|
@code{j*ostride}, respectively. (For a multi-dimensional array,
|
|
@code{j} is the ordinary row-major index.) When combined with the
|
|
@code{k}-th transform in a @code{howmany} loop, from above, this means
|
|
that the (@code{j},@code{k})-th element is at @code{j*stride+k*dist}.
|
|
(The basic @code{fftw_plan_dft} interface corresponds to a stride of 1.)
|
|
@cindex stride
|
|
|
|
|
|
For in-place transforms, the input and output @code{stride} and
|
|
@code{dist} parameters should be the same; otherwise, the planner may
|
|
return @code{NULL}.
|
|
|
|
Arrays @code{n}, @code{inembed}, and @code{onembed} are not used after
|
|
this function returns. You can safely free or reuse them.
|
|
|
|
@strong{Examples}:
|
|
One transform of one 5 by 6 array contiguous in memory:
|
|
@example
|
|
int rank = 2;
|
|
int n[] = @{5, 6@};
|
|
int howmany = 1;
|
|
int idist = odist = 0; /* unused because howmany = 1 */
|
|
int istride = ostride = 1; /* array is contiguous in memory */
|
|
int *inembed = n, *onembed = n;
|
|
@end example
|
|
|
|
Transform of three 5 by 6 arrays, each contiguous in memory,
|
|
stored in memory one after another:
|
|
@example
|
|
int rank = 2;
|
|
int n[] = @{5, 6@};
|
|
int howmany = 3;
|
|
int idist = odist = n[0]*n[1]; /* = 30, the distance in memory
|
|
between the first element
|
|
of the first array and the
|
|
first element of the second array */
|
|
int istride = ostride = 1; /* array is contiguous in memory */
|
|
int *inembed = n, *onembed = n;
|
|
@end example
|
|
|
|
Transform each column of a 2d array with 10 rows and 3 columns:
|
|
@example
|
|
int rank = 1; /* not 2: we are computing 1d transforms */
|
|
int n[] = @{10@}; /* 1d transforms of length 10 */
|
|
int howmany = 3;
|
|
int idist = odist = 1;
|
|
int istride = ostride = 3; /* distance between two elements in
|
|
the same column */
|
|
int *inembed = n, *onembed = n;
|
|
@end example
|
|
|
|
@c =========>
|
|
@node Advanced Real-data DFTs, Advanced Real-to-real Transforms, Advanced Complex DFTs, Advanced Interface
|
|
@subsection Advanced Real-data DFTs
|
|
|
|
@example
|
|
fftw_plan fftw_plan_many_dft_r2c(int rank, const int *n, int howmany,
|
|
double *in, const int *inembed,
|
|
int istride, int idist,
|
|
fftw_complex *out, const int *onembed,
|
|
int ostride, int odist,
|
|
unsigned flags);
|
|
fftw_plan fftw_plan_many_dft_c2r(int rank, const int *n, int howmany,
|
|
fftw_complex *in, const int *inembed,
|
|
int istride, int idist,
|
|
double *out, const int *onembed,
|
|
int ostride, int odist,
|
|
unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_many_dft_r2c
|
|
@findex fftw_plan_many_dft_c2r
|
|
|
|
Like @code{fftw_plan_many_dft}, these two functions add @code{howmany},
|
|
@code{nembed}, @code{stride}, and @code{dist} parameters to the
|
|
@code{fftw_plan_dft_r2c} and @code{fftw_plan_dft_c2r} functions, but
|
|
otherwise behave the same as the basic interface.
|
|
|
|
The interpretation of @code{howmany}, @code{stride}, and @code{dist} are
|
|
the same as for @code{fftw_plan_many_dft}, above. Note that the
|
|
@code{stride} and @code{dist} for the real array are in units of
|
|
@code{double}, and for the complex array are in units of
|
|
@code{fftw_complex}.
|
|
|
|
If an @code{nembed} parameter is @code{NULL}, it is interpreted as what
|
|
it would be in the basic interface, as described in @ref{Real-data DFT
|
|
Array Format}. That is, for the complex array the size is assumed to be
|
|
the same as @code{n}, but with the last dimension cut roughly in half.
|
|
For the real array, the size is assumed to be @code{n} if the transform
|
|
is out-of-place, or @code{n} with the last dimension ``padded'' if the
|
|
transform is in-place.
|
|
|
|
If an @code{nembed} parameter is non-@code{NULL}, it is interpreted as
|
|
the physical size of the corresponding array, in row-major order, just
|
|
as for @code{fftw_plan_many_dft}. In this case, each dimension of
|
|
@code{nembed} should be @code{>=} what it would be in the basic
|
|
interface (e.g. the halved or padded @code{n}).
|
|
|
|
Arrays @code{n}, @code{inembed}, and @code{onembed} are not used after
|
|
this function returns. You can safely free or reuse them.
|
|
|
|
@c =========>
|
|
@node Advanced Real-to-real Transforms, , Advanced Real-data DFTs, Advanced Interface
|
|
@subsection Advanced Real-to-real Transforms
|
|
|
|
@example
|
|
fftw_plan fftw_plan_many_r2r(int rank, const int *n, int howmany,
|
|
double *in, const int *inembed,
|
|
int istride, int idist,
|
|
double *out, const int *onembed,
|
|
int ostride, int odist,
|
|
const fftw_r2r_kind *kind, unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_many_r2r
|
|
|
|
Like @code{fftw_plan_many_dft}, this functions adds @code{howmany},
|
|
@code{nembed}, @code{stride}, and @code{dist} parameters to the
|
|
@code{fftw_plan_r2r} function, but otherwise behave the same as the
|
|
basic interface. The interpretation of those additional parameters are
|
|
the same as for @code{fftw_plan_many_dft}. (Of course, the
|
|
@code{stride} and @code{dist} parameters are now in units of
|
|
@code{double}, not @code{fftw_complex}.)
|
|
|
|
Arrays @code{n}, @code{inembed}, @code{onembed}, and @code{kind} are not
|
|
used after this function returns. You can safely free or reuse them.
|
|
|
|
@c ------------------------------------------------------------
|
|
@node Guru Interface, New-array Execute Functions, Advanced Interface, FFTW Reference
|
|
@section Guru Interface
|
|
@cindex guru interface
|
|
|
|
The ``guru'' interface to FFTW is intended to expose as much as possible
|
|
of the flexibility in the underlying FFTW architecture. It allows one
|
|
to compute multi-dimensional ``vectors'' (loops) of multi-dimensional
|
|
transforms, where each vector/transform dimension has an independent
|
|
size and stride.
|
|
@cindex vector
|
|
One can also use more general complex-number formats, e.g. separate real
|
|
and imaginary arrays.
|
|
|
|
For those users who require the flexibility of the guru interface, it is
|
|
important that they pay special attention to the documentation lest they
|
|
shoot themselves in the foot.
|
|
|
|
@menu
|
|
* Interleaved and split arrays::
|
|
* Guru vector and transform sizes::
|
|
* Guru Complex DFTs::
|
|
* Guru Real-data DFTs::
|
|
* Guru Real-to-real Transforms::
|
|
* 64-bit Guru Interface::
|
|
@end menu
|
|
|
|
@c =========>
|
|
@node Interleaved and split arrays, Guru vector and transform sizes, Guru Interface, Guru Interface
|
|
@subsection Interleaved and split arrays
|
|
|
|
The guru interface supports two representations of complex numbers,
|
|
which we call the interleaved and the split format.
|
|
|
|
The @dfn{interleaved} format is the same one used by the basic and
|
|
advanced interfaces, and it is documented in @ref{Complex numbers}.
|
|
In the interleaved format, you provide pointers to the real part of a
|
|
complex number, and the imaginary part understood to be stored in the
|
|
next memory location.
|
|
@cindex interleaved format
|
|
|
|
|
|
The @dfn{split} format allows separate pointers to the real and
|
|
imaginary parts of a complex array.
|
|
@cindex split format
|
|
|
|
|
|
Technically, the interleaved format is redundant, because you can
|
|
always express an interleaved array in terms of a split array with
|
|
appropriate pointers and strides. On the other hand, the interleaved
|
|
format is simpler to use, and it is common in practice. Hence, FFTW
|
|
supports it as a special case.
|
|
|
|
@c =========>
|
|
@node Guru vector and transform sizes, Guru Complex DFTs, Interleaved and split arrays, Guru Interface
|
|
@subsection Guru vector and transform sizes
|
|
|
|
The guru interface introduces one basic new data structure,
|
|
@code{fftw_iodim}, that is used to specify sizes and strides for
|
|
multi-dimensional transforms and vectors:
|
|
|
|
@example
|
|
typedef struct @{
|
|
int n;
|
|
int is;
|
|
int os;
|
|
@} fftw_iodim;
|
|
@end example
|
|
@tindex fftw_iodim
|
|
|
|
Here, @code{n} is the size of the dimension, and @code{is} and @code{os}
|
|
are the strides of that dimension for the input and output arrays. (The
|
|
stride is the separation of consecutive elements along this dimension.)
|
|
|
|
The meaning of the stride parameter depends on the type of the array
|
|
that the stride refers to. @emph{If the array is interleaved complex,
|
|
strides are expressed in units of complex numbers
|
|
(@code{fftw_complex}). If the array is split complex or real, strides
|
|
are expressed in units of real numbers (@code{double}).} This
|
|
convention is consistent with the usual pointer arithmetic in the C
|
|
language. An interleaved array is denoted by a pointer @code{p} to
|
|
@code{fftw_complex}, so that @code{p+1} points to the next complex
|
|
number. Split arrays are denoted by pointers to @code{double}, in
|
|
which case pointer arithmetic operates in units of
|
|
@code{sizeof(double)}.
|
|
@cindex stride
|
|
|
|
|
|
The guru planner interfaces all take a (@code{rank}, @code{dims[rank]})
|
|
pair describing the transform size, and a (@code{howmany_rank},
|
|
@code{howmany_dims[howmany_rank]}) pair describing the ``vector'' size (a
|
|
multi-dimensional loop of transforms to perform), where @code{dims} and
|
|
@code{howmany_dims} are arrays of @code{fftw_iodim}. Each @code{n} field must
|
|
be positive for @code{dims} and nonnegative for @code{howmany_dims}, while both
|
|
@code{rank} and @code{howmany_rank} must be nonnegative.
|
|
|
|
For example, the @code{howmany} parameter in the advanced complex-DFT
|
|
interface corresponds to @code{howmany_rank} = 1,
|
|
@code{howmany_dims[0].n} = @code{howmany}, @code{howmany_dims[0].is} =
|
|
@code{idist}, and @code{howmany_dims[0].os} = @code{odist}.
|
|
@cindex howmany loop
|
|
@cindex dist
|
|
(To compute a single transform, you can just use @code{howmany_rank} = 0.)
|
|
|
|
|
|
A row-major multidimensional array with dimensions @code{n[rank]}
|
|
(@pxref{Row-major Format}) corresponds to @code{dims[i].n} =
|
|
@code{n[i]} and the recurrence @code{dims[i].is} = @code{n[i+1] *
|
|
dims[i+1].is} (similarly for @code{os}). The stride of the last
|
|
(@code{i=rank-1}) dimension is the overall stride of the array.
|
|
e.g. to be equivalent to the advanced complex-DFT interface, you would
|
|
have @code{dims[rank-1].is} = @code{istride} and
|
|
@code{dims[rank-1].os} = @code{ostride}.
|
|
@cindex row-major
|
|
|
|
|
|
In general, we only guarantee FFTW to return a non-@code{NULL} plan if
|
|
the vector and transform dimensions correspond to a set of distinct
|
|
indices, and for in-place transforms the input/output strides should
|
|
be the same.
|
|
|
|
@c =========>
|
|
@node Guru Complex DFTs, Guru Real-data DFTs, Guru vector and transform sizes, Guru Interface
|
|
@subsection Guru Complex DFTs
|
|
|
|
@example
|
|
fftw_plan fftw_plan_guru_dft(
|
|
int rank, const fftw_iodim *dims,
|
|
int howmany_rank, const fftw_iodim *howmany_dims,
|
|
fftw_complex *in, fftw_complex *out,
|
|
int sign, unsigned flags);
|
|
|
|
fftw_plan fftw_plan_guru_split_dft(
|
|
int rank, const fftw_iodim *dims,
|
|
int howmany_rank, const fftw_iodim *howmany_dims,
|
|
double *ri, double *ii, double *ro, double *io,
|
|
unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_guru_dft
|
|
@findex fftw_plan_guru_split_dft
|
|
|
|
These two functions plan a complex-data, multi-dimensional DFT
|
|
for the interleaved and split format, respectively.
|
|
Transform dimensions are given by (@code{rank}, @code{dims}) over a
|
|
multi-dimensional vector (loop) of dimensions (@code{howmany_rank},
|
|
@code{howmany_dims}). @code{dims} and @code{howmany_dims} should point
|
|
to @code{fftw_iodim} arrays of length @code{rank} and
|
|
@code{howmany_rank}, respectively.
|
|
|
|
@cindex flags
|
|
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
|
|
as defined in @ref{Planner Flags}.
|
|
|
|
In the @code{fftw_plan_guru_dft} function, the pointers @code{in} and
|
|
@code{out} point to the interleaved input and output arrays,
|
|
respectively. The sign can be either @math{-1} (=
|
|
@code{FFTW_FORWARD}) or @math{+1} (= @code{FFTW_BACKWARD}). If the
|
|
pointers are equal, the transform is in-place.
|
|
|
|
In the @code{fftw_plan_guru_split_dft} function,
|
|
@code{ri} and @code{ii} point to the real and imaginary input arrays,
|
|
and @code{ro} and @code{io} point to the real and imaginary output
|
|
arrays. The input and output pointers may be the same, indicating an
|
|
in-place transform. For example, for @code{fftw_complex} pointers
|
|
@code{in} and @code{out}, the corresponding parameters are:
|
|
|
|
@example
|
|
ri = (double *) in;
|
|
ii = (double *) in + 1;
|
|
ro = (double *) out;
|
|
io = (double *) out + 1;
|
|
@end example
|
|
|
|
Because @code{fftw_plan_guru_split_dft} accepts split arrays, strides
|
|
are expressed in units of @code{double}. For a contiguous
|
|
@code{fftw_complex} array, the overall stride of the transform should
|
|
be 2, the distance between consecutive real parts or between
|
|
consecutive imaginary parts; see @ref{Guru vector and transform
|
|
sizes}. Note that the dimension strides are applied equally to the
|
|
real and imaginary parts; real and imaginary arrays with different
|
|
strides are not supported.
|
|
|
|
There is no @code{sign} parameter in @code{fftw_plan_guru_split_dft}.
|
|
This function always plans for an @code{FFTW_FORWARD} transform. To
|
|
plan for an @code{FFTW_BACKWARD} transform, you can exploit the
|
|
identity that the backwards DFT is equal to the forwards DFT with the
|
|
real and imaginary parts swapped. For example, in the case of the
|
|
@code{fftw_complex} arrays above, the @code{FFTW_BACKWARD} transform
|
|
is computed by the parameters:
|
|
|
|
@example
|
|
ri = (double *) in + 1;
|
|
ii = (double *) in;
|
|
ro = (double *) out + 1;
|
|
io = (double *) out;
|
|
@end example
|
|
|
|
@c =========>
|
|
@node Guru Real-data DFTs, Guru Real-to-real Transforms, Guru Complex DFTs, Guru Interface
|
|
@subsection Guru Real-data DFTs
|
|
|
|
@example
|
|
fftw_plan fftw_plan_guru_dft_r2c(
|
|
int rank, const fftw_iodim *dims,
|
|
int howmany_rank, const fftw_iodim *howmany_dims,
|
|
double *in, fftw_complex *out,
|
|
unsigned flags);
|
|
|
|
fftw_plan fftw_plan_guru_split_dft_r2c(
|
|
int rank, const fftw_iodim *dims,
|
|
int howmany_rank, const fftw_iodim *howmany_dims,
|
|
double *in, double *ro, double *io,
|
|
unsigned flags);
|
|
|
|
fftw_plan fftw_plan_guru_dft_c2r(
|
|
int rank, const fftw_iodim *dims,
|
|
int howmany_rank, const fftw_iodim *howmany_dims,
|
|
fftw_complex *in, double *out,
|
|
unsigned flags);
|
|
|
|
fftw_plan fftw_plan_guru_split_dft_c2r(
|
|
int rank, const fftw_iodim *dims,
|
|
int howmany_rank, const fftw_iodim *howmany_dims,
|
|
double *ri, double *ii, double *out,
|
|
unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_guru_dft_r2c
|
|
@findex fftw_plan_guru_split_dft_r2c
|
|
@findex fftw_plan_guru_dft_c2r
|
|
@findex fftw_plan_guru_split_dft_c2r
|
|
|
|
Plan a real-input (r2c) or real-output (c2r), multi-dimensional DFT with
|
|
transform dimensions given by (@code{rank}, @code{dims}) over a
|
|
multi-dimensional vector (loop) of dimensions (@code{howmany_rank},
|
|
@code{howmany_dims}). @code{dims} and @code{howmany_dims} should point
|
|
to @code{fftw_iodim} arrays of length @code{rank} and
|
|
@code{howmany_rank}, respectively. As for the basic and advanced
|
|
interfaces, an r2c transform is @code{FFTW_FORWARD} and a c2r transform
|
|
is @code{FFTW_BACKWARD}.
|
|
|
|
The @emph{last} dimension of @code{dims} is interpreted specially:
|
|
that dimension of the real array has size @code{dims[rank-1].n}, but
|
|
that dimension of the complex array has size @code{dims[rank-1].n/2+1}
|
|
(division rounded down). The strides, on the other hand, are taken to
|
|
be exactly as specified. It is up to the user to specify the strides
|
|
appropriately for the peculiar dimensions of the data, and we do not
|
|
guarantee that the planner will succeed (return non-@code{NULL}) for
|
|
any dimensions other than those described in @ref{Real-data DFT Array
|
|
Format} and generalized in @ref{Advanced Real-data DFTs}. (That is,
|
|
for an in-place transform, each individual dimension should be able to
|
|
operate in place.)
|
|
@cindex in-place
|
|
|
|
|
|
@code{in} and @code{out} point to the input and output arrays for r2c
|
|
and c2r transforms, respectively. For split arrays, @code{ri} and
|
|
@code{ii} point to the real and imaginary input arrays for a c2r
|
|
transform, and @code{ro} and @code{io} point to the real and imaginary
|
|
output arrays for an r2c transform. @code{in} and @code{ro} or
|
|
@code{ri} and @code{out} may be the same, indicating an in-place
|
|
transform. (In-place transforms where @code{in} and @code{io} or
|
|
@code{ii} and @code{out} are the same are not currently supported.)
|
|
|
|
@cindex flags
|
|
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
|
|
as defined in @ref{Planner Flags}.
|
|
|
|
In-place transforms of rank greater than 1 are currently only
|
|
supported for interleaved arrays. For split arrays, the planner will
|
|
return @code{NULL}.
|
|
@cindex in-place
|
|
|
|
@c =========>
|
|
@node Guru Real-to-real Transforms, 64-bit Guru Interface, Guru Real-data DFTs, Guru Interface
|
|
@subsection Guru Real-to-real Transforms
|
|
|
|
@example
|
|
fftw_plan fftw_plan_guru_r2r(int rank, const fftw_iodim *dims,
|
|
int howmany_rank,
|
|
const fftw_iodim *howmany_dims,
|
|
double *in, double *out,
|
|
const fftw_r2r_kind *kind,
|
|
unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_guru_r2r
|
|
|
|
Plan a real-to-real (r2r) multi-dimensional @code{FFTW_FORWARD}
|
|
transform with transform dimensions given by (@code{rank}, @code{dims})
|
|
over a multi-dimensional vector (loop) of dimensions
|
|
(@code{howmany_rank}, @code{howmany_dims}). @code{dims} and
|
|
@code{howmany_dims} should point to @code{fftw_iodim} arrays of length
|
|
@code{rank} and @code{howmany_rank}, respectively.
|
|
|
|
The transform kind of each dimension is given by the @code{kind}
|
|
parameter, which should point to an array of length @code{rank}. Valid
|
|
@code{fftw_r2r_kind} constants are given in @ref{Real-to-Real Transform
|
|
Kinds}.
|
|
|
|
@code{in} and @code{out} point to the real input and output arrays; they
|
|
may be the same, indicating an in-place transform.
|
|
|
|
@cindex flags
|
|
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
|
|
as defined in @ref{Planner Flags}.
|
|
|
|
@c =========>
|
|
@node 64-bit Guru Interface, , Guru Real-to-real Transforms, Guru Interface
|
|
@subsection 64-bit Guru Interface
|
|
@cindex 64-bit architecture
|
|
|
|
When compiled in 64-bit mode on a 64-bit architecture (where addresses
|
|
are 64 bits wide), FFTW uses 64-bit quantities internally for all
|
|
transform sizes, strides, and so on---you don't have to do anything
|
|
special to exploit this. However, in the ordinary FFTW interfaces,
|
|
you specify the transform size by an @code{int} quantity, which is
|
|
normally only 32 bits wide. This means that, even though FFTW is
|
|
using 64-bit sizes internally, you cannot specify a single transform
|
|
dimension larger than
|
|
@ifinfo
|
|
2^31-1
|
|
@end ifinfo
|
|
@html
|
|
2<sup><small>31</small></sup>−1
|
|
@end html
|
|
@tex
|
|
$2^{31}-1$
|
|
@end tex
|
|
numbers.
|
|
|
|
We expect that few users will require transforms larger than this, but,
|
|
for those who do, we provide a 64-bit version of the guru interface in
|
|
which all sizes are specified as integers of type @code{ptrdiff_t}
|
|
instead of @code{int}. (@code{ptrdiff_t} is a signed integer type
|
|
defined by the C standard to be wide enough to represent address
|
|
differences, and thus must be at least 64 bits wide on a 64-bit
|
|
machine.) We stress that there is @emph{no performance advantage} to
|
|
using this interface---the same internal FFTW code is employed
|
|
regardless---and it is only necessary if you want to specify very
|
|
large transform sizes.
|
|
@tindex ptrdiff_t
|
|
|
|
|
|
In particular, the 64-bit guru interface is a set of planner routines
|
|
that are exactly the same as the guru planner routines, except that
|
|
they are named with @samp{guru64} instead of @samp{guru} and they take
|
|
arguments of type @code{fftw_iodim64} instead of @code{fftw_iodim}.
|
|
For example, instead of @code{fftw_plan_guru_dft}, we have
|
|
@code{fftw_plan_guru64_dft}.
|
|
|
|
@example
|
|
fftw_plan fftw_plan_guru64_dft(
|
|
int rank, const fftw_iodim64 *dims,
|
|
int howmany_rank, const fftw_iodim64 *howmany_dims,
|
|
fftw_complex *in, fftw_complex *out,
|
|
int sign, unsigned flags);
|
|
@end example
|
|
@findex fftw_plan_guru64_dft
|
|
|
|
The @code{fftw_iodim64} type is similar to @code{fftw_iodim}, with the
|
|
same interpretation, except that it uses type @code{ptrdiff_t} instead
|
|
of type @code{int}.
|
|
|
|
@example
|
|
typedef struct @{
|
|
ptrdiff_t n;
|
|
ptrdiff_t is;
|
|
ptrdiff_t os;
|
|
@} fftw_iodim64;
|
|
@end example
|
|
@tindex fftw_iodim64
|
|
|
|
Every other @samp{fftw_plan_guru} function also has a
|
|
@samp{fftw_plan_guru64} equivalent, but we do not repeat their
|
|
documentation here since they are identical to the 32-bit versions
|
|
except as noted above.
|
|
|
|
@c -----------------------------------------------------------
|
|
@node New-array Execute Functions, Wisdom, Guru Interface, FFTW Reference
|
|
@section New-array Execute Functions
|
|
@cindex execute
|
|
@cindex new-array execution
|
|
|
|
Normally, one executes a plan for the arrays with which the plan was
|
|
created, by calling @code{fftw_execute(plan)} as described in @ref{Using
|
|
Plans}.
|
|
@findex fftw_execute
|
|
However, it is possible for sophisticated users to apply a given plan
|
|
to a @emph{different} array using the ``new-array execute'' functions
|
|
detailed below, provided that the following conditions are met:
|
|
|
|
@itemize @bullet
|
|
|
|
@item
|
|
The array size, strides, etcetera are the same (since those are set by
|
|
the plan).
|
|
|
|
@item
|
|
The input and output arrays are the same (in-place) or different
|
|
(out-of-place) if the plan was originally created to be in-place or
|
|
out-of-place, respectively.
|
|
|
|
@item
|
|
For split arrays, the separations between the real and imaginary
|
|
parts, @code{ii-ri} and @code{io-ro}, are the same as they were for
|
|
the input and output arrays when the plan was created. (This
|
|
condition is automatically satisfied for interleaved arrays.)
|
|
|
|
@item
|
|
The @dfn{alignment} of the new input/output arrays is the same as that
|
|
of the input/output arrays when the plan was created, unless the plan
|
|
was created with the @code{FFTW_UNALIGNED} flag.
|
|
@ctindex FFTW_UNALIGNED
|
|
Here, the alignment is a platform-dependent quantity (for example, it is
|
|
the address modulo 16 if SSE SIMD instructions are used, but the address
|
|
modulo 4 for non-SIMD single-precision FFTW on the same machine). In
|
|
general, only arrays allocated with @code{fftw_malloc} are guaranteed to
|
|
be equally aligned (@pxref{SIMD alignment and fftw_malloc}).
|
|
|
|
@end itemize
|
|
|
|
@cindex alignment
|
|
The alignment issue is especially critical, because if you don't use
|
|
@code{fftw_malloc} then you may have little control over the alignment
|
|
of arrays in memory. For example, neither the C++ @code{new} function
|
|
nor the Fortran @code{allocate} statement provide strong enough
|
|
guarantees about data alignment. If you don't use @code{fftw_malloc},
|
|
therefore, you probably have to use @code{FFTW_UNALIGNED} (which
|
|
disables most SIMD support). If possible, it is probably better for
|
|
you to simply create multiple plans (creating a new plan is quick once
|
|
one exists for a given size), or better yet re-use the same array for
|
|
your transforms.
|
|
|
|
@findex fftw_alignment_of
|
|
For rare circumstances in which you cannot control the alignment of
|
|
allocated memory, but wish to determine where a given array is
|
|
aligned like the original array for which a plan was created, you can
|
|
use the @code{fftw_alignment_of} function:
|
|
@example
|
|
int fftw_alignment_of(double *p);
|
|
@end example
|
|
Two arrays have equivalent alignment (for the purposes of applying a
|
|
plan) if and only if @code{fftw_alignment_of} returns the same value
|
|
for the corresponding pointers to their data (typecast to @code{double*}
|
|
if necessary).
|
|
|
|
If you are tempted to use the new-array execute interface because you
|
|
want to transform a known bunch of arrays of the same size, you should
|
|
probably go use the advanced interface instead (@pxref{Advanced
|
|
Interface})).
|
|
|
|
The new-array execute functions are:
|
|
|
|
@example
|
|
void fftw_execute_dft(
|
|
const fftw_plan p,
|
|
fftw_complex *in, fftw_complex *out);
|
|
|
|
void fftw_execute_split_dft(
|
|
const fftw_plan p,
|
|
double *ri, double *ii, double *ro, double *io);
|
|
|
|
void fftw_execute_dft_r2c(
|
|
const fftw_plan p,
|
|
double *in, fftw_complex *out);
|
|
|
|
void fftw_execute_split_dft_r2c(
|
|
const fftw_plan p,
|
|
double *in, double *ro, double *io);
|
|
|
|
void fftw_execute_dft_c2r(
|
|
const fftw_plan p,
|
|
fftw_complex *in, double *out);
|
|
|
|
void fftw_execute_split_dft_c2r(
|
|
const fftw_plan p,
|
|
double *ri, double *ii, double *out);
|
|
|
|
void fftw_execute_r2r(
|
|
const fftw_plan p,
|
|
double *in, double *out);
|
|
@end example
|
|
@findex fftw_execute_dft
|
|
@findex fftw_execute_split_dft
|
|
@findex fftw_execute_dft_r2c
|
|
@findex fftw_execute_split_dft_r2c
|
|
@findex fftw_execute_dft_c2r
|
|
@findex fftw_execute_split_dft_c2r
|
|
@findex fftw_execute_r2r
|
|
|
|
These execute the @code{plan} to compute the corresponding transform on
|
|
the input/output arrays specified by the subsequent arguments. The
|
|
input/output array arguments have the same meanings as the ones passed
|
|
to the guru planner routines in the preceding sections. The @code{plan}
|
|
is not modified, and these routines can be called as many times as
|
|
desired, or intermixed with calls to the ordinary @code{fftw_execute}.
|
|
|
|
The @code{plan} @emph{must} have been created for the transform type
|
|
corresponding to the execute function, e.g. it must be a complex-DFT
|
|
plan for @code{fftw_execute_dft}. Any of the planner routines for that
|
|
transform type, from the basic to the guru interface, could have been
|
|
used to create the plan, however.
|
|
|
|
@c ------------------------------------------------------------
|
|
@node Wisdom, What FFTW Really Computes, New-array Execute Functions, FFTW Reference
|
|
@section Wisdom
|
|
@cindex wisdom
|
|
@cindex saving plans to disk
|
|
|
|
This section documents the FFTW mechanism for saving and restoring
|
|
plans from disk. This mechanism is called @dfn{wisdom}.
|
|
|
|
@menu
|
|
* Wisdom Export::
|
|
* Wisdom Import::
|
|
* Forgetting Wisdom::
|
|
* Wisdom Utilities::
|
|
@end menu
|
|
|
|
@c =========>
|
|
@node Wisdom Export, Wisdom Import, Wisdom, Wisdom
|
|
@subsection Wisdom Export
|
|
|
|
@example
|
|
int fftw_export_wisdom_to_filename(const char *filename);
|
|
void fftw_export_wisdom_to_file(FILE *output_file);
|
|
char *fftw_export_wisdom_to_string(void);
|
|
void fftw_export_wisdom(void (*write_char)(char c, void *), void *data);
|
|
@end example
|
|
@findex fftw_export_wisdom
|
|
@findex fftw_export_wisdom_to_filename
|
|
@findex fftw_export_wisdom_to_file
|
|
@findex fftw_export_wisdom_to_string
|
|
|
|
These functions allow you to export all currently accumulated wisdom
|
|
in a form from which it can be later imported and restored, even
|
|
during a separate run of the program. (@xref{Words of Wisdom-Saving
|
|
Plans}.) The current store of wisdom is not affected by calling any
|
|
of these routines.
|
|
|
|
@code{fftw_export_wisdom} exports the wisdom to any output
|
|
medium, as specified by the callback function
|
|
@code{write_char}. @code{write_char} is a @code{putc}-like function that
|
|
writes the character @code{c} to some output; its second parameter is
|
|
the @code{data} pointer passed to @code{fftw_export_wisdom}. For
|
|
convenience, the following three ``wrapper'' routines are provided:
|
|
|
|
@code{fftw_export_wisdom_to_filename} writes wisdom to a file named
|
|
@code{filename} (which is created or overwritten), returning @code{1}
|
|
on success and @code{0} on failure. A lower-level function, which
|
|
requires you to open and close the file yourself (e.g. if you want to
|
|
write wisdom to a portion of a larger file) is
|
|
@code{fftw_export_wisdom_to_file}. This writes the wisdom to the
|
|
current position in @code{output_file}, which should be open with
|
|
write permission; upon exit, the file remains open and is positioned
|
|
at the end of the wisdom data.
|
|
|
|
@code{fftw_export_wisdom_to_string} returns a pointer to a
|
|
@code{NULL}-terminated string holding the wisdom data. This string is
|
|
dynamically allocated, and it is the responsibility of the caller to
|
|
deallocate it with @code{free} when it is no longer needed.
|
|
|
|
All of these routines export the wisdom in the same format, which we
|
|
will not document here except to say that it is LISP-like ASCII text
|
|
that is insensitive to white space.
|
|
|
|
@c =========>
|
|
@node Wisdom Import, Forgetting Wisdom, Wisdom Export, Wisdom
|
|
@subsection Wisdom Import
|
|
|
|
@example
|
|
int fftw_import_system_wisdom(void);
|
|
int fftw_import_wisdom_from_filename(const char *filename);
|
|
int fftw_import_wisdom_from_string(const char *input_string);
|
|
int fftw_import_wisdom(int (*read_char)(void *), void *data);
|
|
@end example
|
|
@findex fftw_import_wisdom
|
|
@findex fftw_import_system_wisdom
|
|
@findex fftw_import_wisdom_from_filename
|
|
@findex fftw_import_wisdom_from_file
|
|
@findex fftw_import_wisdom_from_string
|
|
|
|
These functions import wisdom into a program from data stored by the
|
|
@code{fftw_export_wisdom} functions above. (@xref{Words of
|
|
Wisdom-Saving Plans}.) The imported wisdom replaces any wisdom
|
|
already accumulated by the running program.
|
|
|
|
@code{fftw_import_wisdom} imports wisdom from any input medium, as
|
|
specified by the callback function @code{read_char}. @code{read_char} is
|
|
a @code{getc}-like function that returns the next character in the
|
|
input; its parameter is the @code{data} pointer passed to
|
|
@code{fftw_import_wisdom}. If the end of the input data is reached
|
|
(which should never happen for valid data), @code{read_char} should
|
|
return @code{EOF} (as defined in @code{<stdio.h>}). For convenience,
|
|
the following three ``wrapper'' routines are provided:
|
|
|
|
@code{fftw_import_wisdom_from_filename} reads wisdom from a file named
|
|
@code{filename}. A lower-level function, which requires you to open
|
|
and close the file yourself (e.g. if you want to read wisdom from a
|
|
portion of a larger file) is @code{fftw_import_wisdom_from_file}. This
|
|
reads wisdom from the current position in @code{input_file} (which
|
|
should be open with read permission); upon exit, the file remains
|
|
open, but the position of the read pointer is unspecified.
|
|
|
|
@code{fftw_import_wisdom_from_string} reads wisdom from the
|
|
@code{NULL}-terminated string @code{input_string}.
|
|
|
|
@code{fftw_import_system_wisdom} reads wisdom from an
|
|
implementation-defined standard file (@code{/etc/fftw/wisdom} on Unix
|
|
and GNU systems).
|
|
@cindex wisdom, system-wide
|
|
|
|
|
|
The return value of these import routines is @code{1} if the wisdom was
|
|
read successfully and @code{0} otherwise. Note that, in all of these
|
|
functions, any data in the input stream past the end of the wisdom data
|
|
is simply ignored.
|
|
|
|
@c =========>
|
|
@node Forgetting Wisdom, Wisdom Utilities, Wisdom Import, Wisdom
|
|
@subsection Forgetting Wisdom
|
|
|
|
@example
|
|
void fftw_forget_wisdom(void);
|
|
@end example
|
|
@findex fftw_forget_wisdom
|
|
|
|
Calling @code{fftw_forget_wisdom} causes all accumulated @code{wisdom}
|
|
to be discarded and its associated memory to be freed. (New
|
|
@code{wisdom} can still be gathered subsequently, however.)
|
|
|
|
@c =========>
|
|
@node Wisdom Utilities, , Forgetting Wisdom, Wisdom
|
|
@subsection Wisdom Utilities
|
|
|
|
FFTW includes two standalone utility programs that deal with wisdom. We
|
|
merely summarize them here, since they come with their own @code{man}
|
|
pages for Unix and GNU systems (with HTML versions on our web site).
|
|
|
|
The first program is @code{fftw-wisdom} (or @code{fftwf-wisdom} in
|
|
single precision, etcetera), which can be used to create a wisdom file
|
|
containing plans for any of the transform sizes and types supported by
|
|
FFTW. It is preferable to create wisdom directly from your executable
|
|
(@pxref{Caveats in Using Wisdom}), but this program is useful for
|
|
creating global wisdom files for @code{fftw_import_system_wisdom}.
|
|
@cindex fftw-wisdom utility
|
|
|
|
|
|
The second program is @code{fftw-wisdom-to-conf}, which takes a wisdom
|
|
file as input and produces a @dfn{configuration routine} as output. The
|
|
latter is a C subroutine that you can compile and link into your
|
|
program, replacing a routine of the same name in the FFTW library, that
|
|
determines which parts of FFTW are callable by your program.
|
|
@code{fftw-wisdom-to-conf} produces a configuration routine that links
|
|
to only those parts of FFTW needed by the saved plans in the wisdom,
|
|
greatly reducing the size of statically linked executables (which should
|
|
only attempt to create plans corresponding to those in the wisdom,
|
|
however).
|
|
@cindex fftw-wisdom-to-conf utility
|
|
@cindex configuration routines
|
|
|
|
@c ------------------------------------------------------------
|
|
@node What FFTW Really Computes, , Wisdom, FFTW Reference
|
|
@section What FFTW Really Computes
|
|
|
|
In this section, we provide precise mathematical definitions for the
|
|
transforms that FFTW computes. These transform definitions are fairly
|
|
standard, but some authors follow slightly different conventions for the
|
|
normalization of the transform (the constant factor in front) and the
|
|
sign of the complex exponent. We begin by presenting the
|
|
one-dimensional (1d) transform definitions, and then give the
|
|
straightforward extension to multi-dimensional transforms.
|
|
|
|
@menu
|
|
* The 1d Discrete Fourier Transform (DFT)::
|
|
* The 1d Real-data DFT::
|
|
* 1d Real-even DFTs (DCTs)::
|
|
* 1d Real-odd DFTs (DSTs)::
|
|
* 1d Discrete Hartley Transforms (DHTs)::
|
|
* Multi-dimensional Transforms::
|
|
@end menu
|
|
|
|
@c =========>
|
|
@node The 1d Discrete Fourier Transform (DFT), The 1d Real-data DFT, What FFTW Really Computes, What FFTW Really Computes
|
|
@subsection The 1d Discrete Fourier Transform (DFT)
|
|
|
|
@cindex discrete Fourier transform
|
|
@cindex DFT
|
|
The forward (@code{FFTW_FORWARD}) discrete Fourier transform (DFT) of a
|
|
1d complex array @math{X} of size @math{n} computes an array @math{Y},
|
|
where:
|
|
@tex
|
|
$$
|
|
Y_k = \sum_{j = 0}^{n - 1} X_j e^{-2\pi j k \sqrt{-1}/n} \ .
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi j k sqrt(-1)/n) .
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-dft.png" align="top">.</center>
|
|
@end html
|
|
The backward (@code{FFTW_BACKWARD}) DFT computes:
|
|
@tex
|
|
$$
|
|
Y_k = \sum_{j = 0}^{n - 1} X_j e^{2\pi j k \sqrt{-1}/n} \ .
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi j k sqrt(-1)/n) .
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-idft.png" align="top">.</center>
|
|
@end html
|
|
|
|
@cindex normalization
|
|
FFTW computes an unnormalized transform, in that there is no coefficient
|
|
in front of the summation in the DFT. In other words, applying the
|
|
forward and then the backward transform will multiply the input by
|
|
@math{n}.
|
|
|
|
@cindex frequency
|
|
From above, an @code{FFTW_FORWARD} transform corresponds to a sign of
|
|
@math{-1} in the exponent of the DFT. Note also that we use the
|
|
standard ``in-order'' output ordering---the @math{k}-th output
|
|
corresponds to the frequency @math{k/n} (or @math{k/T}, where @math{T}
|
|
is your total sampling period). For those who like to think in terms of
|
|
positive and negative frequencies, this means that the positive
|
|
frequencies are stored in the first half of the output and the negative
|
|
frequencies are stored in backwards order in the second half of the
|
|
output. (The frequency @math{-k/n} is the same as the frequency
|
|
@math{(n-k)/n}.)
|
|
|
|
@c =========>
|
|
@node The 1d Real-data DFT, 1d Real-even DFTs (DCTs), The 1d Discrete Fourier Transform (DFT), What FFTW Really Computes
|
|
@subsection The 1d Real-data DFT
|
|
|
|
The real-input (r2c) DFT in FFTW computes the @emph{forward} transform
|
|
@math{Y} of the size @code{n} real array @math{X}, exactly as defined
|
|
above, i.e.
|
|
@tex
|
|
$$
|
|
Y_k = \sum_{j = 0}^{n - 1} X_j e^{-2\pi j k \sqrt{-1}/n} \ .
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi j k sqrt(-1)/n) .
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-dft.png" align="top">.</center>
|
|
@end html
|
|
This output array @math{Y} can easily be shown to possess the
|
|
``Hermitian'' symmetry
|
|
@cindex Hermitian
|
|
@tex
|
|
$Y_k = Y_{n-k}^*$,
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = Y[n-k]*,
|
|
@end ifinfo
|
|
@html
|
|
<i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,
|
|
@end html
|
|
where we take @math{Y} to be periodic so that
|
|
@tex
|
|
$Y_n = Y_0$.
|
|
@end tex
|
|
@ifinfo
|
|
Y[n] = Y[0].
|
|
@end ifinfo
|
|
@html
|
|
<i>Y<sub>n</sub> = Y</i><sub>0</sub>.
|
|
@end html
|
|
|
|
As a result of this symmetry, half of the output @math{Y} is redundant
|
|
(being the complex conjugate of the other half), and so the 1d r2c
|
|
transforms only output elements @math{0}@dots{}@math{n/2} of @math{Y}
|
|
(@math{n/2+1} complex numbers), where the division by @math{2} is
|
|
rounded down.
|
|
|
|
Moreover, the Hermitian symmetry implies that
|
|
@tex
|
|
$Y_0$
|
|
@end tex
|
|
@ifinfo
|
|
Y[0]
|
|
@end ifinfo
|
|
@html
|
|
<i>Y</i><sub>0</sub>
|
|
@end html
|
|
and, if @math{n} is even, the
|
|
@tex
|
|
$Y_{n/2}$
|
|
@end tex
|
|
@ifinfo
|
|
Y[n/2]
|
|
@end ifinfo
|
|
@html
|
|
<i>Y</i><sub><i>n</i>/2</sub>
|
|
@end html
|
|
element, are purely real. So, for the @code{R2HC} r2r transform, the
|
|
halfcomplex format does not store the imaginary parts of these elements.
|
|
@cindex r2r
|
|
@ctindex R2HC
|
|
@cindex halfcomplex format
|
|
|
|
|
|
The c2r and @code{H2RC} r2r transforms compute the backward DFT of the
|
|
@emph{complex} array @math{X} with Hermitian symmetry, stored in the
|
|
r2c/@code{R2HC} output formats, respectively, where the backward
|
|
transform is defined exactly as for the complex case:
|
|
@tex
|
|
$$
|
|
Y_k = \sum_{j = 0}^{n - 1} X_j e^{2\pi j k \sqrt{-1}/n} \ .
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi j k sqrt(-1)/n) .
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-idft.png" align="top">.</center>
|
|
@end html
|
|
The outputs @code{Y} of this transform can easily be seen to be purely
|
|
real, and are stored as an array of real numbers.
|
|
|
|
@cindex normalization
|
|
Like FFTW's complex DFT, these transforms are unnormalized. In other
|
|
words, applying the real-to-complex (forward) and then the
|
|
complex-to-real (backward) transform will multiply the input by
|
|
@math{n}.
|
|
|
|
@c =========>
|
|
@node 1d Real-even DFTs (DCTs), 1d Real-odd DFTs (DSTs), The 1d Real-data DFT, What FFTW Really Computes
|
|
@subsection 1d Real-even DFTs (DCTs)
|
|
|
|
The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
|
|
forward (and backward) DFTs as defined above, where the input array
|
|
@math{X} of length @math{N} is purely real and is also @dfn{even} symmetry. In
|
|
this case, the output array is likewise real and even symmetry.
|
|
@cindex real-even DFT
|
|
@cindex REDFT
|
|
|
|
|
|
@ctindex REDFT00
|
|
For the case of @code{REDFT00}, this even symmetry means that
|
|
@tex
|
|
$X_j = X_{N-j}$,
|
|
@end tex
|
|
@ifinfo
|
|
X[j] = X[N-j],
|
|
@end ifinfo
|
|
@html
|
|
<i>X<sub>j</sub> = X<sub>N-j</sub></i>,
|
|
@end html
|
|
where we take @math{X} to be periodic so that
|
|
@tex
|
|
$X_N = X_0$.
|
|
@end tex
|
|
@ifinfo
|
|
X[N] = X[0].
|
|
@end ifinfo
|
|
@html
|
|
<i>X<sub>N</sub> = X</i><sub>0</sub>.
|
|
@end html
|
|
Because of this redundancy, only the first @math{n} real numbers are
|
|
actually stored, where @math{N = 2(n-1)}.
|
|
|
|
The proper definition of even symmetry for @code{REDFT10},
|
|
@code{REDFT01}, and @code{REDFT11} transforms is somewhat more intricate
|
|
because of the shifts by @math{1/2} of the input and/or output, although
|
|
the corresponding boundary conditions are given in @ref{Real even/odd
|
|
DFTs (cosine/sine transforms)}. Because of the even symmetry, however,
|
|
the sine terms in the DFT all cancel and the remaining cosine terms are
|
|
written explicitly below. This formulation often leads people to call
|
|
such a transform a @dfn{discrete cosine transform} (DCT), although it is
|
|
really just a special case of the DFT.
|
|
@cindex discrete cosine transform
|
|
@cindex DCT
|
|
|
|
|
|
In each of the definitions below, we transform a real array @math{X} of
|
|
length @math{n} to a real array @math{Y} of length @math{n}:
|
|
|
|
@subsubheading REDFT00 (DCT-I)
|
|
@ctindex REDFT00
|
|
An @code{REDFT00} transform (type-I DCT) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = X_0 + (-1)^k X_{n-1}
|
|
+ 2 \sum_{j=1}^{n-2} X_j \cos [ \pi j k / (n-1)].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = X[0] + (-1)^k X[n-1] + 2 (sum for j = 1 to n-2 of X[j] cos(pi jk /(n-1))).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-redft00.png" align="top">.</center>
|
|
@end html
|
|
Note that this transform is not defined for @math{n=1}. For @math{n=2},
|
|
the summation term above is dropped as you might expect.
|
|
|
|
@subsubheading REDFT10 (DCT-II)
|
|
@ctindex REDFT10
|
|
An @code{REDFT10} transform (type-II DCT, sometimes called ``the'' DCT) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = 2 \sum_{j=0}^{n-1} X_j \cos [\pi (j+1/2) k / n].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = 2 (sum for j = 0 to n-1 of X[j] cos(pi (j+1/2) k / n)).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-redft10.png" align="top">.</center>
|
|
@end html
|
|
|
|
@subsubheading REDFT01 (DCT-III)
|
|
@ctindex REDFT01
|
|
An @code{REDFT01} transform (type-III DCT) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = X_0 + 2 \sum_{j=1}^{n-1} X_j \cos [\pi j (k+1/2) / n].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = X[0] + 2 (sum for j = 1 to n-1 of X[j] cos(pi j (k+1/2) / n)).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-redft01.png" align="top">.</center>
|
|
@end html
|
|
In the case of @math{n=1}, this reduces to
|
|
@tex
|
|
$Y_0 = X_0$.
|
|
@end tex
|
|
@ifinfo
|
|
Y[0] = X[0].
|
|
@end ifinfo
|
|
@html
|
|
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
|
|
@end html
|
|
Up to a scale factor (see below), this is the inverse of @code{REDFT10} (``the'' DCT), and so the @code{REDFT01} (DCT-III) is sometimes called the ``IDCT''.
|
|
@cindex IDCT
|
|
|
|
@subsubheading REDFT11 (DCT-IV)
|
|
@ctindex REDFT11
|
|
An @code{REDFT11} transform (type-IV DCT) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = 2 \sum_{j=0}^{n-1} X_j \cos [\pi (j+1/2) (k+1/2) / n].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = 2 (sum for j = 0 to n-1 of X[j] cos(pi (j+1/2) (k+1/2) / n)).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-redft11.png" align="top">.</center>
|
|
@end html
|
|
|
|
@subsubheading Inverses and Normalization
|
|
|
|
These definitions correspond directly to the unnormalized DFTs used
|
|
elsewhere in FFTW (hence the factors of @math{2} in front of the
|
|
summations). The unnormalized inverse of @code{REDFT00} is
|
|
@code{REDFT00}, of @code{REDFT10} is @code{REDFT01} and vice versa, and
|
|
of @code{REDFT11} is @code{REDFT11}. Each unnormalized inverse results
|
|
in the original array multiplied by @math{N}, where @math{N} is the
|
|
@emph{logical} DFT size. For @code{REDFT00}, @math{N=2(n-1)} (note that
|
|
@math{n=1} is not defined); otherwise, @math{N=2n}.
|
|
@cindex normalization
|
|
|
|
|
|
In defining the discrete cosine transform, some authors also include
|
|
additional factors of
|
|
@ifinfo
|
|
sqrt(2)
|
|
@end ifinfo
|
|
@html
|
|
√2
|
|
@end html
|
|
@tex
|
|
$\sqrt{2}$
|
|
@end tex
|
|
(or its inverse) multiplying selected inputs and/or outputs. This is a
|
|
mostly cosmetic change that makes the transform orthogonal, but
|
|
sacrifices the direct equivalence to a symmetric DFT.
|
|
|
|
@c =========>
|
|
@node 1d Real-odd DFTs (DSTs), 1d Discrete Hartley Transforms (DHTs), 1d Real-even DFTs (DCTs), What FFTW Really Computes
|
|
@subsection 1d Real-odd DFTs (DSTs)
|
|
|
|
The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
|
|
forward (and backward) DFTs as defined above, where the input array
|
|
@math{X} of length @math{N} is purely real and is also @dfn{odd} symmetry. In
|
|
this case, the output is odd symmetry and purely imaginary.
|
|
@cindex real-odd DFT
|
|
@cindex RODFT
|
|
|
|
|
|
@ctindex RODFT00
|
|
For the case of @code{RODFT00}, this odd symmetry means that
|
|
@tex
|
|
$X_j = -X_{N-j}$,
|
|
@end tex
|
|
@ifinfo
|
|
X[j] = -X[N-j],
|
|
@end ifinfo
|
|
@html
|
|
<i>X<sub>j</sub> = -X<sub>N-j</sub></i>,
|
|
@end html
|
|
where we take @math{X} to be periodic so that
|
|
@tex
|
|
$X_N = X_0$.
|
|
@end tex
|
|
@ifinfo
|
|
X[N] = X[0].
|
|
@end ifinfo
|
|
@html
|
|
<i>X<sub>N</sub> = X</i><sub>0</sub>.
|
|
@end html
|
|
Because of this redundancy, only the first @math{n} real numbers
|
|
starting at @math{j=1} are actually stored (the @math{j=0} element is
|
|
zero), where @math{N = 2(n+1)}.
|
|
|
|
The proper definition of odd symmetry for @code{RODFT10},
|
|
@code{RODFT01}, and @code{RODFT11} transforms is somewhat more intricate
|
|
because of the shifts by @math{1/2} of the input and/or output, although
|
|
the corresponding boundary conditions are given in @ref{Real even/odd
|
|
DFTs (cosine/sine transforms)}. Because of the odd symmetry, however,
|
|
the cosine terms in the DFT all cancel and the remaining sine terms are
|
|
written explicitly below. This formulation often leads people to call
|
|
such a transform a @dfn{discrete sine transform} (DST), although it is
|
|
really just a special case of the DFT.
|
|
@cindex discrete sine transform
|
|
@cindex DST
|
|
|
|
|
|
In each of the definitions below, we transform a real array @math{X} of
|
|
length @math{n} to a real array @math{Y} of length @math{n}:
|
|
|
|
@subsubheading RODFT00 (DST-I)
|
|
@ctindex RODFT00
|
|
An @code{RODFT00} transform (type-I DST) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = 2 \sum_{j=0}^{n-1} X_j \sin [ \pi (j+1) (k+1) / (n+1)].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = 2 (sum for j = 0 to n-1 of X[j] sin(pi (j+1)(k+1) / (n+1))).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-rodft00.png" align="top">.</center>
|
|
@end html
|
|
|
|
@subsubheading RODFT10 (DST-II)
|
|
@ctindex RODFT10
|
|
An @code{RODFT10} transform (type-II DST) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = 2 \sum_{j=0}^{n-1} X_j \sin [\pi (j+1/2) (k+1) / n].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = 2 (sum for j = 0 to n-1 of X[j] sin(pi (j+1/2) (k+1) / n)).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-rodft10.png" align="top">.</center>
|
|
@end html
|
|
|
|
@subsubheading RODFT01 (DST-III)
|
|
@ctindex RODFT01
|
|
An @code{RODFT01} transform (type-III DST) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = (-1)^k X_{n-1} + 2 \sum_{j=0}^{n-2} X_j \sin [\pi (j+1) (k+1/2) / n].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = (-1)^k X[n-1] + 2 (sum for j = 0 to n-2 of X[j] sin(pi (j+1) (k+1/2) / n)).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-rodft01.png" align="top">.</center>
|
|
@end html
|
|
In the case of @math{n=1}, this reduces to
|
|
@tex
|
|
$Y_0 = X_0$.
|
|
@end tex
|
|
@ifinfo
|
|
Y[0] = X[0].
|
|
@end ifinfo
|
|
@html
|
|
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
|
|
@end html
|
|
|
|
@subsubheading RODFT11 (DST-IV)
|
|
@ctindex RODFT11
|
|
An @code{RODFT11} transform (type-IV DST) in FFTW is defined by:
|
|
@tex
|
|
$$
|
|
Y_k = 2 \sum_{j=0}^{n-1} X_j \sin [\pi (j+1/2) (k+1/2) / n].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
Y[k] = 2 (sum for j = 0 to n-1 of X[j] sin(pi (j+1/2) (k+1/2) / n)).
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-rodft11.png" align="top">.</center>
|
|
@end html
|
|
|
|
@subsubheading Inverses and Normalization
|
|
|
|
These definitions correspond directly to the unnormalized DFTs used
|
|
elsewhere in FFTW (hence the factors of @math{2} in front of the
|
|
summations). The unnormalized inverse of @code{RODFT00} is
|
|
@code{RODFT00}, of @code{RODFT10} is @code{RODFT01} and vice versa, and
|
|
of @code{RODFT11} is @code{RODFT11}. Each unnormalized inverse results
|
|
in the original array multiplied by @math{N}, where @math{N} is the
|
|
@emph{logical} DFT size. For @code{RODFT00}, @math{N=2(n+1)};
|
|
otherwise, @math{N=2n}.
|
|
@cindex normalization
|
|
|
|
|
|
In defining the discrete sine transform, some authors also include
|
|
additional factors of
|
|
@ifinfo
|
|
sqrt(2)
|
|
@end ifinfo
|
|
@html
|
|
√2
|
|
@end html
|
|
@tex
|
|
$\sqrt{2}$
|
|
@end tex
|
|
(or its inverse) multiplying selected inputs and/or outputs. This is a
|
|
mostly cosmetic change that makes the transform orthogonal, but
|
|
sacrifices the direct equivalence to an antisymmetric DFT.
|
|
|
|
@c =========>
|
|
@node 1d Discrete Hartley Transforms (DHTs), Multi-dimensional Transforms, 1d Real-odd DFTs (DSTs), What FFTW Really Computes
|
|
@subsection 1d Discrete Hartley Transforms (DHTs)
|
|
|
|
@cindex discrete Hartley transform
|
|
@cindex DHT
|
|
The discrete Hartley transform (DHT) of a 1d real array @math{X} of size
|
|
@math{n} computes a real array @math{Y} of the same size, where:
|
|
@tex
|
|
$$
|
|
Y_k = \sum_{j = 0}^{n - 1} X_j [ \cos(2\pi j k / n) + \sin(2\pi j k / n)].
|
|
$$
|
|
@end tex
|
|
@ifinfo
|
|
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * [cos(2 pi j k / n) + sin(2 pi j k / n)].
|
|
@end ifinfo
|
|
@html
|
|
<center><img src="equation-dht.png" align="top">.</center>
|
|
@end html
|
|
|
|
@cindex normalization
|
|
FFTW computes an unnormalized transform, in that there is no coefficient
|
|
in front of the summation in the DHT. In other words, applying the
|
|
transform twice (the DHT is its own inverse) will multiply the input by
|
|
@math{n}.
|
|
|
|
@c =========>
|
|
@node Multi-dimensional Transforms, , 1d Discrete Hartley Transforms (DHTs), What FFTW Really Computes
|
|
@subsection Multi-dimensional Transforms
|
|
|
|
The multi-dimensional transforms of FFTW, in general, compute simply the
|
|
separable product of the given 1d transform along each dimension of the
|
|
array. Since each of these transforms is unnormalized, computing the
|
|
forward followed by the backward/inverse multi-dimensional transform
|
|
will result in the original array scaled by the product of the
|
|
normalization factors for each dimension (e.g. the product of the
|
|
dimension sizes, for a multi-dimensional DFT).
|
|
|
|
@tex
|
|
As an explicit example, consider the following exact mathematical
|
|
definition of our multi-dimensional DFT. Let $X$ be a $d$-dimensional
|
|
complex array whose elements are $X[j_1, j_2, \ldots, j_d]$, where $0
|
|
\leq j_s < n_s$ for all~$s \in \{ 1, 2, \ldots, d \}$. Let also
|
|
$\omega_s = e^{2\pi \sqrt{-1}/n_s}$, for all ~$s \in \{ 1, 2, \ldots, d
|
|
\}$.
|
|
|
|
The forward transform computes a complex array~$Y$, whose
|
|
structure is the same as that of~$X$, defined by
|
|
|
|
$$
|
|
Y[k_1, k_2, \ldots, k_d] =
|
|
\sum_{j_1 = 0}^{n_1 - 1}
|
|
\sum_{j_2 = 0}^{n_2 - 1}
|
|
\cdots
|
|
\sum_{j_d = 0}^{n_d - 1}
|
|
X[j_1, j_2, \ldots, j_d]
|
|
\omega_1^{-j_1 k_1}
|
|
\omega_2^{-j_2 k_2}
|
|
\cdots
|
|
\omega_d^{-j_d k_d} \ .
|
|
$$
|
|
|
|
The backward transform computes
|
|
$$
|
|
Y[k_1, k_2, \ldots, k_d] =
|
|
\sum_{j_1 = 0}^{n_1 - 1}
|
|
\sum_{j_2 = 0}^{n_2 - 1}
|
|
\cdots
|
|
\sum_{j_d = 0}^{n_d - 1}
|
|
X[j_1, j_2, \ldots, j_d]
|
|
\omega_1^{j_1 k_1}
|
|
\omega_2^{j_2 k_2}
|
|
\cdots
|
|
\omega_d^{j_d k_d} \ .
|
|
$$
|
|
|
|
Computing the forward transform followed by the backward transform
|
|
will multiply the array by $\prod_{s=1}^{d} n_d$.
|
|
@end tex
|
|
|
|
@cindex r2c
|
|
The definition of FFTW's multi-dimensional DFT of real data (r2c)
|
|
deserves special attention. In this case, we logically compute the full
|
|
multi-dimensional DFT of the input data; since the input data are purely
|
|
real, the output data have the Hermitian symmetry and therefore only one
|
|
non-redundant half need be stored. More specifically, for an @ndims multi-dimensional real-input DFT, the full (logical) complex output array
|
|
@tex
|
|
$Y[k_0, k_1, \ldots, k_{d-1}]$
|
|
@end tex
|
|
@html
|
|
<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
|
|
<i>k</i><sub><i>d-1</i></sub>]
|
|
@end html
|
|
@ifinfo
|
|
Y[k[0], k[1], ..., k[d-1]]
|
|
@end ifinfo
|
|
has the symmetry:
|
|
@tex
|
|
$$
|
|
Y[k_0, k_1, \ldots, k_{d-1}] = Y[n_0 - k_0, n_1 - k_1, \ldots, n_{d-1} - k_{d-1}]^*
|
|
$$
|
|
@end tex
|
|
@html
|
|
<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
|
|
<i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
|
|
<i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
|
|
<i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
|
|
@end html
|
|
@ifinfo
|
|
Y[k[0], k[1], ..., k[d-1]] = Y[n[0] - k[0], n[1] - k[1], ..., n[d-1] - k[d-1]]*
|
|
@end ifinfo
|
|
(where each dimension is periodic). Because of this symmetry, we only
|
|
store the
|
|
@tex
|
|
$k_{d-1} = 0 \cdots n_{d-1}/2$
|
|
@end tex
|
|
@html
|
|
<i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
|
|
@end html
|
|
@ifinfo
|
|
k[d-1] = 0...n[d-1]/2
|
|
@end ifinfo
|
|
elements of the @emph{last} dimension (division by @math{2} is rounded
|
|
down). (We could instead have cut any other dimension in half, but the
|
|
last dimension proved computationally convenient.) This results in the
|
|
peculiar array format described in more detail by @ref{Real-data DFT
|
|
Array Format}.
|
|
|
|
The multi-dimensional c2r transform is simply the unnormalized inverse
|
|
of the r2c transform. i.e. it is the same as FFTW's complex backward
|
|
multi-dimensional DFT, operating on a Hermitian input array in the
|
|
peculiar format mentioned above and outputting a real array (since the
|
|
DFT output is purely real).
|
|
|
|
We should remind the user that the separable product of 1d transforms
|
|
along each dimension, as computed by FFTW, is not always the same thing
|
|
as the usual multi-dimensional transform. A multi-dimensional
|
|
@code{R2HC} (or @code{HC2R}) transform is not identical to the
|
|
multi-dimensional DFT, requiring some post-processing to combine the
|
|
requisite real and imaginary parts, as was described in @ref{The
|
|
Halfcomplex-format DFT}. Likewise, FFTW's multidimensional
|
|
@code{FFTW_DHT} r2r transform is not the same thing as the logical
|
|
multi-dimensional discrete Hartley transform defined in the literature,
|
|
as discussed in @ref{The Discrete Hartley Transform}.
|
|
|