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1 <html lang="en">
2 <head>
3 <title>Introduction - FFTW 3.3.3</title>
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12 This manual is for FFTW
13 (version 3.3.3, 25 November 2012).
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15 Copyright (C) 2003 Matteo Frigo.
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17 Copyright (C) 2003 Massachusetts Institute of Technology.
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48 <a name="Introduction"></a>
49 <p>
50 Next:&nbsp;<a rel="next" accesskey="n" href="Tutorial.html#Tutorial">Tutorial</a>,
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55
56 <h2 class="chapter">1 Introduction</h2>
57
58 <p>This manual documents version 3.3.3 of FFTW, the
59 <em>Fastest Fourier Transform in the West</em>. FFTW is a comprehensive
60 collection of fast C routines for computing the discrete Fourier
61 transform (DFT) and various special cases thereof.
62 <a name="index-discrete-Fourier-transform-1"></a><a name="index-DFT-2"></a>
63 <ul>
64 <li>FFTW computes the DFT of complex data, real data, even-
65 or odd-symmetric real data (these symmetric transforms are usually
66 known as the discrete cosine or sine transform, respectively), and the
67 discrete Hartley transform (DHT) of real data.
68
69 <li>The input data can have arbitrary length.
70 FFTW employs <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithms for all lengths, including
71 prime numbers.
72
73 <li>FFTW supports arbitrary multi-dimensional data.
74
75 <li>FFTW supports the SSE, SSE2, AVX, Altivec, and MIPS PS instruction
76 sets.
77
78 <li>FFTW includes parallel (multi-threaded) transforms
79 for shared-memory systems.
80 <li>Starting with version 3.3, FFTW includes distributed-memory parallel
81 transforms using MPI.
82 </ul>
83
84 <p>We assume herein that you are familiar with the properties and uses of
85 the DFT that are relevant to your application. Otherwise, see
86 e.g. <cite>The Fast Fourier Transform and Its Applications</cite> by E. O. Brigham
87 (Prentice-Hall, Englewood Cliffs, NJ, 1988).
88 <a href="http://www.fftw.org">Our web page</a> also has links to FFT-related
89 information online.
90 <a name="index-FFTW-3"></a>
91 <!-- TODO: revise. We don't need to brag any longer -->
92 <!-- FFTW is usually faster (and sometimes much faster) than all other -->
93 <!-- freely-available Fourier transform programs found on the Net. It is -->
94 <!-- competitive with (and often faster than) the FFT codes in Sun's -->
95 <!-- Performance Library, IBM's ESSL library, HP's CXML library, and -->
96 <!-- Intel's MKL library, which are targeted at specific machines. -->
97 <!-- Moreover, FFTW's performance is @emph{portable}. Indeed, FFTW is -->
98 <!-- unique in that it automatically adapts itself to your machine, your -->
99 <!-- cache, the size of your memory, your number of registers, and all the -->
100 <!-- other factors that normally make it impossible to optimize a program -->
101 <!-- for more than one machine. An extensive comparison of FFTW's -->
102 <!-- performance with that of other Fourier transform codes has been made, -->
103 <!-- and the results are available on the Web at -->
104 <!-- @uref{http://fftw.org/benchfft, the benchFFT home page}. -->
105 <!-- @cindex benchmark -->
106 <!-- @fpindex benchfft -->
107
108 <p>In order to use FFTW effectively, you need to learn one basic concept
109 of FFTW's internal structure: FFTW does not use a fixed algorithm for
110 computing the transform, but instead it adapts the DFT algorithm to
111 details of the underlying hardware in order to maximize performance.
112 Hence, the computation of the transform is split into two phases.
113 First, FFTW's <dfn>planner</dfn> &ldquo;learns&rdquo; the fastest way to compute the
114 transform on your machine. The planner
115 <a name="index-planner-4"></a>produces a data structure called a <dfn>plan</dfn> that contains this
116 <a name="index-plan-5"></a>information. Subsequently, the plan is <dfn>executed</dfn>
117 <a name="index-execute-6"></a>to transform the array of input data as dictated by the plan. The
118 plan can be reused as many times as needed. In typical
119 high-performance applications, many transforms of the same size are
120 computed and, consequently, a relatively expensive initialization of
121 this sort is acceptable. On the other hand, if you need a single
122 transform of a given size, the one-time cost of the planner becomes
123 significant. For this case, FFTW provides fast planners based on
124 heuristics or on previously computed plans.
125
126 <p>FFTW supports transforms of data with arbitrary length, rank,
127 multiplicity, and a general memory layout. In simple cases, however,
128 this generality may be unnecessary and confusing. Consequently, we
129 organized the interface to FFTW into three levels of increasing
130 generality.
131 <ul>
132 <li>The <dfn>basic interface</dfn> computes a single
133 transform of contiguous data.
134 <li>The <dfn>advanced interface</dfn> computes transforms
135 of multiple or strided arrays.
136 <li>The <dfn>guru interface</dfn> supports the most general data
137 layouts, multiplicities, and strides.
138 </ul>
139 We expect that most users will be best served by the basic interface,
140 whereas the guru interface requires careful attention to the
141 documentation to avoid problems.
142 <a name="index-basic-interface-7"></a><a name="index-advanced-interface-8"></a><a name="index-guru-interface-9"></a>
143
144 <p>Besides the automatic performance adaptation performed by the planner,
145 it is also possible for advanced users to customize FFTW manually. For
146 example, if code space is a concern, we provide a tool that links only
147 the subset of FFTW needed by your application. Conversely, you may need
148 to extend FFTW because the standard distribution is not sufficient for
149 your needs. For example, the standard FFTW distribution works most
150 efficiently for arrays whose size can be factored into small primes
151 (2, 3, 5, and 7), and otherwise it uses a
152 slower general-purpose routine. If you need efficient transforms of
153 other sizes, you can use FFTW's code generator, which produces fast C
154 programs (&ldquo;codelets&rdquo;) for any particular array size you may care
155 about.
156 <a name="index-code-generator-10"></a><a name="index-codelet-11"></a>For example, if you need transforms of size
157 513&nbsp;=&nbsp;19*3<sup>3</sup>,you can customize FFTW to support the factor 19 efficiently.
158
159 <p>For more information regarding FFTW, see the paper, &ldquo;The Design and
160 Implementation of FFTW3,&rdquo; by M. Frigo and S. G. Johnson, which was an
161 invited paper in <cite>Proc. IEEE</cite> <b>93</b> (2), p. 216 (2005). The
162 code generator is described in the paper &ldquo;A fast Fourier transform
163 compiler&rdquo;,
164 <a name="index-compiler-12"></a>by M. Frigo, in the <cite>Proceedings of the 1999 ACM SIGPLAN Conference
165 on Programming Language Design and Implementation (PLDI), Atlanta,
166 Georgia, May 1999</cite>. These papers, along with the latest version of
167 FFTW, the FAQ, benchmarks, and other links, are available at
168 <a href="http://www.fftw.org">the FFTW home page</a>.
169
170 <p>The current version of FFTW incorporates many good ideas from the past
171 thirty years of FFT literature. In one way or another, FFTW uses the
172 Cooley-Tukey algorithm, the prime factor algorithm, Rader's algorithm
173 for prime sizes, and a split-radix algorithm (with a
174 &ldquo;conjugate-pair&rdquo; variation pointed out to us by Dan Bernstein).
175 FFTW's code generator also produces new algorithms that we do not
176 completely understand.
177 <a name="index-algorithm-13"></a>The reader is referred to the cited papers for the appropriate
178 references.
179
180 <p>The rest of this manual is organized as follows. We first discuss the
181 sequential (single-processor) implementation. We start by describing
182 the basic interface/features of FFTW in <a href="Tutorial.html#Tutorial">Tutorial</a>.
183 Next, <a href="Other-Important-Topics.html#Other-Important-Topics">Other Important Topics</a> discusses data alignment
184 (see <a href="SIMD-alignment-and-fftw_005fmalloc.html#SIMD-alignment-and-fftw_005fmalloc">SIMD alignment and fftw_malloc</a>),
185 the storage scheme of multi-dimensional arrays
186 (see <a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>), and FFTW's mechanism for
187 storing plans on disk (see <a href="Words-of-Wisdom_002dSaving-Plans.html#Words-of-Wisdom_002dSaving-Plans">Words of Wisdom-Saving Plans</a>). Next,
188 <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a> provides comprehensive documentation of all
189 FFTW's features. Parallel transforms are discussed in their own
190 chapters: <a href="Multi_002dthreaded-FFTW.html#Multi_002dthreaded-FFTW">Multi-threaded FFTW</a> and <a href="Distributed_002dmemory-FFTW-with-MPI.html#Distributed_002dmemory-FFTW-with-MPI">Distributed-memory FFTW with MPI</a>. Fortran programmers can also use FFTW, as described in
191 <a href="Calling-FFTW-from-Legacy-Fortran.html#Calling-FFTW-from-Legacy-Fortran">Calling FFTW from Legacy Fortran</a> and <a href="Calling-FFTW-from-Modern-Fortran.html#Calling-FFTW-from-Modern-Fortran">Calling FFTW from Modern Fortran</a>. <a href="Installation-and-Customization.html#Installation-and-Customization">Installation and Customization</a> explains how to
192 install FFTW in your computer system and how to adapt FFTW to your
193 needs. License and copyright information is given in <a href="License-and-Copyright.html#License-and-Copyright">License and Copyright</a>. Finally, we thank all the people who helped us in
194 <a href="Acknowledgments.html#Acknowledgments">Acknowledgments</a>.
195
196 </body></html>
197