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A fast Fourier transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where n is the data size. The difference in speed can be enormous, especially for long data sets where n may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.
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In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample.
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PROJ is a library for performing conversions between cartographic projections. The library is based on the work of Gerald Evenden at the United States Geological Survey (USGS), but since 2019-11-26 is an Open Source Geospatial Foundation (OSGeo) project maintained by the PROJ Project Steering Committee (PSC). The library also ships with executables for performing these transformations from the command line.
References
- Official website
- Swarztrauber, P.N. (1982). "Vectorizing the FFTs". In Rodrigue, Garry (ed.). Parallel Computations. Academic Press. pp. 51–83. ISBN 978-0-12-592101-5.
- Wendykie, Piotr. "JTransforms". Java. Open source.
Multithreaded FFT library
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