3.8.4 Fourier TransformsA common operation in analyzing various kinds of data is to find the Fourier transform, or spectrum, of a list of values. The idea is typically to pick out components of the data with particular frequencies, or ranges of frequencies.
Fourier[{ ,  , ... ,  }] | Fourier transform | InverseFourier[{ ,  , ... ,  }] | inverse Fourier transform |
Fourier transforms. | Here is some data, corresponding to a square pulse. | |
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{-1, -1, -1, -1, 1, 1, 1, 1}
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| Here is the Fourier transform of the data. It involves complex numbers. | |
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| Here is the inverse Fourier transform. | |
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InverseFourier[%]
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| Fourier works whether or not your list of data has a length which is a power of two. | |
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Fourier[{1, -1, 1}]
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| This generates a length-200 list containing a periodic signal with random noise added. | |
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data = Table[ N[Sin[30 2 Pi n/200] + (Random[ ] - 1/2)],
{n, 200} ] ;
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| The data looks fairly random if you plot it directly. | |
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ListPlot[ data, PlotJoined -> True ]
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In Mathematica, the discrete Fourier transform  of a list  of length  is by default defined to be  . Notice that the zero frequency term appears at position 1 in the resulting list. The inverse discrete Fourier transform  of a list  of length  is by default defined to be  . In different scientific and technical fields different conventions are often used for defining discrete Fourier transforms. The option FourierParameters in Mathematica allows you to choose any of these conventions you want.
| common convention | setting | discrete Fourier transform | inverse discrete Fourier transform | | Mathematica default | {0, 1} |  |  | | data analysis | {-1, 1} |  |  | | signal processing | {1, -1} |  |  | | general case | {a, b} |  |  |
Typical settings for FourierParameters with various conventions.
Fourier[{{ ,  , ... }, { ,  , ... }, ... }]
| | two-dimensional Fourier transform |
Two-dimensional Fourier transform. Mathematica can find Fourier transforms for data in any number of dimensions. In  dimensions, the data is specified by a list nested  levels deep. Two-dimensional Fourier transforms are often used in image processing.
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, and a symmetric peak at 
, reflecting the frequency component of the original signal near 
. 
