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Table of common Fourier transforms |
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The most general and natural context for studying the continuous Fourier transform is given by the [[distribution|tempered distributions]]; these include all the integrable functions mentioned above and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution and the rule for the inverse of the Fourier transform is universally valid. Furthermore, the useful [[Dirac delta function|Dirac delta]] is a tempered distribution but not a function; its Fourier transform is the constant function 1. The above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.
The following table records some important Fourier transforms. ''F''(''s'') and ''G''(''s'') denote the Fourier transforms of ''f''(''t'') and ''g''(''t''), respectively. ''f'' and ''g'' may be integrable functions or tempered distributions.
<table border="1">
<tr>
<th align=left>Signal</th>
<th align=left>Fourier transform</th>
<th align=left>Remarks</th>
</tr>
<tr>
<td>''af''(''t'') + ''bg''(''t'')</td>
<td>''aF''(''s'') + ''bG''(''s'')</td>
<td>Linearity</td>
</tr>
<tr>
<td>''f''(''t'' - ''a'')</td>
<td>e<sup>-2π''ias''</sup>'' F''(''s'')</td>
<td>Shift</td>
</tr>
<tr>
<td>e<sup>2π''iat''</sup>''f''(''t'')</td>
<td>''F''(''s''-''a'')</td>
<td>Shift</td>
</tr>
<tr>
<td>''f''(''at'')</td>
<td>1/|''a''| ''F''(''s''/''a'')</td>
<td></td>
</tr>
<tr>
<td><em>f</em>'(''t'')</td>
<td>2π''is'' ''F''(''s'')</td>
<td><em>f</em>'(''t'') is the derivative of ''f''(''t'')</td>
</tr>
<tr>
<td>''t'' ''f''(''t'')</td>
<td>1/(2π''i'') <em>F</em>'(-''s'')</td>
<td></td>
</tr>
<tr>
<td>(''f'' * ''g'')(''t'')</td>
<td>''F''(''s'') ''G''(''s'')</td>
<td>''f'' * ''g'' denotes the convolution of ''f'' and ''g''</td>
</tr>
<tr>
<td>''f''(''t'') ''g''(''t'')</td>
<td>(''F'' * ''G'')(-''s'')</td>
<td></td>
</tr>
<tr>
<td>δ(''t'')</td>
<td>1</td>
<td>δ(''t'') denotes the [[Dirac delta]] distribution</td>
</tr>
<tr>
<td>1</td>
<td>δ(''s'')</td>
<td></td>
</tr>
<tr>
<td>δ(''t''-''a'')</td>
<td>e<sup>-2π''ias''</td>
<td></td>
</tr>
<tr>
<td>''t''</td>
<td>1/(2π''i'') &delta'(''s'')</td>
<td>&delta'(''s'') is the distribution derivative of the Dirac delta</td>
</tr>
<tr>
<td>e<sup>2π''iat''</td>
<td>δ(''s''-''a'')</td>
<td></td>
</tr>
<tr>
<td>cos(2π''at'')</td>
<td>1/2 ( δ (''s'' - ''a'') + δ(''s'' + ''a'') )</td>
<td></td>
</tr>
<tr>
<td>exp(-''a'' ''t''<sup>2</sup>)</td>
<td>(π/''a'')<sup>1/2</sup> exp(-π<sup>2</sup> ''s''<sup>2</sup> / ''a'')</td>
<td></td>
</tr>
See also: [[Fourier transform]], [[Fourier series]], [[Laplace transform]], [[Discrete Fourier transform]]
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