Fourier transform: Difference between revisions

Revision as of 23:23, 19 May 2002

Revision as of 09:41, 20 May 2002

Line 20:

<table border="1">

<tr>

<tdth>&nbsp;</tdth>

<th align=left>Signal</th>

<th align=left>Fourier transform</th>

Line 25 ⟶ 26:

</tr>

<tr>

<td>&nbsp;1.</td>

<td>''af''(''t'') + ''bg''(''t'')</td>

<td>''aF''(''s'') + ''bG''(''s'')</td>

Line 30 ⟶ 32:

</tr>

<tr>

<td>&nbsp;2.</td>

<td>''f''(''t'' - ''a'')</td>

<td>e^{-2&pi;''ias''}'' F''(''s'')</td>

<td>Shift in time ___domain</td>

</tr>

<tr>

<td>&nbsp;3.</td>

<td>e^2&pi;''iat''''f''(''t'')</td>

<td>''F''(''s''-''a'')</td>

<td>Shift in frequency ___domain</td>

</tr>

<tr>

<td>&nbsp;4.</td>

<td>''f''(''at'')</td>

<td>1/|''a''| ''F''(''s''/''a'') / |''a''|</td>

<td>If ''a'' is large, then ''f''(''at'') is concentrated around 0 and ''F''(''s''/''a'')/|''a''| spreads out and flattens</td>

<td>&nbsp;</td>

</tr>

<tr>

<td>5.</td>
<td><em>f</em>&nbsp;'(''t'')</td>

<td>2&pi;''is'' ''F''(''s'')</td>

<td><em>f</em>&nbsp;'(''t'') is the (distribution) derivative of ''f''(''t'')</td>

</tr>

<tr>

<td>6.</td>

<td>''t'' ''f''(''t'')</td>

<td>1/(2&pi;''i'') <em>F</em>'(-''s'')</td>

<td>&nbsp;This is the inverse rule to 5.</td>

</tr>

<tr>

<td>7.</td>

<td>(''f'' * ''g'')(''t'')</td>

<td>''F''(''s'') ''G''(''s'')</td>

Line 60 ⟶ 68:

</tr>

<tr>

<td>8.</td>

<td>''f''(''t'') ''g''(''t'')</td>

<td>(''F'' * ''G'')(-''s'')</td>

<td>&nbsp;This is the inverse of 7.</td>

</tr>

<tr>

<td>9.</td>

<td>&delta;(''t'')</td>

<td>1</td>

<td>&delta;(''t'') denotes the [[Dirac delta]] distribution.</td>

</tr>

<tr>

<td>10.</td>

<td>1</td>

<td>&delta;(''s'')</td>

<td>Inverse of 9. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of everyday functions.</td>

<td>&nbsp;</td>

</tr>

<tr>

<td>&delta;(''t''-''a'')</td>

<td>e^{-2&pi;''ias''}</td>

<td>&nbsp;</td>

</tr>

<tr>

<td>11.</td>

<td>''t''^''n''</td>

<td>(-1)^{''n''(''n''+1)/2}/(-2&pi;''i'')^''n'' &delta^(''n'')((-1)^''n''''s'')</td>

<td>(Here, ''n''needs tois bea checked'')[[natural number]]. &delta^(''n'')(''s'') is the ''n''-th distribution derivative of the Dirac delta. This rule follows from rules 6. and 10. Combining this rule with 1., we can transform all [[polynomial]]s.</td>

</tr>

<tr>

<td>12.</td>

<td>e^2&pi;''iat''</td>

<td>&delta;(''s''-''a'')</td>

<td>&nbsp;This follows from and 3. and 10.</td>

</tr>

<tr>

<td>13.</td>

<td>cos(2&pi;''at'')</td>

<td>1/2 ( &delta; (''s'' - ''a'') + &delta;(''s'' + ''a'') )</td>

<td>Follows from rules 1 and 13 using cos(2&pi;''at'') = 1/2 ( e^2&pi;''iat'' + e^{-2&pi;''iat''} ) ([[Eulers formula in complex analysis|Euler's formula]])</td>

<td>&nbsp;</td>

</tr>

<td>14.</td>

<td>sin(2&deltapi;(''t''-''aat'')</td>

<td>1/(2''i'') ( &delta; (''s'' - ''a'') - &delta;(''s'' + ''a'') )</td>

<td>Also from 1 and 12.</td>

</tr>

<tr>

<td>15.</td>

<td>exp(-''a'' ''t''²)</td>

<td>(&pi;/''a'')^1/2 exp(-&pi;² ''s''² / ''a'')</td>

<td>Shows that the [[Normal distribution|Gaussian function]] exp(-&pi; ''t''²) is its own Fourier-transform</td>

<td>&nbsp;</td>

</tr>

</table>