The Fourier Transform and its applications goals: present the Fourier ...
The Fourier Transform and its applications goals: present the Fourier ...
The Fourier Transform and its applications goals: present the Fourier ...
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Summary:<br />
<strong>The</strong> <strong>Fourier</strong> <strong>Transform</strong> <strong>and</strong> <strong>its</strong> Applications, Jürgen Stutzki, Sommersemester 2007<br />
∞<br />
● <strong>Fourier</strong>-<strong>Transform</strong> F �s�=∫ f �x� e<br />
−∞<br />
−2� i xs ●<br />
dx (“ minus-i ” transform)<br />
∞<br />
<strong>Fourier</strong>-Backtransform f �x �=∫ F �s� e<br />
−∞<br />
2� i xs ●<br />
ds (“ plus-i “-transform)<br />
symmetry relations:<br />
f �x � = e � x � � o� x � = ℜ f � x � � i ℑ f �x � = h � x � � a � x �<br />
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F �s� = E �S � � O �s� = H �s� � A �s� = ℜ F �s� � i ℑ F �s�<br />
f �x � = ℜe � x � � i ℑ e � x � � ℜ o� x � � i ℑo � x �<br />
● Cosine-, <strong>and</strong> Sine-<strong>Transform</strong><br />
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F �s� = ℜ E �s� � i ℑ E �s� � i ℑO �S � � ℜO�s�<br />
∞<br />
CT � f � x ��=2∫ 0<br />
∞<br />
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f � x � cos�2� xs� dx<br />
ST � f � x ��=2∫ f � x� sin�2� xs� dx<br />
0<br />
which are <strong>the</strong>ir own back-transforms<br />
● <strong>Fourier</strong> <strong>Transform</strong> ⇔<br />
Cosine-<strong>Transform</strong> of <strong>the</strong> even <strong>and</strong> Sine-<strong>Transform</strong> of <strong>the</strong> odd part of f �x �<br />
F �s�=FT � f � x ��=CT �e �x �� – i ST �o �x ��<br />
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math_ground_15.odt<br />
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