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The Fourier Transform and its applications goals: present the Fourier ...

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to summarize:<br />

Symmetry properties of FT<br />

or<br />

f �x � = e � x � � o� x � = ℜ f � x � � i ℑ f �x � = h � x � � a � x �<br />

�<br />

�<br />

�<br />

�<br />

<strong>The</strong> <strong>Fourier</strong> <strong>Transform</strong> <strong>and</strong> <strong>its</strong> Applications, Jürgen Stutzki, Sommersemester 2007<br />

�<br />

�<br />

�<br />

�<br />

F �s� = E �S � � O �s� = H �s� � A �s� = ℜ F �s� � i ℑ F �s�<br />

proof: e.g.<br />

f �x � = ℜe � x � � i ℑ e � x � � ℜ o� x � � i ℑo � x �<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

F �s� = ℜ E �s� � i ℑ E �s� � i ℑO �S � � ℜO�s�<br />

FT �e� x��= � e� x�= 1�<br />

1<br />

� f �x�� f �−x��=<br />

2<br />

2 ��f � x���f �−x��<br />

= 1<br />

�∞<br />

2 [∫ e<br />

−∞<br />

−2�i xs �∞<br />

f � x� dx�∫ e<br />

−∞<br />

−2� i xs ] f �−x� dx<br />

= 1 [ 2 F �S ��∫<br />

�∞<br />

e<br />

−∞<br />

−2� i u�−s� u] f �u� d<br />

= 1 [ F �s��F �−s�]=E �s�<br />

2<br />

�<br />

�<br />

�<br />

�<br />

� �<br />

�<br />

�<br />

q.e.d.<br />

�<br />

�<br />

math_ground_8.odt<br />

Page 18

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