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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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 />
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<strong>The</strong> <strong>Fourier</strong> <strong>Transform</strong> <strong>and</strong> <strong>its</strong> Applications, Jürgen Stutzki, Sommersemester 2007<br />
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�<br />
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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 />
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�<br />
�<br />
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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 />
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� �<br />
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q.e.d.<br />
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math_ground_8.odt<br />
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