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

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1.4 Sine- and Cosine-Transformation Recall Euler's relation: e i � =cos ��i sin � . Inserting this into the definition of the FT, we get: �∞ F �s�=FT � f �x��=∫ e −∞ −2� i xs f �x� dx=∫ cos�2� xs� f � x � dx – i ∫ sin�2� xs� f �x� dx −∞ −∞ ∞ =∫ 0 f � x�cos�2� xs� dx�∫ −∞ The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 0 �∞ f � x�cos�2� x s� d x ∞ −i∫ 0 f � x�sin�2� xs� dx−i ∫ −∞ f � x�sin�2� x s� d x Substituting in the second and forth expression: 0 0 | x �−u|∞ ; dx �−du , we get: ∞ =∫ 0 ∞ =∫ 0 f � x�cos�2� xs� dx�∫ 0 ∞ −∞ f �−u�cos�2�u s� d u ∞ −i∫ 0 [ f � x�� f �−x�] cos�2� xs� dx−i∫ 0 ∞ =2∫ 0 e� x� cos�2� xs� dx −i 2∫ 0 ∞ �∞ f � x�sin�2� xs� dx� i∫ 0 ∞ [ f �x�− f �−x�]sin�2� xs� dx o� x� sin�2� xs� dx We define: Cosine-Transformation CT � f � x ��=2∫ 0 Sine-Transformation ∞ ∞ 0 ∞ f �−u�sin�2�u s� d u f � x � cos�2� xs� dx , and ST � f � x��=2∫ f � x� sin�2� xs� dx 0 note: only x�0 counts in CT and ST ! math_ground_10.odt Page 20
F �s�=FT � f � x ��=CT �e �x �� – i ST �o �x �� i.e. Fourier Transform ⇔ Cosine-Transform of the even and Sine-Transform of the odd part of f �x � Note: as e � x � and o � x� need to be defined only for x�0 , the continuation to x�0 being given by symmetry ( e �−x �=e � x � ; o �−x �=−o� x � ), the Fourier transform carries the full information on f �x � , both for x�0 and x�0 , although the integration in the CT and ST only use e � x � and o � x� for x�0 . We can thus, like always, get f �x � back from F �s� through the "+i" transform, giving: f �x�=FT �i � F �s��=CT � E �s��i ST �O�s�� i.e. Fourier Backtransform ⇔ Cosine-Transform of the even and Sine-Transform of the odd part of F �s� Note: in the CT , ST -world (no FT known), given F �s� , calculate E �s� and O�s� , from this, throughCT resp. ST , calculate f �x � Symmetries of Cosine- and Sine-Transform: with F C �s�=CT � f �x �� , F S �s�=ST � f �x�� , we have: F C �−s�=F C �s� Cosine-Transform is even in s, F S �−s�=−F S � s� Sine-Transform is odd in s The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 math_ground_11.odt Page 21
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Page 7 and 8: The Fourier Transform and its Appli
Page 9 and 10: diffraction image of circular apert
Page 11 and 12: Remark: usual abbreviation: �=2
Page 13 and 14: 1.2. Existence of the FT intuitivel
Page 15 and 16: 1.3 properties of the FT 1.3.0 line
Page 17 and 18: Note: any function f �x � can b
Page 19: 1.3.2 FT of the complex conjugate f
Page 23 and 24: FT � f � x � H � x ��=C
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