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

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to summarize: Symmetry properties of FT or f �x � = e � x � � o� x � = ℜ f � x � � i ℑ f �x � = h � x � � a � x � � � � � The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 � � � � F �s� = E �S � � O �s� = H �s� � A �s� = ℜ F �s� � i ℑ F �s� proof: e.g. f �x � = ℜe � x � � i ℑ e � x � � ℜ o� x � � i ℑo � x � � � � � � � F �s� = ℜ E �s� � i ℑ E �s� � i ℑO �S � � ℜO�s� FT �e� x��= � e� x�= 1� 1 � f �x�� f �−x��= 2 2 ��f � x��f �−x�� = 1 �∞ 2 [∫ e −∞ −2�i xs �∞ f � x� dx�∫ e −∞ −2� i xs ] f �−x� dx = 1 [ 2 F �S ��∫ �∞ e −∞ −2� i u�−s� u] f �u� d = 1 [ F �s��F �−s�]=E �s� 2 � � � � � � � � q.e.d. � � math_ground_8.odt Page 18
1.3.2 FT of the complex conjugate function for the Fourier Transform of f * � x � we get: FT � f * � x ��= � f * �x �=∫e −2�i xs f * �x � dx −2� i x �−s =∫ �] * [e f * � x � dx=F * �−s� with this, and with FT � f �−x ��=F �−s� , we can derive all the above symmetry relations: e.g. FT � ℜ f � x ��=FT� 1 2 [ f � x �� f * 1 � x �]� = 2 [ F �S ��F * �−s�]=H �s� , or etc. FT �a� x��=FT � 1 2 [ f � x�− f * �−x�]� =1 2 [ F �S�− F* �s�]=i ℑ F �s� Thus, all symmetry relations above can be derived from the (easy to memorize) three relations: 1) FT � f � x ��=F �s� 2) FT � f �−x ��=F �−s� 3) FT � f * � x ��=F * �−s� The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 math_ground_9.odt Page 19
Page 1 and 2: The Fourier Transform and its appli
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: Note: any function f �x � can b
Page 21 and 22: F �s�=FT � f � x ��=CT
Page 23 and 24: FT � f � x � H � x ��=C
Page 25 and 26: The Fourier Transform and its Appli

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