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

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and last: f �x � even, i.e. f �x�= f �−x� : we get F �s�=∫ e −∞ −2� i x s f � x� d x = � The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 �∞ �∞ =∫ e −∞ −2� i u�−s � Thus: f �u� du=F �−s� f �x � even ⇔ F �s� even and vice versa f �x � odd ⇔ F �s� odd −∞ �∞ �∞| u=−x |−∞ ;du=−dx −∞ −∫ e �∞ 2� i us f �u� d u , math_ground_6.odt Page 16
Note: any function f �x � can be separated into its odd and even parts, o � x� and e � x � : with e � x �=e �−x � , o � x�=−o�−x � we have: f � x� = e� x ��o �x � f �−x� = e� x �−o �x � The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 , and hence e �x � = 1 2 similarly: separation into real and imaginary part: f �x� = ℜ f � x��i ℑ f � x� f * �x� = ℜ f � x�−i ℑ f � x� o � x � = 1 2 � f �x �� f �−x�� . � f �x �− f �−x�� , and ℜ f � x� = 1 2 � f � x�� f * � x�� ℑ f � x� = 1 2i � f � x�− f * � x�� With this, we can separate the hermitean and anti-hermitean parts: so that f �x �=e � x ��o� x �=ℜe � x��i ℑ e � x ��ℜ o� x ��i ℑo � x � =[ ℜe � x ��i ℑo � x �] hermitean anti−hermitean �[ ℜo �x ��i ℑ e � x� ] =h� x ��a � x � f * �−x �= =[ ℜe � x �−�−�i ℑ o �x �]�[−ℜ o� x �−i ℑ e �x �]=h � x �−a �x � h� x� = 1 2 � f � x�� f * �−x�� a� x� = 1 2 � f � x�− f * �−x�� math_ground_7.odt Page 17
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: 1.3 properties of the FT 1.3.0 line
Page 19 and 20: 1.3.2 FT of the complex conjugate f
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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