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

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