8/20/2019 SOLUTION FOURIER PROBLEM

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Question: What is the inverse Fourier transform for )()()(   *21         j X  j X  j  n

?

Solution:

Let )(t  x and )(     j X   be Fourier Transform pairs. So: )()(     j X t  x    

By time reversal property [1]:   )()(     j X t  x    

By complex conjugation property [1]: )()(   **   j X t  x    

For real-valued signal we have: )()(   * t  xt  x   .

Hence we have )()(   *      j X  j X     or )()(*

     j X  j X    .

Thus for any real signal )(t  x , )()(   *   j X t  x   .

Using the convolution theorem [1]:

)()()()( 21*

21   t  xt  x j X  j X        

Finally using the time-derivative property [1]:

)]()([)1()()()( 21*

21   t  xt  xdt 

d  j X  j X  j

n

nnn

       

[1] Web link: http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html