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What is the Fourier Transform of : $\sum_{n=1}^N A_ne^{\large-a_nt} u(t)~?$

This is a time domain function, how can I find its Fourier Transform (continuous not discrete) ?

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    Perfect. That is the correct expression. Thanks2012-06-28

1 Answers 1

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Tips:

  1. The Fourier transform is linear; $\mathcal{F}\left\{\sum_l a_lf_l(t)\right\}=\sum_l a_l\mathcal{F}\{f_l(t)\}.$
  2. Plug $e^{-ct}u(t)$ into $\mathcal{F}$ and then discard part of the region of integration ($u(t)=0$ when $t<0$):

$\int_{-\infty}^\infty e^{-ct}u(t)e^{-2\pi i st}dt=\int_0^\infty e^{(c-2\pi is)t}dt=? $

Now put these two together..

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    Its true, I can not "evaluate" the summation because is until "N" so I have to leave the expression with the integral result. This is just part of one exercise. The whole problem consist in multiply this Fourier Transform with transfer function (in Fourier Transform) that I found. Thanks anon2012-06-28