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Let $f(t)$ be continuous time signal and let $F(w)$be its Fourier Transform defined by

                             $F(\omega )=\displaystyle{}\int_{-\infty }^{\infty }f(t)e^{-j\omega t} dt$

define $g(t)$ by

                              $g(t)=\displaystyle{}\int_{-\infty }^{\infty }F(u)e^{-jut} du$

What is the relationship between $f(t)$ and $g(t)?$

  1. $g(t)$ would always be proportional to $f(t)$.
  2. $g(t)$ would be proportional to $f(t)$ if $f(t)$ is an even function.
  3. $g(t)$ would be proportional to $f(t)$ only if $f(t)$ is a sinusoidal function.
  4. $g(t)$ would never be proportional to $f(t)$.
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