I have to find the fourier transform of $g(t)$. Picture is added at the link below.


I have tried this way \begin{align} G(f)&=\int ^{\infty }_{-\infty } g( t) \ e^{-j\omega t}\, dt\\ &=\int ^{\infty }_{-\infty } e^{-at} rect\left(\frac{t}{T} -\frac{1}{2}\right) e^{-j\omega t}\, dt\\ &=\int ^{T}_{0} e^{-at} e^{-j\omega t}\, dt\\ &=\frac{1-e^{-( a+j\omega ) T}}{a+j\omega } \end{align} where $$ rect(t) ~is ~unit ~rectangular ~function\\ rect(\frac{t}{T}-\frac{1}{2}) = \begin{gather*} \begin{cases} 1 & \text{for }0\le t\le T\\ 0 & \text{for }elsewhere \end{cases} \end{gather*} $$

But i am not sure whether it's right or not.

  • $\begingroup$ What is "rect"? $\endgroup$ Sep 2, 2018 at 6:41
  • $\begingroup$ Then I'd say that both "rect" and the second line of your calculation are unnecessary. $\endgroup$ Sep 2, 2018 at 6:55


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