Question

Find the fourier transform of an exponential signal f(t) = e^-at u(t), a>0.

(a) \(\frac{1}{a+jω}\)

(b) \(\frac{1}{a-jω}\)

(c) \(\frac{1}{-a+jω}\)

(d) \(\frac{1}{-a-jω}\)

I got this question by my college director while I was bunking the class.

This key question is from Fourier Transforms topic in division Fourier Transform of Signals and Systems

Answer 1

Right choice is (a) \(\frac{1}{a+jω}\)

The explanation is: Given  f(t)= e^-at u(t)

We know that  \(

u(t)=\begin{cases}

0 &\text{\(t<0\)} \\

1 &\text{\(t>0\)} \\

\end{cases}\)

Fourier transform,

\(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt = \int_{-∞}^∞ e^{-at} u(t)e^{-jωt} \,dt  = \int_0^∞ e^{-(a+jω)t} \,dt\)

F(ω) = \(\frac{1}{a+jω}\), a>0.