Question

Find the fourier transform of the function f(t) = e^-a|t|, a>0.

(a) \(\frac{2a}{a^2-ω^2}\)

(b) \(\frac{2a}{a^2+ω^2}\)

(c) \(\frac{2a}{ω^2-a^2}\)

(d) \(\frac{a}{a^2+ω^2}\)

This question was addressed to me in examination.

I need to ask this question from Fourier Transforms topic in section Fourier Transform of Signals and Systems

Answer 1

Correct answer is (b) \(\frac{2a}{a^2+ω^2}\)

The explanation is: The given two-sided exponential function f(t) = e^-a|t|, a>0  can be expressed as

\(

f(t)=\begin{cases}

e^{-at} &\text{\(t≥0\)} \\

e^{at} &\text{\(t≤0\)} \\

\end{cases}\)

The Fourier transform is

\(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt = \int_{-∞}^0 f(t)e^{-jωt} \,dt + \int_0^∞ f(t)e^{-jωt} \,dt\)

\(F(ω) = \frac{1}{a+jω} + \frac{1}{a-jω} = \frac{2a}{a^2+ω^2}\).