The Fourier transform of a double-sided exponential signal

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The Fourier transform of a double-sided exponential signal x(t) = e^{– b|t|}

1. 2b
  b² + ω²
2. e^{– j tan– 1} bt
  ω
3. does not exist
4. exists only when it is single sided

Correct Option: A

Fourier transform of signal x(t) is given as
x(jω) = ∫^∞_{– ∞} x(t) e^{– jωt} dt
or x(jω) = ∫^∞_{– ∞} e^{– b|t|} .e^{– jωt} dt
x(jω) = ∫⁰_{– ∞} e^bt.e^{– jωt} dt + ∫^∞₀ e^bt.e^{– jωt} dt

or x(jω) =		e^{(b – jω)t}		⁰	+		e^{– (b + jω)t}		^∞
		b – jω		_{– ∞}			– (b + jω)		₀

or x(jω) =	1	+	1
	b – jω		jω + b

or x(jω) =	2b
	b² + ω²

Hence, alternative (A) is the correct choice.

e^{– j tan– 1}		bt
		ω