$Let X(j ω) denote the Fourier transform of the signal x(t) shown in the figure below, Then the value of x (t) = \int_{- \infty}^{\infty} X (j ω) d ω is_____$

12.57

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Solution

The correct option is A 12.57

$By the definition of Fourier transform, X (j ω) = \int_{- \infty}^{\infty} x (t) . e^{- j ω t} d t x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (j ω) . e^{j ω t} d ω at t = 0, x (0) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (j ω) d ω ∴ \int_{- \infty}^{\infty} X (j ω) d ω = 2 π x (0) = 2 π (2) = 4 π = 12.57$

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