X(\omega) is the Fourier transform of x(t) shown below. The value of \int_{-\infty}^{\infty}{|X(\omega)}|^{2} d \omega (rounded off to two decimal places) is ...........

GATE Electronics and Communications (EC) 2020 Solved Papers

Question : 47 of 65

Marks: +1, -0

X(ω)

is the Fourier transform of

x(t)

shown below. The value of

∫↙{-∞}↖{∞}{|X(ω)}|^{2} d ω

(rounded off to two decimal places) is ...........

Solution:

{E}=\;{1}/{2 π} ∫↙{-∞}↖{∞}{| {X}(ω)|}^{2} {d} ω=∫↙{-∞}↖{∞} {x}^{2}( {t}) {dt}

∫↙{-∞}↖{∞}{| {X}(ω)|}^{2} {d} ω=2 π ∫↙{-∞}↖{∞} {x}^{2}( {t}) {dt}

∫↙{-∞}↖{∞} x^{2}(t) d t=∫↙{-1}↖{0}(t+1)^{2} d t+∫↙{0}↖{1}(2 t+1)^{2} d t+∫↙{1}↖{2}(-2 t+5)^{2} d t+∫↙{2}↖{3}(-t+3)^{2} d t

=9.33

∫↙{-∞}↖{∞}{|X(ω)|}^{2}=2 π × 9.33=2 × 3.14 × 9.33=58.5924

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