GATE Electrical Engineering (EE) 2021 Solved Papers

© examsiri.com

Question : 62 of 65

Marks: +1, -0

Let

{f}( {t})

be an even function, i.e.

{f}(- {t})= {f}( {t})

for all

{t}

. Let the Fourier transform of

{f}( {t})

be defined as

F(ω)=∫↙{-x}↖{∞} f(t) e^{-\;\text "jot "\;} d t .

Suppose

\;{d F(ω)}/{d ω}=-ω F(ω)

for all

ω

, and

F(0)=1 .

Then

$f(0)<1$
${f}(0)>1$
${f}(0)=0$
$f(0)=1$

Solution:

Given,

\;{d}/{d ω} F(ω)=-ω F(ω)

⇒ \;{d F(ω)}/{F(ω)}=-ω d ω

∫ \;{d F(ω)}/{F(ω)}=-∫ ω d ω

⇒ \ln F(ω)=-\;{ω^{2}}/{2}+C

∵ F(0)=1 ⇒ ∴ C=0

\ln F(ω)=-\;{ω^{2}}/{2}+0

F(ω)=e^{-\;{ω^{2}}/{2}}

∵ e^{-a t^{2}} {<────────>}↖{\text "Fourier Transform"}√{\;{π}/{a}} e^{-\;{ω^{2}}/{4 a}}

Put,

a=\;{1}/{2}

e^{-\;{t^{2}}/{2}} {<────────>}↖{\text "Fourier Transform"}√{2 π} e^{-\;{ω^{2}}/{2}}

f(t)=\;{1}/{√{2 π}} e^{-\;{t^{2}}/{2}} {<────────>}↖{\text "Fourier Transform"}F(ω)=e^{-\;{ω^{2}}/{2}}

∴ f(t)=\;{1}/{√{2 π}} e^{-\;{t^{2}}/{2}}

f(0)=\;{1}/{√{2 π}} e^{0}=\;{1}/{√{2 π}}=0.4<1

∴ f(0)<1

© examsiri.com

Go to Question:

Prev Question

Next Question