GATE Electronics and Communications (EC) 2020 Solved Papers

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Question : 16 of 65

Marks: +1, -0

The random variable

{Y}=∫↙{-∞}↖{∞} {W}( {t}) φ( {t}) {dt}

Where

φ(t)=\{\table 1 \text ";" 5 ≤ t ≤ 7 ; 0 \;\text "; otherwise "\;.

And

W(t)

is a real white Gaussian noise process with two-sided power spectral density

{SW}( {f})=3 {W} {/} {Hz}

, for all

{f}

. The variance of

{Y}

is

Your Answer:

Solution:

Given

Y=∫↙{-∞}↖{∞} w(t) s(t) d t

, where

φ(t)=\{\table 1 , 5 ≤ t ≤ 7 ; 0 , \;\text " otherwise "\;.

{S}_{ {w}}( {f})=3 {W} {/} {Hz}

{E}( {Y})=∫↙{-∞}↖{∞} {E}( {w}( {t})) φ( {t}) {dt}=0

{E}[ {Y}^{2}]= {S}_{ {w}}( {f}) \;\; \;\text " energy "\; ϕ( {t})

=6

Var[ {Y}]=6-0=6

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