Consider the signal {f}( {t})=1+2 \cos (\pi {t})+3 \sin (\frac{2 \pi}{3} t)+4 \cos (\frac{\pi}{2} t+\frac{\pi}{4}), where t is in seconds. Its fundamental time period, in seconds, is__________

GATE Electronics and Communications (EC) 2019 Solved Paper

Question : 33 of 65

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Consider the signal

{f}( {t})=1+2 \cos (π {t})+3

\sin (\;{2 π}/{3} t)+4 \cos (\;{π}/{2} t+\;{π}/{4})

, where

t

is in seconds. Its fundamental time period, in seconds, is__________

Solution:

DC value and phase shift does not affect time period of a signal.
So it is equivalent to find time period of

x(t)=2 \cos (π t)+3 \sin (\;{2 π}/{3})+4 \cos (\;{π}/{2} t)

∴ ω_{1}=π \;\; {T}_{1}=\;{2 π}/{ω_{1}}=2

second

ω_{2}=\;{2 π}/{3} \;\; {T}_{2}=\;{2 π}/{ω_{2}}=3 {second}

ω_{3}=\;{π}/{2} \;\; {T}_{3}=\;{2 π}/{ω_{3}}=4

second
Now overall

{T}= {LCM}( {T}_{1}, {T}_{2}, {T}_{3})

= {\LCM}(2,3,4)

∴

overall

{T}=12

seconds

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