GATE-2016 (30)

GATE-2016

30. The time period of the signal s(t)= sin(π/3)t + cos(π/2)t is__________seconds.

Solution:

To identity the period ‘T’ , the frequency f=1/T or the angular

frequency ω = 2*Π*f = (2*Π)/ T of a given sinusoidal or complex

exponential signal, it is always helpful to write it in any of the

follwing

forms.

sin(ωt) = sin( 2*Π*f*t) = sin (2*Π*t/ T) ----------------(1)

The fundamental frequency of a signal is the greatest common

divisor (GCD) of all the frequency components contained in a

siganl, and equivalently, the fundamental period is the least

common multiple (LCM) of all individual periods of the

components

Ex:

S(t) = sin( (Π/3)*t ) + cos ( (Π/2)*t) ---------------(2)

ω₁ = Π/3 ω₂ = Π/2

2*Π*f₁ = Π/3 2*Π*f₂= Π/2

f₁ = 1/6 f₂ = 1/4

The fundamental frequency ‘f₀’ is the GCD of f₁= 1/6 and f₂= 1/4

f₀ = GCD ( 1/6. 1/4 )

= 1/12

Alternatively , the period of the fundamental ‘T₀’ is the LCM of T₁ = 6 and T₂ = 4

T₀= LCM ( 6,4)

= 12

How to find the GCD and LCM:

1. HCF (or) GCD = HCF of numerator/ LCM of denominators

2. LCM = LCM of numerator/ HCF of denominators

GEOPHYSICS