Question

Which one of the following signals is NOT periodic?
1. sin 10πt 
2. sin 31t
3. sin 10πt + sin 13t 
4. sin (10π + 31)t

Answer 1

Correct Answer - Option 3 : sin 10πt + sin 13t 

Concept:

Consider a signal x(t) = A_m sin (ωt)

The time period of the signal T = 2π / ω seconds

Consider a signal with the combination of two signals y(t) = sin ω₁t + sin ω₂t 

The time period of the signal 1 T₁ = 2π / ω₁ seconds

The time period of the signal 2 T₂ = 2π / ω₂ seconds

When both signals are added, take the ratio of T1 and T2, it must be rational in order to result as a periodic signal.

Calculation:

• Signal sin 10πt is periodic with a time period of 0.2 s
• Signal sin 31t is periodic with time period 2π / 31 s
• Signal sin (10πt + 31) is periodic with time period 0.2 s

 

Condition to find time period for a mixed-signal is the ratio of the two time periods must be rational.

For signal sin 10πt + sin 13t 

sin 10πt is periodic with a time period(T₁) of 0.2 s

sin 13t is periodic with a time period(T₂) of 2π / 13 s

T1 / T2 = 13 / 10π  ⇒ Irrational

So the signal sin 10πt + sin 13t  is a non-periodic signal.