Periodicity of sin(15πt) and sin(√2πt)

A continuous-time sinusoid sin(ωt) is periodic with period T = 2π/ω for any real ω ≠ 0.

a) x(t) = sin 15πt

ω = 15π rad/s, so T = 2π / 15π = 2/15 s ≈ 0.133 s.

Periodic, with fundamental period T = 2/15 s.

b) x(t) = sin √2·πt

Reading the argument as √2·π·t: ω = √2 π rad/s,

T = 2π / (√2 π) = 2/√2 = √2 s ≈ 1.414 s.

Periodic, with T = √2 s. The period is irrational, but that does not matter in continuous time: the condition for periodicity is only that a positive T exists such that x(t + T) = x(t).

(Note: if the signal were discrete-time, sin(√2 π n), it would be non-periodic, because 2π/ω₀ = √2 is not a rational number. For the continuous-time signal in the question, both signals are periodic.)