Question

Consider a random process x(t) = 2sin(2πt + ϕ), where the random phase ϕ is uniformly distributed in the interval [0, 2π]. The auto-correlation E[x(t₁)x(t₂)]

(A) cos(2π(t₁ + t₂))

(B) sin(2π(t₁ - t₂))

(C) sin(2π(t₁ + t₂))

(D) cos(2π(t₁ - t₂))

Answer 1

First integral will result into zero as we are integrating from 0 to 2π.

Second integral result into cos{2π(t₁ - t₂)}

⇒ E[X(t₁) X(t₂)] = cos(2π(t₁ - t₂))