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

1. cos(2π(t1 + t2))
2. sin(2π(t1 - t2))
3. sin(2π(t1 + t2))
4. cos(2π(t1 - t2))

Correct answer: cos(2π(t1 - t2))

Solution

The correct option is right because the auto-correlation function for a sinusoidal process with a uniformly distributed random phase results in a cosine function of the difference between the two time points, reflecting the periodic nature of the sine function and the properties of the expected value.

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