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A wide sense stationary random process X(t) passes through the LTI system shown in the figure. If the autocorrelation function of X(t) is R_X(τ), then the autocorrelation function R_Y(τ) of the output Y(t) is equal to

1. 2R_X(τ) + R_X(τ - T₀) + R_X(τ + T₀)
2. 2R_X(τ) - R_X(τ - T₀) - R_X(τ + T₀)
3. 2R_X(τ) + 2R_X(τ - 2T₀)
4. 2R_X(τ) - 2R_X(τ - 2T₀)

Correct answer: 2R_X(τ) + R_X(τ - T₀) + R_X(τ + T₀)

Solution

The correct option reflects the properties of the autocorrelation function when a wide sense stationary process passes through a linear time-invariant (LTI) system, where the output's autocorrelation is influenced by the input's autocorrelation at different time shifts, specifically incorporating contributions from both positive and negative shifts of T₀.

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