Home › Previous year questions › GATE 2016 EC (EC3) › Question 59GATE 2016 EC (EC3) – Question 59 Communications · Random Processes · 2 marks · Multiple choice
A wide-sense stationary process X(t) passes through the shown delay-and-subtract LTI system. If its autocorrelation is $R_X(\tau)$, the output autocorrelation is
$2R_X(\tau)+R_X(\tau-T_0)+R_X(\tau+T_0)$ $2R_X(\tau)-R_X(\tau-T_0)-R_X(\tau+T_0)$ $2R_X(\tau)+2R_X(\tau-2T_0)$ $2R_X(\tau)-2R_X(\tau-2T_0)$ Practise this question in The GATE Grind →
Show answer and explanation Correct answer: (B) $2R_X(\tau)-R_X(\tau-T_0)-R_X(\tau+T_0)$
Explanation The diagram gives $Y(t)=X(t)-X(t-T_0)$. Expand $E[Y(t)Y(t+\tau)]$ into four terms.
The two same-delay products each equal RX(τ) by stationarity. The cross products have lags τ−T0 and τ+T0 and both enter with minus signs.
Thus $R_Y(\tau)=2R_X(\tau)-R_X(\tau-T_0)-R_X(\tau+T_0)$, **B**.
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