Question

Consider that a random process X(t) that contains a DC component equal to '2', then the value of DC component in the auto correlation function of the random process will be equal to _________

• A. 4

Answer 1

The correct option is A 4

$R_x\left(\tau \right)=E\left[X\right(t\left)X\right(t+\tau \left)\right]$

The function X(t) can be written as
$X\left(t\right)=DC+Z\left(t\right)=K+Z\left(t\right)$

$R_x\left(\tau \right)=E\left[\right\{K+Z\left(t\right)\left\}\right\{K+Z(t+\tau )\left\}\right]$

$=E[K^2+KZ(t)+\{Z(t+\tau )+Z\left(t\right)Z(t+\tau )\left\}\right]$
Since the process has zero mean

Thus $E\left[Z\right(t\left)\right]=0$

$R_x\left(\tau \right)=\frac{K^2}{DCcomponent}+R_2\left(\tau \right)$
Thus DC component in the autocorrelation function = $2^2=4$