Exercise 5 of Assignment 4 (due 2/4/08)

Let $\{ \epsilon_t \}$ be a white noise process with mean zero and variance $\sigma^2_\epsilon$. Define the stationary processes $\{ X_t \}$ and $\{ Y_t \}$ by

$$X_t = \frac{2}{9} \epsilon_t + \frac{5}{9} \epsilon_{t-1} + \frac{2}{9} \epsilon_{t-2}$$

and

$$Y_t = \frac{4}{9} \epsilon_t + \frac{4}{9} \epsilon_{t-1} + \frac{1}{9} \epsilon_{t-2}.$$

a)

Let $L_X\{\cdot\}$ and $L_Y\{\cdot\}$ be LTI digital filters such that $L_X\{\epsilon_t \} = X_t$ and $L_Y\{\epsilon_t \} = Y_t$. Show that the gain function $\vert G_X(\cdot)\vert$ corresponding to $L_X\{\cdot\}$ is given by

$$\left\vert G_X(f) \right\vert = \frac{5}{9} + \frac{4}{9} \cos(2\pi f),$$

and find the gain function $\vert G_Y(\cdot)\vert$ corresponding to $L_Y\{\cdot\}$. In what respects do the associated transfer functions $G_X(\cdot)$ and $G_X(\cdot)$ differ?

b)

Compare the sdf of $\{ X_t \}$ with the sdf of $\{ Y_t \}$.

c)

How do $\{ X_t \}$ and $\{ Y_t \}$ differ from a moving average process as defined by Equation (43a)? Are there moving average processes that are equivalent to $\{ X_t \}$ and $\{ Y_t \}$ (i.e., have the same sdfs)?

Return to home page for Stat/EE 520.