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

Suppose that $\{ X_t \}$ is a real-valued stationary process whose sampling interval is $\Delta t$. Assume that its acvs $\{ s_{X,\tau} \}$ and sdf $S_X(\cdot)$ are related as dictated by Equations (134a) and (134b):

$$\begin{eqnarray*} s_{X,\tau} &=& \int_{-f_{(N)}}^{f_{(N)}} S_X(f) e^{i2\pi f\tau... ... t}, \quad \vert f\vert \le f_{(N)}\equiv \frac{1}{2\,\Delta t}, \end{eqnarray*}$$

where $f_{(N)}$ is the Nyquist frequency. Note that we can use the last equation above to define $S_X(\cdot)$ for $\vert f\vert > f_{(N)}$, so that the sdf can be regarded as a periodic function with period $2f_{(N)}$. Let $Y_t = X_{2t}$, $t=\ldots,-1,0,1,\ldots$; i.e., the process $\{ Y_t \}$ is formed by subsampling every other random variable from the process $\{ X_t \}$, and hence the sampling interval for $\{ Y_t \}$ is $\Delta t'\equiv 2 \,\Delta t$.

a)

Show that $\{ Y_t \}$ is a stationary process, and determine its acvs $\{ s_{Y,\tau} \}$ in terms of $\{ s_{X,\tau} \}$.

b)

Show that $\{ Y_t \}$ has an sdf $S_Y(\cdot)$ that is an aliased version of $S_X(\cdot)$, with the two sdfs being related by

$$S_Y(f) = S_X(f) + S_X(f + f_{(N)}).$$

Argue that $S_Y(\cdot)$ is a periodic function with a period of $2f'_{(N)}$, where $f'_{(N)}\equiv \frac{1}{2\,\Delta t'}$ is the Nyquist frequency for $\{ Y_t \}$.

c)

Suppose now that $\{ X_t \}$ is a first order moving average process with mean zero and coefficient $\theta$; i.e., we can write $X_t = \epsilon_t - \theta \epsilon_{t-1}$, where $\{ \epsilon_t \}$ is a white noise process with mean zero and variance $\sigma^2_\epsilon$. The acvs and sdf for this process are given by

$$s_{X,\tau} = \cases{ \sigma^2_\epsilon (1+\theta^2),&$\tau=0... ...ta t \left\vert 1- \theta e^{-i2\pi f\,\Delta t} \right\vert^2$$

(see pages 43 and 167). Use part a of this exercise to show that $\{ Y_t \}$ has an acvs of a white noise process. Use part b to show that $\{ Y_t \}$ has an sdf of a white noise process.

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