Exercise 4 of Assignment 5 (due 2/11/08)

Suppose that $x_0, x_1, \ldots, x_{N-1}$ represents a time series and that we form its periodogram using

$$\hat S^{(p)}_x(f) = {1\over N} \left\vert \sum_{t=0}^{N-1} x_t e^{-i2\pi ft} \right\vert^2$$

(see item [4] on page 205 for a comment on using the indexing $x_0, x_1, \ldots, x_{N-1}$ rather than our usual $x_1, x_2, \ldots, x_{N}$). For any integer $0<m\le N-1$, define $y_t = x_{t+m\bmod N}$; i.e., $y_0, y_1, \ldots, y_{N-1}$ is a circularly shifted version of $x_0, x_1, \ldots, x_{N-1}$ (see Exercise 4 of Assignment 1 for a discussion of the `mod' operator). Let

$$\hat S^{(p)}_y(f) = {1\over N} \left\vert \sum_{t=0}^{N-1} y_t e^{-i2\pi ft} \right\vert^2$$

be the periodogram for $y_t$.

a)

Show that $\hat S^{(p)}_y(f_k) = \hat S^{(p)}_x(f_k)$, where $f_k = k/N$ is any one of the Fourier frequencies.

b)

Prove or disprove the claim that, in general, $\hat S^{(p)}_y(f) = \hat S^{(p)}_x(f)$ at all frequencies (i.e., not necessarily just the Fourier frequencies).

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