Repeat the simulations and analyses in Examples 4.1 and 4.2 with the following changes:

(a) Change the sample size to n = 128 and generate and plot the same series as in Example 4.1:

xt1 = 2 cos(2π .06 t) + 3 sin(2π .06 t), xt2 = 4 cos(2π .10 t) + 5 sin(2π .10 t), xt3 = 6 cos(2π .40 t) + 7 sin(2π .40 t), xt = xt1 + xt2 + xt3.

What is the major difference between these series and the series generated in Example 4.1? (Hint: The answer is fundamental. But if your answer is the series are longer, you may be punished severely.)

(b) As in Example 4.2, compute and plot the periodogram of the series, xt, generated in

(a) and comment.

(c) Repeat the analyses of

(a) and

(b) but with n = 100 (as in Example 4.1), and adding noise to xt; that is xt = xt1 + xt2 + xt3 + wt where wt ∼ iid N(0, 25). That is, you should simulate and plot the data, and then plot the periodogram of xt and comment.