Exercise 3 of Assignment 2 (due 1/23/08)

Consider a periodic function $g_p(\cdot)$ with a period of $T= 2$ such that

$$g_p(t) = e^{a\vert t\vert}, \quad -1 < t \le 1,$$

where $a$ is a nonzero real-valued constant.

a)

What are the Fourier coefficients for this function?

b)

What is its discrete power spectrum?

c)

Determine the $m$th order Fourier series approximation $g_{p,m}(\cdot)$ to $g_{p}(\cdot)$.

d)

(Extra credit)     Create plots (similar to those in Figure 61) showing how well $g_{p,m}(\cdot)$ approximates $g_{p}(\cdot)$ for $m = 2, 4, 8$ and $16$ when $a=-1$.

You might find the following indefinite integral useful:

$$\int e^{ax} \cos(px) \, dx = e^{ax} \frac {a\cos(px) + p\sin(px)} {a^2 + p^2}$$

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