## 1
$$x(t) = \cos(t) + \sin(\pi t)$$

$\cos(t)$: $\omega_1 = 1$, $T_1 = 2\pi$

$\sin(\pi t)$: $\omega_2 = \pi$, $T_2 = 2$

${T_1 \over T_2} = \pi$: Not rational, so $x(t)$ is aperiodic.

## 2
$$x(t) = \cos(t) \cos(2t)$$

### a)
Find the Fourier series

$$\cos(A)\cos(B) = {1\over2}\left[\cos(A+B) + \cos(A-B)\right]$$
$$x(t) = {1\over2}\cos(3t) + {1\over2}\cos(t)$$

$\cos(3t)$: $\omega_1 = 3$, $T_1 = {2\pi \over 3}$

$\cos(t)$: $\omega_2 = 1$, $T_2 = 2\pi$

${T_1 \over T_2} = {1\over3}$: Rational

$$P_x = {\left({1\over2}\right)^2 \over 2} + {\left({1\over2}\right)^2 \over 2} = {1\over8} + {1\over8} = {1\over4}$$

$$x(t) = c_0 + 2 \sum_{k=1}^\infty\left[c_k\cos(kt) + d_k\sin(kt)\right]$$

$$c_0 = {1\over P} \int\limits_{-P/2}^{P/2} x(t)dt$$