Rowan-Classes/5th-Semester-Fall-2023/Signals-and-Systems/Tutorial-Notes/Nov-9.md

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