5th semester files

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

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+## 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
+
+$$P_x = {\left({1\over2}\right)^2 \over 2} + {\left({1\over2}\right)^2 \over 2} = {1\over8} + {1\over8} = {1\over4}$$
+
+$$x(t) = c_0 + 2 \sum_{k=1}^\infty\left[c_k\cos(kt) + d_k\sin(kt)\right]$$
+
+$$c_0 = {1\over P} \int\limits_{-P/2}^{P/2} x(t)dt$$