Continuous Time Fourier Series 02 Class Notes PDF

# Electrical Engineering ## Electronics and Communication Engineering ### SIGNALS & SYSTEMS #### Lecture No. 2 ##### Continuous Time Fourier Series By- Sujal Patel Sir ### Topics to be Covered * Concepts of CTFS * TFS & EFS ### Polar Form or Cosine of Fourier Series $x(t) = a_0 + \sum_{n=1}^\in...

# Electrical Engineering ## Electronics and Communication Engineering ### SIGNALS & SYSTEMS #### Lecture No. 2 ##### Continuous Time Fourier Series By- Sujal Patel Sir ### Topics to be Covered * Concepts of CTFS * TFS & EFS ### Polar Form or Cosine of Fourier Series $x(t) = a_0 + \sum_{n=1}^\infty a_n cos(n \omega_0 t) + b_n sin(n \omega_0 t)$ # Polar form: $x(t) = d_0 + \sum_{n=1}^\infty d_n Cos(n \omega_0 t + \theta_n)$ $\implies x(t) = d_0 + \sum_{n=1}^\infty d_n cos\theta_n cos(n \omega_0 t) - d_n sin\theta_n sin(n \omega_0 t)$ $\implies a_n = d_n cos\theta_n \implies (a_n)^2 + (b_n)^2 = d_n^2cos^2\theta_n + d_n^2sin^2\theta_n = d_n^2$ $\implies -b_n = d_n sin\theta_n \implies d_n^2 = a_n^2 + (-b_n)^2 \implies d_n sin\theta_n/ d_n cos\theta_n = - b_n / a_n$ $\implies |d_n| = \sqrt{(a_n)^2 + (b_n)^2} \implies \theta_n = tan^{-1}(b_n/a_n)$ $\implies d_n = a_n - jb_n$ $\implies \theta_n = tan^{-1}(-b_n/a_n) = -tan^{-1}(b_n/a_n)$ $\implies d_n^* = a_n + jb_n$ # Magnitude Spectrum $x(t) = d_0 + \sum_{n=1}^\infty d_n cos(n \omega_0 t + \theta_n)$ $\theta_n = \angle d_n$ # Phase Spectrum ### Exponential Fourier Series $x(t) = a_0 + \sum_{n=1}^\infty a_n cos(n \omega_0 t) + b_n sin(n \omega_0 t)$ $\implies x(t) = a_0 + \sum_{n=1}^\infty a_n [\frac{e^{jn \omega_0 t} + e^{-jn \omega_0 t}}{2}] + b_n [\frac{e^{jn \omega_0 t} e^{-jn \omega_0 t}}{2j}]$ $\implies x(t) = a_0 + \sum_{n=1}^\infty (\frac{a_n - jb_n}{2}) e^{jn \omega_0 t} + (\frac{a_n + jb_n}{2})e^{-jn \omega_0 t}$ Let, $C_n = \frac{a_n - jb_n}{2}$ & $C_{-n} = \frac{a_n + jb_n}{2} \implies C_{-n} = C_n^* \implies |C_n| = |C_{-n}|$ $\implies x(t) = C_0 + \sum_{n=1}^\infty C_n e^{jn \omega_0 t} + \sum_{n = - \infty}^{-1} C_n e^{-jn \omega_0 t}$ $\implies x(t) = C_0 + \sum_{n=1}^\infty C_n e^{jn \omega_0 t} + \sum_{n= - \infty}^{-1} C_n e^{jn \omega_0 t}$ $\implies x(t) = \sum_{n=- \infty}^\infty C_n e^{jn \omega_0 t} \longrightarrow Exponential Fourier Series of x(t)$ Where, $C_n$ is EFS Coefficient. # $C_n = \frac{a_n - jb_n}{2}$ $\implies C_n = \frac{1}{2} [\frac{1}{T} \int_{T} x(t). Cosn\omega_0 t dt - \frac{j}{2} \frac{1}{T} \int_{T} x(t). Sinn\omega_0 t dt ]$ $\implies C_n = \frac{1}{T} \int_{T} x(t). (Cosn\omega_0 t - jSinn\omega_0 t). dt $ $\implies C_n = \frac{1}{T} \int_{T} x(t). e^{-jn \omega_0 t} dt$ $\implies d_n = a_n - jb_n \longrightarrow |C_n| = \sqrt{\frac{a_n^2 + b_n^2}{2}} = \frac{|d_n|}{2}$ $\implies \theta_n = tan^{-1} (-b_n/ a_n)$ $\implies C_n = \frac{a_n - jb_n}{2} = \frac{d_n}{2}$ $\implies C_{-n} = \frac{a_n + jb_n}{2} = \frac{d_n^*}{2}$ $\implies |C_{-n}| = \sqrt{\frac{a_n^2 + b_n^2}{2}} = \frac{|d_n^*|}{2} = \frac{|d_n|}{2}$. # $|C_n| = |C_{-n}| \longrightarrow Even$ $\implies \angle C_n = \theta_n = tan^{-1} (-b_n/a_n) = -tan^{-1}(b_n/a_n)$ $\implies a_n = C_{-n} + C_n$ $\implies \angle C_{-n} = \theta_{-n} = tan^{-1}(b_n / a_n)$ # $\theta_n = -\theta_{-n} \longrightarrow Odd$ $\implies jb_n = C_{-n} - C_n$ # Magnitude Spectrum # Phase Spectrum # $x(t) = a_0 + \sum_{n=1}^\infty a_n Cosn\omega_0 t + b_n Sinn\omega_0 t$ $\implies P_{total} = \frac{1}{T} \int_{T} |x(t)|^2 dt = a_0^2 + \sum_{n=1}^\infty \frac{a_n^2 + b_n^2}{2}$ # $x(t) = d_0 + \sum_{n=1}^\infty d_n Cos(n\omega_0 t + \theta_n)$ $d_n = a_n - jb_n$ $P_{total} = \frac{1}{T} \int_{T} |x(t)|^2 dt = d_0^2 + \sum_{n=1}^\infty \frac{|d_n|^2}{2}$ $|d_n|^2 = a_n^2 + b_n^2$ # $x(t) = \sum_{n= - \infty}^\infty (C_n e^{jn \omega_0 t})$ $P_{x(t)} = \frac{1}{T} \int_{T} |x(t)|^2 dt = \sum_{n= - \infty}^\infty (\frac{1}{T} \int_{T} |C_n|^2 e^{jn\omega_0 t} dt)$ $P_{x(t)} = \sum_{n = -\infty }^\infty \frac{1}{T} \int_{T} |C_n|^2 dt = \sum_{n = - \infty}^\infty |C_n|^2$ $P_{x(t)} = \frac{1}{T} \int_{T} |x(t)|^2 dt = \sum_{n= - \infty}^\infty |C_n|^2 = |C_0|^2 + \sum_{n= - \infty}^{-1} |C_n|^2 + \sum_{n=1}^\infty |C_n|^2$ $P_{x(t)} = |C_0|^2 + 2 \sum_{n=1}^\infty |C_n|^2$ # $x(t) \longleftarrow FS \longrightarrow C_n, a_0, a_n, b_n, d_n$ $P_{total} = \frac{1}{T} \int_{T} |x(t)|^2 dt = a_0^2 + \sum_{n=1}^\infty \frac{a_n^2 + b_n^2}{2} = d_0^2 + \sum_{n=1}^\infty \frac{|d_n|^2}{2}$ $ = \sum_{n = - \infty}^\infty |C_n|^2 = |C_0|^2 + 2 \sum_{n=1}^\infty |C_n|^2$ # Power up to 5th harmonic $P_{upto 5th} = a_0^2 + \sum_{n=1}^{5}\frac{a_n^2 + b_n^2}{2} = d_0^2 + \sum_{n=1}^ 5\frac{|d_n|^2}{2}$ $= \sum_{n = -5}^5 |C_n|^2 = |C_0|^2 + 2 \sum_{n=1}^5 |C_n|^2$ # Power Spectrum