# Dragon Notes

UNDER CONSTRUCTION
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# Fourier Series

## Fourier representations

 Sine/cosine $$\ds \vpl x(t)=a_0+\sum_{n=1}^{\infty}[a_n\cos{(n\omega_0 t)}+b_n\sin{(n\omega_0 t)}]$$ \begin{align} a_0 \vphantom{\ds \int_{0}^{\int^{\int}}} & =\frac{1}{T_0}\int_{0}^{T_0}x(t)dt \\ a_n & = \frac{2}{T_0}\int_{0}^{T_0}x(t)\cos{(n\omega_0 t)} \\ b_n & = \frac{2}{T_0}\int_{0}^{T_0}x(t)\sin{(n\omega_0 t)}dt \end{align} Amplitude/phase $$\ds \vpl x(t)=c_0+\sum_{n=1}^{\infty}c_n\cos{(n\omega_0 t+\phi_n)}$$ $$\vpL c_n\angle{\phi_n}=a_n-jb_n$$$$\ds c_n=\sqrt{a_n^2+b_n^2}\quad$$ $$\ds {\phi }_n=\left\{ \begin{array}{c} -{{\mathrm{tan}}^{-1} \left(\lfrac{b_n}{a_n}\right)},\ a_n>0 \\ \pi -{{\mathrm{tan}}^{-1} \left(\lfrac{b_n}{a_n}\right)},\ a_n<0 \end{array} \right.$$ Exponential $$\ds \vpl x(t)=\sum_{n=-\infty}^{\infty}\bn{x}_ne^{jn\omega_0 t}$$ $$\ds \bn{x}_n=\frac{1}{T_0}\int_{0}^{T_0}x(t)e^{-jn\omega_0 t}dt$$$$\ds \vphantom{\int_{A^A}^{A^A}} \bn{x}_n=|\bn{x}_n|e^{j\phi_n},\ \ |\bn{x}_n|=c_n/2$$ $$\bn{x}_{-n}=\bn{x}_n^*,\ \ \phi_{-n}=-\phi_n, \ \ x_0=c_0$$
$a_0=c_0=x_0\qquad a_n=c_n\cos{(\phi_n)}\qquad b_n=-c_n\sin{(\phi_n)}\qquad \bn{x}_n=(a_n-jb_n)/2$

## Symmetries

Even

\begin{align} \ds a_0 & =\frac{2}{T_0}\int_{0}^{T_0/2}x(t)dt, \\ a_n & = \frac{4}{T_0}\int_{0}^{T_0/2}x(t)\cos{(n\omega_0 t)},\ \ b_n=0 \\ c_n & = |a_n|, \ds \ {\phi }_n=\left\{ \begin{array}{c} 0,\quad \quad a_n>0 \\ 180^{\t{o}}, \hspace{12px} a_n<0 \end{array} \right. \end{align}
Odd

\begin{align} \ds a_0 & = 0,\ a_n = 0 \\ b_n & = \frac{4}{T_0}\int_{0}^{T_0/2}x(t)\sin{(n\omega_0 t)} \\ c_n & = |b_n|, \ds \ {\phi }_n=\left\{ \begin{array}{c} -90^{\t{o}},\ \ b_n>0 \\ 180^{\t{o}}, \hspace{16px} b_n<0 \end{array} \right. \end{align}
dc

 $$a_0=c_0=0$$
dc only
even only
dc and even

## Waveform examples

 Waveform Fourier Series $$\bb{1}\ \t{Square Wave}$$ $$\ds x(t) = \Frac{A\tau}{T_0}+\sum_{n=1}^{\infty}\frac{4A}{n\pi}\Sinf{n\pi}{2}\Cosf{2n\pi t}{T_0}$$ $$\bb{2}\ \t{Pulse Train}$$ $$\ds x(t) = \frac{A\tau}{T_0}+\sum_{n=1}^{\infty}\frac{2A}{n\pi}\Sinf{n\pi \tau}{T_0}\Cosf{2n\pi t}{T_0}$$ $$\bb{3}\ \t{Sawtoowth}$$ $$\ds x(t) = \sum_{n=1}^{\infty}(-1)^{n+1}\frac{2A}{n\pi}\Sinf{2n\pi t}{T_0}$$ $$\bb{4}\ \t{Full-wave Rectified Sinusoid}$$ $$\ds x(t) = \frac{2A}{\pi}+\sum_{n=1}^{\infty}\frac{4A}{\pi(1-4n^2)}\Cosf{2n\pi t}{T_0}$$ $$\t{[...] More}\ \$$ $$\t{Applied Examples }$$   $$\vpl$$