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  UNDER CONSTRUCTION

Waveform Fourier Series


WaveformFourier Series
\(\bb{1}\ \t{Square Wave}\) \(\ds x(t) = \Frac{A\tau}{T_0}+\sum_{n=1}^{\infty}\frac{4A}{n\pi}\Sinf{n\pi}{2}\Cosf{2n\pi t}{T_0}\)
\(\bb{2}\ \t{Time-shifted Square Wave}\) \(\ds x(t) = \sum_{\substack{n=1 \\ n=\t{odd}}}^{\infty}\frac{4A}{n\pi}\Sinf{2n\pi t}{T_0}\)
\(\bb{3}\ \t{Pulse Train}\) \(\ds x(t) = \frac{A\tau}{T_0}+\sum_{n=1}^{\infty}\frac{2A}{n\pi}\Sinf{n\pi \tau}{T_0}\Cosf{2n\pi t}{T_0}\)
\(\bb{4}\ \t{Triangular Wave}\) \(\ds x(t) = \sum_{\substack{n=1 \\ n=\t{odd}}}^{\infty}\frac{8A}{n^2\pi^2}\Cosf{2n\pi t}{T_0}\)
\(\bb{5}\ \t{Shifted Triangular Wave}\) \(\ds x(t) = \sum_{\substack{n=1 \\ n=\t{odd}}}^{\infty}\frac{8A}{n^2\pi^2}\Sinf{n\pi}{2}\Sinf{2n\pi t}{T_0}\)
\(\bb{6}\ \t{Sawtooth}\) \(\ds x(t) = \sum_{n=1}^{\infty}(-1)^{n+1}\frac{2A}{n\pi}\Sinf{2n\pi t}{T_0}\)
\(\bb{7}\ \t{Backward Sawtooth}\) \(\ds x(t) = \frac{A}{2} + \sum_{n=1}^{\infty}\frac{A}{n\pi}\Sinf{2n\pi}{T_0}\)
\(\bb{8}\ \t{Full-wave Rectified Sinusoid}\) \(\ds x(t) = \frac{2A}{\pi}+\sum_{n=1}^{\infty}\frac{4A}{\pi(1-4n^2)}\Cosf{2n\pi t}{T_0}\)
\(\bb{9}\ \t{Half-wave Rectified Sinusoid}\) \(\ds x(t) = \frac{A}{\pi}+\frac{A}{2}\Sinf{2\pi t}{T_0}+\sum_{\substack{n=2 \\ n=\t{even}}}^{\infty}\frac{2A}{\pi(1-n^2)}\Cosf{2n\pi t}{T_0}\)





Dragon Notes,   Est. 2018     About

By OverLordGoldDragon