Kurt's Assignment

Common Synthesizer Waveforms

$\begin{array}{rl}x(t)& =x(t+T)=a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t)\\ a_0& =\frac{1}{T}\underset{0}{\overset{T}{\int }}f(t)dt\\ a_n& =\frac{2}{T}\underset{0}{\overset{T}{\int }}f(t)\cos (n{\omega }_{0}t)dt\\ b_n& =\frac{2}{T}\underset{0}{\overset{T}{\int }}f(t)\sin (n{\omega }_{0}t)dt\\ \end{array}$

Square Wave

$\begin{array}{rl}{a}_0& =\frac{1}{T}\underset{0}{\overset{\frac{1}{2}T}{\int }}Hdt+\frac{1}{T}\underset{\frac{1}{2}T}{\overset{T}{\int }}-Hdt\\ & =\frac{1}{T}\left[Ht\right]{|}_{t=0}^{\frac{1}{2}T}-\frac{1}{T}\left[Ht\right]{|}_{t=\frac{1}{2}T}^T\\ & =\frac{1}{T}H\frac{1}{2}T-0-[\frac{1}{T}HT-\frac{1}{T}H\frac{1}{2}T]\\ & =\frac{H}{2}-[H-\frac{1}{2}H]\\ & =\frac{H}{2}-\frac{H}{2}\\ & =0\end{array}$

$\begin{array}{rl}{a}_n& =\frac{2}{T}\underset{0}{\overset{\frac{1}{2}T}{\int }}H\cos (n{\omega }_{0}t)dt+\frac{2}{T}\underset{\frac{1}{2}T}{\overset{T}{\int }}-H\cos (n{\omega }_{0}t)dt\\ & =\frac{2}{T}{[\frac{H}{n{\omega }_0}\sin (n{\omega }_{0}t)]}_0^{\frac{1}{2}T}+\frac{2}{T}{[\frac{-H}{n{\omega }_0}\sin (n{\omega }_{0}t)]}_{\frac{1}{2}T}^T\\ & =\frac{2}{T}[\frac{H}{n\frac{2\pi }{T}}\sin (n\pi )]& =0\end{array}$

TODO: finish