Kurt's Assignment

Common Synthesizer Waveforms

${\begin{aligned}x(t)&=x(t+T)=a_{0}+\sum _{n=1}^{\infty }a_{n}\cos(n\omega _{0}t)+b_{n}\sin(n\omega _{0}t)\\a_{0}&={\frac {1}{T}}\int _{0}^{T}f(t)dt\\a_{n}&={\frac {2}{T}}\int _{0}^{T}f(t)\cos(n\omega _{0}t)dt\\b_{n}&={\frac {2}{T}}\int _{0}^{T}f(t)\sin(n\omega _{0}t)dt\\\end{aligned}}$

Square Wave

${\begin{aligned}a_{0}&={\frac {1}{T}}\int _{0}^{{\frac {1}{2}}T}Hdt+{\frac {1}{T}}\int _{{\frac {1}{2}}T}^{T}-Hdt\\&={\frac {1}{T}}\left[Ht\right]{\bigg |}_{t=0}^{{\frac {1}{2}}T}-{\frac {1}{T}}\left[Ht\right]{\bigg |}_{t={{\frac {1}{2}}T}}^{T}\\&={\frac {1}{T}}H{\frac {1}{2}}T-0-\left[{\frac {1}{T}}HT-{\frac {1}{T}}H{\frac {1}{2}}T\right]\\&={\frac {H}{2}}-\left[H-{\frac {1}{2}}H\right]\\&={\frac {H}{2}}-{\frac {H}{2}}\\&=0\end{aligned}}$

${\begin{aligned}a_{n}&={\frac {2}{T}}\int _{0}^{{\frac {1}{2}}T}H\cos(n\omega _{0}t)dt+{\frac {2}{T}}\int _{{\frac {1}{2}}T}^{T}-H\cos(n\omega _{0}t)dt\\&={\frac {2}{T}}\left[{\frac {H}{n\omega _{0}}}\sin(n\omega _{0}t)\right]_{0}^{{\frac {1}{2}}T}+{\frac {2}{T}}\left[{\frac {-H}{n\omega _{0}}}\sin(n\omega _{0}t)\right]_{{\frac {1}{2}}T}^{T}\\&={\frac {2}{T}}\left[{\frac {H}{n{\frac {2\pi }{T}}}}\sin(n\pi )\right]&=0\end{aligned}}$

TODO: finish