Common Synthesizer Waveforms
Square Wave
x ( t ) = x ( t + T ) = a 0 + ∑ n = 1 ∞ a n cos ( n ω 0 t ) + b n sin ( n ω 0 t ) {\displaystyle 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 = 1 T ∫ 0 T f ( t ) d t = 1 T ∫ 0 1 2 T H d t + 1 T ∫ 1 2 T T 0 d t = 1 T [ H t ] {\displaystyle {\begin{aligned}a_{0}&={\frac {1}{T}}\int _{0}^{T}f(t)dt\\&={\frac {1}{T}}\int _{0}^{{\frac {1}{2}}T}Hdt+{\frac {1}{T}}\int _{{\frac {1}{2}}T}^{T}0dt\\&={\frac {1}{T}}[Ht]\end{aligned}}}
TODO: finish
![{\displaystyle {\begin{aligned}a_{0}&={\frac {1}{T}}\int _{0}^{T}f(t)dt\\&={\frac {1}{T}}\int _{0}^{{\frac {1}{2}}T}Hdt+{\frac {1}{T}}\int _{{\frac {1}{2}}T}^{T}0dt\\&={\frac {1}{T}}[Ht]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c90e566989d00ef1051c26476e23d72b3a1b1f5)