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Non-sinusoidal Periodic Waveforms



Overview about non-sinusoidal periodic waveforms.

Sine wave model


$$ \large x(t)=A sin(2\pi f t + \phi)=A sin(\omega t + \phi) $$

where,

1. Square wave


$$ \large x_\textrm{sqr}(t)=A \textrm{sgn}(sin(\omega t + \phi)) \quad ; \quad sgn(x)=\begin{cases}\begin{aligned} -1, x<0 \\\ 0, x=0 \\\ 1, x>0 \end{aligned}\end{cases} $$

1.1. Harmonic form


$$ \large x_\textrm{sqrh}(t)=A\frac{4}{\pi}\sum_{k=1}^{\infty}\frac{sin((2k-1)(\omega t + \phi))}{2k-1} $$

2. Triangular wave


$$ \large x_\textrm{tri}(t)=A\frac{2}{a}\left(t+\frac{\phi}{\omega}-a\left\lfloor\frac{t}{a}+\frac{\phi}{\omega a}+\frac{1}{2}\right\rfloor\right)(-1)^{\left\lfloor\frac{t}{a}+\frac{\phi}{\omega a}+\frac{1}{2}\right\rfloor} \quad ; \quad a=\frac{1}{2f} $$