4.2 $S[x]$: Unit-Step Staircase Function

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The Fourier series representation of the unit-step staircase function $S[x]$ is illustrated in the three plots below at the frequency evaluation limits $f=1$, $f=2$, and $f=3$ respectively. The linear function $x$ (gray) is shown as a reference.

Fourier series representation of $S[x]$ evaluated at $f=1$.
Fourier series representation $S[x]$ evaluated at $f=2$.
Fourier series representation $S[x]$ evaluated at $f=3$.