5.2 $S_b'[x]$: First-Order Derivative of Unit-Step Staircase Function

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The Fourier series for the first-order derivative $S_b'[x]$ of the staircase function evaluates to $2\ f$ at positive integers where $f$ is the evaluation frequency limit. Note in the case of $S_b'[x]$, the evaluation at an integer value of $x$ is always exactly at the peak of the primary lobe associated with the integer $x$. Note  $S'[x]$ is the Fourier series for the Dirac comb.


$S_b'[x]$ evaluated at $f=1$.

$S_b'[x]$ evaluated at $f=1$.
$S_b'[x]$ evaluated at $f=2$.
$S_b'[x]$ evaluated at $f=3$.


$S_b'[x]$ around $x=19$ evaluated at $f={2, 4, 8}$.