6.2 $S”[x]$: Second-Order Derivative of Unit-Step Staircase Function

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The first plot below illustrates $S”[0]=0$ and $S”[x]$ is an odd function of $x$. The next three plots illustrate $S”[x]$ in the right-half plane at $f={1, 2, 3}$. The last plot below focuses around $x=19$ with evaluation frequencies $f={2, 4, 8}$. Note $S”[x]$ always evaluates exactly to zero at integer values of $x$, which is consistent with the evaluation of $S_b'[x]$ at an integer value of $x$ always being exactly at the peak of the primary lobe associated with the integer $x$.

$S”[x]$ evaluated at $f=1$.

$S”[x]$ evaluated at $f=1$.
$S”[x]$ evaluated at $f=2$.
$S”[x]$ evaluated at $f=3$.

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