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On page 2 of the slide, I don't understand the example. Like $j$ and $s$ are both on the x-axis, what does j+s mean? The filter is $(f(−1), f(0), f(1)) = (−0.5, 0, 0.5)$. How can we use it? Could anyone explain how we get $r(j)$ with numbers?

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For the simple $3$-tap filter given in the example, each output value $r(j)$ is computed as

$$\begin{align}r(j)&=f(-1)I(j-1)+f(0)I(j)+f(1)I(j+1)\\&=-\frac12 I(j-1)+\frac12 I(j+1)\tag{1}\end{align}$$

So each input value is replaced by the (scaled) difference between its neighboring values.

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