导数的离散化

From limits to taps

The half-sample shift

The Fourier verdict

Two dimensions and history

The kernel $d_0 = [1, -1]$ compares sample $n$ with sample $n-1$. At which location does its output naturally belong, and what does that imply on a step edge?

The central difference $d_1 = [1,0,-1]/2$ has response $j\sin(2\pi u/N)$. What does this response do at the highest image frequency?

The signal reads 5 then 7 then 2 across three samples. Apply $d_0 = [1, -1]$ and write the first valid output value, the one that compares the second sample against the first.