Book says:
$$\delta[n]=u[n] - u[n-1]$$
Therefore, the unit sample reponse $h[n]$ is related to unit step reponse, $s[n]$, as follows:
$$h[n]=s[n] - s[n-1]$$
My question is, how to prove this relationship?
Here are a few related definitions:
$$n\in\mathbb{Z}$$
Step Function:
$$u[n] = \begin{cases}1&,\text{if }n\ge0\\0&,\text{else}\end{cases}$$
Impulse Function:
$$\delta[n] = \begin{cases}1&,\text{if }n=0\\0&,\text{else}\end{cases}$$
Unit Sample Response $$s[n] = h[n] * u[n] = \sum_{k=0}^{\infty} h[n-k]u[k]$$ ':
wait, I forget that convolution operator is distributive.
$$ s[n] - s[n-1] = h[n] * u[n] - h[n] * u[n-1]$$ $$ s[n] - s[n-1] = h[n] * (u[n] - u[n-1])$$ $$ s[n] - s[n-1] = h[n] * \delta[n]$$ $$ s[n] - s[n-1] = h[n] $$