Temporal Filters and Causality

Time has an arrow

Truncation costs delay

The recursive escape

Stability is one condition

Which of these temporal impulse responses could run live on a streaming camera feed, processing frame $t$ using only frames up to $t$?

The filter $\ell_{out}[t] = \ell_{in}[t] + \alpha\,\ell_{out}[t-1]$ starts from rest with $\alpha = 0.5$ and receives a unit impulse at $t = 0$ only. What are the first three output values?

For the recursion $\ell_{out}[t] = \ell_{in}[t] + \alpha\,\ell_{out}[t-1]$ with $\alpha = 0.5$ starting from rest and a unit impulse at $t = 0$, compute the output value at $t = 2$.