I am trying to understand discrete convolution using scaling $S$ and zero-padding of a vector with the formula given according to:
$$ R_{t} = \left\{ \begin{array}{ll} \frac{1}{S}\sum_{i=0}^{k-1}A_{t-i}K_{i} & \text{if} \; t\geq k-1 \\ \frac{1}{S}\sum_{i=0}^{t}A_{t-i}K_{i} & \text{otherwise} \\ \end{array} \right. $$
Does this formula mean that the elements at the edges of the vector don't need to be padded with zeros, only the end of the vector? Why is this procedure done? What determines the padding?