Although we may have redundant operations with some few zeros for \beta_i , we can rewrite the above sum in a more compact way as
\delta_i = \sum_{k=0}^{k=m-1}\alpha_k\beta_{i-k},where m=3 in our case, the maximum length of the vector \alpha . Note that the vector \boldsymbol{\beta} has length n=4 . Below we will find an even more efficient representation.