Further simplification

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.