Generalizing the above one-dimensional case

In order to align the above simple case with the more general convolution cases, we rename \( \boldsymbol{\alpha} \), whose length is \( m=3 \), with \( \boldsymbol{w} \). We will interpret \( \boldsymbol{w} \) as a weight/filter function with which we want to perform the convolution with an input variable \( \boldsymbol{x} \) of length \( n \). We will assume always that the filter \( \boldsymbol{w} \) has dimensionality \( m \le n \).

We replace thus \( \boldsymbol{\beta} \) with \( \boldsymbol{x} \) and \( \boldsymbol{\delta} \) with \( \boldsymbol{y} \) and have

$$ y(i)= \left(x*w\right)(i)= \sum_{k=0}^{k=m-1}w(k)x(i-k), $$

where \( m=3 \) in our case, the maximum length of the vector \( \boldsymbol{w} \). Here the symbol \( * \) represents the mathematical operation of convolution.