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.

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