First of all, sorry for my lack of terminology. It might be the reason why I'm not able to find the answer.
Let's assume a 1-dimensional discrete signal $y[k]$ consisting of finite number of integer samples. Is it possible to analytically find the set of its prime convolutors? By prime convolutors I mean a set of integer signals $x_1[k], x_2[k], \ldots, x_n[k]$ that fulfill:
$x_1[k] \ast x_2[k] \ast \cdots \ast x_n[k] = y[k]$
(where $\ast$ denotes discrete convolution),
$x_1[k], x_2[k], \ldots, x_n[k] \in \mathbb Z$ for every $k \in \mathbb Z$
and they cannot be further decomposed into smaller all-integer convolutors (similarly to prime numbers, which can't be expressed as a product of two smaller integers).
For example, signal $(1, 0, -1)$ can be deconvolved into $(1,1)\ast(1,-1)$.
Or signal $(4,9,-6,-26,-18,-3)$ into $(1,1)\ast(1,1)\ast(4,1)\ast(1,0,-3)$.
Under what conditions is it possible to find the set of prime convolutors for a given signal?