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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?

MinistrChleba
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1 Answers1

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Stumbling back upon this after some time, I realized the problem is equivalent to factoring a univariate polynomial into a product of irreducible factors over the integer field.

This thread discusses the criteria for reducibility of polynomials.

And the respective Wikipedia article offers a few popular algorithms for reducing polynomials.

MinistrChleba
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