I understand correlation is defined as below and have questions on using the concept for signal pattern detection. I'm relatively new to this topic so appreciate anyone sending me in the best direction.
$$ a,f,g : S \rightarrow Y $$ $$ a(t) = (f \star g)(t) = \int_{-\infty}^{\infty} f(\tau) g(\tau + t) d\tau $$
I would like to find the best shifted matches of $g$ over $f$, so I: $$ F = argmax_{x \in S}(a(t)) $$ Is it an exact match: $$ D(f,g,t) = \forall \tau\in S(\delta_{f(\tau)g(\tau+t)} = 1) $$ $$ MatchesShift = \{t : t \in F \land D(f, g, t)\} $$
Does this look correct to find best correlation and then exact matches?
I'm assuming more complex matching is performed by testing all the subsets of $g$. Appreciate any guidance there