I think you mistake convolution for cross-correlation. They have similar forms, but convolution is more general.
The correlation of two signals $f$ and $g$ could be calculated as:
The convolution of the same signals is:
Convolution could be used to calculate the response of an LTI system, and (normalized) cross-correlation could be used for pattern matching: the maxima of the cross-correlation function is at the offset where pattern g is most likely to be situated in the signal f. If you know this offset you could use a similarity measure (such as the euclidean distance) to quantify similarity.