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We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartiersubalgebra on $R$ as defined by the first two authors. In particular, we showthe existence and positivity of the $F$-signature for any strongly $F$-regularpair. In one application, we answer an open question of I. Aberbach and F.Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure localring is strictly positive. Furthermore, we derive effective methods forcomputing the $F$-signature and the $F$-splitting ratio in the spirit of thework of R. Fedder.

F-signature of pairs and the asymptotic behavior of Frobenius splittings