  We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier
subalgebra on $R$ as defined by the first two authors. In particular, we show
the existence and positivity of the $F$-signature for any strongly $F$-regular
pair. 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 local
ring is strictly positive. Furthermore, we derive effective methods for
computing the $F$-signature and the $F$-splitting ratio in the spirit of the
work of R. Fedder.
