  The conventional characterization of periodically driven systems usually
necessitates the time-domain information beyond Floquet bands, hence lacking
universal and direct schemes of measuring Floquet topological invariants. Here
we propose a unified theory based on quantum quenches to characterize generic
$d$-dimensional ($d$D) Floquet topological phases, in which the topological
invariants are constructed with only minimal information of the static Floquet
bands. For a $d$D phase which is initially static and trivial, we introduce the
quench dynamics by suddenly turning on the periodic driving, and show that the
quench dynamics exhibits emergent topological patterns in ($d-1$)D momentum
subspaces where Floquet bands cross, from which the Floquet topological
invariants are directly obtained. This prediction provides a simple and unified
characterization, in which one can not only extract the number of conventional
and anomalous Floquet boundary modes, but also identify the topologically
protected singularities in the phase bands. The applications are illustrated
with 1D and 2D models which are readily accessible in cold atom experiments.
Our study opens a new framework for the characterization of Floquet topological
phases.
