  We consider a periodic Jacobi operator $H$ with finitely supported
perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove
that the mapping from finitely supported perturbations to the scattering data:
the inverse of the transmission coefficient and the Jost function on the right
half-axis, is one-to-one and onto. We consider the problem of reconstruction of
the scattering data from all eigenvalues, resonances and the set of zeros of
$R_-(\lambda)+1,$ where $R_-$ is the reflection coefficient.
