  We build upon the past studies of inflation with rank-2 antisymmetric tensor
field, including here the tensor perturbations to metric. We perform a
comprehensive analysis of the background dynamics of our model in the presence
of non-minimal coupling curvature terms $R$ and $R_{\mu \nu}$. We find
appropriate conditions on the nonminimal coupling parameters to satisfy the
constraint of speed of propagation of gravitational waves. Including the tensor
perturbations, the model is found to be free from ghost instabilities with
minimal constraints on the parameters. We also study the evolution of
gravitational waves, calculate the power spectrum and the tensor spectral
index.
