  We study a multi-field inflationary theory with separable Lagrangian, which
has different speed of sound for each field. We find that the fields always
coupled at perturbative level through gravitational interaction. We show that
if the coupling terms among the perturbation fields are weak enough, these
fields can be treated as a combination of decoupled fields, which are similar
to normal modes in coupled oscillation. By virtue of such fields, the curvature
perturbation at the horizon-crossing can be calculated up to the leading order
of slow variation parameters via $\delta \mathcal{N}$ formalism. Explicitly, we
consider a model of multi-speed DBI inflation, and calculate the power spectrum
in detail. The result depends on the ratio of different speeds of sound, and
shows an apparent amplification when the ratio deviates from unity.
