  We present an inflationary model preceded by a bounce in a metric theory a
l\'{a} $f(R)$ where $R$ is the scalar curvature of the space-time. The model is
asymptotically de Sitter such that the gravitational action tends
asymptotically to a Hilbert-Einstein action, therefore modified gravity affects
only the early stages of the universe. We then analyse the spectrum of the
gravitational waves through the method of the Bogoliubov coefficients by two
means: taking into account the gravitational perturbations due to the modified
gravitational action in the $f(R)$ setup and by simply considering those
perturbations inherent to the standard Hilbert-Einstein action. We show that
there are distinctive (oscillatory) signals on the spectrum for very low
frequencies; i.e. corresponding to modes that are currently entering the
horizon.
