  A study of Final State Effects (FSE) on the dynamic structure function of
superfluid $^4$He in the Gersch--Rodriguez formalism is presented. The main
ingredients needed in the calculation are the momentum distribution and the
semidiagonal two--body density matrix. The influence of these ground state
quantities on the FSE is analyzed. A variational form of $\rho_2$ is used, even
though simpler forms turn out to give accurate results if properly chosen.
Comparison to the experimental response at high momentum transfer is performed.
The predicted response is quite sensitive to slight variations on the value of
the condensate fraction, the best agreement with experiment being obtained with
$n_0=0.082$. Sum rules of the FSE broadening function are also derived and
commented. Finally, it is shown that Gersch--Rodriguez theory produces results
as accurate as those coming from other more recent FSE theories.
