  In this paper, we focus on a family of generalized Kloosterman sums over the
torus. With a few changes to Haessig and Sperber's construction, we derive some
relative $p$-adic cohomologies corresponding to the $L$-functions. We present
explicit forms of bases of top dimensional cohomology spaces, so to obtain a
concrete method to compute lower bounds of Newton polygons of the
$L$-functions. Using the theory of GKZ system, we derive the Dwork's
deformation equation for our family. Furthermore, with the help of Dwork's dual
theory and deformation theory, the strong Frobenius structure of this equation
is established. Our work adds some new evidences for Dwork's conjecture.
