  If $\beta\in\CC^d$ is integral but not a strongly resonant parameter for the
homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the
associated GKZ-system carries a naturally defined mixed Hodge module structure.
We study here in the normal case the corresponding weight filtration by
computing the intersection complexes with respective multiplicities on the
associated graded parts. We do this by computing the weight filtration of a
Gauss-Manin system with respect to a locally closed embedding of a torus inside
an affine space. We then produce a result, based on a Fourier-Sato transforms,
that allows to port an MHS structure on a monodromic module through a
Fourier-Laplace transform, from the Gauss-Manin system to the GKZ-system. Our
results show that these data, which we express in terms of intersection
cohomology groups on induced toric varieties, are purely combinatorial, and not
arithmetic, in the sense that they only depend on the polytopal structure of
the cone of $A$ but not on the semigroup itself. As a corollary we get a purely
combinatorial formula for the length of the underlying (regular) holonomic
GKZ-system, irrespective of homogeneity. In dimension up to three, and for
simplicial semigroups, we give explicit generators of the weight filtration.
