  Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$
of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and
let $ j: X^{*} = X - Y \to X $ denotes the open embedding. The purpose of this
paper is to describe the weight filtration $W$ on a combinatorial logarithmic
complex computing the (higher) direct image ${\bf j}_{*}{\cal L} $, underlying
a mixed Hodge complex when $X$ is proper, proving in this way the results in
the note [14] generalizing the constant coefficients case. When a morphism $f:
X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration
on the complex of nearby cocycles $\Psi_f ({\cal L})$ on $Y$ can be described
by these logarithmic techniques and a comparison theorem shows that the
filtration coincides with the weight defined by the logarithm of the monodromy
which provides the link with various results on the subject.
