  We show that the extended noncommutative de Rham complex of a cofibrant
resolution, when completed at a certain Hodge filtration, is (reduced)
quasi-isomorphic to the periodic cyclic complex, while each of its filtration
piece is quasi-isomorphic to the negative cyclic complex. This extends a
classical result of Feigin and Tsygan, which corresponds to the Hodge degree
$0$ part of our quasi-isomorphism. This result is applied to the study of
Calabi-Yau categories in \cite{Yeu1, Yeu2}.
