  We prove that pointwise finite-dimensional S^1 persistence modules over an
arbitrary field decompose uniquely, up to isomorphism, into the direct sum of a
bar code and finitely-many Jordan cells. These persistence modules have also
been called angle-valued or circular persistence modules. We allow either a
cyclic order or partial order on S^1 and do not have additional finiteness
requirements on the modules. We also show that a pointwise finite-dimensional
S^1 persistence module is indecomposable if and only if it is a bar or Jordan
cell (a string or a band module, respectively, in representation theory). Along
the way we classify the isomorphism classes of such indecomposable modules.
