  We study the $K$-user, $M$-subchannel parallel multiple-input-single-output
(MISO) broadcast channel (BC) under arbitrary levels of partial channel state
information at the transmitter (CSIT). We show that the parallel subchannels
constituting this setting are separable from a degrees-of-freedom (DoF) region
perspective if and only if the partial CSIT pattern is totally ordered. This
total order condition corresponds to users abiding by the same order, with
respect to their CSIT quality levels, in each of the parallel subchannels. For
instance, let $\alpha_{k}^{[l]}$ and $\alpha_{j}^{[l]}$ be the CSIT quality
parameters for users $k$ and $j$ over subchannel $l$. Under total order, having
$\alpha_{k}^{[l]} \geq \alpha_{j}^{[l]}$ implies that $\alpha_{k}^{[m]} \geq
\alpha_{j}^{[m]}$ holds for every subchannel $m$. In this case, the entire DoF
region is achievable using simple separate coding, where a
single-subchannel-type transmission scheme is employed in each subchannel. To
show this separability result, we first derive an outer bound for the DoF
region by extending the aligned image sets approach of Davoodi and Jafar to the
considered setting. We then show that this outer bound coincides with the inner
bound achieved through separate coding, given by the Minkowski sum of $M$
single-subchannel DoF regions, under the total order condition, hence settling
the if part of the main theorem. To prove the only if part of the theorem, we
identify a set of DoF tuples achievable through joint coding across
subchannels, yet not achievable through separate coding whenever the total
order condition is violated. Moreover, we also highlight the implications of
our main result on the design of CSIT feedback schemes for multi-carrier
multi-antenna wireless networks.
